PMU optimal placement method for distribution network based on improved grey wolf algorithm

Abstract

Based on the improved grey wolf optimization algorithm, the PMU optimal configuration method for distribution network is proposed. Under the condition of system full observability, the PMU optimal configuration mathematical model is established. Considering the influence of zero injection node and the system observability under single PMU interruption and single line outage, the constraint conditions of the established PMU optimal configuration mathematical model are modified. On the basis of grey wolf optimization algorithm, the population initialization, nonlinear convergence factor and position update weight coefficient are improved. The improved grey wolf optimization algorithm is used to solve the PMU configuration model under different scenarios. The method builds the optimal configuration model under different scenarios to achieve the goal of system full observability, and uses the improved grey wolf optimization algorithm to solve the PMU optimal configuration scheme. Compared with other algorithms, the method has better optimization effect.

Description

Technical Field

[0001] This invention relates to the field of PMU optimization configuration technology, and the technical problem it aims to solve is to provide a distribution network PMU optimization configuration method based on an improved gray wolf algorithm. Background Technology

[0002] Renewable energy sources, represented by wind power and photovoltaics, have become the main forms of new energy power generation. In recent years, with the integration of distributed power sources into the distribution network, the operation mode and structure of the distribution network system have changed. Simultaneously, the randomness, volatility, and dispersion of distributed power sources will increase the randomness of power flow in the distribution network, potentially leading to bidirectional power flow and increased voltage fluctuations at some nodes. As the foundation of distribution network situational awareness, state estimation provides more reliable data for the distribution system. Among these, Supervisory Control and Data Acquisition (SCADA) systems and Phasor Measurement Units (PMUs) are the operational basis for state estimation. Phasor Measurement Units (PMUs) can directly measure node voltages and branch current phasors with high data accuracy and low transmission delay, enabling better monitoring of system state changes. However, PMUs are expensive, and installing a PMU at every node in the system could generate massive amounts of data for the dispatch center. Therefore, a more reasonable PMU configuration strategy, i.e., optimized PMU configuration, needs to be considered.

[0003] Current research on PMU (Power Management Unit) optimization configuration largely focuses on achieving network observability as a constraint, optimizing the number and location of PMUs in areas such as measurement redundancy, topology changes, and line faults. Solution methods for PMU optimization configuration models are mainly divided into numerical algorithms and heuristic algorithms. Numerical algorithms are fast but have poor universality; heuristic algorithms have strong global search capabilities and are mainly suitable for solving nonlinear, high-dimensional, and multi-objective models. Therefore, based on existing research on PMU optimization configuration and its solution methods, seeking an effective and reliable method for studying the optimization configuration of distribution network PMUs is a worthwhile issue to consider. Summary of the Invention

[0004] To address the PMU optimization configuration problem in distribution networks caused by configuration constraints and solution issues, this invention provides a distribution network PMU optimization configuration method based on an improved Grey Wolf algorithm. This method constructs optimization configuration models under different scenarios with the goal of achieving complete system observability, and uses the improved Grey Wolf algorithm to solve the PMU optimization configuration scheme, achieving better optimization results compared to other algorithms.

[0005] The technical solution adopted in this invention is as follows:

[0006] The method for optimizing the configuration of distribution network PMUs based on the improved gray wolf algorithm includes the following steps:

[0007] Step 1: Under the condition that the system is fully observable, establish a mathematical model for PMU optimal configuration;

[0008] Step 2: Considering the impact of zero-injection nodes, as well as the system observability issues under single PMU interruption and single line power outage, modify the constraints of the PMU optimization configuration mathematical model established in Step 1.

[0009] Step 3: Improve the population initialization, nonlinear convergence factor, and position update weight coefficient based on the gray wolf optimization algorithm;

[0010] Step 4: Using the improved Grey Wolf optimization algorithm from Step 3, solve for optimized configuration schemes for the PMU configuration model under different scenarios.

[0011] In step 1, in a power system, if the voltage phasor of a node can be directly or indirectly measured, then the node is said to be observable; if all nodes in a system are observable, then the system is said to be completely observable. Whether a system is observable can be analyzed from two perspectives: algebraic observability and topological observability.

[0012] Algebraic observability, from a numerical analysis perspective, establishes the equation z = Hx + v between state variables and measurements in an n-node system, where z is an m-dimensional measurement; x is a 2n-1 dimensional voltage state variable; H is an m×(2n-1) dimensional measurement Jacobian matrix; and v is an m-dimensional measurement error phasor. The observability of the system is determined by whether matrix H has full rank.

[0013] Topological observability, from a topological perspective, views the nodes and branches in a system as sets of vertices V and edges E of a graph G. Assume the measurement network forms a subgraph G' = (V', E'), and has... If satisfied If all vertices of graph G are contained in subgraph G', then the system is topologically observable.

[0014] The computation of high-dimensional matrix H in the definition of algebraic observability is large. In comparison, the topological observability analysis method is easier to implement.

[0015] In step 1, a fully observable power grid is defined when all nodes in the system are monitored directly or indirectly by the PMU. Therefore, the node where the PMU is located can directly monitor that node and the nodes connected to it. In indirect measurements, Kirchhoff's laws can be used to calculate the unknown current and voltage values ​​of the nodes. Therefore, the observability rule can be expressed as:

[0016] Rule 1: The voltage of the busbar where the PMU is installed and the current of the branch connected to it are directly measured by the installed PMU.

[0017] Rule 2: If the voltage and current phasors at one end of a branch are known, the voltage phasor at the other end of the branch can be obtained simply by using Ohm's law.

[0018] Rule 3: If the voltage phasors at both ends of a branch are known, Ohm's law can be used to calculate the current in that branch.

[0019] In step 1, the main objective of the PMU optimization configuration problem is to find the minimum PMU to achieve complete observability of the system. A mathematical model is a scientific or engineering model constructed using mathematical logic and mathematical language. In a narrow sense, a mathematical model refers only to the mathematical relationship structure that reflects a specific problem or a specific system of things; in this sense, it can also be understood as the mathematical expression connecting the variables within a system. Therefore, the mathematical model for PMU optimization configuration is expressed as shown in formula (1):

[0020]

[0021] In equation (1), c i This represents the cost required to install the PMU on node i, without considering the differences in PMU configuration due to installation costs on different nodes, and uniformly selecting c. i =1; x i It is a binary variable representing whether a PMU is installed on node i. If a PMU is installed on node i, then x is... i =1 otherwise 0; min represents the minimum value; ∑ represents the summation; n represents the number of nodes; i represents the i-th node; f represents the constraint expression; st represents the constraint symbol; X = (x1, x2, ..., x... n ) T B represents the configuration of PMUs at each node of the system; B is an n-dimensional column vector with all elements being 1; A is an n×n-dimensional node association matrix, whose elements a ij Represented as

[0022] In step 2, a zero-injection node refers to a node with no load or generator connection; in other words, the power injected into this node is zero. Therefore, according to the KCL, the sum of the branch currents connected to this node is zero. Considering zero-injection nodes reduces the number of PMUs required to achieve full observability, with the observability rules as follows:

[0023] Rule 4: If there exists an indirectly observable zero-injection node, and the voltage phasors of all but one of the adjacent buses are known, then the voltage phasors of the unknown node can be calculated using KCL.

[0024] Rule 5: If the voltage phasor of a zero-injection node is unknown and the voltage phasors of all its neighboring nodes are known, then the voltage phasor of the zero-injection node can be obtained through the node voltage equations.

[0025] Therefore, when considering the impact of zero-injection nodes, the constraint transformation in the PMU optimization configuration mathematical model is shown in Equation (2):

[0026]

[0027] In equation (2), x i It is a binary variable representing whether a PMU is installed on node i. If a PMU is installed on node i, then x is... i =1 if true, 0 otherwise; ∑ is the summation symbol; n represents the number of nodes; i represents the i-th node; f j Let a be the constraint expression symbol for the j-th node; ij It can be represented as Introducing an auxiliary binary variable y ij When the value is 1, it means that node j can be made observable based on the measurement data of the adjacent zero-injection node i; when the value is 0, it means that node j cannot be observed; I is the set of system nodes; Represents any symbol; j represents the j-th node; z i It is also a binary variable. A value of 1 indicates that node i is a zero-injection node, and a value of 0 indicates that node i is a non-zero-injection node.

[0028] In step 2, a single PMU interruption refers to the failure of any PMU in the system. Any PMU failure or communication channel interruption can lead to the loss of measurement data. To further improve system reliability and ensure system observability when a PMU is out of service, each node should be configured to be observable through two methods. Therefore, the constraints are modified, and the model is transformed into the form shown in equation (3):

[0029] f = AX ≥ 2B (3);

[0030] In equation (3), f is the constraint expression symbol; X = (x1, x2, ..., x n ) T This represents the PMU configuration of each node in the system; B is an n-dimensional column vector with all elements being 1. A is an n×n node association matrix, whose elements a ij It can be represented as

[0031] When considering zero-injection nodes, the constraints in the PMU optimization configuration mathematical model should be changed, as shown in formula (4):

[0032]

[0033] In equation (4), x jIt is a binary variable representing whether a PMU is installed on node j. If a PMU is installed on node j, then x is... j The summation is 1 if the summation is 1, otherwise it is 0. ∑ represents the summation symbol; n represents the number of nodes; i represents the i-th node; j represents the j-th node; f j Let a be the constraint expression symbol for the j-th node; ij It can be represented as Introducing an auxiliary binary variable y ij When z is 1, it means that node j can be made observable based on the measurement data of the neighboring zero-injection node i; when z is 0, it means that node j cannot be observed. j It is also a binary variable; a value of 1 indicates that node j is a zero-injection node, and a value of 0 indicates that node i is a non-zero-injection node. B is an n-dimensional column vector with all elements being 1. I represents any symbol; I is the set of system nodes.

[0034] In step 2, a single-line power outage refers to the loss of power to any single line in the system. Without power to any line, no measurement data can be obtained. Adding node measurement sources ensures that the system remains observable even during a single-line fault. Similar to the case of a single PMU interruption, to meet observability requirements, each node in the system:

[0035] 1) The node itself is equipped with a PMU device;

[0036] 2) Or the node has two or more adjacent nodes with PMU devices installed.

[0037] Therefore, the constraints are modified, and the transformation of the constraints in the PMU optimization configuration mathematical model is shown in formula (5):

[0038] f=(A+I)X≥2B (5);

[0039] In equation (5), f is the constraint expression symbol; X = (x1, x2, ..., x n ) T This represents the PMU configuration at each node of the system; B is an n-dimensional column vector with all elements being 1; I is an n×n identity matrix. A is an n×n node incidence matrix, whose elements a ij It can be represented as

[0040] When considering zero-injection nodes, the constraints in the PMU optimization configuration mathematical model should be changed, as shown in formula (6):

[0041]

[0042] In equation (6), x jIt is a binary variable representing whether a PMU is installed on node j. If a PMU is installed on node j, then x is... j The summation is 1 if the summation is 1, otherwise it is 0. ∑ represents the summation symbol; n represents the number of nodes; i represents the i-th node; j represents the j-th node; f j Let a be the constraint expression symbol for the j-th node; ij It can be represented as Introducing an auxiliary binary variable y ij When z is 1, it means that node j can be made observable based on the measurement data of the neighboring zero-injection node i; when z is 0, it means that node j cannot be observed. j It is also a binary variable; a value of 1 indicates that node j is a zero-injection node, and a value of 0 indicates that node i is a non-zero-injection node. B is an n-dimensional column vector with all elements being 1. I represents any symbol; I is the set of system nodes.

[0043] In step 2, the PMU optimization configuration problem under different scenarios is described in detail, and a mathematical model is established for it. Based on the research on PMU optimization configuration, a heuristic algorithm is used to solve the PMU optimization configuration model under various constraints.

[0044] In step 3, the Grey Wolf Optimization (GWO) algorithm is a novel metaheuristic optimization algorithm developed by Mirjalili et al. Inspired by nature, GWO mimics the leadership hierarchy and hunting mechanism of grey wolves in the wild. It involves four types of grey wolves to simulate the leadership hierarchy: alpha wolf, beta wolf, δ wolf, and w wolf. Furthermore, it implements the three main steps of hunting: surrounding the prey, chasing the prey, and attacking the prey. Its mathematical model is shown in formulas (7) and (8):

[0045] D = |C·X l (t)-X(t)| (7)

[0046] X(t+1)=X l (t)-A·D (8)

[0047] In equations (7) and (8), t represents the current iteration number; X(t) represents the position vector of the gray wolf; X l (t) is the position vector of the prey; D represents the step size surrounding the prey; X(t+1) represents the position vector of the gray wolf at time t+1; A and C are coefficient vectors, defined as shown in formulas (9) and (10):

[0048] A = 2a·r1-a (9)

[0049] C = 2·r² (10)

[0050] In equations (9) and (10), A and C are coefficient vectors; r1 and r2 are variables randomly selected within the interval [0, 1]; a is the convergence factor, which decreases linearly from 2 to 0 as t increases, as shown in equation (11):

[0051]

[0052] In equation (11), a is the convergence factor; t represents the current iteration number; T max is a constant representing the maximum number of iterations the algorithm can achieve during convergence.

[0053] In the GWO algorithm, guided by α, β, and δ, the wolf pack approaches the prey from all directions to achieve its hunting objective. The mathematical description of this behavior is shown in equations (12), (13), and (14):

[0054]

[0055]

[0056]

[0057] In equations (12), (13), and (14), D α D β D δ X represents the distance that the bottom wolf w needs to move to α, β, δ; while X α X β X δ X and X represent the current positions of α, β, δ and w, respectively; C1, C2, C3 and A1, A2, A3 represent the coefficient vectors of the movement distance and corresponding position of each wolf pack; X1, X2, X3 represent the position update of the bottom wolf w in the gray wolf pack according to α, β, δ; t represents the current iteration number; X(t+1) represents the last updated position of the wolf pack.

[0058] In step 3, improvements are made to the traditional gray wolf algorithm in terms of population initialization. In the traditional gray wolf algorithm, the distribution of the initialized wolf pack across the entire solution space affects the algorithm's optimization speed and solution accuracy. The traditional GWO algorithm randomly generates the initial positions of the gray wolf pack, but the randomization process often results in uneven population distribution within the search space and a lack of diversity. Therefore, the Bernoulli chaotic mapping is introduced to improve the wolf pack distribution attributes during initialization in the GWO algorithm, increasing the diversity and computational efficiency of the initialized individuals, further enhancing the algorithm's global search capability and avoiding getting trapped in local optima. The Bernoulli chaotic mapping is shown in formula (15):

[0059]

[0060] In equation (15), X n Let X be the nth chaotic variable; n is the dimension size; n+1 X is the (n+1)th chaotic variable; λ is the adjustment factor. n ∈(0,1) and λ=0.5, X0=0.8473.

[0061] As shown in formula (11), the convergence factor a is a straight line that decreases linearly from 2 to 0 with each algorithm iteration. However, due to the complexity of the gray wolf's prey search process, a linear convergence curve cannot accurately reflect the actual optimization search process. Therefore, a nonlinear parameter control strategy is an effective way to prevent premature convergence. Thus, a nonlinear convergence factor update strategy based on a sine function is used, as shown in formula (16):

[0062]

[0063] In equation (16), a is the convergence factor; t represents the current iteration number; T max is a constant representing the maximum number of iterations the algorithm can achieve during convergence. p is a control coefficient, the value of which determines the balance between the early exploration and later development of the algorithm.

[0064] Since the position update formula (14) for the wolf in the traditional GWO is based on the average of the updated positions of the α, β, and δ wolves, the alpha wolf's ability to guide the individual wolves in the pack is the same. However, in order to better mimic the hierarchical system of the gray wolf pack, dynamic weight coefficients are introduced into the position update formula, as shown in formulas (17), (18), (19), and (20):

[0065]

[0066]

[0067]

[0068]

[0069] In equations (17), (18), (19), and (20), w1, w2, and w3 represent the weight coefficients corresponding to α, β, and δ wolves, respectively; X1, X2, and X3 represent the position updates of the bottom wolf w in the gray wolf pack based on α, β, and δ; and X(t+1) represents the last updated position of the wolf pack.

[0070] In step 3, to verify the performance of the improved Grey Wolf Algorithm (IGWO), simulation tests were conducted on the improved Grey Wolf Algorithm, the traditional Grey Wolf Algorithm (GWO), the Particle Swarm Optimization Algorithm (PSO), and the Whale Optimization Algorithm (WOA) using standard functions. Six standard test functions were selected, including four unimodal functions and two multimodal functions. During the test, the dimension of all algorithms was uniformly set to 30, the maximum number of iterations was 500, and each of the four comparison algorithms was tested 30 times. The average and standard deviation of the results were taken as the final results, and convergence curves of different algorithms based on the standard functions were plotted.

[0071] Step 4 includes the following steps:

[0072] S1: Based on the network topology of the IEEE-33 node system and the relationships between the nodes, calculate the node association matrix A of the system.

[0073] S2: Initialize the parameters of the Grey Wolf Algorithm, including population size, maximum number of iterations, and parameters a, A, and C.

[0074] S3: Randomly initialize the location of the PMU in the system.

[0075] S4: Calculate the fitness function and define the prey position based on the current position.

[0076] S5: Updated to the top three best search agents X α X β X δ Update the coefficient vectors a, A, C, and update the gray wolf positions X1, X2, X3.

[0077] S6: Repeat steps S4 and S5 until the maximum number of iterations is reached, and finally output the result.

[0078] In step 4, considering the impact of zero-injection nodes, as well as single PMU interruptions and single-line power outages, the different test scenarios include the following six types:

[0079] Scenario 1 is a simulation analysis without considering zero-injection nodes;

[0080] Scenario 2 considers simulation analysis under zero-injection nodes;

[0081] Scenario 3 is a simulation analysis of a single PMU interrupt without considering zero-injection nodes;

[0082] Scenario 4 is a simulation analysis considering the interruption of a single PMU under a zero-injection node;

[0083] Scenario 5 is a simulation analysis of a single line power outage without considering zero-injection nodes;

[0084] Scenario 6 is a simulation analysis considering a power outage on a single line under zero-injection nodes.

[0085] An improved gray wolf algorithm is used on the IEEE-33 node distribution network to solve the optimal configuration scheme of the PMU configuration model under different scenarios. Simulation comparison analysis is carried out with different algorithms to further verify the feasibility and effectiveness of the proposed model and the improved algorithm.

[0086] This invention provides a method for optimizing the configuration of PMUs in a distribution network based on an improved gray wolf algorithm. The technical effects are as follows:

[0087] 1) Under the condition of complete system observability, this invention establishes a mathematical model for the PMU optimization configuration problem, while considering the impact of zero-injection nodes and the system observability problem under single PMU interruption and single line power outage, and further constructs optimization configuration models under different scenarios with the goal of achieving complete system observability.

[0088] 2) Based on the Grey Wolf optimization algorithm, this invention improves the population initialization, nonlinear convergence factor and position update weight coefficient. Through performance analysis with GWO, PSO and WOA algorithms on different standard test functions, the results show that the proposed improved Grey Wolf algorithm has better optimization accuracy and convergence speed, proving that the proposed algorithm is effective and feasible.

[0089] 3) This invention uses the improved gray wolf algorithm and the three different algorithms mentioned above to solve the optimization configuration scheme of the PMU configuration model. Simulation analysis was carried out on different system networks under different test scenarios. The results show that the present invention has a better optimization effect than other algorithms, which verifies the feasibility and effectiveness of the proposed model and algorithm. Attached Figure Description

[0090] Figure 1 Convergence curves for different p-values.

[0091] Figure 2(a) shows the convergence curves of the four algorithms based on the test function F1;

[0092] Figure 2(b) shows the convergence curves of the four algorithms based on the test function F2;

[0093] Figure 2(c) shows the convergence curves of the four algorithms based on the test function F3;

[0094] Figure 2(d) shows the convergence curves of the four algorithms based on the test function F4;

[0095] Figure 2(e) shows the convergence curves of the four algorithms based on the test function F5;

[0096] Figure 2(f) shows the convergence curves of the four algorithms based on the test function F6.

[0097] Figure 3 This is a diagram of the power distribution network for the IEEE-33 node.

[0098] Figure 4 A diagram showing the number of PMU configurations under different scenarios and algorithms.

[0099] Figure 5 The flowchart shows the overall algorithm solution process. Detailed Implementation

[0100] This paper proposes an optimized PMU configuration method for distribution networks based on an improved Grey Wolf algorithm. First, under the condition of complete system observability, a mathematical model of the PMU optimization configuration problem is established, considering the impact of zero-injection nodes and the system observability issues under single PMU outages and single-line power outages. Second, based on the Grey Wolf optimization algorithm, improvements are made to the population initialization, nonlinear convergence factor, and position update weight coefficient. The improved Grey Wolf algorithm is then used to solve the optimized configuration schemes for PMU configuration models under different scenarios. Finally, simulation calculations verify the feasibility and effectiveness of the proposed model and algorithm. The overall algorithm solution process is as follows: Figure 5 As shown, it includes the following steps:

[0101] Step 1: Under the condition that the system is fully observable, establish a mathematical model for PMU optimal configuration;

[0102] Step 2: Considering the impact of zero-injection nodes, as well as the system observability issues under single PMU interruption and single line power outage, modify the constraints of the PMU optimization configuration mathematical model established in Step 1.

[0103] Step 3: Improve the population initialization, nonlinear convergence factor, and position update weight coefficient based on the gray wolf optimization algorithm;

[0104] Step 4: Using the improved Grey Wolf optimization algorithm from Step 3, solve for optimized configuration schemes for the PMU configuration model under different scenarios.

[0105] The preferred embodiments are described in detail below with reference to the accompanying drawings:

[0106] 1. PMU configuration optimization issues:

[0107] The meaning of PMU optimal configuration is to minimize the number of PMUs installed in the system so that the system is fully observable. The observability of a system can be analyzed from two perspectives: algebraic observability and topological observability. Algebraic observability starts from numerical analysis, establishing the equation z = Hx + v between state variables and measurements in an n-node system, where z is an m-dimensional measurement; x is a 2n-1 dimensional voltage state variable; H is an m×(2n-1) dimensional measurement Jacobian matrix; and v is an m-dimensional measurement error phasor. System observability is determined by whether matrix H has full rank. Topological observability starts from a topological perspective, considering the nodes and branches in the system as a set of vertices V and edges E of a graph G. Assume the measurement network forms a subgraph G' = (V', E'), and... If satisfied If all vertices of graph G are contained within subgraph G', then the system is topologically observable. The high-dimensional matrix H in the definition of algebraic observability involves significant computation; therefore, topological analysis methods are easier to implement.

[0108] A fully observable power grid is defined when all nodes in the system are monitored directly or indirectly by a PMU. Therefore, the node where the PMU is located can be directly monitored, as well as the nodes connected to it. In indirect measurements, Kirchhoff's laws can be used to calculate the unknown current and voltage values ​​of a node. Thus, the observability rule can be expressed as:

[0109] Rule 1: The voltage of the busbar where the PMU is installed and the current of the branch connected to it are directly measured by the installed PMU.

[0110] Rule 2: If the voltage and current phasors at one end of a branch are known, the voltage phasor at the other end of the branch can be obtained simply by using Ohm's law.

[0111] Rule 3: If the voltage phasors at both ends of a branch are known, Ohm's law can be used to calculate the current in that branch.

[0112] The main objective of the PMU optimization configuration problem is to find the minimum PMU to achieve complete observability of the system. Its mathematical model can be expressed as shown in Equation (1):

[0113]

[0114] In equation (1), c i This represents the cost required to install the PMU on node i, without considering the differences in PMU configuration due to installation costs on different nodes, and uniformly selecting c. i =1. x i It is a binary variable representing whether a PMU is installed on node i. If a PMU is installed on node i, then x is... i=1 otherwise =0. min represents the minimum value; ∑ represents the summation; n represents the number of nodes; i represents the i-th node; f represents the constraint expression; st represents the constraint symbol; X = (x1, x2, ..., x... n ) T This represents the PMU configuration of each node in the system; B is an n-dimensional column vector with all elements being 1. A is an n×n-dimensional node association matrix, whose elements a ij It can be represented as

[0115] A zero-injection node is a node with no load or generator connected to it; in other words, the power injected into this node is zero. Therefore, according to the Kirchhoff's Current Law (KCL), the sum of the branch currents connected to this node is zero. Considering zero-injection nodes reduces the number of PMUs required to achieve full observability, with the observability rules as follows:

[0116] Rule 4: If there exists an indirectly observable zero-injection node, and the voltage phasors of all but one of the adjacent buses are known, then the voltage phasors of the unknown node can be calculated using KCL.

[0117] Rule 5: If the voltage phasor of a zero-injection node is unknown and the voltage phasors of all its neighboring nodes are known, then the voltage phasor of the zero-injection node can be obtained through the node voltage equations.

[0118] Therefore, when considering the impact of zero-injection nodes, the constraint transformation in the PMU optimization configuration mathematical model is shown in equation (2):

[0119]

[0120] In equation (2), x i It is a binary variable representing whether a PMU is installed on node i. If a PMU is installed on node i, then x is... i The summation is 1 if the summation is 1, otherwise it is 0. ∑ represents the summation symbol; n represents the number of nodes; i represents the i-th node; f j Let a be the constraint expression symbol for the j-th node; ij It can be represented as Introducing an auxiliary binary variable y ij When the value is 1, it means that node j can be made observable based on the measurement data of the neighboring zero-injection node i; when the value is 0, it means that node j cannot be observed. I is the set of system nodes. Represents any symbol; j represents the j-th node; z i It is also a binary variable. A value of 1 indicates that node i is a zero-injection node, and a value of 0 indicates that node i is a non-zero-injection node.

[0121] Any PMU failure or communication channel interruption may lead to the loss of measurement data. To further improve system reliability and ensure system observability when the PMU is out of service, each node should be configured to be observable through two methods. Therefore, the constraints in the PMU optimization configuration mathematical model are modified as shown in formula (3):

[0122] f=AX≥2B (3)

[0123] In equation (3), f is the constraint expression symbol; X = (x1, x2, ..., x n ) T This represents the PMU configuration of each node in the system; B is an n-dimensional column vector with all elements being 1. A is an n×n node association matrix, whose elements a ij It can be represented as

[0124] When considering zero-injection nodes, the constraints in the PMU optimization configuration mathematical model should be changed, as shown in formula (4):

[0125]

[0126] In equation (4), x j It is a binary variable representing whether a PMU is installed on node j. If a PMU is installed on node j, then x is... j The summation is 1 if the summation is 1, otherwise it is 0. ∑ represents the summation symbol; n represents the number of nodes; i represents the i-th node; j represents the j-th node; f j Let a be the constraint expression symbol for the j-th node; ij It can be represented as Introducing an auxiliary binary variable y ij When z is 1, it means that node j can be made observable based on the measurement data of the neighboring zero-injection node i; when z is 0, it means that node j cannot be observed. j It is also a binary variable; a value of 1 indicates that node j is a zero-injection node, and a value of 0 indicates that node i is a non-zero-injection node. B is an n-dimensional column vector with all elements being 1. I represents any symbol; I is the set of system nodes.

[0127] If any line is de-energized, measurement data cannot be obtained. By increasing the number of measurement sources at each node, the system can still be observed even if a single line fails. Similar to the case of a single PMU interruption, to meet the observability requirements, each node in the system must: 1) have a PMU device installed on it; 2) or have two or more adjacent nodes with PMU devices installed. Therefore, the constraints in the mathematical model for PMU optimization configuration are changed, as shown in formula (5):

[0128] f=(A+I)X≥2B (5)

[0129] In equation (5), f is the constraint expression symbol; X = (x1, x2, ..., x n ) T This represents the PMU configuration at each node of the system; B is an n-dimensional column vector with all elements being 1; I is an n×n identity matrix. A is an n×n node incidence matrix, whose elements a ij It can be represented as

[0130] When considering zero-injection nodes, the constraints in the PMU optimization configuration mathematical model change as shown in Equation (6):

[0131]

[0132] In equation (6), x j It is a binary variable representing whether a PMU is installed on node j. If a PMU is installed on node j, then x is... j The summation is 1 if the summation is 1, otherwise it is 0. ∑ represents the summation symbol; n represents the number of nodes; i represents the i-th node; j represents the j-th node; f j Let a be the constraint expression symbol for the j-th node; ij It can be represented as Introducing an auxiliary binary variable y ij When z is 1, it means that node j can be made observable based on the measurement data of the neighboring zero-injection node i; when z is 0, it means that node j cannot be observed. j It is also a binary variable; a value of 1 indicates that node j is a zero-injection node, and a value of 0 indicates that node i is a non-zero-injection node. B is an n-dimensional column vector with all elements being 1. I represents any symbol; I is the set of system nodes.

[0133] 2. Improved Gray Wolf Algorithm:

[0134] The above content elaborates on the PMU optimization configuration problem under different scenarios and establishes its mathematical model. Based on the research on PMU optimization configuration, a heuristic algorithm is used to solve the PMU optimization configuration model under various constraints. The Grey Wolf Optimization (GWO) algorithm is a new metaheuristic optimization algorithm developed by Mirjalili et al. Inspired by nature, GWO imitates the leadership hierarchy and hunting mechanism of grey wolves in nature. It involves four types of grey wolves to simulate the leadership hierarchy, namely the leader wolf α, the deputy leader wolf β, the ordinary wolf δ, and the bottom wolf w. In addition, it realizes the three main steps of hunting: surrounding the prey, chasing the prey, and attacking the prey. Its mathematical model is shown in formulas (7) and (8):

[0135] D = |C·X l (t)-X(t)| (7)

[0136] X(t+1)=X l (t)-A·D (8)

[0137] In equations (7) and (8), t represents the current iteration number; X(t) represents the position vector of the gray wolf; X l (t) is the position vector of the prey; D represents the step size surrounding the prey; X(t+1) represents the position vector of the gray wolf at time t+1; A and C are coefficient vectors, defined as shown in formulas (9) and (10):

[0138] A = 2a·r1-a (9)

[0139] C = 2·r² (10)

[0140] In equations (9) and (10), A and C are coefficient vectors; r1 and r2 are variables randomly selected within the interval [0, 1]; a is the convergence factor, which decreases linearly from 2 to 0 as t increases, as shown in equation (11):

[0141]

[0142] In equation (11), a is the convergence factor; t represents the current iteration number; T max is a constant representing the maximum number of iterations the algorithm can achieve during convergence.

[0143] In the GWO algorithm, guided by α, β, and δ, the wolf pack approaches the prey from all directions to achieve its hunting objective. The mathematical description of this behavior is shown in equations (12), (13), and (14):

[0144]

[0145]

[0146]

[0147] In equations (12), (13), and (14), D α D β D δ X represents the distance that the bottom wolf w needs to move to α, β, δ; while X α X β X δX and X represent the current positions of α, β, δ and w, respectively; C1, C2, C3 and A1, A2, A3 represent the coefficient vectors of the movement distance and corresponding position of each wolf pack; X1, X2, X3 represent the position update of the bottom wolf w in the gray wolf pack according to α, β, δ; t represents the current iteration number; X(t+1) represents the last updated position of the wolf pack.

[0148] In the traditional GWO algorithm, the distribution of the wolf pack after initialization in the entire solution space has a certain impact on the optimization speed and solution accuracy of the algorithm. The traditional GWO algorithm randomly generates the initial position of the wolf pack, but the randomization process often results in an uneven distribution of the population in the search space and a lack of diversity. Therefore, the Bernoulli chaotic mapping is introduced to improve the distribution properties of the wolf pack during the initialization of the GWO algorithm, increase the diversity of the initial individuals and the computational efficiency, further improve the global search capability of the algorithm and avoid getting trapped in local optima. The Bernoulli chaotic mapping is shown in formula (15):

[0149]

[0150] In equation (15), X n Let X be the nth chaotic variable; n is the dimension size; n+1 X is the (n+1)th chaotic variable; λ is the adjustment factor. n ∈(0,1) and λ=0.5, X0=0.8473.

[0151] As shown in formula (11), the convergence factor a is a straight line that decreases linearly from 2 to 0 with each algorithm iteration. However, due to the complexity of the gray wolf's prey search process, a linear convergence curve cannot accurately reflect the actual optimization search process. Therefore, a nonlinear parameter control strategy is an effective way to prevent premature convergence. Thus, a nonlinear convergence factor update strategy based on a sine function is used, as shown in formula (16):

[0152]

[0153] In equation (16), a is the convergence factor; t represents the current iteration number; T max is a constant representing the maximum number of iterations the algorithm can achieve during convergence. p is a control coefficient, the value of which determines the balance between the early exploration and later development of the algorithm.

[0154] To further determine the p-value, Figure 1 The figure shows convergence curves for different p-values. Figure 1It can be seen that different p values ​​correspond to different curves. When p = 0.2 and p = 0.5, the algorithm converges quickly in the early stage and slowly in the later stage. When p = 1.5 and p = 2, the curve state meets the expected requirements. In order to balance the convergence speed in the early and late stages of iteration, p = 2 is selected.

[0155] Since the position update formula (14) for the wolf in the traditional GWO is based on the average of the updated positions of the α, β, and δ wolves, the alpha wolf's ability to guide the individual wolves in the pack is the same. However, in order to better mimic the hierarchical system of the gray wolf pack, dynamic weight coefficients are introduced into the position update formula, as shown in formulas (17), (18), (19), and (20):

[0156]

[0157]

[0158]

[0159]

[0160] In equations (17), (18), (19), and (20), w1, w2, and w3 represent the weight coefficients corresponding to α, β, and δ wolves, respectively; X1, X2, and X3 represent the position updates of the bottom wolf w in the gray wolf pack based on α, β, and δ; and X(t+1) represents the last updated position of the wolf pack.

[0161] To verify the performance of the improved Grey Wolf algorithm, simulation tests were conducted on standard functions against the traditional Grey Wolf algorithm, Particle Swarm Optimization (PSO), and Whale Optimization (WOA). Six standard test functions were selected, including four unimodal functions and two multimodal functions. During testing, the dimension of all algorithms was uniformly set to 30, and the maximum number of iterations was 500. Each of the four comparison algorithms was run 30 times, and the average and standard deviation of the results were taken as the final results, as shown in Table 1. Table 1 shows that the improved Grey Wolf algorithm achieves better average values ​​on test functions F1, F2, F4, F5, and F6, but lags behind on function F3. However, the improved Grey Wolf algorithm outperforms the improved algorithm in terms of the standard deviation of the test results on functions F1–F6. Convergence curves of different algorithms based on standard functions were also plotted, as shown in Table 1. Figures 2(a) to 2(f) The graphs show the convergence curves of four algorithms based on test functions F1 to F6. Figures 2(a) to 2(f) It can be seen that the IGWO algorithm has a better optimization effect than the other three algorithms. For example, the F1 and F2 curves show that the IGWO algorithm has a faster convergence speed and demonstrates a stronger local search capability. Combined with... Figures 2(a) to 2(f)As can be seen from Table 1, under the six test functions, the IGWO algorithm has better optimization accuracy and convergence speed.

[0162] Table 1. Comparison of test results for GWO, PSO, WOA, and IGWO functions.

[0163]

[0164] The specific steps for solving the PMU optimization configuration problem using the improved Grey Wolf algorithm are as follows:

[0165] S1: Based on the IEEE-33 node network topology and the relationships between the nodes, calculate the system node correlation matrix A.

[0166] S2: Initialize the parameters of the Grey Wolf Algorithm, including population size, maximum number of iterations, and parameters a, A, and C.

[0167] S3: Randomly initialize the location of the PMU in the system.

[0168] S4: Calculate the fitness function and define the prey position based on the current position.

[0169] S5: Updated to the top three best search agents X α X β X δ Update the coefficient vectors a, A, C, and update the gray wolf positions X1, X2, X3.

[0170] S6: Repeat S4 and S5 until the maximum number of iterations is reached, and finally output the result.

[0171] like Figure 3 The IEEE-33 node power distribution network diagram is provided, with 33 network nodes, 32 branches, and zero-injection nodes at positions 4, 5, 13, 14, and 15. Under the condition of complete system observability, a simulation analysis of the standard system is first performed. Further consideration is given to the impact of zero-injection nodes, as well as the scenarios of a single PMU outage and a single line power failure, resulting in the following six test scenarios:

[0172] Scenario 1 is a simulation analysis without considering zero-injection nodes;

[0173] Scenario 2 considers simulation analysis under zero-injection nodes;

[0174] Scenario 3 is a simulation analysis of a single PMU interrupt without considering zero-injection nodes;

[0175] Scenario 4 is a simulation analysis considering the interruption of a single PMU under a zero-injection node;

[0176] Scenario 5 is a simulation analysis of a single line power outage without considering zero-injection nodes;

[0177] Scenario 6 is a simulation analysis considering a power outage on a single line under zero-injection nodes.

[0178] During the simulation, the population size N = 100 and the maximum number of iterations was set to 500 in all four algorithm programs. The inertia weight of the PSO algorithm decreased linearly from 0.9 to 0.2, and the learning factor was 2 for all algorithms. The convergence factors of the GWO and WOA algorithms decreased linearly from 2 to 0.

[0179] The number of PMU configurations obtained in the IEEE-33 node case study under six test scenarios and four different algorithms are as follows: Figure 4 As shown. By Figure 4 It can be seen that, comparing the optimization results of the improved Gray Wolf algorithm with the traditional Gray Wolf algorithm, Particle Swarm Optimization algorithm, and Whale Optimization algorithm, the IGWO algorithm requires the fewest PMUs in each scenario, with 11, 10, 17, 13, 16, and 13 units respectively, further verifying the effectiveness of the improved Gray Wolf algorithm. Comparing Scenario 1 and Scenario 2, it is clear that the required number of PMUs decreases when considering zero-injection nodes. However, when special situations occur in the system, such as Scenario 3 to Scenario 6, the required number of PMUs will increase to better monitor the entire system. But in this case, considering the impact of zero-injection nodes, the change in the number of PMUs is not significant. The specific configuration locations of PMUs under different scenarios and algorithms are shown in Table 2. Figure 4 As shown in Table 2, the feasibility and effectiveness of the proposed model and algorithm are further demonstrated.

[0180] Table 2 PMU Configuration Locations under Different Scenarios and Algorithms

[0181]

[0182]

[0183] In addition, the time errors of PMU optimization configuration under different scenarios and different algorithms were compared, and Table 3 was obtained.

[0184] Table 3 Comparison of IEEE-33 Node Simulation Time

[0185]

[0186] As can be seen from Table 3, the improved Grey Wolf algorithm has a better convergence speed, but overall the optimization algorithm can quickly obtain results in terms of solution time, requiring less time and meeting practical needs.

Claims

1. A method for optimizing the configuration of distribution network PMUs based on an improved gray wolf algorithm, characterized in that... Includes the following steps: Step 1: Under the condition that the system is fully observable, establish a mathematical model for PMU optimal configuration; Step 2: Considering the impact of zero-injection nodes, as well as the system observability issues under single PMU interruption and single line power outage, modify the constraints of the PMU optimization configuration mathematical model established in Step 1. Step 3: Improve the population initialization, nonlinear convergence factor, and position update weight coefficient based on the gray wolf optimization algorithm; Step 4: Using the improved Grey Wolf optimization algorithm from Step 3, solve for optimized configuration schemes for the PMU configuration model under different scenarios; In step 3, a nonlinear convergence factor update strategy based on the sine function is used, as shown in formula (16): (16) ； In equation (16), The convergence factor; Indicates the current iteration number; is a constant representing the maximum number of iterations the algorithm can achieve during convergence; This is a control coefficient, and its value determines the balance between the early exploration and later development of the algorithm. In the gray wolf optimization algorithm The wolf's position update formula is obtained by retrieving the updated position. , , The average position of the gray wolf is used to introduce a dynamic weight coefficient in the position update formula, as shown in formulas (17), (18), (19), and (20): (17)； (18)； (19)； (20)； In equations (17), (18), (19), and (20), , , They represent , , The corresponding weighting coefficient for wolves; , , This refers to the bottom-ranking wolf in a gray wolf pack. according to , , Location update performed; This indicates the last location the wolf pack last updated.

2. The distribution network PMU optimization configuration method based on the improved gray wolf algorithm according to claim 1, characterized in that: In step 1, the mathematical model for PMU optimization configuration is expressed as shown in formula (1): (1)； In equation (1), Indicates at node The cost of installing the PMU is not considered, and the differences in PMU configuration due to installation costs at different nodes are uniformly selected. ; It is a binary variable representing a node. Is PMU installed on the node? If PMU is installed, then It is 1 if it is true, otherwise it is 0; Indicates the sign for finding the minimum value; The summation symbol; Indicates the number of nodes; Indicates the first One node; Symbols for constraint expressions; Represents constraint symbols; This indicates the PMU configuration status of each node in the system; For elements all of 1 3D column vector; for A dimensional node incidence matrix, whose elements Represented as .

3. The distribution network PMU optimization configuration method based on the improved gray wolf algorithm according to claim 2, characterized in that: In step 2, it is considered that zero-injection nodes will reduce the number of PMUs required to achieve full observability, and the observability rule is as follows: If there exists an indirectly observable zero-injection node, and the voltage phasors of all but one of the adjacent buses are known, then the voltage phasors of the unknown node can be calculated using KCL. If the zero-injection node voltage phasor is unknown and the voltage phasors of all its neighboring nodes are known, then the zero-injection node voltage phasor can be obtained through the node voltage equation. Therefore, when considering the impact of zero-injection nodes, the constraint transformation in the PMU optimization configuration mathematical model is shown in formula (2): (2)； In equation (2), It is a binary variable representing a node. Is PMU installed on the node? If PMU is installed, then It is 1 if it is true, otherwise it is 0; The summation symbol; Indicates the number of nodes; Indicates the first One node; For the first Symbols for constraint condition expressions for each node; It can be represented as ; Introducing auxiliary binary variables When the value is 1, it means that it can inject nodes based on adjacent zeros. The measurement data makes the nodes Observable; a value of 0 indicates a node Cannot be observed; For the system node set; Represents any symbol; Indicates the first One node; It is also a binary variable; a value of 1 indicates a node. A node is injected with zero values; a value of 0 indicates a node... This is a non-zero injection node.

4. The distribution network PMU optimization configuration method based on the improved gray wolf algorithm according to claim 2, characterized in that: In step 2, a single PMU interruption refers to any PMU failure in the system. At this time, the model is transformed as shown in formula (3): (3)； In equation (3), Symbols for constraint expressions; This indicates the PMU configuration status of each node in the system; For elements all of 1 3D column vector; for A dimensional node incidence matrix, whose elements It can be represented as ; When considering zero-injection nodes, the constraints in the PMU optimization configuration mathematical model are modified as shown in Equation (4): (4)； In equation (4), It is a binary variable representing a node. Is PMU installed on the node? If PMU is installed, then It is 1 if it is true, otherwise it is 0; The summation symbol; Indicates the number of nodes; Indicates the first One node; Indicates the first One node; For the first Symbols for constraint condition expressions for each node; It can be represented as Introducing auxiliary binary variables When the value is 1, it means that it can inject nodes based on adjacent zeros. The measurement data makes the nodes Observable; a value of 0 indicates a node Cannot be observed; It is also a binary variable; a value of 1 indicates a node. A node is injected with zero values; a value of 0 indicates a node... Non-zero injection nodes; For elements all of 1 3D column vector; Represents any symbol; This is the set of system nodes.

5. The distribution network PMU optimization configuration method based on the improved gray wolf algorithm according to claim 2, characterized in that: In step 2, a single line outage refers to the outage of any line in the system. At this time, the constraint conditions in the PMU optimization configuration mathematical model are transformed as shown in formula (5): (5)； In equation (5), Symbols for constraint expressions; This indicates the PMU configuration status of each node in the system; For elements all of 1 3D column vector; for An identity matrix of order 1; for A dimensional node incidence matrix, whose elements It can be represented as ; When considering zero-injection nodes, the constraints in the PMU optimization configuration mathematical model are modified as shown in Equation (6): (6)； In equation (6), It is a binary variable representing a node. Is PMU installed on the node? If PMU is installed, then It is 1 if it is true, otherwise it is 0; The summation symbol; Indicates the number of nodes; Indicates the first One node; Indicates the first One node; For the first Symbols for constraint condition expressions for each node; It can be represented as ; Introducing auxiliary binary variables When the value is 1, it means that it can inject nodes based on adjacent zeros. The measurement data makes the nodes Observable; a value of 0 indicates a node Cannot be observed; It is also a binary variable; a value of 1 indicates a node. A node is injected with zero values; a value of 0 indicates a node... Non-zero injection nodes; For elements all of 1 3D column vector; Represents any symbol; This is the set of system nodes.

6. The distribution network PMU optimization configuration method based on the improved gray wolf algorithm according to claim 1, characterized in that: In step 3, the Bernoulli chaotic map is introduced to improve the distribution properties of the wolf pack during the initialization of the GWO algorithm. The Bernoulli chaotic map is shown in formula (15): (15) ； In equation (15), For the first One chaotic variable; For dimension size; For the first One chaotic variable; As a regulating factor, .

7. The distribution network PMU optimization configuration method based on the improved gray wolf algorithm according to claim 1, characterized in that: Step 4 includes the following steps: S1: Based on the network topology of the IEEE-33 node system and the relationships between the nodes, determine the system's node... Point Incidence Matrix ; S2: Initialize the parameters of the Grey Wolf Algorithm, including population size, maximum number of iterations, and parameters. , , ; S3: Randomly initialize the location of the PMU in the system; S4: Calculate the fitness function and define the prey position based on the current position; S5: Top Three Search Agents Before the Update , , Update the coefficient vector , , And update the position of the Grey Wolves , , ; S6: Repeat steps S4 and S5 until the maximum number of iterations is reached, and finally output the result.

8. The distribution network PMU optimization configuration method based on the improved gray wolf algorithm according to claim 1, characterized in that: In step 4, considering the impact of zero-injection nodes, as well as single PMU interruptions and single-line power outages, the different test scenarios include the following six types: Scenario 1 is a simulation analysis without considering zero-injection nodes; Scenario 2 considers simulation analysis under zero-injection nodes; Scenario 3 is a simulation analysis of a single PMU interrupt without considering zero-injection nodes; Scenario 4 is a simulation analysis considering the interruption of a single PMU under a zero-injection node; Scenario 5 is a simulation analysis of a single line power outage without considering zero-injection nodes; Scenario 6 is a simulation analysis considering a power outage on a single line under zero-injection nodes.

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