Patent Publication Number: US-11036249-B2

Title: Building energy storage system with peak load contribution cost optimization

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation of U.S. patent application Ser. No. 15/405,234, filed Jan. 12, 2017, the entire disclosure of which is incorporated by reference herein. 
    
    
     BACKGROUND 
     The present disclosure relates generally to an energy storage system or energy cost optimization system for a building. The present disclosure relates more particularly to an energy system or energy cost optimization system which optimally allocates energy assets (e.g., batteries, thermal energy storage, HVAC equipment, etc.) while accounting for peak load contribution charges. 
     Peak load contribution (PLC) is a customer&#39;s contribution to regional demand peaks that occur in geographic area managed by a regional transmission organization (RTO) or independent system operator (ISO) at certain hours within a base period. The regional demand at a given hour may be the summation of the customer&#39;s demand during (i.e., the rate at which the customer purchases electricity or another resource from a utility) as well as the demand of other buildings in the geographic area during that hour. The customer may be billed based on its contribution to the peak regional demand (e.g., $/kW of the customer&#39;s PLC) in addition to energy consumption charges and demand charges. 
     Accounting for PLC costs can be more difficult than accounting for energy consumption costs and demand charges. Unlike demand charge which is calculated based on the customer&#39;s maximum demand during predetermined demand charge periods, the hours over which PLC is calculated may not be known in advance. The hours of peak regional demand (i.e., the coincidental peak (CP) hours) may not be known until the end of the base period over which PLC is calculated. For example, the CP hours for a given base period (e.g., one year) may be determined by a RTO at the end of the base period based on the demand of all the buildings within the geographic area managed by the RTO during the base period (e.g., by selecting the hours with the highest regional demand). The customer&#39;s PLC may then be determined based on the customer&#39;s demand during the designated CP hours and used to calculate a cost of the customer&#39;s PLC. This cost may then be billed to the customer during the next time period (e.g., the next year), referred to as the billing period. It can be difficult and challenging to optimize the allocation of energy assets for a building while accounting for peak load contribution charges. 
     SUMMARY 
     One implementation of the present disclosure is an energy storage system for a building. The system includes a battery and an energy storage controller. The battery is configured to store electrical energy purchased from a utility and to discharge the stored electrical energy for use in satisfying a building energy load. The energy storage controller is configured to generate a cost function including a peak load contribution (PLC) term. The PLC term represents a cost based on an amount of the electrical energy purchased from the utility during coincidental peak (CP) hours in an optimization period. The controller is configured to modify the cost function by applying a peak hours mask to the PLC term. The peak hours mask identifies one or more hours in the optimization period as projected peak hours and causes the energy storage controller to disregard the electrical energy purchased from the utility during any hours not identified as projected peak hours when calculating a value for the PLC term. The controller is configured to allocate, to each of a plurality of time steps within the optimization period, an optimal amount of electrical energy to store in the battery or discharge from the battery during the time step by optimizing the modified cost function. 
     In some embodiments, the CP hours include one or more hours of the optimization period during which regional demand peaks occur for a geographic region in which the building is located. In some embodiments, the CP hours are unknown prior to the optimization period. The projected peak hours may include an estimate of the CP hours and may be identified prior to the optimization period. 
     In some embodiments, the peak hours mask includes a vector of binary values. Each of the binary values may correspond to an hour of the optimization period and may indicate whether the corresponding hour is a projected peak hour. 
     In some embodiments, the energy storage controller is configured to create the peak hours mask based on user input defining the projected peak hours. In other embodiments, the energy storage controller is configured to predict regional demand peaks for a geographic region in which the building is located, define the projected peak hours based on the predicted regional demand peaks, and create the peak hours mask using the projected peak hours. In some embodiments, the energy storage controller is configured to predict the regional demand peaks based on at least one of load history data and weather data for the geographic region in which the building is located. 
     In some embodiments, the PLC term includes a rate factor defining a cost per unit of the electrical energy purchased from the utility during the CP hours. The energy storage controller may be configured to identify a geographic region in which the building is located and calculate the rate factor based on one or more parameters that depend on the identified geographic region. 
     In some embodiments, the energy storage controller is configured to identify the geographic region in which the building is located as a Pennsylvania, Jersey, Maryland (PJM) region. The energy storage controller can calculate the rate factor based on at least one of a capacity loss factor for the PJM region and a PLC rate for the PJM region. 
     In some embodiments, the energy storage controller is configured to identify the geographic region in which the building is located as an Ontario region. The energy storage controller can calculate the rate factor based on at least one of a region-wide global adjustment cost for the Ontario region and a region-wide peak load estimate for the Ontario region. 
     Another implementation of the present disclosure is method for allocating a battery asset in an energy storage system. The method includes generating a cost function including a peak load contribution (PLC) term. The PLC term represents a cost based on an amount of electrical energy purchased from a utility during coincidental peak (CP) hours in an optimization period. The method includes modifying the cost function by applying a peak hours mask to the PLC term. The peak hours mask identifies one or more hours in the optimization period as projected peak hours. The method includes calculating a value for the PLC term in the modified cost function. The peak hours mask causes the electrical energy purchased from the utility during any hours not identified as projected peak hours to be disregarded when calculating a value for the PLC term. The method includes allocating, to each of a plurality of time steps within the optimization period, an optimal amount of electrical energy to store in a battery or discharge from the battery during the time step by optimizing the modified cost function. The method includes operating the battery to store electrical energy purchased from the utility and discharge the stored electrical energy based on the optimal amounts of electrical energy allocated to each time step. 
     In some embodiments, the CP hours include one or more hours of the optimization period during which regional demand peaks occur for a geographic region in which the building is located. In some embodiments, the CP hours are unknown prior to the optimization period and the projected peak hours comprise an estimate of the CP hours and are identified prior to the optimization period. 
     In some embodiments, the peak hours mask includes a vector of binary values. Each of the binary values may correspond to an hour of the optimization period and may indicate whether the corresponding hour is a projected peak hour. 
     In some embodiments, the method includes predicting regional demand peaks for a geographic region in which the building is located, defining the projected peak hours based on the predicted regional demand peaks, and creating the peak hours mask using the projected peak hours. In some embodiments, the regional demand peaks are predicted based on at least one of load history data and weather data for the geographic region in which the building is located. 
     In some embodiments, the PLC term includes a rate factor defining a cost per unit of the electrical energy purchased from the utility during the CP hours. The method may further include identifying a geographic region in which the building is located and calculating the rate factor based on one or more parameters that depend on the identified geographic region. 
     Another implementation of the present disclosure is an energy cost optimization system for a building. The system includes HVAC equipment configured to consume energy purchased from a utility for use in satisfying a building energy load and a controller. The controller is configured to generate a cost function including a peak load contribution (PLC) term. The PLC term represents a cost based on an amount of the energy purchased from the utility during coincidental peak (CP) hours in an optimization period. The controller is configured to modify the cost function by applying a peak hours mask to the PLC term. The peak hours mask identifies one or more hours in the optimization period as projected peak hours and causes the controller to disregard the energy purchased from the utility during any hours not identified as projected peak hours when calculating a value for the PLC term. The controller is configured to allocate, to each of a plurality of time steps within the optimization period, an optimal amount of energy to be consumed by the HVAC equipment during the time step by optimizing the modified cost function. 
     In some embodiments, the peak hours mask includes a vector of binary values. Each of the binary values may correspond to an hour of the optimization period and may indicate whether the corresponding hour is a projected peak hour. 
     In some embodiments, the PLC term includes a rate factor defining a cost per unit of the energy purchased from the utility during the CP hours. The controller may be configured to identify a geographic region in which the building is located and calculate the rate factor based on one or more parameters that depend on the identified geographic region. 
     Another implementation of the present disclosure is an energy storage system for a building, a group of buildings, or a central plant. The energy storage system includes one or more energy storage devices and a controller. The energy storage devices are configured to store one or more energy resources (e.g., electricity, hot or cold water, natural gas, thermal energy, etc.) and to discharge the stored energy resources for use in satisfying an energy load of the building, group of buildings, or the central plant. The energy resources may include one or more energy resource purchased from a utility and/or energy resources generated by the energy storage system or by the central plant. The energy storage controller is configured to generate a cost function including one or more peak load contribution (PLC) terms. Each PLC term represents a cost based on an amount of an energy resource purchased from the utility during coincidental peak (CP) hours in an optimization period. The controller is configured to modify the cost function by applying a peak hours mask to each PLC term. The peak hours mask identifies one or more hours in the optimization period as projected peak hours for a particular energy resource and causes the energy storage controller to disregard any amount of that energy resource purchased from the utility during any hours not identified as projected peak hours when calculating a value for the PLC term. The energy storage controller is configured to allocate, to each of a plurality of time steps within the optimization period, an optimal amount each energy resource to store in the energy storage devices or discharge from the energy storage devices during the time step by optimizing the modified cost function. 
     Another implementation of the present disclosure is a method for allocating one or more energy storage devices in an energy storage system for a building, group of buildings, or a central plant. The method includes generating a cost function including one or more peak load contribution (PLC) terms. Each PLC term represents a cost based on an amount of an energy resource (e.g., electricity, hot or cold water, natural gas, thermal energy, etc.) purchased from a utility during coincidental peak (CP) hours in an optimization period. The method includes modifying the cost function by applying a peak hours mask to each PLC term. Each peak hours mask identifies one or more hours in the optimization period as projected peak hours for one of the energy resources purchased from the utility. The method includes calculating a value for each PLC term in the modified cost function. The peak hours mask applied to a PLC term causes any amount of the corresponding energy resource purchased from the utility during any hours not identified as projected peak hours to be disregarded when calculating a value for the PLC term. The method includes allocating, to each of a plurality of time steps within the optimization period, an optimal amount of each energy resource purchased from the utility to store in an energy storage device or discharge from the energy storage device during the time step by optimizing the modified cost function. The method includes operating the energy storage devices to store the energy resources purchased from the utility and discharge the stored energy resources based on the optimal amounts of the energy resources allocated to each time step. 
     In some embodiments, the method includes allocating one or more other energy resources (e.g., energy resources generated by the central plant) in addition to the energy resources purchased from the utility. The method may include operating the other energy storage devices to store the other energy resources and discharge the other stored energy resources based on the optimal amounts of the other energy resources allocated to each time step. 
     Another implementation of the present disclosure is an energy cost optimization system for a building, a group of buildings, or a central plant. The system includes energy-consuming equipment and a controller. The equipment are configured to consume one or more energy resources (e.g., electricity, hot or cold water, natural gas, thermal energy, etc.) for use in satisfying an energy load of the building, group of buildings, or the central plant. The energy resources may include one or more energy resources purchased from a utility and/or energy resources generated by the system or by the central plant. The controller is configured to generate a cost function including one or more peak load contribution (PLC) terms. Each PLC term represents a cost based on an amount of an energy resource purchased from the utility during coincidental peak (CP) hours in an optimization period. The controller is configured to modify the cost function by applying a peak hours mask to each PLC term. Each peak hours mask identifies one or more hours in the optimization period as projected peak hours for one of the energy resources purchased from the utility and causes the controller to disregard any amount of the energy resource purchased from the utility during any hours not identified as projected peak hours when calculating a value for the PLC term. The controller is configured to allocate, to each of a plurality of time steps within the optimization period, an optimal amount of each energy resource to be consumed by the equipment during the time step by optimizing the modified cost function. 
     Those skilled in the art will appreciate that the summary is illustrative only and is not intended to be in any way limiting. Other aspects, inventive features, and advantages of the devices and/or processes described herein, as defined solely by the claims, will become apparent in the detailed description set forth herein and taken in conjunction with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a frequency response optimization system, according to an exemplary embodiment. 
         FIG. 2  is a graph of a regulation signal which may be provided to the system of  FIG. 1  and a frequency response signal which may be generated by the system of  FIG. 1 , according to an exemplary embodiment. 
         FIG. 3  is a block diagram of a photovoltaic energy system configured to simultaneously perform both ramp rate control and frequency regulation while maintaining the state-of-charge of a battery within a desired range, according to an exemplary embodiment. 
         FIG. 4  is a drawing illustrating the electric supply to an energy grid and electric demand from the energy grid which must be balanced in order to maintain the grid frequency, according to an exemplary embodiment. 
         FIG. 5A  is a block diagram of an energy storage system including thermal energy storage and electrical energy storage, according to an exemplary embodiment. 
         FIG. 5B  is a block diagram of an energy cost optimization system without thermal or electrical energy storage, according to an exemplary embodiment. 
         FIG. 6A  is block diagram of an energy storage controller which may be used to operate the energy storage system of  FIG. 5A , according to an exemplary embodiment. 
         FIG. 6B  is a block diagram of a controller which may be used to operate the energy cost optimization system of  FIG. 5B , according to an exemplary embodiment. 
         FIG. 7  is a block diagram of a planning tool which can be used to determine the benefits of investing in a battery asset and calculate various financial metrics associated with the investment, according to an exemplary embodiment. 
         FIG. 8  is a drawing illustrating the operation of the planning tool of  FIG. 7 , according to an exemplary embodiment. 
         FIG. 9  is a block diagram of a high level optimizer which can be implemented as a component of the controllers of  FIGS. 6A-6B  or the planning tool of  FIG. 7 , according to an exemplary embodiment. 
         FIG. 10  is a block diagram illustrating a peak load contribution (PLC) module of the high level optimizer of  FIG. 9  in greater detail, according to an exemplary embodiment. 
         FIG. 11  is a drawing of a user interface that allows a user to select one or more programs in which to participate, each program adding a corresponding term to a cost function optimizer by the high level optimizer, according to an exemplary embodiment. 
         FIG. 12  is a drawing of a user interface which allows a user to input values for various parameters used by the PLC module of  FIG. 10 , according to an exemplary embodiment. 
         FIG. 13  is a drawing of a user interface which allows a user to enter a percent change in various parameters used by the PLC module of  FIG. 10  from year-to-year, according to an exemplary embodiment. 
         FIG. 14  is a drawing of a user interface which allows a user to define projected coincidental peak (CP) hours that occur during an optimization period, according to an exemplary embodiment. 
         FIG. 15  is a flowchart of a process for allocating energy storage, according to an exemplary embodiment. 
         FIG. 16  is a flowchart of a process for optimizing energy cost including a PLC charge, according to an exemplary embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Overview 
     Referring generally to the FIGURES, an energy storage system with peak load contribution cost optimization and components thereof are shown, according to various exemplary embodiments. The energy storage system can determine an optimal allocation of energy storage assets (e.g., batteries, thermal energy storage, etc.) over an optimization period. The optimal energy allocation may include an amount of energy purchased from utilities, an amount of energy stored or withdrawn from energy storage, and/or an amount of energy sold to energy purchasers or used to participate in incentive-based demand response (IBDR) programs. In some embodiments, the optimal allocation maximizes the economic value of the energy storage assets while satisfying the predicted loads for the building or campus and generating revenue from IBDR programs. 
     In some embodiments, the energy storage system accounts for a cost associated with a customer&#39;s peak load contribution (PLC) when determining the optimal allocation of energy storage assets. PLC is a customer&#39;s contribution to regional demand peaks that occur in geographic area managed by a regional transmission organization (RTO) or independent system operator (ISO) at certain hours within a base period. The regional demand at a given hour may be the summation of the customer&#39;s demand during (i.e., the rate at which the customer purchases electricity or another resource from a utility) as well as the demand of other buildings in the geographic area during that hour. The customer may be billed based on its contribution to the peak regional demand (e.g., $/kW of the customer&#39;s PLC) in addition to energy consumption charges and demand charges. 
     In some embodiments, the energy storage system includes a controller configured to account for a cost associated with the customer&#39;s PLC. By accounting for PLC costs, the controller enables the energy storage system to allocate resource consumption and resource purchases to reduce the customer&#39;s PLC. The energy storage system can reduce PLC costs by shifting the customer&#39;s load to non-peak times or shaving the customer&#39;s peak load. This can be done, for example, by precooling the building during non-peak times, using thermal energy storage, and/or using electrical energy storage such as a battery asset. 
     Accounting for the cost associated with the customer&#39;s PLC can be more difficult than accounting for energy consumption costs and demand charges. Unlike demand charge which is calculated based on the customer&#39;s maximum demand during predetermined demand charge periods, the hours over which PLC is calculated may not be known in advance. The hours of peak regional demand (i.e., the coincidental peak (CP) hours) may not be known until the end of the base period over which PLC is calculated. For example, the CP hours for a given base period (e.g., one year) may be determined by a RTO at the end of the base period based on the demand of all the buildings within the geographic area managed by the RTO during the base period (e.g., by selecting the hours with the highest regional demand). The customer&#39;s PLC may then be determined based on the customer&#39;s demand during the designated CP hours and used to calculate a cost of the customer&#39;s PLC. This cost may then be billed to the customer during the next time period (e.g., the next year), referred to as the billing period. 
     In order to account for PLC costs and allocate resource consumption/purchases in advance, the controller can generate or obtain a projection of the CP hours for an upcoming base period. The projected CP hours can then be used as an estimate of the actual CP hours. In some embodiments, the projected peak hours are used to create a peak hours mask that indicates which of the hours are projected peak hours and which of the hours are not projected peak hours. The controller can use the peak hours mask to modify a cost function that includes a PLC cost term. The controller can optimize the cost function to allocate one or more assets (e.g., a battery, thermal energy storage, HVAC equipment, etc.) and/or to minimize the customer&#39;s demand during the projected CP hours. 
     The following sections of this disclosure describe the PLC cost optimization technique in greater detail as well as several systems which can use the cost optimization technique. For example, the PLC cost optimization can be used in a frequency response optimization system with electrical energy storage (described with reference to  FIGS. 1-2 ), a photovoltaic (PV) energy system that simultaneously performs both frequency regulation and ramp rate control using electrical energy storage (described with reference to  FIGS. 3-4 ), an energy storage system that uses thermal energy storage and/or electrical energy storage to perform load shifting and optimize energy cost (described with reference to  FIGS. 5A and 6A ), an energy cost optimization system without thermal or electrical energy storage (described with reference to  FIGS. 5B and 6B ), and/or a planning tool that determines the benefits of investing in a battery asset and calculates various financial metrics associated with the investment (described with reference to  FIGS. 7-8 ). The PLC cost optimization technique is described in greater detail with reference to  FIGS. 9-10 . 
     Frequency Response Optimization 
     Referring now to  FIG. 1 , a frequency response optimization system  100  is shown, according to an exemplary embodiment. System  100  is shown to include a campus  102  and an energy grid  104 . Campus  102  may include one or more buildings  116  that receive power from energy grid  104 . Buildings  116  may include equipment or devices that consume electricity during operation. For example, buildings  116  may include HVAC equipment, lighting equipment, security equipment, communications equipment, vending machines, computers, electronics, elevators, or other types of building equipment. 
     In some embodiments, buildings  116  are served by a building management system (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, and/or any other system that is capable of managing building functions or devices. An exemplary building management system which may be used to monitor and control buildings  116  is described in U.S. patent application Ser. No. 14/717,593 filed May 20, 2015, the entire disclosure of which is incorporated by reference herein. 
     In some embodiments, campus  102  includes a central plant  118 . Central plant  118  may include one or more subplants that consume resources from utilities (e.g., water, natural gas, electricity, etc.) to satisfy the loads of buildings  116 . For example, central plant  118  may include a heater subplant, a heat recovery chiller subplant, a chiller subplant, a cooling tower subplant, a hot thermal energy storage (TES) subplant, and a cold thermal energy storage (TES) subplant, a steam subplant, and/or any other type of subplant configured to serve buildings  116 . The subplants may be configured to convert input resources (e.g., electricity, water, natural gas, etc.) into output resources (e.g., cold water, hot water, chilled air, heated air, etc.) that are provided to buildings  116 . An exemplary central plant which may be used to satisfy the loads of buildings  116  is described U.S. patent application Ser. No. 14/634,609 filed Feb. 27, 2015, the entire disclosure of which is incorporated by reference herein. 
     In some embodiments, campus  102  includes energy generation  120 . Energy generation  120  may be configured to generate energy that can be used by buildings  116 , used by central plant  118 , and/or provided to energy grid  104 . In some embodiments, energy generation  120  generates electricity. For example, energy generation  120  may include an electric power plant, a photovoltaic energy field, or other types of systems or devices that generate electricity. The electricity generated by energy generation  120  can be used internally by campus  102  (e.g., by buildings  116  and/or central plant  118 ) to decrease the amount of electric power that campus  102  receives from outside sources such as energy grid  104  or battery  108 . If the amount of electricity generated by energy generation  120  exceeds the electric power demand of campus  102 , the excess electric power can be provided to energy grid  104  or stored in battery  108 . The power output of campus  102  is shown in  FIG. 1  as P campus . P campus  may be positive if campus  102  is outputting electric power or negative if campus  102  is receiving electric power. 
     Still referring to  FIG. 1 , system  100  is shown to include a power inverter  106  and a battery  108 . Power inverter  106  may be configured to convert electric power between direct current (DC) and alternating current (AC). For example, battery  108  may be configured to store and output DC power, whereas energy grid  104  and campus  102  may be configured to consume and generate AC power. Power inverter  106  may be used to convert DC power from battery  108  into a sinusoidal AC output synchronized to the grid frequency of energy grid  104 . Power inverter  106  may also be used to convert AC power from campus  102  or energy grid  104  into DC power that can be stored in battery  108 . The power output of battery  108  is shown as P bat . P bat  may be positive if battery  108  is providing power to power inverter  106  or negative if battery  108  is receiving power from power inverter  106 . 
     In some embodiments, power inverter  106  receives a DC power output from battery  108  and converts the DC power output to an AC power output. The AC power output can be used to satisfy the energy load of campus  102  and/or can be provided to energy grid  104 . Power inverter  106  may synchronize the frequency of the AC power output with that of energy grid  104  (e.g., 50 Hz or 60 Hz) using a local oscillator and may limit the voltage of the AC power output to no higher than the grid voltage. In some embodiments, power inverter  106  is a resonant inverter that includes or uses LC circuits to remove the harmonics from a simple square wave in order to achieve a sine wave matching the frequency of energy grid  104 . In various embodiments, power inverter  106  may operate using high-frequency transformers, low-frequency transformers, or without transformers. Low-frequency transformers may convert the DC output from battery  108  directly to the AC output provided to energy grid  104 . High-frequency transformers may employ a multi-step process that involves converting the DC output to high-frequency AC, then back to DC, and then finally to the AC output provided to energy grid  104 . 
     System  100  is shown to include a point of interconnection (POI)  110 . POI  110  is the point at which campus  102 , energy grid  104 , and power inverter  106  are electrically connected. The power supplied to POI  110  from power inverter  106  is shown as P sup . P sup  may be defined as P bat +P loss , where P batt  is the battery power and P loss  is the power loss in the battery system (e.g., losses in power inverter  106  and/or battery  108 ). P bat  and P sup  may be positive if power inverter  106  is providing power to POI  110  or negative if power inverter  106  is receiving power from POI  110 . P campus  and P sup  combine at POI  110  to form P POI . P POI  may be defined as the power provided to energy grid  104  from POI  110 . P POI  may be positive if POI  110  is providing power to energy grid  104  or negative if POI  110  is receiving power from energy grid  104 . 
     Still referring to  FIG. 1 , system  100  is shown to include a frequency response controller  112 . Controller  112  may be configured to generate and provide power setpoints to power inverter  106 . Power inverter  106  may use the power setpoints to control the amount of power P sup  provided to POI  110  or drawn from POI  110 . For example, power inverter  106  may be configured to draw power from POI  110  and store the power in battery  108  in response to receiving a negative power setpoint from controller  112 . Conversely, power inverter  106  may be configured to draw power from battery  108  and provide the power to POI  110  in response to receiving a positive power setpoint from controller  112 . The magnitude of the power setpoint may define the amount of power P sup  provided to or from power inverter  106 . Controller  112  may be configured to generate and provide power setpoints that optimize the value of operating system  100  over a time horizon. 
     In some embodiments, frequency response controller  112  uses power inverter  106  and battery  108  to perform frequency regulation for energy grid  104 . Frequency regulation is the process of maintaining the stability of the grid frequency (e.g., 60 Hz in the United States). The grid frequency may remain stable and balanced as long as the total electric supply and demand of energy grid  104  are balanced. Any deviation from that balance may result in a deviation of the grid frequency from its desirable value. For example, an increase in demand may cause the grid frequency to decrease, whereas an increase in supply may cause the grid frequency to increase. Frequency response controller  112  may be configured to offset a fluctuation in the grid frequency by causing power inverter  106  to supply energy from battery  108  to energy grid  104  (e.g., to offset a decrease in grid frequency) or store energy from energy grid  104  in battery  108  (e.g., to offset an increase in grid frequency). 
     In some embodiments, frequency response controller  112  uses power inverter  106  and battery  108  to perform load shifting for campus  102 . For example, controller  112  may cause power inverter  106  to store energy in battery  108  when energy prices are low and retrieve energy from battery  108  when energy prices are high in order to reduce the cost of electricity required to power campus  102 . Load shifting may also allow system  100  reduce the demand charge incurred. Demand charge is an additional charge imposed by some utility providers based on the maximum power consumption during an applicable demand charge period. For example, a demand charge rate may be specified in terms of dollars per unit of power (e.g., $/kW) and may be multiplied by the peak power usage (e.g., kW) during a demand charge period to calculate the demand charge. Load shifting may allow system  100  to smooth momentary spikes in the electric demand of campus  102  by drawing energy from battery  108  in order to reduce peak power draw from energy grid  104 , thereby decreasing the demand charge incurred. 
     Still referring to  FIG. 1 , system  100  is shown to include an incentive provider  114 . Incentive provider  114  may be a utility (e.g., an electric utility), a regional transmission organization (RTO), an independent system operator (ISO), or any other entity that provides incentives for performing frequency regulation. For example, incentive provider  114  may provide system  100  with monetary incentives for participating in a frequency response program. In order to participate in the frequency response program, system  100  may maintain a reserve capacity of stored energy (e.g., in battery  108 ) that can be provided to energy grid  104 . System  100  may also maintain the capacity to draw energy from energy grid  104  and store the energy in battery  108 . Reserving both of these capacities may be accomplished by managing the state-of-charge of battery  108 . 
     Frequency response controller  112  may provide incentive provider  114  with a price bid and a capability bid. The price bid may include a price per unit power (e.g., $/MW) for reserving or storing power that allows system  100  to participate in a frequency response program offered by incentive provider  114 . The price per unit power bid by frequency response controller  112  is referred to herein as the “capability price.” The price bid may also include a price for actual performance, referred to herein as the “performance price.” The capability bid may define an amount of power (e.g., MW) that system  100  will reserve or store in battery  108  to perform frequency response, referred to herein as the “capability bid.” 
     Incentive provider  114  may provide frequency response controller  112  with a capability clearing price CP cap , a performance clearing price CP perf , and a regulation award Reg award , which correspond to the capability price, the performance price, and the capability bid, respectively. In some embodiments, CP cap , CP perf , and Reg award  are the same as the corresponding bids placed by controller  112 . In other embodiments, CP cap , CP perf , and Reg award  may not be the same as the bids placed by controller  112 . For example, CP cap , CP perf , and Reg award  may be generated by incentive provider  114  based on bids received from multiple participants in the frequency response program. Controller  112  may use CP cap , CP perf , and Reg award  to perform frequency regulation. 
     Frequency response controller  112  is shown receiving a regulation signal from incentive provider  114 . The regulation signal may specify a portion of the regulation award Reg award  that frequency response controller  112  is to add or remove from energy grid  104 . In some embodiments, the regulation signal is a normalized signal (e.g., between −1 and 1) specifying a proportion of Reg award . Positive values of the regulation signal may indicate an amount of power to add to energy grid  104 , whereas negative values of the regulation signal may indicate an amount of power to remove from energy grid  104 . 
     Frequency response controller  112  may respond to the regulation signal by generating an optimal power setpoint for power inverter  106 . The optimal power setpoint may take into account both the potential revenue from participating in the frequency response program and the costs of participation. Costs of participation may include, for example, a monetized cost of battery degradation as well as the energy and demand charges that will be incurred. The optimization may be performed using sequential quadratic programming, dynamic programming, or any other optimization technique. 
     In some embodiments, controller  112  uses a battery life model to quantify and monetize battery degradation as a function of the power setpoints provided to power inverter  106 . Advantageously, the battery life model allows controller  112  to perform an optimization that weighs the revenue generation potential of participating in the frequency response program against the cost of battery degradation and other costs of participation (e.g., less battery power available for campus  102 , increased electricity costs, etc.). An exemplary regulation signal and power response are described in greater detail with reference to  FIG. 2 . 
     Referring now to  FIG. 2 , a pair of frequency response graphs  200  and  250  are shown, according to an exemplary embodiment. Graph  200  illustrates a regulation signal Reg signal    202  as a function of time. Reg signal    202  is shown as a normalized signal ranging from −1 to 1 (i.e., −1≤Reg signal ≤1). Reg signal    202  may be generated by incentive provider  114  and provided to frequency response controller  112 . Reg signal    202  may define a proportion of the regulation award Reg award    254  that controller  112  is to add or remove from energy grid  104 , relative to a baseline value referred to as the midpoint b  256 . For example, if the value of Reg award    254  is 10 MW, a regulation signal value of 0.5 (i.e., Reg signal =0.5) may indicate that system  100  is requested to add 5 MW of power at POI  110  relative to midpoint b (e.g., P POI *=10 MW×0.5+b), whereas a regulation signal value of −0.3 may indicate that system  100  is requested to remove 3 MW of power from POI  110  relative to midpoint b (e.g., P POI *=10 MW×−0.3+b). 
     Graph  250  illustrates the desired interconnection power P POI *  252  as a function of time. P POI *  252  may be calculated by frequency response controller  112  based on Reg signal    202 , Reg award    254 , and a midpoint b  256 . For example, controller  112  may calculate P POI *  252  using the following equation:
 
 P   POI *=Reg award ×Reg signal   +b  
 
where P POI * represents the desired power at POI  110  (e.g., P POI *=P sup +P campus ) and b is the midpoint. Midpoint b may be defined (e.g., set or optimized) by controller  112  and may represent the midpoint of regulation around which the load is modified in response to Reg signal    202 . Optimal adjustment of midpoint b may allow controller  112  to actively participate in the frequency response market while also taking into account the energy and demand charge that will be incurred.
 
     In order to participate in the frequency response market, controller  112  may perform several tasks. Controller  112  may generate a price bid (e.g., $/MW) that includes the capability price and the performance price. In some embodiments, controller  112  sends the price bid to incentive provider  114  at approximately 15:30 each day and the price bid remains in effect for the entirety of the next day. Prior to beginning a frequency response period, controller  112  may generate the capability bid (e.g., MW) and send the capability bid to incentive provider  114 . In some embodiments, controller  112  generates and sends the capability bid to incentive provider  114  approximately 1.5 hours before a frequency response period begins. In an exemplary embodiment, each frequency response period has a duration of one hour; however, it is contemplated that frequency response periods may have any duration. 
     At the start of each frequency response period, controller  112  may generate the midpoint b around which controller  112  plans to perform frequency regulation. In some embodiments, controller  112  generates a midpoint b that will maintain battery  108  at a constant state-of-charge (SOC) (i.e. a midpoint that will result in battery  108  having the same SOC at the beginning and end of the frequency response period). In other embodiments, controller  112  generates midpoint b using an optimization procedure that allows the SOC of battery  108  to have different values at the beginning and end of the frequency response period. For example, controller  112  may use the SOC of battery  108  as a constrained variable that depends on midpoint b in order to optimize a value function that takes into account frequency response revenue, energy costs, and the cost of battery degradation. Exemplary techniques for calculating and/or optimizing midpoint b under both the constant SOC scenario and the variable SOC scenario are described in detail in U.S. patent application Ser. No. 15/247,883 filed Aug. 25, 2016, U.S. patent application Ser. No. 15/247,885 filed Aug. 25, 2016, and U.S. patent application Ser. No. 15/247,886 filed Aug. 25, 2016. The entire disclosure of each of these patent applications is incorporated by reference herein. 
     During each frequency response period, controller  112  may periodically generate a power setpoint for power inverter  106 . For example, controller  112  may generate a power setpoint for each time step in the frequency response period. In some embodiments, controller  112  generates the power setpoints using the equation:
 
 P   POI *=Reg award ×Reg signal   +b  
 
where P POI *=P sup +P campus . Positive values of P POI * indicate energy flow from POI  110  to energy grid  104 . Positive values of P sup  and P campus  indicate energy flow to POI  110  from power inverter  106  and campus  102 , respectively.
 
     In other embodiments, controller  112  generates the power setpoints using the equation:
 
 P   POI *=Reg award ×Res FR   +b  
 
where Res FR  is an optimal frequency response generated by optimizing a value function. Controller  112  may subtract P campus  from P POI * to generate the power setpoint for power inverter  106  (i.e., P sup =P POI *−P campus ). The power setpoint for power inverter  106  indicates the amount of power that power inverter  106  is to add to POI  110  (if the power setpoint is positive) or remove from POI  110  (if the power setpoint is negative). Exemplary techniques which can be used by controller  112  to calculate power inverter setpoints are described in detail in U.S. patent application Ser. No. 15/247,793 filed Aug. 25, 2016, U.S. patent application Ser. No. 15/247,784 filed Aug. 25, 2016, and U.S. patent application Ser. No. 15/247,777 filed Aug. 25, 2016. The entire disclosure of each of these patent applications is incorporated by reference herein.
 
Photovoltaic Energy System with Frequency Regulation and Ramp Rate Control
 
     Referring now to  FIGS. 3-4 , a photovoltaic energy system  300  that uses battery storage to simultaneously perform both ramp rate control and frequency regulation is shown, according to an exemplary embodiment. Ramp rate control is the process of offsetting ramp rates (i.e., increases or decreases in the power output of an energy system such as a photovoltaic energy system) that fall outside of compliance limits determined by the electric power authority overseeing the energy grid. Ramp rate control typically requires the use of an energy source that allows for offsetting ramp rates by either supplying additional power to the grid or consuming more power from the grid. In some instances, a facility is penalized for failing to comply with ramp rate requirements. 
     Frequency regulation is the process of maintaining the stability of the grid frequency (e.g., 60 Hz in the United States). As shown in  FIG. 4 , the grid frequency may remain balanced at 60 Hz as long as there is a balance between the demand from the energy grid and the supply to the energy grid. An increase in demand yields a decrease in grid frequency, whereas an increase in supply yields an increase in grid frequency. During a fluctuation of the grid frequency, system  300  may offset the fluctuation by either drawing more energy from the energy grid (e.g., if the grid frequency is too high) or by providing energy to the energy grid (e.g., if the grid frequency is too low). Advantageously, system  300  may use battery storage in combination with photovoltaic power to perform frequency regulation while simultaneously complying with ramp rate requirements and maintaining the state-of-charge of the battery storage within a predetermined desirable range. 
     Referring particularly to  FIG. 3 , system  300  is shown to include a photovoltaic (PV) field  302 , a PV field power inverter  304 , a battery  306 , a battery power inverter  308 , a point of interconnection (POI)  310 , and an energy grid  312 . PV field  302  may include a collection of photovoltaic cells. The photovoltaic cells are configured to convert solar energy (i.e., sunlight) into electricity using a photovoltaic material such as monocrystalline silicon, polycrystalline silicon, amorphous silicon, cadmium telluride, copper indium gallium selenide/sulfide, or other materials that exhibit the photovoltaic effect. In some embodiments, the photovoltaic cells are contained within packaged assemblies that form solar panels. Each solar panel may include a plurality of linked photovoltaic cells. The solar panels may combine to form a photovoltaic array. 
     PV field  302  may have any of a variety of sizes and/or locations. In some embodiments, PV field  302  is part of a large-scale photovoltaic power station (e.g., a solar park or farm) capable of providing an energy supply to a large number of consumers. When implemented as part of a large-scale system, PV field  302  may cover multiple hectares and may have power outputs of tens or hundreds of megawatts. In other embodiments, PV field  302  may cover a smaller area and may have a relatively lesser power output (e.g., between one and ten megawatts, less than one megawatt, etc.). For example, PV field  302  may be part of a rooftop-mounted system capable of providing enough electricity to power a single home or building. It is contemplated that PV field  302  may have any size, scale, and/or power output, as may be desirable in different implementations. 
     PV field  302  may generate a direct current (DC) output that depends on the intensity and/or directness of the sunlight to which the solar panels are exposed. The directness of the sunlight may depend on the angle of incidence of the sunlight relative to the surfaces of the solar panels. The intensity of the sunlight may be affected by a variety of environmental factors such as the time of day (e.g., sunrises and sunsets) and weather variables such as clouds that cast shadows upon PV field  302 . When PV field  302  is partially or completely covered by shadow, the power output of PV field  302  (i.e., PV field power P PV ) may drop as a result of the decrease in solar intensity. 
     In some embodiments, PV field  302  is configured to maximize solar energy collection. For example, PV field  302  may include a solar tracker (e.g., a GPS tracker, a sunlight sensor, etc.) that adjusts the angle of the solar panels so that the solar panels are aimed directly at the sun throughout the day. The solar tracker may allow the solar panels to receive direct sunlight for a greater portion of the day and may increase the total amount of power produced by PV field  302 . In some embodiments, PV field  302  includes a collection of mirrors, lenses, or solar concentrators configured to direct and/or concentrate sunlight on the solar panels. The energy generated by PV field  302  may be stored in battery  306  or provided to energy grid  312 . 
     Still referring to  FIG. 3 , system  300  is shown to include a PV field power inverter  304 . Power inverter  304  may be configured to convert the DC output of PV field  302  P PV  into an alternating current (AC) output that can be fed into energy grid  312  or used by a local (e.g., off-grid) electrical network. For example, power inverter  304  may be a solar inverter or grid-tie inverter configured to convert the DC output from PV field  302  into a sinusoidal AC output synchronized to the grid frequency of energy grid  312 . In some embodiments, power inverter  304  receives a cumulative DC output from PV field  302 . For example, power inverter  304  may be a string inverter or a central inverter. In other embodiments, power inverter  304  may include a collection of micro-inverters connected to each solar panel or solar cell. PV field power inverter  304  may convert the DC power output P PV  into an AC power output u PV  and provide the AC power output u PV  to POI  310 . 
     Power inverter  304  may receive the DC power output P PV  from PV field  302  and convert the DC power output to an AC power output that can be fed into energy grid  312 . Power inverter  304  may synchronize the frequency of the AC power output with that of energy grid  312  (e.g., 50 Hz or 60 Hz) using a local oscillator and may limit the voltage of the AC power output to no higher than the grid voltage. In some embodiments, power inverter  304  is a resonant inverter that includes or uses LC circuits to remove the harmonics from a simple square wave in order to achieve a sine wave matching the frequency of energy grid  312 . In various embodiments, power inverter  304  may operate using high-frequency transformers, low-frequency transformers, or without transformers. Low-frequency transformers may convert the DC output from PV field  302  directly to the AC output provided to energy grid  312 . High-frequency transformers may employ a multi-step process that involves converting the DC output to high-frequency AC, then back to DC, and then finally to the AC output provided to energy grid  312 . 
     Power inverter  304  may be configured to perform maximum power point tracking and/or anti-islanding. Maximum power point tracking may allow power inverter  304  to produce the maximum possible AC power from PV field  302 . For example, power inverter  304  may sample the DC power output from PV field  302  and apply a variable resistance to find the optimum maximum power point. Anti-islanding is a protection mechanism that immediately shuts down power inverter  304  (i.e., preventing power inverter  304  from generating AC power) when the connection to an electricity-consuming load no longer exists. In some embodiments, PV field power inverter  304  performs ramp rate control by limiting the power generated by PV field  302 . 
     Still referring to  FIG. 3 , system  300  is shown to include a battery power inverter  308 . Battery power inverter  308  may be configured to draw a DC power P bat  from battery  306 , convert the DC power P bat  into an AC power u bat , and provide the AC power u bat  to POI  310 . Battery power inverter  308  may also be configured to draw the AC power u bat  from POI  310 , convert the AC power u bat  into a DC battery power P bat , and store the DC battery power P bat  in battery  306 . The DC battery power P bat  may be positive if battery  306  is providing power to battery power inverter  308  (i.e., if battery  306  is discharging) or negative if battery  306  is receiving power from battery power inverter  308  (i.e., if battery  306  is charging). Similarly, the AC battery power u bat  may be positive if battery power inverter  308  is providing power to POI  310  or negative if battery power inverter  308  is receiving power from POI  310 . 
     The AC battery power u bat  is shown to include an amount of power used for frequency regulation (i.e., u FR ) and an amount of power used for ramp rate control (i.e., u RR ) which together form the AC battery power (i.e., u bat =u FR +u RR ). The DC battery power P bat  is shown to include both u FR  and u RR  as well as an additional term P loss  representing power losses in battery  306  and/or battery power inverter  308  (i.e., P bat =u FR +u RR +P loss ). The PV field power u PV  and the battery power u bat  combine at POI  110  to form P POI  (i.e., P POI =u PV +u bat ), which represents the amount of power provided to energy grid  312 . P POI  may be positive if POI  310  is providing power to energy grid  312  or negative if POI  310  is receiving power from energy grid  312 . 
     Still referring to  FIG. 3 , system  300  is shown to include a controller  314 . Controller  314  may be configured to generate a PV power setpoint u PV  for PV field power inverter  304  and a battery power setpoint u bat  for battery power inverter  308 . Throughout this disclosure, the variable u PV  is used to refer to both the PV power setpoint generated by controller  314  and the AC power output of PV field power inverter  304  since both quantities have the same value. Similarly, the variable u bat  is used to refer to both the battery power setpoint generated by controller  314  and the AC power output/input of battery power inverter  308  since both quantities have the same value. 
     PV field power inverter  304  uses the PV power setpoint u PV  to control an amount of the PV field power P PV  to provide to POI  110 . The magnitude of u PV  may be the same as the magnitude of P PV  or less than the magnitude of P PV . For example, u PV  may be the same as P PV  if controller  314  determines that PV field power inverter  304  is to provide all of the photovoltaic power P PV  to POI  310 . However, u PV  may be less than P PV  if controller  314  determines that PV field power inverter  304  is to provide less than all of the photovoltaic power P PV  to POI  310 . For example, controller  314  may determine that it is desirable for PV field power inverter  304  to provide less than all of the photovoltaic power P PV  to POI  310  to prevent the ramp rate from being exceeded and/or to prevent the power at POI  310  from exceeding a power limit. 
     Battery power inverter  308  uses the battery power setpoint u bat  to control an amount of power charged or discharged by battery  306 . The battery power setpoint u bat  may be positive if controller  314  determines that battery power inverter  308  is to draw power from battery  306  or negative if controller  314  determines that battery power inverter  308  is to store power in battery  306 . The magnitude of u bat  controls the rate at which energy is charged or discharged by battery  306 . 
     Controller  314  may generate u PV  and u bat  based on a variety of different variables including, for example, a power signal from PV field  302  (e.g., current and previous values for P PV ), the current state-of-charge (SOC) of battery  306 , a maximum battery power limit, a maximum power limit at POI  310 , the ramp rate limit, the grid frequency of energy grid  312 , and/or other variables that can be used by controller  314  to perform ramp rate control and/or frequency regulation. Advantageously, controller  314  generates values for u PV  and u bat  that maintain the ramp rate of the PV power within the ramp rate compliance limit while participating in the regulation of grid frequency and maintaining the SOC of battery  306  within a predetermined desirable range. 
     An exemplary controller which can be used as controller  314  and exemplary processes which may be performed by controller  314  to generate the PV power setpoint u PV  and the battery power setpoint u bat  are described in detail in U.S. patent application Ser. No. 15/247,869 filed Aug. 25, 2016, U.S. patent application Ser. No. 15/247,844 filed Aug. 25, 2016, U.S. patent application Ser. No. 15/247,788 filed Aug. 25, 2016, U.S. patent application Ser. No. 15/247,872 filed Aug. 25, 2016, U.S. patent application Ser. No. 15/247,880 filed Aug. 25, 2016, and U.S. patent application Ser. No. 15/247,873 filed Aug. 25, 2016. The entire disclosure of each of these patent applications is incorporated by reference herein. 
     Energy Storage System with Thermal and Electrical Energy Storage 
     Referring now to  FIG. 5A , a block diagram of an energy storage system  500  is shown, according to an exemplary embodiment. Energy storage system  500  is shown to include a building  502 . Building  502  may be the same or similar to buildings  116 , as described with reference to  FIG. 1 . For example, building  502  may be equipped with a HVAC system and/or a building management system that operates to control conditions within building  502 . In some embodiments, building  502  includes multiple buildings (i.e., a campus) served by energy storage system  500 . Building  502  may demand various resources including, for example, hot thermal energy (e.g., hot water), cold thermal energy (e.g., cold water), and/or electrical energy. The resources may be demanded by equipment or subsystems within building  502  or by external systems that provide services for building  502  (e.g., heating, cooling, air circulation, lighting, electricity, etc.). Energy storage system  500  operates to satisfy the resource demand associated with building  502 . 
     Energy storage system  500  is shown to include a plurality of utilities  510 . Utilities  510  may provide energy storage system  500  with resources such as electricity, water, natural gas, or any other resource that can be used by energy storage system  500  to satisfy the demand of building  502 . For example, utilities  510  are shown to include an electric utility  511 , a water utility  512 , a natural gas utility  513 , and utility M  514 , where M is the total number of utilities  510 . In some embodiments, utilities  510  are commodity suppliers from which resources and other types of commodities can be purchased. Resources purchased from utilities  510  can be used by generator subplants  520  to produce generated resources (e.g., hot water, cold water, electricity, steam, etc.), stored in storage subplants  530  for later use, or provided directly to building  502 . For example, utilities  510  are shown providing electricity directly to building  502  and storage subplants  530 . 
     Energy storage system  500  is shown to include a plurality of generator subplants  520 . Generator subplants  520  are shown to include a heater subplant  521 , a chiller subplant  522 , a heat recovery chiller subplant  523 , a steam subplant  524 , an electricity subplant  525 , and subplant N, where N is the total number of generator subplants  520 . Generator subplants  520  may be configured to convert one or more input resources into one or more output resources by operation of the equipment within generator subplants  520 . For example, heater subplant  521  may be configured to generate hot thermal energy (e.g., hot water) by heating water using electricity or natural gas. Chiller subplant  522  may be configured to generate cold thermal energy (e.g., cold water) by chilling water using electricity. Heat recovery chiller subplant  523  may be configured to generate hot thermal energy and cold thermal energy by removing heat from one water supply and adding the heat to another water supply. Steam subplant  524  may be configured to generate steam by boiling water using electricity or natural gas. Electricity subplant  525  may be configured to generate electricity using mechanical generators (e.g., a steam turbine, a gas-powered generator, etc.) or other types of electricity-generating equipment (e.g., photovoltaic equipment, hydroelectric equipment, etc.). 
     The input resources used by generator subplants  520  may be provided by utilities  510 , retrieved from storage subplants  530 , and/or generated by other generator subplants  520 . For example, steam subplant  524  may produce steam as an output resource. Electricity subplant  525  may include a steam turbine that uses the steam generated by steam subplant  524  as an input resource to generate electricity. The output resources produced by generator subplants  520  may be stored in storage subplants  530 , provided to building  502 , sold to energy purchasers  504 , and/or used by other generator subplants  520 . For example, the electricity generated by electricity subplant  525  may be stored in electrical energy storage  533 , used by chiller subplant  522  to generate cold thermal energy, provided to building  502 , and/or sold to energy purchasers  504 . 
     Energy storage system  500  is shown to include storage subplants  530 . Storage subplants  530  may be configured to store energy and other types of resources for later use. Each of storage subplants  530  may be configured to store a different type of resource. For example, storage subplants  530  are shown to include hot thermal energy storage  531  (e.g., one or more hot water storage tanks), cold thermal energy storage  532  (e.g., one or more cold thermal energy storage tanks), electrical energy storage  533  (e.g., one or more batteries), and resource type P storage  534 , where P is the total number of storage subplants  530 . The resources stored in subplants  530  may be purchased directly from utilities  510  or generated by generator subplants  520 . 
     In some embodiments, storage subplants  530  are used by energy storage system  500  to take advantage of price-based demand response (PBDR) programs. PBDR programs encourage consumers to reduce consumption when generation, transmission, and distribution costs are high. PBDR programs are typically implemented (e.g., by utilities  510 ) in the form of energy prices that vary as a function of time. For example, utilities  510  may increase the price per unit of electricity during peak usage hours to encourage customers to reduce electricity consumption during peak times. Some utilities also charge consumers a separate demand charge based on the maximum rate of electricity consumption at any time during a predetermined demand charge period. 
     Advantageously, storing energy and other types of resources in subplants  530  allows for the resources to be purchased at times when the resources are relatively less expensive (e.g., during non-peak electricity hours) and stored for use at times when the resources are relatively more expensive (e.g., during peak electricity hours). Storing resources in subplants  530  also allows the resource demand of building  502  to be shifted in time. For example, resources can be purchased from utilities  510  at times when the demand for heating or cooling is low and immediately converted into hot or cold thermal energy by generator subplants  520 . The thermal energy can be stored in storage subplants  530  and retrieved at times when the demand for heating or cooling is high. This allows energy storage system  500  to smooth the resource demand of building  502  and reduces the maximum required capacity of generator subplants  520 . Smoothing the demand also allows energy storage system  500  to reduce the peak electricity consumption, which results in a lower demand charge. 
     In some embodiments, storage subplants  530  are used by energy storage system  500  to take advantage of incentive-based demand response (IBDR) programs. IBDR programs provide incentives to customers who have the capability to store energy, generate energy, or curtail energy usage upon request. Incentives are typically provided in the form of monetary revenue paid by utilities  510  or by an independent service operator (ISO). IBDR programs supplement traditional utility-owned generation, transmission, and distribution assets with additional options for modifying demand load curves. For example, stored energy can be sold to energy purchasers  504  (e.g., an energy grid) to supplement the energy generated by utilities  510 . In some instances, incentives for participating in an IBDR program vary based on how quickly a system can respond to a request to change power output/consumption. Faster responses may be compensated at a higher level. Advantageously, electrical energy storage  533  allows system  500  to quickly respond to a request for electric power by rapidly discharging stored electrical energy to energy purchasers  504 . 
     Still referring to  FIG. 5A , energy storage system  500  is shown to include an energy storage controller  506 . Energy storage controller  506  may be configured to control the distribution, production, storage, and usage of resources in energy storage system  500 . In some embodiments, energy storage controller  506  performs an optimization process determine an optimal set of control decisions for each time step within an optimization period. The control decisions may include, for example, an optimal amount of each resource to purchase from utilities  510 , an optimal amount of each resource to produce or convert using generator subplants  520 , an optimal amount of each resource to store or remove from storage subplants  530 , an optimal amount of each resource to sell to energy purchasers  504 , and/or an optimal amount of each resource to provide to building  502 . In some embodiments, the control decisions include an optimal amount of each input resource and output resource for each of generator subplants  520 . 
     Controller  506  may be configured to maximize the economic value of operating energy storage system  500  over the duration of the optimization period. The economic value may be defined by a value function that expresses economic value as a function of the control decisions made by controller  506 . The value function may account for the cost of resources purchased from utilities  510 , revenue generated by selling resources to energy purchasers  504 , and the cost of operating energy storage system  500 . In some embodiments, the cost of operating energy storage system  500  includes a cost for losses in battery capacity as a result of the charging and discharging electrical energy storage  533 . The cost of operating energy storage system  500  may also include a cost of excessive equipment start/stops during the optimization period. 
     Each of subplants  520 - 530  may include equipment that can be controlled by energy storage controller  506  to optimize the performance of energy storage system  500 . Subplant equipment may include, for example, heating devices, chillers, heat recovery heat exchangers, cooling towers, energy storage devices, pumps, valves, and/or other devices of subplants  520 - 530 . Individual devices of generator subplants  520  can be turned on or off to adjust the resource production of each generator subplant. In some embodiments, individual devices of generator subplants  520  can be operated at variable capacities (e.g., operating a chiller at 10% capacity or 60% capacity) according to an operating setpoint received from energy storage controller  506 . 
     In some embodiments, one or more of subplants  520 - 530  includes a subplant level controller configured to control the equipment of the corresponding subplant. For example, energy storage controller  506  may determine an on/off configuration and global operating setpoints for the subplant equipment. In response to the on/off configuration and received global operating setpoints, the subplant controllers may turn individual devices of their respective equipment on or off, and implement specific operating setpoints (e.g., damper position, vane position, fan speed, pump speed, etc.) to reach or maintain the global operating setpoints. 
     In some embodiments, controller  506  maximizes the life cycle economic value of energy storage system  500  while participating in PBDR programs, IBDR programs, or simultaneously in both PBDR and IBDR programs. For the IBDR programs, controller  506  may use statistical estimates of past clearing prices, mileage ratios, and event probabilities to determine the revenue generation potential of selling stored energy to energy purchasers  504 . For the PBDR programs, controller  506  may use predictions of ambient conditions, facility thermal loads, and thermodynamic models of installed equipment to estimate the resource consumption of subplants  520 . Controller  506  may use predictions of the resource consumption to monetize the costs of running the equipment. 
     Controller  506  may automatically determine (e.g., without human intervention) a combination of PBDR and/or IBDR programs in which to participate over the optimization period in order to maximize economic value. For example, controller  506  may consider the revenue generation potential of IBDR programs, the cost reduction potential of PBDR programs, and the equipment maintenance/replacement costs that would result from participating in various combinations of the IBDR programs and PBDR programs. Controller  506  may weigh the benefits of participation against the costs of participation to determine an optimal combination of programs in which to participate. Advantageously, this allows controller  506  to determine an optimal set of control decisions that maximize the overall value of operating energy storage system  500 . 
     In some instances, controller  506  may determine that it would be beneficial to participate in an IBDR program when the revenue generation potential is high and/or the costs of participating are low. For example, controller  506  may receive notice of a synchronous reserve event from an IBDR program which requires energy storage system  500  to shed a predetermined amount of power. Controller  506  may determine that it is optimal to participate in the IBDR program if cold thermal energy storage  532  has enough capacity to provide cooling for building  502  while the load on chiller subplant  522  is reduced in order to shed the predetermined amount of power. 
     In other instances, controller  506  may determine that it would not be beneficial to participate in an IBDR program when the resources required to participate are better allocated elsewhere. For example, if building  502  is close to setting a new peak demand that would greatly increase the PBDR costs, controller  506  may determine that only a small portion of the electrical energy stored in electrical energy storage  533  will be sold to energy purchasers  504  in order to participate in a frequency response market. Controller  506  may determine that the remainder of the electrical energy will be used to power chiller subplant  522  to prevent a new peak demand from being set. 
     In some embodiments, energy storage system  500  and controller include some or all of the components and/or features described in U.S. patent application Ser. No. 15/247,875 filed Aug. 25, 2016, U.S. patent application Ser. No. 15/247,879 filed Aug. 25, 2016, and U.S. patent application Ser. No. 15/247,881 filed Aug. 25, 2016. The entire disclosure of each of these patent applications is incorporated by reference herein. 
     Energy Cost Optimization System 
     Referring now to  FIG. 5B , a block diagram of an energy cost optimization system  550  is shown, according to an exemplary embodiment. Energy cost optimization system  550  is shown to include many of the same components as energy storage system  500  (described with reference to  FIG. 5A ) with the exception of storage subplants  530 . System  550  is an example of a system without thermal or electrical energy storage in which the peak load contribution cost optimization techniques can be implemented. 
     Energy cost optimization system  550  is shown to include a building  502 . Building  502  may be the same or similar to buildings  116 , as described with reference to  FIG. 1 . For example, building  502  may be equipped with a HVAC system and/or a building management system that operates to control conditions within building  502 . In some embodiments, building  502  includes multiple buildings (i.e., a campus) served by energy cost optimization system  550 . Building  502  may demand various resources including, for example, hot thermal energy (e.g., hot water), cold thermal energy (e.g., cold water), and/or electrical energy. The resources may be demanded by equipment or subsystems within building  502  or by external systems that provide services for building  502  (e.g., heating, cooling, air circulation, lighting, electricity, etc.). Energy cost optimization system  550  operates to satisfy the resource demand associated with building  502 . 
     Energy cost optimization system  550  is shown to include a plurality of utilities  510 . Utilities  510  may provide system  550  with resources such as electricity, water, natural gas, or any other resource that can be used by system  550  to satisfy the demand of building  502 . For example, utilities  510  are shown to include an electric utility  511 , a water utility  512 , a natural gas utility  513 , and utility M  514 , where M is the total number of utilities  510 . In some embodiments, utilities  510  are commodity suppliers from which resources and other types of commodities can be purchased. Resources purchased from utilities  510  can be used by generator subplants  520  to produce generated resources (e.g., hot water, cold water, electricity, steam, etc.) or provided directly to building  502 . For example, utilities  510  are shown providing electricity directly to building  502 . 
     Energy cost optimization system  550  is shown to include a plurality of generator subplants  520 . Generator subplants  520  are shown to include a heater subplant  521 , a chiller subplant  522 , a heat recovery chiller subplant  523 , a steam subplant  524 , an electricity subplant  525 , and subplant N, where N is the total number of generator subplants  520 . Generator subplants  520  may be configured to convert one or more input resources into one or more output resources by operation of the equipment within generator subplants  520 . For example, heater subplant  521  may be configured to generate hot thermal energy (e.g., hot water) by heating water using electricity or natural gas. Chiller subplant  522  may be configured to generate cold thermal energy (e.g., cold water) by chilling water using electricity. Heat recovery chiller subplant  523  may be configured to generate hot thermal energy and cold thermal energy by removing heat from one water supply and adding the heat to another water supply. Steam subplant  524  may be configured to generate steam by boiling water using electricity or natural gas. Electricity subplant  525  may be configured to generate electricity using mechanical generators (e.g., a steam turbine, a gas-powered generator, etc.) or other types of electricity-generating equipment (e.g., photovoltaic equipment, hydroelectric equipment, etc.). 
     The input resources used by generator subplants  520  may be provided by utilities  510  and/or generated by other generator subplants  520 . For example, steam subplant  524  may produce steam as an output resource. Electricity subplant  525  may include a steam turbine that uses the steam generated by steam subplant  524  as an input resource to generate electricity. The output resources produced by generator subplants  520  may be provided to building  502 , sold to energy purchasers  504 , and/or used by other generator subplants  520 . For example, the electricity generated by electricity subplant  525  may be used by chiller subplant  522  to generate cold thermal energy, provided to building  502 , and/or sold to energy purchasers  504 . 
     Still referring to  FIG. 5B , energy cost optimization system  550  is shown to include a controller  552 . Controller  552  may be configured to control the distribution, production, and usage of resources in system  550 . In some embodiments, controller  552  performs an optimization process determine an optimal set of control decisions for each time step within an optimization period. The control decisions may include, for example, an optimal amount of each resource to purchase from utilities  510 , an optimal amount of each resource to produce or convert using generator subplants  520 , an optimal amount of each resource to sell to energy purchasers  504 , and/or an optimal amount of each resource to provide to building  502 . In some embodiments, the control decisions include an optimal amount of each input resource and output resource for each of generator subplants  520 . 
     Controller  552  may be configured to maximize the economic value of operating energy cost optimization system  550  over the duration of the optimization period. The economic value may be defined by a value function that expresses economic value as a function of the control decisions made by controller  552 . The value function may account for the cost of resources purchased from utilities  510 , revenue generated by selling resources to energy purchasers  504 , and the cost of operating system  550 . In some embodiments, the cost of operating system  550  includes a cost of excessive equipment start/stops during the optimization period. 
     Each of subplants  520  may include equipment that can be controlled by controller  552  to optimize the performance of system  550 . Subplant equipment may include, for example, heating devices, chillers, heat recovery heat exchangers, cooling towers, pumps, valves, and/or other devices of subplants  520 . Individual devices of generator subplants  520  can be turned on or off to adjust the resource production of each generator subplant. In some embodiments, individual devices of generator subplants  520  can be operated at variable capacities (e.g., operating a chiller at 10% capacity or 60% capacity) according to an operating setpoint received from controller  552 . 
     In some embodiments, one or more of subplants  520  includes a subplant level controller configured to control the equipment of the corresponding subplant. For example, controller  552  may determine an on/off configuration and global operating setpoints for the subplant equipment. In response to the on/off configuration and received global operating setpoints, the subplant controllers may turn individual devices of their respective equipment on or off, and implement specific operating setpoints (e.g., damper position, vane position, fan speed, pump speed, etc.) to reach or maintain the global operating setpoints. 
     In some embodiments, energy cost optimization system  550  and controller  552  include some or all of the components and/or features described in U.S. patent application Ser. No. 15/247,875 filed Aug. 25, 2016, U.S. patent application Ser. No. 15/247,879 filed Aug. 25, 2016, and U.S. patent application Ser. No. 15/247,881 filed Aug. 25, 2016. The entire disclosure of each of these patent applications is incorporated by reference herein. 
     Energy Storage Controller 
     Referring now to  FIG. 6A , a block diagram illustrating energy storage controller  506  in greater detail is shown, according to an exemplary embodiment. Energy storage controller  506  is shown providing control decisions to a building management system (BMS)  606 . In some embodiments, BMS  606  is the same or similar the BMS described with reference to  FIG. 1 . The control decisions provided to BMS  606  may include resource purchase amounts for utilities  510 , setpoints for generator subplants  520 , and/or charge/discharge rates for storage subplants  530 . 
     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 energy storage controller  506 . 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 subplants  520 - 530  to affect the monitored conditions within the building and to serve the thermal energy loads of the building. 
     BMS  606  may receive control signals from energy storage controller  506  specifying on/off states, charge/discharge rates, and/or setpoints for the subplant equipment. BMS  606  may control the equipment (e.g., via actuators, power relays, etc.) in accordance with the control signals provided by energy storage controller  506 . For example, BMS  606  may operate the equipment using closed loop control to achieve the setpoints specified by energy storage controller  506 . In various embodiments, BMS  606  may be combined with energy storage controller  506  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. 
     Energy storage controller  506  may monitor the status of the controlled building using information received from BMS  606 . Energy storage controller  506  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 an optimization period (e.g., using weather forecasts from a weather service  604 ). Energy storage controller  506  may also predict the revenue generation potential of IBDR programs using an incentive event history (e.g., past clearing prices, mileage ratios, event probabilities, etc.) from incentive programs  602 . Energy storage controller  506  may generate control decisions that optimize the economic value of operating energy storage system  500  over the duration of the optimization period subject to constraints on the optimization process (e.g., energy balance constraints, load satisfaction constraints, etc.). The optimization process performed by energy storage controller  506  is described in greater detail below. 
     According to an exemplary embodiment, energy storage controller  506  is integrated within a single computer (e.g., one server, one housing, etc.). In various other exemplary embodiments, energy storage controller  506  can be distributed across multiple servers or computers (e.g., that can exist in distributed locations). In another exemplary embodiment, energy storage controller  506  may integrated with a smart building manager that manages multiple building systems and/or combined with BMS  606 . 
     Energy storage controller  506  is shown to include a communications interface  636  and a processing circuit  607 . Communications interface  636  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  636  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  636  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  636  may be a network interface configured to facilitate electronic data communications between energy storage controller  506  and various external systems or devices (e.g., BMS  606 , subplants  520 - 530 , utilities  510 , etc.). For example, energy storage controller  506  may receive information from BMS  606  indicating one or more measured states of the controlled building (e.g., temperature, humidity, electric loads, etc.) and one or more states of subplants  520 - 530  (e.g., equipment status, power consumption, equipment availability, etc.). Communications interface  636  may receive inputs from BMS  606  and/or subplants  520 - 530  and may provide operating parameters (e.g., on/off decisions, setpoints, etc.) to subplants  520 - 530  via BMS  606 . The operating parameters may cause subplants  520 - 530  to activate, deactivate, or adjust a setpoint for various devices thereof. 
     Still referring to  FIG. 6A , processing circuit  607  is shown to include a processor  608  and memory  610 . Processor  608  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  608  may be configured to execute computer code or instructions stored in memory  610  or received from other computer readable media (e.g., CDROM, network storage, a remote server, etc.). 
     Memory  610  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  610  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  610  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  610  may be communicably connected to processor  608  via processing circuit  607  and may include computer code for executing (e.g., by processor  608 ) one or more processes described herein. 
     Memory  610  is shown to include a building status monitor  624 . Energy storage controller  506  may receive data regarding the overall building or building space to be heated or cooled by system  500  via building status monitor  624 . In an exemplary embodiment, building status monitor  624  may include a graphical user interface component configured to provide graphical user interfaces to a user for selecting building requirements (e.g., overall temperature parameters, selecting schedules for the building, selecting different temperature levels for different building zones, etc.). 
     Energy storage controller  506  may determine on/off configurations and operating setpoints to satisfy the building requirements received from building status monitor  624 . In some embodiments, building status monitor  624  receives, collects, stores, and/or transmits cooling load requirements, building temperature setpoints, occupancy data, weather data, energy data, schedule data, and other building parameters. In some embodiments, building status monitor  624  stores data regarding energy costs, such as pricing information available from utilities  510  (energy charge, demand charge, etc.). 
     Still referring to  FIG. 6A , memory  610  is shown to include a load/rate predictor  622 . Load/rate predictor  622  may be configured to predict the thermal energy loads (   k ) of the building or campus for each time step k (e.g., k=1 . . . n) of an optimization period. Load/rate predictor  622  is shown receiving weather forecasts from a weather service  604 . In some embodiments, load/rate predictor  622  predicts the thermal energy loads    k  as a function of the weather forecasts. In some embodiments, load/rate predictor  622  uses feedback from BMS  606  to predict loads    k . 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  622  receives a measured electric load and/or previous measured load data from BMS  606  (e.g., via building status monitor  624 ). Load/rate predictor  622  may predict loads    k  as a function of a given weather forecast ({circumflex over (ϕ)} w ), a day type (clay), the time of day (t), and previous measured load data (Y k-1 ). Such a relationship is expressed in the following equation:
 
   k   =f ({circumflex over (ϕ)} w ,day, t|Y   k-1 )
 
     In some embodiments, load/rate predictor  622  uses a deterministic plus stochastic model trained from historical load data to predict loads    k . Load/rate predictor  622  may use any of a variety of prediction methods to predict loads    k  (e.g., linear regression for the deterministic portion and an AR model for the stochastic portion). Load/rate predictor  622  may predict one or more different types of loads for the building or campus. For example, load/rate predictor  622  may predict a hot water load    Hot,k  and a cold water load    Cold,k  for each time step k within the prediction window. In some embodiments, load/rate predictor  622  makes load/rate predictions using the techniques described in U.S. patent application Ser. No. 14/717,593. 
     Load/rate predictor  622  is shown receiving utility rates from utilities  510 . Utility rates may indicate a cost or price per unit of a resource (e.g., electricity, natural gas, water, etc.) provided by utilities  510  at each time step k in the prediction window. 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  510  or predicted utility rates estimated by load/rate predictor  622 . 
     In some embodiments, the utility rates include demand charges for one or more resources provided by utilities  510 . A demand charge may define a separate cost imposed by utilities  510  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. Advantageously, demand response optimizer  630  may be configured to account for demand charges in the high level optimization process performed by high level optimizer  632 . Utilities  510  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  622  may store the predicted loads    k  and the utility rates in memory  610  and/or provide the predicted loads    k  and the utility rates to demand response optimizer  630 . 
     Still referring to  FIG. 6A , memory  610  is shown to include an incentive estimator  620 . Incentive estimator  620  may be configured to estimate the revenue generation potential of participating in various incentive-based demand response (IBDR) programs. In some embodiments, incentive estimator  620  receives an incentive event history from incentive programs  602 . The incentive event history may include a history of past IBDR events from incentive programs  602 . An IBDR event may include an invitation from incentive programs  602  to participate in an IBDR program in exchange for a monetary incentive. The incentive event history may indicate the times at which the past IBDR events occurred and attributes describing the IBDR events (e.g., clearing prices, mileage ratios, participation requirements, etc.). Incentive estimator  620  may use the incentive event history to estimate IBDR event probabilities during the optimization period. 
     Incentive estimator  620  is shown providing incentive predictions to demand response optimizer  630 . The incentive predictions may include the estimated IBDR probabilities, estimated participation requirements, an estimated amount of revenue from participating in the estimated IBDR events, and/or any other attributes of the predicted IBDR events. Demand response optimizer  630  may use the incentive predictions along with the predicted loads    k  and utility rates from load/rate predictor  622  to determine an optimal set of control decisions for each time step within the optimization period. 
     Still referring to  FIG. 6A , memory  610  is shown to include a demand response optimizer  630 . Demand response optimizer  630  may perform a cascaded optimization process to optimize the performance of energy storage system  500 . For example, demand response optimizer  630  is shown to include a high level optimizer  632  and a low level optimizer  634 . High level optimizer  632  may control an outer (e.g., subplant level) loop of the cascaded optimization. High level optimizer  632  may determine an optimal set of control decisions for each time step in the prediction window in order to optimize (e.g., maximize) the value of operating energy storage system  500 . Control decisions made by high level optimizer  632  may include, for example, load setpoints for each of generator subplants  520 , charge/discharge rates for each of storage subplants  530 , resource purchase amounts for each type of resource purchased from utilities  510 , and/or an amount of each resource sold to energy purchasers  504 . In other words, the control decisions may define resource allocation at each time step. The control decisions made by high level optimizer  632  are based on the statistical estimates of incentive event probabilities and revenue generation potential for various IBDR events as well as the load and rate predictions. 
     Low level optimizer  634  may control an inner (e.g., equipment level) loop of the cascaded optimization. Low level optimizer  634  may determine how to best run each subplant at the load setpoint determined by high level optimizer  632 . For example, low level optimizer  634  may determine on/off states and/or operating setpoints for various devices of the subplant equipment in order to optimize (e.g., minimize) the energy consumption of each subplant while meeting the resource allocation setpoint for the subplant. In some embodiments, low level optimizer  634  receives actual incentive events from incentive programs  602 . Low level optimizer  634  may determine whether to participate in the incentive events based on the resource allocation set by high level optimizer  632 . For example, if insufficient resources have been allocated to a particular IBDR program by high level optimizer  632  or if the allocated resources have already been used, low level optimizer  634  may determine that energy storage system  500  will not participate in the IBDR program and may ignore the IBDR event. However, if the required resources have been allocated to the IBDR program and are available in storage subplants  530 , low level optimizer  634  may determine that system  500  will participate in the IBDR program in response to the IBDR event. The cascaded optimization process performed by demand response optimizer  630  is described in greater detail in U.S. patent application Ser. No. 15/247,885. 
     Still referring to  FIG. 6A , memory  610  is shown to include a subplant control module  628 . Subplant control module  628  may store historical data regarding past operating statuses, past operating setpoints, and instructions for calculating and/or implementing control parameters for subplants  520 - 530 . Subplant control module  628  may also receive, store, and/or transmit data regarding the conditions of individual devices of the subplant equipment, such as operating efficiency, equipment degradation, a date since last service, a lifespan parameter, a condition grade, or other device-specific data. Subplant control module  628  may receive data from subplants  520 - 530  and/or BMS  606  via communications interface  636 . Subplant control module  628  may also receive and store on/off statuses and operating setpoints from low level optimizer  634 . 
     Data and processing results from demand response optimizer  630 , subplant control module  628 , or other modules of energy storage controller  506  may be accessed by (or pushed to) monitoring and reporting applications  626 . Monitoring and reporting applications  626  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  626  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 energy storage 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 energy storage system. 
     Still referring to  FIG. 6A , energy storage controller  506  may include one or more GUI servers, web services  612 , or GUI engines  614  to support monitoring and reporting applications  626 . In various embodiments, applications  626 , web services  612 , and GUI engine  614  may be provided as separate components outside of energy storage controller  506  (e.g., as part of a smart building manager). Energy storage controller  506  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. Energy storage controller  506  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. 
     Energy storage controller  506  is shown to include configuration tools  616 . Configuration tools  616  can allow a user to define (e.g., via graphical user interfaces, via prompt-driven “wizards,” etc.) how energy storage controller  506  should react to changing conditions in the energy storage subsystems. In an exemplary embodiment, configuration tools  616  allow a user to build and store condition-response scenarios that can cross multiple energy storage system devices, multiple building systems, and multiple enterprise control applications (e.g., work order management system applications, entity resource planning applications, etc.). For example, configuration tools  616  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  616  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. 
     Energy Cost Optimization Controller 
     Referring now to  FIG. 6B , a block diagram illustrating controller  552  in greater detail is shown, according to an exemplary embodiment. Controller  552  is shown providing control decisions to a building management system (BMS)  606 . In some embodiments, BMS  606  is the same or similar the BMS described with reference to  FIG. 1 . The control decisions provided to BMS  606  may include resource purchase amounts for utilities  510  and/or setpoints for generator subplants  520 . 
     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 controller  552 . 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 subplants  520  to affect the monitored conditions within the building and to serve the thermal energy loads of the building. 
     BMS  606  may receive control signals from controller  552  specifying on/off states and/or setpoints for the subplant equipment. BMS  606  may control the equipment (e.g., via actuators, power relays, etc.) in accordance with the control signals provided by controller  552 . For example, BMS  606  may operate the equipment using closed loop control to achieve the setpoints specified by energy storage controller  552 . In various embodiments, BMS  606  may be combined with controller  552  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. 
     Controller  552  may monitor the status of the controlled building using information received from BMS  606 . Controller  552  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 an optimization period (e.g., using weather forecasts from a weather service  604 ). Controller  552  may generate control decisions that optimize the economic value of operating system  550  over the duration of the optimization period subject to constraints on the optimization process (e.g., energy balance constraints, load satisfaction constraints, etc.). The optimization process performed by controller  552  is described in greater detail below. 
     Controller  552  is shown to include a communications interface  636  and a processing circuit  607  having a processor  608  and memory  610 . These components may be the same as described with reference to  FIG. 6A . For example, controller  552  is shown to include demand response optimizer  630 . Demand response optimizer  630  may perform a cascaded optimization process to optimize the performance of system  550 . For example, demand response optimizer  630  is shown to include a high level optimizer  632  and a low level optimizer  634 . High level optimizer  632  may control an outer (e.g., subplant level) loop of the cascaded optimization. High level optimizer  632  may determine an optimal set of control decisions for each time step in the prediction window in order to optimize (e.g., maximize) the value of operating energy storage system  500 . Control decisions made by high level optimizer  632  may include, for example, load setpoints for each of generator subplants  520 , resource purchase amounts for each type of resource purchased from utilities  510 , and/or an amount of each resource sold to energy purchasers  504 . In other words, the control decisions may define resource allocation at each time step. 
     Low level optimizer  634  may control an inner (e.g., equipment level) loop of the cascaded optimization. Low level optimizer  634  may determine how to best run each subplant at the load setpoint determined by high level optimizer  632 . For example, low level optimizer  634  may determine on/off states and/or operating setpoints for various devices of the subplant equipment in order to optimize (e.g., minimize) the energy consumption of each subplant while meeting the resource allocation setpoint for the subplant. The cascaded optimization process performed by demand response optimizer  630  is described in greater detail in U.S. patent application Ser. No. 15/247,885. These and other components of controller  552  may be the same as previously described with reference to  FIG. 6A . 
     Planning Tool 
     Referring now to  FIG. 7 , a block diagram of a planning system  700  is shown, according to an exemplary embodiment. Planning system  700  may be configured to use demand response optimizer  630  as part of a planning tool  702  to simulate the operation of a central plant over a predetermined time period (e.g., a day, a month, a week, a year, etc.) for planning, budgeting, and/or design considerations. When implemented in planning tool  702 , demand response optimizer  630  may operate in a similar manner as described with reference to  FIGS. 6A-6B . For example, demand response optimizer  630  may use building loads and utility rates to determine an optimal resource allocation to minimize cost over a simulation period. However, planning tool  702  may not be responsible for real-time control of a building management system or central plant. 
     Planning tool  702  can be configured to determine the benefits of investing in a battery asset and the financial metrics associated with the investment. Such financial metrics can include, for example, the internal rate of return (IRR), net present value (NPV), and/or simple payback period (SPP). Planning tool  702  can also assist a user in determining the size of the battery which yields optimal financial metrics such as maximum NPV or a minimum SPP. In some embodiments, planning tool  702  allows a user to specify a battery size and automatically determines the benefits of the battery asset from participating in selected IBDR programs while performing PBDR, as described with reference to  FIG. 5A . In some embodiments, planning tool  702  is configured to determine the battery size that minimizes SPP given the IBDR programs selected and the requirement of performing PBDR. In some embodiments, planning tool  702  is configured to determine the battery size that maximizes NPV given the IBDR programs selected and the requirement of performing PBDR. 
     In planning tool  702 , high level optimizer  632  may receive planned loads and utility rates for the entire simulation period. The planned loads and utility rates may be defined by input received from a user via a client device  722  (e.g., user-defined, user selected, etc.) and/or retrieved from a plan information database  726 . High level optimizer  632  uses the planned loads and utility rates in conjunction with subplant curves from low level optimizer  634  to determine an optimal resource allocation (i.e., an optimal dispatch schedule) for a portion of the simulation period. 
     The portion of the simulation period over which high level optimizer  632  optimizes the resource allocation may be defined by a prediction window ending at a time horizon. With each iteration of the optimization, the prediction window is shifted forward and the portion of the dispatch schedule no longer in the prediction window is accepted (e.g., stored or output as results of the simulation). Load and rate predictions may be predefined for the entire simulation and may not be subject to adjustments in each iteration. However, shifting the prediction window forward in time may introduce additional plan information (e.g., planned loads and/or utility rates) for the newly-added time slice at the end of the prediction window. The new plan information may not have a significant effect on the optimal dispatch schedule since only a small portion of the prediction window changes with each iteration. 
     In some embodiments, high level optimizer  632  requests all of the subplant curves used in the simulation from low level optimizer  634  at the beginning of the simulation. Since the planned loads and environmental conditions are known for the entire simulation period, high level optimizer  632  may retrieve all of the relevant subplant curves at the beginning of the simulation. In some embodiments, low level optimizer  634  generates functions that map subplant production to equipment level production and resource use when the subplant curves are provided to high level optimizer  632 . These subplant to equipment functions may be used to calculate the individual equipment production and resource use (e.g., in a post-processing module) based on the results of the simulation. 
     Still referring to  FIG. 7 , planning tool  702  is shown to include a communications interface  704  and a processing circuit  706 . Communications interface  704  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  704  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  704  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  704  may be a network interface configured to facilitate electronic data communications between planning tool  702  and various external systems or devices (e.g., client device  722 , results database  728 , plan information database  726 , etc.). For example, planning tool  702  may receive planned loads and utility rates from client device  722  and/or plan information database  726  via communications interface  704 . Planning tool  702  may use communications interface  704  to output results of the simulation to client device  722  and/or to store the results in results database  728 . 
     Still referring to  FIG. 7 , processing circuit  706  is shown to include a processor  710  and memory  712 . Processor  710  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  710  may be configured to execute computer code or instructions stored in memory  712  or received from other computer readable media (e.g., CDROM, network storage, a remote server, etc.). 
     Memory  712  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  712  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  712  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  712  may be communicably connected to processor  710  via processing circuit  706  and may include computer code for executing (e.g., by processor  710 ) one or more processes described herein. 
     Still referring to  FIG. 7 , memory  712  is shown to include a GUI engine  716 , web services  714 , and configuration tools  718 . In an exemplary embodiment, GUI engine  716  includes a graphical user interface component configured to provide graphical user interfaces to a user for selecting or defining plan information for the simulation (e.g., planned loads, utility rates, environmental conditions, etc.). Web services  714  may allow a user to interact with planning tool  702  via a web portal and/or from a remote system or device (e.g., an enterprise control application). 
     Configuration tools  718  can allow a user to define (e.g., via graphical user interfaces, via prompt-driven “wizards,” etc.) various parameters of the simulation such as the number and type of subplants, the devices within each subplant, the subplant curves, device-specific efficiency curves, the duration of the simulation, the duration of the prediction window, the duration of each time step, and/or various other types of plan information related to the simulation. Configuration tools  718  can present user interfaces for building the simulation. The user interfaces may allow users to define simulation parameters graphically. In some embodiments, the user interfaces allow a user to select a pre-stored or pre-constructed simulated plant and/or plan information (e.g., from plan information database  726 ) and adapt it or enable it for use in the simulation. 
     Still referring to  FIG. 7 , memory  712  is shown to include demand response optimizer  630 . Demand response optimizer  630  may use the planned loads and utility rates to determine an optimal resource allocation over a prediction window. The operation of demand response optimizer  630  may be the same or similar as previously described with reference to  FIGS. 6-8 . With each iteration of the optimization process, demand response optimizer  630  may shift the prediction window forward and apply the optimal resource allocation for the portion of the simulation period no longer in the prediction window. Demand response optimizer  630  may use the new plan information at the end of the prediction window to perform the next iteration of the optimization process. Demand response optimizer  630  may output the applied resource allocation to reporting applications  730  for presentation to a client device  722  (e.g., via user interface  724 ) or storage in results database  728 . 
     Still referring to  FIG. 7 , memory  712  is shown to include reporting applications  730 . Reporting applications  730  may receive the optimized resource allocations from demand response optimizer  630  and, in some embodiments, costs associated with the optimized resource allocations. Reporting applications  730  may include a web-based reporting 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 various plants, subplants, or the like. Other GUI elements or reports may be generated and shown based on available data that allow users to assess the results of the simulation. The user interface or report (or underlying data engine) may be configured to aggregate and categorize resource allocation and the costs associated therewith and provide the results to a user via a GUI. The GUI elements may include charts or histograms that allow the user to visually analyze the results of the simulation. An exemplary output that may be generated by reporting applications  730  is shown in  FIG. 8 . 
     Referring now to  FIG. 8 , several graphs  800  illustrating the operation of planning tool  702  are shown, according to an exemplary embodiment. With each iteration of the optimization process, planning tool  702  selects an optimization period (i.e., a portion of the simulation period) over which the optimization is performed. For example, planning tool  702  may select optimization period  802  for use in the first iteration. Once the optimal resource allocation  810  has been determined, planning tool  702  may select a portion  818  of resource allocation  810  to send to plant dispatch  830 . Portion  818  may be the first b time steps of resource allocation  810 . Planning tool  702  may shift the optimization period  802  forward in time, resulting in optimization period  804 . The amount by which the prediction window is shifted may correspond to the duration of time steps b. 
     Planning tool  702  may repeat the optimization process for optimization period  804  to determine the optimal resource allocation  812 . Planning tool  702  may select a portion  820  of resource allocation  812  to send to plant dispatch  830 . Portion  820  may be the first b time steps of resource allocation  812 . Planning tool  702  may then shift the prediction window forward in time, resulting in optimization period  806 . This process may be repeated for each subsequent optimization period (e.g., optimization periods  806 ,  808 , etc.) to generate updated resource allocations (e.g., resource allocations  814 ,  816 , etc.) and to select portions of each resource allocation (e.g., portions  822 ,  824 ) to send to plant dispatch  830 . Plant dispatch  830  includes the first b time steps  818 - 824  from each of optimization periods  802 - 808 . Once the optimal resource allocation is compiled for the entire simulation period, the results may be sent to reporting applications  730 , results database  728 , and/or client device  722 , as described with reference to  FIG. 7 . 
     Resource Allocation Optimization 
     Referring now to  FIG. 9 , a block diagram illustrating high level optimizer  632  in greater detail is shown, according to an exemplary embodiment. In some embodiments, high level optimizer  632  may be implemented as a component of energy storage controller  506 , as described with reference to  FIGS. 5A and 6A . In other embodiments, high level optimizer  632  may be implemented as a component of controller  552 , as described with reference to  FIGS. 5B and 6B . In other embodiments, high level optimizer  632  may be implemented as a component of planning tool  702 , as described with reference to  FIGS. 7-8 . 
     High level optimizer  632  may receive load and rate predictions from load/rate predictor  622 , incentive predictions from incentive estimator  620 , and subplant curves from low level optimizer  634 . High level optimizer  632  may determine an optimal resource allocation across energy storage system  500  as a function of the load and rate predictions, the incentive predictions, and the subplant curves. The optimal resource allocation may include an amount of each resource purchased from utilities  510 , an amount of each input and output resource of generator subplants  520 , an amount of each resource stored or withdrawn from storage subplants  530 , and/or an amount of each resource sold to energy purchasers  504 . In some embodiments, the optimal resource allocation maximizes the economic value of operating energy storage system  500  while satisfying the predicted loads for the building or campus. 
     High level optimizer  632  can be configured to optimize the utilization of a battery asset, such as battery  108 , battery  306 , and/or electrical energy storage subplant  533 . A battery asset can be used to participate in IBDR programs which yield revenue and to reduce the cost of energy and the cost incurred from peak load contribution charges. High level optimizer  632  can use an optimization algorithm to optimally allocate a battery asset (e.g., by optimally charging and discharging the battery) to maximize its total value. In a planning tool framework, high level optimizer  632  can perform the optimization iteratively to determine optimal battery asset allocation for an entire simulation period (e.g., an entire year), as described with reference to  FIG. 8 . The optimization process can be expanded to include economic load demand response (ELDR) and can account for peak load contribution charges. High level optimizer  632  can allocate the battery asset at each time step (e.g., each hour) over a given horizon such that energy and demand costs are minimized and frequency regulation (FR) revenue maximized. These and other features of high level optimizer  632  are described in detail below. 
     Cost Function 
     Still referring to  FIG. 9 , high level optimizer  632  is shown to include a cost function module  902 . Cost function module  902  can generate a cost function or objective function which represents the total operating cost of a system over a time horizon (e.g., one month, one year, one day, etc.). The system can include any of the systems previously described (e.g., frequency response optimization system  100 , photovoltaic energy system  300 , energy storage system  500 , planning system  700 , etc.) or any other system in which high level optimizer  632  is implemented. In some embodiments, the cost function can be expressed generically using the following equation: 
                 arg   ⁢           ⁢   min     x     ⁢           ⁢     J   ⁡     (   x   )             
where J(x) is defined as follows:
 
     
       
         
           
             
               J 
               ⁡ 
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               
                 
                   ∑ 
                   sources 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     horizon 
                   
                   ⁢ 
                   
                     cost 
                     ⁡ 
                     
                       ( 
                       
                         
                           purchase 
                           
                             resource 
                             , 
                             tme 
                           
                         
                         , 
                         time 
                       
                       ) 
                     
                   
                 
               
               - 
               
                 
                   ∑ 
                   incentives 
                 
                 ⁢ 
                 
                   
                     ∑ 
                     horizon 
                   
                   ⁢ 
                   
                     revenue 
                     ⁡ 
                     
                       ( 
                       ReservationAmount 
                       ) 
                     
                   
                 
               
             
           
         
       
     
     The first term in the previous equation represents the total cost of all resources purchased over the optimization horizon. Resources can include, for example, water, electricity, natural gas, or other types of resources purchased from a utility or other outside entity. The second term in the equation represents the total revenue generated by participating in incentive programs (e.g., IBDR programs) over the optimization horizon. The revenue may be based on the amount of power reserved for participating in the incentive programs. Accordingly, the total cost function represents the total cost of resources purchased minus any revenue generated from participating in incentive programs. 
     High level optimizer  632  can optimize the cost function J(x) subject to the following constraint, which guarantees the balance between resources purchased, produced, discharged, consumed, and requested over the optimization horizon: 
                   ∑   sources     ⁢     purchase     resource   ,   tme         +       ∑   subplants     ⁢     produces   ⁡     (       x     internal   ,   time       ,     x     external   ,   time       ,     v     uncontrolled   ,   time         )         -       ∑   subplants     ⁢     consumes   ⁡     (       x     internal   ,   time       ,     x     external   ,   time       ,     v     uncontrolled   ,   time         )         +       ∑   storages     ⁢       discharges   resources     ⁡     (       x     internal   ,   time       ,     x     external   ,   time         )         -       ∑   sinks     ⁢     requests   resource         =   0                       ⁢       ∀   resources     ,     ∀     time   ∈   horizon               
where x internal,time  and x external,time  are internal and external decision variables and v uncontrolled,time  includes uncontrolled variables.
 
     The first term in the previous equation represents the total amount of each resource (e.g., electricity, water, natural gas, etc.) purchased from each source (e.g., utilities  510 ) over the optimization horizon. The second term represents the total consumption of each resource within the system (e.g., by generator subplants  520 ) over the optimization horizon. The third term represents the total amount of each resource discharged from storage (e.g., storage subplants  530 ) over the optimization horizon. Positive values indicate that the resource is discharged from storage, whereas negative values indicate that the resource is charged or stored. The fourth term represents the total amount of each resource requested by various resource sinks (e.g., building  502 , energy purchasers  504 , or other resource consumers) over the optimization horizon. Accordingly, this constraint ensures that the total amount of each resource purchased, produced, or discharged from storage is equal to the amount of each resource consumed, stored, or provided to the resource sinks. 
     In some embodiments, cost function module  902  separates the purchase cost of one or more resources into multiple terms. For example, cost function module  902  can separate the purchase cost of a resource into a first term corresponding to the cost per unit of the resource purchased (e.g., $/kWh of electricity, $/liter of water, etc.) and a second term corresponding to one or more demand charges. A demand charge is a separate charge on the consumption of a resource which depends on the maximum or peak resource consumption over a given period (i.e., a demand charge period). Cost function module  902  can express the cost function using the following equation: 
               J   ⁡     (   x   )       =         ∑     s   ∈   sources       ⁢     [         ∑     q   ∈     demands   s                 ⁢       w     demand   ,   s   ,   q       ⁢     r     demand   ,   s   ,   q       ⁢       max     i   ∈     demand     s   ,   q           ⁢     (     purchase     s   ,   i       )           +       ∑   horizon     ⁢       r     s   ,   i       ⁢     purchase     s   ,   i             ]       -       ∑   incentives     ⁢       ∑   horizon     ⁢     revenue   ⁡     (   ReservationAmount   )                   
where r demand,s,q  is the qth demand charge associated with the peak demand of the resource provided by source s over the demand charge period, w demand,s,q  is the weight adjustment of the qth demand charge associated with source s, and the max( ) term indicates the maximum amount of the resource purchased from source s at any time step i during the demand charge period. The variable r s,i  indicates the cost per unit of the resource purchased from source s and the variable purchase s,i  indicates the amount of the resource purchased from source s during the ith time step of the optimization period.
 
     In some embodiments, the energy system in which high level optimizer  632  is implemented includes a battery asset (e.g., one or more batteries) configured to store and discharge electricity. If the battery asset is the only type of energy storage, cost function module  902  can simplify the cost function J(x) to the following equation: 
               J   ⁡     (   x   )       =       -       ∑     i   =   k       k   +   h   -   1       ⁢       r     e   i       ⁢     P     bat   i             -       ∑     i   =   k       k   +   h   -   1       ⁢       r     FR   i       ⁢     P     FR   i           +       ∑     i   =   k       k   +   h   -   1       ⁢       r     s   i       ⁢            P     bat   i       -     P     bat     i   -   1                    +       w   d     ⁢     r   d     ⁢       max   i     ⁢     (       -     P     bat   i         +     eLoad   i       )                 
where h is the duration of the optimization horizon, P bat     i    is the amount of power (e.g., kW) discharged from the battery asset during the ith time step of the optimization horizon for use in reducing the amount of power purchased from an electric utility, r e     i    is the price of electricity (e.g., $/kWh) at time step i, P FR,i  is the battery power (e.g., kW) committed to frequency regulation participation during time step i, r FR     i    is the incentive rate (e.g., $/kWh) for participating in frequency regulation during time step i, r d  is the applicable demand charge (e.g., $/kWh) associated with the maximum electricity consumption during the corresponding demand charge period, w d  is a weight adjustment of the demand charge over the horizon, and the max( ) term selects the maximum amount electricity purchased from the electric utility (e.g., kW) during any time step i of the applicable demand charge period.
 
     In the previous expression of the cost function J(x), the first term represents the cost savings resulting from the use of battery power to satisfy the electric demand of the facility relative to the cost which would have been incurred if the electricity were purchased from the electric utility. The second term represents the amount of revenue derived from participating in the frequency regulation program. The third term represents a switching penalty imposed for switching the battery power P bat  between consecutive time steps. The fourth term represents the demand charge associated with the maximum amount of electricity purchased from the electric utility. The amount of electricity purchased may be equal to the difference between the electric load of the facility eLoad i  (i.e., the total amount of electricity required) at time step i and the amount of power discharged from the battery asset P bat     i    at time step i. In a planning tool framework, historical data of the electric load eLoad over the horizon can be provided as a known input. In an operational mode, the electric load eLoad can be predicted for each time step of the optimization period. 
     Optimization Constraints 
     Still referring to  FIG. 9 , high level optimizer  632  is shown to include a power constraints module  904 . Power constraints module  904  may be configured to impose one or more power constraints on the objective function J(x). In some embodiments, power constraints module  904  generates and imposes the following constraints:
 
 P   bat     i     +P   FR     i     ≤P   eff  
 
− P   bat     i     +P   FR     i     ≤P   eff  
 
 P   bat     i     +P   FR     i     ≤e Load i  
 
where P bat     i    is the amount of power discharged from the battery at time step i for use in satisfying electric demand and reducing the demand charge, P FR     i    is the amount of battery power committed to frequency regulation at time step i, P eff  is the effective power available (e.g., the maximum rate at which the battery can be charged or discharged), and eLoad i  is the total electric demand at time step i.
 
     The first two power constraints ensure that the battery is not charged or discharged at a rate that exceeds the maximum battery charge/discharge rate P eff . If the system includes photovoltaic (PV) power generation, the effective power available P eff  can be calculated as follows:
 
 P   eff   =P   rated   −P   PV FirmingReserve  
 
where P rated  is the rated capacity of the battery and P PV FirmingReserve  is the PV firming reserve power. The third power constraint ensures that energy stored in the battery is not sold or exported to the energy grid. In some embodiments, power constraints module  904  can remove the third power constraint if selling energy back to the energy grid is a desired feature or behavior of the system.
 
     Still referring to  FIG. 9 , high level optimizer  632  is shown to include a capacity constraints module  906 . Capacity constraints module  906  may be configured to impose one or more capacity constraints on the objective function J(x). The capacity constraints may be used to relate the battery power P bat  charged or discharged during each time step to the capacity and state-of-charge (SOC) of the battery. The capacity constraints may ensure that the SOC of the battery is maintained within acceptable lower and upper bounds and that sufficient battery capacity is available for frequency regulation. In some embodiments, the lower and upper bounds are based on the battery capacity needed to reserve the amount of power committed to frequency regulation P FR     i    during each time step i. 
     In some embodiments, capacity constraints module  906  generates two sets of capacity constraints. One set of capacity constraints may apply to the boundary condition at the end of each time step i, whereas the other set of capacity constraints may apply to the boundary condition at the beginning of the next time step i+1. For example, if a first amount of battery capacity is reserved for frequency regulation during time step i and a second amount of battery capacity is reserved for frequency regulation during time step i+1, the boundary point between time step i and i+1 may be required to satisfy the capacity constraints for both time step i and time step i+1. This ensures that the decisions made for the power committed to frequency regulation during the current time step i and the next time step i+1 represent a continuous change in the SOC of the battery. 
     In some embodiments, capacity constraints module  906  generates the following capacity constraints: 
             {                   C   a     -       ∑     n   =   k     i     ⁢     P     bat   n           ≤       C   eff     -       C   FR     ⁢     P     FR   i                           C   a     -       ∑     n   =   k     i     ⁢     P     bat   n           ≥       C   FR     ⁢     P     FR   i                 ⁢     
     ⁢     ∀   i       =       k   ⁢           ⁢   …   ⁢           ⁢   k     +   h   -     1   ⁢     
     ⁢     {                   C   a     -       ∑     n   =   k     i     ⁢     P     bat   n           ≤       C   eff     -       C   FR     ⁢     P     FR     i   +   1                             C   a     -       ∑     n   =   k     i     ⁢     P     bat   n           ≥       C   FR     ⁢     P     FR     i   +   1                   ⁢     
     ⁢     ∀   i       =       k   ⁢           ⁢   …   ⁢           ⁢   k     +   h   -   2                     
where C a  is the available battery capacity (e.g., kWh), C FR  is the frequency regulation reserve capacity (e.g., kWh/kW) which translates the amount of battery power committed to frequency regulation P FR  into an amount of energy needed to be reserved, and C eff  is the effective capacity of the battery.
 
     The first set of constraints ensures that the battery capacity at the end of each time step i (i.e., available capacity C a  minus the battery power discharged through time step i) is maintained between the lower capacity bound C FR P FR     i    and the upper capacity bound C eff −C FR P FR     i    for time step i. The lower capacity bound C FR P FR     i    represents the minimum capacity required to reserve P FR     i    for frequency regulation during time step i, whereas the upper capacity bound C eff −C FR P FR     i    represents maximum capacity required to reserve P FR     i    for frequency regulation during time step i. Similarly, the second set of constraints ensures that the battery capacity at the end of each time step i (i.e., available capacity C a  minus the battery power discharged through time step i) is maintained between the lower capacity bound C FR P FR     i+1    and the upper capacity bound C eff −C FR P FR     i+1    for time step i+1. The lower capacity bound C FR P FR     i+1    represents the minimum capacity required to reserve P FR     i+1    for frequency regulation during time step i+1, whereas the upper capacity bound C eff −C FR P FR     i+1    represents maximum capacity required to reserve P FR     i+1    for frequency regulation during time step i+1. 
     In some embodiments, capacity constraints module  906  calculates the effective capacity of the battery C eff  as a percentage of the rated capacity of the battery. For example, if frequency regulation and photovoltaic power generation are both enabled and the SOC control margin is non-zero, capacity constraints module  906  can calculate the effective capacity of the battery C eff  using the following equation:
 
 C   eff =(1− C   FR −2 C   socCM ) C   rated   −C   PV FirmingReserve  
 
where C socCM  is the control margin and C PV FirmingReserve  is the capacity reserved for photovoltaic firming.
 
     Still referring to  FIG. 9 , high level optimizer  632  is shown to include a switching constraints module  908 . Switching constraints module  908  may be configured to impose one or more switching constraints on the cost function J(x). As previously described, the cost function J(x) may include the following switching term: 
               ∑     i   =   k       k   +   h   -   1       ⁢       r     s   i       ⁢            P     bat   i       -     P     bat     i   -   1                        
which functions as a penalty for switching the battery power P bat  between consecutive time steps i and i−1. Notably, the switching term is nonlinear as a result of the absolute value function.
 
     Switching constraints module  908  can impose constraints which represent the nonlinear switching term in a linear format. For example, switching constraints module  908  can introduce an auxiliary switching variable s i  and constrain the auxiliary switching variable to be greater than the difference between the battery power P bat     i    at time step i and the battery power P bat     i−1    at time step i−1, as shown in the following equations:
 
 s   i   &gt;P   bat     i     −P   bat     i−1    
 
 s   i   &gt;P   bat     i−1     −P   bat     i     ∀i=k . . . k+h− 1
 
Switching constraints module  908  can replace the nonlinear switching term in the cost function J(x) with the following linearized term:
 
               ∑     i   =   k       k   +   h   -   1       ⁢       r     s   i       ⁢     s   i             
which can be optimized using any of a variety of linear optimization techniques (e.g., linear programming) subject to the constraints on the auxiliary switching variable s i .
 
Demand Charge Incorporation
 
     Still referring to  FIG. 9 , high level optimizer  632  is shown to include a demand charge module  910 . Demand charge module  910  can be configured to modify the cost function J(x) and the optimization constraints to account for one or more demand charges. As previously described, demand charges are costs imposed by utilities  510  based on the peak consumption of a resource from utilities  510  during various demand charge periods (i.e., the peak amount of the resource purchased from the utility during any time step of the applicable demand charge period). For example, an electric utility may define one or more demand charge periods and may impose a separate demand charge based on the peak electric consumption during each demand charge period. Electric energy storage can help reduce peak consumption by storing electricity in a battery when energy consumption is low and discharging the stored electricity from the battery when energy consumption is high, thereby reducing peak electricity purchased from the utility during any time step of the demand charge period. 
     In some instances, one or more of the resources purchased from utilities  510  are subject to a demand charge or multiple demand charges. There are many types of potential demand charges as there are different types of energy rate structures. The most common energy rate structures are constant pricing, time of use (TOU), and real time pricing (RTP). Each demand charge may be associated with a demand charge period during which the demand charge is active. Demand charge periods can overlap partially or completely with each other and/or with the optimization period. Demand charge periods can include relatively long periods (e.g., monthly, seasonal, annual, etc.) or relatively short periods (e.g., days, hours, etc.). Each of these periods can be divided into several sub-periods including off-peak, partial-peak, and/or on-peak. Some demand charge periods are continuous (e.g., beginning Jan. 1, 2017 and ending Jan. 31, 2017), whereas other demand charge periods are non-continuous (e.g., from 11:00 AM-1:00 PM each day of the month). 
     Over a given optimization period, some demand charges may be active during some time steps that occur within the optimization period and inactive during other time steps that occur during the optimization period. Some demand charges may be active over all the time steps that occur within the optimization period. Some demand charges may apply to some time steps that occur during the optimization period and other time steps that occur outside the optimization period (e.g., before or after the optimization period). In some embodiments, the durations of the demand charge periods are significantly different from the duration of the optimization period. 
     Advantageously, demand charge module  910  may be configured to account for demand charges in the high level optimization process performed by high level optimizer  632 . In some embodiments, demand charge module  910  incorporates demand charges into the optimization problem and the cost function J(x) using demand charge masks and demand charge rate weighting factors. Each demand charge mask may correspond to a particular demand charge and may indicate the time steps during which the corresponding demand charge is active and/or the time steps during which the demand charge is inactive. Each rate weighting factor may also correspond to a particular demand charge and may scale the corresponding demand charge rate to the time scale of the optimization period. 
     As described above, the demand charge term of the cost function J(x) can be expressed as: 
               J   ⁡     (   x   )       =     …   ⁢           ⁢       ∑     s   ∈   sources       ⁢       ∑     q   ∈     demands   s                 ⁢       w     demand   ,   s   ,   q       ⁢     r     demand   ,   s   ,   q       ⁢       max     i   ∈     demand     s   ,   q           ⁢       (     purchase     s   ,   i       )     ⁢           ⁢   …                     
where the max( ) function selects the maximum amount of the resource purchased from source s during any time step i that occurs during the optimization period. However, the demand charge period associated with demand charge q may not cover all of the time steps that occur during the optimization period. In order to apply the demand charge q to only the time steps during which the demand charge q is active, demand charge module  910  can add a demand charge mask to the demand charge term as shown in the following equation:
 
               J   ⁡     (   x   )       =     …   ⁢           ⁢       ∑     s   ∈   sources       ⁢       ∑     q   ∈     demands   s                 ⁢       w     demand   ,   s   ,   q       ⁢     r     demand   ,   s   ,   q       ⁢       max     i   ∈     demand     s   ,   q           ⁢       (       g     s   ,   q   ,   i       ⁢     purchase     s   ,   i         )     ⁢           ⁢   …                     
where g s,q,i  is an element of the demand charge mask.
 
     The demand charge mask may be a logical vector including an element g s,q,i  for each time step i that occurs during the optimization period. Each element g s,q,i  of the demand charge mask may include a binary value (e.g., a one or zero) that indicates whether the demand charge q for source s is active during the corresponding time step i of the optimization period. For example, the element g s,q,i  may have a value of one (i.e., g s,q,i =1) if demand charge q is active during time step i and a value of zero (i.e., g s,q,i =0) if demand charge q is inactive during time step i. An example of a demand charge mask is shown in the following equation:
 
 g   s,q =[0,0,0,1,1,1,1,0,0,0,1,1] T  
 
where g s,q,1 , g s,q,2 , g s,q,3 , g s,q,8 , g s,q,9 , and g s,q,10  have values of zero, whereas g s,q,4 , g s,q,5 , g s,q,6 , g s,q,7 , g s,q,11 , and g s,q,12  have values of one. This indicates that the demand charge q is inactive during time steps i=1, 2, 3, 8, 9, 10 (i.e., g s,q,i =0 ∀i=1, 2, 3, 8, 9, 10) and active during time steps i=4, 5, 6, 7, 11, 12 (i.e., g s,q,i =1 ∀i=4, 5, 6, 7, 11, 12). Accordingly, the term g s,q,i purchase s,i  within the max( ) function may have a value of zero for all time steps during which the demand charge q is inactive. This causes the max( ) function to select the maximum purchase from source s that occurs during only the time steps for which the demand charge q is active.
 
     In some embodiments, demand charge module  910  calculates the weighting factor w demand,s,q  for each demand charge q in the cost function J(x). The weighting factor w demand,s,q  may be a ratio of the number of time steps the corresponding demand charge q is active during the optimization period to the number of time steps the corresponding demand charge q is active in the remaining demand charge period (if any) after the end of the optimization period. For example, demand charge module  910  can calculate the weighting factor w demand,s,q  using the following equation: 
     
       
         
           
             
               w 
               
                 demand 
                 , 
                 s 
                 , 
                 q 
               
             
             = 
             
               
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       k 
                     
                   
                   
                     k 
                     + 
                     h 
                     - 
                     1 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   g 
                   
                     s 
                     , 
                     q 
                     , 
                     i 
                   
                 
               
               
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       
                         k 
                         + 
                         1 
                       
                     
                   
                   
                     period 
                     ⁢ 
                     _ 
                     ⁢ 
                     end 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   g 
                   
                     s 
                     , 
                     q 
                     , 
                     i 
                   
                 
               
             
           
         
       
     
     where the numerator is the summation of the number of time steps the demand charge q is active in the optimization period (i.e., from time step k to time step k+h−1) and the denominator is the number of time steps the demand charge q is active in the portion of the demand charge period that occurs after the optimization period (i.e., from time step k+h to the end of the demand charge period). The following example illustrates how demand charge module  910  can incorporate multiple demand charges into the cost function J(x). In this example, a single source of electricity (e.g., an electric grid) is considered with multiple demand charges applicable to the electricity source (i.e., a=1 . . . N, where N is the total number of demand charges). The system includes a battery asset which can be allocated over the optimization period by charging or discharging the battery during various time steps. Charging the battery increases the amount of electricity purchased from the electric grid, whereas discharging the battery decreases the amount of electricity purchased from the electric grid. 
     Demand charge module  910  can modify the cost function J(x) to account for the N demand charges as shown in the following equation: 
               J   ⁡     (   x   )       =     …   +       w     d   1       ⁢     r     d   1       ⁢       max   i     ⁢     (       g     1   i       ⁡     (       -     P     bat   i         +     eLoad   i       )       )         +   …   +       w     d   q       ⁢     r     d   q       ⁢       max   i     ⁢     (       g     q   i       ⁡     (       -     P     bat   i         +     eLoad   i       )       )         +   …   +       w     d   N       ⁢     r     d   N       ⁢       max   i     ⁢     (       g     N   i       ⁡     (       -     P     bat   i         +     eLoad   i       )       )                 
where the term −P bat     i    eLoad i  represents the total amount of electricity purchased from the electric grid during time step i (i.e., the total electric load eLoad i  minus the power discharged from the battery P bat     i   ). Each demand charge a=1 . . . N can be accounted for separately in the cost function J(x) by including a separate max( ) function for each of the N demand charges. The parameter r d     q    indicates the demand charge rate associated with the qth demand charge (e.g., $/kW) and the weighting factor w d     q    indicates the weight applied to the qth demand charge.
 
     Demand charge module  910  can augment each max( ) function with an element g q     i    of the demand charge mask for the corresponding demand charge. Each demand charge mask may be a logical vector of binary values which indicates whether the corresponding demand charge is active or inactive at each time step i of the optimization period. Accordingly, each max( ) function may select the maximum electricity purchase during only the time steps the corresponding demand charge is active. Each max( ) function can be multiplied by the corresponding demand charge rate r d     q    and the corresponding demand charge weighting factor w d     q    to determine the total demand charge resulting from the battery allocation P bat  over the duration of the optimization period. 
     In some embodiments, demand charge module  910  linearizes the demand charge terms of the cost function J(x) by introducing an auxiliary variable d q  for each demand charge q. In the case of the previous example, this will result in N auxiliary variables d 1  . . . d N  being introduced as decision variables in the cost function J(x). Demand charge module  910  can modify the cost function J(x) to include the linearized demand charge terms as shown in the following equation:
 
 J ( x )= . . . + w   d     1     r   d     1     d   1   + . . . +w   d     q     r   d     q     d   q   + . . . +w   d     N     r   d     N     d   N  
 
     Demand charge module  910  can impose the following constraints on the auxiliary demand charge variables d 1  . . . d N  to ensure that each auxiliary demand charge variable represents the maximum amount of electricity purchased from the electric utility during the applicable demand charge period: 
     
       
         
           
             
               
                 
                   
                     d 
                     1 
                   
                   ≥ 
                   
                     
                       g 
                       
                         1 
                         i 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           - 
                           
                             P 
                             
                               bat 
                               i 
                             
                           
                         
                         + 
                         
                           eLoad 
                           i 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   
                     
                       ∀ 
                       i 
                     
                     = 
                     
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         … 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         k 
                       
                       + 
                       h 
                       - 
                       1 
                     
                   
                   , 
                 
               
               
                 
                   
                     g 
                     
                       1 
                       i 
                     
                   
                   ≠ 
                   0 
                 
               
             
             
               
                 
                     
                 
               
               
                 
                   
                     d 
                     1 
                   
                   ≥ 
                   0 
                 
               
               
                 ⋮ 
               
             
             
               
                 
                   
                     d 
                     q 
                   
                   ≥ 
                   
                     
                       g 
                       
                         q 
                         i 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           - 
                           
                             P 
                             
                               bat 
                               i 
                             
                           
                         
                         + 
                         
                           eLoad 
                           i 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   
                     
                       ∀ 
                       i 
                     
                     = 
                     
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         … 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         k 
                       
                       + 
                       h 
                       - 
                       1 
                     
                   
                   , 
                 
               
               
                 
                   
                     g 
                     
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                   ≠ 
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                     d 
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                 ⋮ 
               
             
             
               
                 
                   
                     d 
                     N 
                   
                   ≥ 
                   
                     
                       g 
                       
                         N 
                         i 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           - 
                           
                             P 
                             
                               bat 
                               i 
                             
                           
                         
                         + 
                         
                           eLoad 
                           i 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   
                     
                       ∀ 
                       i 
                     
                     = 
                     
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         … 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         k 
                       
                       + 
                       h 
                       - 
                       1 
                     
                   
                   , 
                 
               
               
                 
                   
                     g 
                     
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                       i 
                     
                   
                   ≠ 
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                     d 
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     In some embodiments, the number of constraints corresponding to each demand charge q is dependent on how many time steps the demand charge q is active during the optimization period. For example, the number of constraints for the demand charge q may be equal to the number of non-zero elements of the demand charge mask g q . Furthermore, the value of the auxiliary demand charge variable d q  at each iteration of the optimization may act as the lower bound of the value of the auxiliary demand charge variable d q  at the following iteration. 
     Consider the following example of a multiple demand charge structure. In this example, an electric utility imposes three monthly demand charges. The first demand charge is an all-time monthly demand charge of 15.86 $/kWh which applies to all hours within the entire month. The second demand charge is an on-peak monthly demand charge of 1.56 $/kWh which applies each day from 12:00-18:00. The third demand charge is a partial-peak monthly demand charge of 0.53 $/kWh which applies each day from 9:00-12:00 and from 18:00-22:00. 
     For an optimization period of one day and a time step of one hour (i.e., i=1 . . . 24), demand charge module  910  may introduce three auxiliary demand charge variables. The first auxiliary demand charge variable d 1  corresponds to the all-time monthly demand charge; the second auxiliary demand charge variable d 2  corresponds to the on-peak monthly demand charge; and the third auxiliary demand charge variable d 3  corresponds to the partial-peak monthly demand charge. Demand charge module  910  can constrain each auxiliary demand charge variable to be greater than or equal to the maximum electricity purchase during the hours the corresponding demand charge is active, using the inequality constraints described above. 
     Demand charge module  910  can generate a demand charge mask g q  for each of the three demand charges (i.e., q=1 . . . 3), where g q  includes an element for each time step of the optimization period (i.e., g q =[g q     1    . . . g q     24   ]). The three demand charge masks can be defined as follows:
 
 g   1     i   =1 ∀ i= 1 . . . 24
 
 g   2     i   =1 ∀ i= 12 . . . 18
 
 g   3     i   =1 ∀ i= 9 . . . 12,18 . . . 22
 
with all other elements of the demand charge masks equal to zero. In this example, it is evident that more than one demand charge constraint will be active during the hours which overlap with multiple demand charge periods. Also, the weight of each demand charge over the optimization period can vary based on the number of hours the demand charge is active, as previously described.
 
     In some embodiments, demand charge module  910  considers several different demand charge structures when incorporating multiple demand charges into the cost function J(x) and optimization constraints. Demand charge structures can vary from one utility to another, or the utility may offer several demand charge options. In order to incorporate the multiple demand charges within the optimization framework, a generally-applicable framework can be defined as previously described. Demand charge module  910  can translate any demand charge structure into this framework. For example, demand charge module  910  can characterize each demand charge by rates, demand charge period start, demand charge period end, and active hours. Advantageously, this allows demand charge module  910  to incorporate multiple demand charges in a generally-applicable format. 
     The following is another example of how demand charge module  910  can incorporate multiple demand charges into the cost function J(x). Consider, for example, monthly demand charges with all-time, on-peak, partial-peak, and off-peak. In this case, there are four demand charge structures, where each demand charge is characterized by twelve monthly rates, twelve demand charge period start (e.g., beginning of each month), twelve demand charge period end (e.g., end of each month), and hoursActive. The hoursActive is a logical vector where the hours over a year where the demand charge is active are set to one. When running the optimization over a given horizon, demand charge module  910  can implement the applicable demand charges using the hoursActive mask, the relevant period, and the corresponding rate. 
     In the case of an annual demand charge, demand charge module  910  can set the demand charge period start and period end to the beginning and end of a year. For the annual demand charge, demand charge module  910  can apply a single annual rate. The hoursActive demand charge mask can represent the hours during which the demand charge is active. For an annual demand charge, if there is an all-time, on-peak, partial-peak, and/or off-peak, this translates into at most four annual demand charges with the same period start and end, but different hoursActive and different rates. 
     In the case of a seasonal demand charge (e.g., a demand charge for which the maximum peak is determined over the indicated season period), demand charge module  910  can represent the demand charge as an annual demand charge. Demand charge module  910  can set the demand charge period start and end to the beginning and end of a year. Demand charge module  910  can set the hoursActive to one during the hours which belong to the season and to zero otherwise. For a seasonal demand charge, if there is an All-time, on-peak, partial, and/or off-peak, this translates into at most four seasonal demand charges with the same period start and end, but different hoursActive and different rates. 
     In the case of the average of the maximum of current month and the average of the maxima of the eleven previous months, demand charge module  910  can translate the demand charge structure into a monthly demand charge and an annual demand charge. The rate of the monthly demand charge may be half of the given monthly rate and the annual rate may be the sum of given monthly rates divided by two. 
     Incentive Program Incorporation 
     Referring again to  FIG. 9 , high level optimizer  632  is shown to include an incentive program module  912 . Incentive program module  912  may modify the optimization problem to account for revenue from participating in an incentive-based demand response (IBDR) program. IBDR programs may include any type of incentive-based program that provides revenue in exchange for resources (e.g., electric power) or a reduction in a demand for such resources. For example, energy storage system  500  may provide electric power to an energy grid or an independent service operator as part of a frequency response program (e.g., PJM frequency response) or a synchronized reserve market. In a frequency response program, a participant contracts with an electrical supplier to maintain reserve power capacity that can be supplied or removed from an energy grid by tracking a supplied signal. The participant is paid by the amount of power capacity required to maintain in reserve. In other types of IBDR programs, energy storage system  500  may reduce its demand for resources from a utility as part of a load shedding program. It is contemplated that energy storage system  500  may participate in any number and/or type of IBDR programs. 
     In some embodiments, incentive program module  912  modifies the cost function J(x) to include revenue generated from participating in an economic load demand response (ELDR) program. ELDR is a type of IBDR program and similar to frequency regulation. In ELDR, the objective is to maximize the revenue generated by the program, while using the battery to participate in other programs and to perform demand management and energy cost reduction. To account for ELDR program participation, incentive program module  912  can modify the cost function J(x) to include the following term: 
               min       b   i     ,     P     bat   i           ⁢     (     -       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       b   i     ⁢       r     ELDR   i       ⁡     (       adjCBL   i     -     (       eLoad   i     -       P   bat     i       )       )             )           
where b i  is a binary decision variable indicating whether to participate in the ELDR program during time step i, r ELDR     i    is the ELDR incentive rate at which participation is compensated, and adjCBL i  is the symmetric additive adjustment (SAA) on the baseline load. The previous expression can be rewritten as:
 
               min       b   i     ,     P     bat   i           ⁢     (     -       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       b   i     ⁢       r     ELDR   i       ⁡     (         ∑     l   =   1     4     ⁢           ⁢       e   li     4       +       ∑     p   =     m   -   4         m   -   2       ⁢           ⁢       1   3     ⁢     (       eLoad   p     -     P     bat   p       -       ∑     l   =   1     4     ⁢           ⁢       e   lp     4         )         -     (       eLoad   i     -     P     bat   i         )       )             )           
where e li  and e lp  are the electric loads at the lth hour of the operating day.
 
     In some embodiments, incentive program module  912  handles the integration of ELDR into the optimization problem as a bilinear problem with two multiplicative decision variables. In order to linearize the cost function J(x) and customize the ELDR problem to the optimization framework, several assumptions may be made. For example, incentive program module  912  can assume that ELDR participation is only in the real-time market, balancing operating reserve charges and make whole payments are ignored, day-ahead prices are used over the horizon, real-time prices are used in calculating the total revenue from ELDR after the decisions are made by the optimization algorithm, and the decision to participate in ELDR is made in advance and passed to the optimization algorithm based on which the battery asset is allocated. 
     In some embodiments, incentive program module  912  calculates the participation vector b i  as follows: 
               b   i     =     {         1         ∀         i   /     r     DA   i         ≥       NBT   i     ⁢           ⁢   and   ⁢           ⁢   i       ∈   S               0       otherwise                 
where r DA     i    is the hourly day-ahead price at the ith hour, NBT i  is the net benefits test value corresponding to the month to which the corresponding hour belongs, and S is the set of nonevent days. Nonevent days can be determined for the year by choosing to participate every x number of days with the highest day-ahead prices out of y number of days for a given day type. This approach may ensure that there are nonevent days in the 45 days prior to a given event day when calculating the CBL for the event day.
 
     Given these assumptions and the approach taken by incentive program module  912  to determine when to participate in ELDR, incentive program module  912  can adjust the cost function J(x) as follows: 
               J   ⁡     (   x   )       =       -       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       r     e   i       ⁢     P     bat   i             -       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       r     FR   i       ⁢     P     FR   i           +       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       r     s   i       ⁢     s   i         +       w   d     ⁢     r   d     ⁢   d     -       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       b   i     ⁢       r     DA   i       ⁡     (         ∑     p   =     m   -   4         m   -   2       ⁢           ⁢       -     1   3       ⁢     P     bat   p           +     P     bat   i         )                   
where b i  and m are known over a given horizon. The resulting term corresponding to ELDR shows that the rates at the ith participation hour are doubled and those corresponding to the SAA are lowered. This means it is expected that high level optimizer  632  will tend to charge the battery during the SAA hours and discharge the battery during the participation hours. Notably, even though a given hour is set to be an ELDR participation hour, high level optimizer  632  may not decide to allocate any of the battery asset during that hour. This is due to the fact that it may be more beneficial at that instant to participate in another incentive program or to perform demand management.
 
Battery Sizing
 
     Still referring to  FIG. 9 , high level optimizer  632  is shown to include a battery sizing module  914 . Battery sizing module  914  can be configured to determine the optimal battery size (e.g., the energy capacity and/or the inverter power) required to minimize or maximize a given financial metric. Such metrics include SPP, IRR, and/or NPV. The effective capacity C eff  and the effective power P eff  parameters of the battery appear linearly in the capacity constraints and the power constraints, previously described. Accordingly, battery sizing module  914  can treat these parameters as variables to be optimized without affecting its linear property. 
     Although this approach is described primarily with respect to the sizing of a battery system, the approach may be extended to size equipment, such as chillers or heaters. The effective power P eff  and capacity C eff  can be interpreted as a demand of an asset, where the cost of the asset is desired to be minimized. The battery sizing approach determines the optimal battery size, while minimizing energy and demand costs, while maximizing revenue from incentive programs. The costs associated with a battery system include an energy cost, a power cost, and a fixed cost. The fixed cost can be any cost related to the battery system that is independent of energy capacity and power capacity. This cost is only incurred when the energy capacity is nonzero. Therefore, an integer decision variable (e.g., having a value of one or zero) can be used when considering battery sizing in the optimization problem. In some embodiments, battery sizing module  914  solves the optimization problem with mixed integer linear programming. 
     The following example illustrates an implementation of the battery sizing technique in which the cost function J(x) includes a single demand charge and FR participation. Battery sizing module  914  can adjust the cost function as shown in the following equation: 
               J   ⁡     (   x   )       =       -       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       r     e   i       ⁢     P     bat   i             -       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       r     FR   i       ⁢     P     FR   i           +       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       r     s   i       ⁢     s   i         +       w   d     ⁢     r   d     ⁢   d     +       w   EUC     ⁢     r   EUC     ⁢     C   eff       +       w   PUC     ⁢     r   PUC     ⁢     P   eff       +       w   bat     ⁢     r   bat     ⁢   v             
where r EUC  and w EUC  are the energy unit cost (e.g., $/kWh) and corresponding weight adjustment over the optimization period, r PUC  and w PUC  are the power unit cost (e.g., $/kW) and corresponding weight adjustment over the optimization period, r bat  and w bat  are the fixed cost (e.g., $) and its corresponding weight adjustment over the optimization period, and v is a logical decision variable.
 
     Battery sizing module  914  can generate and impose the following constraints on the optimization problem:
 
 P   bat     i     ≤P   eff  
 
− P   bat     i     ≤P   eff  
 
 C   eff   −Mv≤ 0
 
 P   eff   −C   rate   C   eff ≤0
 
where M is set to a large number (e.g., ten thousand) and C rate  is the power to energy ratio. The final constraint can be added to limit the size of the battery based on the application desired. In some embodiments, the effective capacity C eff  and effective power P eff  can be treated as demand charges. The values of the capacities can be carried over as the optimization is solved and the horizon is shifted in time.
 
Peak Load Contribution Incorporation
 
     Still referring to  FIG. 9 , high level optimizer  632  is shown to include a peak load contribution module  916 . Peak load contribution (PLC) is a customer&#39;s contribution to regional demand peaks that occur in geographic area managed by a regional transmission organization (RTO) or independent system operator (ISO) at certain hours within a base period. The regional demand at a given hour may be the summation of the customer&#39;s demand during (i.e., the rate at which the customer purchases electricity or another resource from a utility) as well as the demand of other buildings in the geographic area during that hour. The customer may be billed based on its contribution to the peak regional demand (e.g., $/kW of the customer&#39;s PLC) in addition to the energy consumption charges and demand charges previously described. 
     PLC module  916  can be configured to modify the cost function J(x) to account for a cost associated with the customer&#39;s PLC. By incorporating PLC costs into the cost function J(x), PLC module  916  enables high level optimizer  632  to allocate resource consumption and resource purchases to reduce the customer&#39;s PLC. High level optimizer  632  can reduce PLC costs by shifting the customer&#39;s load to non-peak times or shaving the customer&#39;s peak load. This can be done, for example, by precooling the building during non-peak times, using thermal energy storage, and/or using electrical energy storage such as a battery asset. 
     Accounting for the cost associated with the customer&#39;s PLC can be more difficult than accounting for energy consumption costs and demand charges. Unlike demand charge which is calculated based on the customer&#39;s maximum demand during predetermined demand charge periods, the hours over which PLC is calculated may not be known in advance. The hours of peak regional demand (i.e., the coincidental peak (CP) hours) may not be known until the end of the base period over which PLC is calculated. For example, the CP hours for a given base period (e.g., one year) may be determined by a RTO at the end of the base period based on the demand of all the buildings within the geographic area managed by the RTO during the base period (e.g., by selecting the hours with the highest regional demand). The customer&#39;s PLC may then be determined based on the customer&#39;s demand during the designated CP hours and used to calculate a cost of the customer&#39;s PLC. This cost may then be billed to the customer during the next time period (e.g., the next year), referred to as the billing period. 
     Another difficulty in accounting for PLC costs is that the base period, billing period, CP hours, and other factors used to calculate the PLC cost may differ from one RTO to another. For example, a RTO for the Pennsylvania, Jersey, and Maryland (PJM) geographic area may define the base period (i.e., the peak-setting period) as June 1 st  of year Y to May 31 st  of year Y+1. The billing period (i.e., the delivery period) may be defined as June 1 st  of year Y+1 to May 31 st  of year Y+2. PJM may define the CP hours as the five hours with the highest loads over the five highest peak load days across the PJM geographic region. 
     A customer&#39;s PLC in the PJM region may be calculated as the product of the customer&#39;s average electric load during the five CP hours and a capacity loss factor (CLF), as shown in the following equation: 
               PLC   customer     =     CLF   ×       ∑     i   =   1     5     ⁢           ⁢       eLoad     cp   i       5               
where PLC customer  is the customer&#39;s peak load contribution calculated during year Y, CLF is the capacity loss factor (e.g., CLF=1.05), and eLoad cp     i    is the customer&#39;s electric load (e.g., kW) during the ith CP hour.
 
     The customer&#39;s PLC cost in the PJM region can be calculated as the product of the customer&#39;s PLC during year Y and a PLC rate, as shown in the following equation:
 
PLC cost   =r   PLC ×PLC customer  
 
where PLC cost  is the customer&#39;s PLC charge billed over the delivery year Y+1 (e.g., $) and r PLC  is the rate at which the customer is charged for its PLC (e.g., $/kW).
 
     An additional complication in the PJM region relates to the interaction between PLC costs and economic load demand response (ELDR) revenue. In some embodiments, a customer participating in ELDR in the PJM region during one of the CP hours may be prohibited from reducing its PLC while earning ELDR revenue at the same time. Accordingly, a customer wishing to reduce its load during an assumed CP hour for the purpose of reducing its capacity, transmission, and/or demand charge costs may be restricted from making a bid for the same assumed CP hour in the ELDR market. 
     Another example of an organization which imposes PLC costs is the independent electricity system operator (IESO) in Ontario, Canada. Relative to PJM, IESO may use a different base period, billing period, CP hours, and other factors used to calculate the PLC cost. For example, IESO may define the base period or peak-setting period as May 1 st  of year Y to April 30 th  of year Y+1. The billing period or adjustment period for IESO may be defined as July 1 st  of year Y+1 to June 30 th  of year Y+2. IESO may define the CP hours as the five hours with the highest regional demands across the IESO geographic region. 
     At the end of the base period, IESO may calculate the customer&#39;s peak demand factor (θ PDF ). The peak demand factor may be defined as the ratio of the sum of the customer&#39;s peak demand to the sum of the region-wide demand peaks during the five CP hours, as shown in the following equation: 
               θ   PDF     =           ∑     i   =   1       5     ⁢           ⁢     eLoad     cp   i               ∑     i   =   1       5     ⁢           ⁢     sysLoad     cp   i                 
where sysLoad cp     i    is the region-wide peak load during the ith CP hour and eLoad cp     i    is the customer&#39;s peak load during the ith CP hour.
 
     The customer&#39;s PLC cost in the IESO region is known as a global adjustment (GA) charge. The GA charge may be imposed as a monthly charge during the billing period. In some embodiments, the GA charge is calculated by multiplying the customer&#39;s peak demand factor by the monthly region-wide global adjustment costs, as shown in the following equation:
 
GA cost,month =θ PDF ×GA total,month  
 
where GA cost,month  is the customer&#39;s monthly PLC cost (e.g., $) and GA total,month  is the region-wide global adjustment cost (e.g., $). The value of GA total,month  may be specified by IESO. In some embodiments, GA total,month  has a known value. In other embodiments, the value of GA total,month  may not be known until the end of the base period.
 
     In order to incorporate PLC costs into the cost function J(x) and allocate resource consumption/purchases in advance, PLC module  916  can generate or obtain a projection of the CP hours for an upcoming base period. The projected CP hours can then be used by high level optimizer  632  as an estimate of the actual CP hours. High level optimizer  632  can use the projected CP hours to allocate one or more assets (e.g., a battery, thermal energy storage, HVAC equipment, etc.) to minimize the customer&#39;s demand during the projected CP hours. The technique used by PLC module  916  to incorporate PLC costs into the cost function J(x) is described in greater detail with reference to  FIG. 10 . 
     Subplant Curve Incorporation 
     Still referring to  FIG. 9 , high level optimizer  632  is shown to include a subplant curves module  930 . In the simplest case, it can be assumed that the resource consumption of each subplant is a linear function of the thermal energy load produced by the subplant. However, this assumption may not be true for some subplant equipment, much less for an entire subplant. Subplant curves module  930  may be configured to modify the high level optimization problem to account for subplants that have a nonlinear relationship between resource consumption and load production. 
     Subplant curves module  930  is shown to include a subplant curve updater  932 , a subplant curves database  934 , a subplant curve linearizer  936 , and a subplant curves incorporator  938 . Subplant curve updater  932  may be configured to request subplant curves for each of subplants  520 - 530  from low level optimizer  634 . Each subplant curve may indicate an amount of resource consumption by a particular subplant (e.g., electricity use measured in kW, water use measured in L/s, etc.) as a function of the subplant load. 
     In some embodiments, low level optimizer  634  generates the subplant curves by running the low level optimization process for various combinations of subplant loads and weather conditions to generate multiple data points. Low level optimizer  634  may fit a curve to the data points to generate the subplant curves and provide the subplant curves to subplant curve updater  832 . In other embodiments, low level optimizer  634  provides the data points to subplant curve updater  932  and subplant curve updater  932  generates the subplant curves using the data points. Subplant curve updater  932  may store the subplant curves in subplant curves database  934  for use in the high level optimization process. 
     In some embodiments, the subplant curves are generated by combining efficiency curves for individual devices of a subplant. 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  618 . 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 subplant curve for the entire subplant, the resultant subplant 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. 
     Subplant curve linearizer  936  may be configured to convert the subplant curves into convex curves. A convex curve is a curve for which a line connecting any two points on the curve is always above or along the curve (i.e., not below the curve). Convex curves may be advantageous for use in the high level optimization because they allow for an optimization process that is less computationally expensive relative to an optimization process that uses non-convex functions. Subplant curve linearizer  936  may be configured to break the subplant curves into piecewise linear segments that combine to form a piecewise-defined convex curve. Subplant curve linearizer  936  may store the linearized subplant curves in subplant curves database  934 . 
     Subplant curve incorporator  938  may be configured to modify the high level optimization problem to incorporate the subplant curves into the optimization. In some embodiments, subplant curve incorporator  938  modifies the decision variables to include one or more decision vectors representing the resource consumption of each subplant. Subplant curve incorporator  938  may modify the inequality constraints to ensure that the proper amount of each resource is consumed to serve the predicted thermal energy loads. In some embodiments, subplant curve incorporator  938  formulates inequality constraints that force the resource usage for each resource in the epigraph of the corresponding linearized subplant curve. For example, chiller subplant  522  may have a linearized subplant curve that indicates the electricity use of chiller subplant  522  (i.e., input resource in 1 ) as a function of the cold water production of chiller subplant  522  (i.e., output resource out 1 ). The linearized subplant curve may include a first line segment connecting point [u 1 , Q 1 ] to point [u 2 , Q 2 ], a second line segment connecting point [u 2 , Q 2 ] to point [u 3 , Q 3 ], and a third line segment connecting point [u 3 , Q 3 ] to point [u 4 , Q 4 ]. 
     Subplant curve incorporator  938  may formulate an inequality constraint for each piecewise segment of the subplant curve that constrains the value of the decision variable representing chiller electricity use to be greater than or equal to the amount of electricity use defined by the line segment for the corresponding value of the cold water production. Similar inequality constraints can be formulated for other subplant curves. For example, subplant curve incorporator  938  may generate a set of inequality constraints for the water consumption of chiller subplant  522  using the points defining the linearized subplant curve for the water consumption of chiller subplant  522  as a function of cold water production. In some embodiments, the water consumption of chiller subplant  522  is equal to the cold water production and the linearized subplant curve for water consumption includes a single line segment connecting point [u 5 , Q 5 ] to point [u 6 , Q 6 ]. Subplant curve incorporator  938  may repeat this process for each subplant curve for chiller subplant  522  and for the other subplants of the central plant to define a set of inequality constraints for each subplant curve. 
     The inequality constraints generated by subplant curve incorporator  938  ensure that high level optimizer  632  keeps the resource consumption above all of the line segments of the corresponding subplant curve. In most situations, there is no reason for high level optimizer  632  to choose a resource consumption value that lies above the corresponding subplant curve due to the economic cost associated with resource consumption. High level optimizer  632  can therefore be expected to select resource consumption values that lie on the corresponding subplant curve rather than above it. 
     The exception to this general rule is heat recovery chiller subplant  523 . The equality constraints for heat recovery chiller subplant  523  provide that heat recovery chiller subplant  523  produces hot water at a rate equal to the subplant&#39;s cold water production plus the subplant&#39;s electricity use. The inequality constraints generated by subplant curve incorporator  838  for heat recovery chiller subplant  523  allow high level optimizer  632  to overuse electricity to make more hot water without increasing the amount of cold water production. This behavior is extremely inefficient and only becomes a realistic possibility when the demand for hot water is high and cannot be met using more efficient techniques. However, this is not how heat recovery chiller subplant  523  actually operates. 
     To prevent high level optimizer  632  from overusing electricity, subplant curve incorporator  838  may check whether the calculated amount of electricity use (determined by the optimization algorithm) for heat recovery chiller subplant  523  is above the corresponding subplant curve. In some embodiments, the check is performed after each iteration of the optimization algorithm. If the calculated amount of electricity use for heat recovery chiller subplant  523  is above the subplant curve, subplant curve incorporator  838  may determine that high level optimizer  632  is overusing electricity. In response to a determination that high level optimizer  632  is overusing electricity, subplant curve incorporator  838  may constrain the production of heat recovery chiller subplant  523  at its current value and constrain the electricity use of subplant  523  to the corresponding value on the subplant curve. High level optimizer  632  may then rerun the optimization with the new equality constraints. 
     Peak Load Contribution Module 
     Referring now to  FIG. 10 , a block diagram illustrating peak load contribution (PLC) module  916  in greater detail is shown, according to an exemplary embodiment. In some embodiments, PLC module  916  may be implemented as a component of energy storage controller  506 , as described with reference to  FIGS. 5A and 6A . In other embodiments, PLC module  916  may be implemented as a component of controller  552 , as described with reference to  FIGS. 5B and 6B . In other embodiments, PLC module  916  may be implemented as a component of planning tool  702 , as described with reference to  FIGS. 7-8 . 
     As discussed above, PLC is a customer&#39;s contribution to regional demand peaks that occur in geographic area managed by a regional transmission organization (RTO) or independent system operator (ISO) at certain hours within a base period. The customer may be billed based on its contribution to the peak regional demand (e.g., $/kW of the customer&#39;s PLC) in addition to the energy consumption charges and demand charges previously described. The rate at which the customer is billed (i.e., the peak load contribution rate) may be provided as an input to PLC module  916 . For example, the PLC rate can be received as an input from utilities  510  or from load/rate predictor  622 , as shown in  FIG. 10 . In other embodiments, the PLC rate can be defined by a user (e.g., received as an input parameter from user device  1002 ) or received from other data sources. 
     PLC module  916  can be configured to modify the cost function J(x) to account for a cost associated with the customer&#39;s PLC. By incorporating PLC costs into the cost function J(x), PLC module  916  enables high level optimizer  632  to allocate resource consumption and resource purchases to reduce the customer&#39;s PLC. High level optimizer  632  can reduce PLC costs by shifting the customer&#39;s load to non-peak times or shaving the customer&#39;s peak load. This can be done, for example, by precooling the building during non-peak times, using thermal energy storage, and/or using electrical energy storage such as a battery asset. 
     In some embodiments, the coincidental peak (CP) hours over which a customer&#39;s PLC is determined may not be known in advance. In the case of the IESO (Ontario), the system-wide peal loads during the CP hours also may not be known in advance. In order to minimize a customer&#39;s PLC over a given base period, PLC module  916  may determine or predict the CP hours for an upcoming base period. In some embodiments, the CP hours are defined by a user. For example, the CP hours can be received as an input from user device  1002  or from another data source. An example of a user interface which can be used to specify peak hours is described in greater detail with reference to  FIG. 12 . In other embodiments, the CP hours are predicted by PLC module  916 . 
     Still referring to  FIG. 10 , PLC module  916  is shown to include a historical load database  1004  and a peak hour predictor  1012 . Historical load database  1004  may be configured to store load history data for a particular building and/or other buildings in the same geographic region. The load history data may include a history of regional electric loads for the geographic region in which the building is located. In some embodiments, peak hour predictor  1012  uses the load history data from historical load database to predict the CP hours (i.e., the projected peak hours) for the upcoming base period. In some embodiments, peak hour predictor  1012  uses historical weather data and/or weather forecasts from a weather service (e.g., weather service  604 ) in combination with the load history data to predict the CP hours. The peak hours, whether specified by a user or predicted by peak hour predictor  1012 , can be provided as an input to a peak hour mask generator  1014 . 
     Peak hour mask generator  1014  can be configured to generate a peak hours mask based on the projected peak hours. The peak hours mask may be similar to the demand charge mask described with reference to  FIG. 9 . For example, the peak hours mask may be a logical vector including an element λ i  for each hour i that occurs during the optimization period. Each element λ i  of the peak hours mask may include a binary value (e.g., one or zero, true or false, yes or no, etc.) that indicates whether the corresponding hour i is a peak hour or a non-peak hour. For example, the element λ i  may have a value of one (i.e., λ i =1) if the hour i is a peak hour and a value of zero (i.e., λ i =0) if the hour i is a non-peak hour. An example of a peak hours mask for a twelve hour period is shown in the following equation:
 
λ=[0,0,0,0,1,1,1,0,0,0,1,1] T  
 
where λ 1 , λ 2 , λ 3 , λ 4 , λ 8 , λ 9 , and λ 10  have values of zero, and λ 5 , λ 6 , λ 7 , λ 11 , and λ 12  have values of one. This indicates that hours 1-4 and 8-10 are non-peak hours, whereas hours 5-7 and 11-12 are peak hours.
 
     In some embodiments, the peak hours mask defines a predetermined number of hours as projected peak hours. The predetermined number of hours may be the same as the known number of CP hours in the base period (e.g., five hours). For example, if the base period is known to include five CP hours, the peak hours mask may define five of the hours as peak hours. However, any number of hours can be defined as projected peak hours using the peak hours mask, even if the number of projected peak hours is greater than or less than the known number of CP hours in the base period. By defining a greater number of hours as projected peak hours (i.e., greater than the known number of CP hours in the base period), peak hour mask generator  1014  can increase the probability that the projected peak hours include the actual peak hours, which may be unknown until the end of the base period. Peak hour mask generator  1014  can provide the peak hours mask to PLC term generator  1016  for use in generating one or more PLC terms in the cost function J(x). 
     Still referring to  FIG. 10 , PLC module  916  is shown to include a rate factor generator  1008 . Rate factor generator  1008  can be configured to generate (e.g., determine, calculate, identify, etc.) a rate factor r associated with the PLC charge. In some embodiments, the rate factor r is provided as an input from utilities  510 , load/rate predictor  622 , user device  1002 , or another data source. In other embodiments, the rate factor r is calculated by rate factor generator  1008  as a function of one or more parameters. In some embodiments, the parameters specify whether PLC module  916  is implemented in a PJM region or an IESO region. The parameters can be specified by a user or otherwise received as an input to PLC module  916 . For example, user interface generator  1006  can generate a user interface which prompts the user to select between PJM and IESO (shown in  FIG. 12 ). 
     In the PJM region, the rate factor r may be a function of a PLC rate r PLC  and a capacity loss factor CLF. For example, rate factor generator  1008  can calculate the rate factor for the PJM region r PJM  using the following equation: 
               r   PJM     =       CLF   ×     r   PLC       5           
where r PJM  is the rate factor, CLF is the capacity loss factor, and r PLC  is the PLC rate. The capacity loss factor CLF and the PLC rate r PLC  can be retrieved from a database, predicted by load/rate predictor  622 , received from utilities  510 , or specified by a user. For example, the user may be prompted to specify values for the PLC rate parameter r PLC  and the capacity loss factor parameter CLF in response to selecting the PJM region via the user interface.
 
     In the IESO region, the rate factor r may be a function of the region-wide global adjustment (GA) costs and the region-wide peak load during coincidental peaks. For example, rate factor generator  1008  can calculate the rate factor for the IESO region r IESO  using the following equation: 
               r   IESO     =       GA     total   year         sys   ⁢           ⁢     L   ^     ⁢           ⁢     oad   cp               
where r IESO  is the rate factor, GA total     year    is an estimate of the annual GA costs for the region (e.g., Ontario-wide GA costs), and sys{circumflex over (L)}oad cp  is an estimate of the sum of the region-wide peak loads during coincidental peaks. The annual GA costs GA total     year    and the estimated region-wide peak load sys{circumflex over (L)}oad cp  can be retrieved from a database, predicted by load/rate predictor  622 , received from utilities  510 , or specified by a user. For example, the user may be prompted to specify values for the annual GA costs parameter GA total     year    and the estimated region-wide peak load sys{circumflex over (L)}oad cp  parameter in response to selecting the IESO region via the user interface. Rate factor generator  1008  can provide the rate factor r PJM  or r IESO  to PLC term generator  1016  for use in generating one or more PLC terms in the cost function J(x).
 
     Still referring to  FIG. 10 , PLC module  916  is shown to include a decision variable selector  1010 . Decision variable selector  1010  can be configured to select or identify one or more decision variables to include in the PLC term of the cost function J(x). In some embodiments, the decision variables are defined by high level optimizer  632  and the optimal values of the decision variables are determined as a result of the high level optimization process. However, not all of the decision variables may be relevant to the PLC cost incurred by a customer. Decision variable selector  1010  may identify one or more of the decision variables that have an impact on the PLC cost and may provide an indication of the identified decision variables to PLC term generator  1016 . PLC term generator  1016  can then use the identified decision variables to generate one or more PLC terms in the cost function J(x). 
     In some embodiments, the decision variables selected by decision variable selector  1010  include predicted resource purchases from a utility for each hour of the optimization period (e.g., predicted electricity purchases, predicted water purchases, predicted natural gas purchases, etc.). The amount of electricity purchased from utilities  510  may be defined as the difference between the electric load and the electricity discharged from a battery asset, as shown in the following equation:
 
 e Purchase i   =e Load i   −P   bat     i    
 
where ePurchase i  represents the rate at which electricity is purchased from utilities  510  during hour i (e.g., average kW during the hour), eLoad i  is the predicted electric load during hour i (e.g., kW), and P bat     i    is the rate at which electricity is discharged from a battery asset during hour i (e.g., kW). The electric load eLoad i  can be predicted by load/rate predictor  622 , whereas the battery charge/discharge rate P bat     i    may be optimized by high level optimizer  632 .
 
     In some embodiments, the decision variables selected by decision variable selector  1010  include battery power setpoints. For example, the selected decision variables can include a battery power setpoint P bat     i    for each hour i of the optimization period. The battery power setpoints can subtract from the predicted electric load eLoad i  to determine the predicted electricity purchase ePurchase i  or can be independent decision variables in the cost function J(x). In some embodiments, the selected decision variables include subplant loads for generator subplants  520  and/or storage subplants  530 . For example, the selected decision variables can include optimal operating points for generator subplants  520  (e.g., optimal power consumption, optimal resource production, etc.), optimal thermal energy storage/discharge rates for TES subplants  531 - 532 , and/or optimal battery charge/discharge rates for electrical energy storage subplant  533 . Decision variable selector  1010  can provide the selected decision variables to PLC term generator  1016  for use in generating one or more PLC terms in the cost function J(x). 
     PLC term generator  1016  is shown receiving the peak hours mask from peak hour mask generator  1014 , the rate factor from rate factor predictor  1008 , and the selected decision variables from decision variable selector  1010 . PLC term generator  1016  can use the peak hours mask, the rate factor, and the selected decision variables to generate one or more PLC terms. The PLC terms can be included in the cost function J(x) or independently optimized to reduce the customer&#39;s PLC cost. For example, high level optimizer  632  can be configured to minimize PLC cost by minimizing the customer&#39;s electric cost during projected peak hours, as shown in the following equation: 
             arg   ⁢           ⁢       min     P     bat   i         ⁢     (     r   ⁢       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       λ   i     ⁡     (       eLoad   i     -     P     bat   i         )           )             
where h is the duration of the optimization period, r is the rate factor (e.g., r PJM  or r IESO ), λ i  is an element of the peak hours mask for hour i, eLoad i  is the estimated electric load during hour i, and P bat     i    is the rate at which electricity is discharged from a battery asset during hour i. The term to be minimized can be generated by PLC term generator  1016 . The difference between eLoad i  and P bat     i    represents the amount of electricity purchased from utilities  510  during hour i.
 
     In some embodiments, PLC term generator  1016  generates a PLC term which uses the battery power setpoint P bat     i    as the decision variable. For example, PLC term generator  1016  can generate the following PLC term: 
               -   r     ⁢       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       λ   i     ⁢     P     bat   i                 
where the variables r, λ i , and P bat     i    are the same as previously described. This PLC term can be added to the objective function J(x) as shown in the following equation:
 
               J   ⁡     (   x   )       =     …   -     r   ⁢       ∑     i   =   k       k   +   h   -   1       ⁢           ⁢       λ   i     ⁢     P     bat   i             +   …           
where the PLC term represents an additional cost to be minimized over the optimization period. Discharging the battery (i.e., positive values of P bat     i   ) subtracts from the amount of electricity purchased from utilities  510  and reduces the value of the PLC term (i.e., makes the PLC term more negative), whereas charging the battery (i.e., negative values of P bat     i   ) adds to the amount of electricity purchased from utilities  510  and increases the value of the PLC term.
 
     The term λ i P bat     i    in the previous equation may have a value of zero for hours that are not projected peak hours and a value of P bat     i    for hours that are projected peak hours due to the value of λ i  (e.g., λ i =0 for non-peak hours and λ i =1 for peak hours). This causes the PLC term to only sum the customer&#39;s electric load during projected peak hours. If the customer is participating in an ELDR program, the customer may be prevented from submitting a bid during the hours that are projected peak hours. This implies that if the hourly mask has a value of λ i =1 for a given hour, the corresponding ELDR participation mask may be forced to zero. The modified cost function can be optimized by high level optimizer  632  to determine optimal values for the decision variables. 
     User Interfaces 
     Referring now to  FIGS. 11-14 , several user interfaces  1100 - 1400  which can be generated by user interface generator  1006  are shown, according to an exemplary embodiment.  FIG. 11  illustrates an interface  1100  which allows a user to select one or more optional terms to include in the cost function J(x). Interface  1100  is shown to include a frequency regulation option  1102 , economic load demand response option  1104 , and a peak load contribution option  1106 . Based on the selections made via interface  1100 , cost function module  902  may add or remove one or more terms from the cost function J(x). For example, if the user selects peak load contribution option  1106 , cost function module  902  may trigger PLC module  916  to generate the PLC term and may add the PLC term to the cost function J(x). 
       FIG. 12  illustrates an interface  1200  which allows a user to input values for various parameters used by PLC module  916 . Interface  1200  is shown to include a RTO parameter  1202  which allows the user to select between PJM, IESO, or any other RTO. If PJM is selected via RTO parameter  1202 , the user can input values for PJM PLC parameters  1204  such as the PLC rate r PLC  and the capacity loss factor CLF. If IESO is selected via RTO parameter  1202 , the user can input values for IESO parameters  1206  such as the annual region-wide GA cost estimate GA total     year    and the region-wide CP peaks estimate sys{circumflex over (L)}oad cp . In  FIG. 12 , IESO parameters  1206  are marked (e.g., grayed out, struck through, etc.) to indicate that they will not be used because PJM is selected via RTO parameter  1202 . Interface  1200  also allows the user to specify actual CP hours  1208 , which can be used by PLC module  916  to generate the peak hours mask. 
       FIG. 13  illustrates an interface  1300  which allows a user to enter a percent change in various parameters from year-to-year. Each row of interface  1300  may define a percent change in each of the parameters used by PLC module  916  relative to the values from the previous year. In some embodiments, interface  1300  is presented to a user when high level optimizer  632  is implemented as a component of planning tool  702 . Interface  1300  allows a user to specify a number of years over which the optimization will be performed by planning tool  702 . The user can specify percent change values via interface  1300  to predict cost over an optimization period or planning period spanning several years. 
       FIG. 14  illustrates an interface  1400  which allows a user to define projected CP hours for the duration of the optimization period (e.g., a year). Interface  1400  may include a cell for every hour of the optimization period. In some embodiments, the user defines projected CP hours by setting the value of the corresponding cell to 1 or another affirmative binary value (e.g., yes, true, etc.). For example, interface  1400  is shown to include a value of 1 in the cell at the intersection of January 3 rd  and hour ending 17. This indicates that the 17 th  hour of January 3rd (i.e., from 4:00 PM until 5:00 PM) is a projected CP hour. Similarly, interface  1400  is shown to include a value of 1 in the cell at the intersection of January 9 th  and hour ending 18. This indicates that the 18 th  hour of January 9 th  (i.e., from 5:00 PM until 6:00 PM) is a projected CP hour. The rest of the cells in interface  1400  have values of 0 or another negative binary value (e.g., no, false, etc.), which indicates that the remaining hours are not projected CP hours. In some embodiments, the cells that indicate projected CP hours may be highlighted or otherwise marked (e.g., colored, shaded, bolded, etc.) to allow the user to easily spot such cells in interface  1400 . PLC module  916  can use the values of the cells in interface  1400  to generate the peak hours mask. 
     After all of the pertinent parameter values are entered via interfaces  1100 - 1400 , high level optimizer  632  can perform an optimization to generate battery asset allocations for the duration of the optimization period. For implementations in which high level optimizer  632  is a component of planning tool  702 , high level optimizer  632  can calculate the savings resulting from PLC or GA due to optimal allocation of the battery asset during the projected CP hours for each year in the optimization period or planning period. In the planning tool framework, the savings due to PLC/GA allows the user to see the potential cost savings resulting from the peak load reduction when a battery asset is used. 
     Flow Diagrams 
     Referring now to  FIG. 15 , a flowchart of a process  1500  for allocating energy storage is shown, according to an exemplary embodiment. Process  1500  can be performed by one or more components of frequency response optimization system  100 , photovoltaic energy system  300 , energy storage system  500 , planning system  700 , or any other type of energy storage system. In some embodiments, process  1500  is performed by energy storage controller  506 , as described with reference to  FIGS. 5A and 6A . 
     Process  1500  is shown to include generating a cost function including a peak load contribution (PLC) term (step  1502 ). In some embodiments, the PLC term defines a cost based on a customer&#39;s contribution to regional demand peaks that occur in geographic area managed by a regional transmission organization (RTO) or independent system operator (ISO) at certain hours within a base period. The customer may be billed based on its contribution to the peak regional demand (e.g., $/kW of the customer&#39;s PLC) in addition to energy consumption charges and demand charges. The PLC term can be generated by PLC term generator  1016 , as described with reference to  FIG. 10 . In some embodiments, step  1502  is performed by cost function module  902 . The cost function generated in step  1502  can be the same or similar to any of the cost functions J(x) described with reference to  FIGS. 9-10 . 
     Process  1500  is shown to include generating a peak hours mask defining one or more hours in an optimization period as projected peak hours (step  1504 ). In some embodiments, the peak hours mask is generated by peak hour mask generator  1014 , as described with reference to  FIG. 10 . The peak hour mask may include a vector of binary values. Each of the binary values may correspond to an hour that occurs within the optimization period and may indicate whether the hour is a projected peak hour or not a projected peak hour. 
     Process  1500  is shown to include modifying the cost function by applying the peak hours mask to the PLC term (step  1506 ). Applying the peak hours mask may include multiplying the resource purchase during each hour by the corresponding value of the peak hours mask. In some embodiments, projected peak hours are multiplied by a value of one, whereas projected non-peak hours are multiplied by a value of zero. This limits the PLC term to only the hours designated as projected peak hours in the peak hours mask. 
     Process  1500  is shown to include allocating an optimal amount of energy to store or discharge from energy storage to each of a plurality of time steps in the optimization period (step  1508 ). Step  1508  can include optimizing the modified cost function generated in step  1506  to determine the optimal amounts of energy to store or discharge in each time step. In some embodiments, the energy is electrical energy and the energy storage is a battery asset. In other embodiments, the energy is thermal energy and the energy storage includes thermal energy storage tanks. 
     Process  1500  is shown to include operating the energy storage to store and discharge the optimal amount of energy allocated to each time step during the optimization period (step  1510 ). Storing energy and other types of resources allows for the resources to be purchased at times when the resources are relatively less expensive (e.g., during non-peak electricity hours) and stored for use at times when the resources are relatively more expensive (e.g., during peak electricity hours). Storing resources also allows the resource demand of the building to be shifted in time. For example, resources can be purchased from utilities at times when the demand for heating or cooling is low and immediately converted into hot or cold thermal energy (e.g., by generator subplants  520 ). The thermal energy can be stored and retrieved at times when the demand for heating or cooling is high. This allows energy storage controller  506  to smooth the resource demand of the building and reduces the maximum required capacity of generator subplants  520 . Smoothing the demand also allows energy storage controller  506  to reduce the peak amount of energy purchased from the utility during an applicable demand charge period, which results in a lower demand charge. 
     Referring now to  FIG. 16 , a flowchart of a process  1600  for optimizing energy cost with a PLC charge is shown, according to an exemplary embodiment. Process  1600  can be performed by one or more components of frequency response optimization system  100 , photovoltaic energy system  300 , planning system  700 , energy cost optimization system  550 , or any other type of energy system. In some embodiments, process  1600  is performed by controller  552 , as described with reference to  FIGS. 5B and 6B . 
     Process  1600  is shown to include generating a cost function including a peak load contribution (PLC) term (step  1602 ). In some embodiments, the PLC term defines a cost based on a customer&#39;s contribution to regional demand peaks that occur in geographic area managed by a regional transmission organization (RTO) or independent system operator (ISO) at certain hours within a base period. The customer may be billed based on its contribution to the peak regional demand (e.g., $/kW of the customer&#39;s PLC) in addition to energy consumption charges and demand charges. The PLC term can be generated by PLC term generator  1016 , as described with reference to  FIG. 10 . In some embodiments, step  1602  is performed by cost function module  902 . The cost function generated in step  1602  can be the same or similar to any of the cost functions J(x) described with reference to  FIGS. 9-10 . 
     Process  1600  is shown to include generating a peak hours mask defining one or more hours in an optimization period as projected peak hours (step  1604 ). In some embodiments, the peak hours mask is generated by peak hour mask generator  1014 , as described with reference to  FIG. 10 . The peak hour mask may include a vector of binary values. Each of the binary values may correspond to an hour that occurs within the optimization period and may indicate whether the hour is a projected peak hour or not a projected peak hour. 
     Process  1600  is shown to include modifying the cost function by applying the peak hours mask to the PLC term (step  1606 ). Applying the peak hours mask may include multiplying the resource purchase during each hour by the corresponding value of the peak hours mask. In some embodiments, projected peak hours are multiplied by a value of one, whereas projected non-peak hours are multiplied by a value of zero. This limits the PLC term to only the hours designated as projected peak hours in the peak hours mask. 
     Process  1600  is shown to include allocating an optimal amount of energy to be consumed by HVAC equipment to each of a plurality of time steps in the optimization period (step  1608 ). Step  1608  can include optimizing the modified cost function generated in step  1606  to determine the optimal amounts of energy to consume and/or purchase from the utility in each time step. In some embodiments, the amount of energy consumed by the HVAC equipment may be equal to the amount of energy purchased from the utility. In other embodiments, the amount of energy consumed by the HVAC equipment in a given time step may be less than the amount of energy purchased from the utility. For example, some electricity purchased from the utility may be provided directly to the building to satisfy the building&#39;s electric load, whereas the remaining electricity purchased from the utility may be consumed by the HVAC equipment. In some embodiments, step  1608  includes determining an optimal amount of energy to purchase from the utility as well as the optimal amount of energy to be consumed by the HVAC equipment during each time step in the optimization period. 
     Process  1600  is shown to include operating the HVAC equipment to consume the optimal amount of energy allocated to each time step during the optimization period (step  1610 ). Step  1610  can include operating any of generator subplants  520  to consume purchased resources and/or generate resources that can be provided to building  502 . In some embodiments, step  1610  includes providing a portion of the energy purchased from the utility to the building  502 . For example, the amount of energy purchased in each time step can be used by generator subplants  520  to generate various resources (e.g., hot thermal energy, cold thermal energy, etc.) or provided directly to building  502  (e.g., providing purchased electricity to building  502 ). 
     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 can be reversed or otherwise varied and the nature or number of discrete elements or positions can 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 can be varied or re-sequenced according to alternative embodiments. Other substitutions, modifications, changes, and omissions can 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 can 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. 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 can 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.