Using parameterized potential functions for equality constrained power system state estimation

Abstract

Existing state estimation methods based on constraint optimization problems with equality and / or inequality constraints depend on penalty-based heuristic methods that can produce very large weight values, resulting in morbidity of gain matrices. The disclosed state estimation embodiments convert a constrained optimization problem into an unconstrained convex optimization problem in which violated equality and / or inequality constraints are represented as parameterized potential functions, each including an attraction center parameter. This unconstrained convex optimization problem may be iteratively formed, using continuously updated values of attraction center parameters, and solved until no violation of equality and / or inequality constraints 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] The embodiments described herein generally relate to state estimation, and more particularly to power system state estimation using parameterized potential functions for equality constraints (and optionally inequality constraints). Background Art

[0002] State estimation (SE) is used in management systems to estimate the most likely state of a network of power systems (e.g., power grids) from a sparse set of redundant 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 state measurements are susceptible to noise or corruption. For example, the state of each node in the network of a power system can be defined by a node voltage amplitude and a 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 the like.

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

[0004] Thus, a system, method, and non-transient computer readable medium are disclosed for performing at least equality-constrained power system state estimation using a parameterized potential function (e.g., a quadratic function). An object of an embodiment is to constrain the solution of a SE problem to a feasible value of a node metric by satisfying all equality constraints within a tolerance. Another object of an embodiment is to estimate the state of a system that can be used to optimize and control a system such as a power system.

[0005] In an embodiment of a state estimation method for 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 under one or more equality 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 of each of at least one of the one or more equality constraints, wherein each parameterized potential function is defined by an equality function and a center-of-attraction parameter; and performing a solution process, the solution process including solving the unconstrained convex optimization problem by finding an input of 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 equality constraints within a tolerance, and wherein the input represents an estimated state of the electric power system. The method may also include using at least one hardware processor to control the electric power system based on the estimated state. The method may also include providing the estimated state of the electric power system through a human-machine interface of the energy management system. The method may also include using the estimated state in one or more of contingency 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 the value of the measurement function for a given input is subtracted from the value of the system telemetry data.

[0006] Each parameter potential function can be defined as:

[0007] w(c(x)-u) 2

[0008] Where w is the weight, x is the input, c(x) is the equation function, and u is the center of attraction parameter. The second objective function can be defined as:

[0009]

[0010] Wherein f(x) is the first objective function, i is an indexed set of equality constraints, and wherein a parameterized potential function for each equality constraint in the indexed set of equality constraints is included in an unconstrained convex optimization problem.

[0011] The method may also include using at least one hardware processor to perform a first process, the first process including: performing an initial iteration of a solution process to find an initial optimization input, wherein the initial iteration of the solution process also includes initializing the value of each center of attraction parameter in each parameterized potential function in the unconstrained convex optimization problem; determining whether the initial optimization input violates one or more equality constraints; when it is determined that the one or more equality constraints are not violated, outputting the initial optimization input as an output estimated state of the power system; and when it is determined that at least one of the one or more equality constraints is violated, in one or more subsequent iterations, performing the solution process until the one or more equality constraints are not violated, wherein the subsequent iteration of the solution process also includes updating the value of each center of attraction parameter in each parameterized potential function corresponding to the one or more equality constraints before solving the unconstrained convex optimization problem, and outputting the input found by the solution process as the output estimated state of the power system in the last iteration of the one or more subsequent iterations. The method may also include using at least one hardware processor to control the power system based on the output estimated state. In one embodiment, one or more equality constraints are determined to be violated when at least one of the one or more equality constraints is outside of tolerance, and wherein one or more equality constraints are determined not to be violated when all of the one or more equality constraints are within tolerance. In each of the initial iteration and one or more subsequent iterations, the unconstrained convex optimization problem may include parameterized potential functions of all of the one or more equality constraints. In each of the one or more subsequent iterations, the initial input for solving the unconstrained convex optimization problem during the solution process of the subsequent iteration may be the input found by the solution process in the immediately previous iteration.

[0012] In each of the one or more subsequent iterations, updating 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 the center of attraction parameter in an immediately preceding iteration. In each of the one or more subsequent iterations, updating the value of each center of attraction parameter may include calculating the value of each center of attraction parameter further based on a value of an equation function of a corresponding one of the one or more equality constraints given an input found by the solution process in the immediately preceding iteration. The value of each center of attraction parameter in the subsequent iteration k may be calculated as follows:

[0013] u (k) =u (k-1) -c(x (k-1) )

[0014] Among them, u (k) is the value of the attraction center parameter in the subsequent iteration k, u (k-1) is the value of the center of attraction parameter in the immediately preceding iteration k-1, x (k-1)is the input found in the immediately preceding iteration k-1, c(x (k-1) ) is given an input x (k-1) The case includes the value of the equation function in the parameterized potential function of the attraction center parameter.

[0015] The constrained convex optimization problem may further include one or more inequality constraints, and the method may further include using at least one hardware processor to: execute a first process while ignoring one or more inequality constraints to find an output estimated state as a first initial optimization input; determine whether the first initial optimization input violates one or more inequality constraints; when it is determined that the one or more inequality constraints are not violated, use the first initial optimization input as the final estimated state of the power system; when it is determined that at least one of the one or more inequality constraints is violated, in one or more external iterations, convert the constrained optimization problem into a convex optimization problem that is not constrained by inequality until the one or more inequality constraints are not violated. The non-inequality-constrained convex optimization problem includes a first objective function and an inequality parameterized potential function for each of at least one of the one or more inequality constraints, wherein each inequality parameterized potential function is defined by an inequality function and a center of attraction parameter, determining a value of each center of attraction parameter in each inequality parameterized potential function in the non-inequality-constrained convex optimization problem, and performing the first process using the non-inequality-constrained convex optimization problem as a constrained optimization problem in the first process, and outputting an input found by the first process in the last iteration of one or more external iterations as a final estimated state of the power system. In each of the one or more external iterations, for each of the one or more inequality constraints that are violated, rather than for any of the one or more inequality constraints that are not violated, an inequality parameterized potential function can be added to the non-inequality-constrained convex optimization problem. For an inequality constraint represented as g(x)≤0, each inequality parameterized potential function component can be defined as:

[0016] w(g(x)-u) 2

[0017] Among them, w is the weight, x is the input, g(x) is the inequality function, and u is the center of attraction parameter.

[0018] In an embodiment of a state estimation method for an electric power system, the method includes using at least one hardware processor to: obtain a constrained convex optimization problem, the constrained convex optimization problem including a first objective function to be minimized under one or more equality constraints; convert the constrained convex 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 of each of the one or more equality constraints, wherein each parameterized potential function is defined by an equality function and a center of attraction parameter; define a joint set of equations, the joint set of equations including a first order optimality condition of the unconstrained convex optimization problem and one or more equality constraints with associated weights; and determine an estimated state of the electric power system by solving the joint set of equations using a root-finding algorithm. The method may also include using at least one hardware processor to control the electric power system based on the estimated state.

[0019] It should be understood that any feature of the above-described methods may be implemented alone or in any combination with any subset of the other features. Therefore, to the extent that the appended claims imply specific dependencies between features, the disclosed embodiments are not limited to these specific dependencies. On the contrary, any feature described herein may be combined with any other feature described herein, or implemented in the form of any combination of features without one or more of the other features described herein. In addition, any method described above and elsewhere herein may be embodied alone 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

[0020] The details of the structure and operation of the present invention may be derived in part by studying the accompanying drawings, in which like reference numerals refer to like parts, and in which:

[0021] Figure 1 An example infrastructure is shown that may implement one or more of the processes described herein according to one embodiment;

[0022] Figure 2 An example processing system is shown that can perform one or more of the processes described herein according to one embodiment;

[0023] Figure 3 illustrates example data flows between systems and software modules within an example infrastructure according to one embodiment;

[0024] Figure 4 illustrates the operation of virtual measurement using center of attraction parameters according to one embodiment;

[0025] Figure 5 A solution process according to one embodiment is shown;

[0026] Figure 6 An overall algorithm for solving a constrained optimization problem according to one embodiment is shown;

[0027] Figure 7 An overall algorithm for solving a constrained optimization problem including one or more equality constraints and one or more inequality constraints according to one embodiment is shown; and

[0028] Figure 8 An algorithm for solving a constrained optimization problem including equality constraints is shown according to one embodiment. DETAILED DESCRIPTION

[0029] In an embodiment, a system, method, and non-transient computer-readable medium are disclosed, which uses parameterized potential functions for power system state estimation with equality constraints (and possible inequality constraints). It is generally expected that state estimation will be performed for power systems, such as power grids (e.g., power grids of any size, from large utility grids to microgrids or smaller grids). A power system can be any network of electrical components (e.g., power system equipment) configured to generate, store, supply, transmit, distribute, and / or consume electricity, including but not limited to power plants configured to produce electricity from combustible fuels (e.g., coal, natural gas, etc.) and / or renewable resources (e.g., wind energy, solar energy, nuclear energy, etc.), transmission systems configured to transport or transmit electricity from a source (e.g., a generator) to a load, and distribution systems configured to feed the supplied electricity to nearby homes, businesses, and / or other institutions. However, the disclosed method is not limited to power systems. On the contrary, the disclosed method can be applied to any system whose state is estimated using a WLS problem with one or more equality constraints.

[0030] 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 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, etc. It should be understood that other types of systems can be similarly represented as networks.

[0031] As used herein, the term "node" or "bus" generally refers to any point in a network where the state needs to be estimated when solving a SE problem. In the case of a power system, the state of each node can be defined as the voltage magnitude and phase angle at that node. However, the state of a node in a power system can 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.

[0032] After reading this specification, it will be clear 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 rather than by way of limitation. Therefore, the detailed description of various embodiments should not be interpreted as limiting the scope or breadth of the present invention set forth in the appended claims.

[0033] 1. System Overview

[0034] 1.1. Infrastructure

[0035] Figure 1 An example infrastructure is shown that can implement one or more of the disclosed processes according to an embodiment. The infrastructure may include a management system 110 (e.g., including one or more servers) that hosts and / or performs 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 the like. Management system 110 may include dedicated servers, or may alternatively be implemented in a computing cloud, where 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 placed in the same location (e.g., in a single data center) and / or geographically dispersed (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.

[0036] 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), Secure Hypertext Transfer Protocol (HTTPS), File Transfer Protocol (FTP), Secure File Transfer Protocol (FTPS), Secure Shell File Transfer Protocol (SFTP), Extensible Messaging and Presence Protocol (XMPP), Open Field Message Bus (OpenFMB), IEEE Smart Energy Profile Application Protocol (IEEE 2030.5), etc.) as well as proprietary protocols. Although the management system 110 is shown as being connected to various systems via a set of networks 120, it should be understood that the management system 110 may be connected to various systems via a different set of one or more networks. For example, the management system 110 may be connected to a subset of the user systems 130 and / or the power system 140 via the Internet, but may also be connected to one or more other user systems 130 and / or the power system 130 via an intranet. Furthermore, while only a few user systems 130 and power systems 140, one instance of software 112, and a set of (one or more) databases 114 are shown, it should be understood that the infrastructure may include any number of user systems, power systems, software instances, and databases.

[0037] The user system(s) 130 may include any type of computing device capable of wired and / or wireless communication, including, but not limited to, a desktop computer, a laptop computer, a tablet computer, a smartphone or other mobile phone, a server, a game console, a television, a set-top box, an electronic kiosk, a point-of-sale terminal, an embedded controller, a programmable logic controller (PLC), etc. However, it is generally contemplated that the user system(s) 130 will include a personal computer, a mobile device, or a workstation through which an agent of an operator of the power system 140 may interact with the management system 110. These interactions may include inputting data (e.g., parameters for configuring one or more processes described herein) and / or receiving data (e.g., output of one or more 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 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, animations, references (e.g., hyperlinks), frames, inputs (e.g., text boxes, text areas, check boxes, radio buttons, drop-down menus, buttons, forms, etc.), scripts (e.g., JavaScript), etc., including elements that contain or are derived from data stored in one or more databases (e.g., database 114).

[0038] The management system 110 may execute software 112 that includes one or more software modules that implement one or more of the disclosed processes. In addition, the management system 110 may include, be communicatively coupled to, or otherwise have access to one or more databases 114 that store data input to and / or output from one or more of the disclosed processes. Any suitable database may be used, including but not limited to MySQL. TM 、Oracle TM 、IBM TM , Microsoft SQL TM 、Access TM , PostgreSQL TM etc., including cloud-based databases, proprietary databases, and unstructured databases.

[0039] 1.2. Example Processing Equipment

[0040] Figure 2 is 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 functions, processes, or methods (e.g., storing and / or executing software 112) described herein, and may represent a component of a management system 110, (one or more) user systems 130, (one or more) 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. It will be apparent to those skilled in the art that other computer systems and / or architectures may also be used.

[0041] 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 that is slave 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 stand-alone 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 processor available from Intel Corporation of Santa Clara, California (e.g., Pentium TM 、Core i7 TM , Xeon TM, etc.), any processor available from Advanced Micro Devices, Incorporated (AMD) of Santa Clara, California, any processor available from Apple Inc. of Cupertino (e.g., A-series, M-series, etc.), or any processor available from Samsung Electronics Co., Ltd. of Seoul, South Korea (e.g., Exynos TM )wait.

[0042] The processor 210 is preferably connected to the communication bus 205. The communication bus 205 may include a data channel for facilitating the transfer of information between the 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, for example, a bus architecture that complies with the Industry Standard Architecture (ISA), the Extended Industry Standard Architecture (EISA), the Micro Channel Architecture (MCA), the Peripheral Component Interconnect (PCI) local bus, the standards promulgated by the Institute of Electrical and Electronics Engineers (IEEE), including the IEEE 488 General Purpose Interface Bus (GPIB), the IEEE 696 / S-100, etc.

[0043] 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 executed on the processor 210, such as one or more functions and / or modules discussed herein (e.g., software 112). It should be understood that the programs stored in the memory and executed by the processor 210 can be written and / or compiled according to any suitable language, including but not limited to C / C++, Java, JavaScript, Perl, VisualBasic, .NET, etc. 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), etc., including read-only memory (ROM).

[0044] The secondary memory 220 may optionally include internal media 225 and / or removable media 230. The removable media 230 is read and / or written in any known manner. The removable storage medium 230 may be, for example, a tape drive, a compact disk (CD) drive, a digital versatile disk (DVD) drive, other optical disk drives, flash drives, etc. 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.

[0045] 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 to the system 200 from an external storage medium 245. Examples of external storage media 245 may include an external hard drive, an external optical drive, an external magneto-optical drive, etc. 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).

[0046] As described 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 other computing devices. The communication interface 240 preferably implements industry-promulgated protocol standards, such as Ethernet IEEE 802 standards, Fiber 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 customized or non-standard interface protocols.

[0047] 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 one embodiment, communication channel 250 may be a wired or wireless network (e.g., one or more networks 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, for example, wire or cable, optical fiber, conventional telephone line, cellular telephone link, wireless data communication link, radio frequency ("RF") link, or infrared link, etc.

[0048] 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 described elsewhere herein.

[0049] In this specification, the term "computer-readable medium" is used to refer to any non-transitory computer-readable storage medium for providing computer-executable code and / or other data to or within the system 200. Examples of such media include the main memory 215, the secondary memory 220 (including the internal memory 225 and / or the removable medium 230), the external storage medium 245, and any peripheral devices (including network information servers or other network devices) communicatively coupled to the communication interface 240. These non-transitory computer-readable media are means for providing executable code, programming instructions, software, and / or other data to the system 200.

[0050] In an embodiment 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 an embodiment, the software is loaded into the system 200 in the form of an electrical communication signal 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.

[0051] In one embodiment, I / O interface 235 provides an interface between one or more components of 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 the like. 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 the like. 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).

[0052] System 200 may also include optional wireless communication components that facilitate wireless communication over voice networks and / or data networks (e.g., where user system 130 is 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 sent and received over the air by antenna system 270 under the management of radio system 265.

[0053] In one 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.

[0054] In an alternative embodiment, the radio system 265 may include one or more radios configured to communicate on various frequencies. In one embodiment, the radio system 265 may combine a demodulator (not shown) and a modulator (not shown) in an integrated circuit (IC). The demodulator and modulator may also be separate components. In the input path, the demodulator removes the RF carrier signal, leaving a baseband receive audio signal that is sent from the radio system 265 to the baseband system 260.

[0055] If the received signal contains audio information (for example, in the case of user system 130 capable of operating as a telephone), 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, which is 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 (the signal is switched to the antenna port for transmission at the antenna system 270).

[0056] 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. The 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 after being received. Such computer programs, when executed, enable the system 200 to perform the various functions of the disclosed embodiments.

[0057] 1.3. Data flow example

[0058] 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 shown. 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.

[0059] The monitoring module 310 can monitor and collect data output by one or more sensors in the network of the power system 140. The monitoring module 310 can also derive data from the collected data. The monitoring module 310 can send or "push" the collected and / or derived data to the state estimation module 330 as system telemetry data (e.g., via an API of the state estimation module 300). Alternatively, the state estimation module 330 can 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 measurement data at each of one or more nodes or other points within the network of the power system 140. The system telemetry data may be transmitted from the monitoring module 310 to the state estimation module 330 in real time or periodically as the data is collected and / or derived. As used herein, the term "real time" includes simultaneous events, as well as simultaneous events indicated by ordinary delays caused by delays in processing, memory access, communication, etc.

[0060] The state estimation module 330 receives system telemetry data from the monitoring module 310 and uses the system telemetry data in combination with the system model 360 to generate an estimated state of the power system 140. Specifically, the state estimation module 330 can implement one or more state estimation processes described herein to establish an optimization problem based on the system telemetry data and the system model 360 and solve the optimization problem. The state estimation module 330 can 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 can 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 unit 340). Alternatively, the optimization and control module 340 can 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 300). As estimated states are generated, the estimated states may be communicated from the state estimation module 330 to the optimization and control module 340 in real-time or periodically.

[0061] The optimization and control module 340 receives the estimated state from the state estimation module 330, and uses the estimated state in combination with the system model 360 to determine the optimal configuration of one or more components of the power system 140, and then controls the power system 140 to switch to the optimal configuration. Specifically, the optimization and control module 340 can generate control signals, which are transmitted to the control module 320 of the power system 140. For example, the control signal can be sent via the API of the control module 320. When the estimated state is received and analyzed regularly, or in response to user operations, the control signal can be transmitted 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. The optimization and control module 340 can 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.

[0062] The control module 320 of the power system 140 receives the control signal from the optimization and control module 340 and controls one or more components of the power system 140 according to the control signal. Examples of such control include setting a set point (e.g., active power and / or reactive power for distributed energy resources), adjusting the power output of a generator, adjusting the charging or discharging of a BES system, adjusting the power input of a load, closing or opening a switch (e.g., a circuit breaker), etc.

[0063] The human-machine interface 350 may generate a graphical user interface (which is transmitted to the user system 130) and receive input to the graphical user interface via the user system 130. The graphical user interface may provide information about the estimated state of the power system 140 determined by the state estimation module 330, the optimal configuration of the power system 140 determined by the optimization and control module 340, the control decision or recommendation determined by the optimization and control module 340, the visual representation of the system model 360, etc. In addition, the graphical user interface may provide inputs that enable a user of the user system 130 to configure settings of the state estimation module 330, configure settings of 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, etc.

[0064] 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 in any known manner (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).

[0065] 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 350, and / or other modules within the management system 110 or an external system. In the case where 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 implementation module 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 300) or from the optimization and control module 340 (e.g., via the API of the optimization and control module 340). When the estimated state is generated, the estimated state can be transmitted to the implementation module in real time or periodically.

[0066] As an example of downstream functionality, the estimated state may be stored and, in response to a triggering event, displayed to a user within a graphical user interface of the human-machine interface 350. The triggering event may be a user requesting the estimated state, the estimated state satisfying an alarm condition, etc. If the estimated state satisfies the alarm condition, the user may be prompted to perform a preventive or corrective control action (e.g., input via a graphical user interface, manually, etc.) via the graphical user interface or other means (e.g., a notification sent via email, text message, voice message, etc.).

[0067] In general, the estimated state (which may include the voltage magnitude and phase angle of each node in the network of the power system 140) can be used to generate a baseline model for one or more downstream functions. The downstream function can use the baseline model to perform any type of analysis on the power system 140, including optimal power flow, distributed energy (DER) management, emergency analysis, etc. The analysis can be performed in response to user operations, or automatically performed in real time or periodically. In some cases, the analysis results can be provided to the user via the human-machine interface 350. In other cases, the optimization and control module 340 can automatically (i.e., without user intervention) or semi-automatically (e.g., approved or confirmed by the user) initiate a control operation based on the analysis results. The initiation of the control operation may include sending a control command to the control module 320 of the power system 140, which can responsively control the power system 140 according to the control command.

[0068] For example, the optimization and control module 340 can use the estimated state as an input for the contingency analysis. The contingency analysis can utilize 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., failures 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 consumption at the node for the contingency analysis. The contingency analysis can be performed in real time (e.g., when the state estimation module 330 outputs the estimated state), periodically, and / or in response to a triggering event (e.g., user request, one or more monitoring criteria are met, etc.). If the contingency analysis detects a problem, the user can perform preventive or corrective control operations (e.g., manual input through a graphical user interface of the human-machine interface 350, etc.). Alternatively or additionally, the optimization and control module 340 can automatically or semi-automatically perform contingency analysis on one or more hypothetical scenarios, and when a problem is detected, initiate preventive or corrective control operations through communication with the control module 320 of the power system 140.

[0069] As an additional example, the optimization and control module 340 can use the estimated state as an input to Volt-Var optimization. Volt-Var optimization can use the estimated state (e.g., voltage amplitude 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). Volt-Var optimization can be performed in real time (e.g., when the state estimation module 330 outputs the estimated state), periodically, and / or in response to a triggering event (e.g., user request, satisfaction of one or more monitoring criteria, etc.). The optimization and control module 340 can automatically or semi-automatically initiate control operations through communication with the control module 320 of the power system 140 to make the power system 140 comply with the optimal voltage level and reactive power determined by the Volt-Var optimization. The control operations can include controlling switchable capacitors, on-load tap changers, etc.

[0070] As an additional example, the optimization and control module 340 can use the estimated state as an input to the optimal power flow. The optimal power flow can use the estimated state and the system model 360 (or other model) as a baseline to manage power generation within the power system 140. For example, the optimal power flow can determine the optimal set point of 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 can be executed in real time (e.g., when the state estimation module 330 outputs the estimated state), periodically, and / or in response to a triggering event (e.g., a user request, meeting one or more monitoring criteria, etc.). The optimization and control module 340 can automatically or semi-automatically initiate a control operation through 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.

[0071] As an additional example, the optimization and control module 340 can use the estimated state as an input to distributed energy resource (DER) management, which can be considered as an optimal power flow. DER management can utilize the estimated state and the system model 360 (or other models) to manage the distributed energy resources within the power system 140. For example, DER management can manage the set points of active and reactive power, power factor, and / or voltage at the interconnection point with the distributed energy resources or other nodes within the network. DER management can be performed in real time (e.g., when the state estimation module 330 outputs the estimated state), periodically, and / or in response to a triggering event (e.g., a user request, one or more monitoring criteria are met, etc.). The optimization and control module 340 can automatically or semi-automatically initiate control operations through 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.

[0072] 2. Process Overview

[0073] An embodiment of a process for using parameterized potential functions for equality-constrained power system state estimation 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 (one or more) processors 210 of the management system 110. The described process may be implemented as instructions represented in source code, object code, and / or machine code. These instructions may be executed directly by (one or more) hardware processors 210, or may be executed by a virtual machine or container running between the object code and the hardware processor 210. In addition, the disclosed software may be built on or connected to one or more existing systems.

[0074] Alternatively, the described process can 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 herein according to their functions. Whether such functions are implemented as hardware or software depends on the specific application and the design constraints imposed on the entire system. The technician can implement the described functions in different ways for each specific application, but such implementation decisions should not be interpreted as causing deviations from the scope of the present invention. In addition, the functional grouping within the component, block, module, circuit, or step is for ease of description. Without departing from the present invention, a specific function or step can be moved from one component, block, module, circuit, or step to another.

[0075] In addition, although the processes described herein are illustrated with certain arrangements and sequences of subprocesses, each process can be implemented with fewer, more, or different subprocesses and subprocesses in different arrangements and / or sequences. In addition, it should be understood that any subprocess that is not dependent on the completion of another subprocess can be performed before, after, or in parallel with other independent subprocesses, even if the subprocesses are described or illustrated in a specific order.

[0076] 2.1. introduction

[0077] In this disclosure, equality constraints will be expressed in the following form:

[0078] c(x)=0

[0079] Where c(·) is an equation function relating input x to output c(x). The input x represents the state of the system being monitored and controlled (e.g., power system 140). In the case of 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 equality constraint can be expressed in this manner as well as in other ways. Therefore, for ease of understanding and conceptualization, it will be assumed that all equality constraints are expressed in the above form. However, it should be understood that in actual implementations, equality constraints can be expressed in different ways, can be converted to the above form, or can be used in their native form after appropriate modifications to other components of the SE problem.

[0080] The constrained optimization problem can be expressed as:

[0081]

[0082] Submission

[0083] c i (x)=0 for i=1,…,n EC

[0084] where i is the index of the set of equality constraints to which the objective function f(x) obeys, and n EC represents the total number of equality constraints to which the objective function f(x) is subject. It should be understood that an "optimization problem" may include any problem of finding the best solution from among all feasible solutions, and an "objective function" may be any function to be maximized or minimized. Furthermore, the term "constrained" refers to problems in which the objective function is subject to corresponding constraints, while the term "unconstrained" refers to problems in which all corresponding constraints have been transformed into the objective function. The term "convex" refers to any function for which the line segment between any two points on the graph of the function does not lie below the graph between the two points.

[0085] Constrained optimization problems can also include inequality constraints. In this disclosure, inequality constraints will be expressed in the following form:

[0086] g(x)≤0

[0087] Where g(·) is an inequality function relating the input x to the output g(x). It should be understood that any inequality constraint can be expressed in this way as well as 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, inequality constraints can be expressed in different ways, converted to the above form, or used in their native form after appropriate modifications to other components of the SE problem.

[0088] Each equality constraint c(x)=0 indicates that the measurement function corresponding to the input x must be equal to zero. However, in practice, a tolerance ε may be allowed so that -ε≤c(x)≤ε is allowed. The range [-ε,ε] represents a feasible range of values ​​for the equation function c(x). Similarly, the inequality constraint g(x)≤0 represents a possible range of values ​​for the measurement value corresponding to the input x. 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), etc. In practice, a tolerance may be allowed so that the feasible range is represented by g(x)≤ε. The tolerance ε may be the same or different for equality and inequality constraints.

[0089] Existing methods for implementing equality constraints in state estimation include elimination, feasible direction, Lagrangian (e.g., Hachtel's matrix formulation), augmented Lagrangian (using multipliers), and penalty function. In methods using penalty functions, the weight of each equality constraint usually needs to be set to a very large value in order to push violations of the equality constraint toward zero.

[0090] In one embodiment, instead of the above approach, a parameterized potential function is used for each equality 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., like a particle within a potential field is pulled toward a point of zero force). Although not required for this embodiment, the parameterized potential function can be a quadratic function. Each parameterized potential function can be expressed as:

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

[0092] Where u is the center of attraction parameter, which defines the "center of attraction" that conceptually acts like a spring, pulling the output of the equation function c(x) toward zero. When c(x) < -ε or c(x) > ε, the value of c(x) violates its corresponding constraint. Conversely, when -ε ≤ c(x) ≤ ε, the value of c(x) does not violate its corresponding constraint. The tolerance ε is a practical consideration because in numerical (computer) embodiments with finite precision, the equality constraints may not be satisfied exactly. The value of ε can be set using any standard method.

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

[0094]

[0095] in,

[0096]

[0097] Where f(x) is the objective function of the constrained optimization problem to be minimized. It should be understood that f(x) can be a WLS problem:

[0098]

[0099] Among them, z l represents the system telemetry data at point l, where h(·) is the measurement function relating input x to output h(x), represents the predicted measurement value given input x, and w l are the corresponding weights, and m represents the total number of points being measured in the system model 360. More generally, however, f(x) encompasses any convex objective function to be minimized.

[0100] Each parameterized potential function can be considered as a virtual measurement. Figure 4 FIG. 2 shows the operation of virtual measurement using the attraction center parameter u according to an embodiment. The dotted line in the figure represents the gradient of the quadratic equation function c(x). Figure 4 As shown, each virtual measurement utilizes the attraction center parameter u. The value of the attraction center parameter u defines the location (i.e., the attraction center) to which c(x) is attracted, and the weight w represents the attraction strength. In other words, the value of the attraction center parameter u can be likened to the anchor point to which one end of a spring is attached, and the other end of the spring is attached to c(x), and the weight w is likened to the Hooke constant of the spring.

[0101] When iteratively solving UCP(u), the value of each center of attraction parameter u can be updated to "pull" the value of c(x) toward zero and eventually into the feasible range defined by the tolerance, rather than solving the original constrained optimization problem. For example, the value of the center of attraction parameter u can start from zero (i.e., u (0) ), the problem UCP(u (0) ) is solved until convergence, producing c(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, producing c(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, producing c(x (2) ). This iterative process may continue until the value of c(x) is within the feasible range (zero, with a tolerance of ε).

[0102] The force field associated with a potential field P(x,u) is defined by the negative gradient of P(x,u):

[0103]

[0104] The stationary point of the force field is the parameterization of the center of attraction parameter u of the parameterized potential function:

[0105]

[0106] If x(u) satisfies the equality 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:

[0107]

[0108] Among them, λ i is the Lagrange multiplier for the equality constraint with index i.

[0109] The following joint system has the same solution as the KKT condition for constrained optimization problems:

[0110]

[0111] c i (x)=0,i=1,…,n EC

[0112] Parameters of the parameterized potential function and the Lagrange multiplier λ i The following are related:

[0113] λ i =-2w i u i

[0114] For any KKT point in a constrained optimization problem, there exists an associated unconstrained convex optimization of P(x,u) where the center of attraction parameter u of each parameterized potential function is uniquely defined for any non-zero weight w. Therefore, 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.

[0115] 2.2. Equality Constraint Algorithm

[0116] Figure 5 A solution process 500 is shown 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 .

[0117] In subprocess 510, an unconstrained convex optimization problem is prepared based on the equality constraints. In particular, a parameterized potential function can be generated for each equality constraint and summed with the objective function from the constrained optimization problem in the second objective function to be minimized. Similarly, the unconstrained convex optimization problem can be expressed as:

[0118]

[0119] Among them, w i (c i (x)-u i ) 2 Indicates size n EC The parameterized quadratic potential function of the equality constraint with index i in the set of equality constraints .

[0120] In subprocess 520, the center of attraction parameters u in all parameterized potential functions are determined. i In an embodiment of the iterative execution process 500, the center of attraction parameter k-1 determined in the immediately previous iteration may be used to determine the center of attraction parameter k-1. The value of the center of attraction parameter of the current iteration k is calculated The value of the center of attraction parameter u in the parameterized potential function i The value of can also be based on the equation function c in the same parameterized potential function i In other words, the value of the center of attraction parameter u can be adjusted based on the amount of violation.

[0121] In one embodiment, each center of attraction parameter u i The value of can be determined as follows:

[0122] if Empty,

[0123] otherwise,

[0124] 0<α≤1

[0125] Among them, u initial is a predefined value, α is a scaling factor, which can be determined empirically or in any appropriate manner, or can be omitted or set to α=1 to effectively omit. initial =0.

[0126] In subprocess 530, given the center of attraction parameter u determined in subprocess 520, the unconstrained convex optimization problem is solved by finding the input x that minimizes P(x,u) In an embodiment of the iterative execution process 500, the initial input x in iteration k (k),ini Can be set to the final input x (k-1),fin , the final input x (k-1),fin is the output from the immediately previous iteration k-1 that is the optimization input (i.e., the input that minimizes the second objective function). The result of subprocess 530 and process 500 will be the final optimization 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 .

[0127] Figure 6 An overall equality constraint (EC) algorithm 600 for solving a constrained optimization problem including equality constraints is shown according to an embodiment. The EC algorithm 600 may be implemented by the state estimation module 330 in the software 112 of the management system 110. The EC algorithm 600 is used to convert a constrained optimization problem (including an objective function to be minimized under one or more equality constraints) into an unconstrained convex optimization problem and then solve the unconstrained convex optimization problem. When solving the unconstrained convex optimization problem, the EC algorithm 600 may iteratively perform the solution process 500.

[0128] 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 an equality function of an equality constraint to which the objective function is subject. Similarly, the constrained optimization problem may be expressed as:

[0129]

[0130] Submission

[0131] c i (x)=0,i=1,…,n EC

[0132] Components of the constrained optimization problem, e.g., the objective function and any equality constraints or equality functions, may be stored in a memory that persists across iterations of the solution process 500 so that they may be easily accessed in each iteration of the solution process without having to be regenerated or re-determined in each iteration.

[0133] In subprocess 620, the constrained optimization problem is transformed into an unconstrained convex optimization problem, where each equality constraint is expressed as a parameterized potential function. The unconstrained convex optimization problem can be expressed as:

[0134]

[0135] where i is the index of the set of equality constraints, n EC is the size of the set of equality constraints. In an embodiment, u initial =0.

[0136] The solving process 500 is performed to find the optimal input x that minimizes the unconstrained convex optimization problem (0),fin At each subsequent iteration k (if any) of the solution process 500, the optimal input x found in the immediately preceding iteration k-1 is (k -1),fincan be used as the initial input x to the unconstrained convex optimization problem to be solved in iteration k of the solution process 500 (ie, in subprocess 530 ). (k),ini In one embodiment, in each iteration of the solution process 500, a parameterized potential function is included in each equality-constrained unconstrained convex optimization problem:

[0137]

[0138] Each center of attraction parameter u in iteration k of the solution process 500 is (k) The corresponding center of attraction parameter u in the immediately previous iteration k-1 of the solution process 500 may be (k-1) or if there is no previous iteration k-1 can be initialized to u initial (e.g., 0). Given the parameters of the unconstrained convex optimization problem in the current iteration k, the output of each iteration of the solution process 500 will be the final input x (k),fin (denoting optimized input).

[0139] In subprocess 630, the most recently found input x is determined (k),fin Whether any of the (one or more) equality constraints are violated. In one embodiment, an equality constraint is violated when the value of the equality function of the equality constraint is not within a tolerance ε of the constraint value (e.g., c(x)<-ε or c(x)>ε if the constraint value is zero). Conversely, in such an embodiment, an equality constraint is not violated when the value of the equality function of the equality constraint is within a tolerance ε of the constraint value (i.e., -ε≤c(x)≤ε if the constraint value is zero). If no equality constraints are violated (i.e., "No" in subprocess 630), the EC algorithm 600 proceeds to subprocess 640. Otherwise, if at least one equality constraint is violated (i.e., "Yes" in subprocess 630), the algorithm performs another iteration of the solution process 500.

[0140] In subprocess 640, once the equality constraints are not violated (ie, "No" in subprocess 630), the most recently found (ie, last) optimization input is output as the final estimated state of power system 140. This will be the input x (k),fin , output by the last iteration of the solution process 500. It should be appreciated that this final estimated state of the power system 140 output by the subprocess 640 is sent to the optimization and control module 340, which can use the final estimated state to optimize the configuration of the power system 140 and / or control the power system 140.

[0141] An example of EC pseudo-code implementing an embodiment of EC algorithm 600 is provided below:

[0142] 01.k=0.

[0143] 02. Initialize the problem to P (0) .

[0144] 03. Use x (0),ini and Initialize P (0) .

[0145] 04. Solve for P (k) until convergence to find x (k),fin .

[0146] 05. If |c i (x (k),fin )|≤ε, Then output x (k),fin and stop.

[0147] 06.

[0148] 07.x (k+1),ini =x (k),fin

[0149] 08.k=k+1.

[0150] 09. Go to 04.

[0151] It should be understood that how iteration k is defined or interpreted is arbitrary. For example, in the above EC pseudocode, for the future iteration k+1, when solving the unconstrained convex optimization problem P (k) After that, determine the attraction center parameter u i However, iteration k can alternatively be interpreted as including in solving the unconstrained convex optimization problem P (k) Before, determine the attraction center parameter u i In any case, in any of the examples disclosed, the specific explanation of iteration k should not be construed as limiting in any respect.

[0152] 2.3 Inequality Constraint Algorithm

[0153] Figure 7An overall inequality constraint (IC) algorithm 700 for solving a constrained optimization problem including equality constraints (one or more) and inequality constraints (one or more) according to an embodiment is shown. The IC algorithm 700 can be implemented by the state estimation module 330 in the software 112 of the management system 100. The IC algorithm 700 is used to convert an IC-EC optimization problem (including an objective function minimized under one or more equality constraints and one or more inequality constraints) into an EC convex optimization problem (i.e., a problem that obeys equality constraints but does not obey any inequality constraints) over one or more iterations, and then use the EC algorithm 600 to solve the EC convex optimization problem, wherein the EC algorithm 600 converts the EC convex optimization problem into a completely unconstrained convex optimization problem (i.e., a problem that does not obey any inequality or equality constraints).

[0154] Initially, in subprocess 710, an IC-EC optimization problem is obtained. This obtaining process may include retrieving the IC-EC optimization problem or retrieving a representation of one or more components of the IC-EC optimization problem, such as an objective function to be minimized, an inequality function of the inequality constraint(s) to which the objective function is subject, and an equality function of the equality constraint(s) to which the objective function is subject. The IC-EC optimization problem may be expressed as:

[0155]

[0156] Submission

[0157] c i (x)=0,i=1,...,n EC

[0158] g j (x)≤0,j=1,...,n IC

[0159] and is equivalent to:

[0160]

[0161] Submission

[0162] g j (x)≤0,j=1,...,n IC

[0163] where i is the index into the set of equality constraints, j is the index into the set of inequality constraints, and n is the index into the set of equality constraints. EC is the size of the set of equality constraints, n ICis the size of the set of inequality constraints. Components of the constrained optimization problem, such as the objective function and any inequality or equality constraints or inequality and equality functions, can be stored in a memory that persists across iterations of the loop formed by the subprocesses 730-750 of the EC algorithm 600 and the IC algorithm 700 so that they can be easily accessed in each iteration without having to be regenerated or re-determined in each iteration.

[0164] In subprocess 720, a convex optimization problem is generated from a constrained optimization problem without any inequality constraints. In other words, the IC-EC optimization problem is converted into an EC-only optimization problem without any representation of any inequality constraints. That is, the parameterized potential function for any inequality constraints is not included in the EC optimization problem. Therefore, the convex optimization problem can be expressed as:

[0165]

[0166] But it does not obey any inequality constraints. It should be understood that It is an optimization problem without inequality constraints, but it is still an EC optimization problem.

[0167] Next, an initial iteration of the EC algorithm 600 is performed to solve the EC optimization problem by converting the unconstrained optimization problem to a completely unconstrained optimization problem (e.g., in subprocess 510).

[0168]

[0169] does not obey any equality and inequality constraints. Next, Solve until convergence in order to find (in subprocess 530) the minimum P (noIC) The input x (noIC),fin This input x (noIC),fin (e.g., in subprocess 640) is output as a possible solution to the optimization problem that may or may not satisfy the equality and inequality constraints. This solution process 500 can be performed iteratively while updating each center of attraction parameter u (e.g., in subprocess 520) i , until the equality constraint is not violated (ie, "No" in subprocess 630).

[0170] In subprocess 730, it is determined whether the most recently found input x violates any inequality constraints. It should be understood that in the first iteration of subprocess 730, the most recently found input to be evaluated in subprocess 730 will be x. (noIC),fin In all subsequent iterations of subprocess 720, the most recently found input to be evaluated in subprocess 720 will be the final input x found by the most recent iteration k-1 of EC algorithm 600. (k-1),fin. In one embodiment, the 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 an embodiment, 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 the inequality constraint is not violated (i.e., "No" in subprocess 730), the IC algorithm 700 proceeds to subprocess 760. Otherwise, if at least one inequality constraint is violated (i.e., "Yes" in subprocess 730), the IC algorithm 700 performs an iteration of subprocess 740, then executes subprocess 750, and then executes EC algorithm 600.

[0171] In subprocess 740, an unconstrained or EC convex optimization problem is generated based on the set of inequality constraints (one or more inequality constraints) that have been determined to be violated in subprocess 730. In particular, a parameterized potential function can be generated for each inequality constraint in the set of one or more inequality constraints that are violated, and the parameterized potential function can be summed with the objective function from the constrained optimization problem in the second objective function to be minimized. Although not required for this embodiment, the parameterized potential function can be a quadratic function. The EC convex optimization problem can be expressed as:

[0172]

[0173] Among them, w j (g j (x)-u j ) 2 Indicates size n VIC The parameterized quadratic potential function of the inequality constraint j in the set of violated inequality constraints. Note that this convex optimization problem without inequality constraints is still subject to equality constraints.

[0174] In subprocess 750, the center of attraction parameters u in all parameterized potential functions of the violated inequality constraints are determined. j The value of . can be based on the attraction center parameter determined in the immediately previous iteration k-1 The value of the center of attraction parameter of the current iteration k is calculated 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 center of attraction parameter u can be adjusted based on the amount of violation.

[0175] In one embodiment, for a parameterized potential function representing an inequality constraint, each center of attraction parameter u can be determined as follows j Values:

[0176] When the set of inequality constraints violated by the most recently found optimization input is determined to be non-empty,

[0177] and

[0178] and

[0179] When the set of inequality constraints violated by the most recently found optimization input is determined to be empty,

[0180] If g j (x (k-1) )≥0,

[0181] If g j (x (k-1) )<0,

[0182] Among them, 0<α≤1

[0183] Among them, u initial is a predefined value, α is a scaling factor that can be determined empirically or in any suitable manner, or can be omitted or set to α=1 to be effectively ignored. initial =0.

[0184] The EC convex optimization problem P has been defined in subprocesses 740 and 750. EC Afterwards, another iteration of the EC algorithm 600 is performed. However, the EC optimization problem P (noIC) is used as a problem that will be transformed into a completely unconstrained optimization problem in the first iteration of the EC algorithm 600, and the EC optimization problem is used as the fully unconstrained optimization problem P in each subsequent iteration k of the EC algorithm 600. (k) In each subsequent iteration k of the EC algorithm 600, the completely unconstrained convex optimization problem P (k) It can be expressed as:

[0185]

[0186] In subprocess 760, once no inequality constraints are violated (i.e., "No" in subprocess 730), the most recently found (i.e., last) optimization input is output as the final estimated state of power system 140. If no inequality violations are determined in the first iteration and thus there is only one iteration of subprocess 730, then this may be the input x output by the initial iteration of EC algorithm 600. (noIC),fin Otherwise, it will be the input x output by the last iteration of the EC algorithm 600(k),fin It should be appreciated that this final estimated state of power system 140 output by subprocess 760 may be sent to optimization and control module 340, which may use the final estimated state to optimize the configuration of power system 140 and / or control power system 140, as described elsewhere.

[0187] In IC algorithm 700, EC algorithm 600 is used to solve an EC optimization problem that is not subject to inequality constraints or (which is still subject to equality constraints). However, in alternative embodiments, a different algorithm may be used to solve the EC optimization problem and / or to convert the EC optimization problem into a completely unconstrained convex optimization problem. Therefore, EC algorithm 600 in IC algorithm 700 may be replaced by any other suitable algorithm for solving the EC optimization problem.

[0188] An example of IC pseudo-code implementing an embodiment of IC algorithm 700 is provided as follows:

[0189] 01. Initialization problem

[0190] 02. Use x (noIC),ini initialization

[0191] 03. Solve until convergence to find x (noIC),fin (e.g. using EC pseudocode).

[0192] 04. If g j (x (noIC),fin )≤ε, Then output x (noIC),fin and stop.

[0193] 05.k=0.

[0194] 06.V 1c ={j|g j (x (noIC),fin )>0}.

[0195] 07. Prepare and solve until convergence to find x (k),fin (e.g. using EC pseudocode).

[0196] 08. and j (x (k),fin )>0}.

[0197] 09. If A k ={} and then go to 11.

[0198] 10. If A k ≠{}Then renew k=k+1; and go to 7.

[0199] 11. If g j (x (k),fin )≤ε, Then output x (k),fin and stop.

[0200] 12. and j (x (k),fin )>0}.

[0201] 13. If A k ≠{}Then

[0202]

[0203] For j∈V IC ;as well as

[0204] For j∈A k .

[0205] 14. If A k ={}Then

[0206] For g j (x (k) ) ≥ 0; and

[0207] For g j (x (k) )<0.

[0208] 15.x (k+1),ini =x (k),fin

[0209] 16.k=k+1.

[0210] 17. Prepare and solve until convergence to find x (k),fin (e.g. using EC pseudocode).

[0211] 18.Go to 11.

[0212] Note that in the above pseudocode, the set includes all inequality constraints that have been determined to be violated in all previous iterations (including the current iteration k), and the set A (k) This includes any inequality constraints that are determined to be violated by the most recently found optimization input. In one embodiment, the initial value u of each center of attraction parameter u is initialis a predefined value, such as 0. However, it should be understood that other initialization values ​​(ie, non-zero initialization values) may be used for the attraction center parameter u, and the same or different initialization values ​​may be used for the attraction center parameter u. i (i.e., for equality constraints) and u j (i.e., for inequality constraints).

[0213] It should also be understood that how iteration k is defined or interpreted is arbitrary. For example, in the pseudocode above, for the future iteration k+1, in solving the unconstrained convex optimization problem Then, determine the set V IC and A and the attraction center parameter u j However, iteration k can alternatively be interpreted as consisting in solving the fully unconstrained convex optimization problem Before, determine the set V IC and A and the attraction center parameter u j , as shown in EC algorithm 600. In any case, in any example disclosed, the specific interpretation of iteration k should not be understood as limiting in any respect. It should also be understood that in IC algorithm 700, EC algorithm 600 performs iterations of the inner loop, which are referred to herein as iteration k and are different from the outer iteration k performed by IC algorithm 700.

[0214] 2.4 Alternative Embodiment of the Equality Constraint Algorithm

[0215] Figure 8 An EC algorithm 800 for solving a constrained optimization problem (including one or more equality constraints) according to an alternative embodiment is shown. The EC algorithm 800 may be implemented by the state estimation module 330 in the software 112 of the management system 110. In addition, the constrained optimization problem may be expressed as follows:

[0216]

[0217] Submission

[0218] c i (x)=0,i=1,…,n EC

[0219] In subprocess 810, an unconstrained parameterized convex optimization problem is generated based on the equality constraints, thereby converting the constrained optimization problem into an unconstrained convex optimization problem. In particular, a parameterized potential function (e.g., a quadratic function) can be generated for each equality constraint, and the parameterized potential function can be summed with the objective function from the constrained optimization problem in a second objective function to be minimized. The unconstrained convex optimization problem can be expressed as follows:

[0220]

[0221] Among them, w i (c i (x)-u i ) 2 Indicates that the determined size is n EC The parameterized quadratic potential function of the equality constraint indexed i in the set of inequality constraints. UC The solution to (u) is x(u), which is the parameter of u and must also satisfy the equality constraint.

[0222] In subprocess 820, a set of joint equations is defined. The set of joint equations includes the problem P UC First-order optimality conditions and equality constraints for (u):

[0223]

[0224] In subprocess 830, the joint set of equations defined in subprocess 820 is solved to find the optimal input x. The solution can be found using any root-finding algorithm, such as Newton's method (also known as the Newton-Raphson method), to produce a solution that satisfies the KKT conditions for the constrained optimization problem. In Newton's method, the input x is made ini As an initial guess, the function is approximated by its tangent line, and the x-intercept of this tangent line is calculated, which is usually smaller than x. ini The roots of the original function are more approximated. This process is repeated using the computed x-intercept from the previous iteration (as a new guess in each new iteration) until the solution converges to within tolerance.

[0225] In subprocess 840, the solution from subprocess 830 is output. For example, the solution is output as estimated states to optimization and control module 340. Optimization and control module 340 may then utilize the estimated states to optimize and / or control power system 140, as described elsewhere herein.

[0226] Use f(x)=w l (z l -h l (x) 2 , the constrained SE problem can be expressed as follows:

[0227]

[0228] Submission

[0229] c(x)=0

[0230] The unconstrained SE problem generated in subprocess 810 can be expressed as follows:

[0231]

[0232] The joint set of equations defined in subprocess 820 can be expressed as follows:

[0233]

[0234] The Newton-Raphson iteration equation used in subprocess 830 can be expressed for the joint system as:

[0235]

[0236] The upper left corner of the submatrix is ​​H T W h H+C T W c C, which is different from H in the Hachtel matrix in the Lagrangian method T W h H, the gain matrix that produces better conditions.

[0237] 3. Example Usage

[0238] In one 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 algorithms 600 and / or 700. Specifically, the state estimation module 330 can 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 an optimization problem subject to equality constraints and optional inequality constraints as an unconstrained convex optimization problem. By generating a parameterized potential function w(c(x)-u) for each equality constraint 2 , the optimization problem is formulated as an unconstrained optimization problem. The unconstrained optimization problem can also include a parameterized potential function w(g(x)-u) for each inequality constraint 2 The sum of any generated parameterized potential function derived from the system telemetry data and the system model 360 and representing the system (e.g., the power system 140) and the objective function f(x) is then minimized to find the optimization input x. Using the parameterized potential function representing the virtual measurements, the potential field of the resulting unconstrained optimization problem is shaped so that its stationarity conditions are the same as the KKT conditions of the constrained optimization problem.

[0239] The resulting unconstrained optimization problem may be iteratively prepared, using successively updated values ​​of the center of attraction parameter u corresponding to the constraints, and solved until no constraints are violated (i.e., all constraints are satisfied within tolerances) to produce a final estimated state. The final estimated state may be output to an 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 generation of a graphical user interface, contingency analysis, power flow analysis (e.g., scheduler or optimal power flow analysis), Volt-Var optimization, DER management, etc.

[0240] 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, thereby increasing the voltage at the measurement point.

[0241] It is noteworthy that the disclosed algorithms 600 and 700 are robust, easy to implement, and provide fast convergence. In one embodiment, the weights w in the parameterized potential function and / or the objective function f(x) can be adaptively adjusted based on the convergence rate. In addition, although discussed in the context of state estimation, the disclosed algorithms 600 and 700 can be used for other applications. One such application is training machine learning models, such as neural networks, particularly physics-based neural networks (PINNs).

[0242] The above description of the disclosed embodiments is provided to enable those skilled in the art to make or use the present invention. Various modifications to these embodiments will be 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 should be understood that the description and drawings presented herein represent the presently preferred embodiments of the present invention and therefore represent the subject matter broadly contemplated by the present invention. It is further understood that the scope of the present invention fully encompasses other embodiments that may become apparent to those skilled in the art, and therefore the scope of protection of the present invention is not limited.

[0243] 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" may include any combination of A, B, and / or C, and may include multiple A, multiple B, or multiple C. Specifically, "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 include 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 thereof constituting A, B, or C. For example, the combination of A and B may include one A and multiple Bs, multiple A and one B, or multiple A and multiple Bs.

Claims

1. A state estimation method for a 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 under one or more equality constraints; The constrained optimization problem is converted into an unconstrained convex optimization problem, wherein the unconstrained convex optimization problem includes a second objective function, which is the sum of the first objective function and a parameterized potential function of each of at least one equality constraint of the one or more equality constraints, wherein Each parameterized potential function is defined by an equation function and a center of attraction parameter; as well as performing a solution process, the solution process comprising 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 equality 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 according to claim 1, wherein: Each parameterized potential function is defined as: w(c(x)-u) 2 Wherein, w is the weight, x is the input, c(x) is the equation function, and u is the center of attraction parameter.

4. The method according to claim 3, wherein: The second objective function is defined as: Wherein, f(x) is the first objective function, i is an indexed set of equality constraints, and the parameterized potential function of each equality constraint in the indexed set of equality constraints is included in the unconstrained convex optimization problem.

5. The method of claim 1 , further comprising using the at least one hardware processor to perform a first process, the first process comprising: performing an initial iteration of the solution process to find an initial optimization input, wherein the initial iteration of the solution process further comprises initializing a value of each center of attraction parameter in each parameterized potential function in the unconstrained convex optimization problem; determining whether the initial optimization input violates the one or more equality constraints; When it is determined that the one or more equality constraints are not violated, outputting the initial optimization input as an output estimated state of the power system; and When at least one of the one or more equality constraints is determined to be violated, performing the solving process in one or more subsequent iterations until the one or more equality constraints are not violated, wherein the subsequent iterations of the solving process further include updating the value of each center of attraction parameter in each parameterized potential function corresponding to the one or more equality constraints before solving the unconstrained convex optimization problem, and The input found by the solving process is output in a last iteration of the one or more subsequent iterations as the output estimated state of the power system. 6 . The method of claim 5 , further comprising using the at least one hardware processor to control the power system based on the output estimate state.

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

8. The method according to claim 5, wherein: In each of the initial iteration and the one or more subsequent iterations, the unconstrained convex optimization problem includes parameterized potential functions for all of the one or more equality constraints.

9. The method according to claim 5, wherein: In each of the one or more subsequent iterations, initial inputs used to solve the unconstrained convex optimization problem in a solution process in the subsequent iteration are inputs found by the solution process in an immediately previous iteration.

10. The method according to claim 5, wherein: In each of the one or more subsequent iterations, updating 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 in an immediately preceding iteration.

11. The method according to claim 10, wherein: In each of the one or more subsequent iterations, updating the value of each center of attraction parameter, given the input found by the solution process in the immediately previous iteration, includes further calculating the value of each center of attraction parameter based on the value of an equation function of a corresponding one of the one or more equality constraints.

12. The method according to claim 11, wherein: The value of each center of attraction parameter in subsequent iteration k is calculated as follows: u (k) =u (k-1) -c(x (k-1) ) Among them, u (k) is the value of the center of attraction parameter in the subsequent iteration k, u (k-1) is the value of the attraction center parameter in the immediately preceding iteration k-1, x (k-1) is the input found in the immediately preceding iteration k-1, c(x (k-1) ) is given the input x (k-1) In the case of , the value of the equation function in the parameterized potential function including the center of attraction parameter.

13. The method according to claim 5, wherein: The constrained convex optimization problem further includes one or more inequality constraints, and the method further includes using the at least one hardware processor to: executing a first process while ignoring the one or more inequality constraints to find the output estimate state as a first initial optimization input; determining whether the first initial optimization input violates the one or more inequality constraints; When it is determined that the one or more inequality constraints are not violated, using the first initial optimization input as a final estimated state of the power system; When at least one of the one or more inequality constraints is determined to be violated, Until the one or more inequality constraints are not violated, in one or more outer iterations, converting the constrained optimization problem into an inequality-free convex optimization problem, the inequality-free convex optimization problem comprising the first objective function and an inequality parameterized potential function for each of at least one inequality constraint of the one or more inequality constraints, wherein each inequality parameterized potential function is defined by an inequality function and a center of attraction parameter, determining a value of each center of attraction parameter in each inequality parameterized potential function in the convex optimization problem not subject to inequality constraints, and performing the first process using the inequality-free convex optimization problem as the constrained optimization problem in the first process, and The input found by the first process is outputted in a last iteration of the one or more outer iterations as a final estimated state of the power system.

14. The method according to claim 13, wherein: In each of the one or more outer iterations, an inequality parameterized potential function is added to the inequality-free convex optimization problem for each inequality constraint in the one or more inequality constraints that is violated, but not for each inequality constraint in the one or more inequality constraints that is not violated.

15. The method according to claim 13, wherein: For an inequality constraint expressed as g(x)≤0, each inequality parameterized potential function component 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 state in one or more of contingency 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 comprising a convex first objective function to be minimized under one or more equality constraints, converting the constrained optimization problem into 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 of each of at least one equality constraint of the one or more equality constraints, wherein each parameterized potential function is defined by an equality 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 equality constraints within a tolerance, wherein the input represents an estimated state of the power system.

19. A method for state estimation for an electric power system, the method comprising using at least one hardware processor to: Obtaining a constrained convex optimization problem, the convex constrained optimization problem comprising a convex first objective function to be minimized under one or more equality constraints; The constrained convex optimization problem is converted into an unconstrained convex optimization problem, wherein the unconstrained convex optimization problem includes a second objective function, which is the sum of the first objective function and a parameterized potential function of each of the one or more equality constraints, wherein Each parameterized potential function is defined by an equation function and a center of attraction parameter; defining a joint set of equations, the joint set of equations comprising a first-order optimality condition for the unconstrained convex optimization problem and one or more equality constraints with associated weights; as well as An estimated state of the power system is determined by solving the joint set of equations using a root-finding algorithm.

20. The method of claim 19, further comprising using the at least one hardware processor to control the power system based on the estimated state.

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