Patent Publication Number: US-2020301408-A1

Title: Model predictive maintenance system with degradation impact model

Description:
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS 
     This application is a continuation-in-part of U.S. patent application Ser. No. 15/895,836 filed Feb. 13, 2018, which claims the benefit of and priority to U.S. Provisional Patent Application No. 62/511,113 filed May 25, 2017. This application also claims the benefit of and priority to U.S. Provisional Patent Application No. 62/883,508 filed Aug. 6, 2019. The entire disclosures of each of these patent applications are incorporated by reference herein. 
    
    
     BACKGROUND 
     The present disclosure relates generally to control systems for building equipment. The present disclosure relates more particularly to control systems that use predictive modeling to determine an optimal operating strategy and maintenance strategy for building equipment. 
     Building equipment operate to affect various conditions in a building such as temperature, humidity, air quality, lighting, etc. Building equipment degrade over time, as a result of operating the building equipment, which leads to reduced operating efficiency and increased power consumption and cost. Performing maintenance on building equipment can restore the equipment to a less degraded state and improve the operating efficiency and thus reduce operating cost. However, performing maintenance typically incurs a maintenance cost. Therefore, choosing to perform maintenance on the building equipment reduces ongoing operating cost as a result of reduced power consumption, but incurs an additional maintenance cost. Performing maintenance too frequently may result in a low operating cost but a high maintenance cost, whereas performing maintenance too infrequently may result in a low maintenance cost but a higher operating cost. It can be difficult to determine an appropriate maintenance strategy for building equipment in the interest of reducing total life cycle cost. 
     SUMMARY 
     One implementation of the present disclosure is a model predictive maintenance (MPM) system for building equipment. The MPM system includes one or more processing circuits having one or more processors and memory. The memory store instructions that, when executed by the one or more processors, cause the one or more processors to perform operations including estimating a degradation state of the building equipment, using a degradation impact model to predict an amount of one or more input resources consumed by the building equipment to produce one or more output resources based on the degradation state of the building equipment, generating a maintenance schedule for the building equipment based on the amount of the one or more input resources predicted using the degradation impact model, and initiating a maintenance activity for the building equipment in accordance with the maintenance schedule. 
     In some embodiments, using the degradation impact model to predict the amount of the one or more input resources consumed by the building equipment includes using the degradation impact model to generate parameters of a resource consumption model for the building equipment as a function of the degradation state of the building equipment and using the resource consumption model to predict the amount of one or more input resources consumed by the building equipment to produce the one or more output resources as a function of the parameters of the resource consumption model. 
     In some embodiments, the degradation impact model is trained using historical or simulated training data prior to using the degradation impact model to predict the amount of the one or more input resources consumed by the building equipment. Training the degradation impact model may include generating training data for the degradation impact model, the training data comprising a plurality of different values of the degradation state of the building equipment and corresponding values of parameters of a resource consumption model for the building equipment, and using the training data to train the degradation impact model to predict the values of the parameters of the resource consumption model as a function of the degradation state. 
     In some embodiments, generating the training data includes performing a regression process to generate the values of the parameters of the resource consumption model using data associated with a first degradation state of the building equipment and repeating the regression process using data associated with one or more additional degradation states of the building equipment to generate a plurality of different values of the parameters of the resource consumption model, the plurality of different values of the parameters corresponding to a plurality of different degradation states of the building equipment. 
     In some embodiments, the degradation impact model includes a neural network model and using the degradation impact model to predict the amount of the one or more input resources consumed by the building equipment includes providing the degradation state of the building equipment and an amount of the one or more output resources to be produced by the building equipment as inputs to the neural network model and obtaining the amount of one or more input resources consumed by the building equipment as an output of the neural network model. 
     In some embodiments, generating the maintenance schedule for the building equipment includes performing an optimization of an objective function that accounts for both a cost of operating the building equipment and a cost of performing maintenance on the building equipment over a time period and generating a set of maintenance decisions for the building equipment as a result of performing the optimization, the set of maintenance decisions forming the maintenance schedule. 
     In some embodiments, generating the maintenance schedule for the building equipment includes calculating a cost of operating the building equipment over a time period as a function of the degradation state of the building equipment at one or more times within the time period, calculating a cost of performing maintenance on the building equipment over the time period as a function of one or more maintenance activities defined by the maintenance schedule, adjusting the degradation state of the building equipment at one or more times following the one or more maintenance activities defined by the maintenance schedule, and generating the maintenance schedule that results in a lowest total cost comprising the cost of operating the building equipment over the time period and the cost of performing maintenance on the building equipment over the time period. 
     In some embodiments, the degradation state of the building equipment is an initial degradation state. The operations may further include predicting one or more future degradation states of the building equipment as a function of the initial degradation state and using the degradation impact model to predict the amount of the one or more input resources consumed by the building equipment at one or more future times as a function of the one or more future degradation states. 
     Another implementation of the present disclosure is a method for using model predictive maintenance (MPM) to generate a maintenance schedule for building equipment. The method includes estimating a degradation state of the building equipment, using a degradation impact model to predict an amount of one or more input resources consumed by the building equipment to produce one or more output resources based on the degradation state of the building equipment, generating a maintenance schedule for the building equipment based on the amount of the one or more input resources predicted using the degradation impact model, and initiating a maintenance activity for the building equipment in accordance with the maintenance schedule. 
     In some embodiments, using the degradation impact model to predict the amount of the one or more input resources consumed by the building equipment includes using the degradation impact model to generate parameters of a resource consumption model for the building equipment as a function of the degradation state of the building equipment and using the resource consumption model to predict the amount of one or more input resources consumed by the building equipment to produce the one or more output resources as a function of the parameters of the resource consumption model. 
     In some embodiments, the degradation impact model is trained using historical or simulated training data prior to using the degradation impact model to predict the amount of the one or more input resources consumed by the building equipment. Training the degradation impact model may include generating training data for the degradation impact model, the training data comprising a plurality of different values of the degradation state of the building equipment and corresponding values of parameters of a resource consumption model for the building equipment, and using the training data to train the degradation impact model to predict the values of the parameters of the resource consumption model as a function of the degradation state. 
     In some embodiments, generating the training data includes performing a regression process to generate the values of the parameters of the resource consumption model using data associated with a first degradation state of the building equipment and repeating the regression process using data associated with one or more additional degradation states of the building equipment to generate a plurality of different values of the parameters of the resource consumption model, the plurality of different values of the parameters corresponding to a plurality of different degradation states of the building equipment. 
     In some embodiments, the degradation impact model includes a neural network model and using the degradation impact model to predict the amount of the one or more input resources consumed by the building equipment includes providing the degradation state of the building equipment and an amount of the one or more output resources to be produced by the building equipment as inputs to the neural network model and obtaining the amount of one or more input resources consumed by the building equipment as an output of the neural network model. 
     In some embodiments, generating the maintenance schedule for the building equipment includes performing an optimization of an objective function that accounts for both a cost of operating the building equipment and a cost of performing maintenance on the building equipment over a time period and generating a set of maintenance decisions for the building equipment as a result of performing the optimization, the set of maintenance decisions forming the maintenance schedule. 
     In some embodiments, generating the maintenance schedule for the building equipment includes calculating a cost of operating the building equipment over a time period as a function of the degradation state of the building equipment at one or more times within the time period, calculating a cost of performing maintenance on the building equipment over the time period as a function of one or more maintenance activities defined by the maintenance schedule, adjusting the degradation state of the building equipment at one or more times following the one or more maintenance activities defined by the maintenance schedule, and generating the maintenance schedule that results in a lowest total cost comprising the cost of operating the building equipment over the time period and the cost of performing maintenance on the building equipment over the time period. 
     In some embodiments, the degradation state of the building equipment is an initial degradation state. The method may further include predicting one or more future degradation states of the building equipment as a function of the initial degradation state and using the degradation impact model to predict the amount of the one or more input resources consumed by the building equipment at one or more future times as a function of the one or more future degradation states. 
     Another implementation of the present disclosure is a model predictive maintenance (MPM) system for building equipment. The MPM system includes one or more processing circuits having one or more processors and memory. The memory store instructions that, when executed by the one or more processors, cause the one or more processors to perform operations including using a degradation impact model to generate parameters of a resource consumption model for the building equipment based on a degradation state of the building equipment, using the resource consumption model to generate a maintenance schedule for the building equipment that results in a lowest total cost of operating the building equipment and performing maintenance on the building equipment over a time period, and initiating a maintenance activity for the building equipment in accordance with the maintenance schedule. 
     In some embodiments, using the resource consumption model to generate the maintenance schedule includes using the resource consumption model to predict an amount of one or more input resources consumed by the building equipment to produce one or more output resources as a function of the parameters of the resource consumption model and generating the maintenance schedule based on the amount of the one or more input resources consumed by the building equipment to produce the one or more output resources. 
     In some embodiments, the degradation impact model is trained using historical or simulated training data prior to using the degradation impact model to generate the parameters of the resource consumption model. Training the degradation impact model may include generating training data for the degradation impact model, the training data comprising a plurality of different values of the degradation state of the building equipment and corresponding values of the parameters of the resource consumption model, and using the training data to train the degradation impact model to predict the values of the parameters of the resource consumption model as a function of the degradation state. 
     In some embodiments, generating the training data includes performing a regression process to generate the values of the parameters of the resource consumption model using data associated with a first degradation state of the building equipment and repeating the regression process using data associated with one or more additional degradation states of the building equipment to generate a plurality of different values of the parameters of the resource consumption model, the plurality of different values of the parameters corresponding to a plurality of different degradation states of the building equipment. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Various objects, aspects, features, and advantages of the disclosure will become more apparent and better understood by referring to the detailed description taken in conjunction with the accompanying drawings, in which like reference characters identify corresponding elements throughout. In the drawings, like reference numbers generally indicate identical, functionally similar, and/or structurally similar elements. 
         FIG. 1  is an illustration of a building equipped with a HVAC system, according some embodiments. 
         FIG. 2  is a block diagram of a waterside system that may be used in conjunction with the building of  FIG. 1 , according to some embodiments. 
         FIG. 3  is a block diagram of an airside system that may be used in conjunction with the building of  FIG. 1 , according to some embodiments. 
         FIG. 4  is a block diagram of a building management system (BMS) which can be used to monitor and control the building of  FIG. 1 , according to some embodiments. 
         FIG. 5  is a block diagram of another BMS which can be used to monitor and control the building of  FIG. 1 , according to some embodiments. 
         FIG. 6  is a block diagram of a building system including a model predictive maintenance (MPM) system that monitors equipment performance information from connected equipment installed in the building, according to some embodiments. 
         FIG. 7  is a schematic diagram of a chiller which may be a type of connected equipment that provides equipment performance information to the MPM system of  FIG. 6 , according to some embodiments. 
         FIG. 8  is a block diagram illustrating the MPM system of  FIG. 6  in greater detail, according to some embodiments. 
         FIG. 9  is a block diagram illustrating a high level optimizer of the MPM system of  FIG. 6  in greater detail, according to some embodiments. 
         FIG. 10  is a flowchart of a process for operating the MPM system of  FIG. 6 , according to some embodiments. 
         FIG. 11A  is a graph illustrating experimental values of Q HVAC  used as training data for a neural network, according to some embodiments. 
         FIG. 11B  is a graph illustrating how a degradation state of building equipment is affected based on a periodic maintenance strategy, according to some embodiments. 
         FIG. 11C  is a graph illustrating how a degradation state of building equipment is affected based on a run-to-fail maintenance strategy, according to some embodiments. 
         FIG. 12  is an illustration of a progression of maintenance strategies, according to some embodiments. 
         FIG. 13  is another block diagram illustrating the MPM system of  FIG. 6  in greater detail, according to some embodiments. 
         FIG. 14A  is another block diagram illustrating a portion of the MPM system of  FIG. 13  in greater detail, according to some embodiments. 
         FIG. 14B  is a flowchart of a process for generating an optimal maintenance schedule, which can be performed by the MPM system of  FIG. 13 , according to some embodiments. 
         FIG. 15  is a block diagram illustrating the degradation impact modeler of  FIG. 13  in greater detail, according to some embodiments. 
         FIG. 16  is a graph illustrating a weighting function that can be used to weight inputs to a neural network model used by the degradation impact modeler of  FIG. 15 , according to some embodiments. 
         FIG. 17  is an illustration of a neural network model, according to some embodiments. 
         FIG. 18  is an example illustration of a multilayer perceptron (MLP) neural network, according to some embodiments. 
         FIG. 19  is an example illustration of a radial basis function neural network (RBFNN), according to some embodiments. 
         FIG. 20  is a flowchart of a process for generating a maintenance schedule for building equipment, which can be performed by the MPM system of  FIG. 13 , according to some embodiments. 
     
    
    
     DETAILED DESCRIPTION 
     Referring generally to the FIGURES, systems and methods for performing model predictive maintenance (MPM) are shown, according to some embodiments. MPM can be performed for building equipment of a building to determine a maintenance and replacement strategy for the building equipment. 
     In order to optimize the scheduling of maintenance it is necessary to understand how degradation effects the performance of equipment. The mapping between degradation and performance can be nonlinear and have no known model form that would lend itself well to gray-box modeling. The systems and methods described herein provide a model that maps equipment degradation to operating performance using artificial intelligence (AI). Operating performance can be characterized by a model that relates the amount of resources consumed by the equipment (e.g., electricity, water, natural gas, etc.) to the amount of output resources produced by the equipment (e.g., hot water, cold water, heating load, cooling load, etc.) at a given time. Such a model can be characterized by a vector of model coefficients or parameters. The coefficients or parameters of the model may change as the equipment degrades. Accordingly, examining the relationship between degradation and model coefficients may allow for a mapping to be generated therebetween. 
     One example of a system in which the systems and methods of the present disclosure can be implemented is a variable refrigerant flow (VRF) system that consumes electric power to serve a heating or cooling load. A power consumption model can be used to relate the amount of power consumed by the VRF equipment to the amount of heating or cooling produced by the VRF equipment. An artificial neural network model is trained to predict values of coefficients of the power consumption model as a function of degradation state. To generate training data for the neural network model, both the degradation state and the power consumption can be estimated by the measurements collected from the VRF system. Once the neural network has been trained, the neural network can be used to predict power model coefficients as a function of the current degradation state. The power model coefficients are then used to predict the power consumption of the equipment during operation. 
     The predicted power consumption (or other resource consumption) can be used to perform a model predictive maintenance process to determine an optimal set of operating decisions and maintenance decisions for the equipment over a given time period. These and other features of the model predictive maintenance system are described in detail below. 
     Building HVAC Systems and Building Management Systems 
     Referring now to  FIGS. 1-5 , several building management systems (BMS) and HVAC systems in which the systems and methods of the present disclosure can be implemented are shown, according to some embodiments. In brief overview,  FIG. 1  shows a building  10  equipped with a HVAC system  100 .  FIG. 2  is a block diagram of a waterside system  200  which can be used to serve building  10 .  FIG. 3  is a block diagram of an airside system  300  which can be used to serve building  10 .  FIG. 4  is a block diagram of a BMS which can be used to monitor and control building  10 .  FIG. 5  is a block diagram of another BMS which can be used to monitor and control building  10 . 
     Building and HVAC System 
     Referring particularly to  FIG. 1 , a perspective view of a building  10  is shown. Building  10  is served by a BMS. A BMS is, in general, a system of devices configured to control, monitor, and manage equipment in or around a building or building area. A BMS can include, for example, a HVAC system, a security system, a lighting system, a fire alerting system, any other system that is capable of managing building functions or devices, or any combination thereof. 
     The BMS that serves building  10  includes a HVAC system  100 . HVAC system  100  can include a plurality of HVAC devices (e.g., heaters, chillers, air handling units, pumps, fans, thermal energy storage, etc.) configured to provide heating, cooling, ventilation, or other services for building  10 . For example, HVAC system  100  is shown to include a waterside system  120  and an airside system  130 . Waterside system  120  may provide a heated or chilled fluid to an air handling unit of airside system  130 . Airside system  130  may use the heated or chilled fluid to heat or cool an airflow provided to building  10 . An exemplary waterside system and airside system which can be used in HVAC system  100  are described in greater detail with reference to  FIGS. 2-3 . 
     HVAC system  100  is shown to include a chiller  102 , a boiler  104 , and a rooftop air handling unit (AHU)  106 . Waterside system  120  may use boiler  104  and chiller  102  to heat or cool a working fluid (e.g., water, glycol, etc.) and may circulate the working fluid to AHU  106 . In various embodiments, the HVAC devices of waterside system  120  can be located in or around building  10  (as shown in  FIG. 1 ) or at an offsite location such as a central plant (e.g., a chiller plant, a steam plant, a heat plant, etc.). The working fluid can be heated in boiler  104  or cooled in chiller  102 , depending on whether heating or cooling is required in building  10 . Boiler  104  may add heat to the circulated fluid, for example, by burning a combustible material (e.g., natural gas) or using an electric heating element. Chiller  102  may place the circulated fluid in a heat exchange relationship with another fluid (e.g., a refrigerant) in a heat exchanger (e.g., an evaporator) to absorb heat from the circulated fluid. The working fluid from chiller  102  and/or boiler  104  can be transported to AHU  106  via piping  108 . 
     AHU  106  may place the working fluid in a heat exchange relationship with an airflow passing through AHU  106  (e.g., via one or more stages of cooling coils and/or heating coils). The airflow can be, for example, outside air, return air from within building  10 , or a combination of both. AHU  106  may transfer heat between the airflow and the working fluid to provide heating or cooling for the airflow. For example, AHU  106  can include one or more fans or blowers configured to pass the airflow over or through a heat exchanger containing the working fluid. The working fluid may then return to chiller  102  or boiler  104  via piping  110 . 
     Airside system  130  may deliver the airflow supplied by AHU  106  (i.e., the supply airflow) to building  10  via air supply ducts  112  and may provide return air from building  10  to AHU  106  via air return ducts  114 . In some embodiments, airside system  130  includes multiple variable air volume (VAV) units  116 . For example, airside system  130  is shown to include a separate VAV unit  116  on each floor or zone of building  10 . VAV units  116  can include dampers or other flow control elements that can be operated to control an amount of the supply airflow provided to individual zones of building  10 . In other embodiments, airside system  130  delivers the supply airflow into one or more zones of building  10  (e.g., via supply ducts  112 ) without using intermediate VAV units  116  or other flow control elements. AHU  106  can include various sensors (e.g., temperature sensors, pressure sensors, etc.) configured to measure attributes of the supply airflow. AHU  106  may receive input from sensors located within AHU  106  and/or within the building zone and may adjust the flow rate, temperature, or other attributes of the supply airflow through AHU  106  to achieve setpoint conditions for the building zone. 
     Waterside System 
     Referring now to  FIG. 2 , a block diagram of a waterside system  200  is shown, according to some embodiments. In various embodiments, waterside system  200  may supplement or replace waterside system  120  in HVAC system  100  or can be implemented separate from HVAC system  100 . When implemented in HVAC system  100 , waterside system  200  can include a subset of the HVAC devices in HVAC system  100  (e.g., boiler  104 , chiller  102 , pumps, valves, etc.) and may operate to supply a heated or chilled fluid to AHU  106 . The HVAC devices of waterside system  200  can be located within building  10  (e.g., as components of waterside system  120 ) or at an offsite location such as a central plant. 
     In  FIG. 2 , waterside system  200  is shown as a central plant having a plurality of subplants  202 - 212 . Subplants  202 - 212  are shown to include a heater subplant  202 , a heat recovery chiller subplant  204 , a chiller subplant  206 , a cooling tower subplant  208 , a hot thermal energy storage (TES) subplant  210 , and a cold thermal energy storage (TES) subplant  212 . Subplants  202 - 212  consume resources (e.g., water, natural gas, electricity, etc.) from utilities to serve thermal energy loads (e.g., hot water, cold water, heating, cooling, etc.) of a building or campus. For example, heater subplant  202  can be configured to heat water in a hot water loop  214  that circulates the hot water between heater subplant  202  and building  10 . Chiller subplant  206  can be configured to chill water in a cold water loop  216  that circulates the cold water between chiller subplant  206  building  10 . Heat recovery chiller subplant  204  can be configured to transfer heat from cold water loop  216  to hot water loop  214  to provide additional heating for the hot water and additional cooling for the cold water. Condenser water loop  218  may absorb heat from the cold water in chiller subplant  206  and reject the absorbed heat in cooling tower subplant  208  or transfer the absorbed heat to hot water loop  214 . Hot TES subplant  210  and cold TES subplant  212  may store hot and cold thermal energy, respectively, for subsequent use. 
     Hot water loop  214  and cold water loop  216  may deliver the heated and/or chilled water to air handlers located on the rooftop of building  10  (e.g., AHU  106 ) or to individual floors or zones of building  10  (e.g., VAV units  116 ). The air handlers push air past heat exchangers (e.g., heating coils or cooling coils) through which the water flows to provide heating or cooling for the air. The heated or cooled air can be delivered to individual zones of building  10  to serve thermal energy loads of building  10 . The water then returns to subplants  202 - 212  to receive further heating or cooling. 
     Although subplants  202 - 212  are shown and described as heating and cooling water for circulation to a building, it is understood that any other type of working fluid (e.g., glycol, CO2, etc.) can be used in place of or in addition to water to serve thermal energy loads. In other embodiments, subplants  202 - 212  may provide heating and/or cooling directly to the building or campus without requiring an intermediate heat transfer fluid. These and other variations to waterside system  200  are within the teachings of the present disclosure. 
     Each of subplants  202 - 212  can include a variety of equipment configured to facilitate the functions of the subplant. For example, heater subplant  202  is shown to include a plurality of heating elements  220  (e.g., boilers, electric heaters, etc.) configured to add heat to the hot water in hot water loop  214 . Heater subplant  202  is also shown to include several pumps  222  and  224  configured to circulate the hot water in hot water loop  214  and to control the flow rate of the hot water through individual heating elements  220 . Chiller subplant  206  is shown to include a plurality of chillers  232  configured to remove heat from the cold water in cold water loop  216 . Chiller subplant  206  is also shown to include several pumps  234  and  236  configured to circulate the cold water in cold water loop  216  and to control the flow rate of the cold water through individual chillers  232 . 
     Heat recovery chiller subplant  204  is shown to include a plurality of heat recovery heat exchangers  226  (e.g., refrigeration circuits) configured to transfer heat from cold water loop  216  to hot water loop  214 . Heat recovery chiller subplant  204  is also shown to include several pumps  228  and  230  configured to circulate the hot water and/or cold water through heat recovery heat exchangers  226  and to control the flow rate of the water through individual heat recovery heat exchangers  226 . Cooling tower subplant  208  is shown to include a plurality of cooling towers  238  configured to remove heat from the condenser water in condenser water loop  218 . Cooling tower subplant  208  is also shown to include several pumps  240  configured to circulate the condenser water in condenser water loop  218  and to control the flow rate of the condenser water through individual cooling towers  238 . 
     Hot TES subplant  210  is shown to include a hot TES tank  242  configured to store the hot water for later use. Hot TES subplant  210  may also include one or more pumps or valves configured to control the flow rate of the hot water into or out of hot TES tank  242 . Cold TES subplant  212  is shown to include cold TES tanks  244  configured to store the cold water for later use. Cold TES subplant  212  may also include one or more pumps or valves configured to control the flow rate of the cold water into or out of cold TES tanks  244 . 
     In some embodiments, one or more of the pumps in waterside system  200  (e.g., pumps  222 ,  224 ,  228 ,  230 ,  234 ,  236 , and/or  240 ) or pipelines in waterside system  200  include an isolation valve associated therewith. Isolation valves can be integrated with the pumps or positioned upstream or downstream of the pumps to control the fluid flows in waterside system  200 . In various embodiments, waterside system  200  can include more, fewer, or different types of devices and/or subplants based on the particular configuration of waterside system  200  and the types of loads served by waterside system  200 . 
     Airside System 
     Referring now to  FIG. 3 , a block diagram of an airside system  300  is shown, according to some embodiments. In various embodiments, airside system  300  may supplement or replace airside system  130  in HVAC system  100  or can be implemented separate from HVAC system  100 . When implemented in HVAC system  100 , airside system  300  can include a subset of the HVAC devices in HVAC system  100  (e.g., AHU  106 , VAV units  116 , ducts  112 - 114 , fans, dampers, etc.) and can be located in or around building  10 . Airside system  300  may operate to heat or cool an airflow provided to building  10  using a heated or chilled fluid provided by waterside system  200 . 
     In  FIG. 3 , airside system  300  is shown to include an economizer-type air handling unit (AHU)  302 . Economizer-type AHUs vary the amount of outside air and return air used by the air handling unit for heating or cooling. For example, AHU  302  may receive return air  304  from building zone  306  via return air duct  308  and may deliver supply air  310  to building zone  306  via supply air duct  312 . In some embodiments, AHU  302  is a rooftop unit located on the roof of building  10  (e.g., AHU  106  as shown in  FIG. 1 ) or otherwise positioned to receive both return air  304  and outside air  314 . AHU  302  can be configured to operate exhaust air damper  316 , mixing damper  318 , and outside air damper  320  to control an amount of outside air  314  and return air  304  that combine to form supply air  310 . Any return air  304  that does not pass through mixing damper  318  can be exhausted from AHU  302  through exhaust damper  316  as exhaust air  322 . 
     Each of dampers  316 - 320  can be operated by an actuator. For example, exhaust air damper  316  can be operated by actuator  324 , mixing damper  318  can be operated by actuator  326 , and outside air damper  320  can be operated by actuator  328 . Actuators  324 - 328  may communicate with an AHU controller  330  via a communications link  332 . Actuators  324 - 328  may receive control signals from AHU controller  330  and may provide feedback signals to AHU controller  330 . Feedback signals can include, for example, an indication of a current actuator or damper position, an amount of torque or force exerted by the actuator, diagnostic information (e.g., results of diagnostic tests performed by actuators  324 - 328 ), status information, commissioning information, configuration settings, calibration data, and/or other types of information or data that can be collected, stored, or used by actuators  324 - 328 . AHU controller  330  can be an economizer controller configured to use one or more control algorithms (e.g., state-based algorithms, extremum seeking control (ESC) algorithms, proportional-integral (PI) control algorithms, proportional-integral-derivative (PID) control algorithms, model predictive control (MPC) algorithms, feedback control algorithms, etc.) to control actuators  324 - 328 . 
     Still referring to  FIG. 3 , AHU  302  is shown to include a cooling coil  334 , a heating coil  336 , and a fan  338  positioned within supply air duct  312 . Fan  338  can be configured to force supply air  310  through cooling coil  334  and/or heating coil  336  and provide supply air  310  to building zone  306 . AHU controller  330  may communicate with fan  338  via communications link  340  to control a flow rate of supply air  310 . In some embodiments, AHU controller  330  controls an amount of heating or cooling applied to supply air  310  by modulating a speed of fan  338 . 
     Cooling coil  334  may receive a chilled fluid from waterside system  200  (e.g., from cold water loop  216 ) via piping  342  and may return the chilled fluid to waterside system  200  via piping  344 . Valve  346  can be positioned along piping  342  or piping  344  to control a flow rate of the chilled fluid through cooling coil  334 . In some embodiments, cooling coil  334  includes multiple stages of cooling coils that can be independently activated and deactivated (e.g., by AHU controller  330 , by BMS controller  366 , etc.) to modulate an amount of cooling applied to supply air  310 . 
     Heating coil  336  may receive a heated fluid from waterside system  200  (e.g., from hot water loop  214 ) via piping  348  and may return the heated fluid to waterside system  200  via piping  350 . Valve  352  can be positioned along piping  348  or piping  350  to control a flow rate of the heated fluid through heating coil  336 . In some embodiments, heating coil  336  includes multiple stages of heating coils that can be independently activated and deactivated (e.g., by AHU controller  330 , by BMS controller  366 , etc.) to modulate an amount of heating applied to supply air  310 . 
     Each of valves  346  and  352  can be controlled by an actuator. For example, valve  346  can be controlled by actuator  354  and valve  352  can be controlled by actuator  356 . Actuators  354 - 356  may communicate with AHU controller  330  via communications links  358 - 360 . Actuators  354 - 356  may receive control signals from AHU controller  330  and may provide feedback signals to controller  330 . In some embodiments, AHU controller  330  receives a measurement of the supply air temperature from a temperature sensor  362  positioned in supply air duct  312  (e.g., downstream of cooling coil  334  and/or heating coil  336 ). AHU controller  330  may also receive a measurement of the temperature of building zone  306  from a temperature sensor  364  located in building zone  306 . 
     In some embodiments, AHU controller  330  operates valves  346  and  352  via actuators  354 - 356  to modulate an amount of heating or cooling provided to supply air  310  (e.g., to achieve a setpoint temperature for supply air  310  or to maintain the temperature of supply air  310  within a setpoint temperature range). The positions of valves  346  and  352  affect the amount of heating or cooling provided to supply air  310  by cooling coil  334  or heating coil  336  and may correlate with the amount of energy consumed to achieve a desired supply air temperature. AHU  330  may control the temperature of supply air  310  and/or building zone  306  by activating or deactivating coils  334 - 336 , adjusting a speed of fan  338 , or a combination of both. 
     Still referring to  FIG. 3 , airside system  300  is shown to include a building management system (BMS) controller  366  and a client device  368 . BMS controller  366  can include one or more computer systems (e.g., servers, supervisory controllers, subsystem controllers, etc.) that serve as system level controllers, application or data servers, head nodes, or master controllers for airside system  300 , waterside system  200 , HVAC system  100 , and/or other controllable systems that serve building  10 . BMS controller  366  may communicate with multiple downstream building systems or subsystems (e.g., HVAC system  100 , a security system, a lighting system, waterside system  200 , etc.) via a communications link  370  according to like or disparate protocols (e.g., LON, BACnet, etc.). In various embodiments, AHU controller  330  and BMS controller  366  can be separate (as shown in  FIG. 3 ) or integrated. In an integrated implementation, AHU controller  330  can be a software module configured for execution by a processor of BMS controller  366 . 
     In some embodiments, AHU controller  330  receives information from BMS controller  366  (e.g., commands, setpoints, operating boundaries, etc.) and provides information to BMS controller  366  (e.g., temperature measurements, valve or actuator positions, operating statuses, diagnostics, etc.). For example, AHU controller  330  may provide BMS controller  366  with temperature measurements from temperature sensors  362 - 364 , equipment on/off states, equipment operating capacities, and/or any other information that can be used by BMS controller  366  to monitor or control a variable state or condition within building zone  306 . 
     Client device  368  can include one or more human-machine interfaces or client interfaces (e.g., graphical user interfaces, reporting interfaces, text-based computer interfaces, client-facing web services, web servers that provide pages to web clients, etc.) for controlling, viewing, or otherwise interacting with HVAC system  100 , its subsystems, and/or devices. Client device  368  can be a computer workstation, a client terminal, a remote or local interface, or any other type of user interface device. Client device  368  can be a stationary terminal or a mobile device. For example, client device  368  can be a desktop computer, a computer server with a user interface, a laptop computer, a tablet, a smartphone, a PDA, or any other type of mobile or non-mobile device. Client device  368  may communicate with BMS controller  366  and/or AHU controller  330  via communications link  372 . 
     Building Management Systems 
     Referring now to  FIG. 4 , a block diagram of a building management system (BMS)  400  is shown, according to some embodiments. BMS  400  can be implemented in building  10  to automatically monitor and control various building functions. BMS  400  is shown to include BMS controller  366  and a plurality of building subsystems  428 . Building subsystems  428  are shown to include a building electrical subsystem  434 , an information communication technology (ICT) subsystem  436 , a security subsystem  438 , a HVAC subsystem  440 , a lighting subsystem  442 , a lift/escalators subsystem  432 , and a fire safety subsystem  430 . In various embodiments, building subsystems  428  can include fewer, additional, or alternative subsystems. For example, building subsystems  428  may also or alternatively include a refrigeration subsystem, an advertising or signage subsystem, a cooking subsystem, a vending subsystem, a printer or copy service subsystem, or any other type of building subsystem that uses controllable equipment and/or sensors to monitor or control building  10 . In some embodiments, building subsystems  428  include waterside system  200  and/or airside system  300 , as described with reference to  FIGS. 2-3 . 
     Each of building subsystems  428  can include any number of devices, controllers, and connections for completing its individual functions and control activities. HVAC subsystem  440  can include many of the same components as HVAC system  100 , as described with reference to  FIGS. 1-3 . For example, HVAC subsystem  440  can include a chiller, a boiler, any number of air handling units, economizers, field controllers, supervisory controllers, actuators, temperature sensors, and other devices for controlling the temperature, humidity, airflow, or other variable conditions within building  10 . Lighting subsystem  442  can include any number of light fixtures, ballasts, lighting sensors, dimmers, or other devices configured to controllably adjust the amount of light provided to a building space. Security subsystem  438  can include occupancy sensors, video surveillance cameras, digital video recorders, video processing servers, intrusion detection devices, access control devices and servers, or other security-related devices. 
     Still referring to  FIG. 4 , BMS controller  366  is shown to include a communications interface  407  and a BMS interface  409 . Interface  407  may facilitate communications between BMS controller  366  and external applications (e.g., monitoring and reporting applications  422 , enterprise control applications  426 , remote systems and applications  444 , applications residing on client devices  448 , etc.) for allowing user control, monitoring, and adjustment to BMS controller  366  and/or subsystems  428 . Interface  407  may also facilitate communications between BMS controller  366  and client devices  448 . BMS interface  409  may facilitate communications between BMS controller  366  and building subsystems  428  (e.g., HVAC, lighting security, lifts, power distribution, business, etc.). 
     Interfaces  407 ,  409  can be or include wired or wireless communications interfaces (e.g., jacks, antennas, transmitters, receivers, transceivers, wire terminals, etc.) for conducting data communications with building subsystems  428  or other external systems or devices. In various embodiments, communications via interfaces  407 ,  409  can be direct (e.g., local wired or wireless communications) or via a communications network  446  (e.g., a WAN, the Internet, a cellular network, etc.). For example, interfaces  407 ,  409  can include an Ethernet card and port for sending and receiving data via an Ethernet-based communications link or network. In another example, interfaces  407 ,  409  can include a Wi-Fi transceiver for communicating via a wireless communications network. In another example, one or both of interfaces  407 ,  409  can include cellular or mobile phone communications transceivers. In one embodiment, communications interface  407  is a power line communications interface and BMS interface  409  is an Ethernet interface. In other embodiments, both communications interface  407  and BMS interface  409  are Ethernet interfaces or are the same Ethernet interface. 
     Still referring to  FIG. 4 , BMS controller  366  is shown to include a processing circuit  404  including a processor  406  and memory  408 . Processing circuit  404  can be communicably connected to BMS interface  409  and/or communications interface  407  such that processing circuit  404  and the various components thereof can send and receive data via interfaces  407 ,  409 . Processor  406  can be implemented as a general purpose processor, an application specific integrated circuit (ASIC), one or more field programmable gate arrays (FPGAs), a group of processing components, or other suitable electronic processing components. 
     Memory  408  (e.g., memory, memory unit, storage device, etc.) can include one or more devices (e.g., RAM, ROM, Flash memory, hard disk storage, etc.) for storing data and/or computer code for completing or facilitating the various processes, layers and modules described in the present application. Memory  408  can be or include volatile memory or non-volatile memory. Memory  408  can include database components, object code components, script components, or any other type of information structure for supporting the various activities and information structures described in the present application. According to some embodiments, memory  408  is communicably connected to processor  406  via processing circuit  404  and includes computer code for executing (e.g., by processing circuit  404  and/or processor  406 ) one or more processes described herein. 
     In some embodiments, BMS controller  366  is implemented within a single computer (e.g., one server, one housing, etc.). In various other embodiments BMS controller  366  can be distributed across multiple servers or computers (e.g., that can exist in distributed locations). Further, while  FIG. 4  shows applications  422  and  426  as existing outside of BMS controller  366 , in some embodiments, applications  422  and  426  can be hosted within BMS controller  366  (e.g., within memory  408 ). 
     Still referring to  FIG. 4 , memory  408  is shown to include an enterprise integration layer  410 , an automated measurement and validation (AM&amp;V) layer  412 , a demand response (DR) layer  414 , a fault detection and diagnostics (FDD) layer  416 , an integrated control layer  418 , and a building subsystem integration later  420 . Layers  410 - 420  can be configured to receive inputs from building subsystems  428  and other data sources, determine optimal control actions for building subsystems  428  based on the inputs, generate control signals based on the optimal control actions, and provide the generated control signals to building subsystems  428 . The following paragraphs describe some of the general functions performed by each of layers  410 - 420  in BMS  400 . 
     Enterprise integration layer  410  can be configured to serve clients or local applications with information and services to support a variety of enterprise-level applications. For example, enterprise control applications  426  can be configured to provide subsystem-spanning control to a graphical user interface (GUI) or to any number of enterprise-level business applications (e.g., accounting systems, user identification systems, etc.). Enterprise control applications  426  may also or alternatively be configured to provide configuration GUIs for configuring BMS controller  366 . In yet other embodiments, enterprise control applications  426  can work with layers  410 - 420  to optimize building performance (e.g., efficiency, energy use, comfort, or safety) based on inputs received at interface  407  and/or BMS interface  409 . 
     Building subsystem integration layer  420  can be configured to manage communications between BMS controller  366  and building subsystems  428 . For example, building subsystem integration layer  420  may receive sensor data and input signals from building subsystems  428  and provide output data and control signals to building subsystems  428 . Building subsystem integration layer  420  may also be configured to manage communications between building subsystems  428 . Building subsystem integration layer  420  translate communications (e.g., sensor data, input signals, output signals, etc.) across a plurality of multi-vendor/multi-protocol systems. 
     Demand response layer  414  can be configured to optimize resource usage (e.g., electricity use, natural gas use, water use, etc.) and/or the monetary cost of such resource usage in response to satisfy the demand of building  10 . The optimization can be based on time-of-use prices, curtailment signals, energy availability, or other data received from utility providers, distributed energy generation systems  424 , from energy storage  427  (e.g., hot TES  242 , cold TES  244 , etc.), or from other sources. Demand response layer  414  may receive inputs from other layers of BMS controller  366  (e.g., building subsystem integration layer  420 , integrated control layer  418 , etc.). The inputs received from other layers can include environmental or sensor inputs such as temperature, carbon dioxide levels, relative humidity levels, air quality sensor outputs, occupancy sensor outputs, room schedules, and the like. The inputs may also include inputs such as electrical use (e.g., expressed in kWh), thermal load measurements, pricing information, projected pricing, smoothed pricing, curtailment signals from utilities, and the like. 
     According to some embodiments, demand response layer  414  includes control logic for responding to the data and signals it receives. These responses can include communicating with the control algorithms in integrated control layer  418 , changing control strategies, changing setpoints, or activating/deactivating building equipment or subsystems in a controlled manner. Demand response layer  414  may also include control logic configured to determine when to utilize stored energy. For example, demand response layer  414  may determine to begin using energy from energy storage  427  just prior to the beginning of a peak use hour. 
     In some embodiments, demand response layer  414  includes a control module configured to actively initiate control actions (e.g., automatically changing setpoints) which minimize energy costs based on one or more inputs representative of or based on demand (e.g., price, a curtailment signal, a demand level, etc.). In some embodiments, demand response layer  414  uses equipment models to determine an optimal set of control actions. The equipment models can include, for example, thermodynamic models describing the inputs, outputs, and/or functions performed by various sets of building equipment. Equipment models may represent collections of building equipment (e.g., subplants, chiller arrays, etc.) or individual devices (e.g., individual chillers, heaters, pumps, etc.). 
     Demand response layer  414  may further include or draw upon one or more demand response policy definitions (e.g., databases, XML files, etc.). The policy definitions can be edited or adjusted by a user (e.g., via a graphical user interface) so that the control actions initiated in response to demand inputs can be tailored for the user&#39;s application, desired comfort level, particular building equipment, or based on other concerns. For example, the demand response policy definitions can specify which equipment can be turned on or off in response to particular demand inputs, how long a system or piece of equipment should be turned off, what setpoints can be changed, what the allowable set point adjustment range is, how long to hold a high demand setpoint before returning to a normally scheduled setpoint, how close to approach capacity limits, which equipment modes to utilize, the energy transfer rates (e.g., the maximum rate, an alarm rate, other rate boundary information, etc.) into and out of energy storage devices (e.g., thermal storage tanks, battery banks, etc.), and when to dispatch on-site generation of energy (e.g., via fuel cells, a motor generator set, etc.). 
     Integrated control layer  418  can be configured to use the data input or output of building subsystem integration layer  420  and/or demand response later  414  to make control decisions. Due to the subsystem integration provided by building subsystem integration layer  420 , integrated control layer  418  can integrate control activities of the subsystems  428  such that the subsystems  428  behave as a single integrated supersystem. In some embodiments, integrated control layer  418  includes control logic that uses inputs and outputs from a plurality of building subsystems to provide greater comfort and energy savings relative to the comfort and energy savings that separate subsystems could provide alone. For example, integrated control layer  418  can be configured to use an input from a first subsystem to make an energy-saving control decision for a second subsystem. Results of these decisions can be communicated back to building subsystem integration layer  420 . 
     Integrated control layer  418  is shown to be logically below demand response layer  414 . Integrated control layer  418  can be configured to enhance the effectiveness of demand response layer  414  by enabling building subsystems  428  and their respective control loops to be controlled in coordination with demand response layer  414 . This configuration may advantageously reduce disruptive demand response behavior relative to conventional systems. For example, integrated control layer  418  can be configured to assure that a demand response-driven upward adjustment to the setpoint for chilled water temperature (or another component that directly or indirectly affects temperature) does not result in an increase in fan energy (or other energy used to cool a space) that would result in greater total building energy use than was saved at the chiller. 
     Integrated control layer  418  can be configured to provide feedback to demand response layer  414  so that demand response layer  414  checks that constraints (e.g., temperature, lighting levels, etc.) are properly maintained even while demanded load shedding is in progress. The constraints may also include setpoint or sensed boundaries relating to safety, equipment operating limits and performance, comfort, fire codes, electrical codes, energy codes, and the like. Integrated control layer  418  is also logically below fault detection and diagnostics layer  416  and automated measurement and validation layer  412 . Integrated control layer  418  can be configured to provide calculated inputs (e.g., aggregations) to these higher levels based on outputs from more than one building subsystem. 
     Automated measurement and validation (AM&amp;V) layer  412  can be configured to verify that control strategies commanded by integrated control layer  418  or demand response layer  414  are working properly (e.g., using data aggregated by AM&amp;V layer  412 , integrated control layer  418 , building subsystem integration layer  420 , FDD layer  416 , or otherwise). The calculations made by AM&amp;V layer  412  can be based on building system energy models and/or equipment models for individual BMS devices or subsystems. For example, AM&amp;V layer  412  may compare a model-predicted output with an actual output from building subsystems  428  to determine an accuracy of the model. 
     Fault detection and diagnostics (FDD) layer  416  can be configured to provide on-going fault detection for building subsystems  428 , building subsystem devices (i.e., building equipment), and control algorithms used by demand response layer  414  and integrated control layer  418 . FDD layer  416  may receive data inputs from integrated control layer  418 , directly from one or more building subsystems or devices, or from another data source. FDD layer  416  may automatically diagnose and respond to detected faults. The responses to detected or diagnosed faults can include providing an alert message to a user, a maintenance scheduling system, or a control algorithm configured to attempt to repair the fault or to work-around the fault. 
     FDD layer  416  can be configured to output a specific identification of the faulty component or cause of the fault (e.g., loose damper linkage) using detailed subsystem inputs available at building subsystem integration layer  420 . In other exemplary embodiments, FDD layer  416  is configured to provide “fault” events to integrated control layer  418  which executes control strategies and policies in response to the received fault events. According to some embodiments, FDD layer  416  (or a policy executed by an integrated control engine or business rules engine) may shut-down systems or direct control activities around faulty devices or systems to reduce energy waste, extend equipment life, or assure proper control response. 
     FDD layer  416  can be configured to store or access a variety of different system data stores (or data points for live data). FDD layer  416  may use some content of the data stores to identify faults at the equipment level (e.g., specific chiller, specific AHU, specific terminal unit, etc.) and other content to identify faults at component or subsystem levels. For example, building subsystems  428  may generate temporal (i.e., time-series) data indicating the performance of BMS  400  and the various components thereof. The data generated by building subsystems  428  can include measured or calculated values that exhibit statistical characteristics and provide information about how the corresponding system or process (e.g., a temperature control process, a flow control process, etc.) is performing in terms of error from its setpoint. These processes can be examined by FDD layer  416  to expose when the system begins to degrade in performance and alert a user to repair the fault before it becomes more severe. 
     Referring now to  FIG. 5 , a block diagram of another building management system (BMS)  500  is shown, according to some embodiments. BMS  500  can be used to monitor and control the devices of HVAC system  100 , waterside system  200 , airside system  300 , building subsystems  428 , as well as other types of BMS devices (e.g., lighting equipment, security equipment, etc.) and/or HVAC equipment. 
     BMS  500  provides a system architecture that facilitates automatic equipment discovery and equipment model distribution. Equipment discovery can occur on multiple levels of BMS  500  across multiple different communications busses (e.g., a system bus  554 , zone buses  556 - 560  and  564 , sensor/actuator bus  566 , etc.) and across multiple different communications protocols. In some embodiments, equipment discovery is accomplished using active node tables, which provide status information for devices connected to each communications bus. For example, each communications bus can be monitored for new devices by monitoring the corresponding active node table for new nodes. When a new device is detected, BMS  500  can begin interacting with the new device (e.g., sending control signals, using data from the device) without user interaction. 
     Some devices in BMS  500  present themselves to the network using equipment models. An equipment model defines equipment object attributes, view definitions, schedules, trends, and the associated BACnet value objects (e.g., analog value, binary value, multistate value, etc.) that are used for integration with other systems. Some devices in BMS  500  store their own equipment models. Other devices in BMS  500  have equipment models stored externally (e.g., within other devices). For example, a zone coordinator  508  can store the equipment model for a bypass damper  528 . In some embodiments, zone coordinator  508  automatically creates the equipment model for bypass damper  528  or other devices on zone bus  558 . Other zone coordinators can also create equipment models for devices connected to their zone busses. The equipment model for a device can be created automatically based on the types of data points exposed by the device on the zone bus, device type, and/or other device attributes. Several examples of automatic equipment discovery and equipment model distribution are discussed in greater detail below. 
     Still referring to  FIG. 5 , BMS  500  is shown to include a system manager  502 ; several zone coordinators  506 ,  508 ,  510  and  518 ; and several zone controllers  524 ,  530 ,  532 ,  536 ,  548 , and  550 . System manager  502  can monitor data points in BMS  500  and report monitored variables to various monitoring and/or control applications. System manager  502  can communicate with client devices  504  (e.g., user devices, desktop computers, laptop computers, mobile devices, etc.) via a data communications link  574  (e.g., BACnet IP, Ethernet, wired or wireless communications, etc.). System manager  502  can provide a user interface to client devices  504  via data communications link  574 . The user interface may allow users to monitor and/or control BMS  500  via client devices  504 . 
     In some embodiments, system manager  502  is connected with zone coordinators  506 - 510  and  518  via a system bus  554 . System manager  502  can be configured to communicate with zone coordinators  506 - 510  and  518  via system bus  554  using a master-slave token passing (MSTP) protocol or any other communications protocol. System bus  554  can also connect system manager  502  with other devices such as a constant volume (CV) rooftop unit (RTU)  512 , an input/output module (IOM)  514 , a thermostat controller  516  (e.g., a TEC5000 series thermostat controller), and a network automation engine (NAE) or third-party controller  520 . RTU  512  can be configured to communicate directly with system manager  502  and can be connected directly to system bus  554 . Other RTUs can communicate with system manager  502  via an intermediate device. For example, a wired input  562  can connect a third-party RTU  542  to thermostat controller  516 , which connects to system bus  554 . 
     System manager  502  can provide a user interface for any device containing an equipment model. Devices such as zone coordinators  506 - 510  and  518  and thermostat controller  516  can provide their equipment models to system manager  502  via system bus  554 . In some embodiments, system manager  502  automatically creates equipment models for connected devices that do not contain an equipment model (e.g., IOM  514 , third party controller  520 , etc.). For example, system manager  502  can create an equipment model for any device that responds to a device tree request. The equipment models created by system manager  502  can be stored within system manager  502 . System manager  502  can then provide a user interface for devices that do not contain their own equipment models using the equipment models created by system manager  502 . In some embodiments, system manager  502  stores a view definition for each type of equipment connected via system bus  554  and uses the stored view definition to generate a user interface for the equipment. 
     Each zone coordinator  506 - 510  and  518  can be connected with one or more of zone controllers  524 ,  530 - 532 ,  536 , and  548 - 550  via zone buses  556 ,  558 ,  560 , and  564 . Zone coordinators  506 - 510  and  518  can communicate with zone controllers  524 ,  530 - 532 ,  536 , and  548 - 550  via zone busses  556 - 560  and  564  using a MSTP protocol or any other communications protocol. Zone busses  556 - 560  and  564  can also connect zone coordinators  506 - 510  and  518  with other types of devices such as variable air volume (VAV) RTUs  522  and  540 , changeover bypass (COBP) RTUs  526  and  552 , bypass dampers  528  and  546 , and PEAK controllers  534  and  544 . 
     Zone coordinators  506 - 510  and  518  can be configured to monitor and command various zoning systems. In some embodiments, each zone coordinator  506 - 510  and  518  monitors and commands a separate zoning system and is connected to the zoning system via a separate zone bus. For example, zone coordinator  506  can be connected to VAV RTU  522  and zone controller  524  via zone bus  556 . Zone coordinator  508  can be connected to COBP RTU  526 , bypass damper  528 , COBP zone controller  530 , and VAV zone controller  532  via zone bus  558 . Zone coordinator  510  can be connected to PEAK controller  534  and VAV zone controller  536  via zone bus  560 . Zone coordinator  518  can be connected to PEAK controller  544 , bypass damper  546 , COBP zone controller  548 , and VAV zone controller  550  via zone bus  564 . 
     A single model of zone coordinator  506 - 510  and  518  can be configured to handle multiple different types of zoning systems (e.g., a VAV zoning system, a COBP zoning system, etc.). Each zoning system can include a RTU, one or more zone controllers, and/or a bypass damper. For example, zone coordinators  506  and  510  are shown as Verasys VAV engines (VVEs) connected to VAV RTUs  522  and  540 , respectively. Zone coordinator  506  is connected directly to VAV RTU  522  via zone bus  556 , whereas zone coordinator  510  is connected to a third-party VAV RTU  540  via a wired input  568  provided to PEAK controller  534 . Zone coordinators  508  and  518  are shown as Verasys COBP engines (VCEs) connected to COBP RTUs  526  and  552 , respectively. Zone coordinator  508  is connected directly to COBP RTU  526  via zone bus  558 , whereas zone coordinator  518  is connected to a third-party COBP RTU  552  via a wired input  570  provided to PEAK controller  544 . 
     Zone controllers  524 ,  530 - 532 ,  536 , and  548 - 550  can communicate with individual BMS devices (e.g., sensors, actuators, etc.) via sensor/actuator (SA) busses. For example, VAV zone controller  536  is shown connected to networked sensors  538  via SA bus  566 . Zone controller  536  can communicate with networked sensors  538  using a MSTP protocol or any other communications protocol. Although only one SA bus  566  is shown in  FIG. 5 , it should be understood that each zone controller  524 ,  530 - 532 ,  536 , and  548 - 550  can be connected to a different SA bus. Each SA bus can connect a zone controller with various sensors (e.g., temperature sensors, humidity sensors, pressure sensors, light sensors, occupancy sensors, etc.), actuators (e.g., damper actuators, valve actuators, etc.) and/or other types of controllable equipment (e.g., chillers, heaters, fans, pumps, etc.). 
     Each zone controller  524 ,  530 - 532 ,  536 , and  548 - 550  can be configured to monitor and control a different building zone. Zone controllers  524 ,  530 - 532 ,  536 , and  548 - 550  can use the inputs and outputs provided via their SA busses to monitor and control various building zones. For example, a zone controller  536  can use a temperature input received from networked sensors  538  via SA bus  566  (e.g., a measured temperature of a building zone) as feedback in a temperature control algorithm. Zone controllers  524 ,  530 - 532 ,  536 , and  548 - 550  can use various types of control algorithms (e.g., state-based algorithms, extremum seeking control (ESC) algorithms, proportional-integral (PI) control algorithms, proportional-integral-derivative (PID) control algorithms, model predictive control (MPC) algorithms, feedback control algorithms, etc.) to control a variable state or condition (e.g., temperature, humidity, airflow, lighting, etc.) in or around building  10 . 
     Model Predictive Maintenance System 
     Referring now to  FIG. 6 , a block diagram of a building system  600  is shown, according to an exemplary embodiment. System  600  may include many of the same components as BMS  400  and BMS  500  as described with reference to  FIGS. 4-5 . For example, system  600  is shown to include building  10 , network  446 , and client devices  448 . Building  10  is shown to include connected equipment  610 , which can include any type of equipment used to monitor and/or control building  10 . Connected equipment  610  can include connected chillers  612 , connected AHUs  614 , connected boilers  616 , connected batteries  618 , or any other type of equipment in a building system (e.g., heaters, economizers, valves, actuators, dampers, cooling towers, fans, pumps, etc.) or building management system (e.g., lighting equipment, security equipment, refrigeration equipment, etc.). Connected equipment  610  can include any of the equipment of HVAC system  100 , waterside system  200 , airside system  300 , BMS  400 , and/or BMS  500 , as described with reference to  FIGS. 1-5 . 
     Connected equipment  610  can be outfitted with sensors to monitor various conditions of the connected equipment  610  (e.g., power consumption, on/off states, operating efficiency, etc.). For example, chillers  612  can include sensors configured to monitor chiller variables such as chilled water temperature, condensing water temperature, and refrigerant properties (e.g., refrigerant pressure, refrigerant temperature, etc.) at various locations in the refrigeration circuit. An example of a chiller  700  which can be used as one of chillers  612  is shown in  FIG. 7 . Chiller  700  is shown to include a refrigeration circuit having a condenser  702 , an expansion valve  704 , an evaporator  706 , a compressor  708 , and a control panel  710 . In some embodiments, chiller  700  includes sensors that measure a set of monitored variables at various locations along the refrigeration circuit. Similarly, AHUs  614  can be outfitted with sensors to monitor AHU variables such as supply air temperature and humidity, outside air temperature and humidity, return air temperature and humidity, chilled fluid temperature, heated fluid temperature, damper position, etc. In general, connected equipment  610  can monitor and report variables that characterize the performance of the connected equipment  610 . Each monitored variable can be forwarded to building management system  606  as a data point including a point ID and a point value. 
     Monitored variables can include any measured or calculated values indicating the performance of connected equipment  610  and/or the components thereof. For example, monitored variables can include one or more measured or calculated temperatures (e.g., refrigerant temperatures, cold water supply temperatures, hot water supply temperatures, supply air temperatures, zone temperatures, etc.), pressures (e.g., evaporator pressure, condenser pressure, supply air pressure, etc.), flow rates (e.g., cold water flow rates, hot water flow rates, refrigerant flow rates, supply air flow rates, etc.), valve positions, resource consumptions (e.g., power consumption, water consumption, electricity consumption, etc.), control setpoints, model parameters (e.g., regression model coefficients), or any other time-series values that provide information about how the corresponding system, device, or process is performing. Monitored variables can be received from connected equipment  610  and/or from various components thereof. For example, monitored variables can be received from one or more controllers (e.g., BMS controllers, subsystem controllers, HVAC controllers, subplant controllers, AHU controllers, device controllers, etc.), BMS devices (e.g., chillers, cooling towers, pumps, heating elements, etc.), or collections of BMS devices. 
     Connected equipment  610  can also report equipment status information. Equipment status information can include, for example, the operational status of the equipment, an operating mode (e.g., low load, medium load, high load, etc.), an indication of whether the equipment is running under normal or abnormal conditions, the hours during which the equipment is running, a safety fault code, or any other information that indicates the current status of connected equipment  610 . In some embodiments, each device of connected equipment  610  includes a control panel (e.g., control panel  710  shown in  FIG. 7 ). Control panel  710  can be configured to collect monitored variables and equipment status information from connected equipment  610  and provide the collected data to BMS  606 . For example, control panel  710  can compare the sensor data (or a value derived from the sensor data) to predetermined thresholds. If the sensor data or calculated value crosses a safety threshold, control panel  710  can shut down the device. Control panel  710  can generate a data point when a safety shut down occurs. The data point can include a safety fault code which indicates the reason or condition that triggered the shutdown. 
     Connected equipment  610  can provide monitored variables and equipment status information to BMS  606 . BMS  606  can include a building controller (e.g., BMS controller  366 ), a system manager (e.g., system manager  503 ), a network automation engine (e.g., NAE  520 ), or any other system or device of building  10  configured to communicate with connected equipment  610 . BMS  606  may include some or all of the components of BMS  400  or BMS  500 , as described with reference to  FIGS. 4-5 . In some embodiments, the monitored variables and the equipment status information are provided to BMS  606  as data points. Each data point can include a point ID and a point value. The point ID can identify the type of data point or a variable measured by the data point (e.g., condenser pressure, refrigerant temperature, power consumption, etc.). Monitored variables can be identified by name or by an alphanumeric code (e.g., Chilled_Water_Temp, 7694, etc.). The point value can include an alphanumeric value indicating the current value of the data point. 
     BMS  606  can broadcast the monitored variables and the equipment status information to a model predictive maintenance system  602 . In some embodiments, model predictive maintenance system  602  is a component of BMS  606 . For example, model predictive maintenance system  602  can be implemented as part of a METASYS® brand building automation system, as sold by Johnson Controls Inc. In other embodiments, model predictive maintenance system  602  can be a component of a remote computing system or cloud-based computing system configured to receive and process data from one or more building management systems via network  446 . For example, model predictive maintenance system  602  can be implemented as part of a PANOPTIX® brand building efficiency platform, as sold by Johnson Controls Inc. In other embodiments, model predictive maintenance system  602  can be a component of a subsystem level controller (e.g., a HVAC controller), a subplant controller, a device controller (e.g., AHU controller  330 , a chiller controller, etc.), a field controller, a computer workstation, a client device, or any other system or device that receives and processes monitored variables from connected equipment  610 . 
     Model predictive maintenance (MPM) system  602  may use the monitored variables and/or the equipment status information to identify a current operating state of connected equipment  610 . The current operating state can be examined by MPM system  602  to expose when connected equipment  610  begins to degrade in performance and/or to predict when faults will occur. In some embodiments, MPM system  602  uses the information collected from connected equipment  610  to estimate the reliability of connected equipment  610 . For example, MPM system  602  can estimate a likelihood of various types of failures that could potentially occur based on the current operating conditions of connected equipment  610  and an amount of time that has elapsed since connected equipment  610  has been installed and/or since maintenance was last performed. In some embodiments, MPM system  602  estimates an amount of time until each failure is predicted to occur and identifies a financial cost associated with each failure (e.g., maintenance cost, increased operating cost, replacement cost, etc.). MPM system  602  can use the reliability information and the likelihood of potential failures to predict when maintenance will be needed and to estimate the cost of performing such maintenance over a predetermined time period. 
     MPM system  602  can be configured to determine an optimal maintenance strategy for connected equipment  610 . In some embodiments, the optimal maintenance strategy is a set of decisions which optimizes the total cost associated with purchasing, maintaining, and operating connected equipment  610  over the duration of an optimization period (e.g., 30 weeks, 52 weeks, 10 years, 30 years, etc.). The decisions can include, for example, equipment purchase decisions, equipment maintenance decisions, and equipment operating decisions. MPM system  602  can use a model predictive control technique to formulate an objective function which expresses the total cost as a function of these decisions, which can be included as decision variables in the objective function. MPM system  602  can optimize (i.e., minimize) the objective function using any of a variety of optimization techniques to identify the optimal values for each of the decision variables. 
     One example of an objective function which can be optimized by MPM system  602  is shown in the following equation: 
     
       
         
           
             J 
             = 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       op 
                       , 
                       i 
                     
                   
                    
                   
                     P 
                     
                       op 
                       , 
                       i 
                     
                   
                    
                   Δ 
                    
                   
                       
                   
                    
                   t 
                 
               
               + 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       main 
                       , 
                       i 
                     
                   
                    
                   
                     B 
                     
                       main 
                       , 
                       i 
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       cap 
                       , 
                       i 
                     
                   
                    
                   
                     P 
                     
                       cap 
                       , 
                       i 
                     
                   
                 
               
             
           
         
       
     
     where C op,i  is the cost per unit of energy (e.g., $/kWh) consumed by connected equipment  610  at time step i of the optimization period, P op,i  is the power consumption (e.g., kW) of connected equipment  610  at time step i, Δt is the duration of each time step i, C main,i  is the cost of maintenance performed on connected equipment  610  at time step i, B main,i  is a binary variable that indicates whether the maintenance is performed, C cap,i  is the capital cost of purchasing a new device of connected equipment  610  at time step i, B cap,i  is a binary variable that indicates whether the new device is purchased, and h is the duration of the horizon or optimization period over which the optimization is performed. 
     The first term in the objective function J represents the operating cost of connected equipment  610  over the duration of the optimization period. In some embodiments, the cost per unit of energy C op,i  is received from a utility  608  as energy pricing data. The cost C op,i  may be a time-varying cost that depends on the time of day, the day of the week (e.g., weekday vs. weekend), the current season (e.g., summer vs. winter), or other time-based factors. For example, the cost C op,i  may be higher during peak energy consumption periods and lower during off-peak or partial-peak energy consumption periods. 
     In some embodiments, the power consumption P op,i  is based on the heating or cooling load of building  10 . The heating or cooling load can be predicted by MPM system  602  as a function of building occupancy, the time of day, the day of the week, the current season, or other factors that can affect the heating or cooling load. In some embodiments, MPM system  602  uses weather forecasts from a weather service  604  to predict the heating or cooling load. The power consumption P op,i  may also depend on the efficiency η i  of connected equipment  610 . For example, connected equipment  610  that operate at a high efficiency may consume less power P op,i  to satisfy the same heating or cooling load relative to connected equipment  610  that operate at a low efficiency. In general, the power consumption P op,i  of a particular device of connected equipment  610  can be modeled using the following equations: 
     
       
         
           
             
               
                 P 
                 
                   
                     o 
                      
                     p 
                   
                   , 
                   i 
                 
               
               = 
               
                 
                   P 
                   
                     
                       i 
                        
                       d 
                        
                       e 
                        
                       a 
                        
                       l 
                     
                     , 
                     i 
                   
                 
                 
                   η 
                   i 
                 
               
             
              
             
               
 
             
              
             
               
                 P 
                 
                   
                     i 
                      
                     d 
                      
                     e 
                      
                     a 
                      
                     l 
                   
                   , 
                   i 
                 
               
               = 
               
                 f 
                  
                 
                   ( 
                   
                     L 
                      
                     o 
                      
                     a 
                      
                     
                       d 
                       i 
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where Load i  is the heating or cooling load on the device at time step i (e.g., tons cooling, kW heating, etc.), P ideal,i  is the value of the equipment performance curve (e.g., tons cooling, kW heating, etc.) for the device at the corresponding load point Load i , and η i  is the operating efficiency of the device at time step i (e.g., 0≤η i ≤1). The function ƒ(Load i ) may be defined by the equipment performance curve for the device or set of devices represented by the performance curve. 
     In some embodiments, the equipment performance curve is based on manufacturer specifications for the device under ideal operating conditions. For example, the equipment performance curve may define the relationship between power consumption and heating/cooling load for each device of connected equipment  610 . However, the actual performance of the device may vary as a function of the actual operating conditions. MPM system  602  can analyze the equipment performance information provided by connected equipment  610  to determine the operating efficiency η i  for each device of connected equipment  610 . In some embodiments, MPM system  602  uses the equipment performance information from connected equipment  610  to determine the actual operating efficiency η i  for each device of connected equipment  610 . MPM system  602  can use the operating efficiency η i  as an input to the objective function J and/or to calculate the corresponding value of P op,i . 
     Advantageously, MPM system  602  can model the efficiency η i  of connected equipment  610  at each time step i as a function of the maintenance decisions B main,i  and the equipment purchase decisions B cap,i . For example, the efficiency η i  for a particular device may start at an initial value η 0  when the device is purchased and may degrade over time such that the efficiency η i  decreases with each successive time step i. Performing maintenance on a device may reset the efficiency η i  to a higher value immediately after the maintenance is performed. Similarly, purchasing a new device to replace an existing device may reset the efficiency η i  to a higher value immediately after the new device is purchased. After being reset, the efficiency η i  may continue to degrade over time until the next time at which maintenance is performed or a new device is purchased. 
     Performing maintenance or purchasing a new device may result in a relatively lower power consumption P op,i  during operation and therefore a lower operating cost at each time step i after the maintenance is performed or the new device is purchased. In other words, performing maintenance or purchasing a new device may decrease the operating cost represented by the first term of the objective function J. However, performing maintenance may increase the second term of the objective function J and purchasing a new device may increase the third term of the objective function J. The objective function J captures each of these costs and can be optimized by MPM system  602  to determine the optimal set of maintenance and equipment purchase decisions (i.e., optimal values for the binary decision variables B main,i  and B cap,i ) over the duration of the optimization period. 
     In some embodiments, MPM system  602  uses the equipment performance information from connected equipment  610  to estimate the reliability of connected equipment  610 . The reliability may be a statistical measure of the likelihood that connected equipment  610  will continue operating without fault under its current operating conditions. Operating under more strenuous conditions (e.g., high load, high temperatures, etc.) may result in a lower reliability, whereas operating under less strenuous conditions (e.g., low load, moderate temperatures, etc.) may result in a higher reliability. In some embodiments, the reliability is based on an amount of time that has elapsed since connected equipment  610  last received maintenance. 
     MPM system  602  may receive operating data from a plurality of devices of connected equipment  610  distributed across multiple buildings and can use the set of operating data (e.g., operating conditions, fault indications, failure times, etc.) to develop a reliability model for each type of equipment. The reliability models can be used by MPM system  602  to estimate the reliability of any given device of connected equipment  610  as a function of its current operating conditions and/or other extraneous factors (e.g., time since maintenance was last performed, geographic location, water quality, etc.). In some embodiments, MPM system  602  uses the estimated reliability of each device of connected equipment  610  to determine the probability that the device will require maintenance and/or replacement at each time step of the optimization period. MPM system  602  can use these probabilities to determine the optimal set of maintenance and equipment purchase decisions (i.e., optimal values for the binary decision variables B main,i  and B cap,i ) over the duration of the optimization period. 
     In some embodiments, MPM system  602  generates and provides equipment purchase and maintenance recommendations. The equipment purchase and maintenance recommendations may be based on the optimal values for the binary decision variables B main,i  and B cap,i  determined by optimizing the objective function J. For example, a value of B main,25 =1 for a particular device of connected equipment  610  may indicate that maintenance should be performed on that device at the 25 th  time step of the optimization period, whereas a value of B main,25 =0 may indicate that the maintenance should not be performed at that time step. Similarly, a value of B cap,25 =1 may indicate that a new device of connected equipment  610  should be purchased at the 25 th  time step of the optimization period, whereas a value of B cap,25 =0 may indicate that the new device should not be purchased at that time step. 
     Advantageously, the equipment purchase and maintenance recommendations generated by MPM system  602  are predictive recommendations based on the actual operating conditions and actual performance of connected equipment  610 . The optimization performed by MPM system  602  weighs the cost of performing maintenance and the cost of purchasing new equipment against the decrease in operating cost resulting from such maintenance or purchase decisions in order to determine the optimal maintenance strategy that minimizes the total combined cost J. In this way, the equipment purchase and maintenance recommendations generated by MPM system  602  may be specific to each group of connected equipment  610  in order to achieve the optimal cost J for that specific group of connected equipment  610 . The equipment-specific recommendations may result in a lower overall cost J relative to generic preventive maintenance recommendations provided by an equipment manufacturer (e.g., service equipment every year) which may be sub-optimal for some groups of connected equipment  610  and/or some operating conditions. 
     In some embodiments, the equipment purchase and maintenance recommendations are provided to building  10  (e.g., to BMS  606 ) and/or to client devices  448 . An operator or building owner can use the equipment purchase and maintenance recommendations to assess the costs and benefits of performing maintenance and purchasing new devices. In some embodiments, the equipment purchase and maintenance recommendations are provided to service technicians  620 . Service technicians  620  can use the equipment purchase and maintenance recommendations to determine when customers should be contacted to perform service or replace equipment. 
     In some embodiments, MPM system  602  includes a data analytics and visualization platform. MPM system  602  may provide a web interface which can be accessed by service technicians  620 , client devices  448 , and other systems or devices. The web interface can be used to access the equipment performance information, view the results of the optimization, identify which equipment is in need of maintenance, and otherwise interact with MPM system  602 . Service technicians  620  can access the web interface to view a list of equipment for which maintenance is recommended by MPM system  602 . Service technicians  620  can use the equipment purchase and maintenance recommendations to proactively repair or replace connected equipment  610  in order to achieve the optimal cost predicted by the objective function J. These and other features of MPM system  602  are described in greater detail below. 
     Referring now to  FIG. 8 , a block diagram illustrating MPM system  602  in greater detail is shown, according to an exemplary embodiment. MPM system  602  is shown providing optimization results to a building management system (BMS)  606 . BMS  606  can include some or all of the features of BMS  400  and/or BMS  500 , as described with reference to  FIGS. 4-5 . The optimization results provided to BMS  606  may include the optimal values of the decision variables in the objective function J for each time step i in the optimization period. In some embodiments, the optimization results include equipment purchase and maintenance recommendations for each device of connected equipment  610 . 
     BMS  606  may be configured to monitor the operation and performance of connected equipment  610 . BMS  606  may receive monitored variables from connected equipment  610 . Monitored variables can include any measured or calculated values indicating the performance of connected equipment  610  and/or the components thereof. For example, monitored variables can include one or more measured or calculated temperatures, pressures, flow rates, valve positions, resource consumptions (e.g., power consumption, water consumption, electricity consumption, etc.), control setpoints, model parameters (e.g., equipment model coefficients), or any other variables that provide information about how the corresponding system, device, or process is performing. 
     In some embodiments, the monitored variables indicate the operating efficiency η i  of each device of connected equipment  610  or can be used to calculate the operating efficiency η i . For example, the temperature and flow rate of chilled water output by a chiller can be used to calculate the cooling load (e.g., tons cooling) served by the chiller. The cooling load can be used in combination with the power consumption of the chiller to calculate the operating efficiency η i  (e.g., tons cooling per kW of electricity consumed). BMS  606  may report the monitored variables to MPM system  602  for use in calculating the operating efficiency η i  of each device of connected equipment  610 . 
     In some embodiments, BMS  606  monitors the run hours of connected equipment  610 . The run hours may indicate the number of hours within a given time period during which each device of connected equipment  610  is active. For example, the run hours for a chiller may indicate that the chiller is active for approximately eight hours per day. The run hours can be used in combination with the average power consumption of the chiller when active to estimate the total power consumption P op,i  of connected equipment  610  at each time step i. 
     In some embodiments, BMS  606  monitors the equipment failures and fault indications reported by connected equipment  610 . BMS  606  can record the times at which each failure or fault occurs and the operating conditions of connected equipment  610  under which the fault or failure occurred. The operating data collected from connected equipment  610  can be used by BMS  606  and/or MPM system  602  to develop a reliability model for each device of connected equipment  610 . BMS  606  may provide the monitored variables, the equipment run hours, the operating conditions, and the equipment failures and fault indications to MPM system  602  as equipment performance information. 
     BMS  606  may be configured to monitor conditions within a controlled building or building zone. For example, BMS  606  may receive input from various sensors (e.g., temperature sensors, humidity sensors, airflow sensors, voltage sensors, etc.) distributed throughout the building and may report building conditions to MPM system  602 . Building conditions may include, for example, a temperature of the building or a zone of the building, a power consumption (e.g., electric load) of the building, a state of one or more actuators configured to affect a controlled state within the building, or other types of information relating to the controlled building. BMS  606  may operate connected equipment  610  to affect the monitored conditions within the building and to serve the thermal energy loads of the building. 
     BMS  606  may provide control signals to connected equipment  610  specifying on/off states, charge/discharge rates, and/or setpoints for connected equipment  610 . BMS  606  may control the equipment (e.g., via actuators, power relays, etc.) in accordance with the control signals to achieve setpoints for various building zones and/or devices of connected equipment  610 . In various embodiments, BMS  606  may be combined with MPM system  602  or may be part of a separate building management system. According to an exemplary embodiment, BMS  606  is a METASYS® brand building management system, as sold by Johnson Controls, Inc. 
     MPM system  602  may monitor the performance of connected equipment  610  using information received from BMS  606 . MPM system  602  may be configured to predict the thermal energy loads (e.g., heating loads, cooling loads, etc.) of the building for plurality of time steps in the optimization period (e.g., using weather forecasts from a weather service  604 ). MPM system  602  may also predict the cost of electricity or other resources (e.g., water, natural gas, etc.) using pricing data received from utilities  608 . MPM system  602  may generate optimization results that optimize the economic value of operating, maintaining, and purchasing connected equipment  610  over the duration of the optimization period subject to constraints on the optimization process (e.g., load constraints, decision variable constraints, etc.). The optimization process performed by MPM system  602  is described in greater detail below. 
     According to an exemplary embodiment, MPM system  602  can be integrated within a single computer (e.g., one server, one housing, etc.). In various other exemplary embodiments, MPM system  602  can be distributed across multiple servers or computers (e.g., that can exist in distributed locations). In another exemplary embodiment, MPM system  602  may integrated with a smart building manager that manages multiple building systems and/or combined with BMS  606 . 
     MPM system  602  is shown to include a communications interface  804  and a processing circuit  806 . Communications interface  804  may include wired or wireless interfaces (e.g., jacks, antennas, transmitters, receivers, transceivers, wire terminals, etc.) for conducting data communications with various systems, devices, or networks. For example, communications interface  804  may include an Ethernet card and port for sending and receiving data via an Ethernet-based communications network and/or a WiFi transceiver for communicating via a wireless communications network. Communications interface  804  may be configured to communicate via local area networks or wide area networks (e.g., the Internet, a building WAN, etc.) and may use a variety of communications protocols (e.g., BACnet, IP, LON, etc.). 
     Communications interface  804  may be a network interface configured to facilitate electronic data communications between MPM system  602  and various external systems or devices (e.g., BMS  606 , connected equipment  610 , utilities  510 , etc.). For example, MPM system  602  may receive information from BMS  606  indicating one or more measured states of the controlled building (e.g., temperature, humidity, electric loads, etc.) and equipment performance information for connected equipment  610  (e.g., run hours, power consumption, operating efficiency, etc.). Communications interface  804  may receive inputs from BMS  606  and/or connected equipment  610  and may provide optimization results to BMS  606  and/or other external systems or devices. The optimization results may cause BMS  606  to activate, deactivate, or adjust a setpoint for connected equipment  610  in order to achieve the optimal values of the decision variables specified in the optimization results. 
     Still referring to  FIG. 8 , processing circuit  806  is shown to include a processor  808  and memory  810 . Processor  808  may be a general purpose or specific purpose processor, an application specific integrated circuit (ASIC), one or more field programmable gate arrays (FPGAs), a group of processing components, or other suitable processing components. Processor  808  may be configured to execute computer code or instructions stored in memory  810  or received from other computer readable media (e.g., CDROM, network storage, a remote server, etc.). 
     Memory  810  may include one or more devices (e.g., memory units, memory devices, storage devices, etc.) for storing data and/or computer code for completing and/or facilitating the various processes described in the present disclosure. Memory  810  may include random access memory (RAM), read-only memory (ROM), hard drive storage, temporary storage, non-volatile memory, flash memory, optical memory, or any other suitable memory for storing software objects and/or computer instructions. Memory  810  may include database components, object code components, script components, or any other type of information structure for supporting the various activities and information structures described in the present disclosure. Memory  810  may be communicably connected to processor  808  via processing circuit  806  and may include computer code for executing (e.g., by processor  808 ) one or more processes described herein. 
     MPM system  602  is shown to include an equipment performance monitor  824 . Equipment performance monitor  824  can receive equipment performance information from BMS  606  and/or connected equipment  610 . The equipment performance information can include samples of monitored variables (e.g., measured temperature, measured pressure, measured flow rate, power consumption, etc.), current operating conditions (e.g., heating or cooling load, current operating state, etc.), fault indications, or other types of information that characterize the performance of connected equipment  610 . In some embodiments, equipment performance monitor  824  uses the equipment performance information to calculate the current efficiency η i  and reliability of each device of connected equipment  610 . Equipment performance monitor  824  can provide the efficiency η i  and reliability values to model predictive optimizer  830  for use in optimizing the objective function J. 
     Still referring to  FIG. 8 , MPM system  602  is shown to include a load/rate predictor  822 . Load/rate predictor  822  may be configured to predict the energy loads (Load i ) (e.g., heating load, cooling load, electric load, etc.) of the building or campus for each time step i of the optimization period. Load/rate predictor  822  is shown receiving weather forecasts from a weather service  604 . In some embodiments, load/rate predictor  822  predicts the energy loads Load i  as a function of the weather forecasts. In some embodiments, load/rate predictor  822  uses feedback from BMS  606  to predict loads Load i . Feedback from BMS  606  may include various types of sensory inputs (e.g., temperature, flow, humidity, enthalpy, etc.) or other data relating to the controlled building (e.g., inputs from a HVAC system, a lighting control system, a security system, a water system, etc.). 
     In some embodiments, load/rate predictor  822  receives a measured electric load and/or previous measured load data from BMS  606  (e.g., via equipment performance monitor  824 ). Load/rate predictor  822  may predict loads Load i  as a function of a given weather forecast ({circumflex over (ϕ)} w ), a day type (day), the time of day (t), and previous measured load data (Y i-1 ). Such a relationship is expressed in the following equation: 
       Load i =ƒ({circumflex over (ϕ)} w ,day, t|Y   i-1 )
 
     In some embodiments, load/rate predictor  822  uses a deterministic plus stochastic model trained from historical load data to predict loads Load i . Load/rate predictor  822  may use any of a variety of prediction methods to predict loads Load i  (e.g., linear regression for the deterministic portion and an AR model for the stochastic portion). Load/rate predictor  822  may predict one or more different types of loads for the building or campus. For example, load/rate predictor  822  may predict a hot water load Load Hot,i  a cold water load Load Cold,i , and an electric load Load Elec,i  for each time step i within the optimization period. The predicted load values Load i  can include some or all of these types of loads. In some embodiments, load/rate predictor  822  makes load/rate predictions using the techniques described in U.S. patent application Ser. No. 14/717,593. 
     Load/rate predictor  822  is shown receiving utility rates from utilities  608 . Utility rates may indicate a cost or price per unit of a resource (e.g., electricity, natural gas, water, etc.) provided by utilities  608  at each time step i in the optimization period. In some embodiments, the utility rates are time-variable rates. For example, the price of electricity may be higher at certain times of day or days of the week (e.g., during high demand periods) and lower at other times of day or days of the week (e.g., during low demand periods). The utility rates may define various time periods and a cost per unit of a resource during each time period. Utility rates may be actual rates received from utilities  608  or predicted utility rates estimated by load/rate predictor  822 . 
     In some embodiments, the utility rates include demand charges for one or more resources provided by utilities  608 . A demand charge may define a separate cost imposed by utilities  608  based on the maximum usage of a particular resource (e.g., maximum energy consumption) during a demand charge period. The utility rates may define various demand charge periods and one or more demand charges associated with each demand charge period. In some instances, demand charge periods may overlap partially or completely with each other and/or with the prediction window. Model predictive optimizer  830  may be configured to account for demand charges in the high level optimization process performed by high level optimizer  832 . Utilities  608  may be defined by time-variable (e.g., hourly) prices, a maximum service level (e.g., a maximum rate of consumption allowed by the physical infrastructure or by contract) and, in the case of electricity, a demand charge or a charge for the peak rate of consumption within a certain period. Load/rate predictor  822  may store the predicted loads Load and the utility rates in memory  810  and/or provide the predicted loads Load i  and the utility rates to model predictive optimizer  830 . 
     Still referring to  FIG. 8 , MPM system  602  is shown to include a model predictive optimizer  830 . Model predictive optimizer  830  can be configured to perform a multi-level optimization process to optimize the total cost associated with purchasing, maintaining, and operating connected equipment  610 . In some embodiments, model predictive optimizer  830  includes a high level optimizer  832  and a low level optimizer  834 . High level optimizer  832  may optimize the objective function J for an entire set of connected equipment  610  (e.g., all of the devices within a building) or for a subset of connected equipment  610  (e.g., a single device, all of the devices of a subplant or building subsystem, etc.) to determine the optimal values for each of the decision variables (e.g., P op,i , B main,i , and B cap,i ) in the objective function J. The optimization performed by high level optimizer  832  is described in greater detail with reference to  FIG. 9 . 
     In some embodiments, low level optimizer  834  receives the optimization results from high level optimizer  832 . The optimization results may include optimal power consumption values P op,i  and/or load values Load i  for each device or set of devices of connected equipment at each time step i in the optimization period. Low level optimizer  834  may determine how to best run each device or set of devices at the load values determined by high level optimizer  832 . For example, low level optimizer  834  may determine on/off states and/or operating setpoints for various devices of connected equipment  610  in order to optimize (e.g., minimize) the power consumption of connected equipment  610  meeting the corresponding load value Load i . 
     Low level optimizer  834  may be configured to generate equipment performance curves for each device or set of devices of connected equipment  610 . Each performance curve may indicate an amount of resource consumption (e.g., electricity use measured in kW, water use measured in L/s, etc.) by a particular device or set of devices of connected equipment  610  as a function of the load on the device or set of devices. In some embodiments, low level optimizer  834  generates the performance curves by performing a low level optimization process at various combinations of load points (e.g., various values of Load i ) and weather conditions to generate multiple data points. The low level optimization may be used to determine the minimum amount of resource consumption required to satisfy the corresponding heating or cooling load. An example of a low level optimization process which can be performed by low level optimizer  834  is described in detail in U.S. patent application Ser. No. 14/634,615 titled “Low Level Central Plant Optimization” and filed Feb. 27, 2015, the entire disclosure of which is incorporated by reference herein. Low level optimizer  834  may fit a curve to the data points to generate the performance curves. 
     In some embodiments, low level optimizer  834  generates equipment performance curves for a set of connected equipment  610  (e.g., a chiller subplant, a heater subplant, etc.) by combining efficiency curves for individual devices of connected equipment  610 . A device efficiency curve may indicate the amount of resource consumption by the device as a function of load. The device efficiency curves may be provided by a device manufacturer or generated using experimental data. In some embodiments, the device efficiency curves are based on an initial efficiency curve provided by a device manufacturer and updated using experimental data. The device efficiency curves may be stored in equipment models  818 . For some devices, the device efficiency curves may indicate that resource consumption is a U-shaped function of load. Accordingly, when multiple device efficiency curves are combined into a performance curve for multiple devices, the resultant performance curve may be a wavy curve. The waves are caused by a single device loading up before it is more efficient to turn on another device to satisfy the subplant load. Low level optimizer  834  may provide the equipment performance curves to high level optimizer  832  for use in the high level optimization process. 
     Still referring to  FIG. 8 , MPM system  602  is shown to include an equipment controller  828 . Equipment controller  828  can be configured to control connected equipment  610  to affect a variable state or condition in building  10  (e.g., temperature, humidity, etc.). In some embodiments, equipment controller  828  controls connected equipment  610  based on the results of the optimization performed by model predictive optimizer  830 . In some embodiments, equipment controller  828  generates control signals which can be provided to connected equipment  610  via communications interface  804  and/or BMS  606 . The control signals may be based on the optimal values of the decision variables in the objective function J. For example, equipment controller  828  may generate control signals which cause connected equipment  610  to achieve the optimal power consumption values P op,i  for each time step i in the optimization period. 
     Data and processing results from model predictive optimizer  830 , equipment controller  828 , or other modules of MPM system  602  may be accessed by (or pushed to) monitoring and reporting applications  826 . Monitoring and reporting applications  826  may be configured to generate real time “system health” dashboards that can be viewed and navigated by a user (e.g., a system engineer). For example, monitoring and reporting applications  826  may include a web-based monitoring application with several graphical user interface (GUI) elements (e.g., widgets, dashboard controls, windows, etc.) for displaying key performance indicators (KPI) or other information to users of a GUI. In addition, the GUI elements may summarize relative energy use and intensity across building management systems in different buildings (real or modeled), different campuses, or the like. Other GUI elements or reports may be generated and shown based on available data that allow users to assess performance across one or more energy storage systems from one screen. The user interface or report (or underlying data engine) may be configured to aggregate and categorize operating conditions by building, building type, equipment type, and the like. The GUI elements may include charts or histograms that allow the user to visually analyze the operating parameters and power consumption for the devices of the building system. 
     Still referring to  FIG. 8 , MPM system  602  may include one or more GUI servers, web services  812 , or GUI engines  814  to support monitoring and reporting applications  826 . In various embodiments, applications  826 , web services  812 , and GUI engine  814  may be provided as separate components outside of MPM system  602  (e.g., as part of a smart building manager). MPM system  602  may be configured to maintain detailed historical databases (e.g., relational databases, XML databases, etc.) of relevant data and includes computer code modules that continuously, frequently, or infrequently query, aggregate, transform, search, or otherwise process the data maintained in the detailed databases. MPM system  602  may be configured to provide the results of any such processing to other databases, tables, XML files, or other data structures for further querying, calculation, or access by, for example, external monitoring and reporting applications. 
     MPM system  602  is shown to include configuration tools  816 . Configuration tools  816  can allow a user to define (e.g., via graphical user interfaces, via prompt-driven “wizards,” etc.) how MPM system  602  should react to changing conditions in BMS  606  and/or connected equipment  610 . In an exemplary embodiment, configuration tools  816  allow a user to build and store condition-response scenarios that can cross multiple devices of connected equipment  610 , multiple building systems, and multiple enterprise control applications (e.g., work order management system applications, entity resource planning applications, etc.). For example, configuration tools  816  can provide the user with the ability to combine data (e.g., from subsystems, from event histories) using a variety of conditional logic. In varying exemplary embodiments, the conditional logic can range from simple logical operators between conditions (e.g., AND, OR, XOR, etc.) to pseudo-code constructs or complex programming language functions (allowing for more complex interactions, conditional statements, loops, etc.). Configuration tools  816  can present user interfaces for building such conditional logic. The user interfaces may allow users to define policies and responses graphically. In some embodiments, the user interfaces may allow a user to select a pre-stored or pre-constructed policy and adapt it or enable it for use with their system. 
     High Level Optimizer 
     Referring now to  FIG. 9 , a block diagram illustrating high level optimizer  832  in greater detail is shown, according to an exemplary embodiment. High level optimizer  832  can be configured to determine an optimal maintenance strategy for connected equipment  610 . In some embodiments, the optimal maintenance strategy is a set of decisions which optimizes the total cost associated with purchasing, maintaining, and operating connected equipment  610  over the duration of an optimization period (e.g., 30 weeks, 52 weeks, 10 years, 30 years, etc.). The decisions can include, for example, equipment purchase decisions, equipment maintenance decisions, and equipment operating decisions. 
     High level optimizer  832  is shown to include an operational cost predictor  910 , a maintenance cost predictor  920 , a capital cost predictor  930 , an objective function generator  935 , and an objective function optimizer  940 . Cost predictors  910 ,  920 , and  930  can use a model predictive control technique to formulate an objective function which expresses the total cost as a function of several decision variables (e.g., maintenance decisions, equipment purchase decisions, etc.) and input parameters (e.g., energy cost, device efficiency, device reliability). Operational cost predictor  910  can be configured to formulate an operational cost term in the objective function. Similarly, maintenance cost predictor  920  can be configured to formulate a maintenance cost term in the objective function and capital cost predictor  930  can be configured to formulate a capital cost term in the objective function. Objective function optimizer  940  can optimize (i.e., minimize) the objective function using any of a variety of optimization techniques to identify the optimal values for each of the decision variables. 
     One example of an objective function which can be generated by high level optimizer  832  is shown in the following equation: 
     
       
         
           
             J 
             = 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       op 
                       , 
                       i 
                     
                   
                    
                   
                     P 
                     
                       op 
                       , 
                       i 
                     
                   
                    
                   Δ 
                    
                   
                       
                   
                    
                   t 
                 
               
               + 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       main 
                       , 
                       i 
                     
                   
                    
                   
                     B 
                     
                       main 
                       , 
                       i 
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       cap 
                       , 
                       i 
                     
                   
                    
                   
                     P 
                     
                       cap 
                       , 
                       i 
                     
                   
                 
               
             
           
         
       
     
     where C op,i  is the cost per unit of energy (e.g., $/kWh) consumed by connected equipment  610  at time step i of the optimization period, P op,i  is the power consumption (e.g., kW) of connected equipment  610  at time step i, Δt is the duration of each time step i, C main,i  is the cost of maintenance performed on connected equipment  610  at time step i, B main,i  is a binary variable that indicates whether the maintenance is performed, C cap,i  is the capital cost of purchasing a new device of connected equipment  610  at time step i, B cap,i  is a binary variable that indicates whether the new device is purchased, and h is the duration of the horizon or optimization period over which the optimization is performed. 
     Operational Cost Predictor 
     Operational cost predictor  910  can be configured to formulate the first term in the objective function J. The first term in the objective function J represents the operating cost of connected equipment  610  over the duration of the optimization period and is shown to include three variables or parameters (i.e., C op,i , P op,i , and Δt). In some embodiments, the cost per unit of energy C op,i  is determined by energy costs module  915 . Energy costs module  915  can receive a set of energy prices from utility  608  as energy pricing data. In some embodiments, the energy prices are time-varying cost that depend on the time of day, the day of the week (e.g., weekday vs. weekend), the current season (e.g., summer vs. winter), or other time-based factors. For example, the cost of electricity may be higher during peak energy consumption periods and lower during off-peak or partial-peak energy consumption periods. 
     Energy costs module  915  can use the energy costs to define the value of C op,i  for each time step i of the optimization period. In some embodiments, energy costs module  915  stores the energy costs as an array C op  including a cost element for each of the h time steps in the optimization period. For example, energy costs module  915  can generate the following array: 
         C   op =[ C   op,1   C   op,2    . . . C   op,h ] 
     where the array C op  has a size of 1×h and each element of the array C op  includes an energy cost value C op,i  for a particular time step i=1 . . . h of the optimization period. 
     Still referring to  FIG. 9 , operational cost predictor  910  is shown to include an ideal performance calculator  912 . Ideal performance calculator  912  may receive load predictions Load i  from load/rate predictor  822  and may receive performance curves from low level optimizer  834 . As discussed above, the performance curves may define the ideal power consumption P ideal  of a device or set of devices of connected equipment  610  as a function of the heating or cooling load on the device or set of devices. For example, the performance curve one or more devices of connected equipment  610  can be defined by the following equation: 
         P   ideal,i =ƒ(Load i )
 
     where P ideal,i  is the ideal power consumption (e.g., kW) of connected equipment  610  at time step i and Load i  is the load (e.g., tons cooling, kW heating, etc.) on connected equipment  610  at time step i. The ideal power consumption P ideal,i  may represent the power consumption of the one or more devices of connected equipment  610  assuming they operate at perfect efficiency. 
     Ideal performance calculator  912  can use the performance curve for a device or set of devices of connected equipment  610  to identify the value of P ideal,i  that corresponds to the load point Load i  for the device or set of devices at each time step of the optimization period. In some embodiments, ideal performance calculator  912  stores the ideal load values as an array P ideal  including an element for each of the h time steps in the optimization period. For example, ideal performance calculator  912  can generate the following array: 
         P   ideal =[ P   ideal,1   P   ideal,2    . . . P   ideal,h ] T    
     where the array P ideal  has a size of h×1 and each element of the array P ideal  includes an ideal power consumption value P ideal,i  for a particular time step i=1 . . . h of the optimization period. 
     Still referring to  FIG. 9 , operational cost predictor  910  is shown to include an efficiency updater  911  and an efficiency degrader  913 . Efficiency updater  911  can be configured to determine the efficiency η of connected equipment  610  under actual operating conditions. In some embodiments, the efficiency η i  represents the ratio of the ideal power consumption P ideal  of connected equipment to the actual power consumption P actual  of connected equipment  610 , as shown in the following equation: 
     
       
         
           
             η 
             = 
             
               
                 P 
                 ideal 
               
               
                 P 
                 
                   a 
                    
                   c 
                    
                   t 
                    
                   u 
                    
                   a 
                    
                   l 
                 
               
             
           
         
       
     
     where P ideal  is the ideal power consumption of connected equipment  610  as defined by the performance curve for connected equipment  610  and P actual  is the actual power consumption of connected equipment  610 . In some embodiments, efficiency updater  911  uses the equipment performance information collected from connected equipment  610  to identify the actual power consumption value P actual . Efficiency updater  911  can use the actual power consumption P actual  in combination with the ideal power consumption P ideal  to calculate the efficiency η. 
     Efficiency updater  911  can be configured to periodically update the efficiency η to reflect the current operating efficiency of connected equipment  610 . For example, efficiency updater  911  can calculate the efficiency η of connected equipment  610  once per day, once per week, once per year, or at any other interval as may be suitable to capture changes in the efficiency η over time. Each value of the efficiency η may be based on corresponding values of P ideal  and P actual  at the time the efficiency η is calculated. In some embodiments, efficiency updater  911  updates the efficiency η each time the high level optimization process is performed (i.e., each time the objective function J is optimized). The efficiency value calculated by efficiency updater  911  may be stored in memory  810  as an initial efficiency value η 0 , where the subscript 0 denotes the value of the efficiency η at or before the beginning of the optimization period (e.g., at time step 0). 
     In some embodiments, efficiency updater  911  updates the efficiency η i  for one or more time steps during the optimization period to account for increases in the efficiency η of connected equipment  610  that will result from performing maintenance on connected equipment  610  or purchasing new equipment to replace or supplement one or more devices of connected equipment  610 . The time steps i at which the efficiency η i  is updated may correspond to the predicted time steps at which the maintenance will be performed or the equipment will replaced. The predicted time steps at which maintenance will be performed on connected equipment  610  may be defined by the values of the binary decision variables B main,i  in the objective function J. Similarly, the predicted time steps at which the equipment will be replaced may be defined by the values of the binary decision variables B cap,i  in the objective function J. 
     Efficiency updater  911  can be configured to reset the efficiency η i  for a given time step i if the binary decision variables B main,i  and B cap,i  indicate that maintenance will be performed at that time step and/or new equipment will be purchased at that time step (i.e., B main,i =1 and/or B cap,i =1). For example, if B main,i =1, efficiency updater  911  can be configured to reset the value of η i  to η main,i , where η main  is the efficiency value that is expected to result from the maintenance performed at time step i. Similarly, if B cap,i =1, efficiency updater  911  can be configured to reset the value of η i  to η cap,i  where η cap  is the efficiency value that is expected to result from purchasing a new device to supplement or replace one or more devices of connected equipment  610  performed at time step i. Efficiency updater  911  can dynamically reset the efficiency η i  for one or more time steps while the optimization is being performed (e.g., with each iteration of the optimization) based on the values of binary decision variables B main,i  and B cap,i . 
     Efficiency degrader  913  can be configured to predict the efficiency η i  of connected equipment  610  at each time step i of the optimization period. The initial efficiency η 0  at the beginning of the optimization period may degrade over time as connected equipment  610  degrade in performance. For example, the efficiency of a chiller may degrade over time as a result of the chilled water tubes becoming dirty and reducing the heat transfer coefficient of the chiller. Similarly, the efficiency of a battery may decrease over time as a result of degradation in the physical or chemical components of the battery. Efficiency degrader  913  can be configured to account for such degradation by incrementally reducing the efficiency η i  over the duration of the optimization period. 
     In some embodiments, the initial efficiency value η 0  is updated at the beginning of each optimization period. However, the efficiency η may degrade during the optimization period such that the initial efficiency value η 0  becomes increasingly inaccurate over the duration of the optimization period. To account for efficiency degradation during the optimization period, efficiency degrader  913  can decrease the efficiency η by a predetermined amount with each successive time step. For example, efficiency degrader  913  can define the efficiency at each time step i=1 . . . h as follows: 
       η i =η i-1   −Δt  
 
     where η i  is the efficiency at time step i, η i-1  is the efficiency at time step i−1, and Δη is the degradation in efficiency between consecutive time steps. In some embodiments, this definition of η i  is applied to each time step for which B main,i =0 and B cap,i =0. However, if either B main,i =1 or B cap,i =1, the value of η i  may be reset to either η main  or η cap  as previously described. 
     In some embodiments, the value of Δη is based on a time series of efficiency values calculated by efficiency updater  911 . For example, efficiency degrader  913  may record a time series of the initial efficiency values η 0  calculated by efficiency updater  911 , where each of the initial efficiency values η 0  represents the empirically-calculated efficiency of connected equipment  610  at a particular time. Efficiency degrader  913  can examine the time series of initial efficiency values η 0  to determine the rate at which the efficiency degrades. For example, if the initial efficiency η 0  at time t 1  is η 0,1  and the initial efficiency at time t 2  is η 0.2 , efficiency degrader  913  can calculate the rate of efficiency degradation as follows: 
     
       
         
           
             
               
                 Δ 
                  
                 η 
               
               
                 Δ 
                  
                 t 
               
             
             = 
             
               
                 
                   η 
                   
                     0 
                     , 
                     2 
                   
                 
                 - 
                 
                   η 
                   
                     0 
                     , 
                     1 
                   
                 
               
               
                 
                   t 
                   2 
                 
                 - 
                 
                   t 
                   1 
                 
               
             
           
         
       
     
     where 
     
       
         
           
             
               Δ 
                
               η 
             
             
               Δ 
                
               t 
             
           
         
       
     
     is the rate of efficiency degradation. Efficiency degrader  913  can multiply 
     
       
         
           
             
               Δ 
                
               η 
             
             
               Δ 
                
               t 
             
           
         
       
     
     by the duration of each time step Δt to calculate the value of Δη 
     
       
         
           
             
               ( 
               
                 
                   i 
                   . 
                   e 
                   . 
                 
                 , 
                 
                   
                     Δ 
                      
                     η 
                   
                   = 
                   
                     
                       
                         Δ 
                          
                         η 
                       
                       
                         Δ 
                          
                         t 
                       
                     
                     * 
                     Δ 
                      
                     t 
                   
                 
               
               ) 
             
             . 
           
         
       
     
     In some embodiments, efficiency degrader  913  stores the efficiency values over the duration of the optimization period in an array η including an element for each of the h time steps in the optimization period. For example, efficiency degrader  913  can generate the following array: 
       η=[η 1 η 2  . . . η h ]
 
     where the array η has a size of 1×h and each element of the array η includes an efficiency value η i  for a particular time step i=1 . . . h of the optimization period. Each element i of the array η may be calculated based on the value of the previous element and the value of Δη (e.g., if B main,i =0 and B cap,i =0) or may be dynamically reset to either η main  or η cap  (e.g., if B main,i =1 or B cap,i =1. 
     The logic characterizing the efficiency updating and resetting operations performed by efficiency updater  911  and efficiency degrader  913  can be summarized in the following equations: 
       if  B   main,i =1→η i =η main  
 
       if  B   cap,i =1→η i =η cap  
 
       if  B   main,i =0 and  B   cap,i =0→η i =η i-1 −Δη
 
     which can be applied as constraints on the high level optimization performed by objective function optimizer  940 . 
     Advantageously, efficiency updater  911  and efficiency degrader  913  can model the efficiency η i  of connected equipment  610  at each time step i as a function of the maintenance decisions B main,i  and the equipment purchase decisions B cap,i . For example, the efficiency η i  for a particular device may start at an initial value η 0  at the beginning of the optimization period and may degrade over time such that the efficiency η i  decreases with each successive time step i. Performing maintenance on a device may reset the efficiency η i  to a higher value immediately after the maintenance is performed. Similarly, purchasing a new device to replace an existing device may reset the efficiency η i  to a higher value immediately after the new device is purchased. After being reset, the efficiency η i  may continue to degrade over time until the next time at which maintenance is performed or a new device is purchased. 
     Still referring to  FIG. 9 , operational cost predictor  910  is shown to include a power consumption estimator  914  and an operational cost calculator  916 . Power consumption estimator  914  can be configured to estimate the power consumption P op,i  of connected equipment  610  at each time step i of the optimization period. In some embodiments, power consumption estimator  914  estimates the power consumption P op,i  as a function of the ideal power consumption P ideal,i  calculated by ideal performance calculator  912  and the efficiency η i  determined by efficiency degrader  913  and/or efficiency updater  911 . For example, power consumption estimator  914  can calculate the power consumption P op,i  using the following equation: 
     
       
         
           
             
               P 
               
                 
                   o 
                    
                   p 
                 
                 , 
                 i 
               
             
             = 
             
               
                 P 
                 
                   
                     i 
                      
                     d 
                      
                     e 
                      
                     a 
                      
                     l 
                   
                   , 
                   i 
                 
               
               
                 η 
                 i 
               
             
           
         
       
     
     where P ideal,i  is the power consumption calculated by ideal performance calculator  912  based on the equipment performance curve for the device at the corresponding load point Load i , and η i  is the operating efficiency of the device at time step i. 
     In some embodiments, power consumption estimator  914  stores the power consumption values as an array P op  including an element for each of the h time steps in the optimization period. For example, power consumption estimator  914  can generate the following array: 
         P   op =[ P   op,1   P   op,2    . . . P   op,h ] T    
     where the array P op  has a size of h×1 and each element of the array P op  includes a power consumption value P op,i  for a particular time step i=1 . . . h of the optimization period. 
     Operational cost calculator  916  can be configured to estimate the operational cost of connected equipment  610  over the duration of the optimization period. In some embodiments, operational cost calculator  916  calculates the operational cost during each time step i using the following equation: 
       Cost op,i   =C   op,i   P   op,i   Δt    
     where P op,i  is the predicted power consumption at time step i determined by power consumption estimator  914 , C op,i  is the cost per unit of energy at time step i determined by energy costs module  915 , and Δt is the duration of each time step. Operational cost calculator  916  can sum the operational costs over the duration of the optimization period as follows: 
     
       
         
           
             
               C 
                
               o 
                
               s 
                
               
                 t 
                 
                   o 
                    
                   p 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 h 
               
                
               
                 C 
                  
                 o 
                  
                 s 
                  
                 
                   t 
                   
                     
                       o 
                        
                       p 
                     
                     , 
                     i 
                   
                 
               
             
           
         
       
     
     where Cost op  is the operational cost term of the objective function J. 
     In other embodiments, operational cost calculator  916  estimates the operational cost Cost op  by multiplying the cost array C op  by the power consumption array P op  and the duration of each time step Δt as shown in the following equations: 
       Cost op   =C   op   P   op   Δt    
       Cost op =[ C   op,1   C   op,2    . . . C   op,h ][ P   op,1   P   op,2    . . . P   op,h ] T   Δt    
     Maintenance Cost Predictor 
     Maintenance cost predictor  920  can be configured to formulate the second term in the objective function J. The second term in the objective function J represents the cost of performing maintenance on connected equipment  610  over the duration of the optimization period and is shown to include two variables or parameters (i.e., C main,i  and B main,i ). Maintenance cost predictor  920  is shown to include a maintenance estimator  922 , a reliability estimator  924 , a maintenance cost calculator  926 , and a maintenance costs module  928 . 
     Reliability estimator  924  can be configured to estimate the reliability of connected equipment  610  based on the equipment performance information received from connected equipment  610 . The reliability may be a statistical measure of the likelihood that connected equipment  610  will continue operating without fault under its current operating conditions. Operating under more strenuous conditions (e.g., high load, high temperatures, etc.) may result in a lower reliability, whereas operating under less strenuous conditions (e.g., low load, moderate temperatures, etc.) may result in a higher reliability. In some embodiments, the reliability is based on an amount of time that has elapsed since connected equipment  610  last received maintenance and/or an amount of time that has elapsed since connected equipment  610  was purchased or installed. 
     In some embodiments, reliability estimator  924  uses the equipment performance information to identify a current operating state of connected equipment  610 . The current operating state can be examined by reliability estimator  924  to expose when connected equipment  610  begins to degrade in performance and/or to predict when faults will occur. In some embodiments, reliability estimator  924  estimates a likelihood of various types of failures that could potentially occur in connected equipment  610 . The likelihood of each failure may be based on the current operating conditions of connected equipment  610 , an amount of time that has elapsed since connected equipment  610  has been installed, and/or an amount of time that has elapsed since maintenance was last performed. In some embodiments, reliability estimator  924  identifies operating states and predicts the likelihood of various failures using the systems and methods described in U.S. patent application Ser. No. 15/188,824 titled “Building Management System With Predictive Diagnostics” and filed Jun. 21, 2016, the entire disclosure of which is incorporated by reference herein. 
     In some embodiments, reliability estimator  924  receives operating data from a plurality of devices of connected equipment  610  distributed across multiple buildings. The operating data can include, for example, current operating conditions, fault indications, failure times, or other data that characterize the operation and performance of connected equipment  610 . Reliability estimator  924  can use the set of operating data to develop a reliability model for each type of equipment. The reliability models can be used by reliability estimator  924  to estimate the reliability of any given device of connected equipment  610  as a function of its current operating conditions and/or other extraneous factors (e.g., time since maintenance was last performed, time since installation or purchase, geographic location, water quality, etc.). 
     One example of a reliability model which can be used by reliability estimator  924  is shown in the following equation: 
       Reliability i =ƒ(OpCond i   ,Δt   main,i   ,Δt   cap,i )
 
     where Reliability i  is the reliability of connected equipment  610  at time step i, OpCond i  are the operating conditions at time step i, Δt main,i  is the amount of time that has elapsed between the time at which maintenance was last performed and time step i, and Δt cap,i  is the amount of time that has elapsed between the time at which connected equipment  610  was purchased or installed and time step i. Reliability estimator  924  can be configured to identify the current operating conditions OpCond i  based on the equipment performance information received as a feedback from connected equipment  610 . Operating under more strenuous conditions (e.g., high load, extreme temperatures, etc.) may result in a lower reliability, whereas operating under less strenuous conditions (e.g., low load, moderate temperatures, etc.) may result in a higher reliability. 
     Reliability estimator  924  may determine the amount of time Δt main,i  that has elapsed since maintenance was last performed on connected equipment  610  based on the values of the binary decision variables B main,i . For each time step i, reliability estimator  924  can examine the corresponding values of B main  at time step i and each previous time step (e.g., time steps i−1, i−2, . . . , 1). Reliability estimator  924  can calculate the value of Δt main,i  by subtracting the time at which maintenance was last performed (i.e., the most recent time at which B main,i =1) from the time associated with time step i. A long amount of time Δt main,i  since maintenance was last performed may result in a lower reliability, whereas a short amount of time since maintenance was last performed may result in a higher reliability. 
     Similarly, reliability estimator  924  may determine the amount of time Δt cap,i  that has elapsed since connected equipment  610  was purchased or installed based on the values of the binary decision variables B cap,i . For each time step i, reliability estimator  924  can examine the corresponding values of B cap  at time step i and each previous time step (e.g., time steps i−1, i−2, . . . , 1). Reliability estimator  924  can calculate the value of Δt cap,i  by subtracting the time at which connected equipment  610  was purchased or installed (i.e., the most recent time at which B cap,i =1) from the time associated with time step i. A long amount of time Δt cap,i  since connected equipment  610  was purchased or installed may result in a lower reliability, whereas a short amount of time since connected equipment  610  was purchased or installed may result in a higher reliability. 
     Reliability estimator  924  can be configured to reset the reliability for a given time step i if the binary decision variables B main,i  and B cap,i  indicate that maintenance will be performed at that time step and/or new equipment will be purchased at that time step (i.e., B main,i =1 and/or B cap,i =1). For example, if B main,i =1, reliability estimator  924  can be configured to reset the value of Reliability to Reliability main , where Reliability main  is the reliability value that is expected to result from the maintenance performed at time step i. Similarly, if B cap,i =1, reliability estimator  924  can be configured to reset the value of Reliability to Reliability cap , where Reliability cap  is the reliability value that is expected to result from purchasing a new device to supplement or replace one or more devices of connected equipment  610  performed at time step i. Reliability estimator  924  can dynamically reset the reliability for one or more time steps while the optimization is being performed (e.g., with each iteration of the optimization) based on the values of binary decision variables B main,i  and B cap,i . 
     Maintenance estimator  922  can be configured to use the estimated reliability of connected equipment  610  over the duration of the optimization period to determine the probability that connected equipment  610  will require maintenance and/or replacement at each time step of the optimization period. In some embodiments, maintenance estimator  922  is configured to compare the probability that connected equipment  610  will require maintenance at a given time step to a critical value. Maintenance estimator  922  can be configured to set the value of B main,i =1 in response to a determination that the probability that connected equipment  610  will require maintenance at time step i exceeds the critical value. Similarly, maintenance estimator  922  can be configured to compare the probability that connected equipment  610  will require replacement at a given time step to a critical value. Maintenance estimator  922  can be configured to set the value of B cap,i =1 in response to a determination that the probability that connected equipment  610  will require replacement at time step i exceeds the critical value. 
     In some embodiments, a reciprocal relationship exists between the reliability of connected equipment  610  and the values of the binary decision variables B main,i  and B cap,i . In other words, the reliability of connected equipment  610  can affect the values of the binary decision variables B main,i  and B cap,i  selected in the optimization, and the values of the binary decision variables B main,i  and B cap,i  can affect the reliability of connected equipment  610 . Advantageously, the optimization performed by objective function optimizer  940  can identify the optimal values of the binary decision variables B main,i  and B cap,i  while accounting for the reciprocal relationship between the binary decision variables B main,i  and B cap,i  and the reliability of connected equipment  610 . 
     In some embodiments, maintenance estimator  922  generates a matrix B main  of the binary maintenance decision variables. The matrix B main  may include a binary decision variable for each of the different maintenance activities that can be performed at each time step of the optimization period. For example, maintenance estimator  922  can generate the following matrix: 
     
       
         
           
             
               B 
               
                 m 
                  
                 a 
                  
                 i 
                  
                 n 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       B 
                       
                         main 
                         , 
                         1 
                         , 
                         1 
                       
                     
                   
                   
                     
                       B 
                       
                         main 
                         , 
                         1 
                         , 
                         2 
                       
                     
                   
                   
                     
                       … 
                     
                   
                   
                     
                       B 
                       
                         main 
                         , 
                         1 
                         , 
                         h 
                       
                     
                   
                 
                 
                   
                     
                       B 
                       
                         main 
                         , 
                         2 
                         , 
                         1 
                       
                     
                   
                   
                     
                       B 
                       
                         main 
                         , 
                         2 
                         , 
                         2 
                       
                     
                   
                   
                     
                       ⋯ 
                     
                   
                   
                     
                       B 
                       
                         main 
                         , 
                         2 
                         , 
                         h 
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋱ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       B 
                       
                         main 
                         , 
                         m 
                         , 
                         1 
                       
                     
                   
                   
                     
                       B 
                       
                         main 
                         , 
                         m 
                         , 
                         2 
                       
                     
                   
                   
                     
                       … 
                     
                   
                   
                     
                       B 
                       
                         main 
                         , 
                         m 
                         , 
                         h 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     where the matrix B main  has a size of m×h and each element of the matrix B main  includes a binary decision variable for a particular maintenance activity at a particular time step of the optimization period. For example, the value of the binary decision variable B main,j,i  indicates whether the jth maintenance activity will be performed during the ith time step of the optimization period. 
     Still referring to  FIG. 9 , maintenance cost predictor  920  is shown to include a maintenance costs module  928  and a maintenance costs calculator  926 . Maintenance costs module  928  can be configured to determine costs C main,i  associated with performing various types of maintenance on connected equipment  610 . Maintenance costs module  928  can receive a set of maintenance costs from an external system or device (e.g., a database, a user device, etc.). In some embodiments, the maintenance costs define the economic cost (e.g., $) of performing various types of maintenance. Each type of maintenance activity may have a different economic cost associated therewith. For example, the maintenance activity of changing the oil in a chiller compressor may incur a relatively small economic cost, whereas the maintenance activity of completely disassembling the chiller and cleaning all of the chilled water tubes may incur a significantly larger economic cost. 
     Maintenance costs module  928  can use the maintenance costs to define the values of C main,i  in objective function J. In some embodiments, maintenance costs module  928  stores the maintenance costs as an array C main  including a cost element for each of the maintenance activities that can be performed. For example, maintenance costs module  928  can generate the following array: 
         C   main =[ C   main,1   ,C   main,2    . . . C   main,m ] 
     where the array C main  has a size of 1×m and each element of the array C main  includes a maintenance cost value C main,j  for a particular maintenance activity j=1 . . . m. 
     Some maintenance activities may be more expensive than other. However, different types of maintenance activities may result in different levels of improvement to the efficiency η and/or the reliability of connected equipment  610 . For example, merely changing the oil in a chiller may result in a minor improvement in efficiency η and/or a minor improvement in reliability, whereas completely disassembling the chiller and cleaning all of the chilled water tubes may result in a significantly greater improvement to the efficiency η and/or the reliability of connected equipment  610 . Accordingly, multiple different levels of post-maintenance efficiency (i.e., η main ) and post-maintenance reliability (i.e., Reliability main ) may exist. Each level of η main  and Reliability main  may correspond to a different type of maintenance activity. 
     In some embodiments, maintenance estimator  922  stores each of the different levels of η main  and Reliability main  in a corresponding array. For example, the parameter η main  can be defined as an array η main  with an element for each of the m different types of maintenance activities. Similarly, the parameter Reliability main  can be defined as an array Reliability main  with an element for each of the m different types of maintenance activities. Examples of these arrays are shown in the following equations: 
       η main =[η main,1 η main,2  . . . η main,m ]
 
       Reliability main =[Reliability main,1  Reliability main,2  . . . Reliability main,m ] 
     where the array η main  has a size of 1×m and each element of the array η main  includes a post-maintenance efficiency value η main,j  for a particular maintenance activity. Similarly, the array Reliability main  has a size of 1×m and each element of the array Reliability main  includes a post-maintenance reliability value Reliability main,j  for a particular maintenance activity. 
     In some embodiments, efficiency updater  911  identifies the maintenance activity associated with each binary decision variable B main,j,i  and resets the efficiency η to the corresponding post-maintenance efficiency level η main,j  if B main,j,i =1. Similarly, reliability estimator  924  can identify the maintenance activity associated with each binary decision variable B main,j,i  and can reset the reliability to the corresponding post-maintenance reliability level Reliability main,j  if B main,j,i =1. 
     Maintenance cost calculator  926  can be configured to estimate the maintenance cost of connected equipment  610  over the duration of the optimization period. In some embodiments, maintenance cost calculator  926  calculates the maintenance cost during each time step i using the following equation: 
       Cost main,i   =C   main,i   B   main,i    
     where C main,i  is an array of maintenance costs including an element for each of the m different types of maintenance activities that can be performed at time step i and B main,i  is an array of binary decision variables indicating whether each of the m maintenance activities will be performed at time step i. Maintenance cost calculator  926  can sum the maintenance costs over the duration of the optimization period as follows: 
     
       
         
           
             
               C 
                
               o 
                
               s 
                
               
                 t 
                 
                   m 
                    
                   a 
                    
                   i 
                    
                   n 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 h 
               
                
               
                 C 
                  
                 o 
                  
                 s 
                  
                 
                   t 
                   
                     
                       m 
                        
                       a 
                        
                       i 
                        
                       n 
                     
                     , 
                     i 
                   
                 
               
             
           
         
       
     
     where Cost main  is the maintenance cost term of the objective function J. 
     In other embodiments, maintenance cost calculator  926  estimates the maintenance cost Cost main  by multiplying the maintenance cost array C main  by the matrix of binary decision variables B main  as shown in the following equations: 
     
       
         
           
             
                 
             
              
             
               
                 Cost 
                 
                   m 
                    
                   a 
                    
                   i 
                    
                   n 
                 
               
               = 
               
                 
                   C 
                   
                     m 
                      
                     a 
                      
                     i 
                      
                     n 
                   
                 
                  
                 
                   B 
                   
                     m 
                      
                     a 
                      
                     i 
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               C 
                
               o 
                
               s 
                
               
                 t 
                 
                   m 
                    
                   a 
                    
                   i 
                    
                   n 
                 
               
             
             = 
             
               
                 [ 
                 
                   
                     C 
                     
                       
                         m 
                          
                         a 
                          
                         i 
                          
                         n 
                       
                       , 
                       1 
                     
                   
                    
                   
                       
                   
                    
                   
                     C 
                     
                       
                         m 
                          
                         a 
                          
                         i 
                          
                         n 
                       
                       , 
                       2 
                     
                   
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   
                     C 
                     
                       
                         m 
                          
                         a 
                          
                         i 
                          
                         n 
                       
                       , 
                       m 
                     
                   
                 
                 ] 
               
                
               
                 [ 
                 
                   
                     
                       
                         B 
                         
                           main 
                           , 
                           1 
                           , 
                           1 
                         
                       
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           1 
                           , 
                           2 
                         
                       
                     
                     
                       
                         … 
                       
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           1 
                           , 
                           h 
                         
                       
                     
                   
                   
                     
                       
                         B 
                         
                           main 
                           , 
                           2 
                           , 
                           1 
                         
                       
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           2 
                           , 
                           2 
                         
                       
                     
                     
                       
                         ⋯ 
                       
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           2 
                           , 
                           h 
                         
                       
                     
                   
                   
                     
                       ⋮ 
                     
                     
                       ⋮ 
                     
                     
                       ⋱ 
                     
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         B 
                         
                           main 
                           , 
                           m 
                           , 
                           1 
                         
                       
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           m 
                           , 
                           2 
                         
                       
                     
                     
                       
                         … 
                       
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           m 
                           , 
                           h 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Capital Cost Predictor 
     Capital cost predictor  930  can be configured to formulate the third term in the objective function J. The third term in the objective function J represents the cost of purchasing new devices of connected equipment  610  over the duration of the optimization period and is shown to include two variables or parameters (i.e., C cap,i  and B cap,i ). Capital cost predictor  930  is shown to include a purchase estimator  932 , a reliability estimator  934 , a capital cost calculator  936 , and a capital costs module  938 . 
     Reliability estimator  934  can include some or all of the features of reliability estimator  924 , as described with reference to maintenance cost predictor  920 . For example, reliability estimator  934  can be configured to estimate the reliability of connected equipment  610  based on the equipment performance information received from connected equipment  610 . The reliability may be a statistical measure of the likelihood that connected equipment  610  will continue operating without fault under its current operating conditions. Operating under more strenuous conditions (e.g., high load, high temperatures, etc.) may result in a lower reliability, whereas operating under less strenuous conditions (e.g., low load, moderate temperatures, etc.) may result in a higher reliability. In some embodiments, the reliability is based on an amount of time that has elapsed since connected equipment  610  last received maintenance and/or an amount of time that has elapsed since connected equipment  610  was purchased or installed. Reliability estimator  934  can include some or all of the features and/or functionality of reliability estimator  924 , as previously described. 
     Purchase estimator  932  can be configured to use the estimated reliability of connected equipment  610  over the duration of the optimization period to determine the probability that new devices of connected equipment  610  will be purchased at each time step of the optimization period. In some embodiments, purchase estimator  932  is configured to compare the probability that new devices of connected equipment  610  will be purchased at a given time step to a critical value. Purchase estimator  932  can be configured to set the value of B cap,i =1 in response to a determination that the probability that connected equipment  610  will be purchased at time step i exceeds the critical value. 
     In some embodiments, purchase estimator  932  generates a matrix B cap  of the binary capital decision variables. The matrix B cap  may include a binary decision variable for each of the different capital purchases that can be made at each time step of the optimization period. For example, purchase estimator  932  can generate the following matrix: 
     
       
         
           
             
               B 
               
                 c 
                  
                 a 
                  
                 p 
               
             
             = 
             
               [ 
               
                 
                   
                     
                       B 
                       
                         cap 
                         , 
                         1 
                         , 
                         1 
                       
                     
                   
                   
                     
                       B 
                       
                         cap 
                         , 
                         1 
                         , 
                         2 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       B 
                       
                         cap 
                         , 
                         1 
                         , 
                         h 
                       
                     
                   
                 
                 
                   
                     
                       B 
                       
                         cap 
                         , 
                         2 
                         , 
                         1 
                       
                     
                   
                   
                     
                       B 
                       
                         cap 
                         , 
                         2 
                         , 
                         2 
                       
                     
                   
                   
                     ⋯ 
                   
                   
                     
                       B 
                       
                         cap 
                         , 
                         2 
                         , 
                         h 
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋮ 
                   
                   
                     ⋱ 
                   
                   
                     
                         
                     
                   
                 
                 
                   
                     
                       B 
                       
                         cap 
                         , 
                         p 
                         , 
                         1 
                       
                     
                   
                   
                     
                       B 
                       
                         cap 
                         , 
                         p 
                         , 
                         2 
                       
                     
                   
                   
                     … 
                   
                   
                     
                       B 
                       
                         cap 
                         , 
                         p 
                         , 
                         h 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     where the matrix B cap  has a size of p×h and each element of the matrix B cap  includes a binary decision variable for a particular capital purchase at a particular time step of the optimization period. For example, the value of the binary decision variable B cap,k,i  indicates whether the kth capital purchase will be made during the ith time step of the optimization period. 
     Still referring to  FIG. 9 , capital cost predictor  930  is shown to include a capital costs module  938  and a capital cost calculator  936 . Capital costs module  938  can be configured to determine costs C cap,i  associated with various capital purchases (i.e., purchasing one or more new devices of connected equipment  610 ). Capital costs module  938  can receive a set of capital costs from an external system or device (e.g., a database, a user device, etc.). In some embodiments, the capital costs define the economic cost (e.g., $) of making various capital purchases. Each type of capital purchase may have a different economic cost associated therewith. For example, purchasing a new temperature sensor may incur a relatively small economic cost, whereas purchasing a new chiller may incur a significantly larger economic cost. 
     Capital costs module  938  can use the purchase costs to define the values of C cap,i  in objective function J. In some embodiments, capital costs module  938  stores the capital costs as an array C cap  including a cost element for each of the capital purchases that can be made. For example, capital costs module  938  can generate the following array: 
         C   cap =[ C   cap,1   C   cap,2    . . . C   cap,p ] 
     where the array C cap  has a size of 1×p and each element of the array C cap  includes a cost value C cap,k  for a particular capital purchase k=1 . . . p. 
     Some capital purchases may be more expensive than other. However, different types of capital purchases may result in different levels of improvement to the efficiency η and/or the reliability of connected equipment  610 . For example, purchasing a new sensor to replace an existing sensor may result in a minor improvement in efficiency η and/or a minor improvement in reliability, whereas purchasing a new chiller and control system may result in a significantly greater improvement to the efficiency η and/or the reliability of connected equipment  610 . Accordingly, multiple different levels of post-purchase efficiency (i.e., η cap ) and post-purchase reliability (i.e., Reliability cap ) may exist. Each level of η cap  and Reliability cap  may correspond to a different type of capital purchase. 
     In some embodiments, purchase estimator  932  stores each of the different levels of η cap  and Reliability cap  in a corresponding array. For example, the parameter η cap  can be defined as an array η cap  with an element for each of the p different types of capital purchases which can be made. Similarly, the parameter Reliability cap  can be defined as an array Reliability cap  with an element for each of the p different types of capital purchases that can be made. Examples of these arrays are shown in the following equations: 
       η cap =[η cap,1 η cap,2  . . . η cap,p ]
 
       Reliability cap =[Reliability cap,1  Reliability cap,2  . . . Reliability cap,p ] 
     where the array η cap  has a size of 1×p and each element of the array η cap  includes a post-purchase efficiency value η cap,k  for a particular capital purchase k. Similarly, the array Reliability cap  has a size of 1×p and each element of the array Reliability cap  includes a post-purchase reliability value Reliability cap,k  for a particular capital purchase k. 
     In some embodiments, efficiency updater  911  identifies the capital purchase associated with each binary decision variable B main,k,i  and resets the efficiency η to the corresponding post-purchase efficiency level η cap,k  if B cap,k,i =1. Similarly, reliability estimator  924  can identify the capital purchase associated with each binary decision variable B cap,k,i  and can reset the reliability to the corresponding post-purchase reliability level Reliability cap,k  if B main,k,i =1. 
     Capital cost calculator  936  can be configured to estimate the capital cost of connected equipment  610  over the duration of the optimization period. In some embodiments, capital cost calculator  936  calculates the capital cost during each time step i using the following equation: 
       Cost cap,i   =C   cap,i   B   cap,i    
     where C cap,i  is an array of capital purchase costs including an element for each of the p different capital purchases that can be made at time step i and B cap,i  is an array of binary decision variables indicating whether each of the p capital purchases will be made at time step i. Capital cost calculator  936  can sum the capital costs over the duration of the optimization period as follows: 
     
       
         
           
             
               C 
                
               o 
                
               s 
                
               
                 t 
                 
                   c 
                    
                   a 
                    
                   p 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 h 
               
                
               
                 C 
                  
                 o 
                  
                 s 
                  
                 
                   t 
                   
                     
                       c 
                        
                       a 
                        
                       p 
                     
                     , 
                     i 
                   
                 
               
             
           
         
       
     
     where Cost cap  is the capital cost term of the objective function J. 
     In other embodiments, capital cost calculator  936  estimates the capital cost Cost cap  by multiplying the capital cost array C cap  by the matrix of binary decision variables B cap  as shown in the following equations: 
     
       
         
           
             
                 
             
              
             
               
                 Cost 
                 
                   c 
                    
                   a 
                    
                   p 
                 
               
               = 
               
                 
                   C 
                   
                     c 
                      
                     a 
                      
                     p 
                   
                 
                  
                 
                   B 
                   
                     c 
                      
                     a 
                      
                     p 
                   
                 
               
             
           
         
       
       
         
           
             
               C 
                
               o 
                
               s 
                
               
                 t 
                 cap 
               
             
             = 
             
               
                 [ 
                 
                   
                     C 
                     
                       cap 
                       , 
                       1 
                     
                   
                    
                   
                       
                   
                    
                   
                     C 
                     
                       
                         c 
                          
                         a 
                          
                         p 
                       
                       , 
                       2 
                     
                   
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   
                     C 
                     
                       
                         c 
                          
                         a 
                          
                         p 
                       
                       , 
                       p 
                     
                   
                 
                 ] 
               
                
               
                 [ 
                 
                   
                     
                       
                         B 
                         
                           cap 
                           , 
                           1 
                           , 
                           1 
                         
                       
                     
                     
                       
                         B 
                         
                           cap 
                           , 
                           1 
                           , 
                           2 
                         
                       
                     
                     
                       … 
                     
                     
                       
                         B 
                         
                           cap 
                           , 
                           1 
                           , 
                           h 
                         
                       
                     
                   
                   
                     
                       
                         B 
                         
                           cap 
                           , 
                           2 
                           , 
                           1 
                         
                       
                     
                     
                       
                         B 
                         
                           cap 
                           , 
                           2 
                           , 
                           2 
                         
                       
                     
                     
                       ⋯ 
                     
                     
                       
                         B 
                         
                           cap 
                           , 
                           2 
                           , 
                           h 
                         
                       
                     
                   
                   
                     
                       ⋮ 
                     
                     
                       ⋮ 
                     
                     
                       ⋱ 
                     
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         B 
                         
                           cap 
                           , 
                           
                             
                               p 
                               , 
                             
                              
                             1 
                           
                         
                       
                     
                     
                       
                         B 
                         
                           cap 
                           , 
                           p 
                           , 
                           2 
                         
                       
                     
                     
                       … 
                     
                     
                       
                         B 
                         
                           cap 
                           , 
                           p 
                           , 
                           h 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Objective Function Optimizer 
     Still referring to  FIG. 9 , high level optimizer  832  is shown to include an objective function generator  935  and an objective function optimizer  940 . Objective function generator  935  can be configured to generate the objective function J by summing the operational cost term, the maintenance cost term, and the capital cost term formulated by cost predictors  910 ,  920 , and  930 . One example of an objective function which can be generated by objective function generator  935  is shown in the following equation: 
     
       
         
           
             J 
             = 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       op 
                       , 
                       i 
                     
                   
                    
                   
                     P 
                     
                       op 
                       , 
                       i 
                     
                   
                    
                   
                     
                       
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                       
                     
                      
                     
                       Δ 
                        
                       
                           
                       
                        
                       t 
                     
                      
                     
                       
                         
                           
                               
                           
                         
                       
                       
                         
                           
                               
                           
                         
                       
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       main 
                       , 
                       i 
                     
                   
                    
                   
                     B 
                     
                       main 
                       , 
                       i 
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       cap 
                       , 
                       i 
                     
                   
                    
                   
                     P 
                     
                       cap 
                       , 
                       i 
                     
                   
                 
               
             
           
         
       
     
     where C op,i  is the cost per unit of energy (e.g., $/kWh) consumed by connected equipment  610  at time step i of the optimization period, P op,i  is the power consumption (e.g., kW) of connected equipment  610  at time step i, Δt is the duration of each time step i, C main,i  is the cost of maintenance performed on connected equipment  610  at time step i, B main,i  is a binary variable that indicates whether the maintenance is performed, C cap,i  is the capital cost of purchasing a new device of connected equipment  610  at time step i, B cap,i  is a binary variable that indicates whether the new device is purchased, and h is the duration of the horizon or optimization period over which the optimization is performed. 
     Another example of an objective function which can be generated by objective function generator  935  is shown in the following equation: 
     
       
         
           
             
                 
             
              
             
               J 
               = 
               
                 
                   
                     C 
                     op 
                   
                    
                   
                     P 
                     op 
                   
                    
                   Δ 
                    
                   t 
                 
                 + 
                 
                   
                     C 
                     
                       m 
                        
                       a 
                        
                       i 
                        
                       n 
                     
                   
                    
                   
                     B 
                     
                       m 
                        
                       a 
                        
                       i 
                        
                       n 
                     
                   
                 
                 + 
                 
                   
                     C 
                     
                       c 
                        
                       a 
                        
                       p 
                     
                   
                    
                   
                     B 
                     
                       c 
                        
                       a 
                        
                       p 
                     
                   
                 
               
             
           
         
       
       
         
           
             J 
             = 
             
               
                 
                   
                     
                       [ 
                       
                         
                           C 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           C 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             2 
                           
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           C 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             h 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           P 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           P 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             2 
                           
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           P 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             h 
                           
                         
                       
                       ] 
                     
                   
                   T 
                 
                  
                 Δ 
                  
                 
                     
                 
                  
                 t 
               
               + 
               
                   
                 
                   
                     
                       [ 
                       
                         
                           C 
                           
                             
                               m 
                                
                               a 
                                
                               i 
                                
                               n 
                             
                             , 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           C 
                           
                             
                               m 
                                
                               a 
                                
                               i 
                                
                               n 
                             
                             , 
                             2 
                           
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           C 
                           
                             
                               m 
                                
                               a 
                                
                               i 
                                
                               n 
                             
                             , 
                             m 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               B 
                               
                                 main 
                                 , 
                                 1 
                                 , 
                                 1 
                               
                             
                           
                           
                             
                               B 
                               
                                 main 
                                 , 
                                 1 
                                 , 
                                 2 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               B 
                               
                                 main 
                                 , 
                                 1 
                                 , 
                                 h 
                               
                             
                           
                         
                         
                           
                             
                               B 
                               
                                 main 
                                 , 
                                 2 
                                 , 
                                 1 
                               
                             
                           
                           
                             
                               B 
                               
                                 main 
                                 , 
                                 2 
                                 , 
                                 2 
                               
                             
                           
                           
                             ⋯ 
                           
                           
                             
                               B 
                               
                                 main 
                                 , 
                                 2 
                                 , 
                                 h 
                               
                             
                           
                         
                         
                           
                             ⋮ 
                           
                           
                             ⋮ 
                           
                           
                             ⋱ 
                           
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               B 
                               
                                 main 
                                 , 
                                 m 
                                 , 
                                 1 
                               
                             
                           
                           
                             
                               B 
                               
                                 main 
                                 , 
                                 m 
                                 , 
                                 2 
                               
                             
                           
                           
                             … 
                           
                           
                             
                               B 
                               
                                 main 
                                 , 
                                 m 
                                 , 
                                 h 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                       
                     
                       
                         [ 
                         
                           
                             C 
                             
                               
                                 c 
                                  
                                 a 
                                  
                                 p 
                               
                               , 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             
                               C 
                               
                                 
                                   c 
                                    
                                   a 
                                    
                                   p 
                                 
                                 , 
                                 2 
                               
                             
                              
                             
                                 
                             
                             . 
                             . 
                             . 
                             
                                 
                             
                              
                             
                               C 
                               
                                 
                                   c 
                                    
                                   a 
                                    
                                   p 
                                 
                                 , 
                                 p 
                               
                             
                           
                         
                         ] 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 B 
                                 
                                   cap 
                                   , 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             
                               
                                 B 
                                 
                                   cap 
                                   , 
                                   1 
                                   , 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 B 
                                 
                                   cap 
                                   , 
                                   1 
                                   , 
                                   h 
                                 
                               
                             
                           
                           
                             
                               
                                 B 
                                 
                                   cap 
                                   , 
                                   2 
                                   , 
                                   1 
                                 
                               
                             
                             
                               
                                 B 
                                 
                                   cap 
                                   , 
                                   2 
                                   , 
                                   2 
                                 
                               
                             
                             
                               ⋯ 
                             
                             
                               
                                 B 
                                 
                                   cap 
                                   , 
                                   2 
                                   , 
                                   h 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋱ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 B 
                                 
                                   cap 
                                   , 
                                   
                                     
                                       p 
                                       , 
                                     
                                      
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 B 
                                 
                                   cap 
                                   , 
                                   p 
                                   , 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 B 
                                 
                                   cap 
                                   , 
                                   p 
                                   , 
                                   h 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
             
           
         
       
     
     where the array C op  includes an energy cost value C op,i  for a particular time step i=1 . . . h of the optimization period, the array P op  includes a power consumption value P op,i  for a particular time step i=1 . . . h of the optimization period, each element of the array C main  includes a maintenance cost value C main,j  for a particular maintenance activity j=1 . . . m, each element of the matrix B main  includes a binary decision variable for a particular maintenance activity j=1 . . . m at a particular time step i=1 . . . h of the optimization period, each element of the array C cap  includes a capital cost value C cap,k  for a particular capital purchase k=1 . . . p, and each element of the matrix B cap  includes a binary decision variable for a particular capital purchase k=1 . . . p at a particular time step i=1 . . . h of the optimization period. 
     Objective function generator  935  can be configured to impose constraints on one or more variables or parameters in the objective function J. The constraints can include any of the equations or relationships described with reference to operational cost predictor  910 , maintenance cost predictor  920 , and capital cost predictor  930 . For example, objective function generator  935  can impose a constraint which defines the power consumption values P op,i  for one or more devices of connected equipment  610  as a function of the ideal power consumption P ideal,i  and the efficiency η i  (e.g., P op,i =P ideal,i /η i ). Objective function generator  935  can impose a constraint which defines the efficiency η i  as a function of the binary decision variables B main,i  and B cap,i , as described with reference to efficiency updater  911  and efficiency degrader  913 . Objective function generator  935  can impose a constraint which constrains the binary decision variables B main,i  and B cap,i  to a value of either zero or one and defines the binary decision variables B main,i  and B cap,i  as a function of the reliability Reliability i  of connected equipment  610 , as described with reference to maintenance estimator  922  and purchase estimator  932 . Objective function generator  935  can impose a constraint which defines the reliability Reliability i  of connected equipment  610  as a function of the equipment performance information (e.g., operating conditions, run hours, etc.) as described with reference to reliability estimators  924  and  934 . 
     Objective function optimizer  940  can optimize the objective function J to determine the optimal values of the binary decision variables B main,i  and B cap,i  over the duration of the optimization period. Objective function optimizer  940  can use any of a variety of optimization techniques to formulate and optimize the objective function J. For example, objective function optimizer  940  can use integer programming, mixed integer linear programming, stochastic optimization, convex programming, dynamic programming, or any other optimization technique to formulate the objective function J, define the constraints, and perform the optimization. These and other optimization techniques are known in the art and will not be described in detail here. 
     In some embodiments, objective function optimizer  940  uses mixed integer stochastic optimization to optimize the objective function J. In mixed integer stochastic optimization, some of the variables in the objective function J can be defined as functions of random variables or probabilistic variables. For example, the decision variables B main,i  and B cap,i  can be defined as binary variables that have probabilistic values based on the reliability of connected equipment  610 . Low reliability values may increase the probability that the binary decision variables B main,i  and B cap,i  will have a value of one (e.g., B main,i =1 and B cap,i =1), whereas high reliability values may increase the probability that the binary decision variables B main,i  and B cap,i  will have a value of zero (e.g., B main,i =0 and B cap,i =0). In some embodiments, maintenance estimator  922  and purchase estimator  932  use a mixed integer stochastic technique to define the values of the binary decision variables B main,i  and B cap,i  as a probabilistic function of the reliability of connected equipment  610 . 
     As discussed above, the objective function J may represent the predicted cost of operating, maintaining, and purchasing one or more devices of connected equipment  610  over the duration of the optimization period. In some embodiments, objective function optimizer  940  is configured to project these costs back to a particular point in time (e.g., the current time) to determine the net present value (NPV) of the one or more devices of connected equipment  610  at a particular point in time. For example, objective function optimizer  940  can project each of the costs in objective function J back to the current time using the following equation: 
     
       
         
           
             
               N 
                
               P 
                
               
                 V 
                 
                   c 
                    
                   o 
                    
                   s 
                    
                   t 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 h 
               
                
               
                 
                   C 
                    
                   o 
                    
                   s 
                    
                   
                     t 
                     i 
                   
                 
                 
                   
                     ( 
                     
                       1 
                       + 
                       r 
                     
                     ) 
                   
                   i 
                 
               
             
           
         
       
     
     where r is the interest rate, Cost i  is the cost incurred during time step i of the optimization period, and NPV cost  is the net present value (i.e., the present cost) of the total costs incurred over the duration of the optimization period. In some embodiments, objective function optimizer  940  optimizes the net present value NPV cost  to determine the NPV of one or more devices of connected equipment  610  at a particular point in time. 
     As discussed above, one or more variables or parameters in the objective function J can be updated dynamically based on closed-loop feedback from connected equipment  610 . For example, the equipment performance information received from connected equipment  610  can be used to update the reliability and/or the efficiency of connected equipment  610 . Objective function optimizer  940  can be configured to optimize the objective function J periodically (e.g., once per day, once per week, once per month, etc.) to dynamically update the predicted cost and/or the net present value NPV cost  based on the closed-loop feedback from connected equipment  610 . 
     In some embodiments, objective function optimizer  940  generates optimization results. The optimization results may include the optimal values of the decision variables in the objective function J for each time step i in the optimization period. The optimization results include operating decisions, equipment maintenance decisions, and/or equipment purchase decisions for each device of connected equipment  610 . In some embodiments, the optimization results optimize the economic value of operating, maintaining, and purchasing connected equipment  610  over the duration of the optimization period. In some embodiments, the optimization results optimize the net present value of one or more devices of connected equipment  610  at a particular point in time. The optimization results may cause BMS  606  to activate, deactivate, or adjust a setpoint for connected equipment  610  in order to achieve the optimal values of the decision variables specified in the optimization results. 
     In some embodiments, MPM system  602  uses the optimization results to generate equipment purchase and maintenance recommendations. The equipment purchase and maintenance recommendations may be based on the optimal values for the binary decision variables B main,i  and B cap,i  determined by optimizing the objective function J. For example, a value of B main,25 =1 for a particular device of connected equipment  610  may indicate that maintenance should be performed on that device at the 25 th  time step of the optimization period, whereas a value of B main,25 =0 may indicate that the maintenance should not be performed at that time step. Similarly, a value of B cap,25 =1 may indicate that a new device of connected equipment  610  should be purchased at the 25 th  time step of the optimization period, whereas a value of B cap,25 =0 may indicate that the new device should not be purchased at that time step. 
     In some embodiments, the equipment purchase and maintenance recommendations are provided to building  10  (e.g., to BMS  606 ) and/or to client devices  448 . An operator or building owner can use the equipment purchase and maintenance recommendations to assess the costs and benefits of performing maintenance and purchasing new devices. In some embodiments, the equipment purchase and maintenance recommendations are provided to service technicians  620 . Service technicians  620  can use the equipment purchase and maintenance recommendations to determine when customers should be contacted to perform service or replace equipment. 
     Model Predictive Maintenance Process 
     Referring now to  FIG. 10 , a flowchart of a model predictive maintenance process  1000  is shown, according to an exemplary embodiment. Process  1000  can be performed by one or more components of building system  600 . In some embodiments, process  1000  is performed by MPM system  602 , as described with reference to  FIGS. 6-9 . 
     Process  1000  is shown to include operating building equipment to affect a variable state or condition of a building (step  1002 ) and receiving equipment performance information as feedback from the building equipment (step  1004 ). The building equipment can include type of equipment which can be used to monitor and/or control a building (e.g., connected equipment  610 ). For example, the building equipment can include chillers, AHUs, boilers, batteries, heaters, economizers, valves, actuators, dampers, cooling towers, fans, pumps, lighting equipment, security equipment, refrigeration equipment, or any other type of equipment in a building system or building management system. The building equipment can include any of the equipment of HVAC system  100 , waterside system  200 , airside system  300 , BMS  400 , and/or BMS  500 , as described with reference to  FIGS. 1-5 . The equipment performance information can include samples of monitored variables (e.g., measured temperature, measured pressure, measured flow rate, power consumption, etc.), current operating conditions (e.g., heating or cooling load, current operating state, etc.), fault indications, or other types of information that characterize the performance of the building equipment. 
     Process  1000  is shown to include estimating an efficiency and reliability of the building equipment as a function of the equipment performance information (step  1006 ). In some embodiments, step  1006  is performed by efficiency updater  911  and reliability estimators  924 ,  926  as described with reference to  FIG. 9 . Step  1006  can include using the equipment performance information to determine the efficiency η of the building equipment under actual operating conditions. In some embodiments, the efficiency η i  represents the ratio of the ideal power consumption P ideal  of the building equipment to the actual power consumption P actual  of the building equipment, as shown in the following equation: 
     
       
         
           
             η 
             = 
             
               
                 P 
                 
                   i 
                    
                   deal 
                 
               
               
                 P 
                 
                   a 
                    
                   c 
                    
                   t 
                    
                   u 
                    
                   a 
                    
                   l 
                 
               
             
           
         
       
     
     where P ideal  is the ideal power consumption of the building equipment as defined by the performance curve for the building equipment and P actual  is the actual power consumption of the building equipment. In some embodiments, step  1006  includes using the equipment performance information collected in step  1002  to identify the actual power consumption value P actual . Step  1006  can include using the actual power consumption P actual  in combination with the ideal power consumption P ideal  to calculate the efficiency η. 
     Step  1006  can include periodically updating the efficiency η to reflect the current operating efficiency of the building equipment. For example, step  1006  can include calculating the efficiency η of the building equipment once per day, once per week, once per year, or at any other interval as may be suitable to capture changes in the efficiency η over time. Each value of the efficiency η may be based on corresponding values of P ideal  and P actual  at the time the efficiency η is calculated. In some embodiments, step  1006  includes updating the efficiency η each time the high level optimization process is performed (i.e., each time the objective function J is optimized). The efficiency value calculated in step  1006  may be stored in memory  810  as an initial efficiency value η 0 , where the subscript 0 denotes the value of the efficiency η at or before the beginning of the optimization period (e.g., at time step 0). 
     Step  1006  can include predicting the efficiency η i  of the building equipment at each time step i of the optimization period. The initial efficiency η 0  at the beginning of the optimization period may degrade over time as the building equipment degrade in performance. For example, the efficiency of a chiller may degrade over time as a result of the chilled water tubes becoming dirty and reducing the heat transfer coefficient of the chiller. Similarly, the efficiency of a battery may decrease over time as a result of degradation in the physical or chemical components of the battery. Step  1006  can account for such degradation by incrementally reducing the efficiency η i  over the duration of the optimization period. 
     In some embodiments, the initial efficiency value η 0  is updated at the beginning of each optimization period. However, the efficiency η may degrade during the optimization period such that the initial efficiency value η 0  becomes increasingly inaccurate over the duration of the optimization period. To account for efficiency degradation during the optimization period, step  1006  can include decreasing the efficiency η by a predetermined amount with each successive time step. For example, step  1006  can include defining the efficiency at each time step i=1 . . . h as follows: 
       η i =η i-1 −Δη
 
     where η i  is the efficiency at time step i, η i-1  is the efficiency at time step i−1, and Δη is the degradation in efficiency between consecutive time steps. In some embodiments, this definition of η i  is applied to each time step for which B main,i =0 and B cap,i =0. However, if either B main,i =1 or B cap,i =1, the value of η i  may be reset to either η main  or η cap  in step  1018 . 
     In some embodiments, the value of Δη is based on a time series of efficiency values. For example, step  1006  may include recording a time series of the initial efficiency values η 0 , where each of the initial efficiency values η 0  represents the empirically-calculated efficiency of the building equipment at a particular time. Step  1006  can include examining the time series of initial efficiency values η 0  to determine the rate at which the efficiency degrades. For example, if the initial efficiency η 0  at time t 1  is η 0,1  and the initial efficiency at time t 2  is η 0.2 , the rate of efficiency degradation can be calculated as follows: 
     
       
         
           
             
               
                 Δ 
                  
                 η 
               
               
                 Δ 
                  
                 t 
               
             
             = 
             
               
                 
                   η 
                   
                     0 
                     , 
                     2 
                   
                 
                 - 
                 
                   η 
                   
                     0 
                     , 
                     1 
                   
                 
               
               
                 
                   t 
                   2 
                 
                 - 
                 
                   t 
                   1 
                 
               
             
           
         
       
     
     where 
     
       
         
           
             
               Δ 
                
               η 
             
             
               Δ 
                
               t 
             
           
         
       
     
     is the rate of efficiency degradation. Step  1006  can include multiplying 
     
       
         
           
             
               Δ 
                
               η 
             
             
               Δ 
                
               t 
             
           
         
       
     
     by the duration of each time step Δt to calculate the value of Δη 
     
       
         
           
             
               ( 
               
                 
                   i 
                   . 
                   e 
                   . 
                 
                 , 
                 
                     
                 
                  
                 
                   Δη 
                   = 
                   
                     
                       
                         Δ 
                          
                         η 
                       
                       
                         Δ 
                          
                         t 
                       
                     
                     * 
                     Δ 
                      
                     
                         
                     
                      
                     t 
                   
                 
               
               ) 
             
             . 
           
         
       
     
     Step  1006  can include estimating the reliability of the building equipment based on the equipment performance information received in step  1004 . The reliability may be a statistical measure of the likelihood that the building equipment will continue operating without fault under its current operating conditions. Operating under more strenuous conditions (e.g., high load, high temperatures, etc.) may result in a lower reliability, whereas operating under less strenuous conditions (e.g., low load, moderate temperatures, etc.) may result in a higher reliability. In some embodiments, the reliability is based on an amount of time that has elapsed since the building equipment last received maintenance and/or an amount of time that has elapsed since the building equipment were purchased or installed. 
     In some embodiments, step  1006  includes using the equipment performance information to identify a current operating state of the building equipment. The current operating state can be examined to expose when the building equipment begin to degrade in performance and/or to predict when faults will occur. In some embodiments, step  1006  includes estimating a likelihood of various types of failures that could potentially occur the building equipment. The likelihood of each failure may be based on the current operating conditions of the building equipment, an amount of time that has elapsed since the building equipment have been installed, and/or an amount of time that has elapsed since maintenance was last performed. In some embodiments, step  1006  includes identifying operating states and predicts the likelihood of various failures using the systems and methods described in U.S. patent application Ser. No. 15/188,824 titled “Building Management System With Predictive Diagnostics” and filed Jun. 21, 2016, the entire disclosure of which is incorporated by reference herein. 
     In some embodiments, step  1006  includes receiving operating data from building equipment distributed across multiple buildings. The operating data can include, for example, current operating conditions, fault indications, failure times, or other data that characterize the operation and performance of the building equipment. Step  1006  can include using the set of operating data to develop a reliability model for each type of equipment. The reliability models can be used in step  1006  to estimate the reliability of any given device of the building equipment as a function of its current operating conditions and/or other extraneous factors (e.g., time since maintenance was last performed, time since installation or purchase, geographic location, water quality, etc.). 
     One example of a reliability model which can be used in step  1006  is shown in the following equation: 
       Reliability i =ƒ(OpCond i   ,Δt   main,i   ,Δt   cap,i )
 
     where Reliability is the reliability of the building equipment at time step i, OpCond i  are the operating conditions at time step i, Δt main,i  is the amount of time that has elapsed between the time at which maintenance was last performed and time step i, and Δt cap,i  is the amount of time that has elapsed between the time at which the building equipment were purchased or installed and time step i. Step  1006  can include identifying the current operating conditions OpCond i  based on the equipment performance information received as a feedback from the building equipment. Operating under more strenuous conditions (e.g., high load, extreme temperatures, etc.) may result in a lower reliability, whereas operating under less strenuous conditions (e.g., low load, moderate temperatures, etc.) may result in a higher reliability. 
     Still referring to  FIG. 10 , process  1000  is shown to include predicting an energy consumption of the building equipment over an optimization period as a function of the estimated efficiency (step  1008 ). In some embodiments, step  1008  is performed by ideal performance calculator  912  and/or power consumption estimator, as described with reference to  FIG. 9 . Step  1008  can include receiving load predictions Load i  from load/rate predictor  822  and performance curves from low level optimizer  834 . As discussed above, the performance curves may define the ideal power consumption Pea of the building equipment a function of the heating or cooling load on the device or set of devices. For example, the performance curve for the building equipment can be defined by the following equation: 
         P   ideal,i =ƒ(Load i )
 
     where P ideal,i  is the ideal power consumption (e.g., kW) of the building equipment at time step i and Load i  is the load (e.g., tons cooling, kW heating, etc.) on the building equipment at time step i. The ideal power consumption P ideal,i  may represent the power consumption of the building equipment assuming they operate at perfect efficiency. Step  1008  can include using the performance curve for the building equipment to identify the value of P ideal,i  that corresponds to the load point Load i  for the building equipment at each time step of the optimization period. 
     In some embodiments, step  1008  includes estimating the power consumption P op,i  as a function of the ideal power consumption P ideal,i  and the efficiency η i  of the building equipment. For example, step  1008  can include calculating the power consumption P op,i  using the following equation: 
     
       
         
           
             
               P 
               
                 
                   o 
                    
                   p 
                 
                 , 
                 i 
               
             
             = 
             
               
                 P 
                 
                   
                     i 
                      
                     d 
                      
                     e 
                      
                     a 
                      
                     l 
                   
                   , 
                   i 
                 
               
               
                 η 
                 i 
               
             
           
         
       
     
     where P ideal,i  is the power consumption based on the equipment performance curve for the building equipment at the corresponding load point Load i , and η i  is the operating efficiency of the building equipment at time step i. 
     Still referring to  FIG. 10 , process  1000  is shown to include defining a cost Cost op  of operating the building equipment over the optimization period as a function of the predicted energy consumption (step  1010 ). In some embodiments, step  1010  is performed by operational cost calculator  916 , as described with reference to  FIG. 9 . Step  1010  can include calculating the operational cost during each time step i using the following equation: 
       Cost op,i   =C   op,i   P   op,i   Δt    
     where P op,i  is the predicted power consumption at time step i determined in step  1008 , C op,i  is the cost per unit of energy at time step i, and Δt is the duration of each time step. Step  1010  can include summing the operational costs over the duration of the optimization period as follows: 
     
       
         
           
             
               C 
                
               o 
                
               s 
                
               
                 t 
                 
                   o 
                    
                   p 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 h 
               
                
               
                 C 
                  
                 o 
                  
                 s 
                  
                 
                   t 
                   
                     
                       o 
                        
                       p 
                     
                     , 
                     i 
                   
                 
               
             
           
         
       
     
     where Cost op  is the operational cost term of the objective function J. 
     In other embodiments, step  1010  can include calculating the operational cost Cost op  by multiplying the cost array C op  by the power consumption array P op  and the duration of each time step Δt as shown in the following equations: 
       Cost op   =C   op   P   op   Δt    
       Cost op =[ C   op,1   C   op,2    . . . C   op,h ][ P   op,1   P   op,2    . . . P   op,h ] T   Δt    
     where the array C op  includes an energy cost value C op,i  for a particular time step i=1 . . . h of the optimization period, the array P op  includes a power consumption value P op,i  for a particular time step i=1 . . . h of the optimization period. 
     Still referring to  FIG. 10 , process  1000  is shown to include defining a cost of performing maintenance on the building equipment over the optimization period as a function of the estimated reliability (step  1012 ). Step  1012  can be performed by maintenance cost predictor  920 , as described with reference to  FIG. 9 . Step  1012  can include using the estimated reliability of the building equipment over the duration of the optimization period to determine the probability that the building equipment will require maintenance and/or replacement at each time step of the optimization period. In some embodiments, step  1012  includes comparing the probability that the building equipment will require maintenance at a given time step to a critical value. Step  1012  can include setting the value of B main,i =1 in response to a determination that the probability that the building equipment will require maintenance at time step i exceeds the critical value. Similarly, step  1012  can include comparing the probability that the building equipment will require replacement at a given time step to a critical value. Step  1012  can include setting the value of B cap,i =1 in response to a determination that the probability that the building equipment will require replacement at time step i exceeds the critical value. 
     Step  1012  can include determining the costs C main,i  associated with performing various types of maintenance on the building equipment. Step  1012  can include receiving a set of maintenance costs from an external system or device (e.g., a database, a user device, etc.). In some embodiments, the maintenance costs define the economic cost (e.g., $) of performing various types of maintenance. Each type of maintenance activity may have a different economic cost associated therewith. For example, the maintenance activity of changing the oil in a chiller compressor may incur a relatively small economic cost, whereas the maintenance activity of completely disassembling the chiller and cleaning all of the chilled water tubes may incur a significantly larger economic cost. Step  1012  can include using the maintenance costs to define the values of C main,i  in objective function J. 
     Step  1012  can include estimating the maintenance cost of the building equipment over the duration of the optimization period. In some embodiments, step  1012  includes calculating the maintenance cost during each time step i using the following equation: 
       Cost main,i   =C   main,i   B   main,i    
     where C main,i  is an array of maintenance costs including an element for each of the m different types of maintenance activities that can be performed at time step i and B main,i  is an array of binary decision variables indicating whether each of the m maintenance activities will be performed at time step i. Step  1012  can include summing the maintenance costs over the duration of the optimization period as follows: 
     
       
         
           
             
               C 
                
               o 
                
               s 
                
               
                 t 
                 
                   m 
                    
                   a 
                    
                   i 
                    
                   n 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 h 
               
                
               
                 C 
                  
                 o 
                  
                 s 
                  
                 
                   t 
                   
                     
                       m 
                        
                       a 
                        
                       i 
                        
                       n 
                     
                     , 
                     i 
                   
                 
               
             
           
         
       
     
     where Cost main  is the maintenance cost term of the objective function J. 
     In other embodiments, step  1012  includes estimating the maintenance cost Cost main  by multiplying the maintenance cost array C main  by the matrix of binary decision variables B main  as shown in the following equations: 
     
       
         
           
             
                 
             
              
             
               
                 Cost 
                 
                   m 
                    
                   a 
                    
                   i 
                    
                   n 
                 
               
               = 
               
                 
                   C 
                   
                     m 
                      
                     a 
                      
                     i 
                      
                     n 
                   
                 
                  
                 
                   B 
                   
                     m 
                      
                     a 
                      
                     i 
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               C 
                
               o 
                
               s 
                
               
                 t 
                 
                   m 
                    
                   a 
                    
                   i 
                    
                   n 
                 
               
             
             = 
             
               
                 [ 
                 
                   
                     C 
                     
                       
                         m 
                          
                         a 
                          
                         i 
                          
                         n 
                       
                       , 
                       1 
                     
                   
                    
                   
                       
                   
                    
                   
                     C 
                     
                       
                         m 
                          
                         a 
                          
                         i 
                          
                         n 
                       
                       , 
                       2 
                     
                   
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   
                     C 
                     
                       
                         m 
                          
                         a 
                          
                         i 
                          
                         n 
                       
                       , 
                       m 
                     
                   
                 
                 ] 
               
                
               
                 [ 
                 
                   
                     
                       
                         B 
                         
                           main 
                           , 
                           1 
                           , 
                           1 
                         
                       
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           1 
                           , 
                           2 
                         
                       
                     
                     
                       … 
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           1 
                           , 
                           h 
                         
                       
                     
                   
                   
                     
                       
                         B 
                         
                           main 
                           , 
                           2 
                           , 
                           1 
                         
                       
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           2 
                           , 
                           2 
                         
                       
                     
                     
                       ⋯ 
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           2 
                           , 
                           h 
                         
                       
                     
                   
                   
                     
                       ⋮ 
                     
                     
                       ⋮ 
                     
                     
                       ⋱ 
                     
                     
                       ⋮ 
                     
                   
                   
                     
                       
                         B 
                         
                           main 
                           , 
                           m 
                           , 
                           1 
                         
                       
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           m 
                           , 
                           2 
                         
                       
                     
                     
                       … 
                     
                     
                       
                         B 
                         
                           main 
                           , 
                           m 
                           , 
                           h 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
     where each element of the array C main  includes a maintenance cost value C main,j  for a particular maintenance activity j=1 . . . m and each element of the matrix B main  includes a binary decision variable for a particular maintenance activity j=1 . . . m at a particular time step i=1 . . . h of the optimization period. 
     Still referring to  FIG. 10 , process  1000  is shown to include defining a cost Cost cap  of purchasing or replacing the building equipment over the optimization period as a function of the estimated reliability (step  1014 ). Step  1014  can be performed by capital cost predictor  930 , as described with reference to  FIG. 9 . In some embodiments, step  1014  includes using the estimated reliability of the building equipment over the duration of the optimization period to determine the probability that new devices of the building equipment will be purchased at each time step of the optimization period. In some embodiments, step  1014  includes comparing the probability that new devices of the building equipment will be purchased at a given time step to a critical value. Step  1014  can include setting the value of B cap,i =1 in response to a determination that the probability that the building equipment will be purchased at time step i exceeds the critical value. 
     Step  1014  can include determining the costs C cap,i  associated with various capital purchases (i.e., purchasing one or more new devices of the building equipment). Step  1014  can include receiving a set of capital costs from an external system or device (e.g., a database, a user device, etc.). In some embodiments, the capital costs define the economic cost (e.g., $) of making various capital purchases. Each type of capital purchase may have a different economic cost associated therewith. For example, purchasing a new temperature sensor may incur a relatively small economic cost, whereas purchasing a new chiller may incur a significantly larger economic cost. Step  1014  can include using the purchase costs to define the values of C cap,i  in objective function J. 
     Some capital purchases may be more expensive than other. However, different types of capital purchases may result in different levels of improvement to the efficiency η and/or the reliability of the building equipment. For example, purchasing a new sensor to replace an existing sensor may result in a minor improvement in efficiency η and/or a minor improvement in reliability, whereas purchasing a new chiller and control system may result in a significantly greater improvement to the efficiency η and/or the reliability of the building equipment. Accordingly, multiple different levels of post-purchase efficiency (i.e., η cap ) and post-purchase reliability (i.e., Reliability cap ) may exist. Each level of η cap  and Reliability cap  may correspond to a different type of capital purchase. 
     Step  1014  can include estimating the capital cost of the building equipment over the duration of the optimization period. In some embodiments, step  1014  includes calculating the capital cost during each time step i using the following equation: 
       Cost cap,i   =C   cap,i   B   cap,i    
     where C cap,i  is an array of capital purchase costs including an element for each of the p different capital purchases that can be made at time step i and B cap,i  is an array of binary decision variables indicating whether each of the p capital purchases will be made at time step i. Step  1014  can include summing the capital costs over the duration of the optimization period as follows: 
     
       
         
           
             
               C 
                
               o 
                
               s 
                
               
                 t 
                 
                   c 
                    
                   a 
                    
                   p 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 h 
               
                
               
                 C 
                  
                 o 
                  
                 s 
                  
                 
                   t 
                   
                     
                       c 
                        
                       a 
                        
                       p 
                     
                     , 
                     i 
                   
                 
               
             
           
         
       
     
     where Cost cap  is the capital cost term of the objective function J. 
     In other embodiments, step  1014  includes estimating the capital cost Cost cap  by multiplying the capital cost array C cap  by the matrix of binary decision variables B cap  as shown in the following equations: 
     
       
         
           
             
                 
             
              
             
               
                 Cost 
                 
                   c 
                    
                   a 
                    
                   p 
                 
               
               = 
               
                 
                   C 
                   
                     c 
                      
                     a 
                      
                     p 
                   
                 
                  
                 
                   B 
                   
                     c 
                      
                     a 
                      
                     p 
                   
                 
               
             
           
         
       
       
         
           
             
               C 
                
               o 
                
               s 
                
               
                 t 
                 
                   c 
                    
                   a 
                    
                   p 
                 
               
             
             = 
             
               
                 [ 
                 
                   
                     C 
                     
                       
                         c 
                          
                         a 
                          
                         p 
                       
                       , 
                       1 
                     
                   
                    
                   
                       
                   
                    
                   
                     C 
                     
                       cap 
                       , 
                       2 
                     
                   
                    
                   
                       
                   
                    
                   … 
                    
                   
                       
                   
                    
                   
                     C 
                     
                       
                         c 
                          
                         a 
                          
                         p 
                       
                       , 
                       p 
                     
                   
                 
                 ] 
               
                
               
                   
                 
                   [ 
                   
                     
                       
                         
                           B 
                           
                             cap 
                             , 
                             1 
                             , 
                             1 
                           
                         
                       
                       
                         
                           B 
                           
                             
                               c 
                                
                               a 
                                
                               
                                 p 
                                 ′ 
                               
                                
                               1 
                             
                             , 
                             2 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           B 
                           
                             c 
                              
                             a 
                              
                             
                               p 
                               ′ 
                             
                              
                             
                               1 
                               ′ 
                             
                              
                             h 
                           
                         
                       
                     
                     
                       
                         
                           B 
                           
                             cap 
                             , 
                             
                               
                                 2 
                                 ′ 
                               
                                
                               1 
                             
                           
                         
                       
                       
                         
                           B 
                           
                             cap 
                             , 
                             2 
                             , 
                             2 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           B 
                           
                             cap 
                             , 
                             2 
                             , 
                             h 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         ⋱ 
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           B 
                           
                             cap 
                             , 
                             p 
                             , 
                             1 
                           
                         
                       
                       
                         
                           B 
                           
                             cap 
                             , 
                             p 
                             , 
                             2 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           B 
                           
                             cap 
                             , 
                             p 
                             , 
                             h 
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
     where each element of the array C cap  includes a capital cost value C cap,k  for a particular capital purchase k=1 . . . p and each element of the matrix B cap  includes a binary decision variable for a particular capital purchase k=1 . . . p at a particular time step i=1 . . . h of the optimization period. 
     Still referring to  FIG. 10 , process  1000  is shown to include optimizing an objective function including the costs Cost op , Cost main,i  and Cost cap  to determine an optimal maintenance strategy for the building equipment (step  1016 ). Step  1016  can include generating the objective function J by summing the operational cost term, the maintenance cost term, and the capital cost term formulated in steps  1010 - 1014 . One example of an objective function which can be generated in step  1016  is shown in the following equation: 
     
       
         
           
             J 
             = 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       op 
                       , 
                       i 
                     
                   
                    
                   
                     P 
                     
                       op 
                       , 
                       i 
                     
                   
                    
                   Δ 
                    
                   
                       
                   
                    
                   t 
                 
               
               + 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       main 
                       , 
                       i 
                     
                   
                    
                   
                     B 
                     
                       main 
                       , 
                       i 
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   h 
                 
                  
                 
                   
                     C 
                     
                       cap 
                       , 
                       i 
                     
                   
                    
                   
                     P 
                     
                       cap 
                       , 
                       i 
                     
                   
                 
               
             
           
         
       
     
     where C op,i  is the cost per unit of energy (e.g., $/kWh) consumed by connected equipment  610  at time step i of the optimization period, P op,i  is the power consumption (e.g., kW) of connected equipment  610  at time step i, Δt is the duration of each time step i, C main,i  is the cost of maintenance performed on connected equipment  610  at time step i, B main,i  is a binary variable that indicates whether the maintenance is performed, C cap,i  is the capital cost of purchasing a new device of connected equipment  610  at time step i, B cap,i  is a binary variable that indicates whether the new device is purchased, and h is the duration of the horizon or optimization period over which the optimization is performed. 
     Another example of an objective function which can be generated in step  1016  is shown in the following equation: 
     
       
         
           
             
                 
             
              
             
               J 
               = 
               
                 
                   
                     C 
                     op 
                   
                    
                   
                     P 
                     op 
                   
                    
                   Δ 
                    
                   
                       
                   
                    
                   t 
                 
                 + 
                 
                   
                     C 
                     main 
                   
                    
                   
                     B 
                     main 
                   
                 
                 + 
                 
                   
                     C 
                     
                       c 
                        
                       a 
                        
                       p 
                     
                   
                    
                   
                     B 
                     
                       c 
                        
                       a 
                        
                       p 
                     
                   
                 
               
             
           
         
       
       
         
           
             J 
              
             
                 
             
             = 
             
               
                 
                   
                     
                       [ 
                       
                         
                           C 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           C 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             2 
                           
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           C 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             h 
                           
                         
                       
                       ] 
                     
                      
                     
                       [ 
                       
                         
                           P 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           P 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             2 
                           
                         
                          
                         
                             
                         
                          
                         … 
                          
                         
                             
                         
                          
                         
                           P 
                           
                             
                               o 
                                
                               p 
                             
                             , 
                             h 
                           
                         
                       
                       ] 
                     
                   
                   T 
                 
                  
                 Δ 
                  
                 
                     
                 
                  
                 t 
               
               + 
               
                   
                 
                   
                     [ 
                     
                       
                         C 
                         
                           
                             m 
                              
                             a 
                              
                             i 
                              
                             n 
                           
                           , 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         C 
                         
                           
                             m 
                              
                             a 
                              
                             i 
                              
                             n 
                           
                           , 
                           2 
                         
                       
                        
                       
                           
                       
                        
                       … 
                        
                       
                           
                       
                        
                       
                         C 
                         
                           
                             m 
                              
                             a 
                              
                             i 
                              
                             n 
                           
                           , 
                           m 
                         
                       
                     
                     ] 
                   
                    
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 B 
                                 
                                   main 
                                   , 
                                   1 
                                   , 
                                   1 
                                 
                               
                             
                             
                               
                                 B 
                                 
                                   main 
                                   , 
                                   1 
                                   , 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 B 
                                 
                                   main 
                                   , 
                                   1 
                                   , 
                                   h 
                                 
                               
                             
                           
                           
                             
                               
                                 B 
                                 
                                   main 
                                   , 
                                   2 
                                   , 
                                   1 
                                 
                               
                             
                             
                               
                                 B 
                                 
                                   main 
                                   , 
                                   2 
                                   , 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 B 
                                 
                                   main 
                                   , 
                                   2 
                                   , 
                                   h 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋮ 
                             
                             
                               ⋱ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 B 
                                 
                                   main 
                                   , 
                                   m 
                                   , 
                                   1 
                                 
                               
                             
                             
                               
                                 B 
                                 
                                   main 
                                   , 
                                   m 
                                   , 
                                   2 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 B 
                                 
                                   main 
                                   , 
                                   m 
                                   , 
                                   h 
                                 
                               
                             
                           
                         
                         ] 
                       
                       + 
                       
                           
                         
                           
                             [ 
                             
                               
                                 C 
                                 
                                   
                                     c 
                                      
                                     a 
                                      
                                     p 
                                   
                                   , 
                                   1 
                                 
                               
                                
                               
                                   
                               
                                
                               
                                 
                                   C 
                                   
                                     
                                       c 
                                        
                                       a 
                                        
                                       p 
                                     
                                     , 
                                     2 
                                   
                                 
                                  
                                 
                                     
                                 
                                 . 
                                 . 
                                 . 
                                 
                                     
                                 
                                  
                                 
                                   C 
                                   
                                     
                                       c 
                                        
                                       a 
                                        
                                       p 
                                     
                                     , 
                                     p 
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                               
                             
                               [ 
                               
                                 
                                   
                                     
                                       B 
                                       
                                         cap 
                                         , 
                                         1 
                                         , 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       B 
                                       
                                         cap 
                                         , 
                                         1 
                                         , 
                                         2 
                                       
                                     
                                   
                                   
                                     … 
                                   
                                   
                                     
                                       B 
                                       
                                         cap 
                                         , 
                                         1 
                                         , 
                                         h 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       B 
                                       
                                         cap 
                                         , 
                                         2 
                                         , 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       B 
                                       
                                         cap 
                                         , 
                                         2 
                                         , 
                                         2 
                                       
                                     
                                   
                                   
                                     … 
                                   
                                   
                                     
                                       B 
                                       
                                         cap 
                                         , 
                                         2 
                                         , 
                                         h 
                                       
                                     
                                   
                                 
                                 
                                   
                                     ⋮ 
                                   
                                   
                                     ⋮ 
                                   
                                   
                                     ⋱ 
                                   
                                   
                                     ⋮ 
                                   
                                 
                                 
                                   
                                     
                                       B 
                                       
                                         cap 
                                         , 
                                         m 
                                         , 
                                         1 
                                       
                                     
                                   
                                   
                                     
                                       B 
                                       
                                         cap 
                                         , 
                                         m 
                                         , 
                                         2 
                                       
                                     
                                   
                                   
                                     … 
                                   
                                   
                                     
                                       B 
                                       
                                         cap 
                                         , 
                                         p 
                                         , 
                                         h 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     where the array C op  includes an energy cost value C op,i  for a particular time step i=1 . . . h of the optimization period, the array P op  includes a power consumption value P op,i  for a particular time step i=1 . . . h of the optimization period, each element of the array C main  includes a maintenance cost value C main,j  for a particular maintenance activity j=1 . . . m, each element of the matrix B main  includes a binary decision variable for a particular maintenance activity j=1 . . . m at a particular time step i=1 . . . h of the optimization period, each element of the array C cap  includes a capital cost value C cap,k  for a particular capital purchase k=1 . . . p, and each element of the matrix B cap  includes a binary decision variable for a particular capital purchase k=1 . . . p at a particular time step i=1 . . . h of the optimization period. 
     Step  1016  can include imposing constraints on one or more variables or parameters in the objective function J. The constraints can include any of the equations or relationships described with reference to operational cost predictor  910 , maintenance cost predictor  920 , and capital cost predictor  930 . For example, step  1016  can include imposing a constraint which defines the power consumption values P op,i  for one or more devices of the building equipment as a function of the ideal power consumption P ideal,i  and the efficiency η i  (e.g., P op,i =P ideal,i /h). Step  1016  can include imposing a constraint which defines the efficiency η i  as a function of the binary decision variables B main,i  and B cap,i , as described with reference to efficiency updater  911  and efficiency degrader  913 . Step  1016  can include imposing a constraint which constrains the binary decision variables B main,i  and B cap,i  to a value of either zero or one and defines the binary decision variables B main,i  and B cap,i  as a function of the reliability Reliability i  of connected equipment  610 , as described with reference to maintenance estimator  922  and purchase estimator  932 . Step  1016  can include imposing a constraint which defines the reliability Reliability i  of connected equipment  610  as a function of the equipment performance information (e.g., operating conditions, run hours, etc.) as described with reference to reliability estimators  924  and  934 . 
     Step  1016  can include optimizing the objective function J to determine the optimal values of the binary decision variables B main,i  and B cap,i  over the duration of the optimization period. Step  1016  can include using any of a variety of optimization techniques to formulate and optimize the objective function J. For example, step  1016  can include using integer programming, mixed integer linear programming, stochastic optimization, convex programming, dynamic programming, or any other optimization technique to formulate the objective function J, define the constraints, and perform the optimization. These and other optimization techniques are known in the art and will not be described in detail here. 
     In some embodiments, step  1016  includes using mixed integer stochastic optimization to optimize the objective function J. In mixed integer stochastic optimization, some of the variables in the objective function J can be defined as functions of random variables or probabilistic variables. For example, the decision variables B main,i  and B cap,i  can be defined as binary variables that have probabilistic values based on the reliability of the building equipment. Low reliability values may increase the probability that the binary decision variables B main,i  and B cap,i  will have a value of one (e.g., B main,i =1 and B cap,i =1), whereas high reliability values may increase the probability that the binary decision variables B main,i  and B cap,i  will have a value of zero (e.g., B main,i =0 and B cap,i =0). In some embodiments, step  1016  includes using a mixed integer stochastic technique to define the values of the binary decision variables B main,i  and B cap,i  as a probabilistic function of the reliability of the building equipment. 
     As discussed above, the objective function J may represent the predicted cost of operating, maintaining, and purchasing one or more devices of the building equipment over the duration of the optimization period. In some embodiments, step  1016  includes projecting these costs back to a particular point in time (e.g., the current time) to determine the net present value (NPV) of the one or more devices of the building equipment at a particular point in time. For example, step  1016  can include projecting each of the costs in objective function J back to the current time using the following equation: 
     
       
         
           
             
               N 
                
               P 
                
               
                 V 
                 
                   c 
                    
                   o 
                    
                   s 
                    
                   t 
                 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   1 
                 
                 h 
               
                
               
                 
                   C 
                    
                   o 
                    
                   s 
                    
                   
                     t 
                     i 
                   
                 
                 
                   
                     ( 
                     
                       1 
                       + 
                       r 
                     
                     ) 
                   
                   i 
                 
               
             
           
         
       
     
     where r is the interest rate, Cost i  is the cost incurred during time step i of the optimization period, and NPV cost  is the net present value (i.e., the present cost) of the total costs incurred over the duration of the optimization period. In some embodiments, step  1016  includes optimizing the net present value NPV cost  to determine the NPV of the building equipment at a particular point in time. 
     As discussed above, one or more variables or parameters in the objective function J can be updated dynamically based on closed-loop feedback from the building equipment. For example, the equipment performance information received from the building equipment can be used to update the reliability and/or the efficiency of the building equipment. Step  1016  can include optimizing the objective function J periodically (e.g., once per day, once per week, once per month, etc.) to dynamically update the predicted cost and/or the net present value NPV cost  based on the closed-loop feedback from the building equipment. 
     In some embodiments, step  1016  include generating optimization results. The optimization results may include the optimal values of the decision variables in the objective function J for each time step i in the optimization period. The optimization results include operating decisions, equipment maintenance decisions, and/or equipment purchase decisions for each device of the building equipment. In some embodiments, the optimization results optimize the economic value of operating, maintaining, and purchasing the building equipment over the duration of the optimization period. In some embodiments, the optimization results optimize the net present value of one or more devices of the building equipment at a particular point in time. The optimization results may cause BMS  606  to activate, deactivate, or adjust a setpoint for the building equipment in order to achieve the optimal values of the decision variables specified in the optimization results. 
     In some embodiments, process  1000  includes using the optimization results to generate equipment purchase and maintenance recommendations. The equipment purchase and maintenance recommendations may be based on the optimal values for the binary decision variables B main,i  and B cap,i  determined by optimizing the objective function J. For example, a value of B main,25 =1 for a particular device of the building equipment may indicate that maintenance should be performed on that device at the 25 th  time step of the optimization period, whereas a value of B main,25 =0 may indicate that the maintenance should not be performed at that time step. Similarly, a value of B cap,25 =1 may indicate that a new device of the building equipment should be purchased at the 25 th  time step of the optimization period, whereas a value of B cap,25 =0 may indicate that the new device should not be purchased at that time step. 
     In some embodiments, the equipment purchase and maintenance recommendations are provided to building  10  (e.g., to BMS  606 ) and/or to client devices  448 . An operator or building owner can use the equipment purchase and maintenance recommendations to assess the costs and benefits of performing maintenance and purchasing new devices. In some embodiments, the equipment purchase and maintenance recommendations are provided to service technicians  620 . Service technicians  620  can use the equipment purchase and maintenance recommendations to determine when customers should be contacted to perform service or replace equipment. 
     Still referring to  FIG. 10 , process  1000  is shown to include updating the efficiency and the reliability of the building equipment based on the optimal maintenance strategy (step  1018 ). In some embodiments, step  1018  includes updating the efficiency η i  for one or more time steps during the optimization period to account for increases in the efficiency η of the building equipment that will result from performing maintenance on the building equipment or purchasing new equipment to replace or supplement one or more devices of the building equipment. The time steps i at which the efficiency η i  is updated may correspond to the predicted time steps at which the maintenance will be performed or the equipment will replaced. The predicted time steps at which maintenance will be performed on the building equipment may be defined by the values of the binary decision variables B main,i  in the objective function J. Similarly, the predicted time steps at which the building equipment will be replaced may be defined by the values of the binary decision variables B cap,i  in the objective function J. 
     Step  1018  can include resetting the efficiency η i  for a given time step i if the binary decision variables B main,i  and B cap,i  indicate that maintenance will be performed at that time step and/or new equipment will be purchased at that time step (i.e., B main,i =1 and/or B cap,i =1). For example, if B main,i =1, step  1018  can include resetting the value of η i  to η main , where η main  is the efficiency value that is expected to result from the maintenance performed at time step i. Similarly, if B cap,i =1, step  1018  can include resetting the value of η i  to η cap , where η cap  is the efficiency value that is expected to result from purchasing a new device to supplement or replace one or more devices of the building equipment performed at time step i. Step  1018  can include resetting the efficiency η i  for one or more time steps while the optimization is being performed (e.g., with each iteration of the optimization) based on the values of binary decision variables B main,i  and B cap,i . 
     Step  1018  may include determining the amount of time Δt main,i  that has elapsed since maintenance was last performed on the building equipment based on the values of the binary decision variables B main,i . For each time step i, step  1018  can examine the corresponding values of B main  at time step i and each previous time step (e.g., time steps i−1, i−2, . . . , 1). Step  1018  can include calculating the value of Δt main,i  by subtracting the time at which maintenance was last performed (i.e., the most recent time at which B main,i =1) from the time associated with time step i. A long amount of time Δt main,i  since maintenance was last performed may result in a lower reliability, whereas a short amount of time since maintenance was last performed may result in a higher reliability. 
     Similarly, step  1018  may include determining the amount of time Δt cap,i  that has elapsed since the building equipment were purchased or installed based on the values of the binary decision variables B cap,i . For each time step i, step  1018  can examine the corresponding values of B cap  at time step i and each previous time step (e.g., time steps i−1, i−2, . . . , 1). Step  1018  can include calculating the value of Δt cap,i  by subtracting the time at which the building equipment were purchased or installed (i.e., the most recent time at which B cap,i =1) from the time associated with time step i. A long amount of time Δt cap,i  since the building equipment were purchased or installed may result in a lower reliability, whereas a short amount of time since the building equipment were purchased or installed may result in a higher reliability 
     Some maintenance activities may be more expensive than other. However, different types of maintenance activities may result in different levels of improvement to the efficiency η and/or the reliability of the building equipment. For example, merely changing the oil in a chiller may result in a minor improvement in efficiency η and/or a minor improvement in reliability, whereas completely disassembling the chiller and cleaning all of the chilled water tubes may result in a significantly greater improvement to the efficiency η and/or the reliability of the building equipment. Accordingly, multiple different levels of post-maintenance efficiency (i.e., η main ) and post-maintenance reliability (i.e., Reliability main ) may exist. Each level of η main  and Reliability main  may correspond to a different type of maintenance activity. 
     In some embodiments, step  1018  includes identifying the maintenance activity associated with each binary decision variable B main,j,i  and resets the efficiency η to the corresponding post-maintenance efficiency level η main,j  if B main,j,i =1. Similarly, step  1018  may include identifying the maintenance activity associated with each binary decision variable B main,j,i  and can reset the reliability to the corresponding post-maintenance reliability level Reliability main,j  if B main,j,i =1. 
     Some capital purchases may be more expensive than other. However, different types of capital purchases may result in different levels of improvement to the efficiency η and/or the reliability of the building equipment. For example, purchasing a new sensor to replace an existing sensor may result in a minor improvement in efficiency η and/or a minor improvement in reliability, whereas purchasing a new chiller and control system may result in a significantly greater improvement to the efficiency η and/or the reliability of the building equipment. Accordingly, multiple different levels of post-purchase efficiency (i.e., η cap ) and post-purchase reliability (i.e., Reliability cap ) may exist. Each level of η cap  and Reliability cap  may correspond to a different type of capital purchase. 
     In some embodiments, step  1018  includes identifying the capital purchase associated with each binary decision variable B main,k,i  and resetting the efficiency η to the corresponding post-purchase efficiency level η cap,k  if B cap,k,i =1. Similarly, step  1018  may include identifying the capital purchase associated with each binary decision variable B cap,k,i  and can resetting the reliability to the corresponding post-purchase reliability level Reliability cap,k  if B main,k,i =1. 
     Model Predictive Maintenance with Artificial Intelligence Functionality 
     Overview 
     Referring generally to  FIGS. 11A-19 , systems and methods for performing MPM for building equipment of a building (e.g., building  10 ) are shown, according to some embodiments. Performing MPM for the building can allow for generation of maintenance, replacement, and/or upgrade recommendations and decisions. Maintenance of building equipment can refer to various repairs and other activities that can be performed on building equipment to lower a degradation state of the building equipment. Replacement of building equipment can refer to switching out building equipment with new versions of the building equipment. For example, an indoor unit (IDU) of a variable refrigerant flow (VRF) system can be replaced after a degradation state of the IDU exceeds some threshold value as to reset the degradation state of the IDU to an original value (e.g., 0) when the IDU was first installed. Upgrading building equipment can refer to purchasing improved building devices for the building that are different from those currently installed. For example, upgrading a heating unit may include installing a new heating unit made by a different company as compared to the currently installed heating unit. In some embodiments, replacing and upgrading building equipment are considered together rather than separately as described above. 
     In a MPM process, degradation states of the building equipment can be estimated based on various measurements taken during operation of the building equipment. The degradation state of the building equipment can be defined for different building devices of the building equipment by various parameters such as, for example, an amount of resources (e.g., energy, water, etc.) the building equipment consumes during a time period, an amount of time it takes the building equipment to affect a particular change in an environmental condition of the building, a refrigerant charge level, an air flow restriction, power consumption of a compressor, etc. In some embodiments, the degradation state is divided up into elements. In particular, there may be a degradation state associated with maintenance decisions, a degradation state associated with system replacement, and/or a degradation state associated with system upgrades. 
     In some embodiments, artificial intelligence (AI) is integrated with the MPM system to learn how to properly estimate degradation states of the building equipment. The AI can learn mappings between degradation states and parameters of a power consumption model that can be used to predict how the building equipment consumes power over time. If the MPM system with AI is operating online, outputs of the AI can be adjusted if the parameters of the power consumption model are over estimated and/or under estimated. 
     MPM has emerged as an effective approach for enhancing asset reliability and availability by considering a multi-objective function including operational and maintenance (O&amp;M) costs over a planning horizon. Reliability assessment of the complex dynamic systems can provide MPM with high predictive accuracy. Since such a dynamic system may operate under variable operational and environmental conditions, precisely estimating the reliability of an asset through historical information of similar systems can be complex and time consuming. In this way, intrinsic properties for dynamic systems can evolve and change over time. Many traditional methods of reliability estimation highly depend on the failure-time or lifetime data which are not always available, obtainable, and/or trustable. On the other hand, it may not be reasonable to collect the failure or lifetime data through an accelerated life test or censored approaches for the dynamic systems which are highly reliable. 
     Traditional reliability algorithms can be based on recorded lifetime data and an amount of time until physical failure of a given unit or system. Traditional methods of reliability assessment can rely on large sampling experiments in order to obtain insight regarding the failure time of the same unit. The main purpose of these algorithms can be analyzing the lifetime data in order to reach an insight into how the system may fail in the future. Lifetime data can be obtained by 1) the history of the failures in real applications, 2) results of accelerated testing in laboratories, or 3) standards for the common units. In some embodiments, obtaining information regarding the time to physical failure is not always possible. In particular, products are becoming more reliable due to technological developments. As a result, only a few or even zero failures may occur during the monitoring phases, thereby leading to a lack of valuable information. Furthermore, it should be noted that some of the experiments can degrade the units. Therefore, obtaining the information regarding the time to physical failure may not always be an optimal approach. 
     It should be noted that systems may inevitably deteriorate over time with dissimilar rates of deterioration. In some embodiments, the rates of deterioration may not be the same even for similar systems with a certain load due to distinct environmental factors in which the systems operate. Most general degradation models are mainly introduced for systems with a constant degradation rate, which cannot be applied for complex systems that face a time-varying degradation process. Furthermore, obtaining a degradation observation and evolution of the observations over time is still one of the practical challenges for complex systems. 
     In contrast to the failure data, a robust method of reliability assessment that takes into account actual conditions of the system during the operation time can ensure MPM decisions are accurate. This methodology can act as an important part of the MPM system which can significantly affect further maintenance and replacement decisions. In some embodiments, this methodology is based on condition-based maintenance (CBM) rather than the lifetime analyses. Reliability estimates can significantly affect maintenance and replacement decisions derived by an MPM optimization problem. Reliability and probability of failure (PoF) estimates are one of the main elements of the MPM objective function. Intuitively, the ability of a system to consistently perform its intended or required functions over a period of time given certain operational and environmental conditions can highly affect a recommended maintenance and replacement schedule. The MPM optimization problem can seek to constantly trigger an optimum maintenance dispatch to minimize the unscheduled downtime which can lead to optimizing the reliability and availability of the system. 
     Various studies can be conducted in order to enhance real-time reliability estimation and evaluation of complex dynamic systems. Most of the methodologies can be categorized into two categories as 1) regression-based analysis and 2) time-series analysis which performance data is available or obtainable. Degradation estimates can be considered as a key performance index (KPI) such that the predictive models can estimate a KPI value over time. Degradation-based algorithms may rely on an operational and environmental status of components of the system. Information regarding these explanatory variables can be obtained by collecting sensor measurements and/or from simulation results. The degradation models can be projected into the future in order to obtain an estimate of the failure time. 
     Degradation-based models can consider the failure when either an observed or a projected degradation profile hits a threshold value for the first time. Most of the degradation-based reliability algorithms consider a predetermined value for the threshold limit. The threshold can be assumed to be a deterministic fixed value. Failure time, T, can be defined as the first time at which the degradation profile hits the threshold. In some embodiments, a distribution function of the failure times can be determined. Consequently, a probability of failure and reliability can be estimated based on a statistical distribution function of failure times. In some embodiments, the probability of failure is defined as a probability that a degradation state is greater than or equal to a threshold value. In this case, a density function of a new random variable can be derived analytically, which allows for calculating the probability of failure analytically and can be used by an optimizer (e.g., high level optimizer  832 ). It should be considered that the reliability estimates, obtained by the statistical distribution of the failure time, may not be able to accurately describe the behavior of a dynamic system since each system may deteriorate differently. Therefore, a novel robust methodology is desirable in order to analytically estimate the reliability based on the actual condition of the assets. 
     Determining an overall degradation profile can be difficult. For this reason, the degradation profile can be projected into the future with respect to observed degradation estimates and degradation models. In this case, the distribution of degradation may have a larger variance as the prediction time is farther in time compared with a current time. It should be considered that the obtained distribution function of the lifetime based on the degradation methods is based on a unique definition of failure. In degradation-based analysis, failure can be considered to occur if a degradation estimate hits the threshold for the first time. Both of these methodologies can depend on multiple degradation profiles over the time which might not be always available or accurately possible to be estimated. Furthermore, since the reliability assessment is based on the statistical distribution, these methodologies are not robust enough to estimate the reliability based on an actual condition of the assets. Although these approaches are based on the degradation profiles, the approaches still may rely on statistical inference which makes them unable to accurately describe an actual condition of similar assets. In this way, the results of these algorithms may lead to unrealistic judgments regarding a future condition of the assets and ultimately, suboptimal maintenance and replacement schedules. 
     A main feature of an optimum MPM strategy is an ability to predict the future condition of units (e.g., building equipment) or systems. To measure a probability of failure and reliability, judgments can be made about what the units or systems might be like in the future. Reliability can be, therefore, an extension of quality into a time domain. Reliability of a system can be defined as the likelihood that the system will perform required functions under the stated conditions for a specified period of time. These operational or environmental conditions may not always be controllable or predictable in real-world applications. The lifetime of the assets can be determined given specific working conditions that can be considered as nominal values which have been determined by design engineers during design, manufacturing, and test phases of the assets. Consequently, a remaining useful life (RUL) of each asset can be considered a random variable. 
     Reliability analysis can incorporate activities to identify potential failure modes and mechanisms, to make reliability predictions, and to quantify risks for the critical components in order to optimize life-cycle costs. Reliability engineering tries to ensure that a unit is reliable during operation in specific conditions by avoiding any failure. In other words, the purpose of reliability engineering is maximizing reliability while minimizing failure effects. As such, the purpose of reliability analysis may not simply be to describe how, when, and why systems fail, but rather to use information about failures to support decisions that improve the system&#39;s quality, safety and performance to reduce costs. Said aspect is particularly important in areas where failures have serious consequences. For instance, if a component of a variable refrigerant flow (VRF) system fails, then cooling or heating would not be available, thereby resulting in occupant discomfort. Consequences of failure events can be more concerning for commercial buildings due to a large number of the people affected by a shutdown/failure. As a result, a robust methodology for estimating reliability is an essential part of MPM. 
     As described in greater detail below, a novel robust methodology to analytically obtain reliability estimates based on state space models (SSMs) is presented. In other words, an innovative analytical solution technique to estimate dynamic reliability based on the SSMs of degradation is described. In this way, an analytical approach to estimating an age and state-dependent reliability and probability of failure for degrading systems which experience a stochastic degradation process with known properties is described. It should be noted that an effect of the age or wear-out mechanism is considered embedded inside estimates of the rate of events. Furthermore, it can be assumed that for each deteriorating system, a single degradation estimate, which can be obtained by analyzing various dependent variables or KPIs, is available. It can also be assumed that uncertainty in defining the failure threshold can be improved by considering the threshold as a random variable which follows a normal distribution function. 
     Benefits of Model Predictive Maintenance 
     Referring now to  FIGS. 11A-11C , several graphs  1100 - 1120  illustrating disadvantages of traditional maintenance strategies are shown, according to some embodiments. Traditional maintenance strategies can result in increased costs over a time horizon, unnecessary upkeep of the building equipment, and/or repairing or replacing building equipment at suboptimal times. Examples of traditional maintenance strategies include periodic maintenance, run-to-fail, etc. Periodic maintenance may cause building equipment to have maintenance performed on a schedule (e.g., once per month, once per year, etc.). Depending on degradation of the building equipment, periodic maintenance may result in the building equipment being maintenance too much or not enough to optimize (e.g., minimize) costs. Likewise, a run-to-fail strategy may result in costly repairs if the building equipment fails completely without being maintained. 
     Referring particularly to  FIG. 11A , a graph  1100  illustrating life cycle costs of operating and maintaining building equipment is shown, according to some embodiments. Graph  1100  is shown to include a series  1102  illustrating how cost can increase over time for a traditional maintenance strategy. As shown in graph  1100 , series  1102  increases dramatically over a time period. The drastic increase of series  1102  may be due to increased operational costs as a result of the building equipment degrading, too many maintenance/replacement activities being performed, etc. 
     Referring now to  FIG. 11B , a graph  1110  illustrating how a degradation state of building equipment is affected by a periodic maintenance strategy is shown, according to some embodiments. In a periodic maintenance strategy, the building equipment may have maintenance/replacement performed at set intervals (e.g., every week, every month, etc.). Graph  1110  is shown to include a series  1112  illustrating the degradation state of the building equipment over time. The degradation state increases over time as the equipment is used, but is reset or decreased when maintenance occurs or when the equipment is replaced. The times at which maintenance is performed or the equipment is replaced are shown as t 1 , t 2 , t 3 , and t 4 . As such, the degradation state of the building equipment is shown to increase up to the time when maintenance/replacement occurs and decreases as a result of the maintenance/replacement. Typically, periodic maintenance may not be able to restore the equipment to new condition (i.e., zero degradation), which is why the post-maintenance degradation state is still higher than zero. While periodic maintenance can keep the degradation state of the building equipment low, periodic maintenance may result in too many maintenance/replacement activities being performed if the periodic maintenance interval is too short, thus incurring unnecessary maintenance costs. Additionally, periodic maintenance may not keep the degradation state of the building equipment low enough if the periodic maintenance interval is too long, thereby resulting in additional operating costs over time. 
     Referring now to  FIG. 11C , a graph  1120  illustrating how a degradation state of building equipment is affected based on a run-to-fail maintenance strategy is shown, according to some embodiments. Graph  1120  is shown to include a series  1122  illustrating how the degradation state of the building equipment continues to increase over time if the building equipment never has maintenance/replacement performed. As compared to series  1112  as described with reference to  FIG. 11B , series  1122  does not decrease as no maintenance/replacement activities are performed on the building equipment, thus leaving the degradation state of the building equipment to continuously increase. Inevitably, the degradation state indicated by series  1122  a threshold at which the equipment fails or becomes too degraded to use and is thus completely or effectively inoperable. Equipment failure can be described as a point where operating the building equipment is either impossible and/or results in high operating costs as compared to normal operation of the building equipment at a lower degradation state. As such, it should be appreciated that allowing the degradation state of the building equipment to follow series  1120  may not be optimal and can result in unnecessary costs. 
     Referring now to  FIG. 12 , an illustration  1200  of a progression of maintenance strategies is shown, according to some embodiments. Illustration  1200  is shown to include reactive maintenance (e.g., run-to-fail) as a least optimal maintenance strategy for building equipment. In a reactive maintenance strategy, building equipment may only receive maintenance/replacement after an equipment failure. As described above with reference to  FIG. 11C , performing maintenance/replacement after equipment failure may result in high operational costs due to a high degradation state of the building equipment and due to a high cost of performing maintenance/replacement on failed building equipment. 
     Illustration  1200  is also shown to include preventive maintenance as a possible maintenance strategy for the building equipment. In general, a preventive maintenance strategy can provide more cost savings as compared to reactive maintenance strategy (unless maintenance/replacement is performed extremely frequently) as the probability of failure of the building equipment can be kept at a lower value. In some embodiments, the probability of failure is estimated based on a degradation state of the building equipment. By performing maintenance/replacement routinely, the degradation state of the building equipment can be routinely improved (e.g., reduced) such that probability of failure of the building equipment is reduced. However, preventive maintenance may not result in maintenance/replacement occurring at optimal times and therefore may not be a best maintenance strategy. 
     Illustration  1200  is also shown to include predictive maintenance as a possible maintenance strategy for building equipment. Predictive maintenance can utilize predictive models of building equipment to estimate an optimal or near-optimal time to perform maintenance/replacement on the building equipment. Unlike preventive maintenance, predictive maintenance may not require the building equipment to be maintained on a regular basis. Instead, predictive maintenance can be used to determine a recommended time to perform maintenance/replacement of the building equipment such that the cost of the maintenance/replacement and costs related to operating the building equipment are optimized (e.g., reduced). In this way, predictive maintenance can result in lower costs as compared to preventive and reactive maintenance. 
     Model predictive maintenance (MPM) is a type of predictive maintenance strategy for building equipment. In some embodiments, MPM is a maintenance strategy which minimizes a likelihood of failure through monitoring performance and condition of the building equipment during normal operational time. MPM algorithms may seek to determine optimized future maintenance and replacement schedules based on the condition of an in-service component or system. As maintenance activities are performed when warranted by algorithms, the MPM approach can result in cost savings over a time horizon. 
     MPM can be considered a type of condition based maintenance (CBM) which carries out maintenance/replacement activities as suggested by degradation estimators of an asset or system. In some embodiments, a primary purpose of MPM is providing an optimized dispatch of preventive and corrective maintenance to prevent unexpected failure of building equipment. An efficient MPM strategy can be first predict when equipment failure could occur based on KPIs, followed by preventing the failure through corrective maintenance. Condition monitoring may be necessary for MPM to be successfully implemented as to ensure optimized usage of assets. High penetration of smart devices and Internet of Things (IoT) principles can bring condition monitoring strategies to continuous real-time monitoring. IoT, machine learning (ML), cloud computing, and big data analytics can assist in implementation of MPM by providing more information regarding conditions of assets. In this way, MPM can provide benefits from a cost perspective, minimize unexpected downtime, as well as maximize lifespan, availability, reliability, and employee productivity. While implementing MPM may take a large amount of time and budget to develop, implement, and validate the algorithms, once fully implemented the cost savings can help offset any initial costs associated with integrating MPM for a building. MPM algorithms can be applied in many applications which data can be collected for selected KPIs. 
     Model Predictive Maintenance System With Degradation Impact Model 
     Referring now to  FIG. 13 , a model predictive maintenance (MPM) system  1300  is shown, according to some embodiments. In some embodiments, one or more of the components of MPM system  1300  may be the same as or similar to the corresponding components of building system  600  and/or MPM system  602  as described with reference to  FIGS. 6-10 . The components of MPM system  1300  are given new reference numbers in  FIG. 13  for ease of explanation. However, it should be understood that MPM system  1300  may be integrated into building system  600  in the same manner as MPM system  602  and may perform some or all of the functions of MPM system  602  as described with reference to  FIGS. 6-10 . 
     MPM system  1300  is shown to include a MPM controller  1302 , service providers  1330 , connected equipment  1332 , a weather service  1334 , and utilities  1336 . Connected equipment  1332  may be the same as or similar to connected equipment  610 , as described with reference to  FIGS. 6 and 8 . For example, connected equipment  1332  may include one or more chillers, boilers, air handling units, batteries, valves, actuators, thermal energy storage tanks, fans, dampers, or any other type of equipment that can be used to perform the various functions of a building or campus. Connected equipment  1332  may include sensors, local controllers, and/or communications electronics capable of providing performance variables y k  to MPM controller  1302 . 
     The performance variables y k  can include measurements or other performance data characterizing the operating performance of connected equipment  1332 . For example, the performance variables y k  may include an amount of electricity consumed by connected equipment  1332 , an amount of other resources (e.g., water, natural gas, etc.) consumed by connected equipment  1332 , an amount of time it takes connected equipment  1332  to affect a desired change in a zone of the building, an operating efficiency of connected equipment  1332  (e.g., a ratio of resources produced to resources consumed, a coefficient of performance, etc.), a number of run hours of connected equipment, or any other variable that can be used to estimate the degradation state of connected equipment  1332 . The performance variables y k  can be provided to MPM controller  1302  and used by MPM controller  1302  to estimate a degradation state of connected equipment  1332 . In some embodiments, the variable y k  is a vector that includes values for one or more performance variables at time step k. 
     Service providers  1330  may include any entity capable of performing maintenance on connected equipment  1332 , repairing connected equipment  1332 , replacing connected equipment  1332 , or otherwise performing actions in accordance with the maintenance schedule m k  generated by MPM controller  1302 . For example, service providers  1330  may include maintenance personnel who work within the building or campus, external service providers such as contractors, service technicians, or any other person or entity capable of executing the maintenance activities specified by the maintenance schedule m k . Service providers  1330  may receive service requests from MPM controller  1302  and execute the service requests by performing maintenance, repairing, replacing, or otherwise servicing connected equipment  1332 . 
     Weather service  1334  and utilities  1336  may be the same as or similar to weather service  604  and utilities  608 , as described with reference to  FIGS. 6 and 8 . Utilities  1336  may provide utility pricing data (e.g., electricity prices, natural gas prices, water prices, demand charge prices, etc.) to MPM controller  1302 , whereas weather service  1334  may provide weather forecasts (e.g., outdoor air temperature, outdoor air humidity, wind speed, precipitation forecasts, etc.) to MPM controller  1302 . MPM controller  1302  may use the pricing data and weather forecasts to predict the energy loads of the building or campus and utility prices at each time step of an optimization period. 
     MPM controller  1302  is shown to include a communications interface  1304  and a processing circuit  1306 . Communications interface  1304  may include wired or wireless interfaces (e.g., jacks, antennas, transmitters, receivers, transceivers, wire terminals, etc.) for conducting data communications with various systems, devices, or networks. For example, communications interface  1304  may include an Ethernet card and port for sending and receiving data via an Ethernet-based communications network and/or a Wi-Fi transceiver for communicating via a wireless communications network. Communications interface  1304  may be configured to communicate via local area networks or wide area networks (e.g., the Internet, a building WAN, etc.) and may use a variety of communications protocols (e.g., BACnet, IP, LON, etc.). 
     Communications interface  1304  may be a network interface configured to facilitate electronic data communications between MPM controller  1302  and various external systems or devices (e.g., connected equipment  1332 , utilities  1336 , weather service  1334 , service providers  1330 , etc.). For example, MPM controller  1302  may receive performance variables y k  from connected equipment  1332  indicating one or more measured states of the controlled building (e.g., temperature, humidity, electric loads, etc.) and/or equipment performance information (e.g., run hours, power consumption, operating efficiency, etc.). Communications interface  1304  may receive inputs from utilities  1336 , weather service  1334 , connected equipment  1332  and may provide a maintenance schedule m k  or service requests to service providers  1330  or other external systems or devices. 
     Processing circuit  1306  is shown to include a processor  1308  and memory  1310 . Processor  1308  may be a general purpose or specific purpose processor, an application specific integrated circuit (ASIC), one or more field programmable gate arrays (FPGAs), a group of processing components, or other suitable processing components. Processor  1308  may be configured to execute computer code or instructions stored in memory  1310  or received from other computer readable media (e.g., CDROM, network storage, a remote server, etc.). 
     Memory  1310  may include one or more devices (e.g., memory units, memory devices, storage devices, etc.) for storing data and/or computer code for completing and/or facilitating the various processes described in the present disclosure. Memory  1310  may include random access memory (RAM), read-only memory (ROM), hard drive storage, temporary storage, non-volatile memory, flash memory, optical memory, or any other suitable memory for storing software objects and/or computer instructions. Memory  1310  may include database components, object code components, script components, or any other type of information structure for supporting the various activities and information structures described in the present disclosure. Memory  1310  may be communicably connected to processor  1308  via processing circuit  1306  and may include computer code for executing (e.g., by processor  1308 ) one or more processes described herein. 
     Still referring to  FIG. 13 , MPM controller  1302  is shown to include a load/rate predictor  1312 , a degradation impact modeler  1314 , a degradation estimator  1316 , a model predictive optimizer  1320 , and a maintenance scheduler  1318 . Load/rate predictor  1312  may be configured to predict the energy loads (Load i ) (e.g., heating load, cooling load, electric load, etc.) of the building or campus for each time step i of the optimization period. Load/rate predictor  1312  is shown receiving weather forecasts from weather service  1334 . In some embodiments, load/rate predictor  1312  predicts the energy loads Load i  as a function of the weather forecasts. In some embodiments, load/rate predictor  1312  uses feedback from connected equipment  1332  to predict loads Load i . Feedback from connected equipment  1332  may include various types of sensory inputs (e.g., temperature, flow, humidity, enthalpy, etc.) or other data relating to the controlled building (e.g., inputs from a HVAC system, a lighting control system, a security system, a water system, etc.) and may be included in performance variables y k . 
     In some embodiments, load/rate predictor  1312  receives a measured electric load and/or previous measured load data from connected equipment  1332 . Load/rate predictor  1312  may predict loads Load i  as a function of a given weather forecast ({circumflex over (ϕ)} w ), a day type (day), the time of day (t), and previous measured load data (Y i-1 ). Such a relationship is expressed in the following equation: 
       Load i =ƒ({circumflex over (ϕ)} w ,day, t|Y   i-1 )
 
     In some embodiments, load/rate predictor  1312  uses a deterministic plus stochastic model trained from historical load data to predict loads Load i . Load/rate predictor  1312  may use any of a variety of prediction methods to predict loads Load i  (e.g., linear regression for the deterministic portion and an AR model for the stochastic portion). Load/rate predictor  1312  may predict one or more different types of loads for the building or campus. For example, load/rate predictor  1312  may predict a hot water load Load Hot,i , a cold water load Load Cold,i , and an electric load Load Elec,i  for each time step i within the optimization period. The predicted load values Load i  can include some or all of these types of loads. In some embodiments, load/rate predictor  1312  makes load/rate predictions using the techniques described in U.S. patent application Ser. No. 14/717,593. 
     Load/rate predictor  1312  is shown receiving utility rates from utilities  1336 . Utility rates may indicate a cost or price per unit of a resource (e.g., electricity, natural gas, water, etc.) provided by utilities  1336  at each time step i in the optimization period. In some embodiments, the utility rates are time-variable rates. For example, the price of electricity may be higher at certain times of day or days of the week (e.g., during high demand periods) and lower at other times of day or days of the week (e.g., during low demand periods). The utility rates may define various time periods and a cost per unit of a resource during each time period. Utility rates may be actual rates received from utilities  608  or predicted utility rates estimated by load/rate predictor  1312 . 
     In some embodiments, the utility rates include demand charges for one or more resources provided by utilities  1336 . A demand charge may define a separate cost imposed by utilities  608  based on the maximum usage of a particular resource (e.g., maximum energy consumption) during a demand charge period. The utility rates may define various demand charge periods and one or more demand charges associated with each demand charge period. In some instances, demand charge periods may overlap partially or completely with each other and/or with the prediction window. Model predictive optimizer  1320  may be configured to account for demand charges in a high level optimization process performed by model predictive optimizer  1320 . Utilities  1336  may be defined by time-variable (e.g., hourly) prices, a maximum service level (e.g., a maximum rate of consumption allowed by the physical infrastructure or by contract) and, in the case of electricity, a demand charge or a charge for the peak rate of consumption within a certain period. Load/rate predictor  1312  may store the predicted loads Load i  and the utility rates in memory  1310  and/or provide the predicted loads Load i  and the utility rates to model predictive optimizer  1320 . 
     Degradation estimator  1316  can be configured to estimate the degradation states {circumflex over (δ)} k  of connected equipment  1332 . As used herein, the variable {circumflex over (δ)} k  denotes one or more estimated degradation states of connected equipment  1332  at time step k. In some embodiments, the variable {circumflex over (δ)} k  is a vector containing a plurality of degradation state estimates. For example, the variable {circumflex over (δ)} k  may be defined as: 
     
       
         
           
             
               
                 δ 
                 ^ 
               
               k 
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         δ 
                         ^ 
                       
                       
                         1 
                         , 
                         k 
                       
                     
                   
                 
                 
                   
                     
                       
                         δ 
                         ^ 
                       
                       
                         
                           2 
                           ′ 
                         
                          
                         k 
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       
                         δ 
                         ^ 
                       
                       
                         n 
                         , 
                         k 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     where {circumflex over (δ)} 1,k  is a first estimated degradation state of connected equipment  1333  at time step k, {circumflex over (δ)} 2,k  is a second estimated degradation state of connected equipment  1333  at time step k, and {circumflex over (δ)} n,k  is a n th  estimated degradation state of connected equipment  1333  at time step k, where n is the total number of estimated degradation states contained within vector {circumflex over (δ)} k . In various embodiments, the degradation states {circumflex over (δ)} 1,k , {circumflex over (δ)} 2,k  . . . {circumflex over (δ)} n,k  may represent degradation states of different devices of connected equipment  1332  (e.g., the degradation state of a chiller, the degradation state of a boiler, the degradation state of a fan, etc.) and/or degradation states of particular components of a device of connected equipment  1332  (e.g., the degradation state of a chiller&#39;s compressor, the degradation state of the same chiller&#39;s refrigerant tubes, etc.). 
     In some embodiments, degradation estimator  1316  estimates the degradation states {circumflex over (δ)} k  based on the performance variables y k  received from connected equipment  1332 . Values of the performance variables y k  can be gathered by various sensors and/or other devices in a building and provided as inputs to degradation estimator  1316 . For example, Y k  can include information such as an operating temperature of a building device as gathered by a temperature sensor, power consumption of a building device as gathered by an electrical measurement device, a current flowing through building equipment, a pressure of components in a building device, etc. Degradation estimator  1316  can estimate the degradation state {circumflex over (δ)} k  of connected equipment  1332  at time step k as a function of the performance variables y k , as shown in the following equation: 
       {circumflex over (δ)} k =ƒ( y   k )
 
     where the function ƒ( ) is a function that relates the performance variables y k  to the degradation states {circumflex over (δ)} k . 
     It is contemplated that the function ƒ( ) can have any of a variety of forms. For example, the function ƒ( ) may include operations that compare one or more values of the performance variables y k  (or functions thereof) to design parameters of connected equipment  1332  and calculate the degradation states {circumflex over (δ)} k  based on the values of the performance variables y k  relative to the design parameters (e.g., a ratio of operating efficiency at time step k relative to design efficiency). In other embodiments, the function ƒ( ) may represent a degradation estimation model that can be generated empirically by degradation estimator  1316 . For example, degradation estimator  1316  may use a set of historical data from one or more building sites to train the degradation estimation model. The set of historical data may include values of the performance variables y k  and corresponding values of the degradation states {circumflex over (δ)} k  or values representative of the degradation states {circumflex over (δ)} k  (e.g., equipment efficiency, operating cost, etc.). The degradation estimation model may include a regression model, a neural network, or any other type of model that provides a mapping between the performance variables y k  and the degradation states {circumflex over (δ)} k . The estimated degradation state {circumflex over (δ)} k  at time step k can be provided to degradation predictor  1322 . 
     In some embodiments, degradation estimator  1316  generates a raw degradation estimate {circumflex over (δ)} raw,k . The raw degradation estimate {circumflex over (δ)} raw,k  may be a function of the performance variables y k  and can be calculated using the same or similar technique as the estimated degradation states {circumflex over (δ)} k . Like the estimated degradation states {circumflex over (δ)} k , the raw degradation estimate {circumflex over (δ)} raw,k  may be a vector that includes an estimated degradation state for each device of connected equipment  1332  and/or components of the devices of connected equipment  1332 . In some embodiments, the raw degradation estimate {circumflex over (δ)} raw,k  is a function of the values of the performance variables y k  at time step k and one or more previous time steps. For example, the raw degradation estimate {circumflex over (δ)} raw,k  can be defined as: 
       {circumflex over (δ)} raw,k =ƒ( Y   k )
 
     where Y k  is a matrix that includes all of the values of the performance variables y k  over the period of time from k−h b  to k, where k is the time step at which the degradation state is evaluated and h b  is a backward looking time horizon. The matrix Y k  may include a value of each performance variable at each time step from k−h b  to k and may be defined as: 
     
       
         
           
             
               Y 
               k 
             
             = 
             
               [ 
               
                 
                   
                     
                       y 
                       
                         1 
                         , 
                         
                           k 
                           - 
                           
                             h 
                             b 
                           
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       y 
                       
                         1 
                         , 
                         k 
                       
                     
                   
                 
                 
                   
                     ⋮ 
                   
                   
                     ⋱ 
                   
                   
                     ⋮ 
                   
                 
                 
                   
                     
                       y 
                       
                         n 
                         , 
                         
                           k 
                           - 
                           
                             h 
                             b 
                           
                         
                       
                     
                   
                   
                     … 
                   
                   
                     
                       y 
                       
                         n 
                         , 
                         k 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     where y 1  is the first performance variable, y n  is the n th  performance variable, k−h b  is the first time step included in the matrix Y k  (i.e., h b  time steps before time step k), and k is the last time step included in the matrix Y k . The raw degradation state {circumflex over (δ)} k,raw  at time step k can be provided to degradation impact modeler  1314 . 
     In some embodiments, degradation estimator  1316  scales the raw degradation state {circumflex over (δ)} raw,k  by a scaling factor α (e.g., by multiplying {circumflex over (δ)} raw,k  by the scaling factor α) to produce a scaled degradation estimate α{circumflex over (δ)} raw,k . The scaled degradation estimate α{circumflex over (δ)} raw,k  represents a scaled output of degradation estimator  1316  and can be provided to degradation impact modeler  1314 . Scaling the values of {circumflex over (δ)} raw,k  can ensure inputs to a neural network used by degradation impact modeler  1314  are scaled to limit the values between a lower threshold and an upper threshold. Degradation estimator  1316  can provide the scaled values of α{circumflex over (δ)} raw,k  to degradation impact modeler  1314 . If a scale value of {circumflex over (δ)} raw,k  is not calculated, α can effectively be considered one (i.e. 1.0). Degradation impact modeler  1314  can use the values of α{circumflex over (δ)} raw,k  to train a neural network to map degradation states to power model coefficients, described in greater detail below. 
     In some embodiments, degradation estimator  1316  performs an optimization process to generate a value of the scaling factor α. For example, degradation estimator  1316  can find value of the scaling factor α that optimizes the following objective function: 
     
       
         
           
             
               
                 arg 
                  
                 min 
               
               ∝ 
             
              
             
               
                 ∑ 
                 
                   l 
                   = 
                   1 
                 
                 ∞ 
               
                
               
                  
                 
                   
                     P 
                     
                       k 
                       - 
                       l 
                     
                   
                   - 
                   
                     
                       
                         
                           P 
                           ^ 
                         
                         
                           k 
                           - 
                           l 
                         
                       
                       ( 
                       
                         
                           Q 
                           
                             k 
                             - 
                             l 
                           
                         
                         , 
                         
                           ϕ 
                            
                           
                             ( 
                             
                               α 
                                
                               
                                 
                                   δ 
                                   ^ 
                                 
                                 
                                   raw 
                                   , 
                                   k 
                                 
                               
                             
                             ) 
                           
                         
                       
                        
                     
                      
                     
                       e 
                       
                         - 
                         
                           l 
                           τ 
                         
                       
                     
                   
                 
               
             
           
         
       
     
     where P k−l  is the actual power consumption of connected equipment  1332  at time step k−l, {circumflex over (P)} k−l  is a predicted power consumption of connected equipment  1332  at time step k−l, Q k−l  is the heating or cooling load of connected equipment  1332  at time step k−l, φ are coefficients of a power consumption model used to predict {circumflex over (P)} k−l , and e −l/τ  is a weighting factor. The predicted power consumption {circumflex over (P)} k−l  can be predicted using a power model that predicts {circumflex over (P)} k−l  as a function of power model coefficients φ and the heating or cooling load Q k−l . The power model coefficients φ can be generated by degradation impact modeler  1314  as a function of the degradation state α{circumflex over (δ)} raw,k , as described in greater detail below. By optimizing this objective function, degradation estimator  1316  may seek to minimize the difference between the actual power consumption P k−l  and the model predicted power consumption {circumflex over (P)} k−l . 
     In some embodiments, degradation estimator  1316  separates the degradation estimation into two processes: (1) an offline process that trains a degradation estimation model with historical data from various building sites and (2) an online process that uses data from past time horizons from the specific building site at which connected equipment  1332  are located and estimates the current state of degradation {circumflex over (δ)} k . Calculating the values of the degradation states {circumflex over (δ)} k  as a function of the performance variables y k  using the function ƒ( ) can be considered the online portion, whereas generating the degradation estimation model represented by the function ƒ( ) can be considered the offline portion. 
     Degradation impact modeler  1314  can be configured to determine the impact of the estimated degradation state {circumflex over (δ)} k  or scaled degradation estimate α{circumflex over (δ)} raw,k  on the cost of operating connected equipment  1332 . In some embodiments, the cost of operating connected equipment  1332  depends on the amount of electric power or other resource (e.g., water, natural gas, etc.) consumed by connected equipment  1332  during operation, which in turn may be a function of the degradation state. Although degradation impact modeler  1314  is described primarily with reference to electric power consumption, it should be understood that any other resource consumed by connected equipment  1332  can be used instead of electric power or in addition to electric power without departing from the teachings of the present disclosure. 
     Advantageously, degradation impact modeler  1314  can be configured to predict the power consumption of connected equipment  1332  as a function of the estimated degradation state {circumflex over (δ)} k  or scaled degradation estimate α{circumflex over (δ)} raw,k  For ease of explanation, the following description assumes that degradation impact modeler  1314  uses the scaled degradation estimate α{circumflex over (δ)} raw,k . However, it should be understood that the estimated degradation state {circumflex over (δ)} k  can be used in place of or in addition to the scaled degradation estimate α{circumflex over (δ)} raw,k  without departing from the teachings of the present disclosure. The predicted power consumption of connected equipment  1332  can be provided to model predictive optimizer  1320  for use in calculating the cost of operating connected equipment. 
     In some embodiments, degradation impact modeler  1314  is configured to generate power model coefficients φ of connected equipment  1332  as a function of the estimated degradation state {circumflex over (δ)} k  or scaled degradation estimate α{circumflex over (δ)} raw,k . The power model coefficients φ may be coefficients of a power consumption model that is used by model predictive optimizer  1320  to determine that power consumption of connected equipment  1332  as a function of the operating decisions for connected equipment  1332 . For example, the power consumption model may provide a mapping between the amount of power consumed by connected equipment  1332  and the heating or cooling load on connected equipment  1332  (e.g., if connected equipment  1332  is a heater or chiller). More generally, the power consumption model may be a function or curve that defines the relationship between the amount of an input resource (or multiple input resources) consumed by connected equipment  1332  and the corresponding amount of an output resource (or multiple output resources) produced by connected equipment  1332 . In this regard, the power consumption model may be similar to or the same as equipment models  818 , described with reference to  FIG. 8 . As the degradation state of connected equipment  1332  increases, degradation impact modeler  1314  may update the power consumption model to reflect the decreased efficiency of connected equipment  1332  as a result of the degradation. Accordingly, by mapping the scaled degradation estimate α{circumflex over (δ)} raw,k  to the power model coefficients φ, degradation impact modeler  1314  can automatically adjust the power consumption model to account for equipment degradation. The updated values of the power model coefficients φ may be provided as an input to model predictive optimizer  1320 . 
     Still referring to  FIG. 13 , model predictive optimizer  1320  can be configured to perform an optimization process to generate the maintenance schedule m k  for connected equipment  1332  along with operating decisions for connected equipment  1332 . Model predictive optimizer  1320  may receive the degradation estimate {circumflex over (δ)} k  from degradation estimator  1316 , the load and rate predictions from load/rate predictor  1312 , and the power model coefficients φ from degradation impact modeler  1314 . Model predictive optimizer  1320  may use these inputs to perform an optimization process that seeks to optimize (e.g., minimize) the total cost of operating connected equipment  1332  and performing maintenance on connected equipment  1332  over a given time period (i.e., the optimization period). 
     The maintenance schedule m k  may be provided as an output of the optimization process performed by model predictive optimizer  1320 . It should be appreciated that m k  can be likewise referred to as a maintenance schedule, a maintenance and replacement schedule, and/or a maintenance strategy. The maintenance schedule m k  can include various information such as when connected equipment  1332  should have maintenance or replacement performed, specific building devices of connected equipment  1332  to have maintenance or replacement performed, equipment parts required for the maintenance or replacement activities, etc. It should be understood that the maintenance schedule m k  is not limited to maintenance activities and can also include replacement activities, equipment upgrades, adding new equipment that does not replace existing equipment, or any other type of service or modification that alters the set of connected equipment  1332  as a whole. In general, the maintenance schedule m k  can include any information necessary for connected equipment  1332  to be suitably maintained, replaced, upgraded, repaired, and/or otherwise serviced. 
     Model predictive optimizer  1320  is shown to include a degradation predictor  1322  and a cost calculator  1324 . Degradation predictor  1322  can be configured to predict future degradation states {circumflex over (δ)} k+1  of connected equipment  1332  at one or more time steps after time step k. In some embodiments, degradation predictor  1322  uses a degradation prediction model to predict the future degradation states {circumflex over (δ)} k+1  as a function of the degradation states {circumflex over (δ)} k  at time step k and the maintenance schedule m k  for time step k. For example, the future degradation states {circumflex over (δ)} k+1  can be predicted using the following equation: 
       {circumflex over (δ)} k+1 =({circumflex over (δ)} k   ,m   k )
 
     where {circumflex over (δ)} k+1  is a vector of the future degradation states of connected equipment  1332  at a future time step k+1 (i.e., a time step after k) and m k  is the maintenance schedule at time step k. In some embodiments, the maintenance schedule m k  is generated by cost calculator  1324  and provided back to degradation predictor  1322  to predict the future degradation states {circumflex over (δ)} k+1 . 
     In some embodiments, both the maintenance schedule m k  and the future degradation states {circumflex over (δ)} k+1  are generated as results of an optimization process performed by model predictive optimizer  1320 . The optimization process may seek to optimize (e.g., minimize) the total cost of operating connected equipment  1332  and performing maintenance on connected equipment  1332  over a given time horizon. The cost of operating connected equipment  1332  at the future time step k+1 can be defined as a function of the future degradation states {circumflex over (δ)} k+1 . Both the cost of performing maintenance on connected equipment  1332  and the future degradation states {circumflex over (δ)} k+1  can be defined as functions of the maintenance schedule m k . For example, maintenance/replacement activities that occur at time step k can affect (e.g., improve) a degradation state of connected equipment  1332  and therefore can affect a predicted degradation state at time step k+1. Accordingly, the optimization performed by model predictive optimizer  1320  may generate optimal values of the maintenance schedule m k  and the resulting future degradation states {circumflex over (δ)} k+1 . The future degradation states {circumflex over (δ)} k+1  may be provided as an input to degradation impact modeler  1314  and used by degradation impact modeler  1314  to determine the corresponding values of the power model coefficients φ k+1  at the future time step. 
     Cost calculator  1324  is shown to include a reliability model  1326  and a system model  1328 . Reliability model  1326  can be used to estimate projections of reliability forward in time for connected equipment  1332 . In this way, reliability model  1326  can incorporate a risk of failure of connected equipment  1332  into the optimization problem solved by model predictive optimizer  1320 . System model  1328  may model the operating performance of connected equipment  1332  and may include the power consumption model described above (or any other model that relates input resource consumption to output resource generation). In some embodiments, system model  1328  has parameters p as well as the independent variable inputs x. For example, system model  1328  may have the form: 
         p=p   equip (φ; x )
 
     where p is the predicted power consumption of connected equipment  1320 , p equip  is a function that defines power consumption p as a function of the power model parameters p and the independent variables x, φ includes estimated power parameters, and x is a matrix or vector of power estimation predictors (i.e., independent variables). 
     For example, in a variable refrigerant flow (VRF) system, system model  1328  may define the power consumption of VRF equipment (i.e., a type of connected equipment  1332 ) as a function of one or more model parameters φ and a set of independent variable inputs x that represent the heating and cooling loads on the system (i.e., {circumflex over (Q)} h  and {circumflex over (Q)} c ) as well as the temperature lift {circumflex over (T)} lift  (i.e., the difference between outdoor air temperature and a setpoint temperature value). Accordingly, the matrix or vector of independent variable inputs x can be defined as: 
     
       
         
           
             x 
             = 
             
               [ 
               
                 
                   
                     
                       
                         Q 
                         ^ 
                       
                       c 
                     
                   
                 
                 
                   
                     
                       
                         Q 
                         ^ 
                       
                       h 
                     
                   
                 
                 
                   
                     
                       
                         T 
                         ^ 
                       
                       lift 
                     
                   
                 
               
               ] 
             
           
         
       
     
     where {circumflex over (Q)} c  is the estimated cooling load, {circumflex over (Q)} h  is the estimated heating load, and {circumflex over (T)} lift  is a lift temperature. Although x is shown as a vector in the equation above, it should be understood that each of the variables {circumflex over (Q)} c , {circumflex over (Q)} h , and {circumflex over (T)} lift  may include multiple values (e.g., one value for each time step. The multiple values of {circumflex over (Q)} c , {circumflex over (Q)} h , and {circumflex over (T)} Lift  can be included in x by adding another dimension to x, in which case x becomes a 3 by n matrix where n is the total number of time steps included in the matrix x. 
     Continuing the example of the VRF system, the system model  1328  for the VRF system can be defined as: 
         p=p   design (φ 1 ·max( {circumflex over (Q)}   c   ,{circumflex over (Q)}   h )+φ 2   ·|{circumflex over (Q)}   c   −{circumflex over (Q)}   h |+φ 3 ·max( {circumflex over (Q)}   c   ,{circumflex over (Q)}   h )· {circumflex over (T)}   lift )
 
     where p is the power consumption of the VRF equipment, p design  is the design power of the VRF equipment, φ 1 , φ 2 , and φ 3  are parameters of the system model  1328 , and the remaining variables are the same as previously described. 
     Cost calculator  1324  may use the power model coefficients φ k+1  provided by degradation impact modeler  1314  to update system model  1328  and may use the updated system model  1328  to formulate the optimization problem. For example, cost calculator  1324  may use system model  1328  to define a relationship between the power consumption of connected equipment  1332  and the load served by connected equipment  1332 . The relationship between power consumption and load served may be imposed as a constraint on the optimization problem solved by model predictive optimizer  1320 . 
     Cost calculator  1324  can be configured to obtain (e.g., generate, receive, formulate, etc.) an objective function J that is optimized by model predictive optimizer  1320 . An example of such an objective function J is: 
     
       
         
           
             
               J 
                
               
                 ( 
                 
                   m 
                   k 
                 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   i 
                   = 
                   k 
                 
                 
                   
                     h 
                     b 
                   
                   + 
                   k 
                   - 
                   1 
                 
               
                
               
                 { 
                 
                   
                     
                       c 
                       
                         
                           o 
                            
                           p 
                         
                         , 
                         i 
                       
                     
                      
                     
                       ( 
                       
                         δ 
                         i 
                       
                       ) 
                     
                   
                   + 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 c 
                                 
                                   main 
                                   , 
                                   i 
                                 
                               
                             
                           
                           
                             
                               
                                 c 
                                 
                                   replace 
                                   , 
                                   i 
                                 
                               
                             
                           
                         
                         ] 
                       
                       T 
                     
                      
                     
                       m 
                       i 
                     
                   
                   + 
                   
                     
                       c 
                       
                         
                           f 
                            
                           a 
                            
                           il 
                         
                         , 
                         i 
                       
                       T 
                     
                      
                     
                       
                         p 
                         
                           
                             f 
                              
                             a 
                              
                             i 
                              
                             l 
                           
                           , 
                           i 
                         
                       
                        
                       
                         ( 
                         
                           δ 
                           i 
                         
                         ) 
                       
                     
                   
                 
                 } 
               
             
           
         
       
     
     where m k  is a maintenance and replacement schedule, k is a given time step (past, present, or future) h b  is a backward optimization horizon (backward from the time step k), C op,i (δ i ) is an operational cost dependent on a degradation state δ i  at time step i, c main,i  is a cost of maintenance at time step i, C replace,i  is a replacement cost at time step i, m i  is a binary vector representing which maintenance actions are taken at time step i, C fail,i  is a cost of failure of building equipment at time step i, and p fail,i (δ i ) is a vector of probabilities of failure for each component of building equipment dependent on the state of degradation δ i . In the above objective function, the T superscript indicates a transpose of the associated matrix. Values of c fail,k  can include a cost to repair/replace the tracked building equipment and/or any opportunity costs related to failure of the tracked building equipment. 
     It should be appreciated that a first portion of the maintenance vector m i  (i.e., the portion to which maintenance costs c main,i  are applied) includes maintenance decisions, whereas a second portion of the maintenance vector m i  (i.e., the portion to which replacement costs C replace,i  are applied) includes replacement decisions. For example, the maintenance vector m i  can be defined as 
     
       
         
           
             
               m 
               i 
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         m 
                         
                           main 
                           , 
                           i 
                         
                       
                     
                   
                   
                     
                       
                         m 
                         
                           replace 
                           , 
                           i 
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     Each maintenance action m main,i  is associated with a corresponding maintenance cost c main,i , whereas each replacement action m replace,i  is associated with a corresponding replacement cost c replace,i . Further, it should be appreciated that C fail,k   T  p fail,k (δ k ) represents a risk cost term of the objective function. In some embodiments, the probability of failure (PoF) given each degradation state, p fail,i (δ i ), can be an output of reliability model  1326 . 
     The objective function J is shown as a summation of three costs. The first term of the objective function J (i.e., C op,i (δ i )) represents the total cost of operating connected equipment  1332  over the time period from time step k to time step h b +k−1. The second term of the objective function j 
     
       
         
           
             ( 
             
               
                 i 
                 . 
                 e 
                 . 
               
               , 
               
                   
               
                
               
                 
                   
                     [ 
                     
                       
                         
                           
                             c 
                             
                               m 
                                
                               a 
                                
                               i 
                                
                               
                                 n 
                                 ′ 
                               
                                
                               i 
                             
                           
                         
                       
                       
                         
                           
                             c 
                             
                               
                                 replace 
                                 ′ 
                               
                                
                               i 
                             
                           
                         
                       
                     
                     ] 
                   
                   T 
                 
                  
                 
                   m 
                   i 
                 
               
             
             ) 
           
         
       
     
     represents the total cost of performing any of the maintenance or replacement activities defined by the maintenance vector m i  on connected equipment  1332  over the time period from time step k to time step h b +k−1. The third term of the objective function J (i.e., C fail,i   T p fail,i (δ i )) represents the total cost of failure of connected equipment  1332  over the time period from time step k to time step h b +k−1. The time step k can be any time step in the past, present, or future. Accordingly, the time period ranging from time step k to time step h b +k−1 may be entirely in the past; partially in the past and partially in the present; partially in the past, present, and future; partially in the present and partially in the future; or entirely in the future in various embodiments. 
     Model predictive optimizer  1320  can be configured to perform an optimization of the objective function J subject to a set of constraints. The constraints may include the power consumption model or any other type of system model  1328  that defines the relationship between the operating cost C op,i  and the load served by connected equipment  1332 . For example, one constraint on the objective function J may be a power consumption model that defines the amount of power consumed p i  as a function of the load served by connected equipment and the power model parameters φ i . Another constraint on the objective function J may be a cost model that defines the operating cost C op,i  as a function of the amount of power consumed p i  and the pricing data received from utilities  1336 . Another constraint on the objective function J may be a model that defines the relationship between the probability of failure p fail,i  and the degradation state δ i . Another constraint on the objective function J may require connected equipment  1332  to satisfy the predicted heating or cooling load provided by load/rate predictor  1312 . Another constraint on the objective function J may require connected equipment  1332  to operate within their respective capacity limits (e.g., limiting the amount of input resources consumed, output resources produced, or other capacity-related variables at each time step). Other constraints on the objective function J can include any of the equations or relationships described with reference to operational cost predictor  910 , maintenance cost predictor  920 , and capital cost predictor  930 , as described with reference to  FIG. 9 . 
     Model predictive optimizer  1320  can perform an optimization of the objective function J, subject to the constraints, to determine the maintenance schedule m k  as well as operating decisions for connected equipment  1332 . The maintenance schedule m k  can be provided to maintenance scheduler  1318 , which may operate to schedule maintenance/replacement activities to be performed by service providers  1330  at times indicated by m k . In some embodiments, maintenance scheduler  1318  selects a particular service provider  1330  by determining available service providers  1330  that are capable of performing a maintenance/replacement activity indicated by m k  at a particular time. If m k  indicates multiple maintenance/replacement activities to be performed, maintenance scheduler  1318  may schedule each particular activity at an associated time. It should be appreciated that different service providers  1330  can be scheduled for different maintenance/replacement activities. In other words, the same service provider  1330  need not perform all maintenance/replacement activities indicated by m k . 
     Service providers  1330  may receive service requests from maintenance scheduler  1318  and perform the requested maintenance/replacement activities. As a result of the maintenance/replacement activity, the degradation state of connected equipment  1332  can be improved (e.g., reduced). In this way, operational costs associated with connected equipment  1332  can be reduced. The maintenance/replacement activities performed by service providers  1330  can include any number of maintenance/replacement activities as indicated by m k  and scheduled by maintenance scheduler  1318 . 
     Referring now to  FIGS. 14A-14B , a block diagram illustrating a portion of MPM system  1300  in greater detail and a corresponding process  1400  performed by these components of MPM system  1300  are shown, according to some embodiments. Process  1400  can be performed to generate a maintenance and replacement strategy m k  for connected equipment  1332 . Process  1400  further illustrates how degradation estimates and predictions can be used to generate m k . The steps of process  1400  can be performed by various components of MPM system  1300  as shown in  FIG. 14A . 
     Process  1400  is shown to include estimating the degradation state {circumflex over (δ)} k  of connected equipment  1332  at time step k as a function of performance variables y k  (step  1402 ). In some embodiments, step  1402  is performed by degradation estimator  1316  using a degradation estimation model, as described with reference to  FIG. 13 . Step  1402  may include receiving values of the performance variables y k . Values of the performance variables y k  can be gathered by various sensors and/or other devices in a building that can measure performance information associated with connected equipment. For example, y k  can include information such as an operating temperature of a building device as gathered by a temperature sensor, power consumption of a building device as gathered by an electrical measurement device, a current flowing through building equipment, a pressure of components in a building device, etc. Based on Y k , a degradation state {circumflex over (δ)} k  of building equipment at time step k can be estimated by the following function: 
       {circumflex over (δ)} k =ƒ( y   k )
 
     In other words, {circumflex over (δ)} k  can be expressed as a function of sensor measurements/performance variables y k . 
     Process  1400  is shown to include predicting the degradation state {circumflex over (δ)} k+1  of connected equipment  1332  at one or more time steps after time step k as a function of the degradation state {circumflex over (δ)} k  at time step k and a maintenance strategy m k  for connected equipment  1332  at time step k (step  1404 ). In some embodiments, step  1404  is performed by degradation predictor  1332 , as described with reference to  FIG. 13 . The degradation state of connected equipment  1332  can be predicted for a time step k+1 (i.e., a time step after time step k). {circumflex over (δ)} k+1  can be predicted as a function of {circumflex over (δ)} k  and m k  as shown in the following equation: 
       {circumflex over (δ)} k+1 =ƒ({circumflex over (δ)} k   ,m   k )
 
     such that {circumflex over (δ)} k+1  is a function of a state of degradation at time step k and a maintenance and replacement strategy at time step k. As shown in  FIG. 14A , the maintenance strategy m k  may be generated by cost calculator  1324 . In some embodiments, both {circumflex over (δ)} k+1  and m k  are generated by model predictive optimizer  1320  by performing an optimization of an objective function in step  1406 . In other words, m k  can be generated in step  1406  (which may occur concurrently with step  1404 ) and provided as an input to step  1404  for more accurately determining {circumflex over (δ)} k+1  in step  1404 . Of course, maintenance/replacement activities that occur at time step k can affect (e.g., improve) a degradation state of connected equipment  1332  and therefore can affect the predicted degradation state {circumflex over (δ)} k+1  at time step k+1. 
     Process  1400  is shown to include performing an optimization of an objective function using the predicted degradation {circumflex over (δ)} k+1  to generate the maintenance strategy m k  (step  1406 ). In some embodiments, step  1406  is performed by cost calculator  1324 , as described with reference to  FIG. 13 . Based on {circumflex over (δ)} k+1  as predicted in step  1404 , a recommended maintenance and replacement strategy m k  can be determined in step  1406  that optimizes (e.g., minimizes) costs related to performing maintenance/replacement and operating the building equipment. As an example, if {circumflex over (δ)} k+1  is predicted to be extremely high in step  1404 , operational costs at time step k+1 may be determined to be high, thereby necessitating maintenance or replacement to be performed. If so, m k  can be generated such that maintenance or replacement is performed on connected equipment  1332  to improve {circumflex over (δ)} k+1 . The value of m k  can be provided back to step  1404  to predict a new value of {circumflex over (δ)} k+1  as to ensure the degradation state of connected equipment  1332  is improved. As discussed above, both steps  1404  and  1406  may be performed concurrently in some embodiments such that both the predicted degradation state {circumflex over (δ)} k+1  and the maintenance strategy m k  are results of optimizing the objective function. 
     Degradation Impact Modeling 
     Referring now to  FIG. 15 , a block diagram illustrating degradation impact modeler  1314  in greater detail is shown, according to an exemplary embodiment. As discussed above, degradation impact modeler  1314  may be configured to generate power model coefficients φ of connected equipment  1332  as a function of the estimated degradation state {circumflex over (δ)} k . The power model coefficients φ may be coefficients of a power consumption model that is used by model predictive optimizer  1320  to determine that power consumption of connected equipment  1332  as a function of the operating decisions for connected equipment  1332 . For example, the power consumption model may provide a mapping between the amount of power consumed by connected equipment  1332  and the heating or cooling load on connected equipment  1332  (e.g., if connected equipment  1332  is a heater or chiller). 
     Although degradation impact modeler  1314  is described primarily with reference to electric power consumption, it should be understood that any other resource consumed by connected equipment  1332  can be used instead of electric power or in addition to electric power without departing from the teachings of the present disclosure. For example, the power consumption model may be a function or curve that defines the relationship between the amount of an input resource (or multiple input resources) consumed by connected equipment  1332  and the corresponding amount of an output resource (or multiple output resources) produced by connected equipment  1332 , even if none of the input resources or output resources are electric power. In some embodiments, the input resource is electric power and the output resources are heating or cooling load. However, the input resource and output resource can be replaced with any other resources in various embodiments. For example, a gas-fueled boiler may consume natural gas as the input resource instead of electric power. 
     In some embodiments, degradation impact modeler  1314  uses a neural network  1512  to generate the power model coefficients φ NN  as a function of the estimated degradation state {circumflex over (δ)} k . Degradation impact modeler  1314  may train the neural network  1512  using a set of training data that includes input values of the estimated degradation state {circumflex over (δ)} k  and corresponding values of the power model coefficients φ reg . The values of the estimated degradation state {circumflex over (δ)} k  in the training data may be generated by degradation estimator  1316  as described above. The values of the power model coefficients φ reg  in the training data may be generated by performing a regression process, described in greater detail below. As used herein, the variable φ NN  denotes the power model coefficients generated by neural network  1512 , whereas the variable φ reg  denotes the power model coefficients generated by performing the regression process. Although degradation impact modeler  1314  is described primarily as using neural network  1512  to generate the power model coefficients and/or predict the resource consumption as a function of the estimated degradation state, it should be understood that any other type of model (i.e., other than neural network models) can be used in addition to or in place of neural network  1512 . Examples of such models may include regression models, polynomial models, physics-based models, linear or nonlinear models, static or dynamic models, discrete or continuous models, deterministic or stochastic models, or any other type of model that relates the estimated degradation state to the power model coefficients and/or the predicted resource consumption. 
     Degradation impact modeler  1314  is shown to include a data preprocessor  1502 . Data preprocessor  1502  can be configured to associate values of the performance variables y k  with corresponding values of the estimated degradation state {circumflex over (δ)} k . The performance variables y k  may include any of a variety of variables that characterize the performance of connected equipment  1332  including for example, power consumption, natural gas consumption, water consumption, heating load produced, cooling load produced, temperature lift, or any other variable that indicates the resource consumption or production of connected equipment  1332  or characterizes the performance of connected equipment  1332 . In some embodiments, data preprocessor  1502  generates a plurality of different sets of preprocessed data. Each set of preprocessed data may include a value of the estimated degradation state {circumflex over (δ)} k  and corresponding values of the performance variables y k . 
     In some embodiments, data preprocessor  1502  prepares the raw input data to be used by regression power model generator  1504 . For example, data preprocessor  1502  may modify the input data such that it fits an expected form for use in the power regression model. Prior to being processed, a raw dataset can include one or more files (e.g., an Excel file) which are a combination of both cooling and heating mode data. Each file in the raw dataset can be related to a specific degradation case that has been generated by simulation for an amount of time (e.g., one hour, two hours, etc.). Each file can include several feature columns. However, only specific features of the raw dataset may be needed by regression power model generator  1504 . 
     Data preprocessor  1502  can be configured to extract information from the raw data including a degradation state, a power value, a load value, {circumflex over (T)} lift , {circumflex over (P)} lift , etc. Data preprocessor  1502  can also organize the extracted information based on the degradation state. In particular, information related to the same degradation state can be concatenated together. In some embodiments, the processed data is divided into processed data files. Each processed data file can include both heating and cooling mode information. As a result of performing the preprocessing, data preprocessor  1502  can generate one or more data files such that each data file relates to a different degradation case and is ready to feed to regression power model generator  1504 . 
     Regression power model generator  1504  can be configured to perform a regression process to generate a set of power model coefficients φ reg  and related uncertainties based on the preprocessed data. The power model coefficients φ reg  parameters may be used to train neural network  1512 . To obtain the power model coefficients φ reg  and related uncertainties, regression power model generator  1504  can perform a regression process, using the preprocessed data as training data, to generate a power consumption regression model. For example, the preprocessed data may include values of power consumption P, heating load {dot over (Q)} h , cooling load {dot over (Q)} c , temperature lift T lift , or any other variable included in the power consumption regression model. Regression power model generator  1504  can use any of a variety of regression techniques (e.g., ordinary least squares, linear, nonlinear, weighted least squares, ridge regression, etc.) to generate the power model coefficients φ reg . The following equation is one example of the power consumption model for which the power model coefficients φ reg  can be generated: 
         P=φ   1 *max( {dot over (Q)}   c   ,{dot over (Q)}   h )+φ 2 *max( {dot over (Q)}   c   ,{dot over (Q)}   h )* T   lift  
 
     where P is a power value, φ 1  and φ 2  are the power model coefficients, {dot over (Q)} c  is an estimated cooling load, {dot over (Q)} h  is an estimated heating load, and T lift  is the difference between the outside ambient temperature and the predefined setpoint value. 
     In some embodiments, it may be desirable to have uncorrelated predictors in the power consumption model. In other words, it may be desirable that the terms of the power consumption model are not correlated with each other. Regression power model generator  1504  can be configured to reduce or eliminate correlation between the two predictors max({dot over (Q)} c , {dot over (Q)} h ) and max({dot over (Q)} c , {dot over (Q)} h )*T lift . Eliminating the correlation can be achieved using orthogonalization by performing two consecutive regression steps. 
     In some embodiments, regression power model generator  1504  performs the first regression step using the following model: 
       max( {dot over (Q)}   c   ,{dot over (Q)}   h )* T   lift =φ 1 *max( {dot over (Q)}   c   ,{dot over (Q)}   h )+Residual of(max( {dot over (Q)}   c   ,{dot over (Q)}   h )* T   lift )
 
     In the first regression step, a regression model can be constructed for the second predictor (i.e., max({dot over (Q)} c , {dot over (Q)} h )*T lift ) based on the first predictor (i.e., max({dot over (Q)} c , {dot over (Q)} h )). The residual obtained in the first regression step (i.e., Residual of (max({dot over (Q)} c , {dot over (Q)} h )*T lift )) indicates the amount of the second predictor that is orthogonal or uncorrelated with the first predictor. Regression power model generator  1504  can provide the values of heating load {dot over (Q)} h , cooling load {dot over (Q)} c , and temperature lift T lift  as inputs to the regression process to determine the values of φ 1  and the residual Residual of (max({dot over (Q)} c , {dot over (Q)} h )*T lift ). 
     In some embodiments, regression power model generator  1504  performs the second regression step using the following model: 
         P=φ   1 ′*max( {dot over (Q)}   c   ,{dot over (Q)}   h )+φ 2 ′*Residual of(max( {dot over (Q)}   c   ,{dot over (Q)}   h )* T   lift )
 
     where P is the desired variable of power. Regression power model generator  1504  can provide the values of power consumption P, heating load {dot over (Q)} h , cooling load {dot over (Q)} c , and the residual Residual of (max({dot over (Q)} c , {dot over (Q)} h )*T lift ) as inputs to the second regression step to determine the values of φ 1 ′ and φ 2 ′ and their related uncertainties. Accordingly, the final outputs of the regression process are the power model coefficients φ 1 ′ and φ 2 ′ and their related uncertainties (for parameters total). φ 1 ′ and φ 2 ′ are also referred to as φ 1,reg  and φ 2,reg  respectively, or φ reg  collectively, throughout the present disclosure. In some embodiments, regression power model generator  1504  removes outputs which have p-values greater than a threshold value (e.g., 0.1). 
     Regression power model generator  1504  can be configured to repeat the regression process for each set of the preprocessed data to generate a plurality of different sets of power model coefficients φ reg . Each set of the power model coefficients φ reg  may be associated with a corresponding set of estimated degradation states {circumflex over (δ)} k . Regression power model generator  1504  can update the sets of preprocessed data provided by data preprocessor  1502  to include the values of the power model coefficients φ reg  that were generated from the corresponding values of the performance variables y k  and may associate each set of the power model coefficients φ reg  with the degradation states {circumflex over (δ)} k  previously associated with the corresponding values of the performance variables Y k . From a physical standpoint, the set of power model coefficients φ reg  represents the relationship between resource consumption (e.g., power consumption) and resource production (e.g., heating or cooling load) predicted to result from the corresponding degradation states {circumflex over (δ)} k . 
     Degradation impact modeler  1314  is shown to include an input scaler  1506  and an input weighter  1508 . In some embodiments, prior to using the sets of power model coefficients φ reg  and corresponding degradation states {circumflex over (δ)} k  as inputs to train neural network  1512 , input scaler  1506  may scale these inputs to limit their values between a lower threshold and an upper threshold. For example, input scaler  1506  may add or subtract a scaling value from the inputs and/or multiply the inputs by a scaling factor to ensure that each input has a value between the lower and upper thresholds. In some embodiments, input scaler  1506  standardizes (e.g., modifies, adjusts, etc.) the input data such that adjusted values have zero mean and unity variance. 
     Input weighter  1508  can be configured to assign a weight to each set of power model coefficients φ reg  and corresponding degradation states {circumflex over (δ)} k . It may be beneficial in training neural network  1512  if inputs that correspond to more efficient operation of connected equipment  1332  (e.g., higher coefficient of performance (COP) values) have a larger effect on training neural network  1512  as compared to inputs that correspond to less efficient operation of connected equipment  1332  (e.g., lower COP values). Input weighter  1508  can apply a weighting function to the inputs to assign larger weights to inputs with higher COP values and smaller weights to inputs with higher COP values. 
     To generate the weight function, input weighter  1508  can divide the model used in the second regression step described above by the variable max({dot over (Q)} c , {dot over (Q)} h ). This results in the left side of the equation being the inverse of the coefficient of performance (i.e., 1/COP) and the right side of the equation being proportional to φ 1 ′. Accordingly, this relationship is defined as: 
     
       
         
           
             
               ϕ 
               1 
               ′ 
             
             ∝ 
             
               1 
               COP 
             
           
         
       
     
     Due to the inverse relationship between φ 1 ′ and COP, input weighter  1508  can generate a weighting function that assigns weights that are inversely proportional to the value of φ 1 ′. An example of such a weighting function is: 
       weight=10 1−φ     1     ′   
     which is shown graphically in  FIG. 16  as graph  1600 . In graph  1600 , curve  1602  represents the relationship between the weight and the power model coefficient φ 1 ′. 
     Referring again to  FIG. 15 , neural network trainer  1510  is shown receiving scaled inputs input scaler  1506  (e.g., the power model coefficients φ reg ) and the corresponding input weights from input weighter  1508 . Neural network trainer  1510  may also receive the estimated degradation states {circumflex over (δ)} k  from degradation estimator  1316 . Neural network trainer  1510  may use these inputs to train neural network  1512 . In some embodiments, neural network  1512  is a radial basis function neural network (RBFNN). However, other various types of neural networks can be used. Neural network trainer  1510  can train neural network  1512  to map between the degradation states {circumflex over (δ)} k  and the power model parameters φ reg . Accordingly, once neural network  1512  has been trained, the output of neural network  1512  (i.e., φ NN ) may be the same as or similar to the values of φ reg  used to train neural network  1512 . 
     Neural network  1512  can be configured to map degradation states {circumflex over (δ)} k  of connected equipment  1332  to the power model coefficients φ NN , which can be used to calculate predicted operational costs for connected equipment  1332 . The degradation states {circumflex over (δ)} k  can specify which of the degradation indices is contributing to coefficients of power (COP) reduction. For this reason, it can be desirable to have a standard COP calculation from the degradation states {circumflex over (δ)} k  that is consistent with a standard COP calculation from measuring the site data. For example, a standard COP calculation can be given by the following equation: 
           COP   (φ reg   ,x   standard   ,w )=   COP   (φ NN   ,x   standard   ,w )
 
     where φ reg  are the values of the power model coefficients generated by regression power model generator  1504 , x standard  is a standard matrix of power estimation predictors, w is a weight calculated by a weight function, and φ NN  are the power model coefficients generated by neural network  1512 . The previous equation shows that the two COP calculations are equivalent, regardless of whether the power model coefficients φ reg  or φ NN  are used. 
     Advantageously, neural network  1512  benefits the MPM optimization process performed by MPM system  1300 . In some embodiments, neural network  1512  accepts degradation states {circumflex over (δ)} k  as inputs (e.g., refrigerant leakage, compressor power, and airflow restriction) and outputs values of the power model coefficients φ NN  parameters as well as their related uncertainties. The power model coefficients φ NN  generated by neural network  1512  may be used in place of the power model coefficients φ reg  generated by regression power model generator  1504  when calculating the power consumption and resulting operating cost of connected equipment  1332 . 
     In some embodiments, the data used to train neural network  1512  is generated using a simulation framework. The simulation framework can be used to generate a variety of degradation cases that can be used to train neural network  1512 . In some embodiments, a simulation platform such as Simulink is used to generate the operational simulation data of the system. Further, neural network  1512  can be retrained as new data is obtain obtained. Retraining neural network  1512  can ensure neural network  1512  properly maps degradation states {circumflex over (δ)} k  to power model coefficients φ NN  even as the system changes. 
     In some embodiments, degradation impact modeler  1314  trains neural network  1512  to map the degradation state {circumflex over (δ)} k  to power consumption or other resource consumption of connected equipment  1332 . For example, neural network trainer  1510  can receive a set of training data including the estimated degradation states {circumflex over (δ)} k  at each time step k from degradation estimator  1316  along with data indicating the amounts of input resources consumed and output resources produced at each time step k. The amounts of resources consumed and produced at each time step k may be indicated by the performance variables y k . 
     Neural network trainer  1510  can use these training data to train neural network  1512  to predict the amount of one or more input resources consumed by connected equipment  1332  as a function of both the degradation states {circumflex over (δ)} k  and the requested amount(s) of one or more output resources to be produced by connected equipment  1332 . For example, for a VRF system, neural network  1512  can be trained to predict the amount of power consumed at time step k as a function of the degradation states {circumflex over (δ)} k  of the VRF equipment at time step k as well as the requested heating load or cooling load to be served by the VRF equipment at time step k. In this way, neural network  1512  can be trained to predict resource consumption as a function of both the degradation states {circumflex over (δ)} k  and the requested load on connected equipment  1332  without explicitly generating power model coefficients φ NN  in some embodiments. 
     Neural Network Examples 
     Referring now to  FIGS. 17-19 , several examples of neural network architectures which can be used for neural network  1512  are shown, according to some embodiments. Referring particularly to  FIG. 17 , an illustration  1700  of a neural network model is shown, according to some embodiments. As a simple definition for neural networks, neural networks can be considered black boxes that map input samples to desired output values by tuning some parameters that can be referred to as weights. Neural networks can include input layer, hidden layer(s), and output layer and each layer has some nodes or neurons. As such, illustration  1700  is shown to include input nodes  1702  in an input layer, hidden neurons  1704  in a hidden layer, and output neurons  1706  in an output layer. It should be appreciated that while the hidden layer is shown to include one hidden layer, the neural network can include multiple hidden layers if necessary. In a fully connected network, every neuron in each layer is connected to all of the nodes at the next layer as shown in illustration  1700 . 
     A number of neurons in each layer can be determined based on an available data set, a dimension of the data, and what problem is being solved by the neural network. Neural networks can be used for various tasks such as, for example, classification, regression or function approximation, clustering, and so on. Neural networks can be considered universal approximators. In other words, neural networks can approximate any function from simple to complex using some input-output data pairs provided to the neural networks. Different types of neural networks exist and can be used based on the type of the problem. Some examples of neural networks include multilayer perceptron (MLP), radial basis function (RBF) networks, recurrent neural network (RNN), and autoencoder neural networks. In particular, RBF networks can be beneficial for solving regression problems in multidimensional space. 
     Referring now to  FIG. 18 , an example illustration  1800  of an MLP neural network is shown, according to some embodiments. Example illustration  1800  is shown to include an input layer  1802  that includes multiple input nodes (e.g., input nodes  1702  as described with reference to  FIG. 17 ), a first hidden layer  1804 , a second hidden layer  1806 , and an output layer  1808 . Hidden layers  1804  and  1806  are shown to include hidden neurons (e.g., hidden neurons  1704 ). Likewise, output layer  1808  is shown to include output neurons (e.g., output neurons  1706 ). It should be appreciated that the number of input nodes, hidden neurons, and output neurons can vary depending on the problem being solved. In general, an MLP neural network includes one input layer, one output layer, and usually more than one hidden layer. In some embodiments, however, the MLP neural network only has one hidden layer dependent on a problem being solved. Each hidden layer neuron can have nonlinear activation functions (e.g., a sigmoid function). 
     Referring now to  FIG. 19 , an example illustration  1900  of a radial basis function neural network (RBFNN) is shown, according to some embodiments. Example illustration  1900  is shown to include an input layer  1902  of size m 0 , a hidden layer  1904  of size K&lt;N, and an output layer  1906  of size one where m 0  is a size of an input vector x, N is a size of hidden layer  1904 , and K is a number of input neurons. In general, an RBFNN is a three-layer neural network that has only one hidden layer (i.e., hidden layer  1904 ). The activation functions in the hidden layer can be radial basis functions or Gaussian shape functions. Output of the neurons in the hidden layers can, for example, correspond to a distance of their respective inputs from a center of the Gaussian function. The output layer can generate a linear combination of the hidden unit activations. An example of a function which can be used as activation function in RBF networks can be given by the following: 
     
       
         
           
             
               h 
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               exp 
                
               
                 ( 
                 
                   - 
                   
                     
                       
                         ( 
                         
                           x 
                           - 
                           c 
                         
                         ) 
                       
                       2 
                     
                     
                       r 
                       2 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     where c is a center of the radial basis function and r is a measure of a width of the radial basis function. 
     As mentioned above, an output of output layer  1906  can be given as a linear combination of the hidden unit activations. In particular, the output can be given by the following: 
     
       
         
           
             
               f 
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 
                   j 
                   = 
                   1 
                 
                 m 
               
                
               
                 
                   w 
                   j 
                 
                  
                 
                   
                     h 
                     j 
                   
                    
                   
                     ( 
                     x 
                     ) 
                   
                 
               
             
           
         
       
     
     where m is a number of hidden neurons, w j  is a weight associated with hidden node j, and h j (x) is a result of the activation function for hidden node j. 
     Neural network  1512  may include any of a variety of neural network model such as, for example, an MLP, an RBFNN, an RNN, an autoencoder neural network, etc. In some embodiments, neural network  1512  receives a degradation state {circumflex over (δ)} k  as input. The degradation state {circumflex over (δ)} k  can include various degradation metrics such as an air flow restriction, refrigerant loss, compressor degradation, etc. of various components of connected equipment  1332 . In this way, values of {circumflex over (δ)} k  can represent input neurons to neural network  1512 . As described above with reference to  FIGS. 17-19 , inputs of the input neurons can be passed through hidden layers of neural network model until an output layer of the neural network model is reached. The output of an output neural network  1512  can include power model coefficients φ k  for a time step k and an uncertainty σ φ     k    (e.g., a standard deviation, a variance, etc.) of the power model coefficients φ k . In some embodiments, σ φ     k    is an optional output of neural network model  1512  and can be omitted in some implementations. 
     As applied to a VRF system, the power consumption model may be a linear equation that relates power consumption, heating or cooling load, and a lift temperature {circumflex over (T)} lift , where {circumflex over (T)} lift  represents a difference between an outdoor ambient temperature and a predefined setpoint. Coefficients of the power consumption model φ reg  as well as their uncertainty values can be obtained by performing a linear regression. The coefficients φ reg  obtained by the regression can be used as a target of neural network  1512  when trained by neural network trainer  1510 . In this way, neural network  1512  can be trained to predict coefficients φ NN  of the power consumption model as well as their related uncertainties. In some embodiments, an RBFNN is a preferred neural network for mapping degradation indices to power model coefficients φ NN . 
     Degradation Impact Modeling Process 
     Referring now to  FIG. 20 , a flowchart of a process  2000  which can be performed by MPM system  1300  is shown, according to an exemplary embodiment. Process  2000  is shown to include a set of offline steps  2002 - 2006  and a set of online steps  2008 - 2016 . It is contemplated that steps  2002 - 2006  can be performed offline (e.g., prior to operating actual building equipment) using historical operating data and/or simulated operating data for a set of building equipment. Steps  2008 - 2016  may be performed online (e.g., during real-time operation of the building equipment or different building equipment) to generate operating decisions and maintenance decisions for the building equipment over a given time period. The steps of process  2000  may be performed by various components of MPM system  1300  or any of the systems or components previously described herein. 
     Process  2000  is shown to include obtaining training data characterizing an operating performance of building equipment over time (step  2002 ). The training data may indicate an amount of resource consumption of the building equipment (e.g., electricity, natural gas, water, etc.) at a plurality of different times and corresponding amounts of resource production (e.g., hot water, cold water, heating load, cooling load, filtered air, etc.) at each of the plurality of different times. The training data may indicate the relationship between resource consumption and resource production at each of the different times. In some embodiments, the training data can be retrieved from a database of historical operating data for the building equipment (e.g., connected equipment  1332 ) or similar building equipment. In other embodiments, the training data can be generated by running a simulation of the building equipment. 
     Process  2000  is shown to include estimating degradation states of the building equipment as a function of performance variables in the training data (step  2004 ). In some embodiments, step  2004  is performed by degradation estimator  1316  as described with reference to  FIGS. 13-15 . Step  2004  may include estimating one or more degradation states {circumflex over (δ)} k  of the building equipment at each of a plurality of time steps k (e.g., k=1 . . . n). Step  2004  may further include matching each of the degradation states {circumflex over (δ)} k  with corresponding values of resource consumption and resource production for the building equipment at the given time step k. 
     Process  2000  is shown to include training a neural network to predict parameters of a resource consumption model for the building equipment as a function of the degradation state (step  2006 ). Step  2006  may be performed by degradation impact modeler  1314  as described with reference to  FIGS. 13-15 . The resource consumption model may be any type of model that relates the amount of any input resource (or multiple input resources) consumed by building equipment to an amount of any output resource (or multiple output resources) produced by the building equipment. For example, a resource consumption model for VRF equipment may relate power consumption (an input resource) to heating load or cooling load (output resources). As another example, a resource consumption model for a chiller may relate water consumption and/or electricity consumption (input resources) to an amount of chilled water produced and/or cooling load (output resources). 
     In some embodiments, step  2006  includes using the estimated degradation states {circumflex over (δ)} k  and corresponding values of resource consumption and resource production to train a regression model. For example, the regression model may be a power consumption model that predicts the power consumption of the building equipment as a function of the heating or cooling load on the equipment and a set of regression model coefficients φ reg,k  for each of the time steps k. Step  2006  may include performing a regression process to generate values of the power model coefficients φ reg,k  using the corresponding values of resource consumption and resource production at each time step k. 
     In some embodiments, step  2006  includes using the generated power model coefficients φ reg,k  and the corresponding degradation states {circumflex over (δ)} k  to train a neural network. In some embodiments, the neural network is neural network  1512  and is trained by neural network trainer  1510  as described with reference to  FIG. 15 . The neural network can be trained to predict the values of the power model coefficients φ NN ,k as a function of the degradation states {circumflex over (δ)} k . Once the neural network has been trained, it can be used to generate values of the power model coefficients φ NN,k  that are the same as or similar to the regression model coefficients φ reg,k  generated by regression power modeler  1504 . The output of step  2006  may include a set of neural network model parameters (e.g., learned weights between nodes of the neural network) for use in the online portion of process  2000 . 
     Moving into the online portion of process  2000 , process  2000  is shown to include estimating a current degradation state of the building equipment (step  2008 ). In some embodiments, step  2008  is performed by degradation estimator  1316  as described with reference to  FIGS. 13-15 . Step  2008  may be similar to step  2004 , with the exception that the degradation state {circumflex over (δ)} k  estimated in step  2008  is the current degradation state of the building equipment for which operating decisions and maintenance decisions are desired. Process  2000  is shown to include predicting future degradation states of the building equipment (step  2010 ). Step  2010  may be performed by degradation predictor  1322  as described with reference to  FIGS. 13-15  and may include predicting a degradation state {circumflex over (δ)} k+1  for one or more time steps subsequent to the current time step k. In some embodiments, step  2010  includes predicting the future degradation states {circumflex over (δ)} k+1  as a function of the current degradation state {circumflex over (δ)} k  and a set of maintenance decisions m k  for the building equipment. The future degradation states {circumflex over (δ)} k+1  may be predicted for each time step within a given time horizon. 
     Process  2000  is shown to include using the neural network model to predict parameters of the resource consumption model over the length of a time horizon (step  2012 ). Step  2012  may include applying the predicted degradation states {circumflex over (δ)} k+1  as inputs to the neural network model and obtaining the parameters of the resource consumption model φ NN,k  as outputs of the neural network model. Step  2012  may include generating a set of resource consumption model parameters φ NN,k  for each time step within the given time horizon, which may be the same time horizon for which predicted degradation states {circumflex over (δ)} k+1  are generated in step  2010 . 
     Process  2000  is shown to include using the resource consumption model to generate a maintenance schedule for the building equipment that results in a lowest combined operating and maintenance cost (step  2014 ). In some embodiments, step  2014  is performed by model predictive optimizer  1320  as described with reference to  FIGS. 13-15 . Step  2014  may include optimizing an objective function J that accounts for the operating cost of the building equipment and the maintenance cost of the building equipment over the time horizon. Both the operating cost and the maintenance cost may be a function of a set of maintenance decisions defined by the maintenance schedule m k . The maintenance cost may be a direct function of the maintenance decisions because each maintenance activity may incur a corresponding cost when the maintenance activity is performed. The operating cost may be an indirect function of the maintenance decisions because the maintenance activities at a given time step k reduce the predicted degradation states {circumflex over (δ)} k+1  at subsequent time steps, which results in improved operating efficiency and reduced operating cost. 
     Process  2000  is shown to include operating the building equipment in accordance with the maintenance schedule and operating decisions (step  2016 ). In some embodiments, the optimization performed in step  2014  may produce both a set of maintenance decisions and a set of operating decisions for the building equipment. Step  2016  may include executing the decisions generated in step  2014 . Maintenance decisions can be executed by placing service requests with service providers  1330  and performing maintenance, replacement, upgrades, or other activities that result in changes to the set of building equipment. Operating decisions can be executed by providing adjusted setpoints to the building equipment, providing control signals to the building equipment, or otherwise operating the building equipment in accordance with the operating decisions determined in step  2014 . 
     In some embodiments, process  2000  includes initiating a maintenance activity for the building equipment in accordance with the maintenance schedule. The maintenance activity may include performing maintenance on the building equipment, repairing the building equipment, replacing one or more devices of the building equipment, upgrading the building equipment, placing a service request with service providers for the building equipment, scheduling a service appointment, generating a maintenance recommendation for a user to review and approve, or otherwise taking action based on the maintenance schedule generated in step  2014 . 
     Configuration of Exemplary Embodiments 
     The construction and arrangement of the systems and methods as shown in the various exemplary embodiments are illustrative only. Although only a few embodiments have been described in detail in this disclosure, many modifications are possible (e.g., variations in sizes, dimensions, structures, shapes and proportions of the various elements, values of parameters, mounting arrangements, use of materials, colors, orientations, etc.). For example, the position of elements may be reversed or otherwise varied and the nature or number of discrete elements or positions may be altered or varied. Accordingly, all such modifications are intended to be included within the scope of the present disclosure. The order or sequence of any process or method steps may be varied or re-sequenced according to alternative embodiments. Other substitutions, modifications, changes, and omissions may be made in the design, operating conditions and arrangement of the exemplary embodiments without departing from the scope of the present disclosure. 
     The present disclosure contemplates methods, systems and program products on any machine-readable media for accomplishing various operations. The embodiments of the present disclosure may be implemented using existing computer processors, or by a special purpose computer processor for an appropriate system, incorporated for this or another purpose, or by a hardwired system. Embodiments within the scope of the present disclosure include program products comprising machine-readable media for carrying or having machine-executable instructions or data structures stored thereon. Such machine-readable media can be any available media that can be accessed by a general purpose or special purpose computer or other machine with a processor. By way of example, such machine-readable media can comprise RAM, ROM, EPROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which can be used to carry or store desired program code in the form of machine-executable instructions or data structures and which can be accessed by a general purpose or special purpose computer or other machine with a processor. When information is transferred or provided over a network or another communications connection (either hardwired, wireless, or a combination of hardwired or wireless) to a machine, the machine properly views the connection as a machine-readable medium. Thus, any such connection is properly termed a machine-readable medium. Combinations of the above are also included within the scope of machine-readable media. Machine-executable instructions include, for example, instructions and data which cause a general purpose computer, special purpose computer, or special purpose processing machines to perform a certain function or group of functions. 
     Although the figures show a specific order of method steps, the order of the steps may differ from what is depicted. Also two or more steps may be performed concurrently or with partial concurrence. Such variation will depend on the software and hardware systems chosen and on designer choice. All such variations are within the scope of the disclosure. Likewise, software implementations could be accomplished with standard programming techniques with rule based logic and other logic to accomplish the various connection steps, processing steps, comparison steps and decision steps.