A high-voltage transmission line electromagnetic transient process calculation method based on FD-WENO format

Abstract

This invention relates to a method for calculating the electromagnetic transient processes of high-voltage transmission lines based on the FD-WENO scheme. This method employs a high-order, inherently oscillatory-free differential FD-WENO scheme for numerical flux reconstruction in the spatial domain, and a total variation decay (TVD-RK) scheme for time-domain numerical integration, progressively solving for the time-varying curves of electrical quantities at various points in the transmission line. Compared with existing technologies, this invention effectively solves the numerical oscillation problem encountered when using traditional finite-difference time-domain (FDTD) methods to solve high-voltage transmission lines, and the spatial discretization accuracy of high-voltage transmission lines can reach arbitrary precision of order 2 or higher.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic transient processes in power systems, and in particular to a method for calculating electromagnetic transient processes in high-voltage transmission lines based on the FD-WENO format. Background Technology

[0002] The electromagnetic transient processes of high-voltage transmission lines can be reduced to the more general problem of solving the transient response of transmission line systems. This problem is an important research topic in the field of numerical electromagnetic field calculation. When a power system line experiences a short-circuit fault, overvoltage, or other abnormal operating conditions, the voltage and current on the transmission line change rapidly within a short period of time, which may damage electrical equipment and even the entire power system. To avoid such damage, it is necessary to accurately analyze the dynamic changes in voltage and current on the transmission line system, thus requiring accurate solutions for the transmission line system. The mathematical model of a transmission line system is a typical hyperbolic partial differential equation. The commonly used numerical method for solving the electromagnetic transient processes of this system is the time-domain finite-difference (FDTD) method. However, when there are discontinuities in the system, the FDTD method will generate non-physical oscillations at the discontinuity points, which will cause significant errors in the calculation results.

[0003] To address the oscillation phenomenon encountered in electromagnetic transient calculations, researchers have conducted beneficial discussions on applying the finite element method (FEM) to the transient response analysis of multi-conductor transmission lines. The FEM can resolve numerical oscillation phenomena that FDTD (Functional Dynamics Theory) cannot. Researchers employed flux splitting techniques, using different difference schemes based on the sign of eigenvalues ​​to approximate spatial partial differentials, overcoming the non-physical oscillation problem at discontinuous solutions in traditional FDTD and Lax-Wendroff schemes. Researchers proposed a variable-parameter rational fraction fitting method to suppress numerical oscillations. This method achieves L-stability by adjusting parameters, with a computational accuracy of third order. Researchers derived an explicit and L-stable numerical method using the Padé approximation of Taylor series, which avoids numerical oscillations and effectively handles discontinuities. Researchers applied an improved block method to the time-domain simulation of transmission lines, achieving high time-domain simulation accuracy, but with lower spatial discretization accuracy. While the above calculation methods effectively suppress numerical oscillations, they suffer from overall low accuracy. Especially for spatial discretization accuracy, a central difference scheme with second-order accuracy is often used. In fact, cascaded PI-type equivalent circuits used to simulate distributed parameter transmission lines have second-order spatial discretization accuracy. However, most numerical algorithms used to simulate transmission line systems only pursue high accuracy in the time domain simulation, ignoring the impact of spatial discretization accuracy on the overall simulation accuracy. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the existing technology and provide a calculation method for electromagnetic transient processes of high-voltage transmission lines based on the FD-WENO scheme. This method can effectively solve the numerical oscillation problem faced by high-voltage transmission lines when using the traditional FDTD method, and the spatial discretization accuracy of high-voltage transmission lines can reach arbitrary accuracy of second order or higher.

[0005] The objective of this invention can be achieved through the following technical solutions:

[0006] This invention provides a method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO format. The method includes the following steps:

[0007] Step 1, Modeling: Based on the distributed parameters and boundary conditions of high-voltage transmission lines, a multi-conductor transmission line system model is constructed using hyperbolic partial differential equations, i.e., telegraph equations, and rewritten in matrix form;

[0008] Step 2, Spatial Domain Discretization: Numerical flux reconstruction is performed in space using a high-order, inherently non-oscillatory differential FD-WENO scheme, specifically by spatially discretizing the matrix-form multi-conductor transmission line system model;

[0009] Step 3: Initialize simulation settings, detect and input faults or operations of high-voltage transmission lines during electromagnetic transient numerical calculations;

[0010] Step 4, Fault or Operation Judgment: Based on the detection results at time t, determine whether there is a fault or operation in the high-voltage transmission line system. If there is no fault or operation, proceed directly to step 5; otherwise, modify and update the multi-conductor transmission line system model after spatial domain discrete reconstruction.

[0011] Step 5, Time Domain Discretization: In the time domain, the total variation decay (TVD-RK) scheme is used for stepwise integration.

[0012] Step 6: Update simulation time t;

[0013] Step 7: Determine if the numerical integration process terminates: If t < t f Then proceed to step 4 to continue the time-domain numerical integration of the high-voltage transmission line system at the next time step; if t ≥ t f Then proceed to step 8; where t f To set the total simulation time;

[0014] Step 8: Output the electromagnetic transient numerical simulation results of electrical quantities at various points along the high-voltage transmission line.

[0015] Preferably, step 1 specifically comprises:

[0016] Step 1.1: Establish a set of partial differential equations describing the electromagnetic transient process of high-voltage transmission lines. Based on engineering practice, model the boundary conditions of the high-voltage transmission lines, providing the boundary equations for the beginning and end of the transmission line. Use hyperbolic partial differential equations, i.e., telegraph equations, to describe the multi-conductor transmission line system model composed of three-phase overhead transmission lines or power cable lines. The mathematical expression is:

[0017]

[0018] In the formula: ν represents the voltage traveling wave column vector on the high-voltage transmission line; i represents the current traveling wave column vector on the high-voltage transmission line; R represents the distributed parameter resistance matrix of the high-voltage transmission line; L represents the distributed parameter inductance matrix of the high-voltage transmission line; C represents the distributed parameter capacitance matrix of the high-voltage transmission line; G represents the distributed parameter conductance matrix of the high-voltage transmission line.

[0019] Step 1.2, rewrite equation (1) into the following matrix form:

[0020]

[0021] Where: the state variable to be determined High-voltage transmission line system flux f(u) = Au; system coefficient matrix System coefficient matrix

[0022] Preferably, step 2 specifically comprises:

[0023] Step 2.1: Define a high-voltage transmission line system of length l along its spatial arrangement direction, and connect the high-voltage transmission lines [x... a ,x b Divide into N equally spaced intervals I i =[x i-1 / 2 ,x i+1 / 2 ], i = 1, 2, ..., N, satisfying x a =0, x b =l, where l is the total length of the high-voltage transmission line, then the semi-discrete conservative difference scheme of equation (2) is:

[0024]

[0025] In the formula, f i-1 / 2 f i+1 / 2 These are sections I of the high-voltage transmission line system. i =[x i-1 / 2 ,x i+1 / 2 The flux corresponding to the boundary; Δx is the spatial dispersion step size;

[0026] Step 2.2: Use a high-order, inherently non-oscillatory differential FD-WENO scheme to reconstruct the flux values ​​at the boundaries of the interval segment;

[0027] Step 2.3: Use Lax-Friedrichs numerical flux to approximate the transmission system flux f(u), thereby achieving spatial discretization of equation (2);

[0028] Step 2.4: Summarize equation (3) into an initial value problem, expressed as:

[0029]

[0030] In the formula, W represents the spatial difference linear operator.

[0031] Preferably, step 2.2 specifically includes:

[0032] Using the third-order WENO-JS3 format, the numerical flux at the interval boundary i+1 / 2 is calculated. To refactor, the expression is:

[0033]

[0034] Numerical flux is obtained by nonlinearly combining the two low-order fluxes in equation (5).

[0035]

[0036] In the formula, the nonlinear weighting coefficient ω r and smoothing factor β r The expression is as follows:

[0037]

[0038]

[0039] In the formula, d0 = 1 / 3, d1 = 2 / 3, and ε is a very small positive number.

[0040] Preferably, step 2.3 specifically includes:

[0041] The flux f i+1 / 2 It is split into two parts, positive and negative, and the expression is:

[0042] h(a,b)=f + (a)+f - (b) (9)

[0043] Wherein, positive flux f + It satisfies the property of monotonically increasing, and has a negative flux f. - It satisfies the property of monotonically decreasing;

[0044]

[0045] In the formula, In equation (2), A is a Jacobian matrix, and α is taken as the spectral radius of A; the positive flux f + negative flux f - All are reconstructed using Equation (5), and the numerical flux is finally obtained from Equation (6);

[0046] Let the WENO difference operator be a third-order H WENO3 Then there is

[0047]

[0048] The final numerical flux at the boundary of the interval segment i+1 / 2 is... It can be represented as

[0049]

[0050] Preferably, step 2.2 specifically includes:

[0051] Using the fifth-order WENO scheme, the numerical flux at the interval boundary i+1 / 2 is calculated. To refactor, the expression is:

[0052]

[0053] Numerical flux is obtained by nonlinearly combining the three low-order fluxes in equation (8).

[0054]

[0055] In the formula, the nonlinear weighting coefficient ω r and smoothing factor β r The expression is as follows:

[0056]

[0057]

[0058] In the formula, the linear weights and ε is a very small positive number.

[0059] Preferably, step 2.3 specifically includes:

[0060] The flux f i+1 / 2 It is split into two parts, positive and negative, and the expression is:

[0061] h(a,b)=f + (a)+f -(b) (17)

[0062] Wherein, positive flux f + It satisfies the property of monotonically increasing, and has a negative flux f. - It satisfies the property of monotonically decreasing;

[0063]

[0064] In the formula, In equation (2), A is a Jacobian matrix, and α is taken as the spectral radius of A; the positive flux f + negative flux f - All are reconstructed using Equation (13), and the numerical flux is finally obtained from Equation (14);

[0065] Let the WENO difference operator be a fifth-order H WENO5 Then we have:

[0066]

[0067] The final numerical flux at the boundary of the interval segment i+1 / 2 is... for:

[0068]

[0069] Preferably, the initialization of simulation settings in step 3 specifically involves setting the initial simulation time t = 0.0s, the number of integration steps n = 0, the fixed step size h for numerical integration, and the total time t for electromagnetic transient simulation calculation. f ; Set the initial values ​​u0, ... l Among them, u0 and u l Let be the initial value of the state variable to be determined at the starting point of the high-voltage transmission line.

[0070] Preferably, step 5 specifically comprises:

[0071] The discretization of equation (3) in the time domain is performed using the Total Variation Decay (TVD-RK) scheme for time advancement. A third-order TVD-RK method is used. For the initial value problem equation (4), the calculation expression is:

[0072]

[0073] In the formula, h represents the step size of the stepwise time integration, and W represents the spatial difference linear operator; the calculation format includes a three-fold accumulation process.

[0074] By t n The state variable u at time t n Calculate the state variables at t=t n+1 =t n The approximate value u at +h n+1.

[0075] Preferably, step 5 specifically comprises:

[0076] The discretization of equation (3) in the time domain is performed using the Total Variation Decay (TVD-RK) scheme for time advancement. A 4-level, 10th-order TVD-RK method is used. For the initial value problem equation (4), the calculation expression is:

[0077]

[0078] In the formula, h represents the step size of the stepwise time integration, and W represents the spatial difference linear operator; the calculation format includes a 10-fold value accumulation process;

[0079] By t n The state variable u at time t n Calculate the state variables at t=t n+1 =t n The approximate value u at +h n +1 .

[0080] Compared with the prior art, the present invention has the following advantages:

[0081] 1) This invention employs a high-order finite difference scheme based on WENO for spatial discretization and a strongly stable total variation decay (TVD-RK) method for time discretization. Compared to the traditional implicit trapezoidal integral method, this numerical calculation scheme can completely avoid numerical oscillation problems. This method can achieve high-order computational accuracy in both the spatial and time domains, with an overall local truncation error of O(Δx). 3 +h 3 ) or O(Δx 5 +h 10 Compared with the simulation results of commercial PSCAD software, the electromagnetic transient numerical calculation method of high-voltage transmission lines based on the FD-WENO calculation method can not only completely avoid the numerical oscillation problem, but also the electromagnetic transient initialization efficiency is significantly better than the electromagnetic transient initialization method based on the implicit trapezoidal method.

[0082] 2) The electromagnetic transient numerical calculation of high-voltage transmission lines based on FD-WENO used in this invention can flexibly handle complex transmission line systems containing lumped parameter elements and distributed parameter line coupling. Attached Figure Description

[0083] Figure 1 This is a flowchart of the method of the present invention;

[0084] Figure 2 This is a schematic diagram of a spatial grid.

[0085] Figure 3 This is a schematic diagram in WENO format; where, Figure 3 'a' is a format diagram of WENO-JS3. Figure 3 b is a schematic diagram of the WENO-5 format;

[0086] Figure 4 This is a topology diagram of a three-phase overhead transmission line.

[0087] Figure 5 This is a schematic diagram of the voltage and current of a three-phase overhead line; where, Figure 5 'a' represents the no-load voltage at the end of the three-phase overhead line. Figure 5 b represents the no-load current near the end of the three-phase overhead line;

[0088] Figure 6 This refers to the no-load voltage at the end of a three-phase overhead line (TR algorithm).

[0089] Figure 7 This is a topology diagram of a four-node transmission line system;

[0090] Figure 8 For network node voltages (FD-WENO algorithm);

[0091] Figure 9 This refers to the network node voltage (TR algorithm). Detailed Implementation

[0092] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0093] Example 1

[0094] like Figure 1 As shown in the figure, this embodiment presents a method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO (finite difference-weighted essentially nonoscillatory) scheme. The method includes the following steps:

[0095] Step 1, Modeling: Based on the distributed parameters and boundary conditions of high-voltage transmission lines, a multi-conductor transmission line system model is constructed using hyperbolic partial differential equations, i.e., telegraph equations, and rewritten in matrix form, as follows:

[0096] Step 1.1: Establish a set of partial differential equations describing the electromagnetic transient process of high-voltage transmission lines. Based on engineering practice, model the boundary conditions of the high-voltage transmission lines, providing the boundary equations for the beginning and end of the transmission line. Use hyperbolic partial differential equations, i.e., telegraph equations, to describe the multi-conductor transmission line system model composed of three-phase overhead transmission lines or power cable lines. The mathematical expression is:

[0097]

[0098] In the formula: ν represents the voltage traveling wave column vector on the high-voltage transmission line; i represents the current traveling wave column vector on the high-voltage transmission line; R represents the distributed parameter resistance matrix of the high-voltage transmission line; L represents the distributed parameter inductance matrix of the high-voltage transmission line; C represents the distributed parameter capacitance matrix of the high-voltage transmission line; G represents the distributed parameter conductance matrix of the high-voltage transmission line.

[0099] Step 1.2, rewrite equation (1) into the following matrix form:

[0100]

[0101] Where: the state variable to be determined High-voltage transmission line system flux f(u) = Au; system coefficient matrix System coefficient matrix

[0102] Step 2, Spatial Domain Discretization: A high-order WENO reconstruction scheme is used to spatially discretize and reconstruct the matrix-form multi-conductor transmission line system model. Specifically:

[0103] Step 2.1, as follows Figure 2 As shown, a high-voltage transmission line system of length l is defined along its spatial arrangement direction, with the high-voltage transmission lines [x] a ,x b Divide into N equally spaced intervals I i =[x i-1 / 2 ,x i+1 / 2 ], i = 1, 2, ..., N, satisfying x a =0, x b =l, then the semi-discrete conserved difference scheme of equation (2) is:

[0104]

[0105] In the formula, f i-1 / 2 f i+1 / 2 These are sections I of the high-voltage transmission line system. i =[x i-1 / 2 ,x i+1 / 2 The flux corresponding to the boundary; Δx is the spatial dispersion step size;

[0106] Step 2.2: The flux values ​​at the boundaries of the interval segment are reconstructed using the inherently non-oscillating difference scheme FD-WENO reconstruction scheme, specifically as follows:

[0107] like Figure 3 As shown, the numerical flux at the interval boundary i+1 / 2 is obtained using the third-order WENO-JS3 scheme. To refactor, the expression is:

[0108]

[0109] Numerical flux is obtained by nonlinearly combining the two low-order fluxes in equation (5).

[0110]

[0111] In the formula, the nonlinear weighting coefficient ω r and smoothing factor β r The expression is as follows:

[0112]

[0113]

[0114] In the formula, d0 = 1 / 3, d1 = 2 / 3, and ε is a very small positive number, usually taken as ε = 10. -8 .

[0115] Step 2.3: The Lax-Friedrichs numerical flux is used to approximate the transmission system flux f(u), thereby achieving spatial discretization of equation (2), specifically as follows:

[0116] The flux f i+1 / 2 It is split into two parts, positive and negative, and the expression is:

[0117] h(a,b)=f + (a)+f - (b) (9)

[0118] Wherein, positive flux f + It satisfies the property of monotonically increasing, and has a negative flux f. - It satisfies the property of monotonically decreasing;

[0119]

[0120] In the formula, In equation (2), A is a Jacobian matrix, and α is taken as the spectral radius of A; the positive flux f + negative flux f -All are reconstructed using Equation (5), and the numerical flux is finally obtained from Equation (6);

[0121] Let the WENO difference operator be a third-order H WENO3 Then there is

[0122]

[0123] The final numerical flux at the boundary of the interval segment i+1 / 2 is... It can be represented as

[0124]

[0125] Step 2.4: Summarize equation (3) into an initial value problem, expressed as:

[0126]

[0127] Step 3: Initialize the simulation settings. Set the initial simulation time t = 0.0s, the number of integration steps n = 0, the fixed step size h for numerical integration, and the total time t for electromagnetic transient simulation calculation. f ; Set the initial values ​​u0, ... l Among them, u0 and u l The initial value of the state variable to be determined at the starting point of the high-voltage transmission line; the fault or operation of the high-voltage transmission line is detected and input during the electromagnetic transient numerical calculation.

[0128] Step 4: Fault or Operation Judgment: Based on the detection results at time t, determine whether there is a fault or operation in the high-voltage transmission line system. If there is no fault or operation, proceed directly to Step 5; otherwise, modify and update the multi-conductor transmission line system model after spatial domain discrete reconstruction.

[0129] Step 5, Time Domain Discretization: Perform stepwise integration in the time domain using the Total Variation Decay (TVD-RK) scheme;

[0130] The discretization of equation (3) in the time domain is performed using the Total Variation Diminishing (TVD-RK) scheme for time advancement. A third-order TVD-RK method is used. For the initial value problem equation (4), the calculation expression is:

[0131]

[0132] In the formula, h represents the step size of the stepwise time integration;

[0133] By t n The state variable u at time t n Calculate the state variables at t=t n+1 =t nThe approximate value u at +h n+1 .

[0134] Preferably, step 5 specifically comprises:

[0135] Alternatively, the 4th-level, 10th-order TVD-RK (Total Variation Diminishing) method can be used. For the initial value problem (4), the calculation expression is:

[0136]

[0137] In the formula, h represents the step size of the stepwise time integration;

[0138] By t n The state variable u at time t n Calculate the state variables at t=t n+1 =t n The approximate value u at +h n+1 .

[0139] Step 6: Update simulation time.

[0140] Step 7: Determine if the numerical integration process terminates: If t < t f Then proceed to step 4 to continue the time-domain numerical integration of the high-voltage transmission line system at the next time step; if t ≥ t f Then proceed to step 8; where t f To set up electromagnetic transient simulation calculations.

[0141] Step 8: Output the electromagnetic transient numerical simulation results of electrical quantities at various points along the high-voltage transmission line.

[0142] Example 2

[0143] This embodiment presents a method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO format. The method includes the following steps:

[0144] Step 1, Modeling: Based on the electromagnetic transient processes and boundary conditions of high-voltage transmission lines, a multi-conductor transmission line system model is constructed using hyperbolic partial differential equations, i.e., telegraph equations, and rewritten in matrix form, as follows:

[0145] Step 1.1: Establish a set of partial differential equations describing the electromagnetic transient process of high-voltage transmission lines. Based on engineering practice, model the boundary conditions of the high-voltage transmission lines, providing the boundary equations for the beginning and end of the transmission line. Use hyperbolic partial differential equations, i.e., telegraph equations, to describe the multi-conductor transmission line system model composed of three-phase overhead transmission lines or power cable lines. The mathematical expression is:

[0146]

[0147] In the formula: ν represents the voltage traveling wave column vector on the high-voltage transmission line; i represents the current traveling wave column vector on the high-voltage transmission line; R represents the distributed parameter resistance matrix of the high-voltage transmission line; L represents the distributed parameter inductance matrix of the high-voltage transmission line; C represents the distributed parameter capacitance matrix of the high-voltage transmission line; G represents the distributed parameter conductance matrix of the high-voltage transmission line.

[0148] Step 1.2, rewrite equation (1) into the following matrix form:

[0149]

[0150] Where: the state variable to be determined High-voltage transmission line system flux f(u) = Au; system coefficient matrix System coefficient matrix

[0151] Step 2, Spatial Domain Discretization: A high-order WENO reconstruction scheme is used to spatially discretize and reconstruct the matrix-form multi-conductor transmission line system model. Specifically:

[0152] Step 2.1, as follows Figure 2 As shown, a high-voltage transmission line system of length l is defined along its spatial arrangement direction, with the high-voltage transmission lines [x] a ,x b Divide into N equally spaced intervals I i =[x i-1 / 2 ,x i+1 / 2 ], i = 1, 2, ..., N, satisfying x a =0, x b =l, then the semi-discrete conserved difference scheme of equation (2) is:

[0153]

[0154] In the formula, f i-1 / 2 f i+1 / 2 These are sections I of the high-voltage transmission line system. i =[x i-1 / 2 ,x i+1 / 2 The flux corresponding to the boundary; Δx is the difference interval;

[0155] Step 2.2: The flux values ​​at the boundaries of the interval segment are reconstructed using the inherently non-oscillating difference scheme FD-WENO reconstruction scheme, specifically as follows:

[0156] like Figure 4 As shown, the numerical flux at the interval boundary i+1 / 2 is calculated using the fifth-order WENO scheme. To refactor, the expression is:

[0157]

[0158] Numerical flux is obtained by nonlinearly combining the three low-order fluxes in equation (8).

[0159]

[0160] In the formula, the nonlinear weighting coefficient ω r and smoothing factor β r The expression is as follows:

[0161]

[0162]

[0163] In the formula, the linear weights and ε is a very small positive number, usually taken as ε = 10. -8 .

[0164] Step 2.3: The Lax-Friedrichs numerical flux is used to approximate the transmission system flux f(u), thereby achieving spatial discretization of equation (2), specifically as follows:

[0165] The flux f i+1 / 2 It is split into two parts, positive and negative, and the expression is:

[0166] h(a,b)=f + (a)+f - (b) (17)

[0167] Wherein, positive flux f + It satisfies the property of monotonically increasing, and has a negative flux f. - It satisfies the property of monotonically decreasing;

[0168]

[0169] In the formula, In equation (2), A is a Jacobian matrix, and α is taken as the spectral radius of A; the positive flux f + negative flux f - All are reconstructed using Equation (13), and the numerical flux is finally obtained from Equation (14).

[0170] Let the WENO difference operator be a fifth-order H WENO5 Then we have:

[0171]

[0172] The final numerical flux at the boundary of the interval segment i+1 / 2 is... for:

[0173]

[0174] Step 2.4: Summarize equation (3) into an initial value problem, expressed as:

[0175]

[0176] In the formula, W represents the spatial difference linear operator.

[0177] Step 3: Initialize the simulation settings. Set the initial simulation time t = 0.0s, the number of integration steps n = 0, the fixed step size h for numerical integration, and the total time t for electromagnetic transient simulation calculation. f ; Set the initial values ​​u0, ... l Among them, u0 and u l The initial value of the state variable to be determined at the starting point of the high-voltage transmission line; detect and input the fault or operation of the high-voltage transmission line during electromagnetic transient numerical calculation;

[0178] Step 4, Fault or Operation Judgment: Based on the detection results at time t, determine whether there is a fault or operation in the high-voltage transmission line system. If there is no fault or operation, proceed directly to step 5; otherwise, modify and update the multi-conductor transmission line system model after spatial domain discrete reconstruction.

[0179] Step 5, Time Domain Discretization: Perform stepwise integration in the time domain using the Total Variation Decay (TVD-RK) scheme;

[0180] The discretization of equation (3) in the time domain is performed using the Total Variation Diminishing (TVD-RK) scheme for time advancement. A third-order TVD-RK method is used. For the initial value problem equation (4), the calculation expression is:

[0181]

[0182] In the formula, h represents the step size of the stepwise time integration;

[0183] By t n The state variable u at time t n Calculate the state variables at t=t n+1 =t n The approximate value u at +h n+1 .

[0184] Preferably, step 5 specifically comprises:

[0185] Alternatively, the 4th-level, 10th-order TVD-RK (Total Variation Diminishing) method can be used. For the initial value problem (4), the calculation expression is:

[0186]

[0187] In the formula, h represents the step size of the stepwise time integration;

[0188] By t n The state variable u at time t n Calculate the state variables at t=t n+1 =t n The approximate value u at +h n+1 .

[0189] Step 6: Update simulation time.

[0190] Step 7: Determine if the numerical integration process terminates: If t < t f Then proceed to step 4 to continue the time-domain numerical integration of the high-voltage transmission line system at the next time step; if t ≥ t f Then proceed to step 8; where t f To set up electromagnetic transient simulation calculations.

[0191] Step 8: Output the electromagnetic transient numerical simulation results of electrical quantities at various points along the high-voltage transmission line.

[0192] Next, combined Figure 4 The electromagnetic transient simulation calculation of no-load closing under different initial phase angles for a three-phase overhead transmission line, as shown in the example, provides a detailed explanation of the specific implementation steps:

[0193] 1) Input initialization data, establish differential equations for each component of the system, and form the basic mathematical model for electromagnetic transient numerical calculation:

[0194] As is well known, it is used to describe Figure 4 The mathematical model of the electromagnetic transient process of the three-phase overhead transmission line shown is the telegraph equation. A mathematical model of the high-voltage transmission line is performed. The input parameters are the distributed parameters R0, L0, C0, and G0 of the transmission line, the total length of the line is L, and the boundary conditions of the three-phase overhead line are given, such as the three-phase excitation voltage source U at the beginning of the line. a,b,c Its internal resistance, and the end of the line is in an unloaded state.

[0195] 2) Spatial discretization: The number of line segments is N = 30, and the total line length is l = 100 km. For the fifth-order WENO scheme, the WENO difference operator is denoted as the fifth-order H... WENO5 Then there is

[0196]

[0197] The numerical flux at the final cell boundary i+1 / 2 is then... It can be represented as

[0198]

[0199] At this point, the telegraph equations for the line can be reduced to the following initial value problem:

[0200]

[0201] 3) Initialization for electromagnetic transient numerical calculations:

[0202] The simulation initial time is set to t = 0.0s, the integration step number n = 0; the numerical integration step size is set to h = 0.57μs, and the total calculation time for the electromagnetic transient simulation is set to t. f =0.02s. Set the initial values ​​u0, u... of each state variable in the system. l Since it is an unloaded line, u l All elements in the array are 0;

[0203] Inputting faults or operations related to electromagnetic transient numerical calculations:

[0204] t≤0 - When i0(t) = 0; t ≥ 0 + hour,

[0205] 4) At t G Real-time fault or operation judgment:

[0206] When t≤t G- If no operation is performed and the mathematical model (3) for electromagnetic transient numerical calculation remains unchanged, then proceed directly to step 5); when t≥t G+ When there is a switching operation or a line short circuit fault, the correlation coefficient matrix in equation (3) needs to be modified. The specific situation needs to be handled according to the location of the fault or the equipment component being operated.

[0207] Equation (3) is the basic mathematical model for performing electromagnetic transient numerical calculations in this example.

[0208] 5) Time-domain numerical integration:

[0209] The numerical integration of equation (3) is performed using a strongly stable 4-level 10th-order TVD-RK scheme, and the calculation format is as follows:

[0210]

[0211] In the formula, h represents the step size of the stepwise time integration. According to equation (4), t... n The state variable u at time t nCalculate the state variables at t=t n+1 =t n The approximate value u at +h n+1 ;

[0212] 6) t = t n+1 =t n +h; Let n = n+1;

[0213] 7) Determining whether the numerical integration process terminates (t) f (Total simulation time)

[0214] If t < t f If so, proceed to step 4) and continue with the numerical integration at the next time step;

[0215] If t≥t f Then proceed to step 8);

[0216] 8) Output of electromagnetic transient numerical simulation results.

[0217] This calculation example primarily outputs the change curves of voltage and current at the end of the line when the circuit is closed under no-load conditions. Figure 5 a: No-load voltage at the end of a three-phase overhead line Figure 5 b: No-load current near the end of a three-phase overhead line Figure 6 : No-load voltage at the end of a three-phase overhead line (TR algorithm: implicit trapezoidal algorithm, the maximum value of the sinusoidal voltage source at the beginning is 220kV).

[0218] To verify the effectiveness of the algorithm presented in this paper, the results were calculated using PSCAD simulation software. Figure 6 For comparative analysis, the three-phase lines were first decoupled using the Karenbauer transform before simulation. The decoupled three-phase lines can be regarded as independent lines, thus completing the transformation of the transmission line from "phase" to "sequence". After calculating the sequence components of the three-phase overhead transmission line, the three-phase voltage and current were obtained by the inverse Karenbauer transform. Figure 6 The simulation results using PSCAD with no load at the end of the line (300km length) are shown, with a calculation step size h = 1μs and 30 spatial discrete divisions. Similarly, the PSCAD simulation results exhibit severe numerical oscillations compared to the algorithm proposed in this invention. Furthermore, after three cycles of calculation, the electromagnetic transient process still fails to stabilize. This indicates that the implicit trapezoidal integral algorithm for electromagnetic transient initialization is less effective than the algorithm presented in this invention. The algorithm proposed in this invention can achieve electromagnetic transient initialization in approximately half a cycle, demonstrating higher computational efficiency than the electromagnetic transient initialization method in PSCAD simulation software.

[0219] In summary, the FD-WENO algorithm used in this invention calculates the terminal overvoltage waveform without generating numerical oscillations. This is because the algorithm uses the WENO scheme during spatial discretization to capture shock waves at discontinuities, avoiding the non-physical oscillations at discontinuities encountered by the FDTD algorithm. Furthermore, it employs a 4-level, 10th-order TVD-RK scheme for step-by-step integration in the time direction to ensure high accuracy and strong stability in time discretization. This constitutes the theoretical basis of this invention.

[0220] Below is a specific example of using the FD-WENO algorithm to perform electromagnetic transient simulation of a four-node transmission line system without generating numerical oscillations.

[0221] Figure 7 This is a simulation example of a four-node transmission line system. A unit right-angle wave (amplitude 1.0 pu) intrudes at node 1. All lines are lossless transmission lines, and the wave velocity is approximately the speed of light. The wave impedance and propagation delay of each line are shown in the appendix. Figure 7 As shown in the diagram, the FD-WENO algorithm is used to calculate the voltage at each node in the network. Lumped parameters such as inductance and capacitance are handled using the state equation method. Figure 8 Network node voltage (FD-WENO algorithm); Figure 9 Network node voltage (TR algorithm, the line in PSCAD simulation uses the Berylon model). Clearly, from... Figure 8 As can be seen, the voltages at nodes 2, 3, and 4 calculated by the algorithm proposed in this invention did not cause numerical oscillations, which further verifies that the FD-WENO algorithm is effective for simulation results during system mutations. In contrast, the TR algorithm cannot avoid this problem.

[0222] The key feature of this invention is the use of the FD-WENO algorithm, combined with the high-order TVD-RK scheme, which not only solves the problem of non-physical oscillations caused by discontinuities but also ensures high precision in spatiotemporal discretization.

[0223] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any person skilled in the art can easily conceive of various equivalent modifications or substitutions within the technical scope disclosed in the present invention, and these modifications or substitutions should all be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO scheme, characterized in that, The method includes the following steps: Step 1, Modeling: Based on the distributed parameters and boundary conditions of high-voltage transmission lines, a multi-conductor transmission line system model is constructed using hyperbolic partial differential equations, i.e., telegraph equations, and rewritten in matrix form; Step 2, Spatial Domain Discretization: Numerical flux reconstruction is performed using a high-order, inherently non-oscillatory differential FD-WENO scheme in space, and the matrix-form multi-conductor transmission line system model is spatially discretized. The high-order, inherently non-oscillatory differential FD-WENO scheme is used to reconstruct the flux values ​​at the boundaries of the interval segment, specifically as follows: Using the third-order WENO-JS3 format, interval segments are mapped to interval boundaries. Numerical flux at To refactor, the expression is: (5) Numerical flux is obtained by nonlinearly combining the two low-order fluxes in equation (5). : (6) In the formula, the nonlinear weighting coefficient and smoothing factor The expression is as follows: (7) (8) In the formula, , , It is a very small positive number; Alternatively, a fifth-order WENO scheme can be used to define the interval segments relative to their boundaries. Numerical flux at To refactor, the expression is: (13) Numerical flux is obtained by nonlinearly combining the three low-order fluxes in equation (13) according to equation (8). : (14) In the formula, the nonlinear weighting coefficient and smoothing factor The expression is as follows: (15) (16) In the formula, the linear weights , and , It is a very small positive number; Step 3: Initialize simulation settings, detect and input faults or operations of high-voltage transmission lines during electromagnetic transient numerical calculations; Step 4, Fault or Operation Judgment: Based on the time... t Based on the detection results, determine whether there is a fault or operation in the high-voltage transmission line system. If there is no fault or operation, proceed directly to step 5; otherwise, modify and update the multi-conductor transmission line system model after spatial domain discrete reconstruction. Step 5, Time Domain Discretization: In the time domain, the total variation decay (TVD-RK) scheme is used for stepwise integration. Step 6: Update simulation time ; Step 7: Determine if the numerical integration process has terminated: If Then proceed to step 4 to continue the time-domain numerical integration of the high-voltage transmission line system at the next time step; if Then proceed to step 8; where, To set the total simulation time; Step 8: Output the electromagnetic transient numerical simulation results of electrical quantities at various points along the high-voltage transmission line.

2. The method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO format according to claim 1, characterized in that, Step 1 specifically involves: Step 1.1: Establish a set of partial differential equations describing the electromagnetic transient process of high-voltage transmission lines. Based on engineering practice, model the boundary conditions of the high-voltage transmission lines, providing the boundary equations for the beginning and end of the transmission line. Use hyperbolic partial differential equations, i.e., telegraph equations, to describe the multi-conductor transmission line system model composed of three-phase overhead transmission lines or power cable lines. The mathematical expression is: (1) In the formula: This represents the voltage traveling wave column vector on a high-voltage transmission line; i This represents the column vector of the traveling wave of current on a high-voltage transmission line; The distributed resistance matrix represents the parameters of a high-voltage transmission line. The inductance matrix represents the distributed parameters of a high-voltage transmission line. This represents the distributed parameter capacitance matrix of a high-voltage transmission line. The conductance matrix represents the distributed parameters of a high-voltage transmission line. Step 1.2, rewrite equation (1) into the following matrix form: (2) Where: the state variable to be determined High-voltage transmission line system flux System coefficient matrix System coefficient matrix .

3. The method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO format according to claim 2, characterized in that, Step 2 specifically involves: Step 2.1, Define the length as The high-voltage transmission line system, along its spatial layout, connects the high-voltage transmission lines... Divided into equal distances intervals , ,satisfy , , Let be the total length of the high-voltage transmission line. Then, the semi-discrete conserved difference scheme of equation (2) is: (3) In the formula, These are sections within a high-voltage transmission line system. Flux corresponding to the boundary; The space is far from the walking distance; Step 2.2: Use a high-order, inherently non-oscillatory differential FD-WENO scheme to reconstruct the flux values ​​at the boundaries of the interval segment; Step 2.3: Use Lax-Friedrichs numerical flux to analyze the flux of the transmission system. Approximate calculations are performed to achieve spatial discretization of equation (2); Step 2.4: Summarize equation (3) into an initial value problem, expressed as: (4) In the formula, This represents the spatial difference linear operator.

4. The method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO format according to claim 3, characterized in that, Step 2.3 specifically involves: flux It is split into two parts, positive and negative, and the expression is: (9) Among them, positive flux Satisfies monotonically increasing property, negative flux It satisfies the property of monotonically decreasing; (10) In the formula, In equation (2) A It is a Jacobian matrix, take for A Spectral radius; positive flux negative flux All are reconstructed using Equation (5), and the numerical flux is finally obtained from Equation (6); Let the WENO difference operator be a third-order H WENO3 Then there is (11) Then the final interval boundary Numerical flux at Represented as (12) 5. The method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO format according to claim 3, characterized in that, Step 2.3 specifically involves: flux It is split into two parts, positive and negative, and the expression is: (17) Among them, positive flux Satisfies monotonically increasing property, negative flux It satisfies the property of monotonically decreasing; (18) In the formula, In equation (2) A It is a Jacobian matrix, take for A Spectral radius; positive flux negative flux All are reconstructed using Equation (13), and the numerical flux is finally obtained from Equation (14); Let the WENO difference operator be a fifth-order H WENO5 Then we have: (19) Then the final interval boundary Numerical flux at for: (20) 6. The method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO format according to claim 3, characterized in that, Step 3, which initializes the simulation settings, specifically involves setting the initial simulation time. Integration steps n =0, numerical integration with fixed step size h and the total time for electromagnetic transient simulation calculations ; Set the initial values ​​for each state variable in the multi-conductor transmission line system model. ;in, Let be the initial value of the state variable to be determined at the starting point of the high-voltage transmission line.

7. The method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO format according to claim 3, characterized in that, Step 5 specifically involves: The discretization of equation (3) in the time domain is performed using the Total Variation Decay (TVD-RK) scheme for time advancement. A third-order TVD-RK method is used. For the initial value problem equation (4), the calculation expression is: (21) In the formula, h This indicates the step size for gradual time integration. This represents the spatial difference linear operator; the computational format includes a 3rd degree value accumulation process. Depend on State variables at time 1 Calculate the state variables at approximate value at .

8. The method for calculating the electromagnetic transient process of high-voltage transmission lines based on the FD-WENO format according to claim 3, characterized in that, Step 5 specifically involves: The discretization of equation (3) in the time domain is performed using the Total Variation Decay (TVD-RK) scheme for time advancement. A 4-level, 10th-order TVD-RK method is used. For the initial value problem equation (4), the calculation expression is: (22) In the formula, h This indicates the step size for gradual time integration. This represents the spatial difference linear operator; the computational format includes a 10th-order value accumulation process. Depend on State variables at time 1 Calculate the state variables at approximate value at .

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