Parallel computation method for full-pure embedded power system power flow

Abstract

The application discloses a kind of large-scale power system full pure embedding power flow parallel computing method, comprising: using stable biorthogonal conjugate gradient method parallel iteration to solve the linear equation set of high-dimensional power series coefficient after preprocessing;According to the solution vector and residual vector in the output result of iteration solving the linear equation set of high-dimensional power series coefficient after preprocessing, the convergence determination link of stable biorthogonal conjugate gradient method is carried out;When max (|r k |) Less than the maximum error of stable biorthogonal conjugate gradient method, the output solution vector meets the error precision requirement of stable biorthogonal conjugate gradient, and Aitken difference method is used instead of Padé approximation Parallel calculation voltage power series rational approximation value, and the vector containing each node voltage value is output;According to the obtained each node voltage value, the maximum power mismatch value convergence determination link is carried out, and if the maximum power mismatch value is greater than ε eps , the right side column vector of linear equation set of high-dimensional power series coefficient after preprocessing is updated, and the next cycle calculation is carried out.

Description

Technical Field

[0001] This invention relates to the field of power systems, and more particularly to a fully embedded parallel computing method for power flow in large-scale power systems. Background Technology

[0002] Power flow calculation is fundamental to power system security and stability analysis and control, and has significant engineering value in guiding power system planning and operation. However, with the large-scale grid integration and consumption of new energy sources, the increasing scale of AC / DC interconnections, and the growing complexity of power grid structures, the efficiency and robustness of power flow calculation face enormous challenges. [1]-[3] Currently, the Newton-Raphson (NR) method remains the most widely used method for power flow calculation in power systems. However, the convergence of this method depends on the selection of initial values; inappropriate initial values ​​will lead to divergence in the power flow calculation results. [4] To address the problems of the NR method in power system power flow calculation, Dr. Antonio Trias from Spain proposed a Holomorphic Embedding Load Flow Method (HELM) for power system power flow calculation. [5] This method constructs holomorphic functions by embedding complex variables into the power flow equations. Then, based on the expansion properties of these holomorphic functions, it constructs a high-dimensional linear system of power-series coefficients to solve for the holomorphic power-series coefficients of the node voltages. Finally, based on the obtained power-series coefficients, it performs Padé approximation on the holomorphic power-series expansions of the node voltages to obtain rational approximations of the node voltages, thus yielding the power flow solution of the power system. [6] .

[0003] However, HELM requires a high power series order to solve the power flow of large-scale power systems, and usually requires solving the linear equation system with high-dimensional power series coefficients multiple times, which makes HELM computational efficiency low. In addition, after each solution of the linear equation system with high-dimensional power series coefficients to obtain the voltage power series coefficients, HELM needs to perform Padé approximation calculations on all nodes in the large-scale power system to obtain the power series rational approximation value of the voltage of each node, which further reduces the computational efficiency of HELM. To this end, the determinant ratio method is used to replace the traditional calculation form of Padé approximation, but the voltage power series order required by this method is high, and it is difficult to effectively reduce the number of rational approximations of the voltage power series. The Padé approximation variant calculation form is proposed in the literature [7], which can improve the single calculation speed of Padé approximation, but this method cannot guarantee the theoretical convergence and numerical convergence of HELM when applied to the calculation of the rational approximation value of the voltage power series, and the accuracy is low when applied to the power flow solution of large-scale power systems.

[0004] Therefore, it is particularly important to research and develop a fully embedded parallel power flow computing method suitable for large-scale power systems. Summary of the Invention

[0005] This invention provides a parallel computation method for fully embedded power flow in large-scale power systems. Addressing the time-consuming nature of high-dimensional power series coefficient linear equations and Padé approximation calculations in HELM computation, this invention proposes a parallel iterative solution method for high-dimensional power series coefficient linear equations based on approximate inverse preprocessing, and a parallel computation method for voltage power series rational approximation based on Aitken differentials. These two parallel computation methods are combined to achieve accurate and rapid computation of fully embedded power flow in large-scale power systems, exhibiting strong robustness. Details are described below:

[0006] A fully embedded parallel computation method for power flow in large-scale power systems, the method comprising:

[0007] Based on the sparsity and invariance of the coefficient matrix in the linear equation system with high-dimensional power series coefficients, a sparse matrix approximate inverse preprocessor is constructed and applied to its coefficient matrix. The stable biorthogonal conjugate gradient method is used to solve the preprocessed linear equation system with high-dimensional power series coefficients in parallel iteratively.

[0008] The convergence determination step of the stable biorthogonal conjugate gradient method is performed based on the solution vector and residual vector in the output of the preprocessed high-dimensional power series coefficient linear equation system obtained through iterative solution; when max(|r k |) is less than the maximum error of the stable biorthogonal conjugate gradient method, and the output solution vector meets the error accuracy requirements of the stable biorthogonal conjugate gradient. The Aitken difference method is used to replace the Padé approximation to calculate the rational approximation value of the voltage power series in parallel, and the vector containing the voltage values ​​of each node is output.

[0009] Based on the obtained node voltage values, a convergence determination step is performed for the maximum power mismatch value. If the maximum power mismatch value is greater than ε... eps Then update the right-hand column vector of the preprocessed high-dimensional power series coefficient linear equation system and perform the next iteration.

[0010] The system of linear equations with high-dimensional power series coefficients is as follows:

[0011]

[0012] In the formula, Y sys Here, f(U[n-1], Q) represents the coefficient matrix of the high-dimensional power series linear equation system; n is the order of the power series expansion of the node voltage and reactive power; U[n] is the vector of coefficients of the nth power series of the node voltage to be determined; l[n-1]) is a vector expression containing the coefficients of the (n-1)th order voltage power series and the coefficients of the reactive power power series; G and B are the conductance and susceptance in the power system, respectively; and , respectively, are the coefficients of the nth order inverse power series of the PV node and the PQ node; “Re” and “Im” represent taking the real part and imaginary part of the complex number, respectively.

[0013] The sparse matrix approximate inverse preprocessor is as follows:

[0014] The coefficient matrix Y sys Perform factorization:

[0015]

[0016] In the formula, Y D and Y W The coefficient matrix Y is respectively sys The lower triangular matrix and the upper triangular matrix;

[0017] By Y D and Y W Constructing a sparse matrix approximate inverse preprocessor F pre :

[0018]

[0019] In the formula, E Y To be with Y A Identity matrices of the same dimension.

[0020] The beneficial effects of the technical solution provided by this invention are:

[0021] 1. This invention enables accurate and rapid solutions for power flow in large-scale power systems;

[0022] 2. This invention proposes an approximate inverse preprocessor construction method for the coefficient matrix in a high-dimensional power series linear equation system, which effectively concentrates the distribution range of its eigenvalues, thereby realizing the parallel iterative solution of the high-dimensional power series linear equation system based on approximate inverse preprocessing.

[0023] 3. This invention proposes a calculation form for calculating the rational approximation value of the voltage power series based on the Aitken difference method, which effectively reduces the calculation cost of solving the voltage value, and thus realizes parallel calculation of the rational approximation value of the voltage power series using the Aitken difference method.

[0024] 4. This invention proposes a combined mechanism of parallel iterative solution of linear equations with high-dimensional power series coefficients and parallel calculation of rational approximation values ​​of voltage power series, which significantly improves the computational efficiency of power system power flow solution.

[0025] 5. Compared with the NR method, the method proposed in this invention can accurately and quickly calculate the power flow solution of large-scale power systems without relying on initial values, providing a new approach for power flow calculation of large-scale power systems;

[0026] 6. This invention can significantly improve the speed of power flow calculation in large-scale power systems, providing an effective way for dispatchers and operators to fundamentally solve the problem of real-time power system analysis. Attached Figure Description

[0027] Figure 1 A flowchart of a fully embedded parallel computation method for power flow in a large-scale power system;

[0028] Figure 2 A strategy diagram for stabilizing the biorthogonal conjugate gradient method to solve preprocessed high-dimensional power series linear equations in parallel iteratively;

[0029] Figure 3 A diagram illustrating the strategy for parallel computation of rational approximations of voltage power series using the Aitken finite difference method;

[0030] Figure 4 This is a schematic diagram showing the voltage magnitude and phase angle distribution of the maximum error node in the power flow calculation results of power systems of different scales, compared with the NR method.

[0031] Figure 5 This diagram illustrates the comparison of computational time for solving linear equation systems with high-dimensional power series coefficients using different methods. Detailed Implementation

[0032] To make the objectives, technical solutions, and advantages of the present invention clearer, the embodiments of the present invention will be described in further detail below.

[0033] To address the shortcomings and deficiencies of the HELM algorithm in solving power flow problems in large-scale power systems, this invention proposes a parallel computation method for fully embedded power flow in large-scale power systems. First, to address the sparsity, asymmetry, and non-positive definiteness of the coefficient matrix in high-dimensional power series linear equations, a stable biorthogonal conjugate gradient method is employed for parallel iterative solving of the high-dimensional power series linear equations. A sparse matrix approximation inverse preprocessor is constructed and applied to the coefficient matrix in the high-dimensional power series linear equations to concentrate its eigenvalue distribution range, thereby improving the convergence performance of the stable biorthogonal conjugate gradient method and quickly obtaining the coefficients of each order of the voltage power series. Then, to address the high computational complexity and numerous calculations of the Padé approximation, the Aitken difference method is used to achieve parallel computation of the rational approximation value of the voltage power series, quickly obtaining the voltage values ​​of all nodes, thus providing a more computationally efficient tool for solving power flow problems in large-scale power systems.

[0034] Example 1

[0035] A fully embedded parallel computation method for power flow in large-scale power systems, see [link to relevant documentation]. Figure 1 The method includes the following steps:

[0036] 101: Based on the characteristics of different types of nodes in the power system, a holomorphic embedded power flow model is constructed. Based on the expansion property of holomorphic functions and the property that the coefficients of the power series of each term on both sides of the polynomial are completely equal, a high-dimensional linear equation system of power series coefficients is constructed to solve the coefficients of the power series of voltages of each order.

[0037] 102: Based on the sparsity and invariance of the coefficient matrix in the linear equation system with high-dimensional power series coefficients, a sparse matrix approximate inverse preprocessor is constructed and applied to its coefficient matrix. Then, the stable biorthogonal conjugate gradient method is used to solve the preprocessed linear equation system with high-dimensional power series coefficients in parallel iteratively.

[0038] That is, step 102 constructs a preprocessor for the coefficient matrix in the linear equation system with high-dimensional power series coefficients. Since the coefficient matrix itself is a sparse matrix, the constructed preprocessor is called a sparse matrix approximate inverse processor.

[0039] 103: Based on the solution vector and residual vector r in the output of the preprocessed high-dimensional power series linear equation system obtained through iterative solution. k Perform a convergence criterion for the stable biorthogonal conjugate gradient method; if max(|r k |) is less than the maximum error ε of the stable biorthogonal conjugate gradient method. min Execute step 104; otherwise, continue to perform stable biorthogonal conjugate gradient iteration calculation using the current solution vector until the error is satisfied;

[0040] The output solution should mainly include two parts: 1) a solution vector, which contains the coefficients of the voltage power series of all nodes, and the voltage value can be further solved through these coefficients; 2) a residual vector, which is used to determine the convergence of the stable biorthogonal conjugate gradient method.

[0041] 104: If the output solution vector satisfies the error accuracy requirements of the stable biorthogonal conjugate gradient, the Aitken difference method can be used to replace the Padé approximation to calculate the rational approximation value of the voltage power series in parallel, and output a vector containing the voltage values ​​of each node, and then execute step 105.

[0042] 105: Based on the obtained node voltage values, a convergence determination step for the maximum power mismatch value is performed. If the maximum power mismatch value is greater than ε... eps If the condition is met, the right-hand column vector of the preprocessed high-dimensional power series coefficient linear equation system is updated, and the next iteration of the calculation is performed; otherwise, the power flow converges, the power flow distribution of the large-scale power system is obtained, and the process ends.

[0043] Where, ε eps The maximum power mismatch error is calculated based on the current voltage value. Its calculation method is the same as the convergence judgment step of the NR method, and will not be repeated here.

[0044] In summary, the embodiments of the present invention realize parallel computation of fully embedded power flow in large-scale power systems through steps 101-105. This not only effectively avoids the non-convergence phenomenon of power flow in large-scale power systems caused by improper guessing of initial values, but also has higher computational efficiency than traditional HELM, providing staff with a computational tool that combines accuracy and efficiency.

[0045] Example 2

[0046] The following section provides a further explanation of the scheme in Example 1, using specific calculation formulas and examples. See the description below for details:

[0047] 201: Construct a HELM model based on the characteristics of different types of nodes in a power system;

[0048] If there are N nodes in a power system, then according to the holomorphic embedding theory, complex variables s can be embedded into the power balance equations of PQ nodes, PV nodes, and slack nodes (denoted as REF nodes) to construct HELM equations for holomorphic functions of the voltage of three different types of nodes:

[0049] 1) PQ node HELM equations:

[0050]

[0051] In the formula, Inject power into the PQ node; Y iktrans The symmetrical portion after neglecting the ground admittance of the PQ node; Y ishunt Let PQ be the ground admittance of the node in the node admittance matrix; The voltage holomorphic function of the PQ node; "*" is the conjugate operator.

[0052] 2) HELM equations for PV nodes:

[0053]

[0054] In the formula, P l Y represents the active power of the PV node; lktrans Y represents the symmetric part of the nodal admittance matrix after neglecting the PV nodal-to-ground admittance; lshunt This represents the ground branch admittance of the PV node in the node admittance matrix; and Q l (s) are the voltage holomorphic function and the reactive power holomorphic function of the PV node, respectively; "j" represents the voltage amplitude of the PV node; "j" is the imaginary part symbol.

[0055] 3) Equilibrium node HELM equations:

[0056]

[0057] In the formula, The voltage holomorphic function of the balancing node; This represents the voltage amplitude at the balancing node.

[0058] 202: Based on the expansion properties of holomorphic functions and the property that the coefficients of all terms in the power series are completely equal on both sides of a polynomial, a high-dimensional linear system of power series coefficients is constructed to solve for the coefficients of voltage power series of all orders:

[0059] 1) Power series expansion of holomorphic functions:

[0060]

[0061] In the formula, n is the order of the expansion of the voltage power series and the reactive power power series; These are the 0th, 1st, ..., nth order coefficients of the holomorphic power series of the node voltages; Q l [0]~Q l [n] represents the 0th, 1st, ..., nth order coefficients of the pure power series of reactive power at the PV nodes.

[0062] 2) Construction of a system of linear equations with high-dimensional power series coefficients:

[0063] Substituting the expansion shown in equation (4) into equations (1)-(3), and based on the principle that the coefficients of each power series term on both sides of the polynomial are equal, a system of high-dimensional linear equations for solving the coefficients of each order of voltage power series is constructed:

[0064]

[0065] In the formula, Y sys Here, f(U[n-1], Q) represents the coefficient matrix of the high-dimensional power series linear equation system; n is the order of the power series expansion of the node voltage and reactive power; U[n] is the vector of coefficients of the nth power series of the node voltage to be determined; l [n-1]) is a vector expression containing the coefficients of the (n-1)th order voltage power series and the coefficients of the reactive power power series; G and B are the conductance and susceptance in the power system, respectively; and δ represents the coefficients of the nth-order inverse power series for nodes PV and PQ, respectively; “Re” and “Im” represent taking the real and imaginary parts of the complex number, respectively; δ nr To introduce a variable, its value is:

[0066]

[0067] 203: Based on the coefficient matrix Y in the linear equation system of high-dimensional power series coefficients sys Based on the sparsity and invariance of the sparse matrix, a preprocessor F for approximate inverse sparse matrix is ​​constructed. pre It is then applied to its coefficient matrix, and the preprocessed high-dimensional power series coefficient linear equation system is solved in parallel using the stable biorthogonal conjugate gradient method.

[0068] 1) Coefficient matrix preprocessing

[0069] First, the coefficient matrix Y sys Perform factorization:

[0070]

[0071] In the formula, Y D and Y W The coefficient matrix Y is respectively sys The lower triangular matrix and the upper triangular matrix.

[0072] Furthermore, by Y D and Y W Constructing a sparse matrix approximate inverse preprocessor F pre :

[0073]

[0074] In the formula, E Y To be with Y A Identity matrices of the same dimension.

[0075] via preprocessor F pre For Y sys After preprocessing, the linear equation system with higher-order power series coefficients can be further described as follows:

[0076]

[0077] Equation (9) can be further simplified to:

[0078]

[0079] In the formula, the coefficient matrix of the preprocessed high-dimensional power series linear equation system can be obtained from E Y , Y W Calculated.

[0080] Furthermore, referring to the calculation process of the stable biorthogonal conjugate gradient method, the solution vector U[k] of the linear equation system shown in equation (10) for the k-th iteration of the stable biorthogonal conjugate gradient method is:

[0081] U[k]=U[k-1]+α k ρ k +ωk (r k-1 -α k F pre ρ k (11)

[0082] In the formula, U[k-1] and r k-1 These are the solution vector and residual vector, respectively, in the (k-1)th iteration result of the stable biorthogonal conjugate gradient; α k ρ k and ω k All of these are intermediate vectors that need to be calculated in the k-th iteration of the stable biorthogonal conjugate gradient; F pre It is an approximate inverse preprocessor for the sparse matrix constructed by equation (8).

[0083] 2) Parallel solution of linear equation systems with high-dimensional power series coefficients

[0084] The formation of equation (10) first requires Y... sys Decomposition and The construction of the method is relatively simple, so it can be directly obtained and stored through serial computation. Considering that the stable biorthogonal conjugate gradient method requires the use of... and f(U[n-1],Q) l [n-1]) Perform multiple large-scale matrix-vector product calculations to obtain intermediate vectors of different types. The calculation formulas for different types of vectors can be compiled into calculation unit tasks, and then multiple calculation units can be called to perform parallel calculations of all vectors at each node when solving equation (10) using the stable biorthogonal conjugate gradient method. For the specific parallel calculation process strategy, please refer to Figure 2 .

[0085] 204: Based on the residual vector r in the output result k A convergence determination step is performed for the stable biorthogonal conjugate gradient method. To ensure that the coefficients of each order of voltage power series obtained by iteratively solving equation (10) are sufficiently accurate, the error accuracy of the stable biorthogonal conjugate gradient method is usually set to 1e-12. If max(|r k If |) < 1e-12, proceed to step 205; otherwise, continue iterative calculation of the stable biorthogonal conjugate gradient method using the current solution vector U[k] until the error is satisfied.

[0086] 205: After obtaining U[k] that meets the error accuracy requirements of the stable biorthogonal conjugate gradient method according to step 204, the rational approximation value of the voltage power series can be calculated in parallel using the Aitken difference method. Step 205 can be divided into the following parts:

[0087] 1) Calculate the sum of the coefficients of voltage power series of different orders.

[0088] The solution vector U[k] obtained by parallel solving equation (10) using the stable biorthogonal conjugate gradient method contains the coefficients of the k-th order voltage power series at each node:

[0089]

[0090] In the formula, and These are the k-th order coefficients of the holopure power series of the voltages at the balancing node, PV node, and PQ node, respectively.

[0091] Solving equation (10) multiple times yields the voltage power series coefficients for each node at each order:

[0092]

[0093] In the formula, and The zero-order coefficients of the voltage power series for the balancing node, PV node, and PQ node, respectively; U r U l and U i These are vectors containing the coefficients of each order of voltage power series, including the balancing node, PV node, and PQ node.

[0094] The sum of the coefficients of the voltage power series at different orders is calculated using the vector elements of equation (13):

[0095]

[0096] In the formula, These are the zero-order coefficients of the voltage power series of the balancing node, PV node, and PQ node, respectively. These are the sum of the first k terms of the voltage power series of the balancing node, PV node, and PQ node, respectively.

[0097] 2) Calculate the Aitken difference operator:

[0098]

[0099] In the formula, "△ 2 The difference operator defined for the Aitken difference method is calculated from the sum of the coefficients of the first k, k-1, and k-2 terms of the voltage holomorphic power series.

[0100] 3) Aitken difference rational approximation calculation:

[0101]

[0102] In the formula, and These are the rational approximations of the voltage power series for the balancing node, PV node, and PQ node, respectively.

[0103] 4) Improved Aitken difference rational approximation parallel computation formula

[0104] The voltage power series coefficients required for calculation in equation (16) are all double-precision floating-point complex numbers, and each coefficient requires 16 bytes of storage space. To reduce the memory usage of the rational approximation value of the voltage power series in Aitken differential parallel computation and improve its parallel computation efficiency, the Aitken differential operator shown in equation (15) is improved. Taking the PQ node as an example:

[0105]

[0106] In the formula, and In equation (15) and The real part; and In equation (15) and The imaginary part.

[0107] Similarly, the Aitken difference operator for slack nodes and PV nodes can also be decomposed into the above form. Furthermore, substituting the Aitken difference operator shown in equation (17) into equation (16), we obtain the calculation form of the rational approximation value of the power series of voltages for all types of nodes:

[0108]

[0109] 5) Parallel solution of rational approximation values ​​of voltage power series

[0110] Large-scale power systems have a large number of nodes. A unified calculation form for the rational approximation of the power series voltages of all types of nodes, as shown in equation (18), can be compiled on a computing unit. Furthermore, equation (18) can be calculated in parallel using computing units of the same type and scale as the system nodes, thus achieving parallel calculation of the rational approximation values ​​of the power series voltages of all nodes. For specific parallel strategies, please refer to [link to relevant documentation]. Figure 3 .

[0111] 206: A convergence determination step for the maximum power mismatch value is performed based on the obtained node voltage values. If the maximum power mismatch value is greater than ε... eps If the condition is met, the right-hand column vector of the preprocessed high-dimensional power series coefficient linear equation system is updated, and the next iteration of the calculation is performed; otherwise, the power flow converges, the power flow distribution of the large-scale power system is obtained, and the process ends.

[0112] In summary, the embodiments of the present invention, through steps 201-206 above, solve the high-dimensional power series coefficient linear equation system in parallel iteratively based on the stable biorthogonal conjugate gradient method with approximate inverse preprocessing, and further calculate the rational approximation values ​​of the power series of all node voltages in parallel based on the Aitken difference method, thereby realizing accurate and fast calculation of fully embedded power flow in large-scale power systems.

[0113] Example 3

[0114] The following will use specific examples... Figures 4-5 Tables 1-6 are used to verify the feasibility of the solutions in Examples 1 and 2, as detailed below:

[0115] First, to verify the accuracy of the parallel computing method for Holomorphic Embedded Power Flow in Large Scale Power Systems (P-HELM) proposed in this embodiment of the invention in solving power flow problems, the power flow calculation results of P-HELM and the NR method were compared on test systems of different scales. The power flow outer loop error threshold was set to 1e-03, the inner loop error threshold for solving equation (10) in a parallel iterative solution of stable biorthogonal conjugate gradients was set to 1e-12, and the maximum number of iterations for the inner and outer loops were set to 20 and 30, respectively.

[0116] Table 1, based on the power flow calculation results of the NR method, details the maximum errors in voltage amplitude and phase angle in the power flow calculation results of the P-HELM and NR methods. As can be seen from Table 1, for the various test systems in Table 1, the maximum relative error of P-HELM compared to the NR method is only 0.0034% and 0.0053%, respectively. The distribution of voltage amplitude and phase angle at the nodes with the maximum error is as follows: Figure 4 The results show that P-HELM has the same computational accuracy as the NR method when solving power flow problems in large-scale power systems, and can meet the accuracy requirements for power flow calculation in large-scale power systems.

[0117] Table 1. Percentage error of P-HELM relative to NR method / %

[0118]

[0119] Secondly, to verify the effectiveness of the approximate inverse preprocessor construction method proposed in the embodiments of the present invention, Table 2 compares the stable biorthogonal conjugate gradient method with and without preprocessing, and with ILU preprocessing. [9]The average number of computations in the inner and outer loops during convergence of the P-HELM algorithm when using approximate inverse preprocessing is shown in Table 2. In Table 2, the left and right sides of " / " represent the average number of computations in the inner and outer loops of P-HELM, respectively. The average number of computations was taken after ten repeated tests. Table 2 shows that for the various test systems in Table 2, when P-HELM uses the stable biorthogonal conjugate gradient method to solve high-dimensional power series linear equations, applying ILU or approximate inverse preprocessing techniques can achieve better convergence. However, when using ILU preprocessing, the stable biorthogonal conjugate gradient method has a larger number of inner loop iterations, resulting in a longer solution time. In contrast, the preprocessor F proposed in this embodiment of the invention... pre The stable biorthogonal conjugate gradient method can output relatively accurate voltage power series coefficients with only one inner loop calculation, which speeds up the convergence speed of the stable biorthogonal conjugate gradient method and has a significant advantage in solving linear equations with high-dimensional power series coefficients.

[0120] Table 2. The average number of inner and outer cycles of P-HELM.

[0121]

[0122]

[0123] Next, to verify the efficiency of the Aitken difference method in calculating the rational approximation of the power series voltage values ​​of each node compared with the Padé approximation in this embodiment of the invention, Table 3 compares the computation time of P-HELM using equation (16) and Padé approximation in calculating the rational approximation of the power series voltage values ​​of each node. Table 3 shows that for test systems of different sizes, the speedup of the Aitken difference method compared to the Padé approximation is ideal. This indicates that the computational complexity of calculating the rational approximation of the voltage power series based on the Aitken difference method in this embodiment of the invention is low, effectively reducing the computational load for solving power flow problems in large-scale power systems.

[0124] Table 3 Comparison of average calculation times for rational approximations of voltage power series.

[0125] Tab.3 Computational time comparison between Pade approximation and Aitken difference

[0126]

[0127] To evaluate the acceleration effect of the parallel solution of power flow in large-scale power systems and the overall computational efficiency of P-HELM in the embodiments of the present invention, Table 4 first compares the direct solution method based on MATLAB's built-in linear equations and the method based on LU decomposition.

[10] The computation time of solving linear equation systems with high-dimensional power series coefficients using direct methods and parallel iterative methods based on preprocessing stable biorthogonal conjugate gradients proposed in this invention; Figure 5 This demonstrates the speedup effect of the preprocessing stable biorthogonal conjugate gradient parallel iterative method proposed in this embodiment compared to the other two algorithms under different scale test systems. Next, Table 5 tests the speedup ratio of the parallel Aitken difference method implemented based on Equation (18) compared to the serial Aitken difference method and Padé approximation shown in Equation (16). Finally, Table 6 compares the overall computation time of three different algorithms for solving power system power flow: the flat-start NR method, the traditional HELM, and the P-HELM, which directly solve the modified equation system based on LU decomposition, under different scale test systems.

[0128] From Table 4 and Figure 5It can be seen that P-HELM, using the preprocessing stable biorthogonal conjugate gradient method proposed in this embodiment of the invention, achieves better acceleration than the LU direct solution method in solving high-dimensional power series coefficient linear equations in parallel iteratively. However, its computational efficiency improvement is limited compared to MATLAB's built-in linear equation solving algorithm. The main reason for this phenomenon is that in the hybrid programming mode, there are multiple data interactions between the CPU and GPU in the inner and outer double loop computation modes. The large-scale data interaction time is still difficult to reduce under current software and hardware conditions, which limits the improvement of parallel acceleration to a certain extent. It should be noted that MATLAB's built-in linear equation solving algorithm has been greatly optimized and has been widely used in solving large-scale sparse linear equation systems. The preprocessing-stabilized biorthogonal conjugate gradient method and parallel iterative method proposed in this embodiment still have considerable room for optimization, and their parallel computation efficiency can be further improved through optimization. Table 5 shows that P-HELM, using the parallel Aitken difference method, consumes less time to calculate the rational approximation of the node voltage power series. Compared to the Padé approximation and the serial Aitken difference method, the computation time acceleration is ideal, with the highest speedup reaching 58.400 and 22.999, respectively. Table 6 shows that, compared to the traditional HELM, the P-HELM proposed in this embodiment achieves a speedup of over 3.40 on all test systems, with a maximum speedup of 5.64 and an average speedup of 4.85. Compared to the flat-start NR method that directly solves the modified equation system using LU decomposition, P-HELM achieves a speedup of over 54.53, with a maximum speedup of 94.96 and an average speedup of 72.86. Furthermore, for the test systems in Table 6 where the flat-start NR method fails to converge, P-HELM can guarantee convergence and accuracy. This demonstrates that P-HELM not only has high computational efficiency but also strong robustness.

[0129] Table 4 Comparison of computation time for linear equation systems with high-dimensional power series coefficients

[0130] Tab.4Computational time comparison for solving power series linearequations between

[0131] different methods

[0132]

[0133] Table 5. Acceleration ratio of parallel Aitken difference compared with Padé approximation and serial Aitken difference.

[0134] and serial Aitken difference

[0135]

[0136] Table 6. Computational time comparison between P-HELM and other methods

[0137]

[0138]

[0139] Note: In the "Speedup ratio" column, the left and right " / " respectively refer to the speedup ratio of the average computation time of P-HELM relative to the average computation time of the flat-start NR method and the traditional HELM method for solving the modified equation system based on the LU decomposition direct method.

[0140] The above results show that the fully embedded parallel power flow calculation method for large-scale power systems proposed in this embodiment can achieve accurate and fast solution of power flow in large-scale power systems and has strong robustness, providing a new approach for power flow calculation.

[0141] References

[0142] [1] Chen Heng. Steady-state analysis of power systems [M]. China Electric Power Press, 2007.

[0143] [2]Li Xue, Li F.GPU-based two-step preconditioning for conjugategradient method in power flow[C]. IEEE Power&Energy Society General Meeting, America, 2015.

[0144] [3] Liu Chengxi, Lai Qiupin, Yao Liangzhong, et al. Decoupling model of power system based on fully embedded method: weak node identification [J]. Proceedings of the CSEE, 2022, 42(05): 1736-1748.

[0145] [4]Rao S,Tylavsky D.J.Theoretical convergence guarantees versusnumerical convergence behavior of the holomorphically embedded power flowmethod[J].International Journal of Electrical Power&Energy Systems,2018,95(2):166–176.

[0146] [5]Trias A.The holomorphic embedding load flow method[C].IEEE Powerand Energy Society General Meeting,San Diego,2012:1-8.

[0147] [6]Liu C,B.Wang,F.Hu,et al.Online Voltage Stability Assessment forLoad Areas Based on the Holomorphic Embedding Method[J].IEEE Transactions onPower Systems,2018,33(04):3720-3734.

[0148] [7]H.D.Chiang,T.Wang,H.Sheng.A Novel Fast and Flexible HolomorphicEmbedding Power

[0149] Flow Method[J].IEEE Transactions on Power Systems,2018,33(3):2551-2562.

[0150] [8]Trias A,JL Marín.A Padé-Weierstrass technique for the rigorousenforcement of control limits

[0151] in power flow studies[J].International journal of electrical power and energy systems, 2018, 99: 404-418.

[0152] [9] Tang Kunjie, Dong Shufeng, Song Yonghua. Power flow algorithm for power system based on incomplete LU decomposition preprocessing iterative method [J]. Proceedings of the CSEE, 2017, 37(S1): 55-62.

[0153]

[10] Benzi M. Preconditioning Techniques for Large Linear Systems: ASurvey[J]. Journal of Computational Physics, 2002, 182(2): 418-477.

[0154] Those skilled in the art will understand that the accompanying drawings are merely schematic diagrams of a preferred embodiment, and the sequence numbers of the above embodiments of the present invention are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments.

[0155] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A fully embedded parallel computation method for power flow in a large-scale power system, characterized in that, The method includes: Based on the sparsity and invariance of the coefficient matrix in the linear equation system with high-dimensional power series coefficients, a sparse matrix approximate inverse preprocessor is constructed and applied to its coefficient matrix. The stable biorthogonal conjugate gradient method is used to solve the preprocessed linear equation system with high-dimensional power series coefficients in parallel iteratively. The convergence determination step of the stable biorthogonal conjugate gradient method is performed based on the solution vector and residual vector in the output of the preprocessed high-dimensional power series coefficient linear equation system obtained through iterative solution; when max(|r k |) is less than the maximum error of the stable biorthogonal conjugate gradient method, and the output solution vector meets the error accuracy requirements of the stable biorthogonal conjugate gradient. The Aitken difference method is used to replace the Padé approximation to calculate the rational approximation value of the voltage power series in parallel, and the vector containing the voltage values ​​of each node is output. Based on the obtained node voltage values, a convergence determination step is performed for the maximum power mismatch value. If the maximum power mismatch value is greater than ε... eps Then update the right-hand column vector of the preprocessed high-dimensional power series coefficient linear equation system and perform the next iteration calculation. The system of linear equations with high-dimensional power series coefficients is as follows: ； In the formula, Y sys Here, f(U[n-1], Q) represents the coefficient matrix of the high-dimensional power series linear equation system; n is the order of the power series expansion of the node voltage and reactive power; U[n] is the vector of coefficients of the nth power series of the node voltage to be determined; l [n-1]) is a vector expression containing the coefficients of the (n-1)th order voltage power series and the coefficients of the reactive power power series; G and B are the conductance and susceptance in the power system, respectively; and , respectively, are the coefficients of the nth order inverse power series of PV nodes and PQ nodes; "Re" and "Im" represent taking the real and imaginary parts of the complex number, respectively; The sparse matrix approximate inverse preprocessor is: The coefficient matrix Y sys Perform factorization: ； In the formula, Y D and Y W The coefficient matrix Y is respectively sys The lower triangular matrix and the upper triangular matrix; By Y D and Y W Constructing a sparse matrix approximate inverse preprocessor F pre : ； In the formula, E Y To be with Y A Identity matrices of the same dimension.

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