Patent Publication Number: US-2023155383-A1

Title: Measurement-based dynamic modeling of an electrical network

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of U.S. application Ser. No. 17/472,377, filed on Sep. 10, 2021, which is a continuation of U.S. application Ser. No. 16/600,356, filed on Oct. 11, 2019, which is a continuation-in-part of and claims the benefit of international Appl. No. PCT/CA2018/050670, filed Jun. 5, 2018, which claims priority to U.S. Appl. No. 62/517,044 filed Jun. 8, 2017. U.S. application Ser. No. 16/600,356 also claims priority to U.S. Appl. No. 62/744,873, filed on Oct. 12, 2018, and claims priority to U.S. Appl. No. 62/744,874, filed on Oct. 12, 2018. The entire contents of each of the foregoing are hereby expressly incorporated by reference herein in their entireties. To the extent appropriate, a claim of priority is made to each of the above disclosed applications. 
    
    
     FIELD 
     This disclosure relates generally to a system and method for characterizing and controlling aspects of an electrical distribution network including feeder line parameters such as voltage, current, real power, reactive power, and phase angle. 
     BACKGROUND 
     Management of an electrical utility distribution system has become a challenging and potentially costly problem. The widespread and growing installation of intermittent generation (e.g. solar, wind) connected to this system is largely responsible for the challenges because this type of generation was never included in the initial design of the distribution system, and voltage variations caused by the intermittency are lowering power quality of electricity delivered to customers coupled to distribution feeder lines of the system, and lowering the reliability of the system itself under some increasingly common circumstances. 
     SUMMARY 
     According to one aspect, there is provided a method for measuring one or more electricity parameters of a distribution feeder line without directly connecting measurement sensors to it except at the substation, comprising: using a micro phasor measurement unit (μPMU) coupled to the distribution feeder line at a first node at the substation, measuring a voltage phasor at the first node; using a μPMU at a customer&#39;s site coupled through the low voltage side of a voltage-reducing transformer at that customer site that has known, measurable, or calculated properties, wherein the transformer is coupled on its high voltage side to the distribution feeder line at a second node by a drop line and to electrical equipment at the customer&#39;s site on its low voltage side, including the customer site μPMU, by a drop line, where the μPMU measures a voltage phasor and a current phasor at the customer site with respect to a specified time reference; determining the transformer and drop line properties from measurements using environmental sensors and from configured parameters; determining a voltage drop impedance (voltage drop phasor) between the second node and the customer site μPMU from the transformer and drop line properties and a current phasor measured by the customer site μPMU; determining a voltage phasor at the second node by adding the voltage phasor measured by the customer site μPMU and the determined voltage drop impedance between the second node and the customer site μPMU; and determining the one or more electricity parameters of the distribution feeder line from the measured voltage phasor at the first node (substation) and the determined voltage phasor at the second node. 
     The electricity parameters include at least real power flow across a first feeder line sector between the first node and second node, reactive power flow across the first feeder line sector, and current on the first feeder line sector. 
     The voltage drop phasor between the second node and the customer site can be determined by the product of the measured current phasor at the customer site measurement unit and the drop impedance phasor between the second node and the customer site measurement unit, wherein the drop impedance phasor is defined by: 
     
       
         
           
             
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     wherein Z 22  represents the drop impedance between the second node and the customer site measurement unit, X 22  represents the drop reactance between the second node and the customer site measurement unit, and R 22  represents the drop resistance between the second node and the customer site measurement unit. X 22  and R 22  are known or calculated from the transformer properties, the drop line properties, and sensor measurements potentially taken at the location, such as outside air temperature and transformer temperature. The additional measurements may be used to increase the accuracy of the drop impedance calculation as necessary depending on the application to which the electrical feeder line parameters will be put. 
     The real power flow across a feeder line sector (a section of feeder between two adjacent nodes) can be defined by: 
     
       
         
           
             
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     Wherein |V i | is the magnitude of the voltage phasor at the upstream node (node i which is electrically closer to the substation), |V j | is the magnitude of the voltage phasor at the downstream node (node j which is electrically further from the substation), δ is the angle of the voltage phasor at the downstream node, and R and X are a resistance and a reactance of the first feeder line sector. 
     According to another aspect, there is provided a system for measuring one or more electricity parameters of a distribution feeder line, comprising: a μPMU located at a substation and coupled to a distribution feeder line at a first node, for measuring a voltage phasor at the first node; at least one customer site μPMU coupled to a low voltage side of a transformer of at least one customer site, for measuring a voltage phasor and a current phasor at the customer site, and wherein the customer site μPMU is coupled to the distribution feeder line at a second node through a drop line to a transformer that is coupled to the distribution feeder line at the second node by a drop line; and a processor and a memory having encoded thereon program code executable by the processor to:
         (i) determine a voltage drop phasor between the second node and the customer site measurement unit from a measured current phasor and a measured voltage phasor at the customer site measurement unit and a known, measured, or calculated drop impedance between the second node and the customer site;   (ii) determine a voltage phasor at the second node by adding a measured voltage phasor at the customer site measurement unit and the determined voltage drop phasor between the second node and the customer site; and   (iii) determine the one or more electricity parameters of the distribution feeder line from a measured voltage phasor at the first node and a determined voltage phasor at the second node.       

     According to another aspect, there is provided a method for locally controlling delivery of electrical power along a distribution feeder of an electricity grid, wherein the distribution feeder comprises a substation and a plurality of nodes, and the substation has time-referenced voltage and current waveform measurement, and the plurality of nodes comprises connections carrying electrical energy to a plurality of customer sites through drop lines connected to voltage transformers, and at least one customer site contains at least one controllable reactive power source and time-referenced voltage and current waveform measurement wherein the time-reference at the customer site is the same or synchronized to the time reference at the substation. The method comprises: determining voltage and current phasors at each of an upstream node and at a downstream node of the feeder sector, where determining the phasors means computing the phasors at the node from phasor measurements on lines connected to the node through at least a transformer and a drop line; setting a target voltage phasor at the downstream node as a value that maintains the real power value at the upstream node, and when total reactive power injected at the upstream and downstream nodes collectively generates all reactive power consumed by the feeder sector, and adjusting operation of the at least one controllable reactive power resource so that the actual voltage magnitude at the downstream node moves towards the magnitude of the target voltage phasor. Adjusting the operation of the at least one controllable reactive power resource can comprise using a reactive power device that increases reactive power to increase the actual voltage magnitude and using a reactive power device that decreases reactive power to decrease the actual voltage magnitude. 
     The plurality of customer sites can comprise at least one controllable real power resource, in which case the method further comprises: adjusting operation of the at least one controllable real power resource so that the actual phase angle at the downstream node moves towards the phase angle of the target voltage phasor. 
     The plurality of customer sites can include a node having an intermittent power generation source, in which case the method further comprises adjusting the target phasor setting at each node after a change in power generation from the intermittent power generation source. 
     According to another aspect, there is provided a system for determining voltage and current phasors and locally controlling delivery of electrical power along a distribution feeder of an electricity grid. The distribution feeder comprises a substation and a plurality of nodes. The substation has time-referenced voltage and current waveform measurement. The nodes comprise connections to customer sites for delivering electrical energy. The connections comprise drop lines and voltage transformers. A pair of adjacent nodes define a feeder sector of the distribution feeder wherein at least one of the nodes has a connection to a customer site with at least one controllable reactive power resource and time-referenced voltage and current waveform measurement wherein the time-reference at the customer site is the same or synchronized to the time reference at the substation. The system comprises: at least two time-referenced voltage and current waveform measurement units with one of the measurement units at the substation and another at a customer site and the measurement units are have a common time reference or are time synchronized with each other; at least one reactive power resource controller communicative with and programmed to control operation of the at least one reactive power resource; and a server computer communicative with the at least two time-referenced voltage and current measurement units and with at least one reactive power resource controller. The server computer comprises a processor and a memory having encoded thereon program code executable by the processor to:
         (i) receive a voltage and current phasor measurements, or their equivalents, from measurement units at a substation and at customer sites coupled to a distribution feeder;   (ii) determine an actual voltage magnitude at an upstream node and at a downstream node of the feeder sector, and a real power value at the upstream node;   (iii) compute a target voltage phasor at the downstream node as a value that improves the performance of the distribution feeder or the equipment attached to it; and   (iv) transmit the target voltage phasor to the at least one reactive power resource controller, such that the at least one reactive power resource controller operates the at least one controllable reactive power resource so that the actual voltage magnitude at the downstream node moves towards the magnitude of the target voltage phasor; and   (v) transmit a target voltage phase to any real power resource controller, such that the real power resource controller operates the at least on real power resource so that the actual voltage phase at the downstream node moves toward the phase of the target voltage phasor.       

     In one embodiment the target voltage phasor for a downstream node on a feeder sector is determined to maintain the real power value at the upstream node of the feeder sector, and when total reactive power injected at the upstream and downstream nodes collectively generates all reactive power consumed by the feeder sector. 
     Non-transitory computer program products (i.e., physically embodied computer program products) are also described that store instructions, which when executed by one or more data processors of one or more computing systems, causes at least one data processor to perform operations herein. Similarly, computer systems are also described that may include one or more data processors and memory coupled to the one or more data processors. The memory may temporarily or permanently store instructions that cause at least one processor to perform one or more of the operations described herein. In addition, methods can be implemented by one or more data processors either within a single computing system or distributed among two or more computing systems. Such computing systems can be connected and can exchange data and/or commands or other instructions or the like via one or more connections, including a connection over a network (e.g. the Internet, a wireless wide area network, a local area network, a wide area network, a wired network, or the like), via a direct connection between one or more of the multiple computing systems, etc. 
     The details of one or more variations of the subject matter described herein are set forth in the accompanying drawings and the description below. Other features and advantages of the subject matter described herein will be apparent from the description and drawings, and from the claims. 
    
    
     
       DRAWINGS 
         FIG.  1    is a block diagram of an apparatus for providing distributed control to resources on a distribution feeder of an electricity grid according to one embodiment. 
         FIG.  2    is a schematic diagram of the components of the system that allow determination of the voltage phasors at the nodes. 
         FIG.  3    is a schematic of a method and apparatus for measuring certain electricity parameters on a distribution feeder line. 
         FIG.  4    is a schematic diagram of measuring electricity parameters on a distribution feeder line using the system shown in  FIG.  2   . 
         FIG.  5    is a flow chart of steps performed in measuring electricity parameters on a distribution feeder. 
         FIG.  6    is a flowchart illustrating execution of a distributed power delivery control program on the central server computer to generate target phasor instructions for each controlled node on the distribution feeder line according to a first embodiment. 
         FIG.  7    is a schematic illustration of a feeder sector and two adjacent nodes of the distribution feeder. 
         FIG.  8    is a flowchart illustrating execution of a distributed power delivery control program on the central server computer to generate target phasor instructions for each controlled node on the distribution feeder line according to a second embodiment. 
         FIG.  9    is a flowchart illustrating execution of a distributed power delivery control program on the central server computer to generate target phasor instructions for each controlled node on the distribution feeder line according to a third embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     Distribution systems have been largely passive in nature, connecting utility substations to customer sites along a distribution feeder line. The utility has generally managed feeder lines (most typically voltage levels on the lines) using measurements taken at the substation and using controls also at the substation. For example, the utility compensates for low voltage at the end of the feeder by adjusting the substation voltage with a load tap changer (LTC), a purely mechanical device on a transformer in the substation that is designed to correct slowly varying voltage and current changes by moving a physical contact thereby adjusting the turns ratio of the transformer. Additionally, capacitor banks installed along the distribution feeder line can provide reactive power support to maintain customer site voltages within defined limits. But these capacitor banks are usually only grossly controllable by at most a switch per capacitor bank. Intermittent generation connected directly to the distribution system (as opposed to the bulk transmission system), especially from renewable sources, has brought a valuable source of energy to the electricity grid, but the intermittency of generation and extremely coarse control have caused issues, largely because the designers of the distribution systems did not foresee these use cases. To address these issues, attempts have been made to measure and control certain parameters along distribution feeder lines. Referring to  FIG.  3   , it is known to use a phasor measurement unit (PMU)  200  to measure parameters of a distribution feeder line  201  by electrically coupling the PMU  200  to a current transformer  202  (CT) and a potential transformer  203  (PT) that are respectively electrically coupled to the feeder line  201 . A PMU is a device which measures the electrical waves on an electricity grid using a shared time source for synchronization with one or more other PMUs. Time synchronization allows synchronized, real-time measurements of multiple remote measurements points in an electricity grid. In particular, time synchronization allows measurement of the relative angles between voltage waveforms and current waveforms at different points on the distribution system. A PMU can be a dedicated device, or PMU functionality can be incorporated into a protective relay or other device. 
     However, measurements of feeder line parameters with PTs and CTs are costly, typically involving two PTs and three CTs per installation, mounted at intervals along the distribution feeder line, and costing in the order of $30,000-$50,000 per installation. Directly connecting PTs and CTs to the feeder lines requires large, expensive devices because the feeder lines typically operate at 10&#39;s of kilovolts (KV) and carry hundreds of amps (A) of current. This cost practically limits the number of measurement points on the distribution feeder line that can be justified. This scarcity of direct measurements of the voltage and current waveforms has limited the control strategies that utilities have been able to use to manage distribution feeder voltage levels, energy losses, and substation equipment wear-and-tear. Historically this hasn&#39;t been a big cost for utilities, but solar power and other intermittent power generation on distribution feeders is driving up all of these costs. 
     In recent years, an increased penetration of intermittent generation in the electricity grid is causing significant control problems. For example, connected solar capacity as low as 10% of peak capacity on a distribution feeder may result in voltage violations that are beyond ANSI-defined limits. This intermittent generation capacity must be balanced with either load or generation adjustments elsewhere on the electricity grid in order to maintain system frequency. Often, a generation facility used for balancing is located a significant distance from a feeder containing the intermittent generation resulting in significant marginal power losses, which in some cases may exceed 30%. 
     Also, intermittent generation tends to cause voltage changes that can result in poor customer power quality and excess wear-and-tear on substation tap changers. These substation tap changers incur increased maintenance needs and failure rates resulting from increased use caused by the intermittency. To avoid conflict between utility voltage management systems that manage tap changes and voltage regulation capability on solar inverters, as well as to avoid potential poor regulation caused by customer equipment, intermittent generator operators have been forbidden from regulating the system voltage (IEEE 1547 and California Rule 21). Many electrical utilities have been using their historical tools of monitoring line voltages at the substation and at a few points on the feeder and installing some in line capability to selectively manage voltage. This old-school approach tends to be slow in response time, and costly for the utility to implement. 
     Some utilities have made the investment in more comprehensive distribution feeder measurement in order to use optimal power flow control (OPF) algorithms. There are OPF algorithms generally known in the art directed to minimizing loss or cost in an electrical distribution system. There exists optimization algorithms and relaxations which consider constraints such as generation limits, transmission thermal limits, bus voltage limits, number of switching operations etc. These algorithms tend to seek to solve the following non-linear power flow equations: 
     
       
         
           
             
               
                 
                   
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     where P k  and Q k  are real power (P) and reactive power (Q) delivered to bus k in a N bus system defined by Y kn  (Ybus matrix of the system) and V k , δ k  is the voltage magnitude and phase at bus k and θ kn  is the angle of the admittance Y kn . 
     Some OPF approaches lead to complex optimization problems requiring high computational resources, which can result in relatively slow reaction by power control systems executing these algorithms. 
     As it is desirable to respond quickly to intermittent power generation in an electrical distribution system, it is desirable to provide a means for controlling power delivery in an electrical distribution system that improves on prior art approaches. 
     Embodiments of the subject matter described herein relate generally to a system and a method for determining certain electricity parameters of a distribution feeder line using two or more micro phasor measurement units (μPMU) electrically coupled to the feeder line, comprising a substation μPMU electrically coupled to a substation distribution bus at a first node on the feeder line, and at least one customer site μPMU electrically coupled to a low voltage side of a transformer at a customer site. The transformer is coupled by a drop line to a second node on the distribution feeder line and the μPMU is coupled by a drop line to the transformer, and using the determined distribution feeder line electricity parameters in a control system that computes target voltage phasors for each node on the feeder line which are then sent to voltage management resource controllers collocated with the substation μPMU and customer site μPMU to control voltage management resources such that the actual voltage phasor at each node moves toward the target voltage phasor for the node. 
     A μPMU is a type of PMU that is a more appropriate for use on a distribution network because of its ability to measure small phases. PMU&#39;s used on transmission networks often measure phase to within 1 degree which is adequate due to the high voltages and low currents on transmission lines. These transmission line PMUs though must be connected at 100&#39;s or even 1000&#39;s of kV, which is expensive and complex. Distribution lines operate at much lower voltages, and customer sites at lower voltages still. These μPMU&#39;s need to measure phase to within 0.01 degrees because of the higher current but are easier to connect because of the lower voltages. Many μPMUs also include a GPS receiver to act as a time reference for all measurements. 
     In some implementations, the method comprises determining a voltage drop between the second μPMU and the second node on the distribution feeder using voltage and current phasor measurements taken by the second μPMU and knowing, measuring, and/or calculating the impedance (e.g., voltage drop phasor) between the second node and the customer site μPMU given the transformer and the drop line properties and optionally environmental measurements such as air temperature and transformer temperature. Then, the voltage phasor (voltage magnitude and phase angle) at the second node can be calculated by adding the determined voltage drop and a customer site voltage phasor as measured by the second μPMU. Using the voltage phasors (the measured voltage phasor at the substation, or the determined voltage phasor at any other node) for two adjacent nodes on the distribution feeder, electricity parameters such as real power, reactive power, and current across a feeder line sector between the adjacent nodes are determined. As a result, the electricity parameters on the distribution feeder line can be measured without the need to install relatively expensive PTs and CTs on the distribution feeder line. 
     The method also comprises using the electricity parameters so determined to compute loss in the distribution feeder sectors and compute target voltage phasors for each node that improve the performance of the distribution line, where improved performance can include reduce energy loss, reduced mechanical wear-and-tear on substation components (such as tap changes), and/or improved power quality delivered to customers. The target voltage phasors can be used to control voltage management resources electrically coupled to the nodes to cause the actual voltage phasor at each node to change in the direction of the target voltage phasor for that node and thereby improve the performance of the distribution feeder. 
     In one embodiment the target voltage phasors are determined to meet a feeder line power loss threshold while reducing the mechanical movement of voltage management equipment at the substation, such as line tap changers. 
     In another embodiment the target voltage phasors are determined to meet the reactive power consumed by each feeder sector using Optimal Power Flow calculations for a balanced, radial distribution system. 
     Throughout the disclosure where a server or computer is referenced it may include one or more servers or computers in communication with each other through one or more networks or communication mediums. Each server and computer generally comprise one or more processors and one or more non-transitory computer readable mediums in communication with each other through one or more networks or communication mediums. The one or more processors may comprise any suitable processing device known in the art, such as, for example, application specific circuits, programmable logic controllers, field programmable gate arrays, microcontrollers, microprocessors, virtual machines, and electronic circuits. The one or more computer readable mediums may comprise any suitable memory devices known in the art, such as, for example, random access memory, flash memory, read only memory, hard disc drives, optical drives and optical drive media, or flash drives. In addition, where a network is referenced it may include one or more suitable networks known in the art, such as, for example, local area networks, wide area networks, intranets, extranets, virtual private networks, and the Internet. Further, where a communication to a device or a direction of a device is referenced it may be communicated over any suitable electronic communication medium and in any suitable format known to in the art, such as, for example, wired or wireless mediums, compressed or uncompressed formats, encrypted or unencrypted formats. 
     System 
     According to one embodiment and referring to  FIG.  1   , a local power control system  10  for providing local control of power delivery along a distribution feeder  11  comprises a central server computer  12 , μPMUs for measuring voltage and current phasors on low voltage connections  20 , controllers  13  for controlling real power resources  15 , controllers  14  for controlling reactive power resources  16  along the feeder (respectively referred to as “real power resource controllers”  13  and “reactive power resource controllers”  14 ), and controllers  18  for controlling utility voltage management devices  23 . The controllers  13 ,  14 ,  18  and the μPMUs  20  are communicative with the server computer  12  over a network  19  such as the Internet, either directly or with the addition of security tunnelling hardware or software; alternatively, the server computer  12  can be fitted with dedicated communication links to the controllers  13 ,  14 ,  18  and μPMUs  20  such as Frame Relay. 
     The distribution feeder  11  comprises a plurality of nodes  17  that connect to customer sites, wherein some customer sites have one or more controllable reactive power resources  16 , some nodes  17  have one or more controllable real power resources  15  and other nodes  17  have one or more non-controllable resources such as an intermittent power generation source  22 . For the sake of simplicity,  FIG.  1    illustrates a first node  17  having one controllable real power resource  15 , a second node  17  having two controllable reactive power resources  16 , namely a reactive power consuming device and a reactive power generating device, and a third node having a non-controllable solar power generation resource  22 . The distribution feeder also comprises a substation comprising a μPMU  24  coupled to the distribution line  11 , one or more tap changers  23  also coupled to the distribution line  11 , and/or other utility voltage management devices. 
     The real and reactive power resources  15 ,  16  are typically located along the node sites  17  at locations remote from the server computer  12 . The real power resources  15  can be electrical generators having capacity to generate power (“generation resource”), electricity-powered devices having capacity to consume a load (“load resource”), and storage devices having capacity to store energy (“storage resource”) for short periods and later release it back to the grid. Reactive power resources  16  that “generate” reactive power include capacitors, STATCOMs, solar (PV) inverters, and reactive power resources  16  that “consume” reactive power include solar (PV) inverters and inductors. 
     In this example embodiment, the controllable real power resources  15  are all load resources, and in particular comprise multiple single-speed water pumps, analog electrical boilers, and analog electrical blowers. These real power-consuming load resources  15  are normally intended to serve a primary process other than providing local power control to a feeder (herein referred to as “process load resources”), and the server computer  12  is configured to operate these load resources  15  to provide local power control only within the operational constraints defined by the original primary processes of these process load resources  15 . For example, the water pumps are used primarily to regulate the water level in a municipal water supply tank, each electrical boiler is used primarily to provide heat and domestic hot water for a building as part of a hybrid electric-gas heating system, and the blowers are used primarily to aerate a waste water treatment tank. 
     A load resource controller  13  that controls the process load resource  15  and communicates with the remotely located server computer  12  is installed at the customer site which is connected to a node  17  through a transformer  21 . A time-referenced μPMU  20  that communicates with the load resource controller  13  or the server computer  12  is also installed at the customer site and is coupled to the drop line that connects to the low voltage side of the transformer  21 . As will be explained in detail below, each load resource controller  13  receives target phasor setpoints from the server computer  12  comprising a target voltage magnitude and a target phase angle, and is programmed to operate the process load resource  15  at a load setpoint that causes the actual phase angle at the node site  17  to move towards the target phase angle, but is also programmed to only operate the process load resource  15  when the load setpoint is within the operational constraints of the process load resource  15  (typically defined by the load resource&#39;s own control system). In other words, the load resource controller  13  is programmed to allow the load resource&#39;s primary control system to override the load resource controller  13  when the operators of the primary process require the process load resource  15  to be used for its primary processes. For example, a municipal water plant operator may require that a water tank be kept between 10% and 90% full of water, and the load resource controller  13  is programmed to allow the server computer  12  to operate the pumps for this tank while the water level is within this range in order to provide local power control to the feeder  11 . However, when the water level in the tank rises to 90% full, the load resource&#39;s control system will be allowed to turn the pumps on, even if the server computer  12  desires the pumps to be kept off. Controllable process load resources  15  which are being used at a given time to serve their primary process are considered to be “off-line” to the server computer  12  and not available to provide local power control; conversely, controllable load resources  15  which are within their primary operational constraints are considered “on-line” and available to be used to provide local power control. “Off-line” load resources  15  are compensated for by the server computer  12  with other “on-line” load resources  15  so that the overall power control functionality is preserved. 
     The load resource controller  13  in this embodiment is a small rugged computer with capability to connect to the Internet  19 , and to connect to the load resources  15  at their respective resource node sites  17 . The connection between the load resource controller  13  and the server computer  12  is achieved through the internet  19 , using a secure means of communications. The load resource controller  13  is connected to the generation resource, load resource, or storage resource using one of a number of methods, including: direct wiring to controllers or governors of the load resource control system; direct connection to the Supervisory Control and Data Acquisition (SCADA) System used to control the process load resource  15  at the resource node site  17 , or connection to the network  19  used by the control system at the node site  17  that controls the load resource  15 . The real power resource controller  13  is also connected to metering devices (not shown) that measure, to standards required by an appropriate regulatory authority, the power being delivered or consumed by the process load resources  15 . 
     The load resource controller  13  may be connected to additional measurement equipment (not shown) as required to ensure that operating constraints can be properly met, by: direct wiring to controllers or measurement equipment; direct connection to the SCADA System used to measure the process load resources  15  at the resource node site  17 ; or connection to a network  19  used by the load resource&#39;s control system at the node site  17  to measure the process load resource  15 . 
     In operation, the load resource controller  13  will receive a target phasor signal from the server computer  12 , directing a change in consumption or generation from one or more of the process load resources  15  at the node site  17 . The real power resource controller  13  will validate the received signal against the operating constraints of the process load resource  15  and clamp the signal if required. The control system of the load resource  15  will send the setpoint signal to the process load resource  15  identified by the server computer  12 , commanding the requested change. 
     At every update interval (e.g. 2 seconds), the μPMU will send voltage and current phasors or magnitudes and zero-crossing timestamps to the server computer  12  directly or to the load resources controller  13  for local use or retransmission to the server computer  12 , and the load resources controller  13  will send a series of signals to the server computer  12 , specifically:
         The status or level of operation of each process load resource  15  at the resource node site  17  (there may be multiple load resources connected to each load resource control system). The load resource controller  13  will aggregate and send a total power signal to the server computer  12 , reflecting the power generated or consumed at that site;   The load resource controller  13  will send a separate signal to the server computer  12  to define the maximum and minimum power levels that are available for the existing process load resources  15  at the resource node site  17 ;   Any additional state information required by the server computer  12  to execute its costing subroutine, as will be described below; and   An indicator if the load resource controller  13  itself, or the SCADA, or the load resource control system, has suspended server computer  12  control, and the current local control setpoint if the server computer  12  control is suspended.       

     The load resource controller  13  will then store the command status and the power levels measured for every resource at the resource node site  17 . Data storage at the local load resource controller  13  should be sufficient to maintain all records for an extended period of time, for example two years. The server computer  12  and the load resource  15  are time-synchronized so that all time-stamped communications between nodes  17  can be properly interpreted by the server computer  12  and the load resource controller  13 . The control and status protocol between the server computer  12  and the load resource controller  13  insures that network issues (e.g. packet loss or reordering), does not cause incorrect control actions. The system will run continuously, with an intended cycle time between the server computer  12  and the load resource controller  13  of about 5-10 seconds, and 5-60 seconds for larger systems. Local storage of data is maintained, time stamped in the meters, in the server computer  12  and in the control system of the load resources  15 . 
     Like the load resource controllers  13 , the reactive power resource controllers  14  are located at each node site  17  of reactive power resources  16  along with a μPMU  20 , and are operable to control the operation of those reactive power resources  16 . The reactive power resource controller  14  has the same hardware design as the load resource controller, and is programmed to control the reactive power resources  16 . Similarly, the utility resource controller  18  is of the same or similar hardware design as the load and reactive power resource controllers  13 ,  14  with programming adapted to control the utility voltage management devices  23 . 
     The server computer  12  is a redundant server computer system, equipped with a reliable operating system such as Linux, real time software, and a long-term database. The server computer  12  is desirably installed at a secure location, protected from unauthorized physical access, where there is a reliable connection to the internet and a backed-up supply of electricity. For example, the server computer  12  may be installed at the substation. The server computer  12  may be a system that is spread across multiple hardware chassis either to aggregate sufficient processing capability, or to provide redundancy in the event of failure, or both. One chassis can operate as the primary server computer  12 , and another as a backup server computer  12 . Each chassis can run a multi-core capable operating system. The server computer  12  runs a measurement program to process the measurements from the μPMUs and a control program to generate voltage target phasors for the resource controllers. 
     According to another example embodiment, the local power control system  10  comprises controllers  14  for controlling reactive power resources but does not comprise controllers for controlling real power resources. As will be explained in more detail below, each reactive power resource controller  14  receives target voltage phasor setpoints from the server computer  12 , and is programmed to operate the reactive power resource  16  at a setpoint that causes the actual voltage magnitude at the node  17  to move towards the magnitude of the target voltage phasor setpoints. Since this alternative embodiment does not involve controlling real power resources, the phase angle along the distribution line is not controlled. By controlling the reactive power, it is possible to influence the phase angle difference in a system that has an X/R ratio of line impedance of around 1. 
     According to an alternative embodiment the measurement program and the control program are run on different computers. 
     Measurement Program 
     Referring to  FIG.  2   , and according to one example embodiment, the feeder line measurement part of the system,  100 , comprises a programmed computer  102  and multiple μPMUs, namely a substation μPMU  104  coupled to a substation  110  on a distribution feeder line  108 , and one or more additional μPMUs (“customer site μPMU”) which are each coupled to a respective transformer at a customer site, and wherein each transformer is coupled to the distribution feeder line  108  by a respective drop line  109  and  111  and is coupled to the customer site by a respective drop line  116  and  117 . In this embodiment, there are two customer site μPMUs, namely: a first μPMU  106  at a residential site  112  (“residential PMU”) coupled to a transformer  114  on a utility pole at that site (“residential transformer”), which in turn is coupled to the feeder line  108  by a residential drop line  109 ; and, a second μPMU  107  at a commercial site  113  (“commercial μPMU”) coupled to a transformer  115  in a building on the commercial site  113  which in turn is coupled to the feeder line  108  by a commercial drop line  111 . While this embodiment shows two customer site μPMUs at two customer sites, the system  100  can comprise as few as one customer site μPMU at one customer site, or more than two customer site μPMUs at more than two customer sites. 
     The μPMUs  104 ,  106 ,  107  are devices that are capable of measuring electricity parameters including voltage (V), current (I), real power (P) and reactive power (Q) in each phase. In addition to measuring these electricity parameters, the μPMUs  104 ,  106 ,  107  can measure the difference in the voltage phase angle between two points on a distribution feeder line  108  by referencing a common or shared time reference or can measure the amplitude and zero-crossing timestamps of voltage and current waveforms. While μPMUs are used in this embodiment, other types of PMU as known in the art can be substituted. 
     Referring now to  FIGS.  2  and  3    together, substation  110  is considered the start of the distribution feeder line  108  and is herein referred to as the “Node 1” of the distribution feeder line  108 . The substation μPMU  104  is installed in the substation  110 , and thus provides the capability for power, voltage and current measurements at Node  1 , wherein measurements at Node  1  will be denoted by the subscript “1”. The connection point of residential drop line  109  to the distribution feeder line  108  is herein referred to as “Node 2” of the distribution feeder line  108  wherein measurements at Node  2  will be denoted by the subscript “2”, and the connection point of the commercial drop line  111  to the distribution feeder line  108  is herein referred to as “Node 3” of the distribution feeder line  108  wherein measurements at Node  3  will be denoted by the subscript “3”. Although only three nodes are shown in  FIGS.  2  and  3    for the sake of simplicity, the system  100  can comprises additional nodes with corresponding drop lines and customer sites. 
     The substation μPMU  104  will provide the reference angle for the distribution feeder line  108 . The phase angle of the voltage at the substation  110  is herein defined as δ 1 =0. The current in the feeder line  108  may have an angle ϕ 12 , representing the power factor (PF) of a feeder line sector between the first and second nodes. The known electricity parameters at the substation (Node  1 ) are thus: voltage magnitude |V 1 |, voltage angle (which is 0 by definition, i.e. δ 1 =0), current magnitude and phase angle (|I 12 |, ϕ 12 ), and real and reactive power leaving Node  1  (substation  110 ) for Node  2  (P 12  and Q 12 ). 
     The portion of the distribution feeder line  108  between Node  1  and Node  2  is herein referred to as Feeder Sector 1, and the portion of the distribution feeder line  108  between Node  2  and Node  3  is herein referred to Feeder Sector 2. 
     At Node  2 , the electricity parameters are referenced as: voltage (|V 2 |,δ 2 ), current along Feeder Sector 1 (|I 21 |,ϕ 21 ), and real and reactive power leaving Node  2  for Node  1  (P 21  and Q 21 ). None of these parameters are measured directly at Node  2 , but can be determined from measurements taken by the substation and residential μPMUs  104 ,  106  respectively as will be discussed in detail below. Similarly, the electricity parameters at Node  3  are referenced as: voltage (|V 3 |,δ 3 ), current along Feeder Sector 2 (|I 32 |,ϕ 32 ), and real and reactive power leaving Node  3  for Node  2  (P 32  and Q 32 ); none of these parameters are measured directly at Node  3 , but can be determined from measurements taken by the Node  1  substation  110  and commercial μPMUs  104 ,  107  respectively in a similar manner as used to determine the parameters at Node  2 . 
     At the residential site  112 , power is delivered from the feeder line  108  to a single customer that sees a distribution voltage drop ΔVacross the residential drop line  109 , the residential transformer  114 , and drop line  116 . The electricity parameter values at the residential site are herein denoted by the subscript “d2”. The residential μPMU  106  is connected on the low voltage side of the residential transformer  114 , and is configured to measure the consumption at the residential site  112 . This measurement may involve measurement of the voltage phasor (|V d2 |,δ d2 ) and the current phasor (| d2 |,ϕ d2 ) directly if the residential μPMU  106  has access to the timing reference measurements of the substation μPMU  104  if the residential μPMU  106  is communicative with substation μPMU  104 . The measurement alternatively may involve measurement of the voltage magnitude and zero-crossing timestamps and the current magnitude and current zero-crossing timestamps if the residential μPMU  106  is not communicative directly with the substation μPMU  104 . In this latter case, the calculation of the voltage phasor (|V d2 |,δ d2 ) and the current phasor (| d2 |,ϕ d2 ) at the residential site  112  will be performed in the computer  102  which is communicative with both the residential μPMU  106  and the substation μPMU  104 . Of note, the voltage and current magnitudes are measured at the residential site  112 , and the voltage and current angles δ d2  and ϕ d2  are measured or calculated against the reference of the voltage phasor at the substation μPMU  104 . Consequently, the real power P d2 , reactive power Q d2 , voltage V d2  and current I d2  values from the residential μPMU  106  readings measure the consumption at the residential site  112 , and not the electricity parameter values in the distribution feeder line  108 . Similarly, the electricity parameter values at the commercial site  113  are herein denoted by the subscript “d3”, and the commercial μPMU  107  can measure the consumption at the commercial site  113  to obtain and parameter values P d3 , Q d3 , |V d3 |, δ d3 , |I d3 |, and ϕ d3 . 
     The computer  102  in this embodiment is a computer server having a processor and a memory having encoded thereon a feeder line parameter measurement program executable by the processor. The computer  102  is communicative with the substation μPMU  104  and with the customer site μPMUs  106 ,  107  via a communications network in a manner known in the art. While in this embodiment the computer is shown at a location remote from and directly communicative with each of the substation μPMU  104  and customer site μPMUs  106 ,  107 , the computer  102  and μPMUs  104 ,  106 ,  107  can communicate in a different manner in the communication network as would be known to one skilled in the art. For example, the computer  102  can comprise multiple computer servers each directly coupled to a customer site μPMU, and each customer site μPMU can be configured to communicate with the substation μPMU to receive measurement data from the substation μPMU, such that each computer server would receive the substation and customer site measurement data directly via its connection to the customer site μPMU. In another example, the functionality of computer  102  may be performed by one or more of the μPMUs. In another example, the μPMUs  104 ,  106 ,  107  may communicate with the server computer  12  through gateways, such as local resource controllers ( FIGS.  1     13 ,  14 , and  18 ). 
     The feeder line parameter measurement program when executed by the computer  102  can determine the phasor on the distribution feeder line  108  at the connection points of each drop line. i.e. a second node phasor (|V 2 |,δ 2 ) at Node  2 , and a third node phasor (|V 3 |,δ 3 ) at Node  3 . As described previously, it may determine intermediate values, such as (|V d2 |,δ d2 ). (|I d2 |, ϕ d2 ), (|V d3 |,δ d3 ). (|I d3 |,ϕ d3 ), first in order to calculate the second node phasor and third node phasor, as required. With the determined phasors at Node  2  and Node  3 , the feeder line parameter measurement program can calculate the parameter values in a first sector of the feeder line  108  connecting the substation  110  to the second node (Feeder Sector 1) and the parameter values in a second sector of the feeder line  108  connecting the second node to the third node (Feeder sector 2). For each subsequent node (not shown), the voltages and angles from the previous node can be used to calculate the power flowing in the feeder line  108  between the previous and current nodes. 
     The steps carried out by the feeder line parameter measurement program to calculate the feeder line parameters are now described with reference to the diagram shown in  FIG.  4    and the flowchart shown in  FIG.  5   . The feeder line parameter measurement program uses measurement data taken by the substation μPMU  104  and the customer site μPMUs  106 ,  107 . For the sake of simplicity, the execution of the feeder line parameter measurement program is described with reference to the measurements taken by the residential μPMU  106  to determine the feeder line parameter values, with the understanding that the steps to determine the feeder line parameter values using the commercial μPMU  107  are similar. 
     The voltage and current phasors at the residential site (|V d2 |,δ d2 ), (|I d2 |,ϕ d2 ) are measured by the residential μPMU  106  or calculated by the computer  102 . The impedance between the residential site  112  (i.e. at the location of the residential μPMU  106 ) and Node  2  is also used as an input by the feeder line parameter measurement program, and can be represented as: 
     
       
      
       Z 
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       22 
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       22  
      
     
     wherein a resistance R 22  and a reactance X 22  (which includes the resistances and reactances of the drop lines  109 ,  116  and transformer  114 ) are: known values from the properties of the drop lines  109 ,  116  and transformer  114  at the residential site  112 , measured by sensors at the residential site  112 , or calculated from known properties and sensor measurements. 
     The voltage drop phasor (expressed as vector ΔV d2 ) from the second node (Node  2 ) to the residential site  112  is calculated using (step  401 ): 
     
       
         
           
             
               
                 
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     wherein current magnitude  122  is the current along the residential drop line  109  and is assumed to be equal to the measured residential site current magnitude I d2 , or calculated from the residential site current magnitude from a known or calculated current loss for the residential transformer  114  (step  402 ). 
     The feeder line parameter measurement program then calculates the voltage phasor at Node  2  (|V 2 |,δ 2 ) by adding the measured residential site voltage and the calculated voltage drop between Node  2  and the residential  112  site (Step  403 ), which is expressed mathematically as: 
       (| V   2 |,δ 2 )=(| V   d2 |,δ d2 )+Δ V   d2 &lt;Δδ d2  
 
     Once the voltage phasor at Node  1  is measured and the voltage phasor at Node  2  is determined, and the distribution feeder line impedance is known, all other electricity parameter values along the distribution feeder line  108  can be calculated by the feeder line parameter measurement program. In particular, the real power flow P ij  (in Watts) and reactive power Q ij  (in VARs) flow between two nodes i, j, in the feeder line  108  are calculated by the feeder line parameter measurement program (step  404  for Node  2  or  405  for Node  3  or greater) by applying the following general equations: 
     
       
         
           
             
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     wherein R and X are the resistance and reactance of the feeder line sector between nodes i and j, respectively and here i=1 and j=2. 
     The feeder line parameter measurement program then calculates the apparent power S ij  between Nodes i and j, in kilo volt-amps kVA (step  406 ), by applying the following equations: 
         kVA   ij   =S   ij =√{square root over ( P   ij   2   +Q   ij   2 )}
 
     And line currents are calculated by: 
     
       
         
           
             
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     wherein I ij * is the complex conjugate of I ij . 
     The feeder line parameter measurement program then repeats the calculations for each subsequent node, based on the voltages at the current and subsequent nodes, and the angular difference between them. By executing this program, the system  100  provides a full set of parameter values for the operation of the feeder line  108 . 
     Where the feeder line is mixed with single phase loads, the method may be applied to the single phase sectors of the feeder line. 
     While the illustrative embodiments of the present invention are described in detail, it is not the intention of the applicant to restrict or in any way limit the scope of the appended claims to such detail. Additional advantages and modifications within the scope of the appended claims will readily be apparent to those skilled in the art. The invention in its broader aspects is therefore not limited to the specific details, representative apparatus and methods, and illustrative examples shown and described. Accordingly, departures may be made from such details without departing from the spirit or scope of the general concept. 
     Power Delivery Control Program 
     Referring to  FIG.  1   , the server computer  12  has a processor and a memory on which is stored a power delivery control program which when executed by the processor processes the distribution line electrical parameters and controls the utility voltage management devices  23  and the real and reactive power resources  15 ,  16 . 
     The power delivery control program controls the voltage along the feeder line by controlling the operation of the reactive power resources  16  and the utility voltage management devices  23  (collectively “voltage management devices”) and controls the phase angle S along the feeder line by controlling the operation of the real power resources  15 . 
     Generally, the voltage magnitude between two adjacent nodes is similar, and the phase angle will generally be small. Under these conditions, the real power flow in the feeder line will tend to track the phase angle δ and the reactive power flow will track the difference in voltage between the sending and receiving nodes. Using these principles and as will be described in detail below, the power control program is able to determine the target phasor (voltage magnitude and phase angle) at each node that is required to deliver power to the feeder line at a defined feeder line power loss. The power control program also includes a voltage management device optimization module that can preferentially select certain voltage management devices over others, by assigning an operating cost to each voltage management device. In particular, the voltage management device optimization module assigns a relatively high operating cost to the substation tap changer  23  compared to the reactive power resources  16  in order to minimize the use of the tap changer  23  when controlling the voltage magnitude at each node along the feeder line. 
     The power control program also includes a load resource management module which comprises program code for determining which process load resources are available to provide power control, and also to select a cost-effective combination of available process load resources to provide this control. 
     According to one embodiment, the power delivery control program controls the voltage management resources to deliver a required amount of power to the feeder line  11  at an allowable voltage, while ensuring that this power is delivered with a feeder line power loss that is below a defined threshold and while minimizing the operation of the utility voltage management devices  23  and in particular, the substation tap changer. 
     Referring now to  FIG.  5   , the measurement program determines the electrical parameters of the distribution line. 
     Referring now to  FIG.  6   , the power control program when executed determines the target phasor of each node that is required to deliver power to the feeder line under the desired conditions. e.g., with minimal substation tap changer operation and minimal feeder line power loss, and selects the real and reactive power resources to meet these determined target phasors. The total real and reactive power loss on the feeder line  11  is calculated (step  500 ), by summing the real loss (i.e. I L   2 R L ) and reactive power loss (i.e. I L   2 X L ) on each line sector between the nodes. 
     The real and reactive power being removed or injected at each node  17  is calculated using the measured actual phasor at the nodes  17  (step  502 ). The real and reactive power must sum to zero at each node  17 , so power in from an upstream location is equal to power removed at the node  17  plus power flowing down the next line sector. The real and reactive power at the sending (upstream) end of the line sector are determined by: 
     
       
         
           
             
               
                 
                   
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                           ⁢ 
                              
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                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
       
         
           
             	 
             
               and 
               , 
             
           
         
       
       
         
           
             
               
                 
                   
                     Reactive 
                     ⁢ 
                         
                     Power 
                     ⁢ 
                         
                     
                       ( 
                       VARs 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           R 
                           2 
                         
                         + 
                         
                           X 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
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                           ⁢ 
                              
                           sin 
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                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     wherein VS and VR are the sending and receiving end voltages of the line sector. 
     The real power and reactive power at the receiving (downstream) end of the line sector is: 
     
       
         
           
             
               
                 
                   
                     Real 
                     ⁢ 
                         
                     Power 
                     ⁢ 
                         
                     
                       ( 
                       Watts 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           R 
                           2 
                         
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                           X 
                           2 
                         
                       
                     
                     ⁢ 
                     
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                           R 
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                           ⁢ 
                              
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                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
       
         
           
             	 
             and 
           
         
       
       
         
           
             
               
                 
                   
                     Reactive 
                     ⁢ 
                         
                     Power 
                     ⁢ 
                         
                     
                       ( 
                       VARs 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           R 
                           2 
                         
                         + 
                         
                           X 
                           2 
                         
                       
                     
                     ⁢ 
                     
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                           ⁢ 
                              
                           sin 
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                           δ 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Of note, the real and reactive power going in at the sending end of the line sector is different than the power flowing out of the receiving end of the line sector because of real and reactive power loss in the line sector. In other words. Power In−Line Loss=Power Out for both real and reactive power on each line sector. 
     Next, the minimal power loss when delivering the required power to the feeder line is calculated (step  502 ). To minimize power loss along the feeder line  11 , the power delivery should be controlled to minimize current flow along the feeder line  11  while still meeting the power delivery and voltage requirements. This can be determined by selecting the power flow leaving each node (sending node) so that the real power leaving the sending node and going down the line sector is only enough to provide for the load at the adjacent downstream node (receiving node) and the line loss along the line sector between the sending and receiving nodes. Also, the reactive power on the line sector leaving each node should be zero; in other words, a reactive power resource at a node should inject only enough reactive power to supply the reactive power loss from the upstream line sector so that the outgoing line sector at the node carries no reactive power. Based on these principles, the allowable feeder line power loss threshold is calculated by:
         a) starting at the end of the feeder line  11 , calculate the current I L , real power loss and reactive power loss on the last line sector using the measurements of the load and actual phasor at the last line sector;   b) calculate the real power delivered by the next upstream line sector to be the real power required by nodes on the last line sector plus the real power loss on that line sector, and   c) calculate the reactive power required from a reactive power resource at the upstream node of the last line sector to be the reactive power required to replace the reactive power used by the last line sector.       

     Steps (a) to (c) are repeated at each upstream line sector for the entire feeder line  11  (i.e. back to the substation). The minimal total feeder line power loss is determined to be the sum of all of the determined line sector power losses. 
     Now that the reactive and real power at each node  17  to achieve the minimum total feeder line power loss have been determined, it is possible to determine the voltage magnitude and phase angle settings at each node  17  from equations (1) to (4). That is, equations (1) to (4) can be solved for |V S |, |V R | and δ for each line sector L of the feeder line  11 , working upwards from the last line sector and to the first line sector coupled to the substation. 
     An operational constraint is then assigned that represents the maximum allowable feeder line loss an operator will permit when controlling delivery of power to the feeder line  11  (herein referred to as “allowable feeder line power loss threshold”). The allowable feeder line power loss threshold can be set as the minimum feeder line power loss, in which case the determined voltage magnitudes and phase angles represent the target phasor for each node that must be met in order to achieve the minimum total feeder line power loss (step  503 ). Alternatively, the allowable feeder line power loss threshold can be a higher value as selected by an operator, in which case the target voltage magnitude and target phase angle are adjusted accordingly. 
     Next, the voltage management device optimization module is executed to select the voltage management devices that will be used to meet the target voltage magnitude at each node (step  504 ). As noted previously, the voltage management devices include the controllable reactive power resources  16  (i.e. capacitors, inductors, voltage inverters) at node sites  17  and the utility voltage management devices  23  at the substation, and these devices  16 ,  23  can be used to control the reactive power flows at each node  17  and the substation. As is well understood by those skilled in the at, capacitive reactive power resources  16  increase reactive power and consequently increase voltage magnitude at a node  17  and can be selected when the actual voltage magnitude is lower than the target voltage magnitude. Conversely, inductive reactive power resources  16  consume reactive power and consequently decrease voltage magnitude at a node  17  and thus can be selected when the actual voltage magnitude is higher than the target voltage magnitude. 
     The voltage management device optimization module determines which voltage management devices  16 ,  23  are available to achieve the target voltage magnitudes at each node  17 , selects a cost effective combination of available voltage management devices  16 ,  23 , then sends control signals to controllers of those selected voltage management devices  16 ,  23  to operate those devices accordingly. The selected combination can be the combination that provides the lowest operating cost, or any one of a number of combinations with have an operating cost below a selected threshold. Because not all nodes  17  may have a reactive power resource  16  that can be controlled by the system  10 , it may not be possible to achieve the target voltage magnitudes at each node  17 , in which case, the power delivery control program selects the available reactive power resources  16 ,  23  to come as close as possible to the target voltage magnitude. 
     Because frequent use of the substation tap changer is generally undesirable, the voltage management device optimization module assigns a comparatively higher operating cost to using the utility voltage management devices  23  and a comparatively lower operating cost to using the capacitors, inductors, and invertors  16  at the customer sites connected to nodes  17 . The cost function for each reactive device  16 ,  23  is assigned based on actual cost. For example, a smart inverter can react quickly with little cost, and as a result is assigned a relatively low operating cost. Conversely, resources such as transformer tap changers that have life limits based on operations, are assigned a relatively high operating cost. Once the operating cost is assigned to each voltage management device  16 ,  23 , a costing subroutine is executed to determine the available voltage management devices  16 ,  23  and their respective voltage settings. 
     Next, the power delivery control program executes a process load resource management module to select the real power resources  15  that will be used to meet the target phase angle at each node  17  (step  505 ). As noted previously, the real power resources  15  include controllable process load resources  15  that serve a primary process, and can be used by the system  10  to control phase angles along the feeder line  11  provided that the usage does not exceed the operational constraints dictated by the load resource&#39;s primary process. The use and selection of such process load resources  15  to provide load is disclosed in co-owned PCT application publication no. WO 2011/085477, and is hereby incorporated by reference. 
     The process load resource management module includes program code which determines which process load resources  14  are the most cost-effective to operate at any given time, then selects those process load resources  15  to meet the target phasor angle at each node along the feeder line. In order to determine the relative cost to operate a process load resource  15  at a particular point in time, the site control module programming includes a costing sub-routine which attributes a cost for operating each process load resource  15  at a particular point in time. The costing subroutine takes into consideration factors such as the cost that must be paid to the primary process operator of the device  16  for using the resource  16  at that time instance. The aggregated cost is then multiplied by a risk factor allocated to each resource  16  at that time instance; this risk factor takes into consideration the risk that over the period of time the resource  16  will be used to provide power delivery control, the primary process operator will override feeder line power control and use the resource  16  for its primary purpose. The costing sub-routine then selects a cost effective combination of process load resources to be operated; a cost effective combination can be the combination of on-line load resources having the lowest operational cost, or any one of a combination of load resources which fall within a defined operational cost budget. 
     Once the real power resources  15  and the reactive power resources  16  are selected, the system  10  transmits a control signal to the controller  13 ,  14  at each real and reactive power resource  15 ,  16  that contains the target phasor for the node of the real and reactive power resource  15 ,  16 . The controllers  13 ,  14  then operate their associated real and reactive power resource  15 ,  16  to achieve the target phasor. That is, the load resource controller  13  will increase the load of its load resource when the measured phase angle at the node is lower than the target phase angle and decrease the load when the measured phase angle is higher than the target phase angle. The reactive power resource controller  14  will engage a capacitive resource  15  to generate reactive power at a node  17  when the measured voltage magnitude at the node is below the target voltage magnitude, and will engage an inductive resource  15  to consume reactive power at a node  17  when the measured voltage magnitude at the node  17  is below the target voltage magnitude. In this manner, the system  10  can provide localized control of the delivery of power to each node  17  along the feeder line  11 , at a desirably low feeder line power loss (assuming the allowable feeder line power loss threshold is set at or near the minimum feeder line power loss), while keeping the substation tap changer operation at a minimum (assuming the tap changer  23  is assigned a relatively high operational cost). 
     Alternatively, the real power resources  15  can include generation resources, in which case, a process generation resource management module is provided to select the generation resource that will be used to meet the target phase angle at each node. Like the load resources, the generation resources can include resources which serve a primary process, in which case the system only controls those generation resources that are on-line, i.e. within the operational constraints of their primary process. In a manner similar to selecting a cost-effective combination of load resources, a costing sub-routine is executed and each available generation resource is assigned a relative operating cost, and the most cost-effective combination of generation resources is selected to meet the target phasor angle at each node along the feeder line. Once the generation resources are selected, the system  10  sends a control signal to each controller of the selected generation resource that contains the target phasor for the node of the generation resource. The controllers then operate their associated generation resource to achieve the target phasor. That is, the generation resource controller will increase the generation of its load resource when the measured phase angle at the node is higher than the target phase angle, and decrease the generation when the measured phase angle is lower than the target phase angle. 
     According to another embodiment, the power delivery control program uses optimal power flow (OPF) algorithms that are based on a closed form solution for radial distribution systems to control the utility voltage management devices  23  and at least the reactive power resources  16  to deliver a required amount of reactive power to the feeder  11  to bring the voltage to acceptable levels, while minimizing the power loss and minimizing the operation of the utility voltage management devices  23  and in particular, the substation tap changer. 
     Such a closed form solution is expected to be more computationally tractable than complex generic algorithms which tend to be relatively computationally demanding; as a result, the power delivery control program is expected to be able to react more quickly to changes in the distribution feeder than a program executing complex optimization problems, which is desirable for distribution feeders containing intermittent generation sources such as solar and wind power generators. 
     An embodiment of the closed form solution used by the power delivery control program to set the target phasor for each node will now be described. This embodiment assumes a balanced, radial distribution system that can be reduced to a single-phase system. This embodiment is expressed in polar form, and allows for a variable ratio of reactive power injection at each of the upstream and downstream nodes i,j. 
     Referring to  FIG.  7   , each feeder sector extends from an upstream node i to a downstream node j individually and assumes that the shunt capacitance can be neglected. The voltage phasor at sending end V ij , the line current phasor I ij  and the line admittance Y ij  comprising admittance G ij +j B ij , are defined as follows: 
     
       
         
           
             
               
                 
                   
                     V 
                     i 
                   
                   = 
                   
                     
                       
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                         V 
                         i 
                       
                       
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                     ⁢ 
                     
                       ∢δ 
                       i 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     V 
                     j 
                   
                   = 
                   
                     
                       
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                         V 
                         j 
                       
                       
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                     ⁢ 
                     
                       ∢δ 
                       j 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     I 
                     ij 
                   
                   = 
                   
                     
                       
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                         I 
                         ij 
                       
                       
                         ❘ 
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                     ⁢ 
                     
                       ∢δ 
                       
                         I 
                         ij 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     Y 
                     ij 
                   
                   = 
                   
                     
                       
                         G 
                         ij 
                       
                       + 
                       
                         jB 
                         ij 
                       
                     
                     = 
                     
                       1 
                       
                         
                           R 
                           ij 
                         
                         + 
                         
                           jX 
                           ij 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     wherein δ i  is the phase angle at node i. δ j  is the phase angle at node j, and δ l     ij    the phase angle of the line current in the line sector ij. 
     The current (I ij ) in the feeder sector is defined as: 
     
       
         
           
             
               I 
               ij 
             
             = 
             
               
                 
                   ( 
                   
                     
                       V 
                       
                         i 
                           
                       
                     
                     - 
                     
                       V 
                       j 
                     
                   
                   ) 
                 
                 ⁢ 
                 
                   Y 
                   ij 
                 
               
               = 
               
                 
                   
                     P 
                     ij 
                   
                   - 
                   
                     jQ 
                     ij 
                   
                 
                 
                   V 
                   i 
                   * 
                 
               
             
           
         
       
     
     where V i  and V j  are the voltage phasors at sending and receiving end and Y ij  is the line admittance, P ij  is the real power flow from node i to node j and Q ij  is the reactive power flow from node i to node j. 
     P ij  in the feeder sector is expressed in equation (8) as: 
         P   ij =Re{(( V   i   −V   j )( Y   ij ))* V   i }  (10)
 
     The power loss P loss ij  in the feeder sector is: 
     
       
         
           
             
               
                 
                   
                     P 
                     
                       loss 
                       ⁢ 
                          
                       ij 
                     
                   
                   = 
                   
                     
                       
                         R 
                         ij 
                       
                       ⁢ 
                       
                         
                           
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                             I 
                             ij 
                           
                           
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                         2 
                       
                     
                     = 
                     
                       
                         R 
                         ij 
                       
                       ⁢ 
                       
                         
                           
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                                 P 
                                 ij 
                               
                               - 
                               
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                                 ij 
                               
                             
                             
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                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The voltage drop from node i to node j ΔV ij  as defined in equation (10) is: 
     
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       V 
                     
                     ij 
                   
                   = 
                   
                     
                       
                         V 
                         i 
                       
                       - 
                       
                         V 
                         j 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               R 
                               ij 
                             
                             ⁢ 
                             
                               P 
                               ij 
                             
                           
                           + 
                           
                             
                               X 
                               ij 
                             
                             ⁢ 
                             
                               Q 
                               ij 
                             
                           
                         
                         
                           V 
                           i 
                           * 
                         
                       
                       + 
                       
                         j 
                         ⁢ 
                         
                           
                             
                               
                                 X 
                                 ij 
                               
                               ⁢ 
                               
                                 P 
                                 
                                   i 
                                   ⁢ 
                                   j 
                                 
                               
                             
                             + 
                             
                               
                                 R 
                                 
                                   i 
                                   ⁢ 
                                   j 
                                 
                               
                               ⁢ 
                               
                                 Q 
                                 
                                   i 
                                   ⁢ 
                                   j 
                                 
                               
                             
                           
                           
                             V 
                             i 
                             * 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     If P ij »Q ij  in (8) and (9), these equations reduce to equations (11) and (12): 
     
       
         
           
             
               
                 
                   
                     P 
                     
                       loss 
                       ⁢ 
                          
                       ij 
                     
                   
                   = 
                   
                     
                       R 
                       ij 
                     
                     ⁢ 
                     
                       
                         
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                             P 
                             ij 
                           
                           
                             V 
                             i 
                             * 
                           
                         
                         
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                       2 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       V 
                     
                     ij 
                   
                   = 
                   
                     
                       
                         
                           R 
                           ij 
                         
                         ⁢ 
                         
                           P 
                           ij 
                         
                       
                       
                         V 
                         i 
                         * 
                       
                     
                     + 
                     
                       j 
                       ⁢ 
                       
                         
                           
                             X 
                             ij 
                           
                           ⁢ 
                           
                             P 
                             
                               i 
                               ⁢ 
                               j 
                             
                           
                         
                         
                           V 
                           i 
                           * 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     Equations (13) and (14) suggest that reducing the reactance flow Q ij  from node i to node j significantly decreases the power loss P loss ij  and the voltage drop ΔV ij , for equal P ij . In the proposed closed form solution, the goal is to drive the voltage at the receiving end V j  to prevent reactive power flow Q ij  in the feeder sector while maintaining real power flow P ij  through the feeder sector. The consumed reactive power by the line (Q line =|I ij | 2  X ij ) will be supplied from the two adjacent nodes according to equation (13): 
     
       
         
           
             
               
                 
                   
                     aQ 
                     ij 
                   
                   = 
                   
                     
                       Q 
                       
                         j 
                         ⁢ 
                         i 
                       
                     
                     = 
                     
                       
                         Q 
                         line 
                       
                       
                         a 
                         + 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     where a is the ratio between the reactive power supply from the sending end Q ij  and receiving end Q ji  (“Q-ratio”). The two conditions lead to equations (27) where P ij  is the real power flow before optimization that shall be maintained and therefore is a constant obtained by equation (8): 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             aQ 
                             ij 
                           
                           = 
                           
                             Q 
                             ji 
                           
                         
                       
                     
                     
                       
                         
                           
                             P 
                             
                               ij 
                               ⁢ 
                                  
                               new 
                             
                           
                           = 
                           
                             P 
                             ij 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Using the equation for complex power (S=VI*), equation (15) can be formulated: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               a 
                               · 
                               Im 
                             
                             ⁢ 
                                
                             
                               ( 
                               
                                 
                                   V 
                                   i 
                                 
                                 ⁢ 
                                 
                                   I 
                                   ij 
                                   * 
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             Im 
                             ⁢ 
                                
                             
                               ( 
                               
                                 
                                   
                                     V 
                                     j 
                                   
                                   ( 
                                   
                                     - 
                                     
                                       I 
                                       ij 
                                     
                                   
                                   ) 
                                 
                                 * 
                               
                               ) 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             Re 
                             ⁢ 
                                
                             
                               ( 
                               
                                 
                                   V 
                                   i 
                                 
                                 ⁢ 
                                 
                                   I 
                                   i 
                                   * 
                                 
                               
                               ) 
                             
                           
                           = 
                           
                             P 
                             ij 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Substituting equations (3), (4) (5) in (14) and rearranging yields equation (16) with unknown |V j |: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           ( 
                           
                             P 
                             + 
                             
                               
                                 G 
                                 ij 
                               
                               ( 
                               
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             
                                               
                                                 V 
                                                 ⁢ 
                                                 i 
                                               
                                               2 
                                             
                                             ⁢ 
                                             a 
                                           
                                           - 
                                           
                                             
                                               V 
                                               ⁢ 
                                               j 
                                             
                                             2 
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         B 
                                         ij 
                                         2 
                                       
                                     
                                     - 
                                     
                                       
                                         
                                           G 
                                           ij 
                                         
                                         ( 
                                         
                                           P 
                                           - 
                                           
                                             
                                               G 
                                               ij 
                                             
                                             ⁢ 
                                             
                                               
                                                 V 
                                                 ⁢ 
                                                 i 
                                               
                                               2 
                                             
                                           
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         ( 
                                         
                                           a 
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                   
                                     
                                       
                                         ( 
                                         
                                           a 
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         B 
                                         ij 
                                         2 
                                       
                                     
                                     + 
                                     
                                       
                                         ( 
                                         
                                           a 
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       
                                         G 
                                         ij 
                                         2 
                                       
                                     
                                   
                                 
                                 - 
                                 
                                   
                                     V 
                                     ⁢ 
                                     i 
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                         2 
                       
                       
                         B 
                         ij 
                         2 
                       
                     
                     - 
                     
                       
                         ( 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         V 
                                         ⁢ 
                                         i 
                                       
                                       2 
                                     
                                     ⁢ 
                                     a 
                                   
                                   - 
                                   
                                     
                                       V 
                                       ⁢ 
                                       j 
                                     
                                     2 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 B 
                                 ij 
                                 2 
                               
                             
                             - 
                             
                               
                                 
                                   G 
                                   ij 
                                 
                                 ( 
                                 
                                   P 
                                   - 
                                   
                                     
                                       G 
                                       ij 
                                     
                                     ⁢ 
                                     
                                       
                                         V 
                                         ⁢ 
                                         i 
                                       
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   a 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   a 
                                   + 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 B 
                                 ij 
                                 2 
                               
                             
                             + 
                             
                               
                                 ( 
                                 
                                   a 
                                   + 
                                   1 
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 G 
                                 ij 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                       2 
                     
                     - 
                     
                       
                         
                           V 
                           ⁢ 
                           i 
                         
                         2 
                       
                       ⁢ 
                       
                         
                           V 
                           ⁢ 
                           j 
                         
                         2 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     wherein each reference to “G”, “P”, “B” in equation (16) respectively means “G ij ”, “P ij ” and “B ij ” 
     From equation (16), the voltage magnitude at the receiving end |V j |, can be expressed as a function of the voltage magnitude at sending end |V 1 |, the line admittance G ij , B ij  and power flow from node i to node j P ij  and Q-ratio α to produce equation (17): 
       | V   j   |=f (| V   i   |,G   ij   ,B   ij   ,P   ij   ,a )  (19)
 
     Similarly, the voltage phase angle at receiving end δ i  can be expressed as a function of the voltage magnitude at sending and receiving end |V i |, |V j |, the line impedance G ij , B ij , and phase angle at sending end δ i  and Q-ratio α to produce equation (18): 
       δ j   =f (| V   i   |,|V   j   |,G   ij   ,B   ij ,δ i   ,a )  (20)
 
     Solving equation (18) leads to a closed form solution for |V j | in equation (21) and δ j  in equation (22): 
     
       
         
           
             
               
                 
                   
                     V 
                     i 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   B 
                                   ij 
                                   2 
                                 
                                 ⁢ 
                                 M 
                               
                               - 
                               
                                 
                                   G 
                                   ij 
                                   2 
                                 
                                 ⁢ 
                                 M 
                               
                               + 
                               
                                 
                                   B 
                                   ij 
                                   4 
                                 
                                 ⁢ 
                                 
                                   V 
                                   i 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   G 
                                   ij 
                                   4 
                                 
                                 ⁢ 
                                 
                                   V 
                                   i 
                                   2 
                                 
                               
                               - 
                               
                                 
                                   B 
                                   ij 
                                   2 
                                 
                                 ⁢ 
                                 aM 
                               
                               - 
                               
                                 
                                   G 
                                   ij 
                                   2 
                                 
                                 ⁢ 
                                 aM 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 
                                   G 
                                   ij 
                                   4 
                                 
                                 ⁢ 
                                 
                                   V 
                                   i 
                                   2 
                                 
                                 ⁢ 
                                 a 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   B 
                                   ij 
                                   2 
                                 
                                 ⁢ 
                                 
                                   G 
                                   ij 
                                   2 
                                 
                                 ⁢ 
                                 
                                   V 
                                   i 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   B 
                                   ij 
                                   4 
                                 
                                 ⁢ 
                                 
                                   V 
                                   i 
                                   2 
                                 
                                 ⁢ 
                                 
                                   a 
                                   2 
                                 
                               
                               + 
                               
                                 
                                   G 
                                   ij 
                                   4 
                                 
                                 ⁢ 
                                 
                                   V 
                                   i 
                                   2 
                                 
                                 ⁢ 
                                 
                                   a 
                                   2 
                                 
                               
                               - 
                               
                                 4 
                                 ⁢ 
                                 
                                   B 
                                   ij 
                                   2 
                                 
                                 ⁢ 
                                 
                                   G 
                                   ij 
                                 
                                 ⁢ 
                                 
                                   P 
                                   ij 
                                 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 
                                   B 
                                   ij 
                                   2 
                                 
                                 ⁢ 
                                 
                                   G 
                                   ij 
                                   2 
                                 
                                 ⁢ 
                                 
                                   V 
                                   i 
                                   2 
                                 
                                 ⁢ 
                                 
                                   a 
                                   2 
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               2 
                               ⁢ 
                               
                                 B 
                                 ij 
                                 2 
                               
                               ⁢ 
                               
                                 G 
                                 ij 
                                 2 
                               
                               ⁢ 
                               
                                 V 
                                 i 
                                 2 
                               
                               ⁢ 
                               a 
                             
                           
                         
                       
                       
                         
                           2 
                           ⁢ 
                           
                             B 
                             ij 
                             4 
                           
                         
                         + 
                         
                           2 
                           ⁢ 
                           
                             B 
                             ij 
                             2 
                           
                           ⁢ 
                           
                             G 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
       
         
           
             	 
             where 
           
         
       
       
         
           
             	 
             
               M 
               = 
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               B 
                               ij 
                               2 
                             
                             ⁢ 
                             
                               V 
                               i 
                               2 
                             
                           
                           + 
                           
                             
                               G 
                               ij 
                               2 
                             
                             ⁢ 
                             
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                               i 
                               2 
                             
                           
                           - 
                           
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                               B 
                               ij 
                             
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                               ij 
                             
                           
                           + 
                           
                             
                               B 
                               ij 
                               2 
                             
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                               i 
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                               G 
                               ij 
                               2 
                             
                             ⁢ 
                             
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                               i 
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                             ⁢ 
                             a 
                           
                         
                         ) 
                       
                     
                   
                   
                     
                       
                         ( 
                         
                           
                             
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                             2 
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                           + 
                           
                             
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                               ij 
                               2 
                             
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                               i 
                               2 
                             
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                           + 
                           
                             
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                               ij 
                               2 
                             
                             ⁢ 
                             
                               V 
                               i 
                               2 
                             
                             ⁢ 
                             a 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                                    
                   
                     
                       δ 
                       j 
                     
                     = 
                     
                       
                         δ 
                         j 
                       
                       - 
                       
                         
                           cos 
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                            
                         
                           ( 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         a 
                                         ⁢ 
                                         V 
                                       
                                       i 
                                       2 
                                     
                                     - 
                                     
                                       V 
                                       j 
                                       2 
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   B 
                                   ij 
                                   2 
                                 
                               
                               - 
                               
                                 
                                   
                                     G 
                                     ij 
                                   
                                   ( 
                                   
                                     P 
                                     - 
                                     
                                       
                                         G 
                                         ij 
                                       
                                       ⁢ 
                                       
                                         V 
                                         ij 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     a 
                                     + 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 V 
                                 i 
                               
                               ⁢ 
                               
                                 
                                   V 
                                   j 
                                 
                                 ( 
                                 
                                   
                                     
                                       ( 
                                       
                                         a 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       B 
                                       ij 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       ( 
                                       
                                         a 
                                         + 
                                         1 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     
                                       G 
                                       ij 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     Equations (21) and (22) provide a target voltage phasor at the receiving end of a feeder sector that assures that there is no Q flow while the real power flow P ij  is maintained with respect to a given sending end voltage phasor V i . 
     For a radial distribution system, the OPF solution provided in equations (21), (22) at each node can be calculated starting from the substation, successively node by node downstream to the feeder end. The advantage of the proposed closed-form OPF is its fast computation and that it does not require iterative power flow algorithms. Furthermore, it works for bi-directional power flow. 
       FIG.  8    is a flowchart illustrating a method for locally controlling delivery of electrical power along a distribution feeder  11  using a system  10  that executes the power delivery control program based on an embodiment wherein equal reactive power is injected at each node to consume all the reactive power in the feeder sector. The system  10  comprises controllers  13 ,  14  for controlling both controllable real and reactive power resources on the distribution feeder  11 . 
     As noted above, the power delivery control program calculates the target voltage phasor for the downstream node  17  in each feeder sector of the feeder distribution line  11  in succession, starting from the known voltage phasor at the substation  23  and working downstream one feeder sector at a time (the phase angle at the substation is defined to be at angle zero). Thus, the first feeder sector in the distribution line  11  uses the voltage phasor at the substation  23  as the input values for the real and imaginary parts of the voltage at the upstream node v ire , v iim . 
     The power delivery control program is provided with or determines the admittance value Y ij  of the feeder sector (step  700 ). The admittance can be determined from the known resistance and reactance of the feeder sector. 
     The real power at the upstream node i is available from the measurement program. As noted above, this value is input into the power delivery control program as the power flow P ij  across the feeder sector (step  701 ). 
     The inputted real and imaginary parts of the voltage at the upstream node v ire , v iim , the determined admittance value Y ij  and the measured power flow value P ij  are used by the power delivery control program in equations (14) and (15) to solve for the real and imaginary parts of the voltage at the downstream node v jre , v jim . The target voltage phasor (comprising the target voltage magnitude and phase angle) at the downstream node is then determined by solving equations (16) and (17) (step  702 ). 
     The results of the measurement program also provide the actual phasor at the downstream node  17 . The actual phasor measurements consists of the voltage magnitude |V| and angle δ at the downstream node  17 . The real and/or reactive power controllers  13 ,  14  at the downstream node receive their target voltage phasor from the power delivery control program, and receive the actual voltage phasor measurements from the measurement program. With these inputs, the real and/or reactive power controllers  13 ,  14  can determine the difference between the actual voltage magnitude and phase angle and the target voltage angle and phase angle. 
     The real and/or reactive power controllers  13 ,  14  then selects one or more real and/or reactive power resources  15 ,  16  to control to cause the actual voltage magnitude and phase angle at the downstream node to move towards its target voltage magnitude and phase angle, then operates their selected real and/or reactive power resources  15 ,  16  accordingly (step  703 ). As noted previously, the controllable reactive power resources  16  at node sites  17  can be used to control the reactive power injection at each node  17  and the substation. As is well understood by those skilled in the art, reactive power resources  16  such as capacitors increase reactive power and consequently increase voltage magnitude at a node  17  and can be selected when the actual voltage magnitude is lower than the target voltage magnitude. Conversely, reactive power resources  16  that consume reactive power and consequently decrease voltage magnitude at a node  17  can be selected when the actual voltage magnitude is higher than the target voltage magnitude; examples of such reactive power resources include PV inverters and static synchronous compensators (STATSCOMs), which can be operated in inductive mode to lower the voltage as required. 
     The above steps are repeated for each line sector between two nodes along the entire distribution feeder  11 . For the first line sector, the reference voltage at the upstream node will be the voltage phasor at the substation  23 . Applying steps  700  to  703  will provide a voltage target at the downstream node (first node  17  after the substation  23 ). For the second line sector, the voltage phasor at the upstream node will be the target voltage phasor at the downstream node of the first line sector. This sequence is repeated for each feeder sector until the last node is reached within the distribution feeder  11 . 
       FIG.  9    is a flowchart illustrating a method for locally controlling delivery of electrical power along a distribution feeder  11  using a system  10  that executes the power delivery control program based on another embodiment of the closed-form solution, which is implemented in the central control server  12  within a repeated routine. The central control server  12  obtains voltage phasors V i  and V j  at the upstream and downstream nodes using the results of the measurement program at each node i, j (step  800 ); determines the admittance of the feeder Y ij  then calculates the actual power flow P ij  through each feeder sector using equation (23) (step  801 ); determines the line admittance values B ij  and G ij  then calculates the target voltage phasor of the downstream node j using the closed-form equations (21, 22) (step  802 ); ensures that target phasor is within the voltage limits of 0.95 and 1.05 pu (steps  803 - 806 ); and dispatches target voltage phasor to the resource controller at the downstream node (step  807 ). This routine is repeated continually for each successive pair of adjacent nodes. 
     In some embodiments, the real and/or reactive power resources  15 ,  16  do not have any operational constraints, and the power delivery control program should be able to control the power delivery along the feeder sector with a minimum power loss. In other embodiments, the real and/or reactive power resources can be provided with operational constraints. For example, an operational constraint can be assigned that represents the maximum available reactive power resources. If the maximum available reactive power resources are not sufficient to track the voltage phasor targets, a new set of target voltage phasors for the entire feeder line should be computed considering the reactive power constraints. In another example, an operational constraint can be assigned that represents a maximum threshold on the line current in each sector, as the feeders have a maximum current constraint. 
     There is provided, for managing an electrical grid distribution feeder line responsively to characterized electrical parameters thereat, the line having a first line node being a substation with an associated voltage controller that is adapted to control the voltage thereat, and, downstream thereof, a second line node that is electrically coupled to a customer&#39;s energy resource (that generates or consumes electricity), a method of characterizing the voltage phasor at the second line node for a period of time, comprising the steps: (i) coupling a first μPMU at said first line node that is adapted to timestamp measure the electrical voltage and current waveforms thereat; (ii) (a) coupling a voltage transformer between said second line node and said customer&#39;s energy resource with drop lines, and (b) coupling a second μPMU on the drop between the transformer and said customer&#39;s energy resource, that is adapted to timestamp measure the electrical voltage and current waveforms thereat; (iii) making said first μPMU timestamped measurements and making said second μPMU timestamped measurements and time-synchronizing one of said first μPMU timestamped measurements with one of said second μPMU timestamped measurements; (iv) determining, for the period, the impedances of said drop lines and said transformer between said second line node and said customer&#39;s energy resource; (v) determining, for the period, the voltage at the second line node based on said drop line impedances and said transformer impedance and said second μPMU measurements; and (vi) informing said determined second line node voltage, to said voltage controller. 
     There is further provided the preceding wherein the customer&#39;s energy resource has a resource controller that is adapted to control the energy resource to change its real or reactive power consumption or generation to predictably cause a change in the voltage waveforms at the second line node, and the voltage controller is adapted to determine voltage control targets for the substation and the customer&#39;s energy resource, and additionally comprises the steps: (vii) determining a voltage target for the customer&#39;s energy resource coupled to the second line node; and (vii) determining a voltage target for the substation; and (viii) informing said determined voltage target at the second line node to customer&#39;s energy resource controller to adjust the voltage thereat. 
     There is further provided the preceding wherein the voltage target for the substation control and the voltage target for the customer&#39;s energy resource control are determined to reduce the amount of substation control needed. 
     There is further provided the preceding wherein the voltage target for the substation control and the voltage target for the customer&#39;s energy resource control are determined to reduce the amount of energy lost in the feeder line. 
     There is further provided the preceding wherein the voltage target for the substation control and the voltage target for the customer&#39;s energy resource control are determined to trade off between the amount of substation control and the amount of energy loss in the feeder line. 
     There is further provided the preceding wherein the trade off between substation control and energy loss is based on the cost of wear and tear on the substation equipment and the value of the energy lost in the feeder line. 
     There is further provided the preceding wherein the voltage target for the substation control and the voltage target for the customer&#39;s energy resource control are determined to optimally minimize reactive power flow in the feeder line subject to the constraints of the substation control and the customer&#39;s energy resource and the costs of the wear-and-tear on the substation control and the impact to the customer of controlling its energy resource. 
     One or more aspects or features of the subject matter described herein can be realized in digital electronic circuitry, integrated circuitry, specially designed application specific integrated circuits (ASICs), field programmable gate arrays (FPGAs) computer hardware, firmware, software, and/or combinations thereof. These various aspects or features can include implementation in one or more computer programs that are executable and/or interpretable on a programmable system including at least one programmable processor, which can be special or general purpose, coupled to receive data and instructions from, and to transmit data and instructions to, a storage system, at least one input device, and at least one output device. The programmable system or computing system may include clients and servers. A client and server are generally remote from each other and typically interact through a communication network. The relationship of client and server arises by virtue of computer programs running on the respective computers and having a client-server relationship to each other. 
     These computer programs, which can also be referred to as programs, software, software applications, applications, components, or code, include machine instructions for a programmable processor, and can be implemented in a high-level procedural language, an object-oriented programming language, a functional programming language, a logical programming language, and/or in assembly/machine language. As used herein, the term “machine-readable medium” refers to any computer program product, apparatus and/or device, such as for example magnetic discs, optical disks, memory, and Programmable Logic Devices (PLDs), used to provide machine instructions and/or data to a programmable processor, including a machine-readable medium that receives machine instructions as a machine-readable signal. The term “machine-readable signal” refers to any signal used to provide machine instructions and/or data to a programmable processor. The machine-readable medium can store such machine instructions non-transitorily, such as for example as would a non-transient solid-state memory or a magnetic hard drive or any equivalent storage medium. The machine-readable medium can alternatively or additionally store such machine instructions in a transient manner, such as for example as would a processor cache or other random access memory associated with one or more physical processor cores. 
     To provide for interaction with a user, one or more aspects or features of the subject matter described herein can be implemented on a computer having a display device, such as for example a cathode ray tube (CRT) or a liquid crystal display (LCD) or a light emitting diode (LED) monitor for displaying information to the user and a keyboard and a pointing device, such as for example a mouse or a trackball, by which the user may provide input to the computer. Other kinds of devices can be used to provide for interaction with a user as well. For example, feedback provided to the user can be any form of sensory feedback, such as for example visual feedback, auditory feedback, or tactile feedback; and input from the user may be received in any form, including acoustic, speech, or tactile input. Other possible input devices include touch screens or other touch-sensitive devices such as single or multi-point resistive or capacitive trackpads, voice recognition hardware and software, optical scanners, optical pointers, digital image capture devices and associated interpretation software, and the like. 
     In the descriptions above and in the claims, phrases such as “at least one of” or “one or more of” may occur followed by a conjunctive list of elements or features. The term “and/or” may also occur in a list of two or more elements or features. Unless otherwise implicitly or explicitly contradicted by the context in which it is used, such a phrase is intended to mean any of the listed elements or features individually or any of the recited elements or features in combination with any of the other recited elements or features. For example, the phrases “at least one of A and B;” “one or more of A and B;” and “A and/or B” are each intended to mean “A alone, B alone, or A and B together.” A similar interpretation is also intended for lists including three or more items. For example, the phrases “at least one of A, B, and C;” “one or more of A, B, and C;” and “A. B, and/or C” are each intended to mean “A alone, B alone, C alone. A and B together. A and C together. B and C together, or A and B and C together.” In addition, use of the term “based on,” above and in the claims is intended to mean, “based at least in part on,” such that an unrecited feature or element is also permissible. 
     Although specific embodiments have been illustrated and described herein, it should be appreciated that any arrangement which achieves the same or similar purpose may be substituted for the embodiments described or shown by the subject disclosure. The subject disclosure is intended to cover any and all adaptations or variations of various embodiments. Combinations of the above embodiments, and other embodiments not specifically described herein, can be used in the subject disclosure. For instance, one or more features from one or more embodiments can be combined with one or more features of one or more other embodiments. In one or more embodiments, features that are positively recited can also be negatively recited and excluded from the embodiment with or without replacement by another structural and/or functional feature. The steps or functions described with respect to the embodiments of the subject disclosure can be performed in any order. The steps or functions described with respect to the embodiments of the subject disclosure can be performed alone or in combination with other steps or functions of the subject disclosure, as well as from other embodiments or from other steps that have not been described in the subject disclosure. Further, more than or less than all of the features described with respect to an embodiment can also be utilized.