Method for establishing small-signal mathematical model of train network system during dynamic change of train operation

Abstract

The invention discloses a method for establishing a small-signal mathematical model of a train network system during dynamic change of train operation, which comprises the following steps of: firstly, dividing the system into four sub-modules, namely a train, a traction network, a traction transformer and a power grid, and dividing an uplink traction network and a downlink traction network into n + 2 sections according to the maximum number n of trains operating on a power supply arm, the traction network is formed by cascading the impedance of each section of equivalent line of uplink and downlink, then establishing a small-signal mathematical model of each sub-module, and then forming a system small-signal mathematical model when the number of trains capable of running on a power supply arm is maximum through a relational expression between the voltage of each node and the current of each branch of the network according to the actual topology of the system. When the train operation number changes, state variables related to the trains and coefficient matrix elements related to the state variables are deleted, and when the train operation position changes, the specific value of the impedance of each section of the traction network is changed, so that dynamic modeling of the train network coupling system can be realized. According to the invention, the stability of the vehicle network system can be evaluated online.

Description

Technical Field

[0001] The present invention belongs to the field of broadband oscillation analysis of the coupling system between electrified railway trains and the traction network, and particularly relates to a method for dynamically establishing a small-signal mathematical model of the vehicle-network coupling system when the train operation changes dynamically. Background Art

[0002] In recent years, the electrified railway in China has developed rapidly. Broadband oscillations caused by the interaction between the power electronic converters of AC-DC-AC trains and the traction power supply system occur frequently. This problem belongs to the stability problem of the power electronic converter grid-connected system. The impedance analysis method based on the frequency-domain impedance model and the eigenvalue analysis method based on the state-space equation are widely used to analyze the broadband oscillation of the power electronic grid-connected system. After the train is connected to the traction power supply system, due to the different numbers and positions of its operation, the system will form different network topologies. However, in the modeling research of the vehicle-network coupling system, the current situation mostly considers the case where multiple trains are grid-connected at the same point of the traction network, without considering the modeling of the network topology change caused by the dynamic changes in the number of trains and the access positions of each train in actual operation. Therefore, the existing modeling methods cannot be used to realize the dynamic modeling of the vehicle-network coupling system and the online evaluation of its stability. Summary of the Invention

[0003] The present invention aims to provide a small-signal analysis mathematical model that can dynamically change according to the number and position of trains.

[0004] A method for establishing a small-signal mathematical model of a vehicle-network system when the train operation changes dynamically according to the present invention includes the following steps:

[0005] Step 1: Collect the original data of the model.

[0006] According to the short-circuit parameters of the power system and the transformer, the types and sizes of each conductor of the traction network, and the train operation diagram, obtain the original data required for modeling; including the equivalent inductance L of the power grid S , the transformation ratio k of the traction transformer 2 , the winding resistance and leakage inductance R of the traction transformer referred to the secondary side T and L T , the unit resistance r of the traction network q and inductance L q , the length l of the power supply arm, the maximum number n of trains running on the power supply arm, and the lengths of each section of the traction network divided.

[0007] Step 2: Establish a modular small-signal mathematical model.

[0008] Divide the system into train, traction network, traction transformer, and power grid modules, and establish the linearized state equations of each sub-module according to Kirchhoff's theorem.

[0009] Step 3: In order to obtain the relationship between the voltage at each node of the network and the branch current, a virtual resistor R is introduced between each voltage node and the ground. N , write the relationship between the voltage disturbance Δu of each node in the network and the branch current disturbance Δi, and then use the relationship to combine the small signal state equations of each module to obtain the small signal mathematical model of the entire system when the maximum number of trains is n.

[0010] Step 4: According to the real-time train operation diagram, the changes in the number and position of trains on the power supply arm are obtained, the matrix elements of the state variable disturbance and coefficient matrix obtained in step 3 are deleted, and the impedance values ​​of each section of the traction network are changed to obtain the small signal mathematical model of the train-grid system based on the real-time number and position of trains.

[0011] Furthermore, the process of establishing the small signal mathematical model of each submodule in step 2 is specifically as follows:

[0012] (1) Small signal mathematical model of train module:

[0013] According to the main circuit and control circuit structure of the train, the differential equation of each dynamic element is established and linearized:

[0014]

[0015] In the formula, Δx tk is the state variable disturbance of the main circuit and control circuit of each train, A tk , B tk , C tk is the coefficient matrix of the linearized state space model of the train, k 1 is the train's onboard transformer ratio, u k is the pantograph voltage of the locomotive at the network node k, i n,k is the AC side current of the train rectifier at node k, k = 1, 2, 3…n.

[0016] (2) Small signal mathematical model of traction network module:

[0017] The traction network is divided according to the maximum number of running trains n and their positions, where the uplink n u Taiwan, downlink nn u The uplink and downlink traction networks are divided into n+2 sections, and the uplink is divided into n u +1 segment, the descending is divided into nn u +1 section, the length of each section of the traction network is l 1 , l 2 …l n+2; According to Kirchhoff's voltage law (KVL), differential equations for the current in each section of the traction network are established. In the order from left to right and first for the up-line and then for the down-line, the differential equations for the current in each section of the traction network are successively combined to obtain the linearized state equation of the traction network module as follows:

[0018]

[0019] Taking the case where there are only two traction substations in the system as an example, Δi q , Δu, A q1 and B q1 are expressed as follows: If the power supply mode of the system is single-sided power supply, the unit resistance r q of the traction network at the position of the up-down sectionalizing post is set to infinity. If the system is double-sided power supply, the unit resistance r q of the entire section of the traction network is a normal value.

[0020]

[0021]

[0022] In the formula, I (n+2)×(n+2) is an (n + 2)×(n + 2) order identity matrix, and Δi q is a matrix of disturbance quantities of the impedance current in n + 2 sections of the traction network, formed successively according to the order from left to right and first for the up-line and then for the down-line; Δu is the disturbance quantity of the node voltage of the traction network. Excluding the node voltages at the head and tail ends of the traction network, it is a column vector formed successively by the disturbance quantities of the node voltages at the head and tail ends of the traction network and the disturbance quantities of the three-phase node voltages of the power grid in the order from left to right and first for the up-line and then for the down-line; A q1 and B q1 are the coefficient matrices of the linearized state equation. According to the topology of the traction network, A q1 is an identity matrix related to the unit traction network impedance, and each row of B q1 has only two non-zero elements. The two non-zero elements are opposite to each other and their absolute values are the reciprocal of the length of each section of the traction network. Their positive and negative signs are related to the positions of each section of the traction network. The 0 q1 in B i×j represents an i×j order zero matrix.

[0023] (3) Small-signal mathematical model of the traction transformer module:

[0024] According to KVL, differential equations for the current on the secondary side of each traction transformer are established to obtain the linearized state equation composed of the currents on the secondary side of each traction transformer as state variables:

[0025]

[0026] Δi T = [Δi T1 … ΔiTm T

[0027]

[0028] B T =[[0 m×n B 12 B 13

[0029]

[0030] In the formula, I m×m is an m-order identity matrix, 0 m×n is an m×n-order zero matrix, Δi T is a perturbation matrix composed of the secondary-side currents of m traction transformers in sequence from left to right, A T , B T are coefficient matrices of the small-signal differential equations of the secondary-side currents of the traction transformers, and the values of the non-zero elements are only related to the transformation ratio, winding resistance, and leakage inductance of the traction transformers.

[0031] (4) Small-signal mathematical model of the power grid module:

[0032] According to KVL, establish the current differential equation of the equivalent impedance of the high-voltage side three-phase power grid and perform small-signal linearization to obtain the specific small-signal mathematical model of the three-phase power grid module as follows:

[0033]

[0034] Δi G =[Δi A Δi B Δi C T

[0035] B S =[0 3×(n+m) G 12

[0036]

[0037] In the formula, Δi G is the state variable composed of the perturbation amounts of the impedance currents of the three phases A, B, and C of the power grid in sequence, B S is the coefficient matrix, and the columns where the non-zero elements of each row of B S are located are only related to the rows where the voltage nodes of the three phases A, B, and C of the power grid are located in Δu; 0 3×(n+m) is a 3×(n + m)-order zero matrix.

[0038] Furthermore, step 3 is specifically as follows:

[0039] ​​​​According to Kirchhoff's current theorem KCL, the current flowing through R at each node is obtained N The current in the branch and the voltage at each node are equal to R N With the flow through R N The product of the current on the N L CON Δi;

[0040]

[0041] Where Δi is the current disturbance on the AC side of the n train rectifiers: Δi n,1 ,Δi n,2 ,…Δi n,n , current disturbance of each traction network section Δi q , traction transformer secondary side current disturbance Δi T And the three-phase current disturbance Δi of the power grid A, B, and C G The following is a column vector composed of two traction transformers in the system. CON matrix:

[0042]

[0043] According to the actual system topology, the small signal mathematical models of each submodule are combined, that is, the linearized state equations of the train, traction network, traction transformer and power grid submodules obtained in step 2, and Δu = R N L CON Δi is combined to obtain the small signal mathematical model of the system with the maximum number of running trains n:

[0044]

[0045] A total =diag(A t1 ,A t2 …A tn ,A q1 ,A T ,0 3×3 );

[0046] C total =diag(C t1 ,C t2 ,…C tn ,I (n+2)×(n+2) ,I m×m ,I 3×3 );

[0047] Where I is the unit matrix, Δx totalIt is composed of the disturbance quantity matrices of the state variables of each sub-module of the system when the maximum number of trains is n, and is composed of the disturbance quantities Δx of the state variables of each train when the maximum number of trains is n t1 , Δx t2 , …Δx tn , the disturbance quantity Δi of the impedance current of each section of the traction network q , the disturbance quantity Δi of the secondary side current of the traction transformer T and the disturbance quantity Δi of the three-phase line current of the power grid G are column vectors formed in sequence, and R N is a relatively large impedance, which is set to 100000 Ω; according to the arrangement order of the state variables of the sub-modules, A total is a diagonal matrix composed of the coefficient matrices A of the linearized state equations of the train, traction network, and traction transformer sub-modules in sequence, and B total is composed of the coefficient matrices B of the linearized state equations of the train, traction network, traction transformer, and three-phase power grid sub-modules in sequence, where 0 j×4 is a j×4 zero matrix, and j is equal to the number of state variables of n trains. C total is composed of the coefficient matrices C of the linearized state equations of each train sub-module: C t1 , C t2 …C tn and three identity matrices with different orders are formed into a diagonal matrix in sequence. The orders of the three identity matrices are determined by the number of sections of the traction network, the number of traction transformers, and the number of phases of the three-phase power grid respectively.

[0048] Furthermore, step 4 is specifically as follows:

[0049] Step 4.1: According to the train position situation, change the traction network parameters, that is, change the specific values of l 1 , l 2 …l n+2 .

[0050] Step 4.2: Update the state variable Δx: According to the reduction of the number of trains, sequentially delete the column matrix elements of the corresponding train state variables in Δx total ; if the total number of down trains is reduced by i, i ≤ n - n u trains, then Δx total sequentially deletes the matrix elements Δx t(n+1-i) , when the total number of up trains is reduced by i, i ≤ n u trains, then Δx total sequentially deletes the matrix elements

[0051] Step 4.3: Update the coefficient matrix A′ sys : According to the reduction of the number of down and up trains, sequentially delete A total , B total , Ctotal and L CON and the coefficient matrix elements of the corresponding train state variables in L; for example, when the number of down trains decreases by i in total, i ≤ (n - n u ), then A total The matrix elements to be deleted in sequence are the rows and columns corresponding to A t(n+1-i) , B total The matrix elements to be deleted in sequence are the rows corresponding to total The matrix elements to be deleted in sequence are the rows and columns corresponding to C t(n+1-i) , L CON Delete the (n + 1 - i)-th column in sequence; when the number of up trains decreases by i in total, i ≤ n u trains, A total The matrix elements to be deleted in sequence are the rows and columns corresponding to total The matrix elements to be deleted in sequence are the rows corresponding to total The matrix elements to be deleted in sequence are the rows and columns corresponding to CON Delete the n u + 1 - i-th column.

[0052] The beneficial technical effects of the present invention are as follows:

[0053] 1. The present invention takes into account the changes in the number of trains connected to the traction network and the real-time positions of the trains, establishes a small-signal analysis mathematical model for multiple trains connected to the traction network at different positions in the abc stationary coordinate system, and verifies the accuracy of the model on the Matlab / Simulink simulation platform. Based on this model, the eigenvalue method can be used to realize the online evaluation of the stability of the vehicle-network coupling system.

[0054] 2. Compared with the existing model, the breakthrough and advantage of the present invention are: when the number and position of the connected trains change in real time, there is no need to re-derive the state equation of the vehicle-network system. Only by deleting the corresponding position matrix elements and changing the traction network parameter values on the system model with the maximum number of trains n connected to the grid, the calculation process is simple and time-consuming, and based on the train operation diagram, the dynamic modeling of the vehicle-network coupling system can be realized, and the stability of the vehicle-network coupling system can be evaluated online. Description of the Drawings

[0055] Figure 1 It is the equivalent circuit diagram of the main circuit of the train.

[0056] Figure 2 It is the equivalent circuit diagram of the traction network division considering two substations and two adjacent power supply arms in the system.

[0057] Figure 3It is the equivalent circuit diagram of the traction transformer module.

[0058] Figure 4 It is the equivalent circuit of the power grid module.

[0059] Figure 5 It is the system topology diagram of the system with only two traction transformers and their adjacent power supply arms in the system.

[0060] Figure 6 It is the system topology structure diagram of the embodiment.

[0061] Figure 7 It is the characteristic root locus diagram of the system in the embodiment.

[0062] Figure 8 It is the MATLAB / Simulink simulation waveform of the system in the embodiment. Detailed implementation manners

[0063] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0064] A method for establishing a small-signal mathematical model of a vehicle-grid system when the train operation changes dynamically according to the present invention considers that when the maximum number of trains is n, a small-signal mathematical model of four sub-modules including trains, traction networks, traction transformers, and power grids is established. When the number of operating trains changes, the state variables related to the trains and the elements of the coefficient matrix related to the state variables are deleted. When the train operation position changes, the specific values of the impedances of each section of the traction network are changed, and dynamic modeling of the vehicle-grid coupling system can be realized. Based on this model, the stability problem of the vehicle-grid system can be evaluated online by using the characteristic root method.

[0065] A method for establishing a small-signal mathematical model of a vehicle-grid system when the train operation changes dynamically according to the present invention is specifically as follows:

[0066] Step 1: Collect the original data of the model.

[0067] According to the short-circuit parameters of the power system and transformers, the types and sizes of each conductor of the traction network, and the train operation diagram, the original data required for modeling is obtained; including the equivalent inductance L of the power grid S , the transformation ratio k of the traction transformer 2 , the winding resistance and leakage inductance R of the traction transformer referred to the secondary side T and L T , the unit resistance r of the traction network q and inductance L q , the length l of the power supply arm, the maximum number n of trains running on the power supply arm, and the lengths of each section of the traction network divided.

[0068] Step 2: Establish a modular small-signal mathematical model.

[0069] The system is divided into train, traction network, traction transformer and power grid modules, and the linearized state equations of each sub-module are established according to Kirchhoff's theorem.

[0070] (1) Small-signal mathematical model of the train module:

[0071] The equivalent circuit diagram of the main circuit of the train is as Figure 1 shown. The control circuits of trains of different models may be different, which will not be elaborated here. The small-signal mathematical models of the main circuit and control circuit of the train are:

[0072]

[0073] In the formula, Δx tk is the disturbance quantity of the state variable composed of the main circuit and control circuit of each train. A tk , B tk , C tk are the coefficient matrices of the train linearized state space model. k 1 is the turns ratio of the on-board transformer of the train. u k is the pantograph voltage of the locomotive at network node k. i n,k is the AC side current of the train rectifier at node k, where k = 1, 2, 3... n.

[0074] (2) Small-signal mathematical model of the traction network module:

[0075] According to the maximum number of operating trains n (n u trains going uphill, n - n u trains going downhill) and their positions, the traction network is segmented. The total traction network for uphill and downhill is divided into n + 2 segments, with n u +1 segments for uphill and n - n u +1 segments for downhill. The lengths of each segment of the traction network are l 1 , l 2 ... l n+2 respectively. According to Kirchhoff's voltage law (KVL), the differential equations of the current of each segment of the traction network are established. In the order from left to right and first uphill then downhill, the differential equations of the current of each segment of the traction network are merged in sequence to obtain the linearized state equation of the traction network module as:

[0076]

[0077] Taking the case where there are only two traction substations in the system as an example, the equivalent circuit diagram of the traction network division of two adjacent power supply arms is as Figure 2 shown. Δi q , Δu, A q1 and B q1The expression is specifically as follows. It should be noted that if the system power supply mode is single-sided power supply, the unit resistance r of the traction network at the positions of the upper and lower line sections is set to infinity respectively. If the system is double-sided power supply, the unit resistance r of the entire traction network section is the normal value. q Let it be infinity. If the system is double-sided power supply, the unit resistance r of the entire traction network section q is the normal value.

[0078]

[0079]

[0080] In the formula, I (n+2)×(n+2) is an (n + 2)×(n + 2) order identity matrix, and Δi q is the perturbation matrix of the impedance current of the traction network with n + 2 sections; it is composed in sequence from left to right, first for the upper line and then for the lower line. Δu is the column vector composed of the perturbation of the node voltage of the traction network (excluding the node voltages at the beginning and end of the traction network, in the order from left to right, first for the upper line and then for the lower line), the perturbation of the node voltages at the beginning and end of the traction network, and the perturbation of the three-phase node voltages of the power grid in sequence; A q1 and B q1 are the coefficient matrices of the linearized state equation. According to the topology of the traction network, A q1 is the identity matrix related to the unit traction network impedance, and B q1 has only two non-zero elements in each row. The two non-zero elements are opposite to each other and the absolute value is the reciprocal of the length of each traction network section. Their positive and negative are related to the positions of each traction network section. The 0 in B q1 represents the i×j order zero matrix. i×j represents the i×j order zero matrix.

[0081] (3) Small-signal mathematical model of the traction transformer module:

[0082] Since there is a phase separation at the traction substation in the traction network, when only considering one power supply arm of the traction substation, the V / v traction transformer can only consider one single-phase winding. Therefore, the equivalent circuit of the traction transformer module is as Figure 3 shown. According to KVL, the differential equation of the secondary side current of each traction transformer is established, and the linearized state equation composed of the secondary side currents of each traction transformer as state variables is obtained:

[0083]

[0084] Δi T = [Δi T1 … Δi Tm T

[0085]

[0086] B T = [0​m×n B 12 B 13

[0087]

[0088] In the formula, I m×m is the m-order identity matrix, 0 m×n is the m×n-order zero matrix, Δi T is the disturbance quantity matrix composed of the secondary-side currents of m traction transformers in sequence from left to right, A T , B T are the coefficient matrices of the small-signal differential equations of the secondary-side currents of the traction transformers, and the values of the non-zero elements are only related to the turns ratio, winding resistance, and leakage inductance of the traction transformers.

[0089] (4) Small-signal mathematical model of the power grid module:

[0090] The equivalent circuit of the power grid module is as Figure 4 shown. According to KVL, the current differential equation of the equivalent impedance of the high-voltage side three-phase power grid is established and small-signal linearized to obtain the specific small-signal mathematical model of the three-phase power grid module as follows:

[0091]

[0092] Δi G = [Δi A Δi B Δi C T

[0093] B S = [0 3×(n+m) G 12

[0094]

[0095] In the formula, Δi G is the state variable composed of the disturbance quantities of the impedance currents of the three phases A, B, and C of the power grid in sequence, B S is the coefficient matrix, and the columns where the non-zero elements of each row of B S are located are only related to the rows where the voltage nodes of the three phases A, B, and C of the power grid are located in Δu; 0 3×(n+m) is the 3×(n + m)-order zero matrix.

[0096] Step 3: In order to obtain the relationship between the voltages of each node in the network and the branch currents, a virtual resistor R N (R N ​​​can be infinite), write down the relationship between the voltage disturbance Δu of each node in the network and the branch current disturbance Δi, and then combine the small signal state equations of each module according to the relationship to obtain the small signal mathematical model of the entire system when the maximum number of trains is n.

[0097] According to Kirchhoff's current theorem (KCL), the current flowing through R at each node is obtained N The current in the branch and the voltage at each node are equal to R N With the flow through R N The product of the current on the upper circuit is the relationship between the voltage disturbance Δu of each node in the network and the branch current disturbance Δi: Δu=R N L CON Δi;

[0098]

[0099] Where Δi is the current disturbance on the AC side of the n train rectifiers: Δi n,1 ,Δi n,2 ,…Δi n,n , current disturbance of each traction network section Δi q , traction transformer secondary side current disturbance Δi T And the three-phase current disturbance Δi of the power grid A, B, and C G The following is a column vector composed of two traction transformers in the system. CON The matrix, its equivalent circuit diagram is as follows Figure 5 shown.

[0100]

[0101] According to the actual system topology, the small signal mathematical models of each submodule are combined, that is, the linearized state equations of the train, traction network, traction transformer and power grid submodules obtained in step 2, and Δu = R N L CON Δi is combined to obtain the small signal mathematical model of the system with the maximum number of running trains n:

[0102]

[0103] A total =diag(A t1 ,A t2 …A tn ,A q1 ,A T ,0 3×3 );

[0104] C total =diag(Ct1 , C t2 , … C tn , I (n+2)×(n+2) , I m×m , I 3×3 );

[0105] Wherein, I is the identity matrix, and Δx total is composed of the disturbance amount matrices of the state variables of each sub-module of the system at the maximum train number n, and is composed of the disturbance amounts of the state variables Δx t1 , Δx t2 , … Δx tn of each train, the disturbance amount of the impedance current of each section of the traction network Δi q , the disturbance amount of the secondary side current of the traction transformer Δi T and the disturbance amount of the three-phase line current of the power grid Δi G forming a column vector in sequence, and R N is a relatively large impedance, which is set to 100000 Ω; according to the arrangement order of the sub-module state variables, A total is a diagonal matrix composed of the A coefficient matrices of the linearized state equations of the train, traction network, and traction transformer sub-modules in sequence, and B total is composed of the B coefficient matrices of the linearized state equations of the train, traction network, traction transformer, and three-phase power grid sub-modules in sequence, where 0 j×4 is a j×4 zero matrix, and j is equal to the number of state variables of n trains, and C total is composed of the C coefficient matrices of the linearized state equations of each train sub-module: C t1 , C t2 … C tn and three identity matrices with different orders forming a diagonal matrix in sequence, and the orders of the three identity matrices are determined by the number of sections of the traction network, the number of traction transformers, and the number of phases of the three-phase power grid respectively.

[0106] Step 4: According to the real-time train operation diagram, obtain the changes in the number and position of trains on the power supply arm, and perform matrix element deletion on the disturbance amount of the state variable Δx total and the coefficient matrices A total , B total , C total , L CON obtained in step three, and change the impedance values of each section of the traction network, and the small-signal mathematical model of the vehicle-grid system based on the real-time operation number and position of the train can be obtained:

[0107] Step 4.1: When the train position changes, change the traction network parameters, that is, according to the train position, change the traction network parameters, that is, change the specific values of l 1 , l 2 … l n+2 .

[0108] Step 4.2: When the number of trains changes, matrix elements of the state variables and coefficient matrices in the small-signal mathematical model obtained in Step 3 are deleted. The processes of traction network parameter change and matrix element deletion are as follows:

[0109] (1) Update the state variable Δx: According to the reduction of the number of trains, the column matrix elements corresponding to the train state variables in Δx are deleted in sequence. For example, if the number of down trains is reduced by a total of i (i ≤ n - n total ) trains, then Δx u deletes the matrix elements Δx total in sequence. When the number of up trains is reduced by a total of i (i ≤ n t(n+1-i) ) trains, then Δx u deletes the matrix elements total in sequence

[0110] (2) Update the coefficient matrix A′ sys : According to the reduction of the number of down and up trains, the coefficient matrix elements corresponding to the train state variables in A total , B total , C total and L CON are deleted in sequence. For example, when the number of down trains is reduced by a total of i (i ≤ (n - n u )) trains, the matrix elements deleted by A total in sequence are the rows and columns corresponding to A t(n+1-i) . The matrix elements deleted by B total in sequence are the rows corresponding to . The matrix elements deleted by C total in sequence are the rows and columns corresponding to C t(n+1-i) . The (n + 1 - i)-th column is deleted by L CON in sequence; when the number of up trains is reduced by a total of i (i ≤ n u ) trains, the matrix elements deleted by A total in sequence are the rows and columns corresponding to . The matrix elements deleted by B total in sequence are the rows corresponding to . The matrix elements deleted by C total in sequence are the rows and columns corresponding to . The (n CON + 1 - i)-th column is deleted by L u .

[0111] Example:

[0112] In this embodiment, the power supply mode of the traction power supply system is bilateral power supply. There are only two traction substations in the system. The length of the power supply arm of the high-speed railway traction substation is 50 km. The maximum number of trains running on the power supply arm is 6 (3 trains in the up direction and 3 trains in the down direction). The train type is CRH5 EMU. A dynamic model is established for the process of the number of trains running on the power supply arm changing from 6 to 5 (3 trains in the up direction and 2 trains in the down direction) and then to 4 (2 trains in the up direction and 2 trains in the down direction) during the operation of the train. The modeling process is as follows:

[0113] Step 1: Collect the original data of the model: According to the short-circuit parameters of the power system and the transformer, the types and dimensions of each conductor of the traction network, and the train operation diagram, obtain the original data required for modeling, including the equivalent inductance of the power grid 0.24 H, the transformation ratio of the traction transformer 220 / 27.5, the winding resistance of the traction transformer referred to the secondary side 0.3972 Ω and the leakage inductance 12.636 mH, the unit resistance of the traction network 0.1312 Ω / km and the inductance 1.27 mH / km, the length of the power supply arm 50 km, the maximum number of trains running on the power supply arm 6, and the lengths of each section of the traction network divided.

[0114] Step 2: Establish a modular small-signal mathematical model: Divide the system into train, traction network, traction transformer, and power grid modules, and establish the linearized state equations of each sub-module according to Kirchhoff's theorem.

[0115] 2.1 Train module: The CRH5 EMU adopts dq decoupling control. The small-signal mathematical model of the train is:

[0116]

[0117] Δx tk =[Δi n,k Δu dck Δu αk s′ Δu βk s′ Δi αk s′ Δi βk s′ Δx pk Δθ k Δx dk Δx qk Δx vdck T

[0118] In the formula, Δx tk is the disturbance quantity of the state variables composed of the main circuit and control circuit of each train. A tk , B tk , C tk are the coefficient matrices of the train linearized state space model, and k 1 ​= 25 / 1.77, u k is the pantograph voltage of the locomotive at network node k, i n,k is the AC side current of the train rectifier at node k, k = 1, 2, 3…6.

[0119] 2.2 Traction network module:

[0120] The traction network is segmented according to the maximum number of operating trains 6 (3 on the up line and 3 on the down line) and their positions. The traction networks on the up and down lines are divided into 8 segments in total, 4 segments on the up line and 4 segments on the down line. The lengths of each segment of the traction network are successively l 1 、l 2 …l 8 . According to Kirchhoff's voltage law (KVL), the differential equations of the current in each segment of the traction network are established. In the order from left to right and first up line then down line, the differential equations of the current in each segment of the traction network are successively combined to obtain the linearized state equation of the traction network module as:

[0121]

[0122] Δi q = [Δi q1 Δi q2 Δi q3 Δi q4 Δi q5 Δi q6 Δi q7 Δi q8 T

[0123] Δu = [Δu 1 Δu 2 Δu 3 Δu 4 Δu 5 Δu 6 Δu 7 Δu 8 Δu A Δu B Δu C T

[0124]

[0125] In the formula, I 8×8 is an 8×8 order identity matrix, Δi q ​​is the impedance current disturbance matrix of the 8-section traction network; it is formed in sequence from left to right, first ascending and then descending. Δu is a column vector composed of the traction network node voltage disturbances (excluding the first and last nodes of the traction network, in the order from left to right, first ascending and then descending), the voltage disturbances of the first and last nodes of the traction network, and the three-phase node voltage disturbances of the power grid; B q1 the 0 in i×j represents an i×j order zero matrix.

[0126] 2.3 Traction Transformer Module:

[0127] According to KVL, establish the differential equation of the secondary side current of each traction transformer, and obtain the linearized state equation composed of the secondary side currents of each traction transformer as state variables:

[0128]

[0129] Δi T =[Δi T1 Δi T2 T ; B T =[0 2×6 B 12 B 13 ;

[0130]

[0131] In the formula, I 2×2 is a 2-order identity matrix, 0 2×6 is a 2×6 order zero matrix, Δi T is the disturbance matrix composed of the secondary side currents of 2 traction transformers in sequence from left to right.

[0132] 2.4 Power Grid Module:

[0133] According to KVL, establish the current differential equation of the equivalent impedance of the high-voltage side three-phase power grid and perform small-signal linearization to obtain the small-signal mathematical model of the three-phase power grid module specifically as:

[0134]

[0135] Δi G =[Δi A Δi B Δi C T ; B S =[0 3×8 G 12 ;

[0136] In the formula, Δi​​G is the state variable composed of the three-phase impedance current disturbance of the power grid A, B, and C. 3×8 is a 3×8 order zero matrix.

[0137] Step 3: In order to obtain the relationship between the voltage of each node in the network and the branch current, a virtual resistor R is introduced between each voltage node and the ground. N (R N Assuming 100000Ω), the relationship between the voltage disturbance Δu of each node in the network and the branch current disturbance Δi can be written as Δu=100000L CON Δi, and then according to this relationship, the small signal state equations of each module can be combined to obtain the small signal mathematical model of the entire system when the maximum number of trains is 6:

[0138]

[0139] 3.1 According to Kirchhoff's current theorem (KCL), the current flowing through R at each node is obtained N The current in the branch and the voltage at each node are equal to R N With the flow through R N The product of the current on the network node voltage disturbance Δu and the branch current disturbance Δi is obtained:

[0140] Δu=100000L CON Δi(5)

[0141] Δu=[Δu 1 Δu 2 Δu 3 Δu 4 Δu 5 Δu 6 Δu 7 Δu 8 Δu A Δu B Δu C ] T

[0142]

[0143]

[0144] Where Δi is the current disturbance on the AC side of the rectifiers of the six trains (Δi n,1 ,Δi n,2 ,…Δi n,6 ), 8-segment traction network current disturbance (Δi q ), traction transformer secondary side current disturbance (Δi T ) and the three-phase current disturbance of the power grid A, B, and C (Δi G) in sequence.

[0145] 3.2 According to the embodiment, the actual system topology is as follows Figure 6 As shown, the small signal mathematical model of each submodule is combined, that is, the equations (1), (2), (3), (4) obtained in step 2 and the equation (5) obtained in step 3 are combined to obtain the small signal mathematical model of the system with the maximum number of running trains 6:

[0146]

[0147] A total =diag(A t1 ,A t2 …A t6 ,A q1 ,A T ,0 3×3 );

[0148] C total =diag(C t1 ,C t2 ,…C t6 ,I 8×8 ,I 2×2 ,I 3×3 );

[0149] Where I is the unit matrix, Δx total It is composed of the disturbance matrix of the state variables of each submodule of the system when the maximum number of trains is n, and is composed of the disturbance matrix of the state variables of each train when the maximum number of trains is 6 (Δx t1 ,Δx t2 ,…Δx t6 ), impedance current disturbance of each section of traction network (Δi q ), traction transformer secondary side current disturbance (Δi T ) and the three-phase current disturbance of the power grid (Δi G ) in sequence.

[0150] Step 4: According to the train operation diagram, obtain the change in the number of trains on the power supply arm from 6 to 5 (3 up and 2 down) and then to 4 (2 up and 2 down). The state variable disturbance Δx in equation (6) obtained in step 3 is total And the coefficient matrix A total , B total , C total , L CON By deleting matrix elements and changing the impedance values ​​of each section of the traction network, the small signal mathematical model of the train-network system based on the real-time number and position of trains can be obtained:

[0151]

[0152] 4.1 When the train position changes, change the traction network parameters, that is, according to the train position situation, change the traction network parameters, that is, change l 1 、l 2 …l 8 specific values.

[0153] 4.2 The number of trains is 5 (3 going up and 2 going down)

[0154] (1) Update the state variable Δx: The number of trains going down decreases by 1 in total, then Δx total Delete the matrix element Δx t6 .

[0155] (2) Update the coefficient matrix A′ sys : The number of trains going down decreases by 1 in total, then A total The deleted matrix element is the row and column corresponding to A t6 , B total The deleted matrix element is the row corresponding to it, C total The deleted matrix element is the row and column corresponding to C t6 , L CON Delete the 6th column.

[0156] 4.3 The number of trains is 4 (2 going up and 2 going down)

[0157] Based on 4.2,

[0158] (1) Update the state variable Δx: The number of trains going up decreases by 1 in total, then Δx total Delete the matrix element Δx t3 .

[0159] (2) Update the coefficient matrix A′ sys : The number of trains going up decreases by 1 in total, A total The deleted matrix element is the row and column corresponding to A t3 , B total The deleted matrix element is the row corresponding to it, C total The deleted matrix element is the row and column corresponding to C t3 , L CON Delete the 3rd column.

[0160] So far, the small-signal mathematical model of the vehicle-network system considering the dynamic changes of train operation is constructed. By transforming Equation (7) into the frequency domain based on the harmonic state space modeling principle and solving the characteristic roots, we get Figure 7The root locus diagram of the shown characteristics is analyzed to determine whether low-frequency oscillations occur in this system. From the results, when the number of trains is 6, the characteristic roots are in the right half-plane, indicating that the system is unstable. Comparing this result with the electromagnetic transient simulation results using MATLAB / Simulink, such as Figure 8 the waveforms of the traction network voltage and the DC-side voltage of the train network-side rectifier shown, the stability analysis results are consistent, proving the correctness of this modeling method.

Claims

1. A method for establishing a small signal mathematical model of a train-grid system when train operation changes dynamically, characterized in that: The following steps are involved: Step 1: Collect model raw data; According to the short-circuit parameters of the power system and transformer, the types and sizes of the conductors in the traction network, and the train operation diagram, the original data required for modeling are obtained; including the equivalent inductance L of the power grid S , traction transformer ratio k2, traction transformer winding resistance and leakage inductance R converted to the secondary side T and L T , traction network unit resistance r q and inductor L q , the length of the power supply arm l, the maximum number of trains running on the power supply arm n and the length of each section of the traction network; Step 2: Establish a modular small signal mathematical model; The system is divided into train, traction network, traction transformer and power grid modules, and the linearized state equations of each submodule are established according to Kirchhoff's theorem. Step 3: In order to obtain the relationship between the voltage at each node of the network and the branch current, a virtual resistor R is introduced between each voltage node and the ground. N , write out the relationship between the voltage disturbance Δu of each node in the network and the branch current disturbance Δi, and then combine the small signal state equations of each module according to the relationship to obtain the small signal mathematical model of the entire system when the maximum number of trains is n; Step 4: According to the real-time train operation diagram, the changes in the number and position of trains on the power supply arm are obtained, the matrix elements of the state variable disturbance and coefficient matrix obtained in step 3 are deleted, and the impedance values ​​of each section of the traction network are changed to obtain the small signal mathematical model of the train-grid system based on the real-time number and position of trains.

2. The method for establishing a small signal mathematical model of a train-grid system when train operation changes dynamically according to claim 1, characterized in that: The specific process of establishing the small signal mathematical model of each submodule in step 2 is as follows: (1) Small signal mathematical model of train module: According to the main circuit and control circuit structure of the train, the differential equation of each dynamic element is established and linearized: In the formula, Δx tk is the state variable disturbance of the main circuit and control circuit of each train, A tk , B tk , C tk is the coefficient matrix of the linearized state space model of the train, k1 is the transformer ratio of the train, u k is the pantograph voltage of the locomotive at the network node k, i n,k is the AC side current of the train rectifier at node k, k = 1, 2, 3…n; (2) Small signal mathematical model of traction network module: The traction network is divided according to the maximum number of running trains n and their positions, where the uplink n u Taiwan, downlink nn u The uplink and downlink traction networks are divided into n+2 sections, and the uplink is divided into n u +1 segment, the descending is divided into nn u +1 section, the length of each section of the traction network is l1, l2...l n+2 ; According to Kirchhoff's voltage theorem KVL, the differential equation of each section of the traction network current is established. From left to right and first up and then down, the differential equation of each section of the traction network current is merged in sequence to obtain the linearized state equation of the traction network module: Taking the system with only two traction substations as an example, Δi q , Δu, A q1 and B q1 The specific expression is as follows: If the system power supply mode is unilateral power supply, the traction network unit resistance r at the location of the upstream and downstream partitions is q Assumed to be infinite, if the system is bilaterally powered, then the unit resistance of the entire traction network is r q is the normal value; In the formula, I (n+2)×(n+2) is the (n+2)×(n+2)-order unit matrix, Δi q is the impedance current disturbance matrix of the n+2 traction network; it is composed in sequence from left to right, first up and then down, Δu is the voltage disturbance of the traction network node, ignoring the nodes at the beginning and end of the traction network, and the column vector composed of the voltage disturbance of the nodes at the beginning and end of the traction network and the voltage disturbance of the three-phase nodes of the power grid in sequence from left to right, first up and then down; A q1 and B q1 is the coefficient matrix of the linearized state equation. According to the topology of the traction network, A q1 is the unit matrix related to the unit traction network impedance, B q1 Each row has only two non-zero elements. The two non-zero elements are opposite to each other and their absolute values ​​are the reciprocal of the length of each traction network. The positive and negative values ​​are related to the position of each traction network. q1 0 i×j represents the i×j-order zero matrix; (3) Small signal mathematical model of traction transformer module: According to KVL, the differential equation of the secondary current of each traction transformer is established, and the linearized state equation composed of the secondary current of each traction transformer as the state variable is obtained: Δi T =[Δi T1 … Δi Tm ] T B T =[0 m×n B 12 B 13 ] In the formula, I m×m is the m-order identity matrix, 0 m×n is an m×n-order zero matrix, Δi T A is the disturbance matrix composed of m secondary currents of traction transformers from left to right. T , B T is the coefficient matrix of the small signal differential equation of the secondary current of the traction transformer. The values ​​of the non-zero elements are only related to the transformation ratio, winding resistance and leakage inductance of the traction transformer. (4) Small signal mathematical model of the power grid module: According to KVL, the current differential equation of the equivalent impedance of the three-phase power grid on the high-voltage side is established and small signal linearization is performed to obtain the small signal mathematical model of the three-phase power grid module: Δi G =[Δi A Δi B Δi C ] T B S =[0 3×(n+m) G 12 ] In the formula, Δi G is the state variable composed of the three-phase impedance current disturbance of the power grid A, B, and C, B S is the coefficient matrix, B S The column where the non-zero elements in each row are located is only related to the row where the three-phase voltage nodes A, B, and C of the power grid in Δu are located; 0 3×(n+m) It is a 3×(n+m)-order zero matrix.

3. The method for establishing a small signal mathematical model of a train-grid system when train operation changes dynamically according to claim 2, characterized in that: The step 3 is specifically as follows: According to Kirchhoff's current theorem KCL, the current flowing through R at each node is obtained N The current in the branch and the voltage at each node are equal to R N With the flow through R N The product of the current on the N L CON Δi; Where Δi is the current disturbance on the AC side of the n train rectifiers: Δi n,1 ,Δi n,2 ,…Δi n,n , current disturbance of each traction network section Δi q , traction transformer secondary side current disturbance Δi T And the three-phase current disturbance Δi of the power grid A, B, and C G The following is a column vector composed of two traction transformers in the system. CON matrix: According to the actual system topology, the small signal mathematical models of each submodule are combined, that is, the linearized state equations of the train, traction network, traction transformer and power grid submodules obtained in step 2, and Δu = R N L CON Δi is combined to obtain the small signal mathematical model of the system with the maximum number of running trains n: THE total =diag(A t1 ,THE t2 …THE tn ,THE q1 ,THE T ,0 3×3 ); C total =diag(C t1 ,C t2 ,…C tn ,I (n+2)×(n+2) ,I m×m ,I 3×3 ); Where I is the unit matrix, Δx total It is composed of the disturbance matrix of the state variables of each submodule of the system when the maximum number of trains is n, and is composed of the disturbance matrix of the state variables of each train when the maximum number of trains is n. t1 ,Δx t2 ,…Δx tn , impedance current disturbance of each section of traction network Δi q , traction transformer secondary side current disturbance Δi T And the three-phase current disturbance Δi of the power grid G The column vector composed of N is a larger impedance, set it to 100000Ω; according to the order of submodule state variables, A total It is a diagonal matrix composed of the coefficient matrix of the linearized state equation A of the train, traction network, and traction transformer submodules, B total It is composed of the linearized state equation B coefficient matrix of the train, traction network, traction transformer and three-phase power grid submodules in sequence, where 0 j×4 is a j×4-order zero matrix, j is equal to the number of state variables of n trains, C total The linear state equation C coefficient matrix of each train submodule is: C t1 , C t2 …C tn A diagonal matrix composed of three unit matrices of different orders, the orders of the three unit matrices are determined by the number of traction network sections, the number of traction transformers and the number of phases of the three-phase power grid.

4. The method for establishing a small signal mathematical model of a train-grid system when train operation changes dynamically according to claim 3, characterized in that: The step 4 is specifically as follows: Step 4.1: Change the traction network parameters according to the train position, that is, change l1, l2…l n+2 Specific value of Step 4.2: Update the state variable Δx: Delete Δx in turn according to the decrease in the number of trains total The column matrix elements in correspond to the train state variables; If the number of down trains decreases by i, i≤nn u Taiwan, then Δx total Delete matrix elements Δx one by one t(n+1-i) , when the number of up trains decreases by i, i≤n u Taiwan, then Δx total Remove matrix elements one by one Step 4.3: Update the coefficient matrix A s ' ys :According to the decrease in the number of downlink and uplink trains, A is deleted in turn. total , B total , C total and L CON The coefficient matrix elements corresponding to the train state variables; If the number of trains going down decreases by i, i≤(nn u ) , then A total The matrix elements deleted in sequence are A t(n+1-i) The corresponding row and column, B total The matrix elements deleted in sequence are The corresponding row, C total The matrix elements deleted in sequence are C t(n+1-i) The corresponding row and column, L CON Delete the n+1-i trains in sequence; when the number of up trains decreases by i, i≤n u Taiwan time, A total The matrix elements deleted in sequence are The corresponding row and column, B total The matrix elements deleted in sequence are The corresponding row, C total The matrix elements deleted in sequence are The corresponding row and column, L CON Delete the nth u +1-i column.

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