Improved polynomial chaos expansion based optimal scheduling method of cc-tscopf

Abstract

The CC-TSCOPF optimal scheduling method based on improved polynomial chaotic expansion includes: Step 1: Establishing an optimal power flow model with opportunity constraints and transient stability constraints that take into account the uncertainty of power generation at both the source and load ends; Step 2: Based on the Wiener-Askey polynomial chaotic approximation theorem, selecting orthogonal polynomial basis functions corresponding to the prediction deviation to construct the PCE expression, and simultaneously using a hyperbolic truncation scheme and a minimum absolute contraction and selection operator regression scheme to simplify and sparsify the basis functions and verify the fitting accuracy of the truncated PCE expression; Step 3: Realizing the deterministic transformation of opportunity constraints in the CC-TSCOPF model; Step 4: Solving the CC-TSCOPF model after the chaotic expansion of the above steps; This invention considers both the uncertainty of power generation at the source end and power consumption at the load end, as well as the analysis of transient stability, enabling the power system to operate safely and efficiently.

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 optimization techniques, specifically to a power system optimal power flow optimization method based on improved polynomial chaotic expansion. Background Technology

[0002] With the rapid increase in installed capacity of new energy sources such as wind power and photovoltaic power, the intermittency and volatility of wind and solar power generation have significantly increased the uncertainty of power system operation, posing new challenges to the safe and stable operation of the power grid. Therefore, conducting research on transient stability-constrained optimal power flow with wind and solar grid integration is helpful to improve the safety and efficiency of the power system in an environment with a high proportion of new energy sources, and is of great significance to the stable operation of the system.

[0003] Patent application CN117977554A discloses an optimization method for handling transient stability and uncertainty in power systems. This method considers the impact of uncertainties in system operating parameters and incorporates transient stability constraints into the optimization model. However, this method has limitations when dealing with multidimensional uncertainty sources. When facing high-dimensional systems, its uncertainty handling method often struggles to efficiently analyze the complex differential-algebraic equations describing the transient dynamic characteristics of the power system into low-order algebraic constraints, resulting in limited computational efficiency and discrepancies between the calculated results and the actual requirements of high-proportion renewable energy operation. Patent application CN119004780A discloses a method for handling system uncertainties based on polynomial chaotic expansion. This method uses PCE to replace complex stochastic simulations, improving mathematical solution efficiency to some extent. However, this method has significant shortcomings when dealing with large-scale power systems: as the dimension of uncertain input variables increases, the number of basis function terms in its PCE scheme grows exponentially, easily leading to the "curse of dimensionality," resulting in a dramatic increase in computational complexity when solving multivariable systems.

[0004] In summary, the applicant proposes a CC-TSCOPF optimal scheduling method based on improved polynomial chaotic expansion, aiming to provide transient stability assurance for power systems under large-scale grid connection of new energy sources. This method can not only improve the safety of system operation, but also provide efficient decision-making basis for optimizing generator output, which has important guiding significance for promoting the construction of new power systems. Summary of the Invention

[0005] The purpose of this invention is to address the problems of existing power flow calculation methods in power systems, such as the exponential growth of basis functions leading to the curse of dimensionality when dealing with high-dimensional uncertainties, and the difficulty in efficiently analyzing complex transient dynamic differential-algebraic equations. As a result, optimal power flow optimization control methods face technical problems such as computational dimensionality explosion, low solution efficiency, and difficulty in accurately balancing high energy efficiency and transient safety dynamic defenses in scenarios with a high proportion of renewable energy grid connection. 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: The CC-TSCOPF optimization scheduling method based on improved polynomial chaotic expansion includes the following steps: Step 1: To address the impact of uncertainties in source and load operating parameters, reconstruct the objective function of the transient stability-constrained optimal power flow model and embed opportunity constraints. Establish an opportunity-constrained transient stability-constrained optimal power flow model that takes into account the uncertainty of power at both ends of the source and load, so as to transform the complex system dynamic calculus process into a strict probabilistic safety constraint and achieve rapid and accurate quantification of transient stability. Step 2: Select orthogonal polynomial basis functions to construct a polynomial chaotic expansion expression. At the same time, use the hyperbolic truncation scheme and the minimum absolute contraction and selection operator regression scheme to simplify and sparsify the basis functions. This effectively eliminates redundant higher-order terms and reduces the computational complexity of solving high-dimensional systems while ensuring the model fitting accuracy. Step 3: Calculate the undetermined coefficients of the polynomial chaotic expansion expression based on the least squares principle, and use the polynomial chaotic expansion expression to calculate the expected value and standard deviation of each running variable, realizing the transformation of chance constraints into deterministic forms, thereby reducing the complex probabilistic constraints to deterministic algebraic constraints that can be directly solved. Step 4: Based on the improved alternating direction multiplier method, solve the opportunity-constrained transient stability-constrained optimal power flow model after polynomial chaotic expansion and transformation, iteratively optimize to obtain the optimal power flow scheduling strategy for the entire network, use distributed collaborative processing of nonlinear constraints, reduce the amount of communication data and ensure that the algorithm converges quickly and stably to the global optimal solution, and achieve the optimal balance between high energy efficiency and security of system operation. Step 5: Generate underlying hardware drive instructions based on the optimal power flow scheduling strategy to adjust the active and reactive power of each conventional generator unit and new energy unit in the power grid in real time, so as to maintain the transient stable operation of the power grid physical system under the expected fault, thereby ensuring the safety and reliability of the real power grid physical scheduling under the high proportion of new energy access.

[0007] Step 1 specifically includes the following sub-steps: Step 1-1: Based on the power, voltage, and phase angle parameters of conventional generators and new energy generators at the source end and various loads at the load end during operation, analyze the impact of parameter uncertainty on the power system, reconstruct the objective function of the model, and obtain the extended Transient Stability-Constrained Optimal Power Flow (TSCOPF) model; Step 1-2: Perform single-machine infinite equalization on the system, and analyze the transient stability constraints of the system based on the trajectory sensitivity method. Simplify it into the sensitivity constraint of the synchronous generator's active power output to the power angle after the system fault, thereby analyzing the transient stability constraints into low-order algebraic inequality constraints. Steps 1-3: Introduce the power prediction bias at both ends of the source and load to characterize the power prediction bias, embed the chance constraint into the transient stability constraint optimal power flow model, and establish the chance-constrained CC-TSCOPF (Chance-Constrained-TSCOPF) model that takes into account the uncertainty of power at both ends of the source and load.

[0008] Step 2 specifically includes the following sub-steps: Step 2-1: Based on the polynomial chaos approximation theorem, select orthogonal polynomial basis functions corresponding to the prediction bias according to different random distributions of the prediction bias input to construct the polynomial chaos expansion (PCE) expression. Step 2-2: Simplify the basis function expressions based on the Hyperbolic Truncation Scheme (HTS), introduce the q norm and determine the total number of basis functions to be retained, and remove unimportant higher-order cross terms; Steps 2-3: Generate sample points using Latin hypercube sampling (LHS), calculate model response values, and construct regression matrices; Steps 2-4: Based on the Least Absolute Shrinkage and Selection Operator (LASSO), regress the sparsified basis function expression, introduce L1 regularization constraints, further filter the truncated basis function set, and retain only the basis functions that have a greater impact on the output response, thus completing the sparsity screening.

[0009] Step 3 specifically includes the following sub-steps: Step 3-1: Calculate the undetermined coefficients of the PCE expression based on the least squares principle, and solve to obtain the final basis function coefficient vector; Step 3-2: Estimate the posterior error based on leave-one-out cross-validation, and calculate the error definition to measure the generalization accuracy of the model; Step 3-3: Calculate the expected value and standard deviation of each running variable using the PCE expression, re-express the objective function and each constraint condition, and complete the transformation of the chance constraints in the model into a deterministic form.

[0010] Step 4 specifically includes the following sub-steps: Step 4-1: Based on the deterministic optimal power flow model transformed from Step 3, input the network power parameters, voltage magnitude parameters, and phase angle parameters required by the model; Step 4-2: Based on the distribution of the source and load at both ends, select a reasonable and effective partitioning method to partition the physical network of the system, and decompose the physical connection data of the nodes, generators and lines of the power grid system to each control sub-region accordingly; Step 4-3: Each control sub-region independently performs local optimization and then updates the global variables, and exchanges information between adjacent sub-regions to exchange boundary state parameters; Step 4-4: Evaluate the dual residuals and original residuals of the Alternating Direction Method of Multipliers (ADMM), and determine whether the residuals are all less than the preset convergence threshold. If the threshold is met, the iteration stops and the joint strategy of each control sub-region is output as the optimal power flow scheduling strategy. Otherwise, the Lagrange multipliers are updated and the distributed iteration is returned to step 4-3 until the convergence condition is met.

[0011] In step 1-1, the formula set used to establish the extended TSCOPF model is expressed as follows: (1); In the formula: Equation (1a) is the optimization objective of the TSCOPF model, i.e., the objective function that minimizes the total economic cost. Let the total cost function be... , , and These are the conventional generator unit control cost function, excess reserve cost function, wind and solar curtailment penalty cost function, and load switching cost function, respectively. , and Let represent the active power of the conventional generator set, the active power of the new energy generator set, and the active power on the load side, respectively; Equation (1b) is the power balance equation of the system. Let be the power balance equation function of the system. and These represent the net injected active and reactive power at the node, respectively. and These represent the node voltage magnitude and the node voltage phase angle, respectively; Equation (1c) represents the upper and lower limits of the system constraints. A set of system parameter constraints. and These are the lower and upper limits of the system constraints, respectively. Line current margin; Equation (1d) is the differential equation describing the dynamic characteristics of the power system during the transient period; Equation (1e) is the algebraic equation representing the power balance constraints that must be satisfied during the transient process; Equation (1f) is the constraint limiting the maximum rotor angle deviation between any two generators after the disturbance. , , These are system state variables, algebraic variables, and control variables, respectively. For system state variables in time t The first derivative; The duration of the system transient process; t For any time during the duration of the system transient process; For transient dynamic equation functions, Algebraic constraint functions that must be satisfied during transient processes; This is the constraint function for the transient safe operation inequality.

[0012] In steps 1-2, the set of formulas used to analyze the transient stability constraints into low-order algebraic inequality constraints is expressed as follows: (2); In the formula: To anticipate the moment when the system will become unstable under a fault; The critical power angle of the system under the anticipated fault; and These are the system equivalent power angles before and after the active power adjustment of the conventional generator set, respectively. and These are the active power adjustment of a conventional generator and the corresponding equivalent power angle change, respectively. The trajectory sensitivity of the active power of a synchronous generator unit to the power angle; In steps 1-3, the formula set used to establish the CC-TSCOPF model considering the power uncertainty at both ends of the source and load is expressed as follows: (3); In the formula: Let be the expected value of the objective function; The probability that the condition in parentheses is true; Let be the probability of violation of the chance constraint for the corresponding variable in the system, where ; ; The probability of violating the opportunity constraint for the change in equivalent power angle; the random variable of power prediction deviation. Introducing the TSCOPF model, the system runtime variables are all represented as... Related functions.

[0013] In step 2-1, the formula used to construct the PCE expression is as follows: (4); In the formula: This is the actual output of the system; This is an approximate output of the PCE method model; For a certain deviation from the prediction Related functions; For the first n A number of orthogonal polynomial basis functions; These are the undetermined coefficients of the basis functions; The total number of polynomial basis functions; In step 2-2, the formula used to simplify the basis function expression based on HTS is as follows: (5); In the formula: Let the function that calculates the number of elements in a set be denoted as the total number of polynomial basis functions after HTS truncation. ; It is a multi-exponential vector, where each element is a non-negative integer; p The maximum order of the polynomial; In order to seek q Norm operations; when q When = 1, HTS degenerates into a full-order truncation scheme; when 0 < q When <1, HTS will retain the interaction terms of the lower-order polynomials.

[0014] In steps 2-3, the specific calculations for calculating the model response values ​​and constructing the regression matrix include: 1) Based on LHS to predict bias ω Stratified sampling is performed on the probability space to generate N For each sample point, the calculation formula is: (6); In the formula: For the first k The specific values ​​of each sample point; N The total number of samples; For the first k The probability value of each layer, ; For prediction bias The inverse cumulative distribution function; 2) Obtain the response value of the model calculation. The calculation formula is as follows: (7); In the formula: This is the original system model; For the first k The actual system output corresponding to each sample point; A column vector consisting of the response values ​​of all samples; 3) Construct the design matrix and describe the expression in matrix form: (8); In the formula: for The design matrix has row vectors corresponding to samples and column vectors corresponding to basis functions; To make the first k Sample Substitute the first n The value obtained from an orthogonal polynomial; , which is the vector of undetermined coefficients of the basis functions; The residual vector; In steps 2-4, the formula used for sparsity screening based on LASSO regression is expressed as follows: (9); In the formula: This is a vector of estimated basis function coefficients; The vector of undetermined coefficients of the basis functions is obtained by solving... Minimize the target vector value; For designing the matrix; For regularization hyperparameters; This is an L1 regularization penalty term; Let be the square of the second norm of the vector; This is the set of active basis function numbers, i.e., the set of numbers of basis functions that have a significant impact; This represents the total number of active basis functions. The filtering threshold is usually set to 10. -4 ~10 -3 ; The LASSO solution is the first n One coefficient; To find the function that minimizes the target vector value.

[0015] In step 3-1, the formula used to solve for the undetermined coefficients is expressed as follows: (10); In the formula: Design matrices for sparseness, from China retains The corresponding column; This is the final vector of basis function coefficients obtained; In step 3-2, the formula used to define the error is expressed as follows: (11); In the formula: Let the hat matrix satisfy... ; For the first k The actual output value of each sample; For the PCE model, the first k Predicted values ​​for each sample; Hat matrix The k The diagonal element represents the th element. k The leverage value predicted by each sample for itself; ; This is the output sample mean; In step 3-3, the set of formulas used to transform the chance constraints in the model into a deterministic form is expressed as follows: (12); In the formula: This is the deterministic form of the objective function; , and They are respectively , and The corresponding set of active basis function numbers, and the sets corresponding to each variable are independent of each other; , and They are respectively , and The corresponding PCE expression; This is the deterministic form of the power balance equation; , , and They are respectively , , and The corresponding set of active basis function numbers, and the sets corresponding to each variable are independent of each other; , , and They are respectively , , and The corresponding PCE expression; The boundary chance constraints and the work angle chance constraints are as follows: (13); In the formula: This is the expected value; Standard deviation; It is the inverse cumulative distribution function of the violation probability.

[0016] Compared with the prior art, the present invention has the following technical effects: 1) To address the technical problem of exponential growth of basis functions in high-dimensional random fields, the present invention combines HTS and LASSO regression to sparsely simplify the PCE expression. This technique eliminates unimportant high-order cross terms at the structural level and further filters key coefficients through data-driven methods, which greatly reduces computational complexity while accurately preserving key spatiotemporal features. 2) To address the significant computational burden caused by repeated numerical integration during optimization in traditional methods, this invention uses the trajectory sensitivity method to analytically simplify complex transient stability constraints into sensitivity constraints on the power angle of the synchronous generator unit's active power output after a system fault. This technique transforms high-dimensional, complex stochastic transient stability constraints into deterministic algebraic inequalities, effectively avoiding oscillations in extreme value solving and improving the model's solution speed. 3) To address the problem of high communication load in traditional centralized optimization algorithms when dealing with large-scale systems, this invention proposes an improved ADMM algorithm. This technique decomposes the optimization problem of the entire system into sub-optimization problems in each region. By solving the sub-problems in a distributed and collaborative manner, the global optimal solution of the system is obtained, which significantly reduces the amount of data interaction between system regions and improves the energy efficiency and robustness of the power grid operation under high-proportion renewable energy access. 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 improved IEEE 39-node system in an example of the present invention; Figure 3 This is a dynamic response curve of the generator power angle of the improved IEEE 39-bus system in this invention example without using the method of this invention; Figure 4 This is the generator power angle dynamic response curve obtained after solving the problem using the optimization algorithm proposed in this invention example; Figure 5 This is a flowchart of the algorithm for solving CC-TSCOPF in this invention. Detailed Implementation

[0018] like Figure 1 As shown, the method includes the following steps: In step 1, uncertainty modeling is performed on the power generation at the source and the power consumption at the load based on historical data, and opportunity constraints are embedded.

[0019] 1) Based on the source and load dual-end operating parameters, reconstruct the TSCOPF type objective function: Based on the power, voltage, and phase angle parameters of conventional and renewable energy generators at the source and various loads at the load end, the impact of their uncertainties on the power system is analyzed, the objective function of the model is reconstructed, and the extended TSCOPF model is obtained: (1); In the formula: Equation (1a) is the objective function of the TSCOPF model, which aims to minimize the total economic cost. Let the total cost function be... , , and These are the conventional generator unit control cost function, excess reserve cost function, wind and solar curtailment penalty cost function, and load switching cost function, respectively. , and Let represent the active power of the conventional generator set, the active power of the new energy generator set, and the active power on the load side, respectively; Equation (1b) is the power balance equation of the system. The power balance equation function of the system is... and These represent the net injected active and reactive power at the node, respectively. and These represent the node voltage magnitude and the node voltage phase angle, respectively; Equation (1c) represents the upper and lower limits of the system constraints. A set of system parameter constraints. and These are the lower and upper limits of the system constraints, respectively. Line current margin; Equation (1d) is the differential equation describing the dynamic characteristics of the power system during the transient period; Equation (1e) is the algebraic equation representing the power balance constraints that must be satisfied during the transient process; Equation (1f) is the constraint limiting the maximum rotor angle deviation between any two generators after the disturbance. , , These are system state variables, algebraic variables, and control variables, respectively. For system state variables in time t The first derivative; The duration of the system transient process; t For any time during the duration of the system transient process; For transient dynamic equation functions, Algebraic constraint functions that must be satisfied during transient processes; For transient safe operation inequality constraint functions; 2) The system is subjected to single-machine infinite constant equivalence. Based on the trajectory sensitivity method, the complex transient stability constraints of the system are simplified to the sensitivity constraints of the synchronous generator's active power output to the power angle after a system fault. The transient stability constraints are then analyzed into low-order algebraic inequality constraints. The transient stability constraint can be replaced by the following: (2); In the formula: To anticipate the moment when the system will become unstable under a fault; The critical power angle of the system under the anticipated fault; and These are the system equivalent power angles before and after the active power adjustment of the conventional generator set, respectively. and These are the active power adjustment of a conventional generator and the corresponding equivalent power angle change, respectively. The trajectory sensitivity of the active power of a synchronous generator unit to the power angle; 3) Introducing source-load two-end power prediction bias To characterize power prediction bias and embed chance constraints into the TSCOPF model, a CC-TSCOPF model considering power uncertainty at both the source and load ends is established: (3); In the formula: Let be the expected value of the objective function; The probability that the condition in parentheses is true; Let be the probability of violation of the chance constraint for the corresponding variable in the system, where ; ; The probability of violating the opportunity constraint for the change in equivalent power angle; the random variable of power prediction deviation. Introducing the TSCOPF model, the system runtime variables are all represented as... Related functions; In step 2, HTS and LASSO regression are used to simplify and sparsify the PCE surrogate model.

[0020] The specific steps for simplifying the model function using HTS are as follows: Step 2-1: Based on the Wiener-Askey polynomial chaotic approximation theorem, select orthogonal polynomial basis functions corresponding to the prediction bias according to different random distributions of the prediction bias input to construct the PCE expression: (4); In the formula: This is the actual output of the system; This is an approximate output of the PCE method model; For a certain deviation from the prediction Related functions; For the first n A number of orthogonal polynomial basis functions; These are the undetermined coefficients of the basis functions; The total number of polynomial basis functions; Step 2-2: Based on the simplified basis function expression of HTS, introduce... q Norm and determine the total number of basis functions to be retained : (5); In the formula: Let the function that calculates the number of elements in a set be denoted as the total number of polynomial basis functions after HTS truncation. ; It is a multi-exponential vector, where each element is a non-negative integer; p The maximum order of the polynomial; In order to seek q Norm operations; when q When = 1, HTS degenerates into a full-order truncation scheme; when 0 < q When <1, HTS will retain the interaction terms of the lower-order polynomials; The specific steps for sparsifying the model function using LASSO are as follows: Steps 2-3: Generate sample points using LHS, calculate model response values, and construct the regression matrix: 1) Based on LHS to predict bias Stratified sampling is performed on the probability space to generate N One sample point: (6); In the formula: For the first k The specific values ​​of each sample point; N The total number of samples; For the first k The probability value of each layer, ; For prediction bias The inverse cumulative distribution function; 2) Obtain the response value of the model calculation. The calculation formula is as follows: (7); In the formula: This is the original system model; For the first k The actual system output corresponding to each sample point; A column vector consisting of the response values ​​of all samples; 3) Construct the design matrix and describe the PCE expression in matrix form: (8); In the formula: for The design matrix has row vectors corresponding to samples and column vectors corresponding to basis functions; To make the first k Sample Substitute the first n The value obtained from an orthogonal polynomial; , which is the vector of undetermined coefficients of the basis functions; The residual vector; Steps 2-4: Based on the sparsified basis function expression of LASSO regression, L1 regularization constraints are introduced to further filter the truncated basis function set, retaining only the basis functions that have a significant impact on the output response, and performing sparsity filtering based on LASSO regression: (9); In the formula: This is a vector of estimated basis function coefficients; The vector of undetermined coefficients of the basis functions is obtained by solving... Minimize the target vector value; For designing the matrix; For regularization hyperparameters; This is an L1 regularization penalty term; Let be the square of the second norm of the vector; This is the set of active basis function numbers, i.e., the set of numbers of basis functions that have a significant impact; This represents the total number of active basis functions. The filtering threshold is usually set to 10. -4 ~10 -3 ; The LASSO solution is the first n One coefficient; To find the function that minimizes the target vector value; In step 3, the undetermined coefficients of each PCE expression are calculated based on the least squares principle, and the expected value and standard deviation of each operating variable are calculated using the PCE expression, thus realizing the deterministic transformation of the chance constraint in the CC-TSCOPF model. The steps are as follows: The specific steps for solving the coefficients of the expression based on the least squares principle are as follows: Step 3-1: Calculate the undetermined coefficients in the PCE expression based on the least squares principle. The undetermined coefficients in the PCE expansion are solved as follows: (10); In the formula: Design matrices for sparseness, from China retains The corresponding column; This is the final vector of basis function coefficients obtained; Step 3-2: Posterior error estimation is performed using leave-one-out (LOO) cross-validation, where the error is defined as: (11); In the formula: Let the hat matrix satisfy... ; For the first k The actual output value of each sample; For the PCE model, the first k Predicted values ​​for each sample; Hat matrix The k The diagonal element represents the th element. k The leverage value predicted by each sample for itself; ; This is the output sample mean; Step 3-3: Based on the above results, achieve the deterministic transformation of the chance constraints in the CC-TSCOPF model: 1) The deterministic forms of the objective function and the power balance equation in the CC-TSCOPF model are: (12); In the formula: This is the deterministic form of the objective function; , and They are respectively , and The corresponding set of active basis function numbers, and the sets corresponding to each variable are independent of each other; , and They are respectively , and The corresponding PCE expression; This is the deterministic form of the power balance equation; , , and They are respectively , , and The corresponding set of active basis function numbers, and the sets corresponding to each variable are independent of each other; , , and They are respectively , , and The corresponding PCE expression; 2) Using the PCE expression to calculate the expected value and standard deviation of each running variable, equations (3c) and (3d) can be restated as follows: (13); In the formula: This is the expected value; Standard deviation; It is the inverse cumulative distribution function of the violation probability.

[0021] In step 4, the specific steps for solving the CC-TSCOPF model using the improved ADMM are as follows: Step 4-1: Based on the CC-TSCOPF model transformed by the above steps, input the network power parameters, voltage amplitude parameters, and phase angle parameters required by the model; Step 4-2: Based on the distribution of the source and load at both ends, select a reasonable and effective partitioning method to partition the system, decomposing the node, generator, and line data of the system; Step 4-3: After optimizing each sub-region independently, update the global variables and perform information exchange between adjacent sub-regions; Step 4-4: Determine if ADMM converges. If both the dual and the original residuals are less than the threshold, the iteration stops. Otherwise, update the Lagrange multipliers and return to step 4-3 until ADMM converges, at which point the loop ends.

[0022] Example: The method described in this invention was tested and verified in an improved IEEE 39-bus system. This system comprises 39 buses, 8 synchronous generators, 1 wind farm, and 1 photovoltaic power station. The generator model adopts a second-order classical model. To simulate a high proportion of renewable energy integration, wind turbines and a photovoltaic power station are connected to the system, such as... Figure 2 As shown, the thermal generator at bus 33 is replaced with a wind farm, and the thermal generator at bus 37 is replaced with a photovoltaic power station. The simulated load level varies in 5% steps within the range of 80% to 120% of the baseline value. This embodiment selects 80% load condition as representative condition to explain the source-load uncertainty modeling, PCE proxy construction, power angle opportunity constraint transformation, and optimized scheduling results. A source-load random operation scenario is generated through Latin hypercube sampling to simulate the impact of new energy output and load fluctuations on the system's transient stability and active power scheduling results; the system parameter settings are shown in Table 1.

[0023] Table 1 IEEE 39-node system and implementation status

[0024] The predicted combined output of wind and solar power under 80% load conditions is 270MW, while conventional units G1 to G8 need to handle 4607.84MW. The demand is located between the upper and lower limits of the total output of conventional units, with sufficient room for upward and downward adjustments. Therefore, it meets the capacity feasibility requirements for subsequent random sample calculations and chance-constrained scheduling. The parameter settings for uncertainty modeling are shown in Table 2.

[0025] Table 2 Source load uncertainty and PCE modeling parameters:

[0026] Random inputs simultaneously cover uncertainties on both the new energy and load sides. PCE is used to fit the maximum rotor angular deviation of eight synchronous machines relative to the center of inertia. High-order cross terms are reduced using HTS and LASSO, and the final coefficients are calculated using ordinary least squares. Leave-one-out method is used to evaluate the posterior error of the surrogate model, and relevant sample-by-sample results are retained along with the formal experimental data. Simulation parameter settings are shown in Table 3.

[0027] Table 3 Transient simulation and optimization solution parameters:

[0028] The training samples, trajectory sensitivity calculations, optimized time-domain verification, and proportional scheduling benchmarks all use the transient model shown in Table 3. The worst response of the reference scheduling is located at approximately 1.30 s in G5. The trajectory sensitivity is used to describe the impact of synchronous machine output adjustment on the power angle response, enabling the power angle opportunity constraint to enter the ADMM coordination solution in a low-order algebraic form. The simulation results of the unit output are shown in Table 4.

[0029] Table 4. Unit output based on opportunity-constrained scheduling and proportional scheduling:

[0030] Both scheduling methods meet the total power demand of 4607.84MW of conventional generating units. Opportunity-constrained scheduling reduces the output of G1, G3, and G4, and appropriately increases the output of the remaining units, adjusting the system operating point in a direction that minimizes the most unfavorable power angle deviation. Calculated using the quadratic cost function adopted in the embodiment, the cost of opportunity-constrained scheduling is approximately RMB 44006.62, while the cost of capacity-proportional active power scheduling is approximately RMB 44583.19, representing a reduction of approximately 1.29%. Simulation results for transient stability constraints and probabilistic constraints are shown in Table 5.

[0031] Table 5. Main probabilistic constraints and transient stability results:

[0032] The corrected 95th percentile of single-sided power angle is significantly lower than the 120° critical angle, satisfying the upper limit opportunity constraint of power angle. Compared with the proportional scheduling benchmark under the same transient parameters, the method of this invention reduces the maximum COI relative power angle by approximately 20.844% and increases the minimum stability margin by approximately 7.564°. The uncontrolled scenario is used to illustrate the power angle instability trend when no regulation measures are available, and is not used as a quantitative comparison benchmark with the previous two scheduling methods with the same parameters.

[0033] To verify the improvement effect of the method of this invention on the transient stability of the system, a three-phase short-circuit fault was set at 0.5 seconds on line 14-15, with a fault duration of 0.5 seconds, and time-domain simulation was performed. When scheduling is performed using a conventional deterministic optimal power flow method, the generator-to-rotor angle curve of the system after the fault is as follows: Figure 3 As shown in the figure, the rotor angle of some generators continues to increase, and the system experiences transient instability, indicating that traditional methods are insufficient to guarantee the dynamic safety of the system when dealing with the uncertainties of wind and solar power. However, after optimizing the scheduling using the method proposed in this invention, the generator relative rotor angle curves under the same fault are as follows: Figure 4 As shown, the power angles of each generator remain synchronized after a fault, the system remains stable, and the transient stability margin is significantly improved.

[0034] Compared with existing technologies, the CC-TSCOPF optimization scheduling method based on improved polynomial chaotic expansion proposed in this invention has the following advantages: In handling transient stability constraints, this invention adopts a multi-machine classical second-order swing equation in the simulation calculation, supplemented by COI drift correction to eliminate the overall system drift, transforming the high-dimensional complex stochastic transient stability constraints into deterministic algebraic inequality constraints based on PCE statistical moments. This avoids the huge computational burden caused by repeated numerical integration in the optimization process of traditional time-domain simulation methods, significantly reducing the difficulty of model solution. Combining chance-constrained programming theory, by transforming the violation probability into standard normal quantiles, the power angle safety constraint limit is determined analytically, allowing the system to flexibly adjust the operating point within a very small risk threshold, making the optimization results more efficient while ensuring transient safety, especially suitable for the power grid environment with increasingly higher requirements for both operational safety and efficiency under the background of high-proportion wind and solar integration. At the same time, HTS and LASSO regression are used together to sparsely simplify the PCE expansion coefficients: HTS uses... q Norm constraints eliminate unimportant higher-order cross terms at the structural level. LASSO further filters key coefficients through data-driven selection and uses LOO cross-validation to ensure the generalization accuracy of the surrogate model. This achieves accurate characterization of complex stochastic responses with minimal basis function combinations, resulting in high fitting accuracy and significantly improved computational efficiency when dealing with high-dimensional, strongly coupled power system uncertainty propagation problems. In model solving, this invention employs an improved ADMM for distributed solving of the reconstructed CC-TSCOPF model. x- Updates andz- The alternating iterative update mechanism can dynamically decouple the local optimization and global power balance constraints of each control region: x- During the update phase, each conventional unit G1 to G8 independently solves for the locally optimal output with a secondary cost term, achieving parallel computing in each region; z- During the update phase, projection operations and refined power balance corrections unify the decisions of each region into the global constraint set, ensuring a precise match between the total system output and load demand. In the dual variable update phase, constraint violation information is continuously accumulated, guiding each region to gradually reach consensus. Furthermore, the algorithm incorporates an over-relaxation strategy to accelerate the convergence of the original residuals, and an adaptive penalty coefficient mechanism dynamically adjusts every 10 iterations based on the ratio of the original residuals to the dual residuals, effectively balancing convergence speed and numerical stability. This enhances the robustness of optimization solutions in large-scale systems, thereby better ensuring the safe, stable, and efficient operation of the power system under wind and solar grid-connected conditions.

Claims

1. A CC-TSCOPF optimization scheduling method based on improved polynomial chaotic expansion, characterized in that, Includes the following steps: Step 1: To address the impact of uncertainties in the operating parameters of the source and load, reconstruct the objective function of the transient stability-constrained optimal power flow model and embed opportunity constraints to establish an opportunity-constrained transient stability-constrained optimal power flow model that takes into account the power uncertainties at both ends of the source and load. Step 2: Select orthogonal polynomial basis functions to construct a polynomial chaotic expansion expression, and at the same time use the hyperbolic truncation scheme and the minimum absolute contraction and selection operator regression scheme to simplify and sparsify the basis functions; Step 3: Calculate the undetermined coefficients of the polynomial chaotic expansion expression based on the least squares principle, and use the polynomial chaotic expansion expression to calculate the expected value and standard deviation of each running variable; Step 4: Solve the opportunity-constrained transient stability-constrained optimal power flow model after polynomial chaotic expansion and transformation based on the improved alternating direction multiplier method, and iteratively optimize to obtain the optimal power flow scheduling strategy for the entire network. Step 5: Generate underlying hardware drive instructions based on the optimal power flow scheduling strategy to adjust the active and reactive power of each conventional generator set and new energy generator set in the power grid in real time, so as to maintain the transient stable operation of the power grid physical system under the expected fault.

2. The method according to claim 1, characterized in that, Step 1 specifically includes the following sub-steps: Step 1-1: Based on the power, voltage, and phase angle parameters of conventional generators and new energy generators at the source end and various loads at the load end during operation, analyze the impact of parameter uncertainty on the power system, reconstruct the objective function of the model, and obtain the extended transient stability constrained optimal power flow TSCOPF model; Step 1-2: Perform single-machine infinite equalization on the system, and analyze the transient stability constraints of the system based on the trajectory sensitivity method. Simplify it into the sensitivity constraint of the synchronous generator's active power output to the power angle after the system fault, thereby analyzing the transient stability constraints into low-order algebraic inequality constraints. Steps 1-3: Introduce the power prediction bias at both the source and load ends to characterize the power prediction bias, embed the chance constraint into the transient stability constraint optimal power flow model, and establish the chance constraint CC-TSCOPF model that takes into account the power uncertainty at both the source and load ends.

3. The method according to claim 1, characterized in that, Step 2 specifically includes the following sub-steps: Step 2-1: Select orthogonal polynomial basis functions corresponding to the prediction bias based on different random distributions of the prediction bias input to construct a polynomial chaotic expansion PCE expression; Step 2-2: Simplify the basis function expression based on the hyperbolic truncation scheme HTS, introduce the q norm and determine the total number of basis functions to be retained, and remove unimportant higher-order cross terms; Steps 2-3: Generate sample points using Latin hypercube sampling (LHS), calculate the model response values, and construct the regression matrix; Steps 2-4: Based on the LASSO regression sparsification of basis function expressions using the minimum absolute contraction and selection operator, L1 regularization constraints are introduced to further filter the truncated basis function set, retaining only the basis functions that have a greater impact on the output response, thus completing the sparsity filtering.

4. The method according to claim 1, characterized in that, Step 3 specifically includes the following sub-steps: Step 3-1: Calculate the undetermined coefficients of the PCE expression based on the least squares principle, and solve to obtain the final basis function coefficient vector; Step 3-2: Estimate the posterior error based on leave-one-out cross-validation, and calculate the error definition to measure the generalization accuracy of the model; Step 3-3: Calculate the expected value and standard deviation of each running variable using the PCE expression, re-express the objective function and each constraint condition, and complete the transformation of the chance constraints in the model into a deterministic form.

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 optimal power flow model transformed from Step 3, input the network power parameters, voltage magnitude parameters, and phase angle parameters required by the model; Step 4-2: Based on the distribution of the source and load at both ends, select a reasonable and effective partitioning method to partition the physical network of the system, and decompose the physical connection data of the nodes, generators and lines of the power grid system to each control sub-region accordingly; Step 4-3: Each control sub-region independently performs local optimization and then updates the global variables, and exchanges information between adjacent sub-regions to exchange boundary state parameters; Step 4-4: Evaluate the dual residuals and original residuals of the Improved Alternating Direction Multiplier Method (ADMM), and determine whether the residuals are all less than the preset convergence threshold. If the threshold is met, the iteration stops and the joint strategy of each control sub-region is output as the optimal power flow scheduling strategy. Otherwise, the Lagrange multipliers are updated and the distributed iteration is returned to step 4-3 until the convergence condition is met.

6. The method according to claim 2, characterized in that, In step 1-1, the formula set used to establish the extended TSCOPF model is expressed as follows: （1）； In the formula: Equation (1a) is the optimization objective of the TSCOPF model, i.e., the objective function that minimizes the total economic cost. Let the total cost function be... , , and These are the conventional generator unit control cost function, excess reserve cost function, wind and solar curtailment penalty cost function, and load switching cost function, respectively. , and Let represent the active power of the conventional generator set, the active power of the new energy generator set, and the active power on the load side, respectively; Equation (1b) is the power balance equation of the system. The power balance equation function of the system is... and These represent the net injected active and reactive power at the node, respectively. and These represent the node voltage magnitude and the node voltage phase angle, respectively; Equation (1c) represents the upper and lower limits of the system constraints. A set of system parameter constraints. and These are the lower and upper limits of the system constraints, respectively. Line current margin; Equation (1d) is the differential equation describing the dynamic characteristics of the power system during the transient period; Equation (1e) is the algebraic equation representing the power balance constraints that must be satisfied during the transient process; Equation (1f) is the constraint limiting the maximum rotor angle deviation between any two generators after the disturbance. , , These are system state variables, algebraic variables, and control variables, respectively. For system state variables in relation to time t The first derivative; The duration of the system transient process; t For any time within the duration of the system transient process; For transient dynamic equation functions, Algebraic constraint functions that must be satisfied during transient processes; This is the constraint function for the transient safe operation inequality.

7. The method according to claim 2, characterized in that, In steps 1-2, the set of formulas used to analyze the transient stability constraints into low-order algebraic inequality constraints is expressed as follows: （2）； In the formula: To anticipate the moment when the system will become unstable under a fault; The critical power angle of the system under the anticipated fault; and These are the system equivalent power angles before and after the active power adjustment of the conventional generator set, respectively. and These are the active power adjustment of a conventional generator and the corresponding equivalent power angle change, respectively. The trajectory sensitivity of the active power of a synchronous generator unit to the power angle; In steps 1-3, the formula set used to establish the CC-TSCOPF model considering the power uncertainty at both ends of the source and load is expressed as follows: （3）； In the formula: Let be the expected value of the objective function; The probability that the condition in parentheses is true; Let be the probability of violation of the chance constraint for the corresponding variable in the system, where ; ; The probability of violating the opportunity constraint for the change in equivalent power angle; the random variable of power prediction deviation. Introducing the TSCOPF model, the system runtime variables are all represented as... Related functions.

8. The method according to claim 3, characterized in that, In step 2-1, the formula used to construct the PCE expression is as follows: （4）； In the formula: This is the actual output of the system; This is an approximate output of the PCE method model; For a certain deviation from the prediction Related functions; For the first n A number of orthogonal polynomial basis functions; These are the undetermined coefficients of the basis functions; The total number of polynomial basis functions; In step 2-2, the formula used to simplify the basis function expression based on HTS is as follows: （5）； In the formula: Let the function that calculates the number of elements in a set be denoted as the total number of polynomial basis functions after HTS truncation. ; It is a multi-exponential vector, where each element is a non-negative integer; p The maximum order of the polynomial; In order to seek q Norm operations; when q When = 1, HTS degenerates into a full-order truncation scheme; when 0 < q When <1, HTS will retain the interaction terms of the lower-order polynomials.

9. The method according to claim 3, characterized in that, In steps 2-3, the specific calculations for calculating the model response values ​​and constructing the regression matrix include: 1) Based on LHS to predict bias ω Stratified sampling is performed on the probability space to generate N For each sample point, the calculation formula is: （6）； In the formula: For the first k The specific values ​​of each sample point; N The total number of samples; For the first k The probability value of each layer, ; For prediction bias The inverse cumulative distribution function; 2) Obtain the response value of the model calculation. The calculation formula is as follows: （7）； In the formula: This is the original system model; For the first k The actual system output corresponding to each sample point; A column vector consisting of the response values ​​of all samples; 3) Construct the design matrix and describe the expression in matrix form: （8）； In the formula: for The design matrix has row vectors corresponding to samples and column vectors corresponding to basis functions; To make the first k Sample Substitute the first n The value obtained from an orthogonal polynomial; , which is the vector of undetermined coefficients of the basis functions; The residual vector; In steps 2-4, the formula used for sparsity screening based on LASSO regression is expressed as follows: （9）； In the formula: This is a vector of estimated basis function coefficients; Let the vector of undetermined coefficients of the basis functions be solved. Minimize the target vector value; For designing the matrix; For regularization hyperparameters; This is an L1 regularization penalty term; Let be the square of the second norm of the vector; This is the set of active basis function numbers, i.e., the set of numbers of basis functions that have a significant impact; This represents the total number of active basis functions. The filtering threshold is usually set to 10. -4 ~10 -3 ; The LASSO solution is the first n One coefficient; To find the function that minimizes the target vector value.

10. The method according to claim 4, characterized in that, In step 3-1, the formula used to solve for the undetermined coefficients is expressed as follows: （10）； In the formula: Design matrices for sparseness, from China retains The corresponding column; This is the final vector of basis function coefficients. In step 3-2, the formula used to define the error is expressed as follows: （11）； In the formula: Let the hat matrix satisfy... ; For the first k The actual output value of each sample; For the PCE model, the first k Predicted values ​​for each sample; Hat matrix The k The diagonal element represents the th element. k The leverage value predicted by each sample for itself; ; This is the output sample mean; In step 3-3, the set of formulas used to transform the chance constraints in the model into a deterministic form is expressed as follows: （12）； In the formula: This is the deterministic form of the objective function; , and They are respectively , and The corresponding set of active basis function numbers, and the sets corresponding to each variable are independent of each other; , and They are respectively , and The corresponding PCE expression; This is the deterministic form of the power balance equation; , , and They are respectively , , and The corresponding set of active basis function numbers, and the sets corresponding to each variable are independent of each other; , , and They are respectively , , and The corresponding PCE expression; The boundary chance constraints and the work angle chance constraints are as follows: （13）； In the formula: This is the expected value; Standard deviation; It is the inverse cumulative distribution function of the violation probability.

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