Power distribution network line loss probability distribution calculation method based on improved Gaussian mixture model and linear power flow

Abstract

The invention relates to a distribution network line loss probability distribution calculation method based on an improved Gaussian mixture model and linear power flow, and belongs to the field of distribution network line loss probability distribution calculation. According to the method, the optimized Gaussian mixture model is utilized to accurately describe the non-Gaussian characteristic and the multi-peak characteristic of photovoltaic output, and the defect that a traditional parameter estimation method is prone to falling into local optimum is overcome; the analysis mapping relation between the photovoltaic output and the system line loss is established through the linearization power flow model, a large amount of repeated calculation of the Monte Carlo method is avoided, the calculation efficiency is remarkably improved while the calculation precision is guaranteed, and the real-time requirement of engineering practice for rapid line loss analysis can be met.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of power distribution network line loss probability distribution calculation, and particularly relates to a power distribution network line loss probability distribution calculation method based on an improved Gaussian mixture model and linearized power flow. BACKGROUND

[0002] With the continuous promotion of energy structure transformation, the penetration rate of distributed photovoltaics in the power distribution network is continuously increasing. Photovoltaic output is affected by many factors such as sunshine intensity and cloud cover, and has strong intermittency and randomness. The traditional line loss calculation based on deterministic power flow cannot effectively take into account the influence of photovoltaic output randomness, and is only suitable for analysis of specific working conditions, and cannot guide new energy operation optimization in complete scenarios.

[0003] At present, the probability power flow method is an effective tool for analyzing such problems. However, the Monte Carlo simulation method has high precision but low computational efficiency, and the point estimation method and other analytical methods are not accurate in dealing with the non-Gaussian nature of photovoltaic output. Although the Gaussian mixture model (GMM) can effectively fit any distribution, the traditional parameter estimation method, the maximum expectation algorithm, is prone to local optimal solution, resulting in low model accuracy, which in turn affects the accuracy of subsequent line loss probability calculation. Therefore, how to quickly and accurately obtain the probability distribution and expectation of line loss under limited measurement points is a problem to be solved.

[0004] The existing technical defects are as follows:

[0005] (1) The existing method is complex when mapping the multi-node photovoltaic output to the line loss calculation through a nonlinear power flow model. Analytical methods such as point estimation method and semi-invariant method are fast but lose accuracy, and Monte Carlo method is accurate but low in efficiency, and there is a lack of a practical calculation method that can map the influence of photovoltaic output change on line loss.

[0006] (2) Some studies use GMM to describe the uncertainty of new energy output, but the parameter estimation of GMM is mostly dependent on the traditional expectation maximization algorithm (EM). This algorithm is sensitive to initial values and is prone to local optimal solution, resulting in low accuracy and poor robustness of the established probability model. SUMMARY

[0007] The purpose of this invention is to provide a method for calculating the probability distribution of line losses in distribution networks based on an improved Gaussian mixture model and linearized power flow. This method uses the optimized Gaussian mixture model to accurately characterize the non-Gaussian and multi-peak characteristics of photovoltaic power output, overcoming the shortcomings of traditional parameter estimation methods that are prone to getting trapped in local optima. By establishing an analytical mapping relationship between photovoltaic power output and system line losses through a linearized power flow model, the method avoids a large amount of repetitive calculations in the Monte Carlo method, significantly improving computational efficiency while ensuring computational accuracy, and meeting the real-time requirements of rapid line loss analysis in engineering practice.

[0008] To achieve the above objectives, the technical solution of this invention is: a method for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, comprising:

[0009] Based on historical power output data of photovoltaic power plants in the distribution network, a Gaussian mixture model (GMM) is used to characterize the randomness, non-Gaussianity, and multi-peak characteristics of single and multiple photovoltaic power outputs. A particle swarm optimization algorithm is introduced to calculate the Gaussian component parameters in the GMM.

[0010] Based on the topology and baseline operating point of the distribution network, linearized power flow theory is used to derive an approximate linear mapping relationship between the probability distribution of photovoltaic injected power and the probability distribution of line loss, thereby realizing the calculation of the probability distribution of line loss in the distribution network.

[0011] Furthermore, a Gaussian mixture model (GMM) is used to characterize the randomness, non-Gaussianity, and multi-peak characteristics of single and multiple photovoltaic outputs. The specific implementation method is as follows:

[0012] Let the random variable of photovoltaic power output be... Where D is the number of photovoltaic power stations, and its probability density function PDF consists of M multivariate Gaussian distribution components, its expression is:

[0013]

[0014]

[0015] In the formula: It is the mean of the m-th Gaussian component; It is the variance of the m-th Gaussian component; It is the mixing weight of the m-th Gaussian component and satisfies ; The probability density function representing the m-th Gaussian component

[0016] The complete set of parameters to be estimated in GMM is: The likelihood function and logarithmic representation of the photovoltaic power output sample are obtained by the maximum likelihood estimation method (MLE).

[0017]

[0018]

[0019] In the formula: N represents the number of samples of historical photovoltaic power output data.

[0020] The goal of GMM parameter estimation is to find a set of parameters. This maximizes the value of the log-likelihood function.

[0021] Furthermore, a particle swarm optimization algorithm is introduced to calculate the Gaussian component parameters in the Gaussian model; that is, the particle swarm optimization algorithm (PSO) is used to optimize the parameters of the Gaussian model. Perform global optimization to maximize the log-likelihood function. .

[0022] Furthermore, the particle swarm optimization algorithm is introduced to calculate the Gaussian component parameters in the GMM. The specific implementation method is as follows:

[0023] (1) Encoding

[0024] All the parameters to be estimated in the GMM are encoded as the positions of a particle in the search space, a position vector. The structure is as follows:

[0025]

[0026] In the formula, vec(·) stacks the lower triangular parts of the symmetric covariance matrix into a vector, with weights... Depend on Determine the constraints to reduce the dimensionality of the optimization problem;

[0027] (2) Fitness function

[0028] The fitness value, or the value of the objective function, represents the quality of a particle's position and determines whether to update the individual best position and the group best position of the particle. The algorithm aims to find the Gaussian component parameters when the log-likelihood function is maximized; therefore, the log-likelihood function is directly used as the fitness function.

[0029]

[0030] (3) Particle Swarm Optimization Algorithm Flow

[0031] Initialization: Randomly initialize a particle swarm of size S within the feasible region of parameters, where each particle i has a random initial position in the search space. and initial velocity ;

[0032] Optimization: In the t-th iteration, calculate the fitness function for each particle. For each particle, compare its current fitness with its historical best fitness. If the current fitness is better, then update. Find the particle with the highest fitness among all particles. m Compare its fitness with the global historical best fitness. Compare the results; if the current fitness is better, then update. ;

[0033] Update: Update the velocity and position of each particle according to the following formula:

[0034]

[0035] In the formula , For learning factors; , A random number within the range [0,1]; The inertia factor represents the particle's global optimization capability;

[0036] Output: The algorithm stops when the number of iterations reaches the set maximum or the fitness function converges; the global optimum position is then determined. As the optimal parameter set of GMM Output the results.

[0037] Furthermore, using linearized power flow theory, an approximate linear mapping relationship between the photovoltaic injected power probability distribution and the line loss probability distribution is derived, including:

[0038] (1) Constructing a linearized power flow model:

[0039] The nonlinear equation for the alternating current flow is as follows:

[0040]

[0041] In the formula: It is the injected active power of node i; It is the injected reactive power at node i; , These are the voltages at nodes i and j; It is the equivalent conductance between nodes i and j. It is the equivalent susceptance between nodes i and j; It is the voltage phase angle difference between nodes i and j; N is the total number of nodes in the system;

[0042] At the reference operating point Performing a first-order Taylor expansion at the given point, and ignoring all higher-order terms, we get:

[0043]

[0044] Furthermore, rewrite all nodal equations in matrix form:

[0045]

[0046] In the formula Here is the Jacobian matrix calculated at the reference point; represents the state variable change caused by the power change, multiplied by both sides. The reverse of:

[0047] (1)

[0048] In the formula This is the sensitivity matrix.

[0049] (2) Construct a linear relationship between line power flow and node injected power:

[0050] The power flow equations of the line are linearized, and the connection nodes are linearized. and nodes The route Its meritorious trend The expression is:

[0051]

[0052] At its reference operating point Place, to Perform a first-stage Taylor expansion:

[0053]

[0054] In the formula: It is the change in active power on line ij; = This represents the change in the voltage phase angle at node k relative to the voltage phase angle at the reference point. This represents the change in the voltage amplitude at node k relative to the voltage amplitude at the reference point.

[0055] Rewritten in matrix form:

[0056] (2)

[0057] In the formula The sensitivity of the line power flow to state variables; substituting equation (1) into equation (2) yields:

[0058]

[0059] To focus on the impact of photovoltaic output on the system state, we assume the change in system reactive power. The above formula can be simplified as follows:

[0060]

[0061] In the formula Sensitivity matrix The corresponding change in active power The first N columns, It comprehensively reflects the sensitivity of line power flow to state variables and active power injection;

[0062] line The active power flow is represented as:

[0063]

[0064] In the formula: This is the line active power under reference conditions. This represents the injected active power vector of all nodes under the baseline condition;

[0065] Will denoted as a constant Thus, an approximate linear relationship is obtained between line power flow and node injected power:

[0066] .

[0067] (3) Constructing a linear relationship between line loss and photovoltaic output:

[0068] The system bus loss is the sum of the losses of each branch. For any branch Its losses It is a function of node voltage magnitude and phase angle; therefore, system bus loss is a function of system operating state. The complex nonlinear function, and the system operating state It is the node injected power The function;

[0069]

[0070] In the formula: h(.) is the mapping from node state variables to branch power losses; g(.) is the mapping from node injected power to node state variables;

[0071] At the reference operating point At this point, performing a first-order Taylor expansion on the above composite function and neglecting higher-order terms, we obtain the change in system bus loss:

[0072]

[0073] In the formula: It is the change in active power injected into the node;

[0074] Rewriting the above equation in matrix form yields the following:

[0075]

[0076] In the formula The sensitivity of line loss to various state variables is denoted as . ; Right now This represents the change of the state variable with the injected active power. The above equation simplifies to:

[0077]

[0078] In the formula This represents the sensitivity of the system bus loss to injected active power, with the final system bus loss serving as a baseline value. Add the change:

[0079] (3)

[0080] In the formula: Inject active power into the nodes under the baseline state; = This is the constant term in the linear expression for line loss;

[0081] Node Injection Power Powered by photovoltaics and load constitute:

[0082] (4)

[0083] Substituting equation (3) into equation (3), we get:

[0084]

[0085] In the formula: = This is a constant term in the approximate linear relationship between line loss and photovoltaic output.

[0086] The above formula establishes an approximately linear relationship between system line loss and photovoltaic output, transforming complex nonlinear power flow calculations into a linear transformation.

[0087] Furthermore, the probability distribution of distribution network line losses is calculated as follows:

[0088] Photovoltaic power output The probability distribution is described by GMM.

[0089]

[0090] The m-th Gaussian component in GMM Follow the mean The variance is Given a Gaussian distribution, the corresponding linear transformation yields the line loss as... It still follows a Gaussian distribution, with a mean and variance of:

[0091]

[0092]

[0093] Line loss The probability density function is

[0094]

[0095] The PDF of the system bus loss is a weighted sum of the PDFs of the corresponding line losses obtained by linear transformation of each Gaussian component in the GMM, where the weights are... Remain unchanged:

[0096]

[0097] The cumulative distribution function (CDF) of system bus losses is the weighted sum of the CDFs of each component:

[0098]

[0099] The expected system bus loss is:

[0100]

[0101] By changing the order of integral and summation, we get:

[0102]

[0103] Therefore, it is only necessary to solve for the GMM parameters and sensitivity coefficients. , The probability distribution and expected value of system bus loss can be quickly calculated using a weighted summation analytical calculation method.

[0104] The present invention also provides a distribution network line loss probability distribution calculation system based on an improved Gaussian mixture model and linearized power flow, including a memory, a processor, and computer program instructions stored in the memory and executable by the processor. When the processor executes the computer program instructions, it can implement the steps of the method described above.

[0105] The present invention also provides a computer-readable storage medium having stored thereon computer program instructions that can be executed by a processor, wherein when the processor executes the computer program instructions, it can implement the steps of the method as described above.

[0106] Compared with existing technologies, the present invention has the following advantages: The method of the present invention uses an optimized Gaussian mixture model to accurately characterize the non-Gaussian and multi-peak characteristics of photovoltaic power output, overcoming the defect that traditional parameter estimation methods are prone to getting trapped in local optima; by establishing an analytical mapping relationship between photovoltaic power output and system line loss through a linearized power flow model, the method avoids a large amount of repetitive calculations in the Monte Carlo method, significantly improving calculation efficiency while ensuring calculation accuracy, and meeting the real-time requirements of rapid line loss analysis in engineering practice. Attached Figure Description

[0107] Figure 1 This is a technical roadmap for the present invention. Detailed Implementation

[0108] The technical solution of the present invention will now be described in detail with reference to the accompanying drawings.

[0109] This invention provides a method for calculating the probability distribution of line losses in distribution networks based on an improved Gaussian mixture model and linearized power flow, including:

[0110] Based on historical power output data of photovoltaic power plants in the distribution network, a Gaussian mixture model (GMM) is used to characterize the randomness, non-Gaussianity, and multi-peak characteristics of single and multiple photovoltaic power outputs. A particle swarm optimization algorithm is introduced to calculate the Gaussian component parameters in the GMM.

[0111] Based on the topology and baseline operating point of the distribution network, linearized power flow theory is used to derive an approximate linear mapping relationship between the probability distribution of photovoltaic injected power and the probability distribution of line loss, thereby realizing the calculation of the probability distribution of line loss in the distribution network.

[0112] The following is a detailed implementation process of the present invention.

[0113] This invention provides a method for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, implemented as follows:

[0114] (1) Based on the historical output data of photovoltaic power plants in the distribution network, the Gaussian Component Model (GMM) is used to characterize the randomness of single and multiple photovoltaic outputs in order to accurately describe their non-Gaussian and multi-peak characteristics. In view of the problem that the traditional expectation-maximization algorithm is sensitive to the initial value and is prone to getting trapped in local optima in the solution of GMM parameters, the particle swarm optimization algorithm is introduced to calculate the Gaussian component parameters to improve the fitting accuracy of the probability function.

[0115] (2) Based on the topology and reference operating point of the distribution network, the linearized power flow theory is adopted to derive the approximate linear mapping relationship between the probability distribution of photovoltaic injected power and the probability distribution of line loss, thereby reducing the complex nonlinear power flow calculation process.

[0116] The method specifically includes the following steps:

[0117] S1, Gaussian mixture model of photovoltaic output

[0118] S11, Model Definition

[0119] Gaussian mixture model is a probabilistic model used to describe mixture density. It simulates complex non-Gaussian continuous probability distributions by weighted sum of multiple Gaussian distribution components. In this invention, GMM is used to accurately characterize the randomness, non-Gaussianity, and multi-peak characteristics of photovoltaic power output.

[0120] Let the random variable of photovoltaic power output be... Where D is the number of photovoltaic power plants, and its probability density function (PDF) consists of M multivariate Gaussian distribution components, expressed as follows:

[0121]

[0122]

[0123] In the formula: It is the mean of the m-th Gaussian component; It is the variance of the m-th Gaussian component; It is the mixing weight of the m-th Gaussian component and satisfies ;

[0124] The complete set of parameters to be estimated in GMM is: The likelihood function and logarithmic representation of the photovoltaic output sample are usually solved using the Maximum Likelihood Estimation (MLE) method. with formula As shown:

[0125]

[0126]

[0127] The goal of GMM parameter estimation is to find a set of parameters. This maximizes the value of the log-likelihood function.

[0128] S22. Optimization and solution of GMM parameters based on particle swarm optimization algorithm

[0129] Traditional EM algorithms are sensitive to initial values ​​when solving for GMM parameters, and are prone to getting trapped in local optima, thus affecting model accuracy. To address this issue, this invention employs Particle Swarm Optimization (PSO) to solve for the GMM parameters. Perform global optimization to maximize the log-likelihood function. .

[0130] (1) Encoding

[0131] All the parameters to be estimated in the GMM are encoded as the positions of a particle in the search space, a position vector. It can be constructed as the formula

[0132]

[0133] In the formula, vec(·) stacks the lower triangular parts of the symmetric covariance matrix into a vector, with weights... Depend on Determine the constraints to reduce the dimensionality of the optimization problem.

[0134] (2) Fitness function

[0135] The fitness value, or the value of the objective function, represents the quality of a particle's position and determines whether to update the individual best position and the group best position of the particle. The algorithm aims to find the parameters of each Gaussian component when the log-likelihood function is maximized, so the log-likelihood function is directly used as the fitness function.

[0136]

[0137] (3) Particle Swarm Optimization Algorithm Flow

[0138] Initialization: Randomly initialize a particle swarm of size S within the feasible region of parameters, where each particle i has a random initial position in the search space. and initial velocity

[0139] Optimization: In the t-th iteration, calculate the fitness function for each particle. For each particle, compare its current fitness with its historical best fitness. If the current fitness is better, then update. Find the particle with the highest fitness among all particles. m Compare its fitness with the global historical best fitness. Compare the results; if the current fitness is better, then update. .

[0140] Update: According to the formula Update the velocity and position of each particle

[0141]

[0142] In the formula , The learning factor is usually taken as... ; , A random number within the range [0,1]; The inertia factor represents the particle's global optimization capability.

[0143] Output: The algorithm stops when the number of iterations reaches the set maximum or the fitness function converges. The global optimum position is then determined. As the optimal parameter set of GMM Output the results.

[0144] S2, the mapping relationship between line loss and photovoltaic output

[0145] S21, Linearized Power Flow Model

[0146] The nonlinear equation for the alternating current flow is as follows:

[0147]

[0148] At the reference operating point Performing a first-order Taylor expansion at the point and ignoring all higher-order terms, we obtain:

[0149]

[0150] Furthermore, all nodal equations are rewritten in matrix form.

[0151]

[0152] In the formula This is the Jacobian matrix calculated at the reference point. To represent the state changes caused by power variations, multiply both sides by... The inverse can be obtained as:

[0153]

[0154] In the formula Sensitivity matrix

[0155] S22. Linear relationship between line power flow and node injected power

[0156] To further establish a direct link between nodal injected power and line power flow, the line power flow equations need to be linearized, especially for connecting nodes. and nodes The route Its meritorious trend The expression is

[0157]

[0158] At its reference operating point Place, to Perform a first-stage Taylor expansion:

[0159]

[0160] Rewrite in matrix form

[0161]

[0162] In the formula This represents the sensitivity of the power flow to state variables. The equation... Attached achievable

[0163]

[0164] To focus on the impact of photovoltaic output on the system state, we assume the change in system reactive power. The above formula can be simplified.

[0165]

[0166] In the formula Sensitivity matrix The corresponding change in active power The first N columns, It comprehensively reflects the sensitivity of line power flow to state variables and active power injection.

[0167] line The active power flow can be represented as:

[0168]

[0169] Will denoted as a constant An approximate linear relationship can be obtained between line power flow and node injected power.

[0170]

[0171] S23. Linear relationship between system line loss and photovoltaic output

[0172] The system bus loss is the sum of the losses of each branch. For any branch Its losses It is a function of node voltage magnitude and phase angle; therefore, system bus loss is a function of system operating state. The complex nonlinear function, and the system operating state It is the node injected power The function.

[0173]

[0174] At the reference operating point At this point, a first-order Taylor expansion of the composite function is performed, ignoring higher-order terms, to obtain the system bus loss change, which is then rewritten in matrix form.

[0175]

[0176]

[0177] In the formula The sensitivity of line loss to various state variables is denoted as . ; That is, the above , representing the change of state variables with injected active power, equation It can be simplified to:

[0178]

[0179] In the formula This represents the sensitivity of the system bus loss to injected active power; the final system bus loss is the baseline value plus the change.

[0180]

[0181] Node Injection Power Powered by photovoltaics and load constitute

[0182]

[0183] The formula Substitution

[0184]

[0185] Mode An approximate linear relationship between system line loss and photovoltaic output was established, transforming complex nonlinear power flow calculations into linear transformations. This avoids a large amount of repetitive power flow calculations in traditional probabilistic methods such as Monte Carlo simulation, thereby improving computational efficiency.

[0186] S3, Calculation of line loss probability distribution

[0187] Photovoltaic power output The probability distribution can be described by GMM.

[0188]

[0189] The m-th Gaussian component in this GMM Follow the mean The variance is The Gaussian distribution is given. Performing the corresponding linear transformation yields the line loss as... It still follows a Gaussian distribution with a mean and variance of .

[0190]

[0191]

[0192] Line loss The probability density function is

[0193]

[0194] The PDF of the system bus loss is a weighted sum of the PDFs of the corresponding line losses obtained by linear transformation of each Gaussian component in the GMM, where the weights are... remain unchanged

[0195]

[0196] Similarly, the cumulative distribution function (CDF) of system bus loss is the weighted sum of the CDFs of each component.

[0197]

[0198] The expected system bus loss is

[0199]

[0200] By changing the order of integral and summation, we get

[0201]

[0202] Only the GMM parameters and sensitivity coefficients need to be solved. , The probability distribution and expected value of system bus loss can be quickly calculated using a weighted summation analytical calculation method.

[0203] The present invention also provides a distribution network line loss probability distribution calculation system based on an improved Gaussian mixture model and linearized power flow, including a memory, a processor, and computer program instructions stored in the memory and executable by the processor. When the processor executes the computer program instructions, it can implement the steps of the method described above.

[0204] The present invention also provides a computer-readable storage medium having stored thereon computer program instructions that can be executed by a processor, wherein when the processor executes the computer program instructions, it can implement the steps of the method as described above.

[0205] The above are preferred embodiments of the present invention. Any changes made to the technical solution of the present invention that do not exceed the scope of the technical solution of the present invention shall fall within the protection scope of the present invention.

Claims

1. A method for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, characterized in that, include: Based on historical power output data of photovoltaic power plants in the distribution network, a Gaussian mixture model (GMM) is used to characterize the randomness, non-Gaussianity, and multi-peak characteristics of single and multiple photovoltaic power outputs. A particle swarm optimization algorithm is introduced to calculate the Gaussian component parameters in the GMM. Based on the topology and baseline operating point of the distribution network, linearized power flow theory is used to derive an approximate linear mapping relationship between the probability distribution of photovoltaic injected power and the probability distribution of line loss, thereby realizing the calculation of the probability distribution of line loss in the distribution network.

2. The method for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, as described in claim 1, is characterized in that... The Gaussian mixture model (GMM) is used to characterize the randomness, non-Gaussianity, and multi-peak characteristics of single and multiple photovoltaic outputs. The specific implementation method is as follows: Let the random variable of photovoltaic power output be... Where D is the number of photovoltaic power plants, and its probability density function PDF consists of M multivariate Gaussian distribution components. Composition, its expression is In the formula: It is the mean of the m-th Gaussian component; It is the variance of the m-th Gaussian component; It is the mixing weight of the m-th Gaussian component and satisfies ; The probability density function representing the m-th Gaussian component; The complete set of parameters to be estimated in GMM is: The likelihood function and logarithmic representation of the photovoltaic power output sample are obtained by the maximum likelihood estimation method (MLE). In the formula: N represents the number of samples of historical photovoltaic power output data; the goal of GMM parameter estimation is to find a set of parameters. This maximizes the value of the log-likelihood function.

3. The method for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, as described in claim 2, is characterized in that... The particle swarm optimization algorithm is introduced to calculate the Gaussian component parameters in the Gaussian model, that is, the particle swarm optimization algorithm (PSO) is used to optimize the parameters of the Gaussian model. Perform global optimization to maximize the log-likelihood function. .

4. The method for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, as described in claim 2, is characterized by introducing... The particle swarm optimization algorithm is used to calculate the Gaussian component parameters in a Gaussian model. The specific implementation is as follows: (1) Encoding All the parameters to be estimated in the GMM are encoded as the positions of a particle in the search space, a position vector. The structure is as follows: In the formula, vec(·) stacks the lower triangular parts of the symmetric covariance matrix into a vector, with weights... Depend on Determine the constraints to reduce the dimensionality of the optimization problem; (2) Fitness function The fitness value, or the value of the objective function, represents the quality of a particle's position and determines whether to update the individual best position and the group best position of the particle. The algorithm aims to find the Gaussian component parameters when the log-likelihood function is maximized; therefore, the log-likelihood function is directly used as the fitness function. (3) Particle Swarm Optimization Algorithm Flow Initialization: Randomly initialize a particle swarm of size S within the feasible region of parameters, where each particle i has a random initial position in the search space. and initial velocity ; Optimization: In the t-th iteration, calculate the fitness function for each particle. For each particle, compare its current fitness with its historical best fitness. If the current fitness is better, then update. Find the particle with the highest fitness among all particles. m Compare its fitness with the global historical best fitness. Compare the results; if the current fitness is better, then update. ; Update: Update the velocity and position of each particle according to the following formula: In the formula , For learning factors; , A random number within the range [0,1]; The inertia factor represents the particle's global optimization capability; Output: The algorithm stops when the number of iterations reaches the set maximum or the fitness function converges; the global optimum position is then determined. As the optimal parameter set of GMM Output the results.

5. The method for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, as described in claim 1, is characterized in that... Using linearized power flow theory, an approximate linear mapping relationship between the probability distribution of photovoltaic injected power and the probability distribution of line loss is derived, including: (1) Constructing a linearized power flow model: The nonlinear equation for the alternating current flow is as follows: In the formula: It is the injected active power of node i; It is the injected reactive power at node i; , These are the voltages at nodes i and j; It is the equivalent conductance between nodes i and j. It is the equivalent susceptance between nodes i and j; It is the voltage phase angle difference between nodes i and j; N is the total number of nodes in the system; At the reference operating point Performing a first-order Taylor expansion at the given point, and ignoring all higher-order terms, we get: In the formula: , These are the voltage magnitudes and voltage phase angles of the system nodes under the reference state, obtained from deterministic power flow calculations. = This represents the change in the phase angle of the voltage at node k relative to the reference point. This represents the change in the voltage magnitude at node k relative to the reference point. Furthermore, rewrite all nodal equations in matrix form: In the formula This is the Jacobian matrix calculated at the reference point; , Inject changes in active and reactive power into the nodes; , This refers to the changes in the phase angle and magnitude of the node voltage; To represent the change in state variables caused by a change in power, multiply both sides by... The reverse of: （1） In the formula: This is the sensitivity matrix.

6. The method for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, as described in claim 5, is characterized in that... Using linearized power flow theory, an approximate linear mapping relationship between the probability distribution of photovoltaic injected power and the probability distribution of line loss is derived, which also includes: (2) Construct a linear relationship between line power flow and node injected power: The power flow equations of the line are linearized, and the connection nodes are linearized. and nodes The route Its meritorious trend The expression is: At its reference operating point Place, to Perform a first-stage Taylor expansion: In the formula: It is the change in active power on line ij; = This represents the change in the voltage phase angle at node k relative to the voltage phase angle at the reference point. This represents the change in the voltage amplitude at node k relative to the voltage amplitude at the reference point. Rewritten in matrix form: （2） In the formula The sensitivity of the line power flow to state variables; substituting equation (1) into equation (2) yields: To focus on the impact of photovoltaic output on the system state, we assume the change in system reactive power. The above formula can be simplified as follows: In the formula Sensitivity matrix The corresponding change in active power The first N columns, It comprehensively reflects the sensitivity of line power flow to state variables and active power injection; line The active power flow is represented as: In the formula: This is the line active power under reference conditions. This represents the injected active power vector of all nodes under the baseline condition; Will denoted as a constant Thus, an approximate linear relationship is obtained between line power flow and node injected power: 。 7. The method for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, as described in claim 6, is characterized in that... Using linearized power flow theory, an approximate linear mapping relationship between the probability distribution of photovoltaic injected power and the probability distribution of line loss is derived, which also includes: (3) Constructing a linear relationship between line loss and photovoltaic output: The system bus loss is the sum of the losses of each branch. For any branch Its losses It is a function of node voltage magnitude and phase angle; therefore, system bus loss is a function of system operating state. The complex nonlinear function, and the system operating state It is the node injected power The function; In the formula: h(.) is the mapping from node state variables to branch power losses; g(.) is the mapping from node injected power to node state variables; At the reference operating point At this point, performing a first-order Taylor expansion on the above composite function and neglecting higher-order terms, we obtain the change in system bus loss: In the formula: It is the change in active power injected into the node; Rewriting the above equation in matrix form yields the following: In the formula The sensitivity of line loss to various state variables is denoted as . ; Right now This represents the change of the state variable with the injected active power. The above equation simplifies to: In the formula This represents the sensitivity of the system bus loss to injected active power, with the final system bus loss serving as a baseline value. Add the change: （3） In the formula: Inject active power into the nodes under the baseline state; = This is the constant term in the linear expression for line loss; Node Injection Power Powered by photovoltaics and load constitute: （4） Substituting equation (3) into equation (3), we get: In the formula: = This is the constant term in the approximate linear relationship between line loss and photovoltaic output; the above formula establishes an approximate linear relationship between system line loss and photovoltaic output, transforming complex nonlinear power flow calculations into a linear transformation.

8. The method for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, as described in claim 7, is characterized in that... The probability distribution of power distribution network line losses is calculated as follows: Photovoltaic power output The probability distribution is described by GMM. The m-th Gaussian component in GMM Follow the mean The variance is Given a Gaussian distribution, the corresponding linear transformation yields the line loss as... It still follows a Gaussian distribution, with a mean and variance of: Line loss The probability density function is The PDF of the system bus loss is a weighted sum of the PDFs of the corresponding line losses obtained by linear transformation of each Gaussian component in the GMM, where the weights are... Remain unchanged: The cumulative distribution function (CDF) of system bus losses is the weighted sum of the CDFs of each component: The expected system bus loss is: By changing the order of integral and summation, we get: Therefore, it is only necessary to solve for the GMM parameters and sensitivity coefficients. , The probability distribution and expected value of system bus loss can be quickly calculated using a weighted summation analytical calculation method.

9. A system for calculating the probability distribution of line losses in a distribution network based on an improved Gaussian mixture model and linearized power flow, characterized in that, It includes a memory, a processor, and computer program instructions stored in the memory and executable by the processor, which, when executed by the processor, enable the implementation of the steps of the method as described in any one of claims 1-8.

10. A computer-readable storage medium having stored thereon computer program instructions executable by a processor, wherein when the processor executes the computer program instructions, it is able to implement the steps of the method as described in any one of claims 1-8.

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