An analytical method for fault current based on first-order Taylor expansion

Abstract

This invention relates to a fault current analysis method based on first-order Taylor expansion, comprising the following steps: Step 1: Determine the system control strategy, which is a network-type control strategy, including a power loop, a virtual impedance element, and a voltage-current loop; the power loop includes an active power control loop and a reactive power control loop; Step 2: Construct a differential-algebraic equation model that characterizes the system dynamics; Step 3: Based on the characteristic that the integral value does not change instantaneously before and after the fault, use the steady state before the fault to solve for the instantaneous state variable values ​​after the fault, and then obtain the dq-axis current value and the instantaneous current amplitude after the fault; Step 4: At the solved instantaneous state variable values ​​and dq-axis current values ​​after the fault, perform a first-order Taylor expansion on the constructed differential-algebraic equation model, transforming the nonlinear differential-algebraic equation model into a linear differential-algebraic equation model; Step 5: Implement the analysis of the fault current.

Description

Technical Field

[0001] This invention belongs to the field of fault current analysis of grid converters, and particularly relates to a fault current analysis method based on first-order Taylor expansion. Background Technology

[0002] The large-scale grid connection of new energy sources such as wind and solar through power electronic converters is one of the main technical characteristics of the new generation of power systems[1]. The proportion of traditional synchronous generators is declining, and the power system exhibits the characteristics of "double high"[2,3]. At present, the converters widely used in actual engineering adopt the grid-following control architecture, that is, the vector current control is used to control the output current of the converter, and then control the active / reactive power fed into the grid. At the same time, the grid connection point voltage is sampled, and the synchronization between the converter and the grid is achieved through the phase-locked loop[4]. However, large-scale new energy power plants are generally located in remote areas. Due to the decline in the performance of the phase-locked loop and the lack of inertia under weak grids, grid-following control poses a huge challenge to the stable operation of the system. Grid-type control can autonomously establish system voltage and frequency, and has higher stability and robustness in weak grids. It is gradually becoming a better control method for the penetration of new energy power plants. Due to the control characteristics of the grid-type converter itself, the grid-type converter exhibits the characteristics of a controlled voltage source. Its output current is closely related to the equivalent grid voltage and grid impedance[5]. Reference [6] indicates that when a short-circuit fault occurs, in order to maintain the stability of the output voltage of the grid-type converter, a current value of up to 6.7 pu will be injected into the grid-type converter. However, unlike synchronous generators that can withstand currents of 5 to 8 pu, grid-type converters can usually only withstand currents of 1.2 to 2 pu due to limitations in semiconductor devices. Therefore, in order to protect the grid-type converter, it is necessary to analyze the fault current characteristics of the grid-type converter to avoid damaging the converter.

[0003] Some references [7-9], when analyzing the fault characteristics of grid-connected converters, often directly assume that after a fault, the current reaches the current limit value, and the control strategy switches to a current-limiting control strategy, ignoring the dynamic process of the current rising to the limit value after a fault. Reference

[10] proposes a full-state variable sequential solution method suitable for describing the time-domain response of the system, but this method cannot give an approximate expression for the fault current, and can only give a numerical solution. References [11-12] mainly use methods such as solving differential equations and Laplace transform to model the fault current of grid-connected converters, and the calculation process is relatively complex. Reference

[13] models the fault current characteristics of grid-connected converters, and its control structure is relatively simple. The inverse Laplace transform method can be used to solve the fault current expression. However, for more complex control strategies, such as the active power loop that considers both frequency and active power and the reactive power control loop that considers both voltage and reactive power proposed in this invention, it is difficult to use the scheme proposed in Reference

[13] to solve the fault current.

[0004] Due to the high-order nature of system modeling, the aforementioned literature struggles to provide an expression for the time-varying fault current. Generally, Laplace transforms are used, involving complex calculations of complex variables, resulting in a complicated solution process. Furthermore, when complex control strategies are involved, solving the problem using Laplace transforms and differential equations presents certain difficulties. To address these issues, this invention proposes an approximate fault current calculation method based on first-order Taylor expansion, which offers advantages such as simple calculation process and the ability to solve fault currents under complex control strategies.

[0005] References

[0006] [1] Zhou Xiaoxin, Chen Shuyong, Lu Zongxiang, et al. Technical characteristics of my country's new generation power system in energy transition [J]. Proceedings of the CSEE, 2018, 38(7): 1893-1904.

[0007] [2] Lu Zongxiang, Jiang Jiheng, Qiao Ying, et al. A review of research on generalized inertia analysis and optimization of new power systems [J]. Proceedings of the CSEE, 2023, 43(05): 1754-1776.

[0008] [3] Kang Chongqing, Du Ershun, Guo Hongye, et al. Analysis of six elements of new power system [J]. Power System Technology, 2023, 47(05): 1741-1750.

[0009] [4] Xu Jieyi, Liu Wei, Liu Shu, et al. Current status and development trend of power system converter grid control technology [J]. Power System Technology, 2022, 46(09): 3586-3595.

[0010] [5] Zhan Changjiang, Wu Heng, Wang Xiongfei, et al. A review of stability studies of grid converters [J]. Proceedings of the CSEE, 2023, 43(6): 2339-2358.

[0011] [6] TAUL GM, WANG

[0012] [7] Sun P, Tian Z, Huang M, et al. Additional Kinetic Energy Injection and Piecewise Damping Based Postfault Anti-Windup and Transient StabilityEnhanced Control for Grid-Forming Inverter[J]. IEEE Transactions on PowerElectronics, 2024.

[0013] [8] Rokrok E, Qoria T, Bruyere A, et al. Transient stability assessment and enhancement of grid-forming converters embedding current references as current limiting strategy [J]. IEEE Transactions on PowerSystems, 2021, 37(2): 1519-1531.

[0014] [9]Li Y, Lu Y, Yang J, et al. Transient stability of powersynchronization loop based grid forming converter[J]. IEEE Transactions onEnergy Conversion, 2023, 38(4): 2843-2859.

[0015]

[10] Ma Hongyue, Zheng Tao, Zhou Hao, et al. Fault calculation method for inverter power supply grid-connected system considering control switching [J]. Power System Technology, 2024, 48(07):2985-2994.

[0016]

[11] Jia Ke, Li Ying, Kong Jiajing, et al. Transient analysis of wind and solar power faults and practical calculation of peak short-circuit current[J]. Power System Technology, 2024, 48(02):830-840.

[0017]

[12] Jia K, Hou L, Liu Q, et al. Analytical calculation of transientcurrent from an inverter-interfaced renewable energy [J]. IEEE Transactions on Power Systems, 2021, 37(2): 1554-1563.

[0018]

[13] Zheng Tao, Wang Ziming, Zou Pengying, et al. Analytical calculation of virtual synchronous machine current and analysis of influencing factors during symmetrical short-circuit faults in power grids [J]. Power System Technology, 2023, 47(04): 1460-1475. Summary of the Invention

[0019] This invention proposes a fault current analysis method applicable to grid-connected converters. First, a mathematical model is established that ignores the dynamics of the current loop and the electromagnetic transients of the AC line, while considering the dynamics of the power loop and voltage loop. The effectiveness of the mathematical model is verified by comparing it with an electromagnetic transient simulation model. Second, a method for calculating the instantaneous fault current is obtained using this mathematical model. Furthermore, using the initial values ​​at the moment of the fault, a first-order Taylor expansion is performed on the constructed mathematical model, ultimately yielding an analytical expression for the fault current of the grid-connected converter. The technical solution of this invention is as follows:

[0020] A fault current analysis method based on first-order Taylor expansion includes the following steps:

[0021] Step 1: Determine the system control strategy

[0022] The adopted control strategy is a grid-type control strategy, mainly consisting of a power loop, a virtual impedance loop, and a voltage and current loop. The power loop includes two parts: an active power control loop and a reactive power control loop. The voltage loop reference voltage is obtained by subtracting the voltage drop of the virtual impedance from the converter's internal potential. The voltage and current loops use vector control. The inputs to the active power control loop are the actual active power value and the actual angular frequency value. The error between the actual and reference values ​​is passed through a drooping element and summed to obtain the input of the PI regulator in the active power control loop. After passing through the PI regulator, the angular frequency of the grid-type converter is obtained. Integrating the angular frequency of the grid-type converter yields the phase angle of the synchronous rotating coordinate system corresponding to the converter. The inputs to the reactive power control loop are the actual reactive power value and the actual converter terminal voltage value. The error between the actual and reference values ​​is passed through a drooping element and summed to obtain the input of the PI regulator in the reactive power control loop. After passing through the PI regulator, the internal potential of the grid-type converter is obtained.

[0023] Step 2: Construct a system of differential-algebraic equations that can characterize the dynamics of the system.

[0024] Ignoring the dynamics of the inner current loop and the electromagnetic transients of the AC line, and considering the dynamics of the power loop and voltage loop, a coordinate system is constructed with the grid converter's own synchronous rotating coordinate system relative to the grid voltage, and the integral value of the active power loop input error x. AP The integral value of the reactive power loop input error x RP The integral value x of the voltage loop d-axis input error ud The integral value x of the voltage loop q-axis input error uq As a state variable, the dq-axis current I d and I q Algebraic equations with algebraic variables;

[0025] Step 3: Based on the differential algebraic equation system model constructed in Step 2, and taking advantage of the characteristic that the integral value does not change instantaneously before and after the fault, the steady state before the fault is utilized. All differential terms in the differential algebraic equation system in Step 2 are set to 0, and the values ​​of the state variables instantaneously after the fault are obtained. The obtained values ​​of the state variables instantaneously after the fault are substituted into the algebraic equation to obtain the dq-axis current value and the current amplitude instantaneously after the fault.

[0026] Step 4: At the instantaneous state variable values ​​and dq-axis current values ​​obtained in Step 3 after the fault, perform a first-order Taylor expansion on the differential-algebraic equations constructed in Step 2, transforming the nonlinear differential-algebraic equations model constructed in Step 2 into a model with δ and x... AP x RP x ud and x uq The system of linear differential algebraic equations with state variables can be rearranged into the form dx / dt = Ax + b;

[0027] Step 5: Based on the system of linear differential algebraic equations obtained in Step 4, use mathematical methods to obtain δ and x. AP x RP x ud and x uq By substituting the expressions for the five state variables as a function of time into the algebraic equations of the differential-algebraic equation system constructed in the second step, we can obtain the expressions for the changes in the dq-axis current and current amplitude as a function of time after the fault, thus realizing the analysis of the fault current.

[0028] Furthermore, according to the control strategy described in the first step, the expression for the active power loop is shown in equation (1);

[0029]

[0030] In the formula, ω and P represent the actual values ​​of the converter's angular frequency and output active power, respectively. set P setK represents the converter angular frequency reference value and the output active power reference value, respectively. p K f K represents the active power droop factor and the frequency droop factor, respectively. pAPC and K iAPC ω represents the proportional coefficient and integral coefficient of the PI regulator in the active power control loop, respectively. B θ represents the reference value of angular frequency, θ represents the phase angle of the synchronous rotating coordinate system corresponding to the converter, and s is the Laplace transform operator;

[0031] The expression for the reactive power loop is shown in equation (2);

[0032]

[0033] In the formula, U and Q represent the actual value of the converter terminal voltage and the actual value of the output reactive power, respectively. set Q set K represents the converter terminal voltage reference value and the output reactive power reference value, respectively. q K v K represents the reactive power droop factor and the voltage droop factor, respectively. pRPC and K iRPC U represents the proportional and integral coefficients of the PI regulator in the reactive power control loop, respectively. ref represents the internal potential of the converter, and s is the Laplace transform operator;

[0034] In the virtual impedance circuit, the internal potential U of the converter is... ref Subtracting the voltage drop across the virtual impedance yields the dq-axis reference voltage U of the voltage loop. dref and U qref Its expression is shown in equation (3);

[0035]

[0036] In the formula, I d and I q Representing the dq-axis currents, R v and X v These represent virtual resistance and virtual reactance, respectively.

[0037] In the voltage loop, the voltage loop input is the error between the voltage reference and the actual voltage value. This error is passed through the PI link and summed with the current feedforward term to obtain the current reference value of the current inner loop. Its expression is shown in equation (4).

[0038]

[0039] In the formula, U dref and U qref U represents the dq axis reference voltage, respectively.d and U q Representing the actual voltages along the d and q axes, I gd and I gq K represents the dq-axis current on the grid side of the filter circuit, respectively. pu and K iu K represents the proportional gain and integral gain of the voltage loop, respectively. if I represents the current feedforward coefficient. dref and I qref These represent the reference values ​​for the dq-axis current of the voltage loop, respectively.

[0040] Furthermore, the differential-algebraic equation model constructed in the second step, which characterizes the system dynamics, neglects the dynamics of the inner current loop and the electromagnetic transients of the AC circuit. It assumes that the actual current value of the current loop can track the current reference value in real time, and ignores the filter capacitor current, assuming that I... td =I gd =I tdref I tq =I gq =I tqref Considering the dynamics of the power loop and voltage loop, the system of differential algebraic equations that can characterize the dynamics of the system is shown in equation (5).

[0041]

[0042] Where δ represents the phase of the grid-connected converter's own synchronous rotating coordinate system relative to the grid voltage, and ω B K represents the reference value for angular frequency. p K f These represent active power and angular frequency droop coefficient, respectively, K pAPC and K iAPC These are the proportional and integral coefficients of the PI element in the active power loop, respectively. set ω set These represent the converter output active power reference value and angular frequency reference value, respectively. AP The integral value of the active power loop input error is x. AP =∫[K f (ω set -ω)+K p (P set -P)]dt。 K q and K v K represents reactive power and voltage droop coefficient, respectively. pRPC and K iRPC These are the proportional and integral coefficients of the PI element in the reactive power loop, respectively, Q set and U setThese represent the reference values ​​for the converter's output reactive power and terminal voltage, respectively; Q and U represent the actual values ​​for the converter's output reactive power and terminal voltage, respectively; x RP The integral value representing the reactive power loop input error is x. RP =∫[K v (U set -U)+K p (Q set -Q)]dt。 U ref U represents the internal potential of the converter. s Indicates the mains voltage, I d I q Representing the dq-axis currents, R v and X v R represents virtual resistance and virtual reactance, respectively. g and X g R represents the filter resistance and filter reactance between the converter and the power grid, respectively. s and X s x represents the line resistance and line reactance between the converter and the power grid, respectively. ud and x uq K represents the integral value of the voltage loop dq-axis input error, respectively. pu K represents the proportional gain of the PI circuit in the voltage loop. iu K represents the integral coefficient. if This represents the feedforward coefficient of the grid-side current.

[0043] Furthermore, in the third step, during the calculation of the instantaneous current after the fault, the state variables δ and x... AP x RP x ud and x uq It is obtained through integration. The values ​​of these state variables remain unchanged instantaneously before and after the fault occurs. The values ​​of the above state variables are calculated from the steady state before the fault. In the steady state before the fault, the above five state variables do not change, and their first derivatives are 0, that is, all the differential terms in equation (5) are 0. The solution expression is shown in equation (6):

[0044]

[0045] The values ​​of the five state variables immediately after the fault are obtained by solving equation (6); immediately after the fault, due to the grid voltage U s A sudden decrease causes changes in the reactive power Q output by the converter and the converter terminal voltage U. Due to the effects of the droop factor and proportional coefficient in the reactive power loop, this in turn leads to changes in the internal electromotive force U of the converter. ref Changes occur, causing changes in the output current. However, these changes must satisfy the equality constraints shown in equation (7). The dq-axis current I instantaneously after the fault can be obtained using the equation set shown in equation (7).d and I q and current amplitude I mag ;

[0046]

[0047] Equation (6) is used to calculate the instantaneous state variable value after the fault, and Equation (7) is used to calculate the current value after the fault using the result of Equation (6). The grid voltage U is substituted into Equation (6). s It should be the grid voltage value before the fault, and the grid voltage U should be substituted into equation (7). s It should be the grid voltage value after the fault.

[0048] Furthermore, in the fourth step, after obtaining the instantaneous state variable value and instantaneous current value after the fault in the third step, a first-order Taylor expansion is performed on the differential-algebraic equation model constructed in the second step. The expression after the first-order Taylor expansion is shown in equation (8):

[0049]

[0050] In equation (8), I d and I q Eliminating and rearranging, we obtain the result shown in equation (9).

[0051]

[0052] in,

[0053]

[0054] Furthermore, in the fifth step, the mathematical model obtained from the first-order Taylor expansion in the fourth step is rearranged into the form shown in equation (11):

[0055]

[0056] In the formula, A is the coefficient matrix and b is the bias vector; by solving a system of linear differential equations, the state variables δ and x are obtained. AP x RP x ud x uq The analytical expression for is shown in the form of equation (11):

[0057]

[0058] In the formula, the eigenvalues ​​and eigenvectors of the coefficient matrix A are denoted as v. i and λ i (i = 1, 2, 3, 4, 5), δ p x APp x RPpx udp x uqp For equation A[δ p ,x APp ,x RPp ,x udp ,x uqp ] T The solution is +b=0; then, using equation (8), the current amplitude I can be obtained. mag An expression that changes over time.

[0059] The fault current analysis method for grid converters proposed in this invention is as follows: 1) Construct a mathematical model that ignores the dynamics of the current loop and the electromagnetic transients of the AC line, and considers the dynamics of the power loop and the voltage loop, so as to facilitate direct mathematical analysis of the system fault current; 2) Solve for the current at the moment of fault based on the mathematical model constructed in 1); 3) Based on the calculation results in 2), the analytical expression of the fault current can be obtained by using a first-order Taylor expansion. Attached Figure Description

[0060] Figure 1 Block diagram of grid converter control strategy

[0061] Figure 2 Schematic diagram showing the relationship between the DQ coordinate system and the dq coordinate system

[0062] Figure 3 Comparison between the mathematical model of the entire process of power grid voltage dip fault (8) and the electromagnetic transient simulation model

[0063] Figure 4 Instantaneous current after grid voltage drop and fault

[0064] Figure 5 Comparison of fifth-order mathematical model, first-order Taylor expansion model and electromagnetic transient simulation model when grid voltage drops to 0.4 pu Detailed Implementation

[0065] The present invention will now be described in conjunction with the accompanying drawings and embodiments.

[0066] (1) System control strategy

[0067] The grid-connected control strategy for grid-connected converters considered in this invention is as follows: Figure 1 As shown. In the grid-connected circuit of the power electronic converter, L f and L g C represents the filter inductance on the converter side and the grid side, respectively. f For the filter capacitor, U s and L s These are the inductance and voltage of the equivalent Thevenin circuit of the power grid, I. tabc and I gabcThe currents on the converter side and the grid side are U, respectively. tabc This represents the voltage across the converter's filter capacitor. After Park transform, the corresponding dq-axis voltage component U is obtained. dq and current component I tdq and I gdq P set ω set Q set and U set K represents the reference values ​​for the active power, angular frequency, reactive power, and terminal voltage of the converter output, respectively. P, ω, Q, and U represent the actual values ​​for the active power, angular frequency, reactive power, and terminal voltage of the converter output, respectively. p K f K q and K v These represent active power, angular frequency, reactive power, and voltage droop coefficient, respectively, K. pAPC and K iAPC These are the proportional and integral coefficients of the PI element in the active power loop, respectively, K. pRPC and K iRPC These are the proportional and integral coefficients of the PI element in the reactive power loop, respectively. K represents the active power error. p K times the angular frequency error f The converter's angular frequency ω is obtained by multiplying the active power loop PI element, and then the converter's phase θ is obtained by integrating the elements. Where ω... B This represents the reference value for angular frequency, which is taken here as 314.15926 rad / s. K represents the reactive power error. q K times the voltage error v The reactive power loop PI element is multiplied by U. set The voltage reference U is obtained by performing the operation. ref To suppress inrush current and prevent excessive current during faults, a virtual impedance Z is introduced. v =R v +jω B L v R v The resistor L represents the virtual impedance. v Inductance representing virtual impedance. U ref Subtracting the voltage drop across the virtual impedance yields the reference value U of the dq-axis voltage at the converter grid connection point. dref and U qref dq axis voltage reference value U dref and U qref The actual value of the dq axis voltage U d U q The error is passed through a PI circuit and then combined with the dq-axis current I on the grid side. gd I gq With Kif The sum of the products yields the dq-axis current reference value I. dref and I qref The proportional gain of the voltage loop PI element is K. pu The integral coefficient is K iu K if This represents the feedforward coefficient for the grid-side current. The dq-axis current reference value I. tdref and I tqref The actual value of the dq axis current I td I tq The error is passed through the PI circuit and compared with I... tq ω B L f -I td ω B L f and dq axis voltage U d U q The dq-axis modulation voltage U is obtained by performing the operation. dpwm with U qpwm dq axis modulation voltage U dpwm U qpwm The phase θ of the converter is input to the PWM generator, which can generate modulation signals for each switch.

[0068] (2) Establishment of system mathematical model

[0069] In the process of establishing the mathematical model, the following assumptions are assumed to hold:

[0070] (a) Ignore the electromagnetic transient processes of AC lines;

[0071] (b) Ignoring the dynamics of the inner current loop, it is assumed that the current can track the reference value in real time, i.e., I is assumed to be... td =I tdref I tq =I tqref ;

[0072] (c) Since the capacitance of the filter capacitor is small, the current in the filter capacitor in the filter is ignored, that is, I is considered to be... td =I gd I tq =I gq .

[0073] To simplify the expression, I will be referred to as I from now on. td I tdref I gd Written as I d , will I tq I tqref I gq Written as I q .

[0074] In the modeling process, this invention employs, for example... Figure 2 The synchronous rotating coordinate system shown is a q-axis that lags behind the d-axis. The synchronous rotating coordinate system with the grid voltage d-axis voltage orientation is defined as the DQ-axis, and the synchronous rotating coordinate system with the converter terminal voltage d-axis voltage orientation is defined as the dq-axis. δ represents the phase difference between the d-axis and the D-axis, i.e.

[0075] Neglecting the electromagnetic transients of the AC line and the converter filter capacitor current, according to Kirchhoff's voltage law, the dq-axis component U of the converter terminal voltage... d and U q With grid voltage U s The relationship is expressed as shown in equation (1).

[0076]

[0077] In the formula, R g and X g The filter resistor and filter inductor L represent the filter resistor and filter inductor between the converter and the power grid, respectively. g The corresponding reactance, R s and X s These represent the line resistance and line reactance between the converter and the power grid, respectively.

[0078] The expressions for the active power and reactive power output of the converter are shown in equation (2).

[0079]

[0080] Substituting equation (1) into equation (2), we can obtain the expressions for the active and reactive power output of the converter as shown in equation (3).

[0081]

[0082] according to Figure 1 The control strategy of the grid converter shown can be expressed by the power control loop as shown in equation (4).

[0083]

[0084] Where δ represents the phase of the grid-connected converter's own synchronous rotating coordinate system relative to the grid voltage. AP and x RP Let x represent the integral values ​​of the input errors of the active power loop and the reactive power loop, respectively. AP =∫[K f (ω set -ω)+K p (P set -P)]dt,x RP =∫[K v (U set-U)+K p (Q set -Q)]dt, where U represents the converter terminal voltage, U=(U)]dt, U=(U) d 2 +U q 2 ) 1 / 2 .

[0085] After rearranging equation (4), the dynamics of the power loop can be expressed as shown in equation (5).

[0086]

[0087] according to Figure 1 The control strategy of the grid converter shown can be expressed as follows: the dynamics of the virtual impedance link and the voltage loop can be expressed as shown in Equation (6) and Equation (7), respectively.

[0088]

[0089]

[0090] In equation (6), X v Indicates virtual inductance L v The corresponding reactance. In equation (7), x ud and x uq These represent the integral values ​​of the voltage loop dq-axis input error, i.e., x ud =∫(U dref -U d )dt,x uq =∫(U qref -U q )dt.

[0091] By rearranging the above equations (1)-(7), we can obtain the system of differential algebraic equations describing the dynamics of the system, as shown in equation (8).

[0092]

[0093] (3) Validation of mathematical model

[0094] Here, by setting up a grid voltage drop condition, the results of the mathematical model corresponding to Equation (8) and the electromagnetic transient simulation model are compared to verify the effectiveness of the mathematical model shown in Equation (8).

[0095] by Figure 1 Taking the grid-connected converter as an example, analysis and simulation verification are performed. The parameters of the electromagnetic transient simulation model of the grid-connected converter built in the electromagnetic transient simulation software PSCAD / EMTDC are shown in Table 1.

[0096] When a symmetrical three-phase short-circuit fault occurs in the power grid at 3.5 seconds, the voltage Us The voltage dropped to 0.4 pu, the fault was cleared in 3.6 seconds, and the grid voltage recovered to 1.0 pu. From the occurrence of a grid fault to the clearing of the grid fault, and then to the converter returning to normal operation, the mathematical model shown in equation (8) and the process according to... Figure 1 The simulation results of the electromagnetic transient simulation model built with the structure and parameters in Table 1 are as follows: Figure 3 As shown, in Figure 3 In the figure, the red solid line represents the calculation result of the mathematical model shown in equation (8), and the blue dashed line represents the simulation result of the electromagnetic transient model.

[0097] according to Figure 3 The δ and x shown in the middle AP x RP x ud x uq and dq axis current I d I q With current amplitude I mag The results show that the mathematical model shown in equation (8) can effectively describe Figure 1 The diagram shows the dynamics of a grid-connected converter system.

[0098] Table 1 Parameters for grid-connected converters

[0099]

[0100] (4) Method for calculating instantaneous current after a fault

[0101] Due to the current-carrying capacity limitations of the switching transistors in grid-connected converters, a limiting circuit is typically added to the inner current loop control strategy to restrict the current during faults. When the instantaneous current after a fault exceeds the limit value, the converter enters current-limiting mode, and the output current follows the current limit value. At this time, the outer voltage loop and other loops become ineffective. When the instantaneous current after a fault is within the limit value, the converter's inner current loop does not enter the current-limiting model, and the voltage and power loops still affect the current dynamics. Therefore, it is necessary to analyze whether the converter current is overcurrent instantaneously after a fault. The following mainly introduces the solution method for calculating the instantaneous current after a converter fault.

[0102] In the instant after the fault occurs, due to δ, x AP x RP x ud x uq These five variables are all obtained through integration, and changes require a certain amount of time to accumulate. Therefore, immediately after the fault, δ and x... AP x RP x ud x uqThese five variables remain unchanged, and their values ​​remain consistent with the steady-state values ​​before the fault. Before the fault, when the system reached steady state, the values ​​of δ and x in equation (8) were... AP x RP x ud x uq All differential terms are zero, meaning equation (9) holds. Equation (9) can be used to obtain δ and x instantaneously after the fault. AP x RP x ud x uq The value.

[0103]

[0104] When a grid voltage dip fault occurs on the grid side, it will cause the grid voltage amplitude U to drop. s Changes have occurred; note the reactive power loop generating U. ref The process involves a proportional element (the proportionality coefficient is K). pRPC This will cause a momentary malfunction (U). ref Changes occur, resulting in U dref and U qref Changes occur, and simultaneously, the proportional element in the voltage loop (with a proportionality coefficient of K) changes. pu This will also cause the dq axis current I to... d and I q Changes occur, which in turn lead to U d U q The reactive power Q changes, and this cycle repeats continuously. After a fault, the converter output current is constantly adjusted instantaneously to achieve balance, which is the instantaneous current after the fault. The reactive power Q, converter terminal voltage U, and converter internal potential U are mentioned above. ref dq axis voltage reference value U dref and U qref dq axis current I d and I q The process of finally reaching a steady state can be represented by a set of nonlinear equations as shown in equation (10).

[0105]

[0106] Solving equation (9) yields δ and x. RP x ud x uq Substituting this into equation (10), we can obtain the instantaneous dq-axis current I after the fault. d and I q and current amplitude I mag .

[0107] Using the above method, the verification was conducted under the condition that the grid voltage drops to 0.4 pu at 3.5 seconds. The results calculated by the mathematical model and the electromagnetic transient simulation results are as follows: Figure 4 As shown, in Figure 4 In the diagram, the red dots represent the instantaneous dq-axis current I after the fault calculated by the mathematical model shown in equation (10). d and I q and current amplitude I mag The blue dashed line represents the electromagnetic transient simulation results, and the errors between the mathematical model and the electromagnetic transient simulation model are shown in Table 2. Figure 4 Compared with the results in Table 2, the proposed calculation method can effectively calculate the initial value of the current after a fault within a certain error range.

[0108] Table 2. Error Table between Mathematical Model Calculation Results of Instantaneous Current After Fault and Electromagnetic Transient Simulation Results

[0109]

[0110] (5) Fault Current Calculation Method

[0111] from Figure 3 A comparison of the results of the mathematical model shown in Equation (8) and the electromagnetic transient model shows that the mathematical model shown in Equation (8) can be used to describe Figure 1 The system shown illustrates the dynamics of the fault current after a grid voltage drop fault. However, for determining whether the fault current reaches the current limit after a period of time following the fault, or whether the fault current will reach the current limit under a certain set of control parameters, the mathematical model shown in equation (8) can only be solved step by step from the fault start time according to the time sequence using the numerical solution method of differential equations, and cannot quickly obtain the current value at a certain moment. To solve this problem, this invention proposes a fault current solution method based on first-order Taylor expansion.

[0112] In δ, x AP x RP x ud x uq and dq axis current I d I q The values ​​immediately after the fault are δ0 and x, respectively. AP0 x RP0 x ud0 x uq0 I d0 I q0 For the mathematical model shown in equation (8), at δ0, x AP0 x RP0 x ud0 x uq0 I d0 I q0By performing a first-order Taylor expansion, equation (8) can be transformed into the form shown in equation (11).

[0113]

[0114] For equation (11), I can be eliminated using the voltage loop equation. d and I q This can be transformed into using δ and x AP x RP x ud x uq The system of differential equations for state variables can be rearranged as shown in equation (12).

[0115]

[0116] In equation (12),

[0117]

[0118] Equation (12) can be expressed in the form shown in equation (13).

[0119]

[0120] Where A is the coefficient matrix and b is the bias vector. According to equation (13), the state variables δ and x can be obtained. AP x RP x ud x uq The analytical expression of is shown in Equation (14).

[0121]

[0122] Wherein, the eigenvalues ​​and eigenvectors of the coefficient matrix A are denoted as v i and λ i (i = 1, 2, 3, 4, 5), δ p x APp x RPp x udp x uqp For equation A[δ p ,x APp ,x RPp ,x udp ,x uqp ] T The solution is +b=0. Then, using the equation in (12), the current amplitude I can be obtained. mag An expression that changes over time.

[0123] The following section verifies the fault current solution method based on first-order Taylor expansion proposed in this invention by setting a grid voltage dip condition. The condition is set as follows: when a symmetrical three-phase short-circuit fault occurs in the grid at 3.5 seconds, the voltage U s The voltage dropped to 0.4 pu, the fault was cleared in 3.6 seconds, and the grid voltage recovered to 1.0 pu. Within 3.5-3.6 seconds, the fault current curves obtained from the electromagnetic transient simulation model, the mathematical model corresponding to equation (8), and the proposed method based on first-order Taylor expansion are respectively as follows: Figure 5 Midpoint lines, solid lines, and dashed lines are used to indicate this. From... Figure 5 The comparison results show that the fault current solution method based on first-order Taylor expansion proposed in this invention can effectively reflect the dynamics of the fault current.

Claims

1. A fault current analysis method based on first-order Taylor expansion, comprising the following steps: Step 1: Determine the system control strategy The adopted control strategy is a network-based control strategy, mainly comprising a power loop, a virtual impedance loop, and a voltage and current loop. The power loop consists of two parts: an active power control loop and a reactive power control loop. The reference voltage for the voltage loop is obtained by subtracting the voltage drop across the virtual impedance from the converter's internal potential. The voltage loop and the inner current loop employ vector control. The inputs to the active power control loop are the actual value of active power and the actual value of angular frequency. The error between the actual value and the reference value is passed through a drooping element and summed to obtain the input of the PI regulator in the active power control loop. After passing through the PI regulator, the angular frequency of the grid converter is obtained. Then, the phase angle of the synchronous rotating coordinate system corresponding to the converter is obtained by integrating the angular frequency of the grid converter. The inputs to the reactive power control loop are the actual value of reactive power and the actual value of converter terminal voltage. The error between the actual value and the reference value is passed through a drooping element and summed to obtain the input of the PI regulator in the reactive power control loop. After passing through the PI regulator, the internal potential of the grid converter is obtained. Step 2: Construct a system of differential-algebraic equations that can characterize the dynamics of the system. Ignoring the dynamics of the inner current loop and the electromagnetic transients of the AC line, and considering the dynamics of the power loop and voltage loop, a coordinate system is constructed with the grid converter's own synchronous rotating coordinate system relative to the grid voltage, and the integral value of the active power loop input error x. AP The integral value of the reactive power loop input error x RP The integral value x of the voltage loop d-axis input error ud The integral value x of the voltage loop q-axis input error uq As a state variable, the dq-axis current I d and I q Algebraic equations with algebraic variables; Step 3: Based on the differential algebraic equation system model constructed in Step 2, and taking advantage of the characteristic that the integral value does not change instantaneously before and after the fault, the steady state before the fault is utilized. All differential terms in the differential algebraic equation system in Step 2 are set to 0, and the values ​​of the state variables instantaneously after the fault are obtained. The obtained values ​​of the state variables instantaneously after the fault are substituted into the algebraic equation to obtain the dq-axis current value and the current amplitude instantaneously after the fault. Step 4: At the instantaneous state variable values ​​and dq-axis current values ​​obtained in Step 3 after the fault, perform a first-order Taylor expansion on the differential-algebraic equations constructed in Step 2, transforming the nonlinear differential-algebraic equations model constructed in Step 2 into a model with δ and x... AP x RP x ud and x uq The system of linear differential algebraic equations with state variables can be rearranged into the form dx / dt = Ax + b; Step 5: Based on the system of linear differential algebraic equations obtained in Step 4, use mathematical methods to obtain δ and x. AP x RP x ud and x uq By substituting the expressions for the five state variables as a function of time into the algebraic equations of the differential-algebraic equation system constructed in the second step, we can obtain the expressions for the changes in the dq-axis current and current amplitude as a function of time after the fault, thus realizing the analysis of the fault current.

2. The grid converter control method according to claim 1, characterized in that, According to the control strategy described in the first step, the expression for the active power loop is shown in equation (1); In the formula, ω and P represent the actual values ​​of the converter's angular frequency and output active power, respectively. set P set K represents the converter angular frequency reference value and the output active power reference value, respectively. p K f K represents the active power droop factor and the frequency droop factor, respectively. pAPC and K iAPC ω represents the proportional coefficient and integral coefficient of the PI regulator in the active power control loop, respectively. B θ represents the reference value of angular frequency, θ represents the phase angle of the synchronous rotating coordinate system corresponding to the converter, and s is the Laplace transform operator; The expression for the reactive power loop is shown in equation (2); In the formula, U and Q represent the actual value of the converter terminal voltage and the actual value of the output reactive power, respectively. set Q set K represents the converter terminal voltage reference value and the output reactive power reference value, respectively. q K v K represents the reactive power droop factor and the voltage droop factor, respectively. pRPC and K iRPC U represents the proportional and integral coefficients of the PI regulator in the reactive power control loop, respectively. ref represents the internal potential of the converter, and s is the Laplace transform operator; In the virtual impedance circuit, the internal potential U of the converter is... ref Subtracting the voltage drop across the virtual impedance yields the dq-axis reference voltage U of the voltage loop. dref and U qref Its expression is shown in equation (3); In the formula, I d and I q Representing the dq-axis currents, R v and X v These represent virtual resistance and virtual reactance, respectively. In the voltage loop, the voltage loop input is the error between the voltage reference and the actual voltage value. This error is passed through the PI link and summed with the current feedforward term to obtain the current reference value of the current inner loop. Its expression is shown in equation (4). In the formula, U dref and U qref U represents the dq axis reference voltage, respectively. d and U q Representing the actual voltages along the d and q axes, I gd and I gq K represents the dq-axis current on the grid side of the filter circuit, respectively. pu and K iu K represents the proportional gain and integral gain of the voltage loop, respectively. if I represents the current feedforward coefficient. dref and I qref These represent the reference values ​​for the dq-axis current of the voltage loop, respectively.

3. The fault current analysis method according to claim 2, characterized in that, The differential-algebraic equation model constructed in the second step, which characterizes the system dynamics, neglects the dynamics of the inner current loop and the electromagnetic transients of the AC line. It assumes that the actual current value of the current loop can track the current reference value in real time, and ignores the filter capacitor current, assuming that I... td =I gd =I tdref I tq =I gq =I tqref Considering the dynamics of the power loop and voltage loop, the system of differential-algebraic equations that characterize the system dynamics is shown in equation (5): Where δ represents the phase of the grid-connected converter's own synchronous rotating coordinate system relative to the grid voltage, and ω B K represents the reference value for angular frequency. p K f These represent active power and angular frequency droop coefficient, respectively, K pAPC and K iAPC These are the proportional and integral coefficients of the PI element in the active power loop, respectively. set ω set These represent the converter output active power reference value and angular frequency reference value, respectively. AP The integral value of the active power loop input error is x. AP =∫[K f (ω set -ω)+K p (P set -P)]dt;K q and K v K represents reactive power and voltage droop coefficient, respectively. pRPC and K iRPC These are the proportional and integral coefficients of the PI element in the reactive power loop, respectively, Q set and U set These represent the reference values ​​for the converter's output reactive power and terminal voltage, respectively; Q and U represent the actual values ​​for the converter's output reactive power and terminal voltage, respectively; x RP The integral value representing the reactive power loop input error is x. RP =∫[K v (U set -U)+K p (Q set -Q)]dt;U ref U represents the internal potential of the converter. s Indicates the mains voltage, I d I q Representing the dq-axis currents, R v and X v R represents virtual resistance and virtual reactance, respectively. g and X g R represents the filter resistance and filter reactance between the converter and the power grid, respectively. s and X s x represents the line resistance and line reactance between the converter and the power grid, respectively. ud and x uq K represents the integral value of the voltage loop dq-axis input error, respectively; pu K represents the proportional gain of the PI circuit in the voltage loop. iu K represents the integral coefficient. if This represents the feedforward coefficient of the grid-side current.

4. The fault current analysis method according to claim 3, characterized in that, In the third step, during the instantaneous current calculation after the fault, the state variables δ and x AP x RP x ud and x uq It is obtained through integration. The values ​​of these state variables remain unchanged instantaneously before and after the fault occurs. The values ​​of the above state variables are calculated from the steady state before the fault. In the steady state before the fault, the above five state variables do not change, and their first derivatives are 0, that is, all the differential terms in equation (5) are 0. The solution expression is shown in equation (6): The values ​​of the five state variables immediately after the fault are obtained by solving equation (6); immediately after the fault, due to the grid voltage U s A sudden decrease causes changes in the reactive power Q output by the converter and the converter terminal voltage U. Due to the effects of the droop factor and proportional coefficient in the reactive power loop, this in turn leads to changes in the internal electromotive force U of the converter. ref Changes occur, causing changes in the output current. However, these changes must satisfy the equality constraints shown in equation (7). The dq-axis current I instantaneously after the fault can be obtained using the equation set shown in equation (7). d and I q and current amplitude I mag ; Equation (6) is used to calculate the instantaneous state variable value after the fault, and Equation (7) is used to calculate the current value after the fault using the result of Equation (6). The grid voltage U is substituted into Equation (6). s It should be the grid voltage value before the fault, and the grid voltage U should be substituted into equation (7). s It should be the grid voltage value after the fault.

5. The fault current analysis method according to claim 1, characterized in that, In the fourth step, after obtaining the instantaneous state variable value and instantaneous current value after the fault in the third step, the differential algebraic equation model constructed in the second step is expanded by first-order Taylor expansion. The expression after the first-order Taylor expansion is shown in equation (8). In equation (8), I d and I q Eliminating and rearranging, we obtain the result shown in equation (9); in, 。 6. The fault current analysis method according to claim 5, characterized in that, In the fifth step, the mathematical model obtained from the first-order Taylor expansion in the fourth step is rearranged into the form shown in equation (11): In the formula, A is the coefficient matrix and b is the bias vector; by solving a system of linear differential equations, the state variables δ and x are obtained. AP x RP x ud x uq The analytical expression of is shown in the form of equation (12); In the formula, the eigenvalues ​​and eigenvectors of the coefficient matrix A are denoted as v. i and λ i (i = 1, 2, 3, 4, 5), δ p x APp x RPp x udp x uqp For equation A[δ p ,x APp ,x RPp ,x udp ,x uqp ] T The solution is +b=0; then, using equation (8), the current amplitude I can be obtained. mag An expression that changes over time.

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