A transient stability constraint optimal power flow optimization scheduling method containing wind and grid-connected light

Abstract

A kind of wind and light grid-connected transient stability constraint optimal power flow optimization scheduling method, including step 1: based on historical data, combined with Vine-Copula and Weibull distribution, Beta distribution is constructed under the joint probability distribution of wind and light grid connection;Step 2: based on opportunity constraint optimization method (Chance-constrained optimization, CCP), constructs opportunity constraint transient stability constraint optimal power flow (Chance-constrained Transient Stability Constrained Optimal Power Flow, CC-TSCOPF) model, combined with convolutional neural network and transient stability index (Transient Stability Index, TSI) completes transient stability determination;Step 3: based on multivariate sensitivity adaptive (Multivariate Sensitivity-Adaptive, MVSA) constructs sparse polynomial chaos expansion (Sparse Polynomial Chaos Expansion, SPCE) proxy model, high-dimensional random variable is vectorized processing, realize the certainty transformation of CC-TSCOPF;Step 4: based on norm-relaxed sequential quadratic programming (Norm-Relaxed Sequential Quadratic Programming, NSQP) and Peaceman-Rachford split algorithm solve the TSCOPF model after transformation.The present application considers the uncertainty of wind and light and load, and also considers the analysis of transient stability, so that the power system can be safely and efficiently operated.

Description

Technical Field

[0001] This invention relates to the fields of power systems and new energy technologies, and particularly to power system transient stability analysis and optimal power flow acquisition technologies, specifically to a power system transient stability constrained optimal power flow optimization scheduling method that takes into account wind and solar grid connection. Background Technology

[0002] In recent years, the installed capacity of new energy sources such as wind power and photovoltaics has continued to expand, and their grid connection penetration rate has been increasing, making them the core support of new power systems. However, the power system with wind and solar grid connection is greatly affected by natural conditions, such as wind speed and solar radiation, while the load side exhibits significant intermittency, volatility, and randomness due to flexible loads, which greatly increases the uncertainty at both the source and load ends of the power system, posing a severe challenge to the transient stability and safe operation of the system and the optimization of power flow control. Therefore, conducting research on transient stability-constrained optimal power flow under wind and solar grid connection can effectively balance the relationship between new energy consumption, system operation efficiency, and transient safety, and has profound significance for ensuring the safe operation of the power grid and helping to improve the control capabilities of high-proportion new power systems.

[0003] Patent document CN121623892A discloses a transient stability-constrained optimal power flow calculation method based on generalized polynomial chaotic expansion. This method first addresses the uncertainties of the wind-solar-load system by employing generalized polynomial chaotic expansion to perform non-Gaussian probabilistic modeling of wind power, solar power output, and load fluctuations, constructing a TSCOPF model with stochastic parameters. Second, it combines the stochastic Galerkin method to transform the original differential-algebraic equation system of transient stability constraints into algebraic constraints, achieving a dimensionality reduction transformation of PTSCOPF into a parametric nonlinear programming (NLP) problem. Finally, it uses the primordial-dual interior-point method for solution, simultaneously satisfying the dual objectives of high efficiency and transient safety. This method effectively solves the problems of traditional TSCOPF's inability to account for non-Gaussian uncertainties and the explosion of computational dimensions, significantly improving the optimization efficiency in high-proportion renewable energy scenarios. However, this method has certain limitations. It uses a generalized multi-chaotic expansion model, which is insufficient for dimensionality reduction and fitting accuracy of high-dimensional random variables. When facing high-dimensional uncertainties in wind-solar-load, the basis function redundancy is high and the computational efficiency is low. At the same time, it does not clearly construct a confidence constraint system for CC-TSCOPF, but only transforms transient constraints through the stochastic Galerkin method, resulting in insufficient accuracy in quantifying transient stability risks. Patent document CN121715476A discloses a PCE-TSCOPF two-layer optimization scheduling method that takes into account the correlation between wind and solar power. This method first describes the spatiotemporal correlation of wind and solar power output based on Nataf transformation and Frank-Copula, and uses multinomial chaotic expansion to construct a joint probability distribution model to generate uncertainty samples in multiple scenarios. Secondly, it constructs a two-layer optimization framework. The upper layer solves TSCOPF with the goal of minimizing power generation cost and embeds a PCE surrogate model to quantify the impact of uncertainty on transient stability margin. The lower layer handles power flow over-limit risks based on chance constraints and transforms transient power angle and voltage stability indicators into probabilistic safety constraints. Finally, it achieves collaborative solution of the two-layer model through the alternating direction multiplier method. This method effectively addresses the issues of traditional TSCOPF neglecting wind-solar correlation and insufficient accuracy in probabilistic modeling, achieving a balance between safety and efficiency under uncertainty. However, this method has certain limitations. The order and basis function selection of the PCE surrogate model rely on experience, limiting the approximation accuracy in complex scenarios. Furthermore, it does not explicitly construct a confidence constraint system for CC-TSCOPF, merely transforming transient stability indicators into general probabilistic constraints, which leads to some bias in quantifying transient stability risks.

[0004] In summary, a transient stability-constrained optimal power flow scheduling method with wind and solar grid integration is proposed, primarily to provide transient stability assurance for power systems under wind and solar grid integration conditions. This research balances system operational stability and efficiency, providing generator units with better output strategies, and has guiding significance for promoting the construction and development of new power systems. Summary of the Invention

[0005] The purpose of this invention is to address the problems of existing transient stability-constrained optimal power flow scheduling techniques based on polynomial chaotic expansion (PCE). These techniques suffer from insufficient dimensionality reduction and fitting accuracy, high redundancy of basis functions, and inaccurate transient risk quantification due to the lack of a clearly defined confidence constraint system when dealing with high-dimensional random variables. Consequently, optimal power flow scheduling methods face technical challenges such as computational dimensionality explosion, low solution efficiency, and difficulty in balancing system efficiency and transient stability in high-dimensional uncertainties of wind, solar, and load scenarios. This invention is proposed to address these issues.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows: An optimal power flow scheduling method with transient stability constraints involving wind and solar grid integration includes the following steps: Step 1: Based on historical data, a joint probability distribution under wind and solar grid connection is constructed by combining Vine-Copula, Weibull distribution, and Beta distribution. Finally, a three-dimensional joint distribution of wind, solar and load is obtained, thereby restoring the complex nonlinear correlation between wind speed, light intensity and load, and providing a highly reliable input scenario that fits the actual operating conditions for system optimization. Step 2: Construct an optimal power flow model with opportunistic constraints and transient stability constraints based on the opportunistic constraint optimization method. Combine convolutional neural networks and transient stability indices to complete the transient stability determination. Finally, obtain a comprehensive objective function expression that includes boundary opportunity constraints, power angle opportunity constraints and power balance inequality opportunity constraints. This transforms the complex dynamic calculus process of the system into strict probabilistic safety constraints, and realizes the rapid and accurate quantification and determination of transient stability. Step 3: Construct a sparse polynomial chaotic expansion surrogate model based on multivariate sensitivity adaptive model, vectorize the high-dimensional random variables, realize the deterministic transformation of the comprehensive objective function expression, and finally obtain the deterministic objective function and decision variables. This effectively achieves adaptive dimensionality reduction and decoupling of high-dimensional variables while ensuring the model fitting accuracy, thereby reducing the solution dimension and computational complexity. Step 4: Solve the deterministic objective function and decision variables based on modular relaxation sequence quadratic programming and the Peaceman-Rachford splitting algorithm to obtain the optimal power flow strategy, improve the solution stability of the nonlinear constraint optimization problem, and obtain the optimized scheduling solution that meets the preset convergence accuracy and safety constraints. Step 5: Generate control commands based on the optimal power flow strategy to adjust the active and reactive power output of generators in the power grid, so as to maintain the transient stability of the physical operation of the system and thus ensure the safety and reliability of the physical dispatch of the real power grid under the high proportion of new energy access.

[0007] Step 1 specifically includes the following sub-steps: Step 1-1: Collect historical power output data of wind and solar power stations for preprocessing. Use Weibull distribution and Beta distribution to fit the marginal distribution of wind speed and light intensity respectively. Use wind power curve to describe wind power output. Calculate photovoltaic power output from the light intensity conversion formula. And use the maximum likelihood estimation method and parameter calculation to solve for the cumulative distribution function to establish the independent probability distribution characteristics of wind and solar resources. Steps 1-2: To effectively address load uncertainty caused by various factors, a load probability model is constructed, assuming that the load uncertainty follows a normal distribution, and the probability density function of the load is determined, providing basic data for uncertainty analysis at both ends of the source load. Steps 1-3: Combine Vine-Copula to perform correlation analysis on wind speed, light intensity and load data, complete joint distribution modeling based on Sklar's theorem, and finally obtain the three-dimensional joint distribution of wind, light and load.

[0008] Step 2 specifically includes the following sub-steps: Step 2-1: Based on the three-dimensional joint distribution of wind, solar and load generated in Step 1, a three-dimensional joint sample is formed by inverse transformation of the three-dimensional uniform sample, thereby achieving efficient generation of uncertain samples in multiple scenarios; Step 2-2: Construct the system's comprehensive operation objective function based on the three-dimensional joint sample, and list the power balance equations according to the node states; Steps 2-3: Combine convolutional neural networks to mine the mapping relationship between system parameters and transient stability indices to construct transient stability constraints, and use the generator's maximum power angle difference to complete the transient stability determination. This avoids the drawback of traditional time-domain simulation being time-consuming and improves the evaluation efficiency of the system's transient response after a fault. Steps 2-4: Introduce the prediction deviation between the source and load ends under wind and solar grid connection into the boundary opportunity constraint, power angle opportunity constraint and power balance inequality opportunity constraint, and complete the coupling and embedding of opportunity constraints in the model, which effectively improves the model's fault tolerance and robustness to fluctuations in actual environmental parameters. Steps 2-5: Integrate the comprehensive operational objective function, power balance equation, and various opportunity constraints to finally obtain a comprehensive objective function expression that includes boundary opportunity constraints, power angle opportunity constraints, and power balance inequality opportunity constraints.

[0009] Step 3 specifically includes the following sub-steps: Step 3-1: Select the uncertainty source and transform it into an independent uniform variable using the Rosenblatt transformation; Step 3-2: Sample and train data for independent uniform variables to obtain a training sample set, substitute it into the comprehensive objective function expression, obtain a high-dimensional response vector, and organize it into matrix form; Step 3-3: Input variables that follow a standard normal distribution, select the corresponding multinomial basis functions, and construct an initial multi-index set containing zero-order basis terms; Steps 3-4: Based on multivariate sensitivity-driven adaptive basis expansion, calculate the forward neighbor set for the current multi-index set, thereby filtering out the acceptable candidate set; Steps 3-5: For each candidate index in the acceptable candidate set, form a temporary base set, and construct an extended design matrix based on the temporary base set and the input sample matrix; Steps 3-6: Solve for the coefficient matrix of the temporary basis set corresponding to the extended design matrix using vectorized weighted least squares; Steps 3-7: Calculate the multivariate sensitivity contribution index of candidate basis terms based on the coefficient matrix of the temporary basis set; Steps 3-8: Select the candidate with the largest contribution based on the multivariate sensitivity contribution index, and iteratively add the candidate with the largest contribution to the temporary base set to obtain the updated set; Steps 3-9: The minimum contribution variance term in the update set is removed by basis pruning and iterative optimization is performed to obtain the final multi-index set and the corresponding final design matrix. Vectorized weighted least squares is used to solve the optimization coefficient matrix corresponding to the final multi-index set, thereby constructing a high-dimensional surrogate model, eliminating redundant basis functions, and overcoming the dimensionality curse and computational expansion problems caused by high-dimensional uncertainty. Step 3-10: Estimate the response statistical moments using a high-dimensional surrogate model, and use the zero-order coefficients to approximate the mean, expectation, and standard deviation of the output response using polynomial orthogonality; Step 3-11: Use a high-dimensional surrogate model to predict the expectation and standard deviation of random variables, transform the work angle, complete the deterministic transformation, and finally obtain the deterministic objective function and decision variables, realizing the efficient substitution of complex chance constraints into deterministic solvable forms.

[0010] Step 4 specifically includes the following sub-steps: Step 4-1: Based on the deterministic objective function and decision variables, set the initialization parameters and initialize the symmetric positive definite matrix; Step 4-2: At the selected substitution point, the deterministic objective function is transformed into a modular slack problem. Then, through a splitting process, the modular slack problem is further split into two smaller subproblems, thus achieving an effective decomposition of the nonlinear optimization problem. Step 4-3: Solve the subproblem using the effective set method to obtain the search direction and slack variables. Use the augmented Lagrange function as the benefit function to perform a line search along the search direction, complete variable updates and Lagrange multiplier updates, and ensure the strong robustness and efficient convergence of the search process. Step 4-4: Check the relative change and constraint violation to complete the convergence judgment. If convergence is achieved, output the optimal running strategy to obtain the optimal power flow strategy as the basis data for generating control instructions in step 5. If convergence is not achieved, return to step 4-2 to continue iterating.

[0011] In step 1-1, the probability density function used to calculate the marginal distributions of wind speed and light intensity is expressed as: (1); In the formula: Let be the wind speed probability density function; For wind speed, and These are the shape and scale parameters of wind speed, respectively. Let be the probability density function of light intensity; Normalized light intensity; and All are shape parameters of photovoltaic power output; For gamma function; To establish the relationship between wind speed and the physical parameters of the generator set, the formula used to describe wind power output using the wind power curve is as follows: (2); In the formula: , These are the wind speed at which the fan cuts in and the wind speed at which it cuts out; The rated wind speed of the fan; This refers to the rated output of the fan; This refers to the output power of wind power. Combining environmental physical quantities, the formula used to derive photovoltaic output from the light intensity conversion formula is expressed as follows: (3); In the formula: The ambient temperature; The photoelectric conversion coefficient; Rated output for photovoltaic power; Photovoltaic output power; Given wind speed samples and normalized light intensity samples, the likelihood functions of wind power output and photovoltaic power output are solved using the maximum likelihood estimation method. The formula used is as follows: (4); In the formula: and These are random wind speed sample variables and random photovoltaic power output sample variables, respectively. Let Weibull be the likelihood function; Let be the likelihood function of the Beta distribution; n The total number of samples, The sign of consecutive products; By calculating the parameters of the likelihood function, the final cumulative distribution function is expressed as: (5); In the formula: This is a Beta function; , These are the cumulative distribution functions of the Weibull distribution and the Beta distribution, respectively. In steps 1-2, the probability density function for determining the load is expressed by the following formula: (6); In the formula: The active power of the load; The standard deviation of the active power of the load; This represents the average active power of the load. Let be the probability density function of the normal distribution of the load; In steps 1-3, the optimal Pair-Copula is selected based on the C-Vine as the main structure to complete the joint distribution modeling, and the final three-dimensional joint distribution representation of wind, solar and load is obtained as follows: (7); In the formula: and These are the joint cumulative distribution function and the joint probability density function, respectively. The cumulative distribution function of the load is a normal distribution. , , It is a three-dimensional pseudo-uniform sample; These are the parameters of the Copula function; Let Copula be the cumulative distribution function. Let Copula be the probability density function; It is the inverse function of the corresponding function.

[0012] In step 2-2, the system's integrated operation objective function and power balance equation are expressed in the following forms: ; The power balance equations include: (8); In the formula: For expectation operators; Includes conventional generator control cost function Excess Reserve Cost Function Cost function of wind and solar power curtailment penalties and load switching cost function ; and These are the active power and reactive power of conventional generator sets, respectively. This represents the net active power of the node. and These are the active power of the grid-connected wind and solar power units and the active power at the load end, respectively. and These are the predicted active power values ​​and the predicted deviation values ​​of active power for wind and solar grid-connected units, respectively. and These are the predicted active power load and the predicted active power load deviation, respectively. Net reactive power at the node; and These are the reactive power of the grid-connected wind and solar power units and the reactive power at the load end, respectively. and These are the predicted reactive power values ​​of wind and solar grid-connected generating units and the predicted reactive power deviation values ​​of wind and solar grid-connected generating units, respectively. and These are the predicted reactive power load value and the predicted reactive power load deviation value, respectively. To inject power flow balance equations; and These represent the magnitude and phase angle of the node voltage, respectively.

[0013] In steps 2-3, transient stability is determined using the generator's maximum power angle difference; the physical quantity for this determination is the transient stability index. The formula used is expressed as: (9); In the formula: To define the maximum power angle difference between any two generators during a transient process; when The system is considered transiently stable if it is stable, otherwise it is considered unstable. In steps 2-4, the coupling embedding of chance constraints in the model is completed, using the following expression: (10); In the formula: ; Indicates the anticipated moment of system instability under a fault; The fault equivalent power angle; To predict the critical power angle under fault conditions; For the corresponding variable, the probability of violation, , ; The probability of violating the work angle constraint. ; It is a probability symbol. As a lower bound constraint for variables, As an upper limit constraint for variables, For the source of uncertainty, the prediction bias variable; These are active power, reactive power, voltage amplitude, and voltage phase angle under variable constraints, respectively. In steps 2-5, by integrating all constraints, the final comprehensive objective function is expressed as the set expression of the comprehensive operational objective function X, boundary chance constraints, work angle chance constraints, and inequality chance constraints obtained in the above steps: (11).

[0014] In steps 3-6, the vectorized weighted least squares formula used to solve for the coefficient matrix of the temporary basis set is as follows: (12); In the formula: It is a coefficient matrix with dimension 1. ; It is the Frobenius norm; For regularization parameters; Here is the L2 norm regularization term, where It is a coefficient matrix No. line, number Column elements, and These are the row and column dimensions of the system matrix, respectively. To output the response sample matrix; For extended design matrix; In steps 3-7, the formula used to calculate the multivariate sensitivity contribution index of the candidate basis terms is expressed as follows: (13); In the formula: It is a coefficient matrix No. line, number Column elements, for Corresponding basis functions; Quantify the contribution of candidate basis terms to the total variance of all output responses; The dimension is the total number of base items; For the summation index; In section 3-9, the expression for the constructed high-dimensional proxy model is: (14); In the formula: For corresponding multiple indexes coefficient, ; It is a polynomial basis of anisotropic polynomials; Output response values ​​for the high-dimensional proxy model; Input variables are randomly distributed according to a standard normal distribution; For multiple indexes; This is the final multi-index set.

[0015] In step 3-11, after transforming the work angle using the surrogate model, the final deterministic objective function, transformation coefficients, decision variables, expected constraints, and upper and lower bound constraints are expressed as follows: (15); In the formula: As decision variables, ; The lower bounds for decision variables include the generator's minimum active power output, minimum reactive power output, and minimum voltage limit. The upper limit of the decision variables includes the generator's maximum active power output, maximum reactive power output, and maximum voltage limit, etc. Confidence level; It follows a standard normal distribution; The correlation coefficient represents the confidence level. It is a function of total cost; and These are the expected value of the maximum angle difference and the standard deviation of the maximum angle difference, respectively. This is the critical value for the maximum power angle difference; Correlation coefficient at the confidence level of the maximum power angle difference; It is the inverse function of the standard normal distribution.

[0016] Compared with the prior art, the present invention has the following technical effects: 1) To address the problem of computational dimensionality explosion in existing technologies when facing high-dimensional uncertainties of wind, solar and load, this invention constructs a sparse multinomial chaotic expansion SPCE surrogate model through multivariate sensitivity adaptive MVSA. This feature achieves adaptive dimensionality reduction and decoupling of high-dimensional random variables through vectorization processing, effectively eliminating redundant basis functions. Thus, while ensuring fitting accuracy, it significantly reduces computational complexity and solves the technical problems of difficulty and low efficiency in solving traditional optimization problems. 2) To address the problem that the inaccurate quantification of transient stability risks in existing systems leads to scheduling schemes deviating from the actual safety boundary, this invention constructs the Opportunity-Constrained Transient Stability-Constrained Optimal Power Flow (CC-TSCOPF) system and deeply integrates it with a Convolutional Neural Network (CNN). This feature transforms complex nonlinear differential-algebraic equations into strict probabilistic safety constraints, enabling rapid and accurate determination of the transient stability of the system after a fault, effectively avoiding the risk of system instability. 3) To address the shortcomings of traditional methods that assume independent wind and solar power outputs, which leads to scene distortion, this invention adopts a Vine-Copula structure combined with Weibull and Beta distributions. This feature accurately reflects the edge distribution characteristics of wind speed and illumination, deeply explores the complex nonlinear correlation between the source and load ends, and provides a reliable input scenario that closely approximates actual operating conditions for subsequent optimal power flow optimization. 4) To address the technical challenge of traditional optimization algorithms easily getting trapped in local optima when dealing with nonlinear programming problems, this invention introduces a splitting algorithm based on modular relaxation sequential quadratic programming (NSQP) and Peaceman-Rachford (PR). This feature, combining modular relaxation technology with a parallel splitting strategy, can efficiently handle nonlinear constraints, ensuring that the algorithm converges quickly and stably to the preset optimal solution, thus achieving the optimal balance between efficiency and safety of high-proportion new energy power systems under uncertainty. Attached Figure Description

[0017] The present invention will be further described below with reference to the accompanying drawings and embodiments: Figure 1 This is a flowchart of the method of the present invention; Figure 2 This is a schematic diagram of the IEEE 39-node system in an example of the present invention; Figure 3 This is a dynamic response curve of the power angle of a conventional generator in the IEEE 39-bus system without using the method of this invention, as shown in the example of this invention. Figure 4 This is the generator power angle dynamic response curve obtained after solving the problem using the method proposed in this invention example; Figure 5 This is a flowchart illustrating the steps involved in constructing the MVSA-SPCE in this invention. Figure 6 This is a flowchart of the algorithm for solving TSCOPF in this invention; Figure 7 This is a comparison chart of the generator output before and after optimization in this invention. Detailed Implementation

[0018] like Figure 1 As shown, an optimal power flow scheduling method with transient stability constraints involving wind and solar grid integration includes the following steps: Step 1: Based on historical data, construct a joint probability distribution for wind and solar grid connection by combining Vine-Copula, Weibull distribution, and Beta distribution; Step 2: Based on the Chance-constrained optimization (CCP) method, construct the Chance-constrained Transient Stability Constrained Optimal Power Flow (CC-TSCOPF) model, and combine convolutional neural networks and the Transient Stability Index (TSI) to complete the transient stability determination; Step 3: Construct a surrogate model of sparse polynomial chaos expansion (SPCE) based on multivariate sensitivity-adaptive (MVSA) to vectorize high-dimensional random variables and achieve deterministic transformation of CC-TSCOPF. Step 4: Solve the transformed TSCOPF model based on Norm-Relaxed Sequential Quadratic Programming (NSQP) and the Peaceman-Rachford splitting algorithm.

[0019] In step 1, based on historical data, a joint probability distribution for wind-solar grid connection is constructed by combining Vine-Copula, Weibull distribution, and Beta distribution. The specific process is as follows: 1) Fitting of marginal distribution of wind and light Historical power output data from wind and solar power plants were collected and preprocessed. Weibull and Beta distributions were used to fit the marginal distributions of wind speed and solar irradiance, respectively, with their probability density functions (PDFs) as follows: (1); In the formula: Let be the wind speed probability density function; For wind speed, and These are the shape and scale parameters of wind speed, respectively. Let be the probability density function of light intensity; Normalized light intensity; and All are shape parameters of photovoltaic power output; This is the gamma function.

[0020] The wind power output can be described from the wind power curve as follows: (2); In the formula: , These are the wind speed at which the fan cuts in and the wind speed at which it cuts out; The rated wind speed of the fan; This refers to the rated output of the fan; This refers to the output power of wind power.

[0021] Photovoltaic output can be obtained from the formula for converting light intensity: (3); In the formula: The ambient temperature; The photoelectric conversion coefficient; Rated output for photovoltaic power.

[0022] Given wind speed samples Normalized light intensity samples Using the maximum likelihood estimation method, the likelihood functions for wind power output and solar power output are solved as follows: (4); In the formula: and These are random wind speed sample variables and randomized normalized photovoltaic power output sample variables, respectively.

[0023] By calculating the parameters of the likelihood function, the following cumulative distribution function is obtained: (5); In the formula: This is a Beta function; , These are the cumulative distribution functions of the Weibull distribution and the Beta distribution, respectively.

[0024] 2) Load probability model To effectively address load uncertainty caused by various factors, it is approximated that the load uncertainty follows a normal distribution: (6); In the formula: The active power of the load; The standard deviation of the active power of the load; This represents the average active power of the load. Let be the probability density function of the normal distribution of the load.

[0025] 3) Joint probability distribution model Correlation analysis of the three-dimensional joint distribution of wind, solar, and load was conducted using Vine-Copula. Based on Sklar's theorem, and with C-Vine as the main structure, the optimal Pair-Copula was selected using the AIC criterion to complete the joint distribution modeling. (7); In the formula: and These are the joint cumulative distribution function and the joint probability density function, respectively. The cumulative distribution function of the load is a normal distribution. , , It is a three-dimensional pseudo-uniform sample; These are the parameters of the Copula function; Let Copula be the cumulative distribution function. Let Copula be the probability density function; It is the inverse function of the corresponding function.

[0026] In step 2, the chance-constrained optimization method is used to construct CC-TSCOPF. The specific steps are as follows: Step 2-1: Based on the joint cumulative distribution function and probability density function generated in Step 1, a three-dimensional joint sample is formed by performing an inverse transformation on the three-dimensional pseudo-uniform sample.

[0027] Step 2-2: Based on the three-dimensional joint sample, construct the system synthesis objective function and derive the power balance equations according to the power flow equations: (8); In the formula: For expectation operators; Includes "conventional generator control cost function" Excess Reserve Cost Function Cost function of wind and solar power curtailment penalties and load switching cost function ”; and These are the active power and reactive power of conventional generator sets, respectively. This represents the net active power of the node. and These are the active power of the grid-connected wind and solar power units and the active power at the load end, respectively. and These are the predicted active power values ​​and the predicted deviation values ​​of active power for wind and solar grid-connected units, respectively. and These are the predicted active power load and the predicted active power load deviation, respectively. Net reactive power at the node; and These are the reactive power of the grid-connected wind and solar power units and the reactive power at the load end, respectively. and These are the predicted reactive power values ​​of wind and solar grid-connected generating units and the predicted reactive power deviation values ​​of wind and solar grid-connected generating units, respectively. and These are the predicted reactive power load value and the predicted reactive power load deviation value, respectively. To inject power flow balance equations; and These represent the magnitude and phase angle of the node voltage, respectively.

[0028] Steps 2-3: Combining convolutional neural networks to mine the mapping relationship between power system parameters and TSI to construct transient stability constraints. TSI uses the maximum power angle difference of generators to complete the transient stability determination, and its physical quantities can be obtained from... It means that, among them For the normalized transient stability index, when The system is considered transiently stable if it exhibits a transient state of stability, and unstable otherwise, as expressed below: (9); In the formula: To define the maximum power angle difference between any two generators during the transient process.

[0029] Steps 2-4: Prediction bias of uncertainties under wind and solar grid integration By introducing boundary opportunity constraints, power angle opportunity constraints, and power balance inequality opportunity constraints, the coupling and embedding of opportunity constraints in TSCOPF is completed: (10); In the formula: ; Indicates the anticipated moment of system instability under a fault; The fault equivalent power angle; To predict the critical power angle under fault conditions; For the variable, the probability of violation, , ; The probability of violating the work angle constraint. ; It is a probability symbol. As a lower bound constraint for variables, Upper limit of the variable constraint; These represent active power, reactive power, voltage amplitude, and voltage phase angle under variable constraints.

[0030] Steps 2-5: In summary, the CC-TSCOPF model is constructed as follows: (11); In step 3, the SPCE surrogate model is constructed based on MVSA, and the high-dimensional random variables are vectorized to achieve the deterministic transformation of CC-TSCOPF.

[0031] Step 3-1: Select the uncertainty source from Step 2 By using Rosenblatt transformation Transform into independent uniform variables ,in Dimensions of uncertainty sources; Step 3-2: Based on the joint distribution in Step 1, perform data sampling and training to obtain the training sample set. Substituting the values ​​into the CC-TSCOPF model from step 2, we obtain the high-dimensional response vector. ,in The total number of training samples is the number of input samples. With response samples Organize into matrix form , .

[0032] Step 3-3: Based on the above steps, input the variables. ,in Following a standard normal distribution, select the corresponding multinomial basis functions and construct an initial multi-index set. It contains zero-order basis terms. .

[0033] Steps 3-4: Based on multivariate sensitivity-driven adaptive basis expansion, apply the following to the current multi-index set. The forward neighbor set is obtained by calculating the forward neighbor set. ,in Given a unit vector, filter the acceptable candidate set. .

[0034] Steps 3-5: For each candidate index Forming temporary basis sets ,based on and input sample matrix Constructing an extended design matrix .

[0035] Steps 3-6: Solve for the coefficient matrix of the temporary basis set using Vectorized Weighted Least Squares (VWLS): (12); In the formula: It is a coefficient matrix with dimension 1. ; It is the Frobenius norm; For regularization parameters; Here is the L2 norm regularization term, where It is a coefficient matrix No. line, number Column elements, and These represent the row and column dimensions of the system matrix, respectively. To output the response sample matrix; For extended design matrix.

[0036] Step 3-7: Calculate the multivariate sensitivity contribution index of candidate basis terms: (13); In the formula: It is a coefficient matrix No. line, number Column elements, for Corresponding basis functions; Quantify the contribution of candidate basis terms to the total variance of all output responses; The dimension is the total number of base items; For the summation index.

[0037] Steps 3-8: According to Select the candidate with the largest multivariate sensitivity contribution index, and iteratively add the best candidate to the base set to obtain the updated set. .

[0038] Steps 3-9: Following the above steps, the final multi-index set is obtained through base pruning, removal of the minimum contribution variance term, and iterative optimization. The corresponding design matrix is ​​obtained. Using vectorized weighted least squares Solving for the optimization coefficient matrix yields the solution. For a standard normally distributed random input variable The constructed SPCE high-dimensional proxy model is as follows: (14); In the formula: For corresponding multiple indexes coefficient, ; It is a polynomial basis of anisotropic polynomials; Output response values ​​for the high-dimensional proxy model.

[0039] Steps 3-10: Estimate the response statistical moments using SPCE. Based on the orthogonality of PCE, all output responses... The mean and expected value can be obtained from the zeroth-order coefficients. Given, among which The coefficients corresponding to the zero-order basis terms are given, and their standard deviations can be approximated as... .

[0040] Step 3-11: Use the PCE surrogate model to predict random variables in the CC-TSCOPF model Expectations and standard deviation The objective function expectation can be directly approximated using the zeroth-order coefficients of the SPCE, and then the power angle can be transformed through the inverse operation of the standard normal cumulative distribution. The deterministic transformation is as follows: (15); In the formula: As decision variables, ; The lower bounds for decision variables include the generator's minimum active power output, minimum reactive power output, and minimum voltage limit. The upper limit of the decision variables includes the generator's maximum active power output, maximum reactive power output, and maximum voltage limit, etc. Confidence level; It follows a standard normal distribution; The correlation coefficient represents the confidence level. It is a function of total cost; and These are the expected value of the maximum angle difference and the standard deviation of the maximum angle difference, respectively. This is the critical value for the maximum power angle difference; Correlation coefficient at the confidence level of the maximum power angle difference; It is the inverse function of the standard normal distribution.

[0041] The transformed TSCOPF model is solved using Norm-Relaxed Sequential Quadratic Programming (NSQP) and the Peaceman-Rachford splitting algorithm. The specific steps are as follows: Step 4-1: Based on the deterministic objective function generated in Step 3 and decision variables Set initialization parameters and initialize the symmetric positive definite matrix.

[0042] Step 4-2: Iteration Point The deterministic TSCOPF model is transformed into a modular relaxation QP subproblem, and then the decision variables are processed through PR splitting. This further splits the aforementioned QP subproblem into two smaller QP subproblems.

[0043] Step 4-3: Solve the QP subproblem using the effective set method to obtain the search direction. and slack variables Using the augmented Lagrangian function as the benefit function, which includes the objective function and constraint penalty terms, along the direction Perform an Armijo line search to complete variable updates and Lagrange multiplier updates.

[0044] Step 4-4: Check the relative change and constraint violation degree to complete the convergence judgment. If convergence is achieved, output the optimal iteration strategy and verify whether its transient stability probability is satisfied. Otherwise, return to step 4-2 and continue iterating.

[0045] Example: To verify the effectiveness of the proposed transient stability-constrained optimal power flow scheduling method with wind and solar grid integration, it was applied to an improved IEEE 39-bus test system, which includes 10 synchronous generators, 39 buses, and a base capacity of 100 MVA. The basic parameters and model parameters of the test system are shown in Table 1.

[0046] Uncertainty modeling for wind-solar grid connection is performed based on historical wind speed, solar power output, and load data for a certain region. The marginal distribution of wind speed is fitted using a Weibull distribution, and the marginal distribution of solar power output is fitted using a Beta distribution. The load power and active power follow a normal distribution. For MW, a Vine-Copula (specifically a C-Vine structure) is used to model the correlation among wind, solar, and load. The optimal Pair-Copula is selected using the AIC criterion to obtain the correlation coefficient matrix. Based on the joint distribution, 1000 representative wind-solar-load joint output scenarios are generated through sampling for subsequent CC-TSCOPF modeling and optimization. A transient stability-constrained optimal power flow model is constructed based on the chance constraint method. The objective function comprehensively considers the conventional power generation cost, the cost of wind and solar curtailment penalties, and the load switching cost, while introducing chance constraints to handle wind-solar-load prediction bias. Transient stability determination adopts a combination of convolutional neural networks and transient stability indices: the CNN model takes the time-frequency characteristics of the generator power angle trajectory as input and outputs the probability of system instability. As a transient stability index, it is defined as follows: ,in The maximum power angle difference between any two generators. The time is stable. An agent model is constructed based on the MVSA-SPCE method, with random variables as inputs, including the total system cost and... To output the response, an SPCE surrogate model is constructed using 500 training samples and adaptive base expansion.

[0047] To verify the effectiveness of the method of the present invention under transient faults, the following fault scenarios were set as shown in Table 2:

[0048] A three-phase short-circuit fault occurred on the branch between busbars 34 and 35. The fault initiation time was 1.0 s, and the faulty line was disconnected 0.15 s later. The total simulation time was 10 s. First, transient simulation was performed using the system's initial operating point, and the generator power angle response curve was obtained as follows: Figure 3 As shown. By Figure 3 As can be seen, after the fault occurs, the power angles of each generator diverge rapidly, and the system loses transient stability. The method and steps proposed in this invention are then applied for optimization. Transient simulations are performed on the optimized scheduling scheme using the same fault scenario, and the resulting generator power angle response curves are shown below. Figure 4 As shown. From Figure 4 It can be seen that although the power angles of each generator fluctuated after the fault occurred, the maximum power angle difference was less than 180°, and the power angle trajectory gradually converged, and the system remained transiently stable.

[0049] The specific experimental data obtained after the transient simulation process are shown in Table 3:

[0050] In the improved IEEE 39-node system, the proposed transient stability-constrained optimal power flow scheduling method based on the MVSA-SPCE surrogate model meets the preset efficiency and security requirements. As shown in Table 3, the MVSA-SPCE surrogate model ultimately retains 20 effective basis functions. The TSI surrogate model has a root mean square error (RMSE) of 0.0281 and a coefficient of determination (R²) of 0.8366 on the validation samples, indicating that the constructed surrogate model can accurately approximate the transient stable response, providing reliable mathematical support for the deterministic transformation of the chance-constrained model. Furthermore, the optimization algorithm converges after 56 iterations, reducing the objective function value from 31604.39 to 31410.30. This further optimizes the operating cost while satisfying system constraints, demonstrating that the proposed solution strategy has good convergence performance and optimization capability.

[0051] Combination Figure 7It can be seen that, under the premise of satisfying power flow balance and transient stability constraints, the active power output of each conventional generator unit has been redistributed. Specifically, the output of generators 1 and 2 increased from 250MW and 150MW to approximately 275MW and 200MW, respectively, while the output of generators 4, 5, 6, and 7 decreased accordingly, and the output of the remaining units only underwent minor adjustments. These results indicate that this invention does not simply reduce the output of all conventional units, but rather, based on a comprehensive consideration of random fluctuations in wind and solar power output, network power flow constraints, and post-fault transient stability constraints, reconfigures the regulation tasks undertaken by different units. This allows units with stronger regulation capabilities or greater contributions to system stability to appropriately increase their output, while some units appropriately decrease their output, thereby achieving optimized adjustment of the system's operating mode.

[0052] Furthermore, as shown in Table 3, the proportion of renewable energy generation reached 24.05%, the optimized system's actual TSI was 0.818, and the maximum power angle difference was reduced to 36.1°, all meeting the requirements for transient stability operation. This demonstrates that the present invention, by constructing a joint probability model of wind, solar, and load, establishing a chance-constrained transient stability optimal power flow model, and utilizing the MVSA-SPCE surrogate model to achieve efficient deterministic transformation of stochastic constraints, fully considers the stochasticity of renewable energy while achieving coordinated optimization between efficient operation and transient stability, thus improving the safe operation capability of grid-connected power systems with wind and solar power under uncertain conditions.

[0053] Compared with existing technologies, the proposed research on transient stability-constrained optimal power flow with wind and solar grid connection has the following advantages: In terms of uncertainty modeling, the Vine-Copula structure and Weibull and Beta distributions are adopted, which not only characterize the marginal distribution features of wind speed and photovoltaic power, but also capture the complex correlation between them, providing a reliable input scenario for subsequent optimization; In terms of transient stability determination, the nonlinear mapping between power system operating parameters and transient stability indices is learned through a CNN network, replacing the traditional time-consuming time-domain simulation; In terms of high-dimensional uncertainty decoupling, the MVSA-SPCE surrogate model achieves dimensionality reduction and decoupling of high-dimensional random variables through multivariate sensitivity adaptation, efficiently transforming the chance-constrained problem into a deterministic optimization problem, solving the bottleneck of the difficulty in solving traditional chance-constrained programming; In terms of optimization solution, the NSQP-PR split algorithm is adopted. This algorithm combines modular relaxation technology and parallel splitting strategy, which can effectively handle nonlinear programming problems and quickly converge to the preset optimal solution, avoiding the defect of traditional algorithms being prone to getting trapped in local optima.

Claims

1. A method for optimal power flow scheduling with transient stability constraints involving wind and solar grid integration, characterized in that, Includes the following steps: Step 1: Based on historical data, construct the joint probability distribution under wind and solar grid connection, and finally obtain the three-dimensional joint distribution of wind, solar and load. Step 2: Construct an optimal power flow model with opportunity constraints and transient stability constraints, complete the transient stability determination, and finally obtain a comprehensive objective function expression that includes boundary opportunity constraints, power angle opportunity constraints, and power balance inequality opportunity constraints; Step 3: Construct a sparse polynomial chaotic expansion surrogate model, vectorize the high-dimensional random variables, realize the deterministic transformation of the comprehensive objective function expression, and finally obtain the deterministic objective function and decision variables; Step 4: Solve for the deterministic objective function and decision variables to obtain the optimal power flow strategy; Step 5: Generate control commands based on the optimal power flow strategy to adjust the active and reactive power output of generators in the power grid in order to maintain the transient stability of the system's physical operation.

2. The method according to claim 1, characterized in that, Step 1 specifically includes the following sub-steps: Step 1-1: Collect historical power output data of wind and solar power stations for preprocessing. Use Weibull distribution and Beta distribution to fit the marginal distribution of wind speed and light intensity respectively. Use wind power curve to describe wind power output. Calculate photovoltaic power output from the light intensity conversion formula. And obtain the cumulative distribution function by maximum likelihood estimation method and parameter calculation. Step 1-2: To effectively address load uncertainty caused by various factors, a load probability model is constructed, assuming that the load uncertainty follows a normal distribution, and the probability density function of the load is determined. Steps 1-3: Combine Vine-Copula to perform correlation analysis on wind speed, light intensity and load data, complete joint distribution modeling based on Sklar's theorem, and finally obtain the three-dimensional joint distribution of wind, light and load.

3. The method according to claim 2, characterized in that, Step 2 specifically includes the following sub-steps: Step 2-1: Based on the three-dimensional joint distribution of wind, solar and load generated in Step 1, a three-dimensional joint sample is formed by inverse transformation of the three-dimensional uniform sample; Step 2-2: Construct the system's comprehensive operation objective function based on the three-dimensional joint sample, and list the power balance equations according to the node states; Steps 2-3: Combine convolutional neural networks to mine the mapping relationship between system parameters and transient stability indices to construct transient stability constraints, and use the generator's maximum power angle difference to complete the transient stability determination; Steps 2-4: Introduce the prediction deviation between the source and load ends under wind and solar grid connection into the boundary opportunity constraint, power angle opportunity constraint and power balance inequality opportunity constraint to complete the coupling and embedding of opportunity constraints in the model; Steps 2-5: Integrate the comprehensive operational objective function, power balance equation, and various opportunity constraints to finally obtain a comprehensive objective function expression that includes boundary opportunity constraints, power angle opportunity constraints, and power balance inequality opportunity constraints.

4. The method according to any one of claims 1 to 3, characterized in that, Step 3 specifically includes the following sub-steps: Step 3-1: Select the uncertainty source and transform it into an independent uniform variable using the Rosenblatt transformation; Step 3-2: Sample and train data for independent uniform variables to obtain a training sample set, substitute it into the comprehensive objective function expression, obtain a high-dimensional response vector, and organize it into matrix form; Step 3-3: Input variables that follow a standard normal distribution, select the corresponding multinomial basis functions, and construct an initial multi-index set containing zero-order basis terms; Steps 3-4: Based on multivariate sensitivity-driven adaptive basis expansion, calculate the forward neighbor set for the current multi-index set, thereby filtering out the acceptable candidate set; Steps 3-5: For each candidate index in the acceptable candidate set, form a temporary base set, and construct an extended design matrix based on the temporary base set and the input sample matrix; Steps 3-6: Solve for the coefficient matrix of the temporary basis set corresponding to the extended design matrix using vectorized weighted least squares; Steps 3-7: Calculate the multivariate sensitivity contribution index of candidate basis terms based on the coefficient matrix of the temporary basis set; Steps 3-8: Select the candidate with the largest contribution based on the multivariate sensitivity contribution index, and iteratively add the candidate with the largest contribution to the temporary base set to obtain the updated set; Steps 3-9: Remove the minimum contribution variance term from the update set through base pruning and iteratively optimize to obtain the final multi-index set and the corresponding final design matrix. Use vectorized weighted least squares to solve the optimization coefficient matrix corresponding to the final multi-index set, thereby constructing a high-dimensional surrogate model. Step 3-10: Estimate the response statistical moments using a high-dimensional surrogate model, and use the zero-order coefficients to approximate the mean, expectation, and standard deviation of the output response using polynomial orthogonality; Step 3-11: Use a high-dimensional surrogate model to predict the expected value and standard deviation of the random variables, transform the work angle, complete the deterministic transformation, and finally obtain the deterministic objective function and decision variables.

5. The method according to claim 1, characterized in that, Step 4 specifically includes the following sub-steps: Step 4-1: Based on the deterministic objective function and decision variables, set the initialization parameters and initialize the symmetric positive definite matrix; Step 4-2: At the selected substitution point, the deterministic objective function is transformed into a modular slack problem, and then the modular slack problem is further split into two subproblems through a splitting process; Step 4-3: Solve the subproblem using the effective set method to obtain the search direction and slack variables. Use the augmented Lagrange function as the benefit function to perform a line search along the search direction, and complete the variable update and Lagrange multiplier update. Step 4-4: Check the relative change and constraint violation to complete the convergence judgment. If convergence is achieved, output the optimal running strategy to obtain the optimal power flow strategy as the basis data for generating control instructions in step 5. If convergence is not achieved, return to step 4-2 to continue iterating.

6. The method according to claim 2, characterized in that, In step 1-1, the probability density function used to calculate the marginal distributions of wind speed and light intensity is expressed as: （1）； In the formula: Let be the wind speed probability density function; For wind speed, and These are the shape and scale parameters of wind speed, respectively. Let be the probability density function of light intensity; Normalized light intensity; and All are shape parameters of photovoltaic power output; For gamma function; To establish the relationship between wind speed and the physical parameters of the generator set, the formula used to describe wind power output using the wind power curve is as follows: （2）； In the formula: , These are the wind speed at which the fan cuts in and the wind speed at which it cuts out; The rated wind speed of the fan; This refers to the rated output of the fan; This refers to the output power of wind power. Combining environmental physical quantities, the formula used to derive photovoltaic output from the light intensity conversion formula is expressed as follows: （3）； In the formula: The ambient temperature; The photoelectric conversion coefficient; Rated output for photovoltaic power; Photovoltaic output power; Given wind speed samples and normalized light intensity samples, the likelihood functions of wind power output and photovoltaic power output are solved using the maximum likelihood estimation method. The formula used is as follows: （4）； In the formula: and These are random wind speed sample variables and random photovoltaic power output sample variables, respectively. Let Weibull be the likelihood function; Let be the likelihood function of the Beta distribution; n The total number of samples, The sign of consecutive products; By calculating the parameters of the likelihood function, the final cumulative distribution function is expressed as: （5）； In the formula: This is a Beta function; , These are the cumulative distribution functions of the Weibull distribution and the Beta distribution, respectively. In steps 1-2, the probability density function for determining the load is expressed by the following formula: （6）； In the formula: The active power of the load; The standard deviation of the active power of the load; This represents the average active power of the load. Let be the probability density function of the normal distribution of the load; In steps 1-3, the optimal Pair-Copula is selected based on the C-Vine as the main structure to complete the joint distribution modeling, and the final three-dimensional joint distribution representation of wind, solar and load is obtained as follows: （7）； In the formula: and These are the joint cumulative distribution function and the joint probability density function, respectively. The cumulative distribution function is the normal distribution of the load. , , It is a three-dimensional pseudo-uniform sample; These are the parameters of the Copula function; Let Copula be the cumulative distribution function. Let Copula be the probability density function; It is the inverse function of the corresponding function.

7. The method according to claim 3, characterized in that, In step 2-2, the system's integrated operation objective function and power balance equation are expressed in the following forms: ； The power balance equations include: （8）； In the formula: For expectation operators; Includes conventional generator control cost function Excess Reserve Cost Function Cost function of wind and solar power curtailment penalty and load switching cost function ; and These are the active power and reactive power of conventional generator sets, respectively. This represents the net active power of the node. and These are the active power of the grid-connected wind and solar power units and the active power at the load end, respectively. and These are the predicted active power values ​​and the predicted deviation values ​​of active power for wind and solar grid-connected units, respectively. and These are the predicted active power load and the predicted active power load deviation, respectively. Net reactive power at the node; and These are the reactive power of the grid-connected wind and solar power units and the reactive power at the load end, respectively. and These are the predicted reactive power values ​​of wind and solar grid-connected generating units and the predicted reactive power deviation values ​​of wind and solar grid-connected generating units, respectively. and These are the predicted reactive power load value and the predicted reactive power load deviation value, respectively. To inject power flow balance equations; and These represent the magnitude and phase angle of the node voltage, respectively.

8. The method according to claim 7, characterized in that, In steps 2-3, transient stability is determined using the generator's maximum power angle difference; the physical quantity for this determination is the transient stability index. The formula used is expressed as: （9）； In the formula: To define the maximum power angle difference between any two generators during the transient process; when The system is considered transiently stable if it is stable, otherwise it is considered unstable. In steps 2-4, the chance constraints are coupled and embedded in the model, using the following expression: （10）； In the formula: ; Indicates the anticipated moment of system instability under a fault; The fault equivalent power angle; To predict the critical power angle under fault conditions; For the corresponding variable, the probability of violation, , ; The probability of violating the work angle constraint. ; It is a probability symbol. As a lower bound constraint for variables, As an upper limit constraint for variables, For the source of uncertainty, the prediction bias variable; These are active power, reactive power, voltage amplitude, and voltage phase angle under variable constraints, respectively. In steps 2-5, by integrating all constraints, the final comprehensive objective function is expressed as the set expression of the comprehensive operational objective function X, boundary chance constraints, work angle chance constraints, and inequality chance constraints obtained in the above steps: （11）。 9. The method according to claim 4, characterized in that, In steps 3-6, the vectorized weighted least squares formula used to solve for the coefficient matrix of the temporary basis set is as follows: （12）； In the formula: It is a coefficient matrix with dimension 1. ; It is the Frobenius norm; For regularization parameters; Here is the L2 norm regularization term, where It is a coefficient matrix No. line, number Column elements, and These are the row and column dimensions of the system matrix, respectively. To output the response sample matrix; For extended design matrix; In steps 3-7, the formula used to calculate the multivariate sensitivity contribution index of candidate basis terms is expressed as follows: （13）； In the formula: It is a coefficient matrix No. line, number Column elements, for Corresponding basis functions; Quantify the contribution of candidate basis terms to the total variance of all output responses; The dimension is the total number of base items; For the summation index; In section 3-9, the expression for the constructed high-dimensional proxy model is: （14）； In the formula: For corresponding multiple indexes coefficient, ; It is a polynomial basis of anisotropic polynomials; Output response values ​​for the high-dimensional proxy model; Input variables are randomly distributed according to a standard normal distribution; For multiple indexes; This is the final multi-index set.

10. The method according to claim 4, characterized in that, In step 3-11, after transforming the work angle using the surrogate model, the final deterministic objective function, transformation coefficients, decision variables, expectation constraints, and upper and lower bound constraints are expressed as follows: （15）； In the formula: As decision variables, ; The lower bounds for decision variables include the generator's minimum active power output, minimum reactive power output, and minimum voltage limit. The upper limit of the decision variables includes the generator's maximum active power output, maximum reactive power output, and maximum voltage limit, etc. Confidence level; It follows a standard normal distribution; The correlation coefficient represents the confidence level. It is a function of total cost; and These are the expected value of the maximum angle difference and the standard deviation of the maximum angle difference, respectively. This is the critical value for the maximum power angle difference; Correlation coefficient at the confidence level of the maximum power angle difference; It is the inverse function of the standard normal distribution.

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