A method for electromagnetic transient simulation of transmission lines with lossy line extension bergeron model

Abstract

The application relates to a power transmission line electromagnetic transient simulation modeling method of a lossy line extended Bergeron model, which comprises the following steps: obtaining a line to be modeled, equally dividing the line to be modeled into several parts, and connecting lumped resistances in series at two ends of each part of the line, modeling each part of the line by using a Bergeron model, constructing an extended Bergeron model of the whole power transmission line according to each Bergeron model obtained by modeling, correcting model coefficients of the extended Bergeron model by proportionally setting q values, selecting an optimal q value, and obtaining a final power transmission line model. Compared with the prior art, the application has the advantages of higher calculation precision compared with an original extended Bergeron model and higher calculation efficiency compared with a state space model.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic transient simulation modeling technology for transmission lines, and in particular to a method for electromagnetic transient simulation modeling of transmission lines using the extended Bergeron model for lossy lines. Background Technology

[0002] With the rapid development of power systems, transmission lines play a crucial role in both AC and DC systems. The switching operations of these lines involve complex electromagnetic transient processes, and accurate simulation of these processes is essential for power grid planning, design, and safe operation. Electromagnetic transient simulation has become a fundamental tool in modern power system planning and operation, effectively simulating transmission lines. However, the accuracy of the electromagnetic transient (EMT) model of transmission lines significantly impacts the overall accuracy of the power system model; therefore, establishing an accurate time-domain model of transmission line electromagnetic transients is imperative.

[0003] There are many electromagnetic transient simulation models for transmission lines, mainly divided into lumped parameter lines and distributed parameter lines. Four commonly used line models are the constant lumped parameter π-type cascade model, the frequency-dependent distributed parameter JMarti model, the distributed parameter Bergeron model based on lossless lines, and the Dommel model, which approximates the series resistance loss of the line. The π-type cascade model does not consider the distributed characteristics of the transmission line parameters, and according to traditional thinking, this model is considered to have low computational efficiency. The JMarti model is similar to the Bergeron and Dommel models, but for multiphase parameter asymmetric transmission lines, the phase mode transformation matrix is ​​a frequency-dependent complex matrix. The JMarti model uses a simplified real matrix transformation technique, thus slightly reducing the model's accuracy. The Bergeron model is a very accurate transmission line model that fully considers the influence of distributed capacitance and the wave process of voltage and current transmission. It can avoid the influence of capacitive current on differential protection when constructing longitudinal differential protection for lines, but it does not take into account line losses and parameter frequency dependence. The Dommel model, based on the Bergeron model for lossless lines, considers the influence of resistance to a certain extent by embedding lumped resistors at both ends of the line, but it cannot well characterize the distributed characteristics of resistance.

[0004] To address the aforementioned issues, researchers derived an extended Bergeron model for electromagnetic transient calculations of transmission lines, based on a single lossless wire model, which considers resistance losses. However, this extended Bergeron model only concentrates the uniformly distributed resistance along the line to both ends of the line, which can significantly impact the calculation accuracy for long transmission lines. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the existing extended Bergeron model in ensuring the accuracy of calculations for long transmission lines, and to provide a lossy extended Bergeron model for electromagnetic transient simulation modeling of transmission lines.

[0006] The objective of this invention can be achieved through the following technical solutions:

[0007] A method for electromagnetic transient simulation modeling of transmission lines using a lossy extended Bergeron model includes the following steps:

[0008] Obtain the line to be modeled, divide it into several equal parts, and connect the lumped resistance in series at both ends of each section. Model each section using the Bergeron model. Based on the Bergeron model obtained for each section, construct an extended Bergeron model of the entire transmission line. Adjust the model coefficients of the extended Bergeron model by setting a q value proportionally, select the optimal q value, and obtain the final transmission line model.

[0009] Furthermore, the process of constructing the extended Bergeron model of the entire transmission line is as follows:

[0010] Bergeron models of new nodes formed by equally dividing the line to be modeled are constructed, thereby gradually deriving Bergeron models of the end nodes of the line to be modeled. The Bergeron model of the end node is decoupled into two historical current source terms to obtain the extended Bergeron model of the entire transmission line.

[0011] Furthermore, if the line to be modeled is divided into two equal parts, the expression for the Bergeron model of each new node is:

[0012] Node 2:

[0013] Node 3:

[0014] Node 4:

[0015] Node 5:

[0016] The corresponding boundary point equations are:

[0017]

[0018] In the formula, Z0 is the wave impedance of the line; R represents the total resistance of the line; τ represents the time delay of the traveling wave from the beginning 1 to the end 6 of the line; U k(t) represents the voltage at node k at time t, where k is a number from 1 to 6, and represents the node number; I k (t-τ) represents the historical current source of node k at time t-τ, i mn (t) represents the current generated at time t when the reverse or forward traveling wave moves from node m to node n.

[0019] Furthermore, through stepwise simplification and derivation, the expression for the Bergeron model of the terminal node of the line to be modeled is obtained as follows:

[0020]

[0021]

[0022] In the formula, the corrected wave impedance of the two-segment line is...

[0023] Furthermore, The expression for the historical current sources of the Bergeron model of the first and last nodes of the line after decoupling into two historical current source terms at the first and last points of the line is as follows:

[0024]

[0025]

[0026] In the formula, α and β are both coefficients of the modified extended Bergeron model.

[0027] Furthermore, the expression for the modified extended Bergeron model coefficients is as follows:

[0028]

[0029]

[0030] In the formula, q is any non-zero number.

[0031] Furthermore, the optimal q value is determined by using an ATP-EMTP-based segmented insertion model of the line resistance.

[0032] Furthermore, if the circuit to be modeled is divided into two equal parts, the calculation expressions for parameters Z and h are as follows:

[0033]

[0034]

[0035] Furthermore, if the circuit to be modeled is divided into N equal parts, the calculation expressions for parameters Z and h are as follows:

[0036]

[0037]

[0038] Furthermore, the line to be modeled is a single-phase transmission line, but after phase mode transformation, the model is also applicable to three-phase transmission lines.

[0039] Compared with the prior art, the present invention has the following advantages:

[0040] (1) In order to approximate the distributed parameter characteristics of the resistance of the transmission line, this invention adopts the extended Bergeron model combined with the node analysis method to model the electromagnetic transient simulation of the transmission line. After dividing the line into several equal parts, the lumped resistance is connected in series at both ends of each segment of the line. Each segment of the line still adopts the Bergeron model, and the extended Bergeron model of the whole transmission line is derived. The equivalent calculation formula coefficient of the modified extended Bergeron model is proposed. Appropriate calculation parameters can be selected according to the actual calculation results, and the network equivalence requirements are accurately met.

[0041] (2) The modified extended Bergeron model combined with the node analysis method is used to model the electromagnetic transient simulation of the transmission line. The modified extended Bergeron model has higher calculation accuracy than the original extended Bergeron model through typical high-voltage transmission line examples. The calculation model is also applicable when the number of line segments is N, and it has higher calculation efficiency than the state space model of the line. Attached Figure Description

[0042] Figure 1 This is a schematic diagram of a single-phase lossless transmission line provided in an embodiment of the present invention;

[0043] Figure 2 This invention provides an equivalent calculation circuit for a single-phase lossless transmission line in an embodiment of the invention.

[0044] Figure 3 This is a schematic diagram of an equivalent calculation circuit for an extended Bergeron model provided in an embodiment of the present invention, in which regions a, b and c are sequentially progressive.

[0045] Figure 4 This is a schematic diagram of an equivalent calculation circuit for a single-phase lossy transmission line provided in an embodiment of the present invention;

[0046] Figure 5 This is a schematic diagram of a single-phase transmission line with a nonlinear load provided in an embodiment of the present invention;

[0047] Figure 6This is a schematic diagram of an ATP-EMTP simulation calculation model for a lossy transmission line provided in an embodiment of the present invention;

[0048] Figure 7a This is a schematic diagram of the line terminal voltage under two line models, M1 and M2, provided in an embodiment of the present invention;

[0049] Figure 7b This is a schematic diagram showing a partial comparison of line end voltages under two line models, M1 and M2, provided in an embodiment of the present invention.

[0050] Figure 8a This is a schematic diagram of the line head voltage under an M3 line model provided in an embodiment of the present invention;

[0051] Figure 8b This is a schematic diagram showing a partial comparison of the voltage at the beginning of a line under an M3 line model provided in an embodiment of the present invention;

[0052] Figure 9a This is a schematic diagram of the line head voltage under an M3 line model (q = -1.9) provided in an embodiment of the present invention;

[0053] Figure 9b This is a schematic diagram of the line terminal voltage under an M3 line model (q = -1.9) provided in an embodiment of the present invention;

[0054] Figure 10 This is a schematic diagram of the line terminal voltage (N=2, q=-1.9) under two models, M0 and M3, provided in an embodiment of the present invention.

[0055] Figure 11 This is a flowchart illustrating a transmission line modeling method using a lossy line extension Bergeron model provided in an embodiment of the present invention. Detailed Implementation

[0056] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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 embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0057] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0058] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0059] Example 1

[0060] like Figure 11 As shown, this embodiment provides a method for electromagnetic transient simulation modeling of transmission lines using a lossy line extended Bergeron model, including the following steps:

[0061] S1: Obtain the circuit to be modeled, divide the circuit into several equal parts, and connect the lumped resistance in series at both ends of each segment;

[0062] S2: The Bergeron model is used to model each section of the line;

[0063] S3: Based on the Bergeron model obtained from each segment of the model, construct an extended Bergeron model of the entire transmission line. The model coefficients of this extended Bergeron model are modified by setting a q value proportionally. The optimal q value is selected to obtain the final electromagnetic transient simulation model of the transmission line, which is the modified extended Bergeron model.

[0064] The process of constructing the extended Bergeron model for overall transmission line correction is as follows:

[0065] Bergeron models of the new nodes formed by dividing the line to be modeled into equal parts are constructed, thereby deriving the Bergeron model of the end node of the line to be modeled. The Bergeron model of the end node is decoupled into two historical current source terms, resulting in a modified extended Bergeron model of the entire transmission line.

[0066] The above scheme will be described in detail below.

[0067] 1. Construction of the Bergeron model for lossless transmission lines

[0068] Neglecting line losses, a single-phase distributed parameter transmission line can be described by the following partial differential equation:

[0069]

[0070] In equation (1), t represents the time variable; x is the spatial distance; U(x,t) and i(x,t) are one-dimensional voltage and current traveling wave functions with respect to time t and spatial distance x; L0 and C0 are the inductance and capacitance per unit length of the line, and both are constants, neglecting the influence of frequency.

[0071] Solving equation (1) using the method of characteristics of the wave equation, we can obtain:

[0072]

[0073] In equation (2) above, These represent one-dimensional forward and reverse voltage or current traveling waves with respect to time t and spatial distance x, respectively. This indicates the propagation speed of a traveling wave along a line; voltage traveling waves and current traveling waves are transmitted through the wave impedance. They are mutually convertible.

[0074] like Figure 1 As shown, based on the characteristic line equation of the wave equation, assuming the anti-traveling wave moves from the end m of the line to the beginning k, we can obtain:

[0075]

[0076] Similarly, the traveling wave moving from the beginning k to the end m of the line can be obtained as follows:

[0077]

[0078] Combining equations (3) and (4) yields the following result: Figure 2 The Bergeron model of the lossless transmission line is shown.

[0079] 2. Construction of the Bergeron model after segmentation

[0080] To approximate the distributed parameter characteristics of transmission line resistance, the extended Bergeron model is used by dividing the line into several equal parts and connecting the lumped resistance in series at both ends of each section, while each section still uses the Bergeron model.

[0081] As attached Figure 3 As shown, the line is divided into two equal segments, Z0 is the wave impedance of the line; R represents the total resistance of the line; τ represents the time delay of the traveling wave propagating from the first end 1 to the last end 6. The calculation formulas using the Bergeron model for lines 23 and 45 are as follows:

[0082] Node 2:

[0083] Node 3:

[0084] Node 4:

[0085] Node 5:

[0086] The boundary point equations are:

[0087]

[0088] Based on the above system of equations, we can derive:

[0089]

[0090]

[0091] In the above formula, If equation (11) is used directly for electromagnetic transient calculation, it will cause numerical instability. Therefore, equation (11) is decoupled into two historical current source terms at the beginning and end of the line:

[0092]

[0093]

[0094] In the above formula, According to equation (10-13), the following can be obtained: Figure 4 An extended Bergeron model of lossy single-phase transmission lines.

[0095] However, adding equation (13) to equation (12) yields:

[0096]

[0097] Equation (14) is only completely equivalent to equation (11) after the line is infinitely segmented and h→1, which is obviously unnecessary in practical applications. Therefore, the following modified extended Bergeron model coefficients are proposed in this paper:

[0098]

[0099] Where q can take any non-zero number, the optimal q value is determined by using the segmented insertion model of the line resistance built by ATP-EMTP. It is not difficult to verify that the modified extended Bergeron model completely and accurately satisfies equation (11), thus the calculation accuracy is higher. It is worth mentioning that when the number of line segments is N, the above formula is also applicable. At this time, Z and h in equations (12) and (13) become:

[0100]

[0101] Specific implementation process:

[0102] With attachment Figure 5Taking a single-phase transmission line with nonlinear loads as an example, the system parameters for the example are shown in Appendix Table 1. To demonstrate the superiority of the extended Bergeron model for lossy transmission line correction proposed in this invention, the calculation results of the state-space model (denoted as M0) will be compared with the numerical calculation results of the extended Bergeron model (denoted as M3) proposed in this paper. The original extended Bergeron model is denoted as M2. Furthermore, to increase persuasiveness, as shown in Appendix Table 1... Figure 6 The image shows a lossless distributed parameter line model (Bergeron model, denoted as M1) for a two-segment line in ATP-EMTP. The line resistance is achieved by segmenting the line using lumped resistance, and the total line resistance is shown. The nonlinear inductance is modeled using the Type93 component from the ATP-EMTP library.

[0103] Appendix 1 System Parameter Table

[0104] <![CDATA[R S ]]> 2Ω <![CDATA[L S ]]> 0.06H <![CDATA[R0]]> 0.07Ω / km <![CDATA[L0]]> <![CDATA[2.08×10 -3 H / km]]> <![CDATA[C0]]> <![CDATA[12×10 -9 F / km]]> <![CDATA[R L ]]> 96Ω a <![CDATA[8.40×10 2 V·s]]> b <![CDATA[5.95×10 -3 A -1 ]]>

[0105] The specific steps for comparing each circuit model are as follows:

[0106] I. Example of State-Space Model M0

[0107] 1) Establish a state-space model. A discrete-time model of the line space is used to analyze the adjacent... Figure 5 Electromagnetic transient simulation modeling was performed on the transmission line, with the number of discrete intervals M = 30. The state-space model of the line is as follows:

[0108]

[0109] After adding the line boundary equations to the spatial discrete model of the line, the new equation set contains 2M+3 unsolved state variables. The corrected matrix form is as follows:

[0110]

[0111] in,

[0112]

[0113]

[0114]

[0115]

[0116] When using the explicit Euler method to assign initial values ​​for iteration, and when using Newton's iteration, it is necessary to calculate the Jacobian matrix J. This is done on the i... L When taking the partial derivative, there exists a nonlinear inductance L. L Its relationship with state variable i LRelated, that is

[0117]

[0118] For the state variable i of the right-hand function in the above equation L The partial derivatives are as follows:

[0119]

[0120] Then the Jacobian matrix in Newton's iteration is:

[0121]

[0122]

[0123] II. Establishing the modified extended Bergeron model

[0124] Divide the line into two equal segments, where Z0 is the wave impedance of the line; R represents the total resistance of the line; and τ represents the time delay of the traveling wave propagating from the first end 1 to the last end 6. The calculation formulas using the Bergeron model for lines 23 and 45 are as follows:

[0125] Node 2:

[0126] Node 3:

[0127] Node 4:

[0128] Node 5:

[0129] The boundary point equations are:

[0130]

[0131] Based on the above system of equations, we can derive:

[0132]

[0133]

[0134] In the above formula,

[0135] When the number of line segments is N, the above formula also applies. In this case, Z and h in equations (26) and (27) become:

[0136]

[0137] Decouple equation (27) into two historical current source terms at the beginning and end of the line:

[0138]

[0139]

[0140] According to equations (26-29), the following can be obtained: Figure 4 The extended Bergeron model of the lossy single-phase transmission line shown in this invention has the following coefficients:

[0141]

[0142] The optimal q value was determined by constructing a segmented insertion model of the line resistance using ATP-EMTP.

[0143] III. Simulation Calculation

[0144] In the simulation, the simulation step size was uniformly set to 1μs. The electromagnetic transient simulation model of the line using models M0 to M3 was written using MATLAB software. In the M0 model, the convergence accuracy of Newton's iteration was ε = 1e-6.

[0145] This calculation example primarily outputs the voltage waveforms at the beginning and end of a transmission line under different models. (Attached) Figure 7a and 7b Line terminal voltages under two line models, M1 and M2; Appendix Figure 8a and 8b Line head voltage under the M3 line model; Appendix Figure 9a and 9b Line voltage under the M3 line model (q = -1.9); Appendix Figure 10 Line terminal voltage under models M0 and M3 (N=2, q=-1.9).

[0146] As shown in Figure 7, icons M1 and M2 represent the line terminal voltages calculated using models M1 and M2, respectively, and icons N6 and N7 represent the line terminal voltages calculated using the ATP-EMTP model, respectively. Figure 6 The voltages at nodes 6 and 7 are shown in Figure 7. As can be seen from Figure 7, the calculation results of model M1 have a large error, and the voltage at the end of the line calculated by model M2 also differs significantly from the voltage at node 7 in the ATP-EMTP model.

[0147] As shown in Figure 8, in the M3 model, the number of line segments N = 2. By changing the parameter q, it was found that when q = -1.9 and -2.2, the improved model proposed in this paper basically matches the ATP-EMTP model, while the original extended Bergeron model M2 differs significantly from the calculation results of the ATP-EMTP model. This proves that as long as the parameter q is chosen appropriately, the improved model proposed in this paper has higher simulation accuracy than the original extended Bergeron model. Figure 9 shows the comparison results of the voltage at the beginning and end of the line calculated by the improved model M3 (q = -1.9) and the ATP-EMTP model. It can be seen from Figure 9 that from the transient to the steady-state process, the calculation results of the M3 model are consistent with those of the ATP-EMTP model.

[0148] Appendix Figure 10 This is a comparison of simulation results using state-space model M0 and model M3 (with N line segments) proposed in this paper. It can be seen that the transient results calculated by the two models differ significantly in detail, but their change trajectories are basically consistent. Using the line state-space model M0 involves Newton iterations and has many state variables, making the calculation process more cumbersome. However, its advantage is that it can calculate the voltage at any point along the line. In practice, a suitable line model should be selected according to the needs. The calculation results of M0 can also be used as a reference for finding the optimal parameter q.

[0149] In summary, addressing the issue of low accuracy in electromagnetic transient simulation of the original extended Bergeron model for lossy long-distance transmission lines, this invention modifies the coefficients of the equivalent calculation formula for the extended Bergeron model, proposing new extended Bergeron model coefficients. The parameters of the modified extended Bergeron equivalent calculation formula can be selected with appropriate values ​​based on actual calculation results. The modified extended Bergeron model equivalent calculation formula is combined with nodal analysis to model and simulate the electromagnetic transients of transmission lines. A typical high-voltage transmission line example verifies that the modified extended Bergeron model has higher calculation accuracy than the original extended Bergeron model, and it is also applicable when the number of line segments is N, exhibiting higher computational efficiency compared to the state-space model of the line.

[0150] The preferred embodiments of the present invention have been described in detail above. It should be understood that those skilled in the art can make numerous modifications and variations based on the concept of the present invention without creative effort. Therefore, all technical solutions that can be obtained by those skilled in the art based on the concept of the present invention through logical analysis, reasoning, or limited experimentation on the basis of existing technology should be within the scope of protection defined by the claims.

Claims

1. A method for electromagnetic transient simulation modeling of transmission lines using a lossy extended Bergeron model, characterized in that, Includes the following steps: Obtain the line to be modeled, divide it into several equal parts, and connect the lumped resistance in series at both ends of each segment. Model each segment using the Bergeron model. Based on the Bergeron models of each segment, construct an extended Bergeron model of the entire transmission line. The model coefficients of this extended Bergeron model are set proportionally. q The value is adjusted, and the optimal one is selected. q The values ​​are used to obtain the final transmission line model; The process of constructing the extended Bergeron model of the entire transmission line is as follows: Bergeron models of new nodes formed by dividing the line to be modeled into equal parts are constructed, thereby deriving Bergeron models of the end nodes of the line to be modeled. The Bergeron model of the end node is decoupled into two historical current source terms, resulting in the extended Bergeron model of the entire transmission line, i.e., the modified extended Bergeron model. Will The Bergeron model expression for the end node of the line to be modeled after decoupling to two historical current source terms at the beginning and end points of the line is: In the formula, and All are corrected extended Bergeron model coefficients; The expression for the modified extended Bergeron model coefficients is as follows: In the formula, It can be any non-zero number.

2. The method for modeling transmission lines using a lossy line extension Bergeron model according to claim 1, characterized in that, If the line to be modeled is divided into two equal parts, the expression for the Bergeron model of each new node is: The corresponding boundary point equations are: In the formula, , It is the wave impedance of the line; Indicates the total resistance of the circuit; This represents the time delay of the traveling wave propagating from the beginning 1 of the line to the end 6. for k Node at t Voltage at time, k The numbers are 1-6, representing the node sequence number. for k Node at Time-based historical current source For reverse traveling wave or forward traveling wave from node m Move to node n exist t The current generated at any given moment.

3. The method for electromagnetic transient simulation modeling of transmission lines using a lossy line extended Bergeron model according to claim 2, characterized in that, The Bergeron model expression for the terminal node of the line to be modeled is derived as follows: In the formula, the corrected wave impedance of the two-segment line is... , .

4. The method for electromagnetic transient simulation modeling of transmission lines using a lossy line extended Bergeron model according to claim 1, characterized in that, The q The optimal value was determined by constructing a piecewise interpolation model of the line resistance using ATP-EMTP. q value.

5. The method for electromagnetic transient simulation modeling of transmission lines using a lossy line extension Bergeron model according to claim 3, characterized in that, If the line to be modeled is divided into two equal parts, then the parameters and parameters The calculation expression is: 。 6. The method for electromagnetic transient simulation modeling of transmission lines using a lossy line extended Bergeron model according to claim 3, characterized in that, If the circuit to be modeled is divided into equal parts N Next, then the parameter and parameters The calculation expression is: 。 7. The method for electromagnetic transient simulation modeling of transmission lines using a lossy line extended Bergeron model according to claim 1, characterized in that, The line to be modeled is a single-phase transmission line.

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