State estimation for power systems using parameterized potential functions for inequality constraints

Abstract

The previous method of state estimation relies on penalty-based heuristic to enforce inequality constraints, which results in very large weight values, resulting in morbidity of the gain matrix. The disclosed embodiments of state estimation convert an inequality constraint optimization problem into an unconstrained optimization problem, where violated inequality constraints are represented as parameterized potential functions, each parameterized potential function including an attraction center parameter. The unconstrained convex optimization problem may be iteratively prepared and solved using successive updated values of attraction center parameters until no inequality constraints are violated to produce a final estimated state. The final estimated state may then be used to control a monitored system, such as a power system.

Description

Technical Field

[0001] Embodiments described herein relate generally to state estimation, and more particularly, to state estimation for power systems using parameterized potential functions for inequality constraints. Background Art

[0002] State estimation (SE) is used in management systems to estimate the most likely state of a power system (e.g., power grid) network based on a sparse and redundant set of measurements of node and branch quantities (e.g., node voltage phasors, power injection, and branch power flows). State estimation is necessary for complex systems where the state cannot be directly observed and / or the measurements of the state are prone to noise or corruption. The state of each node in the power system network can be defined, for example, by the node voltage amplitude and the node phase angle. Examples of management systems that utilize state estimation include, but are not limited to, supervisory control and data acquisition (SCADA) systems, energy management systems (EMS), distribution management systems (DMS), advanced DMS (ADMS), and others.

[0003] Conventionally, state estimation is solved either as an unconstrained nonlinear weighted least squares (WLS) problem or as an equality constrained nonlinear WLS problem to avoid numerical ill-conditioning of nodes (e.g., buses) with zero power injection. In these cases, the solution does not enforce feasibility constraints on power injection. For example, in the solution, a distributed energy resource (DER) with a maximum output of 10 kilowatts (KW) may be estimated to produce 12KW. Such conditions may occur in state estimates for both transmission and distribution systems, but are more likely to occur in state estimates for distribution systems due to lower telemetry redundancy and lower data quality. Therefore, the management system will benefit from state estimates that constrain node metrics (e.g., power output) to feasible values. Summary of the invention

[0004] Thus, systems, methods, and non-transitory computer-readable media for state estimation of power systems using parameterized potential functions (e.g., quadratic functions) for inequality constraints are disclosed. An object of an embodiment is to constrain the solution of a SE problem to a feasible value by satisfying all inequality constraints within a tolerance. A further object of an embodiment is to eliminate the need for very large weight values, thereby avoiding ill-conditioning of the gain matrix of the objective function. A further object of an embodiment is to estimate the state of a system (such as a power system) that can be used to optimize and control the system.

[0005] In an embodiment of a method for state estimation of an electric power system, the method includes using at least one hardware processor to: obtain a constrained optimization problem, the constrained optimization problem including a convex first objective function to be minimized and satisfying one or more inequality constraints; converting the constrained optimization problem into an unconstrained convex optimization problem, the unconstrained convex optimization problem including a second objective function, the second objective function being the sum of the first objective function and a parameterized potential function for each of at least a subset of the one or more inequality constraints, wherein each parameterized potential function is defined by an inequality function and a center of attraction parameter; and performing a solution process, including solving the unconstrained convex optimization problem by finding an input to the unconstrained convex optimization problem that minimizes an output of the unconstrained convex optimization problem, wherein one or more center of attraction parameters are updated when solving the unconstrained convex optimization problem to ensure that the input satisfies the one or more inequality constraints within a tolerance, and wherein the input represents an estimated state of the electric power system. The method may further include using the at least one hardware processor to control the electric power system based on the estimated state. The method may further include providing the estimated state of the electric power system through a human-machine interface of the energy management system. The method may further include using the estimated state in one or more of emergency analysis, volt-var optimization, or optimal power flow (such as distributed energy resource management). The first objective function may include an error calculation in which a value of the measurement function is subtracted from a value of system telemetry given an input.

[0006] The method may further include using the at least one hardware processor to: solve the first objective function without any inequality constraints by finding an initial optimized input to the first objective function that minimizes the output of the first objective function regardless of the one or more inequality constraints; determine whether the initial optimized input violates the one or more inequality constraints; when it has been determined that the one or more inequality constraints are not violated, use the initial optimized input as an estimated state of the output of the power system; and when it has been determined that at least one of the one or more inequality constraints is violated: perform a solution process through one or more iterations until the one or more inequality constraints are not violated, wherein the solution process further includes determining the value of each center of attraction parameter in each parametric potential function in the unconstrained convex optimization problem; and use the input found by the solution process in the last iteration of the one or more iterations as the estimated state of the output of the power system. The method may further include using the at least one hardware processor to control the power system based on the final estimated state. In an embodiment, when at least one of the one or more inequality constraints is outside the tolerance, the one or more inequality constraints are determined to be violated, and wherein, when the one or more inequality constraints are all within the tolerance, the one or more inequality constraints are determined to be not violated. During the solution process in each of the one or more iterations, for each of the one or more inequality constraints that are violated and none of the one or more inequality constraints are not violated, a parameterized potential function may be included in the unconstrained convex optimization problem. Determining the value of each center of attraction parameter may include: calculating the value of the center of attraction parameter based on a previous value of each center of attraction parameter. Determining the value of each center of attraction parameter may include: calculating the value of each center of attraction parameter in the parameterized potential function based on the value of the inequality function in each parameterized potential function.

[0007] For inequality constraints expressed as g(x)≤0, each parameterized potential function can be defined as:

[0008] w(g(x)-u) 2

[0009] Where w is the weight, x is the input, g(x) is the inequality function, and u is the center of attraction parameter. The second objective function can be defined as:

[0010]

[0011] Where f(x) is the first objective function and j is the index of a set of inequality constraints for which a parameterized potential function is included in the unconstrained convex optimization problem.

[0012] Determine the parameter u of each center of attraction in iteration k j The values ​​can include: When the center of attraction parameter When there is no previous value, the center parameter is drawn The value is set to the predefined value u 初始 ; and when the center of attraction parameter With previous value When, according to the previous value and the inequality function g j (x (k-1) ) value to update the center of attraction parameter The value of . Predefined value u 初始 Can be equal to zero.

[0013] Determining the center of attraction parameters in iteration k can be performed as follows Value: Determine the set V (k) , which consists of all inequality constraints that have been determined to be violated in all previous iterations up to and including iteration k; determine the set A (k) , which is determined by the optimal input x found in iteration k-1 (k-1) (i.e., the input that minimizes the second objective function) violates any inequality constraint that was not in the set of inequality constraints in any previous iteration; when A (k) When not empty,

[0014] j∈V (k) and

[0015] j∈A (k) ;and

[0016] When A (k) When empty,

[0017] For j, g j (x (k-1) )≥0,

[0018] For j, g j (x (k-1) )<0,

[0019] Among them, u 初始 is a predefined value, α is a scaling factor, 0<α≤1, and ε is a tolerance.

[0020] For inequality constraints expressed as g(x)≤0, each parameterized potential function can be defined as:

[0021] w(g(x)-u) 2

[0022] Where w is the weight, x is the input, g(x) is the inequality function, and u is the center of attraction parameter.

[0023] It should be understood that any of the features in the above methods may be implemented individually or in any combination with any subset of the other features. Therefore, to the extent that the attached claims would imply specific dependencies between features, the disclosed embodiments are not limited to these specific dependencies. On the contrary, any of the features described herein may be combined with any other feature described herein, or implemented in any combination of any features without any one or more of the other features described herein. In addition, any of the methods described above and elsewhere herein may be embodied individually or in any combination in an executable software module of a processor-based system (such as a server) and / or in executable instructions stored in a non-transitory computer-readable medium. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] The details of the invention, both as to its structure and operation, may be gleaned in part by examination of the accompanying drawings, in which like reference numerals refer to like parts, and in which:

[0025] Figure 1 illustrates an example infrastructure in which one or more of the processes described herein may be implemented, according to an embodiment;

[0026] Figure 2 An example processing system is illustrated by which one or more of the processes described herein may be performed, according to an embodiment;

[0027] Figure 3 illustrates example data flows between systems and software modules within an example infrastructure according to an embodiment;

[0028] Figure 4A and Figure 4B illustrates the difference between a pseudo measurement weight adjustment operation and a virtual measurement operation using a center of attraction parameter according to an embodiment;

[0029] Figure 5 illustrates a solution process according to an embodiment; and

[0030] Figure 6 An overall algorithm for solving a constrained optimization problem according to an embodiment is illustrated. DETAILED DESCRIPTION

[0031] In an embodiment, a system, method, and non-transitory computer-readable medium for state estimation of an electric power system using a parameterized potential function for inequality constraints are disclosed. It is generally contemplated that state estimation will be performed for an electric power system, such as an electric grid (e.g., at any scale, from a large utility grid to a microgrid or smaller). An electric power system can be any network of electrical components (e.g., electric system equipment) that are configured to generate, store, supply, transmit, distribute, and / or consume electricity, including, but not limited to, a power station configured to generate electricity from combustible fuels (e.g., coal, natural gas, etc.) and / or renewable resources (e.g., wind energy, solar energy, nuclear energy, etc.), a transmission system configured to carry or transmit electricity from a source (e.g., a generator) to a load, and a distribution system configured to feed the supplied electricity to nearby homes, businesses, and / or other institutions. However, the disclosed method is not limited to electric power systems. Instead, the disclosed method can be applied to any system whose state is estimated using a WLS problem with one or more inequality constraints.

[0032] As used herein, the term "network" refers to the interconnection of components within a system whose state is to be estimated. In the case of an electric power system, these components will typically include electrical components such as generators, distributed energy resources (e.g., renewable energy, battery energy storage (BES) systems, etc.), loads, transformers, transmission and distribution lines, and / or others. It should be understood that other types of systems may be similarly represented as networks.

[0033] As used herein, the term "node" or "bus" generally refers to any point in a network at which the state is to be estimated when solving a SE problem. In the case of a power system, the state of each node may be defined as the voltage magnitude and phase angle at that node. However, the state of a node in a power system may be defined in other ways. It should be understood that the state of a node in other types of systems will be defined in a manner appropriate to that type of system.

[0034] After reading this description, it will become apparent to those skilled in the art how to implement the present invention in various alternative embodiments and alternative applications. However, although various embodiments of the present invention will be described herein, it should be understood that these embodiments are presented only by way of example and illustration and not by way of limitation. Thus, this detailed description of various embodiments should not be construed as limiting the scope or breadth of the present invention as set forth in the appended claims.

[0035] 1. System Overview

[0036] 1.1. Infrastructure

[0037] Figure 1An example infrastructure according to an embodiment is illustrated, in which one or more of the disclosed processes may be implemented. The infrastructure may include a management system 110 (e.g., including one or more servers) that hosts and / or executes one or more of the various functions, processes, methods, and / or software modules described herein. Examples of management systems 110 include, but are not limited to, EMS, DMS, ADMS, SCADA systems, and others. Management system 110 may include dedicated servers, or may be implemented in a computing cloud instead, in which the resources of one or more servers are dynamically and elastically allocated to multiple tenants based on demand. In either case, the servers may be collocated (e.g., in a single data center) and / or geographically distributed (e.g., across multiple data centers). Management system 110 may also include or be communicatively connected to software 112 and / or one or more databases 114. In addition, management system 110 may be communicatively connected to one or more user systems 130 and / or power systems 140 (e.g., power grids) via one or more networks 120.

[0038] The network(s) 120 may include the Internet, and the EMS 110 may communicate with the user system(s) 130 and / or the power system(s) 140 via the Internet using standard transfer protocols such as Hypertext Transfer Protocol (HTTP), HTTP Secure (HTTPS), File Transfer Protocol (FTP), FTP Secure (FTPS), Secure Shell FTP (SFTP), Extensible Messaging and Presence Protocol (XMPP), Open Field Message Bus (OpenFMB), IEEE Smart Energy Profile Application Protocol (IEEE 2030.5), and others, as well as proprietary protocols. Although the management system 110 is illustrated as being connected to various systems via a single set of network(s) 120, it should be understood that the management system 110 may be connected to various systems via different sets of one or more networks. For example, the management system 110 may be connected to a subset of the user systems 130 and / or power systems 140 via the Internet, but may be connected to one or more other user systems 130 and / or power systems 140 via an intranet. Furthermore, while only a few user systems 130 and external systems 140, one instance of software 112, and a set of database(s) 114 are illustrated, it should be understood that the infrastructure may include any number of user systems, power systems, software instances, and databases.

[0039] The user system(s) 130 may include any one or more types of computing devices capable of wired and / or wireless communication, including but not limited to desktop computers, laptop computers, tablet computers, smart phones or other mobile phones, servers, game consoles, televisions, set-top boxes, electronic kiosks, point-of-sale terminals, embedded controllers, programmable logic controllers (PLCs), and / or others. However, it is generally contemplated that the user system(s) 130 will include personal computers, mobile devices, or workstations through which agents of operators of the power system 140 can interact with the management system 110. These interactions may include inputting data (e.g., for configuring parameters of one or more of the processes described herein) and / or receiving data (e.g., output of one or more of the processes described herein) via a graphical user interface provided by the management system 110 or a system between the management system 110 and the user system(s) 130. A graphical user interface may include a screen (e.g., a web page) that includes a combination of content and elements such as text, images, video, animation, references (e.g., hyperlinks), frames, inputs (e.g., text boxes, text areas, check boxes, radio buttons, drop-down menus, buttons, tables, etc.), scripts (e.g., JavaScript), and others, including elements that include or are derived from data stored in one or more databases (e.g., database(s) 114).

[0040] Management system 110 may include, be communicatively coupled to, or otherwise have access to, one or more databases 114 that store data input to one or more of the disclosed processes and / or data output from one or more of the disclosed processes. Any suitable database may be utilized, including, but not limited to, MySQL. TM 、Oracle TM 、IBM TM , Microsoft SQL TM 、Access TM , PostgreSQL TM and others, including cloud-based databases, proprietary databases, and unstructured databases.

[0041] 1.2. Example Processing Device

[0042] Figure 2is a block diagram illustrating an example wired or wireless system 200 that may be used in conjunction with the various embodiments described herein. For example, the system 200 may be used as or in conjunction with one or more of the functions, processes, or methods described herein (e.g., to store and / or execute software 112), and may represent a component of a management system 110, (multiple) user systems 130, (multiple) power systems 140, and / or other processing devices described herein. The system 200 may be a server or any conventional personal computer, or any other processor-enabled device capable of wired or wireless data communications. Other computer systems and / or architectures may also be used, as will be apparent to those skilled in the art.

[0043] The system 200 preferably includes one or more processors 210. The processor(s) 210 may include a central processing unit (CPU). Additional processors may be provided, such as a graphics processing unit (GPU), an auxiliary processor to manage input / output, an auxiliary processor to perform floating point math operations, a dedicated microprocessor with an architecture suitable for fast execution of signal processing algorithms (e.g., a digital signal processor), a processor subordinate to the main processing system (e.g., a back-end processor), an additional microprocessor or controller for a dual-processor or multi-processor system, and / or a coprocessor. Such auxiliary processors may be discrete processors or may be integrated with the processor 210. Examples of processors that may be used with the system 200 include, but are not limited to, any of the processors available from Intel Corporation of Santa Clara, California (e.g., Pentium TM 、Corei7 TM , Xeon TM any of the processors available from Advanced Micro Devices, Inc. (AMD) of Santa Clara, California, any of the processors available from Apple Inc. of Cupertino (e.g., A-series, M-series, etc.), any of the processors available from Samsung Electronics Co., Ltd. of Seoul, South Korea (e.g., Exynos TM ) and / or others.

[0044] The processor 210 is preferably connected to a communication bus 205. The communication bus 205 may include a data channel for facilitating information transfer between storage devices and other peripheral components of the system 200. In addition, the communication bus 205 may provide a set of signals for communicating with the processor 210, including a data bus, an address bus, and / or a control bus (not shown). The communication bus 205 may include any standard or non-standard bus architecture, such as, for example, a bus architecture that conforms to the following: Industry Standard Architecture (ISA), Extended Industry Standard Architecture (EISA), Micro Channel Architecture (MCA), Peripheral Component Interconnect (PCI) local bus, standards published by the Institute of Electrical and Electronics Engineers (IEEE) (including IEEE 488 General Purpose Interface Bus (GPIB), IEEE 696 / S-100, and / or others).

[0045] The system 200 preferably includes a main memory 215 and may also include a secondary memory 220. The main memory 215 provides storage of instructions and data for programs (such as one or more of the functions and / or modules discussed herein) (e.g., software 112) executed on the processor 210. It should be understood that the programs stored in the memory and executed by the processor 210 may be written and / or compiled according to any suitable language (including, but not limited to, C / C++, Java, JavaScript, Perl, Visual Basic, .NET, and others). The main memory 215 is typically a semiconductor-based memory, such as a dynamic random access memory (DRAM) and / or a static random access memory (SRAM). Other semiconductor-based memory types include, for example, synchronous dynamic random access memory (SDRAM), Rambus dynamic random access memory (RDRAM), ferroelectric random access memory (FRAM), and others, including read-only memory (ROM).

[0046] The secondary memory 220 may optionally include internal media 225 and / or removable media 230. Reading from and / or writing to the removable media 230 is performed in any well-known manner. The removable storage medium 230 may be, for example, a tape drive, a compact disc (CD) drive, a digital versatile disc (DVD) drive, other optical drives, flash memory drives, and / or other. The secondary memory 220 is a non-transitory computer-readable medium on which computer executable code (e.g., software 112) and / or other data is stored. The computer software or data stored on the secondary memory 220 is read into the main memory 215 for execution by the processor 210.

[0047] In alternative embodiments, the secondary memory 220 may include other similar devices for allowing computer programs or other data or instructions to be loaded into the system 200. Such devices may include, for example, a communication interface 240 that allows software and data to be transferred from an external storage medium 245 to the system 200. Examples of the external storage medium 245 may include an external hard drive, an external optical drive, an external magneto-optical drive, and / or others. Other examples of the secondary memory 220 may include semiconductor-based memories such as programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), electrically erasable read-only memory (EEPROM), and flash memory (a block-oriented memory similar to EEPROM).

[0048] As mentioned above, the system 200 may include a communication interface 240. The communication interface 240 allows software and data to be transferred between the system 200 and an external device (e.g., a printer), a network, or other information source. For example, computer software or executable code may be transferred from a network server (e.g., platform 110) to the system 200 via the communication interface 240. Examples of the communication interface 240 include a built-in network adapter, a network interface card (NIC), a Personal Computer Memory Card International Association (PCMCIA) network card, a cardbus network adapter, a wireless network adapter, a universal serial bus (USB) network adapter, a modem, a wireless data card, a communication port, an infrared interface, an IEEE 1394 FireWire, and any other device capable of interfacing the system 200 with a network (e.g., network(s) 120) or another computing device. The communication interface 240 preferably implements industry-published protocol standards, such as Ethernet IEEE 802 standards, Fibre Channel, Digital Subscriber Line (DSL), Asynchronous Digital Subscriber Line (ADSL), Frame Relay, Asynchronous Transfer Mode (ATM), Integrated Services Digital Network (ISDN), Personal Communications Service (PCS), Transmission Control Protocol / Internet Protocol (TCP / IP), Serial Line Internet Protocol / Point-to-Point Protocol (SLIP / PPP), etc., but may also implement custom or non-standard interface protocols.

[0049] Software and data transmitted via communication interface 240 are typically in the form of electrical communication signals 255. These signals 255 may be provided to communication interface 240 via communication channel 250. In embodiments, communication channel 250 may be a wired or wireless network (e.g., network(s) 120) or any other communication link of any kind. Communication channel 250 carries signals 255 and may be implemented using a variety of wired or wireless communication means, including wire or cable, fiber optics, a conventional telephone line, a cellular telephone link, a wireless data communication link, a radio frequency ("RF") link, or an infrared link, to name a few.

[0050] Computer executable code (e.g., computer programs, such as software 112) is stored in the main memory 215 and / or the secondary memory 220. Computer programs may also be received via the communication interface 240 and stored in the main memory 215 and / or the secondary memory 220. When executed, such computer programs enable the system 200 to perform various functions of the disclosed embodiments as described elsewhere herein.

[0051] In this description, the term "computer-readable medium" is used to refer to any non-transitory computer-readable storage medium used to provide computer-executable code and / or other data to or within the system 200. Examples of such media include main memory 215, secondary memory 220 (including internal memory 225 and / or removable media 230), external storage media 245, and any peripheral devices (including network information servers or other network devices) communicatively coupled to communication interface 240. These non-transitory computer-readable media are devices used to provide executable code, programming instructions, software and / or other data to the system 200.

[0052] In embodiments implemented using software, the software may be stored on a computer-readable medium and loaded into the system 200 via the removable medium 230, the I / O interface 235, or the communication interface 240. In such embodiments, the software is loaded into the system 200 in the form of electrical communication signals 255. When executed by the processor 210, the software preferably causes the processor 210 to perform one or more of the processes and functions described elsewhere herein.

[0053] In an embodiment, the I / O interface 235 provides an interface between one or more components of the system 200 and one or more input and / or output devices. Example input devices include, but are not limited to, sensors, keyboards, touch screens or other touch-sensitive devices, cameras, biometric sensing devices, computer mice, trackballs, pen-based pointing devices, and / or other. Examples of output devices include, but are not limited to, other processing devices, cathode ray tubes (CRTs), plasma displays, light emitting diode (LED) displays, liquid crystal displays (LCDs), printers, vacuum fluorescent displays (VFDs), surface conduction electron emitter displays (SEDs), field emission displays (FEDs), and / or other. In some cases, input and output devices may be combined, such as in the case of a touch panel display (e.g., in a smartphone, tablet, or other mobile device).

[0054] System 200 may also include optional wireless communication components that facilitate wireless communication over voice networks and / or data networks (e.g., in the case of user system 130 being a smartphone or other mobile device). The wireless communication components include antenna system 270, radio system 265, and baseband system 260. In system 200, radio frequency (RF) signals are transmitted and received over the air by antenna system 270 under the management of radio system 265.

[0055] In an embodiment, antenna system 270 may include one or more antennas and one or more multiplexers (not shown) that perform switching functions to provide transmit and receive signal paths for antenna system 270. In the receive path, a received RF signal may be coupled from the multiplexer to a low noise amplifier (not shown) that amplifies the received RF signal and sends the amplified signal to radio system 265.

[0056] In an alternative embodiment, the radio system 265 may include one or more radios configured to communicate over various frequencies. In an embodiment, the radio system 265 may combine a demodulator (not shown) and a modulator (not shown) in one integrated circuit (IC). The demodulator and modulator may also be separate components. In the incoming path, the demodulator strips off the RF carrier signal, leaving behind a baseband receive audio signal that is sent from the radio system 265 to the baseband system 260.

[0057] If the received signal contains audio information (for example, in the case of user system 130 capable of operating as a phone), the baseband system 260 decodes the signal and converts it into an analog signal. Then, the signal is amplified and sent to a speaker. The baseband system 260 also receives analog audio signals from a microphone. These analog audio signals are converted into digital signals and encoded by the baseband system 260. The baseband system 260 also encodes the digital signal for transmission and generates a baseband transmission audio signal routed to the modulator part of the radio system 265. The modulator mixes the baseband transmission audio signal with an RF carrier signal to generate an RF transmission signal, which is routed to the antenna system 270 and can pass through a power amplifier (not shown). The power amplifier amplifies the RF transmission signal and routes it to the antenna system 270, where the signal is switched to the antenna port for transmission.

[0058] The baseband system 260 is also communicatively coupled to the processor(s) 210. The processor(s) 210 may access data storage areas 215 and 220. The processor(s) 210 are preferably configured to execute instructions (i.e., computer programs, such as the disclosed software) which may be stored in the main memory 215 or the secondary memory 220. Computer programs may also be received from the baseband processor 260 and stored in the main memory 210 or the secondary memory 220, or executed upon receipt. When executed, such computer programs enable the system 200 to perform the various functions of the disclosed embodiments.

[0059] 1.3. Example Data Flow

[0060] Figure 3 An example data flow between a management system 110, a user system 130, and a power system 140 according to an embodiment is illustrated. The power system 140 may include a monitoring module 310 and a control module 320. The software 112 of the management system 110 may include a state estimation module 330, a control and optimization module 340, and a human-machine interface (HMI) 350. The database 114 of the management system 110 may store a system model 360. It should be understood that the communication between the various systems may be performed via the network (s) 120. In addition, the communication between a pair of modules may be performed via an application programming interface (API) provided by one of the modules.

[0061] The monitoring module 310 may monitor and collect data output by one or more sensors in the network of the power system 140. The monitoring module 310 may also derive data from the collected data. The monitoring module 310 may transmit or "push" the collected and / or derived data as system telemetry data to the state estimation module 330 (e.g., via an API of the state estimation module 330). Alternatively, the state estimation module 330 may retrieve or "pull" the system telemetry data from the monitoring module 310 (e.g., via an API of the monitoring module 310). The system telemetry data may include measurements at each of one or more nodes and / or other points within the network of the power system 140. The system telemetry data may be communicated from the monitoring module 310 to the state estimation module 330 in real time (when the data is collected and / or derived) or periodically. As used herein, the term "real time" includes events that occur simultaneously, as well as events that occur concurrently, as determined by ordinary delays caused by potential factors in processing, memory access, communication, and / or others.

[0062] The state estimation module 330 receives system telemetry data from the monitoring module 310 and uses the system telemetry data in conjunction with the system model 360 to generate an estimated state of the power system 140. In particular, the state estimation module 330 may implement one or more of the processes for state estimation described herein to prepare and solve an optimization problem based on the system telemetry data and the system model 360. The state estimation module 330 may generate an estimated state of the power system 140 in real time or periodically (e.g., whenever new system telemetry data is received). The estimated state may include an estimated voltage magnitude and phase angle for each node within the network of the power system 140. The state estimation module 330 may send or "push" the estimated state of the power system 140 to the optimization and control module 340 (e.g., via an API of the optimization and control module 340). Alternatively, the optimization and control module 340 may retrieve or "pull" the estimated state of the power system 140 from the state estimation module 330 (e.g., via an API of the state estimation module 330). The estimated states may be communicated from the state estimation module 330 to the optimization and control module 340 in real time (as the estimated states are generated) or periodically.

[0063] The optimization and control module 340 receives the estimated state from the state estimation module 330 and uses the estimated state in conjunction with the system model 360 to determine an optimal configuration of one or more components of the power system 140, and then controls the power system 140 to transition to the optimal configuration. In particular, the optimization and control module 340 may generate control signals that are transmitted to the control module 320 of the power system 140. For example, these control signals may be sent via an API of the control module 320. These control signals may be communicated from the optimization and control module 340 of the management system 110 to the control module 320 of the power system 140 in real time (when the estimated state is received and analyzed), periodically, or in response to user operations. The optimization and control module 340 may control the power system 140 automatically (e.g., without any user intervention), semi-automatically (e.g., requiring user approval or confirmation), and / or in response to manual user input.

[0064] The control module 320 of the power system 140 receives control signals from the optimization and control module 340 and controls one or more components of the power system 140 based on these control signals. Examples of such control include setting set points (e.g., for active power and / or reactive power for distributed energy resources), adjusting power output of generators, adjusting charging or discharging of BES systems, adjusting power input of loads, closing or opening switches (e.g., circuit breakers), and / or the like.

[0065] The human-machine interface 350 may generate a graphical user interface for transmission to the user system 130 and receive inputs to the graphical user interface via the user system 130. The graphical user interface may provide information about an estimated state of the power system 140 determined by the state estimation module 330, an optimal configuration of the power system 140 determined by the optimization and control module 340, a control decision or recommendation determined by the optimization and control module 340, a visual representation of the system model 360, and / or other information. In addition, the graphical user interface may provide inputs that enable a user of the user system 130 to configure settings for the state estimation module 330, configure settings for the optimization and control module 340, configure the system model 360, specify or approve controls of the control module 320 to be transmitted to the power system 140, analyze the power system 140, and / or other information.

[0066] The system models 360 may be stored in the database 114 and accessed by modules such as the state estimation module 330 and the optimization and control module 340 via any known means (e.g., via an API of the database 114, a direct query of the database 114, etc.). The database 114 may store a system model 360 for each power system 140 managed by the management system 110. Each system model 360 models the network of the power system 140 in any suitable manner. For example, the system model 360 may include a single-line diagram representing the components of the network and their relationships to each other. It should be understood that the single-line diagram may be implemented as a data structure that can be automatically analyzed by software modules including the state estimation module 330 and the optimization and control module 340.

[0067] The estimated state output by the state estimation module 330 can be used as an input to any downstream function that can benefit from the estimated state of the power system 140. These downstream functions can be implemented by the optimization and control module 340, the human-machine interface 130, and / or other modules within the management system 110 or external systems. If the downstream function is implemented by another module, the state estimation module 330 can send or "push" the estimated state of the power system 140 to the other module (e.g., via the API of the module being implemented or relayed by the optimization and control module 340). Alternatively, the other module can retrieve or "pull" the estimated state of the power system 140 from the state estimation module 330 (e.g., via the API of the state estimation module 330) or from the optimization and control module 340 (e.g., via the API of the optimization and control module 340). The estimated state can be communicated to the module being implemented in real time (when the estimated state is generated) or periodically.

[0068] As an example of a downstream function, the estimated state may be stored and displayed to the user within the graphical user interface of the human-machine interface 350 in response to a triggering event. The triggering event may be a user requesting the estimated state, the estimated state satisfying an alarm condition, and / or other. If the estimated state satisfies the alarm condition, the user may be prompted to perform a preventive or corrective control action (e.g., via input in the graphical user interface, manually, etc.) via the graphical user interface or other means (e.g., a notification sent via an email message, a text message, a voice message, etc.).

[0069] In general, the estimated state (which may include the voltage magnitude and phase angle of each of the nodes in the network in the power system 140) may be a baseline model generated for one or more downstream functions. The downstream functions may utilize the baseline model to perform any type of analysis on the power system 140, including optimal power flow, distributed energy resource (DER) management, emergency analysis, and / or other. The analysis may be performed in response to a user operation, or automatically performed in real time or periodically. In some cases, the analysis results may be provided to the user via the human-machine interface 350. In other cases, the optimization and control module 340 may automatically (i.e., without user intervention) or semi-automatically (e.g., by user approval or confirmation) initiate a control operation based on the analysis results. The initiation of the control operation may include transmitting a control command to the control module 320 of the power system 140, which may responsively control the power system 140 according to the control command.

[0070] As an example, the optimization and control module 340 can use the estimated state as an input to the emergency analysis. The emergency analysis can use the estimated state of the power system 140 in the baseline model of the power system 140 (e.g., the system model 360) to perform a "what-if" analysis on various hypothetical scenarios (e.g., failure of components of the power system 140). For example, the estimated state at each node (e.g., voltage amplitude and phase angle) can be used to calculate the power generation or power consumption at the node for use in the emergency analysis. The emergency analysis can be performed in real time (e.g., when the estimated state is output by the state estimation module 330), periodically and / or in response to a triggering event (e.g., a user request, meeting one or more monitored criteria, etc.). If the emergency analysis detects a problem, the user can perform preventive or corrective control operations (e.g., via input of a graphical user interface of the man-machine interface 350, manually, etc.). Alternatively or additionally, optimization and control module 340 may automatically or semi-automatically perform contingency analysis for one or more what-if scenarios and, when a problem is detected, initiate preventive or corrective control actions via communication with control module 320 of power system 140 .

[0071] As an additional example, the optimization and control module 340 may use the estimated state as an input for voltage-reactive optimization. The voltage-reactive optimization may utilize the estimated state (e.g., voltage magnitude and phase angle) and the system model 360 (or other model) as a baseline to determine the optimal voltage level and reactive power to achieve efficient operation of the power system 140 (e.g., by reducing system losses, peak demand, and / or energy consumption). The voltage-reactive optimization may be performed in real time (e.g., when the estimated state is output by the state estimation module 330), periodically, and / or in response to a triggering event (e.g., a user request, meeting one or more monitored criteria, etc.). The optimization and control module 340 may automatically or semi-automatically initiate control operations via communication with the control module 320 of the power system 140 to cause the power system 140 to comply with the optimal voltage level and reactive power as determined by the voltage-reactive optimization. The control operations may include controlling switchable capacitors, on-load tap changers, and / or others.

[0072] As an additional example, the optimization and control module 340 may use the estimated state as an input to the optimal power flow. The optimal power flow may manage the power generation within the power system 140 using the estimated state and the system model 360 (or other model) as a baseline. For example, the optimal power flow may determine the optimal set point for the generator within the power system 140 to meet the demand on the power system 140 while minimizing operating costs and / or meeting one or more other criteria. The optimal power flow may be executed in real time (e.g., when the estimated state is output by the state estimation module 330), periodically, and / or in response to a triggering event (e.g., a user request, meeting one or more monitored criteria, etc.). The optimization and control module 340 may automatically or semi-automatically initiate control operations via communication with the control module 320 of the power system 140 to make the power system 140 comply with the set point determined by the optimal power flow.

[0073] As an additional example, the optimization and control module 340 may use the estimated state as an input to distributed energy resource (DER) management, which may be viewed as an optimal power flow. DER management may utilize the estimated state and the system model 360 (or other models) to manage distributed energy resources within the power system 140. For example, DER management may manage set points for active and reactive power, power factor, and / or voltage at points of interconnection with distributed energy resources or other nodes within the network. DER management may be performed in real time (e.g., when the estimated state is output by the state estimation module 330), periodically, and / or in response to a triggering event (e.g., a user request, satisfaction of one or more monitored criteria, etc.). The optimization and control module 340 may automatically or semi-automatically initiate control operations via communication with the control module 320 of the power system 140 to bring the power system 140 into compliance with the set points determined by the DER management.

[0074] 2. Process Overview

[0075] An embodiment of a process for state estimation of an electric power system using a parameterized potential function for inequality constraints will now be described in detail. It should be understood that the described process may be embodied in one or more software modules executed by one or more hardware processors, for example, as software 112 executed by (multiple) processors 210 of a management system 110. The described process may be implemented as instructions represented by source code, object code, and / or machine code. These instructions may be executed directly by (multiple) hardware processors 210, or alternatively, may be executed by a virtual machine or container operating between the object code and the hardware processor 210. In addition, the disclosed software may be built on or interfaced with one or more existing systems.

[0076] Alternatively, the described process may be implemented as a hardware component (e.g., a general purpose processor, an integrated circuit (IC), an application specific integrated circuit (ASIC), a digital signal processor (DSP), a field programmable gate array (FPGA) or other programmable logic device, a discrete gate or transistor logic, etc.), a combination of hardware components, or a combination of hardware and software components. In order to clearly illustrate the interchangeability of hardware and software, various illustrative components, blocks, modules, circuits and steps are generally described in terms of their functionality herein. Whether such functionality is implemented as hardware or software depends on a specific application and the design constraints imposed on the entire system. The technician may implement the described functions in various ways for each specific application, but such implementation decisions will not cause deviations from the scope of the present invention. In addition, the functional grouping within components, blocks, modules, circuits or steps is for ease of description. Without departing from the present invention, a specific function or step may be moved from one component, block, module, circuit or step to another component, block, module, circuit or step.

[0077] In addition, although the processes described herein are illustrated using certain arrangements and sequencing of subprocesses, each process may be implemented with fewer, more, or different subprocesses and different arrangements and / or sequencing of subprocesses. In addition, it should be understood that any subprocess that is not dependent on the completion of another subprocess may be performed before, after, or in parallel with other independent subprocesses, even if the subprocesses are described or illustrated in a particular order.

[0078] Throughout this disclosure, inequality constraints will be expressed in the following form:

[0079] g(x)≤0

[0080] Wherein, g(·) is an inequality function relating input x to output g(x). The input x represents the state of the system being monitored and controlled, such as the power system 140. In the case of the power system 140, the state can be defined as the voltage magnitude and phase angle at each node in the network. It should be understood that any inequality constraint can be expressed in this manner and in other ways. Therefore, for ease of understanding and conceptualization, it will be assumed that all inequality constraints are expressed in the above form. However, it should be understood that in actual implementations, the inequality constraints may be expressed in different ways and either converted to the above form or used in their native form with appropriate modifications to other components of the SE problem.

[0081] The constrained optimization problem can be expressed as:

[0082]

[0083] Satisfaction

[0084] g(x)≤0,j=1,...,n,

[0085] Wherein, n represents the total number of inequality constraints satisfied by the objective function f(x). It should be understood that an "optimization problem" may include any problem of finding the best solution from all feasible solutions, and an "objective function" may be any function to be maximized or minimized. In addition, the term "constrained" refers to a problem in which the objective function satisfies the corresponding constraints, while the term "unconstrained" refers to a problem in which all corresponding constraints have been converted to the objective function. The term "convex" refers to any function in which a line segment between any two points on the graph of the function is not located below the graph between the two points.

[0086] It should be understood that the inequality constraint g(x)≤0 represents a range of possible values ​​of the measurement function corresponding to the input x. For example, in the context of the power system 140, the range may be the capacity of the generator (e.g., from zero to maximum power output), the capacity of the BES system (e.g., based on the current charge level of the BES system), and / or other.

[0087] One approach for enforcing inequality constraints in state estimation is to treat the violated bounds as pseudo-measurements in the WLS problem:

[0088]

[0089] where each w is a weight, j is an index into a set of inequality constraints, and n represents the total number of inequality constraints. j (g j (x) 2Each instance of corresponds to a pseudo-measurement with value zero, which corresponds to the inequality constraint j. In order to drive the violation of the inequality constraint to zero, it is usually necessary to set the weight w of each violated inequality constraint j to j Set to a very large value.

[0090] In an embodiment, instead of the pseudo-measurement method described above, a parameterized potential function is used for each violated inequality constraint. In the context of parameterized potential functions, the term "parameterized" refers to the inclusion of adjustable parameters in the potential function, and the term "potential" refers to the analogy between an unconstrained objective function and a generalized potential field (e.g., such as a particle in a potential field being pulled toward a zero-force point). Although not necessary for an embodiment, the parameterized potential function can be a quadratic function. For example, each parameterized potential function can be expressed as:

[0091] w(g(x)-u) 2

[0092] Where u is a center of attraction parameter that defines a "center of attraction" that conceptually behaves like a spring to pull the output of the inequality function g(x) toward a feasible range. A "feasible range" is a set of values ​​of g(x) that do not violate the corresponding constraints. When g(x)>ε, the value of g(x) violates its corresponding constraint, where ε is a tolerance greater than or and / or equal to zero. Conversely, when g(x)≤ε, the value of g(x) does not violate its corresponding constraint. The tolerance ε is a practical consideration because it may not be possible to perfectly satisfy every inequality constraint in reality. Any standard means can be used to set the value of ε.

[0093] Using the above parameterized potential function, the SE problem becomes an unconstrained convex optimization problem UCP(u):

[0094]

[0095] in,

[0096]

[0097] Wherein, f(x) is the objective function to be minimized of the constrained optimization problem. In an embodiment, f(x) is a WLS problem:

[0098]

[0099] Among them, z i represents the system telemetry data at point i, where h(·) is the measurement function that relates input x to output h(x), which represents the predicted measurement value given input x, and w iare the corresponding weights, and m represents the total number of points measured in the system model 360. However, more generally, f(x) encompasses any convex objective function to be minimized. Each parameterized potential function can be viewed as a virtual measurement, rather than a pseudo-measurement. Figure 4A and Figure 4B The difference between the operation of the pseudo measurement method and the operation of the virtual measurement method using the attraction center parameter u is illustrated. The dotted line in each figure represents the gradient of the quadratic inequality function g(x).

[0100] like Figure 4A As illustrated in , each pseudo-measurement acts as a penalty, with the weights increasing as needed to cause the WLS problem to drive g(x) toward zero (e.g., from g(x) (0) ) to g(x (1) ) and then to g(x (2) ), and so on), in order to minimize the violation. This will lead to very large values ​​of the corresponding weights w in the pseudo-measurements.

[0101] In contrast, Figure 4B As illustrated in , each virtual measurement utilizes a center of attraction parameter u. The value of the center of attraction parameter u defines the location toward which g(x) is attracted (i.e., the center of attraction), with the weight w representing the strength of attraction. In other words, the value of the center of attraction parameter u can be likened to an anchor point to which one end of a spring is attached, with the other end of the spring attached to g(x), and the weight w is similar to the Hook's constant of the spring. It is noteworthy that, in contrast to the use of pseudo-measurements, the use of virtual measurements does not require increasing the weight w to a very large value.

[0102] When solving UCP(u) iteratively, instead of solving the original constrained optimization problem, the value of each center of attraction parameter u may be updated to "pull" the value of g(x) toward and eventually into the feasible range. For example, the value of the center of attraction parameter u may start at zero (i.e., u (0) ), and the problem UCP(u (0) ) is solved until convergence, thus obtaining g(x (0) ). Then, the value of the attraction center parameter u can be updated to u (1) , and the problem UCP(u (1) ) is solved until convergence, thus obtaining g(x (1) ). Then, the value of the attraction center parameter u can be updated to u (2) , and the problem UCP(u (2) ) is solved until convergence, thus obtaining g(x (2) ). This iterative process can continue until the value of g(x) is within a feasible range (e.g., within a given tolerance ε).

[0103] The force field associated with a potential field P(x,u) is defined by the negative gradient of P(x,u):

[0104]

[0105] The stationary point of the force field is parametric with respect to the center of attraction parameter u of the parameterized potential function:

[0106]

[0107] If x(u) satisfies the inequality constraint, then x(u) satisfies the Karush-Kuhn-Tucker (KKT) condition for constrained optimization problems. In particular, the KKT condition for constrained optimization problems is:

[0108]

[0109] Among them, λ j is a Lagrange multiplier, and B represents a set of binding constraints:

[0110] B={j|g j (x)≤0}

[0111] The following joint system has the same solution as the KKT condition for constrained optimization problems:

[0112]

[0113] g j (x)=0,j∈B

[0114] The parameters of the parameterized potential function are as follows and the Lagrange multiplier λ j Related:

[0115] λ j =-2w j u j

[0116] For any KKT point in a constrained optimization problem, there exists a related unconstrained convex optimization problem P(x,u) where the center of attraction parameter u of each parameterized potential function is uniquely defined for any non-zero weight w. Thus, the constrained optimization problem is transformed into the problem of placing the center of attraction parameter u such that the KKT condition of the original constrained optimization problem is satisfied.

[0117] Figure 5 A solution process 500 is illustrated according to an embodiment. The solution process 500 may be implemented by the state estimation module 330 in the software 112 of the management system 110 .

[0118] In subprocess 510, a set (multiple) of inequality constraints satisfied by the objective function f(x) is determined. In embodiments described elsewhere herein (where the solution process 500 is performed iteratively), the set (multiple) of inequality constraints may consist of all inequality constraints that were violated in the immediately preceding iteration. In alternative embodiments, one or more other criteria may be used to determine the set of inequality constraints, or the set of inequality constraints may be determined to include all inequality constraints.

[0119] In subprocess 520, an unconstrained convex optimization problem is prepared based on the set(s) of inequality constraints determined in subprocess 510. In particular, a parameterized potential function may be generated for each inequality constraint in the determined set(s) of inequality constraints and summed with the objective function from the constrained optimization problem in the second objective function to be minimized. Again, the unconstrained convex optimization problem may be expressed as:

[0120]

[0121] Among them, w j (g j (x)-u j ) 2 Represents a parameterized quadratic potential function for inequality constraint j in the set of inequality constraints of size n determined in subprocess 510 .

[0122] In subprocess 530, the center of attraction parameters u in all parameterized potential functions are determined. j In an embodiment where process 500 is performed iteratively, the center of attraction parameter of the current iteration k is The value of may be based on the center of attraction parameter determined in the immediately preceding iteration k-1 The value of the center of attraction parameter u in the parameterized potential function j The value of can also be based on the inequality function g in the same parameterized potential function j In other words, the value of the attraction center parameter u can be adjusted based on the amount of violation.

[0123] In an embodiment, each center of attraction parameter u may be determined as follows j Values:

[0124] When the set of inequality constraints determined to be violated by the most recently found optimization input is not empty,

[0125] j∈V (k) and

[0126] j∈A (k) ;and

[0127] When the set of inequality constraints determined to be violated by the most recently found optimization input is empty,

[0128] If g j (x (k-1) )≥0, then

[0129] If g j (x (k-1) )<0, then

[0130] in,

[0131] 0<α≤1

[0132] Among them, u 初始 is a predefined value, and α is a scaling factor, which may be determined empirically or in any suitable manner, or may be omitted or set to α=1 so as to be effectively omitted. 初始 =0.

[0133] In subprocess 540, P(x,u) is solved by finding an input x that minimizes the unconstrained convex optimization problem P(x,u) given the center of attraction parameter u determined in subprocess 530. In embodiments where process 500 is performed iteratively, the initial input x in iteration k may be (k)，ini Set to the final input x output from the immediately preceding iteration k-1 as the optimization input (k-1)，fin The result of subprocess 540 and process 500 will be the final and optimized input x for the current iteration k of process 500 (k)，fin The final input x (k)，fin represents an estimated state of the power system 140 .

[0134] Figure 6 An overall algorithm 600 for solving a constrained optimization problem according to an embodiment is illustrated. The algorithm 600 may be implemented by the state estimation module 330 in the software 112 of the management system 110. The algorithm 600 operates to convert a constrained optimization problem (including an objective function to be minimized and satisfying one or more inequality constraints) into an unconstrained convex optimization problem, and then solve the unconstrained convex optimization problem. When solving the unconstrained convex optimization problem, the algorithm 600 may iteratively perform the solution process 500.

[0135] Initially, in subprocess 610, a constrained optimization problem is obtained. The obtaining process may include retrieving a representation of the constrained optimization problem or retrieving a representation of one or more components of the constrained optimization problem, such as an objective function to be minimized and at least a subset of inequality functions of (multiple) inequality constraints satisfied by the objective function. Again, the constrained optimization problem may be expressed as:

[0136]

[0137] Satisfaction

[0138] g j (x)≤0, j=1, ..., n

[0139] The components of the constrained optimization problem (such as the objective function and any inequality constraints or inequality functions) can be stored in a memory that persists across iterations of the solution process 500 so that these components can be easily accessed in each iteration of the solution process 500 without having to be regenerated or re-determined in each iteration.

[0140] In subprocess 620, the constrained optimization problem is converted to an unconstrained convex optimization problem without the representation of any inequality constraints. In other words, for any inequality constraints satisfied by the constrained optimization problem, no parameterized potential function is included in the unconstrained convex optimization problem. Therefore, the unconstrained convex optimization problem can be expressed as:

[0141]

[0142] So that no inequality is satisfied.

[0143] In subprocess 630, by finding (noIC) Minimize the input x (noIC)，fin To solve the unconstrained convex optimization problem P (noIC) Solve. The input x (noIC)，fin represents the potential solution to the optimization problem, which may or may not satisfy the inequality constraint(s).

[0144] In subprocess 640, it is determined whether the most recently found input x violates any of the inequality constraints. In the first iteration of subprocess 630, the most recently found input to be evaluated in subprocess 640 will be x. (noIC)，fin In all subsequent iterations of subprocess 640, the most recently found input to be evaluated in subprocess 640 will be the final input x found by the most recent iteration k-1 of solution process 500. (k-1)，fin. In an embodiment, an inequality constraint is violated when the value of the inequality function of the inequality constraint is not within the tolerance ε of the feasible range (i.e., g(x)>ε). Conversely, in such embodiments, the inequality constraint is not violated when the value of the inequality function of the inequality constraint is within the tolerance ε of the feasible range (i.e., g(x)≤ε). If no inequality constraint is violated (i.e., "No" in subprocess 640), algorithm 600 proceeds to subprocess 650. If at least one inequality constraint is violated (i.e., "Yes" in subprocess 640), algorithm 600 performs an iteration of solution process 500.

[0145] In an embodiment, at each iteration of the solution process 500, the set of inequality constraints determined in subprocess 510 consists only of those inequality constraints that have been violated in any previous iteration. In other words, inequality constraints that have never been violated are not included in the set of inequality constraints determined in subprocess 510. Therefore, the unconstrained convex optimization problem generated in subprocess 520 and solved in subprocess 540 will consist only of the parameterized potential functions of the inequality constraints that were previously violated. The unconstrained convex optimization problem generated and solved in iteration k may be referred to herein as P (k) :

[0146]

[0147] where j is the index of the set of violated inequality constraints and n is the size of the set of violated inequality constraints.

[0148] Additionally, as discussed with respect to subprocess 530, each center of attraction parameter u in iteration k of solution process 500 is (k) The corresponding center of attraction parameter u in the immediately preceding iteration k-1 of the solution process 500 may be (k-1) or if there is no corresponding center of attraction parameter u in the immediately preceding iteration k-1 of the solution process 500 (k-1) The value of (e.g., because the corresponding inequality constraint was not previously violated) is initialized to u 初始 (e.g., zero). The output of each iteration of the solution process 500 will be the final input x (k)，fin , which represents the optimization input (given the parameters of the unconstrained convex optimization problem in the current iteration k).

[0149] In subprocess 650, once no inequality constraints are violated (i.e., “No” in subprocess 640), the most recently found (i.e., last) optimization input is output as the final estimated state of power system 140. If no violation is determined in the first and therefore only iteration of subprocess 640, this may be the input x found in subprocess 630. (noIC)，fin Otherwise, the final estimated state will be the input x (IC)，fin , which is output by the last iteration of the solution process 500. It should be understood that the final estimated state of the power system 140 output by the subprocess 650 may be sent to the optimization and control module 340, which may use the final estimated state to optimize the configuration of the power system 140 and / or otherwise control the power system 140, as discussed elsewhere herein.

[0150] An example of pseudo code implementing an embodiment of algorithm 600 is provided below.

[0151]

[0152] It is worth noting that in the above pseudo code, the set consists of all inequality constraints that have been determined to be violated in all previous iterations up to and including the current iteration k, and the set consists of any inequality constraints determined to be violated by the most recently found optimization input. In an embodiment, each center of attraction parameter u j The initialization value u 初始 is a predefined value, such as zero. However, it should be understood that other initialization values ​​(ie, non-zero initialization values) may be used for the center of attraction parameter u j .

[0153] It should also be understood that how iteration k is defined or depicted is arbitrary. For example, in the pseudocode above, for the future iteration k+1, in the unconstrained convex optimization problem P (k) After solving, determine the set V IC and A and the attraction center parameter u j However, in the unconstrained convex optimization problem P (k) Before solving, we can alternatively depict iteration k as a function of the set V IC and A and the attraction center parameter u j , as illustrated in solution process 500 .

[0154] In any case, the specific depiction of iteration k in any of the disclosed examples should not be construed as limiting in any respect.

[0155] 3. Example Uses

[0156] In an embodiment, the state estimation module 330 of the management system 110 is used to estimate the state of the system (such as the power system 140) using the algorithm 600. In particular, the state estimation module 330 may receive system telemetry data from the monitoring module 310 of the system, and use the system telemetry data and the system model 360 (e.g., retrieved from the database 114) to define the optimization problem such that the inequality constraints are satisfied as an unconstrained convex optimization problem. The optimization problem is formed as an unconstrained optimization problem by the following steps: generating a parameterized potential function w(g(x)-u) for at least a subset of the inequality constraints that are violated 2 , and minimizes the sum of any generated parameterized potential function and the objective function f(x). The objective function f(x) may be derived from the system telemetry data and the system model 360 and represents the system. The use of parameterized potential functions (representing virtual measurements in the case of state estimation problems) shapes the potential field of the resulting unconstrained optimization problem so that its static stationarity conditions are the same as the KKT conditions of the constrained optimization problem.

[0157] The resulting unconstrained optimization problem may be iteratively prepared using successively updated values ​​of the center of attraction parameter u corresponding to the violated inequality constraints and solved until no inequality constraints are violated (i.e., all inequality constraints are satisfied within tolerance) to produce a final estimated state. The final estimated state may be output to the optimization and control module 340 of the management system 110, which may input the final estimated state into one or more downstream functions, such as generating a graphical user interface, emergency analysis, power flow analysis (e.g., scheduler or optimal power flow analysis), voltage-var optimization, DER management, and / or other.

[0158] In some cases, one or more of these downstream functions may automatically (e.g., without user intervention) or semi-automatically (e.g., after user approval or confirmation) issue control commands to the system's control module 320, thereby controlling the system. Thus, state estimation may be used to monitor and control systems, such as the power system 140. For example, if the voltage at a measurement point is too low, the corresponding generator in the power system 140 may be controlled to increase its reactive power output to increase the voltage at the measurement point.

[0159] Conventional WLS state estimation methods either ignore inequality constraints or enforce them by using penalty-based heuristics in which weights (e.g., weights of pseudo-measurements representing inequality constraints) are increased. These penalty-based heuristics eliminate violations by increasing the corresponding weights to the extent that they approach infinity (which can cause ill-conditioning of the gain matrix). The disclosed process enforces inequality constraints in state estimation without having to use large weights. In particular, the disclosed process will work for any non-zero weights. Therefore, they do not require very large weight values, and thus avoid ill-conditioning of the gain matrix.

[0160] Notably, the disclosed algorithm 600 is robust and easy to implement, and provides fast convergence. In embodiments, the weights w in the parameterized potential function and / or objective function f(x) may be adaptively adjusted based on the convergence rate. Additionally, while discussed in the context of state estimation, the disclosed algorithm 600 may be used in other applications. One such application is training machine learning models, such as neural networks, and in particular, physically-informed neural networks (PINNs) or physically-constrained neural networks.

[0161] It should be understood that in addition to the (multiple) inequality constraints, the constrained optimization problem may also include one or more equality constraints. In this case, the equality constraints may be solved in any manner, including by generating a parameterized potential function for each equality constraint in a manner similar or the same as for the (multiple) inequality constraints but using a corresponding equality function instead of an inequality function. For example, for the equality constraint c(x)=0, the parameterized potential function may be expressed as w(c(x)-u) 2 .

[0162] The above description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles described herein may be applied to other embodiments without departing from the spirit or scope of the present invention. Therefore, it will be understood that the description and drawings presented herein represent currently preferred embodiments of the present invention, and therefore represent the subject matter broadly contemplated by the present invention. It should be further understood that the scope of the present invention fully encompasses other embodiments that may become apparent to those skilled in the art, and that the scope of the present invention is accordingly not limited.

[0163] The combinations described herein (such as, "at least one of A, B, or C", "one or more of A, B, or C", "at least one of A, B and C", "one or more of A, B, and C", and "A, B, C, or any combination thereof") include any combination of A, B, and / or C, and may include multiple A's, multiple B's, or multiple C's. Specifically, phrases such as "at least one of A, B, or C", "one or more of A, B, or C", "at least one of A, B, and C", "one or more of A, B, and C", and "A, B, C, or any combination thereof" may be only A, only B, only C, A and B, A and C, B and C, or A, B, and C, and any such combination may include one or more members of its constituent parts A, B, and / or C. For example, the combination of A and B may include one A and multiple B's, multiple A's and one B's, or multiple A's and multiple B's.

Claims

1. A method for state estimation of an electric power system, the method comprising using at least one hardware processor to: Obtaining a constrained optimization problem, the constrained optimization problem comprising a convex first objective function to be minimized and satisfying one or more inequality constraints; The constrained optimization problem is converted to an unconstrained convex optimization problem, the unconstrained convex optimization problem comprising a second objective function, the second objective function being the sum of the first objective function and a parameterized potential function for each of at least a subset of the one or more inequality constraints, wherein Each parameterized potential function is defined by an inequality function and a center of attraction parameter; as well as performing a solution process including solving the unconstrained convex optimization problem by finding an input to the unconstrained convex optimization problem that minimizes an output of the unconstrained convex optimization problem, wherein one or more center of attraction parameters are updated when solving the unconstrained convex optimization problem to ensure that the input satisfies the one or more inequality constraints within a tolerance, and Wherein the input represents an estimated state of the power system. 2 . The method of claim 1 , further comprising using the at least one hardware processor to control the power system based on the estimated state.

3. The method of claim 1 , further comprising using the at least one hardware processor to: solving the first objective function without any inequality constraints by finding an initial optimization input to the first objective function that minimizes an output of the first objective function regardless of the one or more inequality constraints; determining whether the initial optimization input violates the one or more inequality constraints; When it has been determined that the one or more inequality constraints are not violated, using the initial optimization input as an estimated state of an output of the power system; as well as When it has been determined that at least one of the one or more inequality constraints is violated: The solving process is performed through one or more iterations until the one or more inequality constraints are not violated, wherein the solving process further includes determining a value of each center of attraction parameter in each parametric potential function in the unconstrained convex optimization problem, and An input found by the solution process in a last iteration of the one or more iterations is used as an estimated state of an output of the power system. 4 . The method of claim 3 , further comprising using the at least one hardware processor to control the power system based on the final estimated state.

5. The method according to claim 3, wherein: The one or more inequality constraints are determined to be violated when at least one of the one or more inequality constraints is outside the tolerance, and wherein the one or more inequality constraints are determined to be not violated when all of the one or more inequality constraints are within the tolerance.

6. The method according to claim 3, wherein: During the solving process in each of the one or more iterations, for each of the one or more inequality constraints that are violated and none of the one or more inequality constraints that are not violated, a parameterized potential function is included in the unconstrained convex optimization problem.

7. The method according to claim 3, wherein: Determining the value of each center of attraction parameter includes calculating the value of each center of attraction parameter based on a previous value of the center of attraction parameter.

8. The method according to claim 3, wherein: Determining the value of each center of attraction parameter includes calculating the value of each center of attraction parameter in each parameterized potential function based on the value of the inequality function in the parameterized potential function.

9. The method according to claim 3, wherein: For inequality constraints expressed as g(x)≤0, each parameterized potential function is defined as: w(g(x)-u) 2 Wherein, w is the weight, x is the input, g(x) is the inequality function, and u is the center of attraction parameter.

10. The method according to claim 9, wherein: The second objective function is defined as: Wherein, f(x) is the first objective function, and j is the index of a set of inequality constraints for which parameterized potential functions are included in the unconstrained convex optimization problem.

11. The method according to claim 10, wherein: Determine the parameters u of each center of attraction in iteration k j The values ​​are: When the center of attraction parameter When there is no previous value, the center of attraction parameter The value is set to the predefined value u 初始 ;and When the center of attraction parameter With previous value When, according to the previous value and the inequality function g j (x (k-1) ) value to update the attraction center parameter The value of .

12. The method according to claim 11, wherein: The predefined value u 初始 is zero.

13. The method according to claim 10, wherein: Determining the center of attraction parameter in iteration k is performed as follows Values: Determine the set V (k) , the set consisting of all inequality constraints that have been determined to be violated in all previous iterations up to and including iteration k; Determine the set A (k) , the set is determined by the optimal input x found in iteration k-1 (k-1) any inequality constraint that is violated and was not in the set of inequality constraints in any previous iteration; When A (k) When not empty, j∈V (k) And j∈A (k) j∈A (k) ;and When A (k) When empty, For j, g j (x (k-1) )≥0, For j, g j (x (k-1) )<0, Among them, u 初始 is a predefined value, α is a scaling factor, 0<α≤1, and ε is the tolerance.

14. The method according to claim 13, wherein: The predefined value u 初始 is zero.

15. The method according to claim 1, wherein: For inequality constraints expressed as g(x)≤0, each parameterized potential function is defined as: w(g(x)-u) 2 Wherein, w is the weight, x is the input, g(x) is the inequality function, and u is the center of attraction parameter.

16. The method of claim 1, further comprising using the estimated status in one or more of emergency analysis, volt-var optimization, or optimal power flow.

17. The method according to claim 1, wherein: The first objective function comprises an error calculation in which a value of a measurement function is subtracted from a value of the system telemetry data given the input.

18. A system comprising: at least one hardware processor; as well as software, configured to, when executed by the at least one hardware processor: Obtaining a constrained optimization problem, the constrained optimization problem comprising a convex first objective function to be minimized and satisfying one or more inequality constraints, converting the constrained optimization problem to an unconstrained convex optimization problem, the unconstrained convex optimization problem comprising a second objective function, the second objective function being a sum of the first objective function and a parameterized potential function for each of at least a subset of the one or more inequality constraints, wherein each parameterized potential function is defined by an inequality function and a center of attraction parameter, and The unconstrained convex optimization problem is solved by finding an input to the unconstrained convex optimization problem that minimizes an output of the unconstrained convex optimization problem, wherein one or more center of attraction parameters are updated when solving the unconstrained convex optimization problem to ensure that the input satisfies the one or more inequality constraints within a tolerance, and wherein the input represents an estimated state of the power system.

19. The method according to claim 18, wherein: The software is further configured to control the power system based on the estimated state.

20. A non-transitory computer readable medium having stored thereon instructions, wherein: When executed by a processor, the instructions cause the processor to: Obtaining a constrained optimization problem, the constrained optimization problem comprising a convex first objective function to be minimized and satisfying one or more inequality constraints; converting the constrained optimization problem to an unconstrained convex optimization problem, the unconstrained convex optimization problem comprising a second objective function, the second objective function being a sum of the first objective function and a parameterized potential function for each of at least a subset of the one or more inequality constraints, wherein each parameterized potential function is defined by an inequality function and a center of attraction parameter; and The unconstrained convex optimization problem is solved by finding an input to the unconstrained convex optimization problem that minimizes an output of the unconstrained convex optimization problem, wherein one or more center of attraction parameters are updated when solving the unconstrained convex optimization problem to ensure that the input satisfies the one or more inequality constraints within a tolerance, and wherein the input represents an estimated state of the power system.

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