Calculation Method and Characteristic Analysis for Fault Current of Permanent Magnet Direct-Drive Wind Power System considering Positive and Negative Sequence Decomposition

. In view of the fact that the infuence of positive and negative sequence decomposition, which is widely used in positive and negative sequence decoupling control in control system, on the fault current calculation process is not deeply considered in the existing transient analysis methods of permanent magnet direct-drive wind farm short circuit current, this paper proposes a transient short circuit current calculation model that takes into account positive and negative sequence decomposition. Te infuence of the transient characteristics of positive and negative sequence decomposition on the control system is studied, and the mechanism of its action on the transient change of short circuit current is revealed. Te positive and negative sequence decoupling processes of the circuit equation are modifed, and the characteristics of the coupling equation are analyzed. Te diference in the converter output voltage between the circuit equation and the control equation and the depth of its infuence on the calculation process is revealed. On the basis of quantifying the diference at the converter output voltage, the circuit equation and the control equation are combined to form a short-circuit current calculation model with positive and negative sequence decomposition, which accurately characterizes the transient characteristics of fault current under diferent voltage drops and efectively improves the accuracy of the calculation results.


Introduction
Since the proposal of the dual-carbon target, various types of clean energy generation methods have received widespread attention in the industry [1,2].Among them, the permanent magnet direct-drive wind power system has developed rapidly in recent years due to its outstanding advantages of abundant wind energy resources, stable wind speed, and so on [3].In order to ensure that the phase angle and frequency of the voltage at the point of common coupling (PCC) can be detected quickly and accurately when the permanent magnet direct-drive wind power system is in the unbalanced operation state, the low-voltage ride-through control, including the positive and negative sequence decomposition link, is widely used in the permanent magnet direct-drive wind power system at present, so as to eliminate the disturbance caused by the negative sequence component to the system and realize the accurate phase lock based on the positive sequence component [4,5].Based on the above, the positive and negative sequence decomposition link has become an important part of the wind farm control strategy.At present, the most common methods of phase-locked loop based on positive sequence components are the dual second-order, generalised integrator phase-locked loop (DSOGI-PLL) [6], and the decoupled double synchronous reference frame phase-locked loop (DDSRF-PLL) [7].Compared with DSOGI-PLL, the algorithm structure of DDSRF-PLL is more complex, and the decoupling structure of DDSRF-PLL is mainly aimed at eliminating the infuence of the negative sequence fundamental frequency component, while the ability to resist low harmonic disturbance is relatively weak [8,9].By contrast, the modelling process of frequency characteristics of the positive and negative sequence decomposition based on DSOGI-PLL is not only simple but can also eliminate the disturbance of the negative sequence fundamental voltage.Terefore, DSOGI-PLL is more widely used in practical applications [10].
When a fault occurs in the permanent magnet direct-drive wind power system, which leads to a three-phase symmetric drop of the voltage at the PCC point, it is of great signifcance to accurately analyze the transient characteristics of the fault current for the research of relay protection and fault treatment.Te analysis methods of fault current are usually divided into steady-state analysis and transient analysis.Steadystate fault current analysis generally considers the role of lowvoltage ride-through control and calculates steady-state current through sequence network analysis, and many research studies have been carried out in the industry [11].Most literature equivalents the wind farm in steady state after a short-circuit fault to a controlled voltage source or a controlled current source and establishes a composite network under diferent fault conditions to simplify the modelling program of fault analysis and calculation [12].
Te analysis of transient short-circuit current considers the infuence of converter control, PLL, and other links and calculates the transient current value by writing down control equations and equivalent circuit diferential equations.Compared with steady-state current calculation, transient current calculation is more complex.Te industry has also carried out extensive research and achieved some results in consideration of its important role in studying the action behavior of relay protection.On the basis of the comprehensive analysis of the converter power supply control system, a calculation model of the fault current of the converter power supply with a low-voltage ride-through control strategy is proposed in [13], and the transient short-circuit current of the converter power supply is calculated by using the transfer functions in the complex frequency domain of both the control system and converter [14], starting with the mathematical model of the control system, which describes the output current of the inverter after the fault in the form of second-order, constant coefcient diferential equations, and the current expression in the transient process is obtained analytically based on the steady-state value before and after the fault and the transient change law afected by the control parameters.Te authors of [15] describe the positive and negative sequence current after fault as second-order, constant coefcient diferential equations, and the short-circuit current expressions based on the independent control strategy of positive and negative sequence are derived.At last, it is concluded in [15] that the transient current is independent of the parameters of the outer power loop and is only afected by the control parameters of the inner current loop.Both [14,15] regard the equation of the control system and the circuit equation as constant coefcient equations, which do not consider the infuence of the nonlinear links in the control system on the transient characteristics of fault current.In view of the infuence of nonlinear links in the control system on the transient current, the study [16] describes the output current as a second-order diferential equation set with variable coefcients when there is an error between PLL output angular frequency and frequency compensation angular frequency and proposes a mathematical method based on the form of decoupling complex domain to solve the transient current of dq-axis [17].It puts forward the conclusion that the dynamic process of PLL and its inductance connected to the power grid makes the control system of the grid-side converter of the wind farm produce positive feedback and amplify the oscillation of subsynchronous frequency.By considering the saturation characteristics of the current inner loop, the nonlinear diferential equations of symmetric transient fault current are deduced, and an analysis method based on the phase plane is proposed to obtain the analytical expression of transient current in [18,19] uses linear control theory to solve the calculation problems caused by nonlinear links in the converter control system.By setting PI parameters, the second-order system is simplifed into the frst-order system, and then, the fault current of the dq-axis is calculated by the time-frequency domain transformation.Te study [20] proposes a calculation method of instantaneous asymmetric fault current of converter, which analyzes compositions and characteristics of the fault current and then concludes that fault current is afected by a control loop, fault type, fault distance, and a nonlinear limiter.None of the above studies on transient fault current take into account the infuence of positive and negative sequence decomposition in the lowvoltage ride-through control strategy.
In view of the infuence of positive and negative sequence decomposition on transient characteristics of fault current [21,22], we propose a short-circuit analysis model of renewable energy converter under the control of positive and negative sequence decoupling based on DSOGI.Te fault transient current is analyzed by using the model in the complex frequency domain, and the conclusion that the transient characteristics of fault current are related to the delay characteristics of positive and negative sequence decomposition is presented.In [21,22], both of them take the positive and negative sequence decomposition link into account in the analysis process of fault current transient characteristics.However, they do not consider the infuence of the voltage drop process and the possible transient response of the negative sequence current suppression strategy caused by the positive and negative sequence decomposition in the transient process.
In fact, the positive and negative sequence decomposition links increase the nonlinear links in the control system, resulting in more complex transient characteristics of the converter output transient current; therefore, the infuence on not only the nonsymmetrical fault current transient characteristics but also the symmetrical fault current transient characteristics cannot be ignored.In view of such problems, the analysis method of the transient current fault characteristics considering the positive and negative sequence decomposition link is deeply studied in this paper, and mainly aiming at the situation of voltage symmetric drop at the PCC point caused by external faults of the permanent magnet direct-drive wind power system in order to accurately study the efect of positive and negative sequence decomposition on symmetric fault.Tis paper is organized as follows: Section 2

Control Strategy of a Grid-Side Converter of Permanent Magnet Direct-Drive Wind Turbine with Positive-Negative Sequence Decomposition
Te grid-connected delivery system of the permanent magnet direct-drive wind power system is shown in Figure 1.
Te stator windings of the permanent magnet wind generator are collected by the converter and then connected to the transmission network by the transformer [23].
When the three-phase fault of the remote AC system leads to voltage drop at the point of common coupling (PCC), if the voltage drop is deep enough to cause the power imbalance between the two ends of the DC side of the converter, the chopper circuit on the DC side will be input to keep the DC voltage stable.Terefore, the machine-side converter has little infuence on the transient current output of the AC system.At this time, the transient process of the fault current output from the wind farm mainly depends on the response characteristics of the control system of the grid-side converter (GSC) during the fault.Tus, this paper mainly analyzes the infuence of the GSC control strategy on the transient characteristics of fault current.

Te Positive and Negative Sequence Decomposition of a Grid-Side Converter.
In order to ensure that the PLL can accurately lock the phase of the fundamental positive sequence component when the power grid is in a three-phase unbalanced state, the voltage at the PCC point u abc will be decomposed into positive and negative sequence at frst.Te basic structure of the DSOGI-PLL which is commonly used is shown in Figure 2 [24,25].
In the structure of Figure 2, "C" is the Clark transformation matrix, and the "P" is the Parker transformation matrix.
It can be learned from Figure 2 that the voltages at the PCC point u a , u b , and u c are translated from the abc threephase coordinate frame to the αβtwo-phase stationary coordinate frame to obtain u α ′ and u β ′ : By using the second-order generalized integrator which has the characteristic of frequency selection and can extract the fundamental voltage component of the input signal u α ′ , the output signal qu α ′ which has the same amplitude with u α ′ but the phase lags u α ′ 90 °, and the output signal u a which has the same amplitude and phase with u α ′ can be obtained.Teir relationships can be expressed as where k is the damping factor.In order to give consideration to fltering efect and response speed, the value of k is usually 2. q is a phase shift operator with a lag of 90 degrees.qu β ′ and u β ′ can also be obtained by equation (2).Te positive sequence voltages in the two-phase stationary coordinate frame u + α and u + β can be obtained by the following equation: Te parker transformation is used to transform u + α and u + β to the dq-axis coordinate frame of positive sequence which rotates synchronously with the three-phase coordinate frame, and the dq-axis voltage components of positive sequence at the PCC point u + d and u + q are obtained.Te process of the Parker transformation can be expressed as Te produce process of θ PLL is as follows: the PLL regulates u + q to zero by using the PI controller and error adjustment which can obtain the angular frequency ω PLL , and then feedback the value to DSOGI.After ω PLL is integrated, the phase angle θ PLL can be obtained and fed back into the Parker transformation matrix: In equation (7), k pPLL and k iPLL represent the proportional and integral coefcients of the phase-locked loop PI controller, respectively.Te value of ω 0 is typically set to 100π rad/s.

Low-Voltage Ride-Trough Control Strategy of a Permanent Magnet Direct-Drive Wind Power System.
According to the standard of the low-voltage ride-through capability of a grid-connected wind farm, the permanent magnet directdrive wind power system should not only keep connected to the power grid during the low-voltage ride-through but also provide reactive current to support the grid voltage [26,27].Te control strategy of GSC after failure is shown in Figure 3.
Te basic principle of the low-voltage ride-through control strategy is when u abc drops, the control system should detect the drop magnitude of positive sequence voltage at the PCC point u PCC and determine the reference value of active current i * d and reactive current i * q according to the drop magnitude of u PCC .When the magnitude of u PCC does not drop to 90%, the control strategy remains unchanged; it means that the dq-axis current reference values of current inner loop control i * d and i * q are generated by the voltage outer loop control which are the same as the normal control strategy.When the magnitude of u PCC drops to 90%, the GSC is required to provide a certain reactive power support.In order to improve the speed of reactive response, i * q is generated without the outer loop control, but the outer loop control of reactive power is switched to the direct current inner loop control, and i * d is decided by i * q .Te fnal control target is the output fault current of GSC i abc after external fault does not exceed 1.2 times the rated current according to the tolerance ability of short-circuit current of converter [28,29].
Based on the low-voltage ride-through control strategy, i * d and i * q can be regarded as functions about u PCC ; thus, their relationships can be expressed as 4 International Transactions on Electrical Energy Systems where ε(u PCC ) is the step function; and i * d−pi and i * q−pi are the current inner loop reference values generated by the voltage outer loop, and can be expressed as u PCC ′ represents the PCC point voltage when the d-axis control strategy switches to direct current control and can be expressed as According to (5), taking the system working in rated state during normal operation as an example, the change process of i * d and i * q is shown in Figure 4 when the magnitude of u PCC drops by more than 10% after failure.
It can be learned from Figure 4. Tat before the failure i * d is 1 p.u. and i * q is 0 p.u. when the system is running in the rated state.When the magnitude of u PCC is more than 90%, i * d and i * q will still be determined by the outer loop control; when the magnitude of u PCC is less than 90%, the control of q-axis will switch to current inner loop control.At this point, i * q changes with the magnitude of u PCC , while i * d is still generated by outer loop control.When the magnitude of u PCC is less than u PCC ′ , the control of d-axis will be switched to direct current control, and i * d is calculated according to the root (1.2 2 -i * q 2 ).When the magnitude of u PCC is less than 10%, i * q reaches the limiting value of 1.2 p.u., and i * d is 0 p.u.

Transient Current Calculation Based on Existing Research Methods
Te existing method of calculating the GSC output currents i a , i b , and i c do not consider the positive and negative sequence decomposition.In the process of solving, the circuit equation between the voltages at the PCC point u a , u b , and u c and the output voltages of GSC v a , v b , and v c are transformed to the dq-axis coordinate frame of positive sequence, and then the equation is solved simultaneously with the equation of inner loop control [12].Te circuit equation in abc three-phase coordinate frame can be expressed as

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In equation ( 9), L is the equivalent inductance of the flter between v a , v b , and v c and u a , u b , and u c .When the circuit equation in three-phase coordinate frame is transformed to the dq-axis coordinate frame of positive sequence, the infuence of positive and negative sequence decomposition link on the process of coordinate transformation is not taken into account in the existing research method since the research object is the three-phase symmetrical fault.Terefore, the circuit equation in the dq-axis coordinate frame of the positive sequence is obtained by multiplying both sides of equation ( 9) by the Parker transformation matrix P directly, which can be expressed as where i + d and i + q are the dq-axis output currents of positive sequence by GSC; and v + d and v + q are the dq-axis output voltages of positive sequence by GSC.
Te equation of current inner loop control can be expressed as [22] where v + d−pi and v + q−pi are the output voltage modulation signals of GSC generated by the inner loop of the control system.k ip is the proportional gain and k ii is the integral time constant of the inner loop PI controller.
Te existing method suggest that v + d � v + d−pi and v + q � v + q−pi .By combining equation (10) with equation ( 11), taking v + d and v + q as intermediate variables and eliminating them, the differential equations of i + d and i + q can be obtained: where i + d and i + q can be obtained by solving (9) with the Runge-Kutta method.Te object of study is a three-phase short-circuit fault.Terefore, the transient component of negative sequence is very small and is generally not considered.i a , i b , and i c can be obtained by directly performing the Parker inverse transformation on the dq-axis current of positive sequence.
It is assumed that the magnitude of u PCC drop to 7% of the rated voltage when an external fault occurs, and the calculation results of equation ( 12) are compared with the simulation results of the control system containing positive and negative sequence decomposition link.Te results are shown in Figures 5(a It can be seen from Figures 5(a) and 5(b) that due to the absence of the infuence of positive and negative sequence decomposition link, i + d and i + q obtained by the existing research method have a large error in comparison with the simulation result, and the overshoot and oscillation attenuation characteristics of their transient process cannot be refected.Terefore, the existing research method is no longer suitable for studying the transient characteristics of fault current with positive and negative sequence decomposition.
According to the analysis in Section 3, in order to accurately analyze the transient characteristics of the fault current containing the positive and negative sequence decomposition link, the transient process of the positive and negative sequence decomposition link should also be taken into account.

Study on Transient Process of Positive and
Negative Sequence Decomposition

Analysis of the Transient Response of DSOGI-PLL.
Te transient response process of DSOGI-PLL is studied by taking the symmetrical fault of the AC system as an example.It is supposed that a three-phase fault occur in the AC system at 2 s, and at this time, the magnitude of u PCC will drop to 7% of the rated voltage.When there is a symmetrical fault in the line, the voltage at the fault point can be regarded as the voltage will change to 0 in a very short time.Due to the short line of the permanent magnet direct drive wind power system, the voltage at the PCC point can also be regarded as the instantaneous sudden change voltage, thus simplifying the calculation of subsequent fault current.Te waveform of the voltage at the PCC point in the two-phase stationary coordinate frame u α ′ is shown by the blue curve in Figure 5.It can be considered that the change of u α ′ caused by external fault is basically seen as an instantaneous drop, and the specifc expression can be expressed as 6 International Transactions on Electrical Energy Systems After the SOGI link, the output waveforms of u α and qu α are shown by the red and yellow curves in Figure 6, respectively.
It can be learned from Figure 6 that the transient process of the output signal u α of the SOGI link after failure is not exactly the same as the input signal u α .Te amplitude and phase of u α cannot track the changes in the amplitude and phase of u α ′ .Under the simulation parameters in this paper, about 20 ms of transient process are needed to achieve steady-state tracking.qu α also tends to be stable after a transient process of about 20 ms.
u α , qu α , u β , and qu β generated by the SOGI link are used to calculate the dq-axis voltage components of positive sequence at the PCC point u d + and u q + by (4), and the comparison results of u d + , u q + and the voltages of dq-axis u d , u q that do not include the positive and negative sequence decomposition link are shown in Figures 7(a It can be learned from Figures 7(a) and 7(b) that after the positive and negative sequence decomposition, both u d + and u q + have transient processes for a certain period of time, and the amplitudes change greatly.However, u d and u q obtained without positive and negative sequence decomposition have small amplitude changes and short durations in the transient processes.
It can be learned from the analysis of the transient characteristics of DSOGI-PLL that when the voltage at the PCC point drops, the output signal of the SOGI link will go through a transient process to track the amplitude and phase of the changed actual three-phase voltage, although the actual three-phase voltage at the PCC point can be approximately regarded as instantaneous drop.As a result, u d + and u q + also need to go through a transient process to reach the steady state.It can be learned from equation (4) that u d + and u q + will afect the drop process of the positive sequence voltage u PCC at the PCC point and then afect the change processes of current reference values of dq-axis i * d and i * q through equation ( 6).
According to the analysis in Section 4.1, the transient fuctuation processes of u d + and u q + calculated after the positive and negative sequence decomposition is obvious.Terefore, the infuence of this decomposition process should be taken into account in the transient analysis of symmetrical fault current.

Circuit Equation in dq-Axis Coordinate Frame of Positive Sequence including the Positive and Negative Sequence
Decomposition.Since the control of the GSC is based on the dq-axis coordinate frame of positive sequence, in order to study the transient characteristics of the output fault currents i a , i b , and i c of GSC, it is necessary to transform the circuit equation between the voltages u a , u b , and u c at the PCC point and the output voltages of GSC v a , v b , and v c from the three-phase coordinate frame to the dq-axis coordinate frame of positive sequence.Due to space limitations, this paper analyzes and calculates the positive sequence current under symmetric faults and verifes the accuracy of the calculation model.According to the analysis in Section 2.1, the transformation process of the circuit equation is changed after the positive and negative sequence decomposition is added to the control strategy.Terefore, the coordinate transformation process of the voltage equation should be analyzed in detail.
Te equation in the αβ two-phase stationary coordinate frame can be obtained from equation ( 6) by the Clarke transformation: where v α ′ and v β ′ are the output voltages of GSC in the twophase stationary coordinate frame; i α ′ and i β ′ are the output currents of GSC in the two-phase stationary coordinate International Transactions on Electrical Energy Systems frame.According to the analysis in Section 2.1, the SOGI link can use the transfer functions in the complex frequency domain to represent the relationship between the input signal and the output signal.Terefore, before the positive and negative sequence decomposition, the Laplace transform of equation ( 14) is frst carried out to obtain the circuit equation in the complex frequency domain of the two-phase stationary coordinate frame: where i α ′ (t 0 ) and i β ′ (t 0 ) are the initial values of the output currents of GSC in the two-phase stationary coordinate frame when the fault occurs.Te coordinate transformation method adopted by the SOGI link is applied to equation (15), and the GSC output voltages v + α (s) and v + β (s) of positive sequence in the twophase stationary coordinate frame are Te transfer functions G 1 (s) and G 2 (s) in equation ( 2) are substituted into (13) and simplifed to obtain  8 International Transactions on Electrical Energy Systems where i + α (t 0 ) and i + β (t 0 ) are the initial values of positive sequence currents output by GSC in the two-phase stationary coordinate frame at the fault moment; and f 1 (s) and f 2 (s) are the oscillation attenuation components generated in the simplifcation process, and the specifc expressions can be expressed as Transform ( 14) into the time domain: where i + α and i + β are the GSC output currents of positive sequence in the two-phase stationary coordinate frame.
Parker transformation is applied to equation ( 19) to obtain the circuit equations in the dq-axis coordinate frame of positive sequence: where v + d and v + q are the calculated output voltages of GSC in the dq-axis coordinate frame of positive sequence by the DSOGI-PLL link.In the low-voltage ride-through control strategy, the output voltage modulation signals of GSC generated by the inner loop of the control system in the dqaxis coordinate frame of positive sequence v + d−pi and v + q−pi are determined by the output results of the current inner loop of the positive sequence.If the transient response of the positive and negative sequence decomposition leads to v + d ≠ v + d−pi and v + q ≠ v + q−pi , v + d and v + q will not be used as intermediate variables to eliminate and simplify the equations when the circuit equation of the dq-axis coordinate frame of the positive sequence and the equation of the low-voltage ride-through control strategy are combined.Terefore, the following mainly focuses on the relationships between v + d and v + d−pi as well as v + q and v + q−pi .

Analysis of Transient Characteristics of GSC Output
Current in the dq-Axis of Positive Sequence.Te inner loop of the control system is often designed with the steady-state circuit equation of the main loop.When the positive and negative sequence decomposition link is added to the control system, the dq-axis output currents of positive sequence by GSC i d + , i q + , and the dq-axis voltages of positive sequence at the PCC point u d + and u q + are used for the current inner loop control to calculate the output voltage modulation signals v + d−pi and v + q−pi of GSC in the dq-axis coordinate frame of positive sequence.Te response equations of current inner loop control can be expressed as In the control system, the modulation method of output voltages of GSC v a , v b , and v c is the dq-axis voltages of positive and negative sequence v obtained from the current inner loop of the positive and negative sequence are transformed to abc three-phase coordinate frame by parker inverse transformation, and then superimpose them to generate the three-phase modulation wave, whose values are equal to the output voltages of GSC v a , v b , and v c .Terefore, the relationship can be expressed as where "P θPLL −1 " is the Parker inverse transformation matrix of positive components, and "P −θPLL −1 " is the Parker inverse transformation matrix of negative components.
According to the coordinate transformation process described in Section 4.2, coordinate transformation, including positive and negative sequence decomposition link, is performed on equation (22).After the positive and negative sequence decomposition link is applied to equation (22), the decoupling of positive and negative sequence equation can be obtained.With the positive order equation being taken into consideration only, the relations of v + d , v + q , v + d−pi , and v + q−pi can be expressed as International Transactions on Electrical Energy Systems where " * " stands for convolution, "•" stands for the product, and g 1 (t) and g 2 (t) are the expressions of SOGI transfer functions G 1 (s) and G 2 (s) reduced to the time domain, respectively.
According to equations ( 23) and (24), by the coordinate transformation, including the positive and negative sequence decomposition, the circuit equation on the left side of the equal sign can obtain the positive sequence components v + d and v + q , which are the same as those in equation (20), and the specifc relationships between v + d and v + q and v + d−pi and v + q−pi are obtained.It can also be learned from equations ( 23) and (24) In order to verify the accuracy of the conclusion of , the AC system is set to have a threephase fault at 2s in the simulation, and the magnitude of positive sequence voltage at the PCC point u PCC drops to about 7% when it reaches the steady state.
Te parameters used in the simulation are shown in Table 1: Te simulation waveforms of v + d , v + q , v + d−pi , and v + q−pi are obtained by PSCAD/EMTDC as shown in Figures 8(a It can be learned from Figures 8(a) and 8(b) that the dynamic processes of v + d and v + d−pi , v + q and v + q−pi are signifcantly diferent, which verifes the correctness of the analysis results of v + d ≠ v + d−pi and v + q ≠ v + q−pi obtained from equations ( 23) and ( 24) in Section 4.3.
It can be learned that the positive and negative sequence decomposition link makes the circuit equation transformed to the dq-axis coordinate frame of positive sequence have great changes compared with the existing research method.
(1) Te positive and negative sequence decomposition link makes the circuit equation have two more transient attenuation components f 1 (t) and f 2 (t) in the process of coordinate transformation.(2) Te positive and negative sequence decomposition link makes the output voltage modulation signals of GSC in the dq-axis coordinate frame of positive sequence v + d−pi and v + q−pi no longer equal to the calculated output voltage of GSC in the dq-axis coordinate frame of positive sequence v + d and v + q by the positive and negative sequence decomposition link.
Terefore, it is necessary to propose a research method that can take (1) and ( 2) into consideration in order to accurately analyze the transient characteristics of i d + and i q + .

Te Solution Method of dq-Axis Output Current of Positive
Sequence by GSC.According to the analysis in Section 4, there are diferences between v + d and v + d−pi and v + q and v + q−pi due to the transient responses of positive and negative sequence decomposition link.Although (20) and ( 21) can represent their relations, it is difcult to obtain the specifc diferences of their transient processes according to the equation.Terefore, in order to fnd out whether there is a transient process expression with obvious law between v + d and v + d−pi and v + q and v + q−pi , the waveforms of should be observed frst, which are shown in Figure 9.
According to Figure 9, when the magnitude of positive sequence voltage at the PCC point u PCC drops to about 7%, can be approximately regarded as attenuating and oscillating waveforms, and they tend to be stable after a transient process which means the diferences of both v + d − v + d−pi and v + q − v + q−pi are 0. Tese transient processes fuctuate greatly and last for a long time, the durations are about 80 ms.
According to the characteristics of waveform shown in Figure 9, MATLAB is used to ft the oscillation attenuation waveforms v + d − v + d−pi and v + q − v + q−pi with the third-order and the fourth-order oscillation attenuation function, respectively, when the magnitude of u PCC drops to about 7%, which can be expressed as Te circuit equation ( 17) and the inner loop control equation ( 18) are combined and reduced to the equation group shown in equation (24): International Transactions on Electrical Energy Systems In order to facilitate the solution, the integral terms are eliminated by taking the derivative of both sides of (34) and simplifed into the nonhomogeneous second-order diferential equations with variable coefcients about i d + and i q + , which are shown in equation ( 25): ,     Equations ( 25) and ( 26) are substituted into equation (28) as known quantities, and the fourth-order Runge-Kutta method is taken as an example to solve the dq-axis transient current of positive sequence i d + and i q + .Te specifc solution process is shown in Figure 10 [30]: In Figure 10, the basic idea of the Runge-Kutta method is to use a linear combination of function values at several points to replace the derivatives of the Taylor expansion, and then determine the coefcients according to the Taylor series expansion, so that we can not only avoid calculating higher derivatives but also improve the accuracy of the integral and the order of truncation error.

Solution and Simulation of Transient Current Verifcation
under Diferent Voltage Drops.When the magnitude of positive sequence voltage at the PCC point u PCC drops to about 7%, the comparisons between the calculation results and simulation values solved by (25) of the dq-axis transient currents of positive sequence i d + and i q + are shown in Figures 11(a In order to verify whether the solution method proposed in Section 5.1 is suitable for diferent voltage drops, the magnitude of positive sequence voltage at the PCC point u PCC is set to drop to 30% and 60%, respectively, when it reaches the steady state in the simulation.Te waveforms of v + d − v + d−pi and v + q − v + q−pi are obtained as shown in Figures 12(a Set matrix A as "n×4" matrix, and store the calculation results in each column respectively: i di , i qi , di di /dt, di qi /dt i=1 i=i+1 Output A(i,:) Figure 10: Numerical solution fowchart of the Runge-Kutta method.
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It can be learned from Figures 12(a) and 12(b) that the deeper the voltage drop of u PCC is, the smaller the oscillation amplitudes of v + d − v + d−pi and v + q − v + q−pi are, but the oscillation trends of are similar.
Te same ftting method as ( 22) and ( 23) is adopted to ft when the magnitude of positive sequence voltage at the PCC point u PCC reaches 30% and 60%, which are shown in ( 26) and ( 27): When the magnitude of u PCC drops to about 60%, the expressions of v Equations ( 29) and ( 30), as well as equations ( 31) and (32), are substituted into equation ( 28) as known quantities, respectively, and the dq-axis transient currents of positive sequence i d + and i q + when voltage drop reaches 30% and 60% are calculated as shown in Figures 13(a)-13(d): It can be learned from Figure 13 that the calculated current waveforms are basically consistent with the simulation waveforms, and the maximum error is less than 5%, which verifes the accuracy of the proposed analysis method of transient characteristics.Te error between the calculation result and the simulation result is due to the approximate simplifcation of the inverter outlet voltage variation process in the calculation process to facilitate the solution, while the simulation variation process is more complex, and the accuracy of the calculation result is closely related to the simplifed result.However, through comparison, it can be seen that the proposed method signifcantly improves the accuracy of the calculation results.

Conclusions
Tis paper focuses on the scenario of a voltage-symmetric drop at the PCC point caused by a three-phase fault of the permanent magnet direct-drive wind power system, proposing a fault analysis method for the output transient current of the wind farm, which takes into account the positive and negative sequence decomposition link of the control system because it is part of the control system and has an obvious transient response with a long duration.Ten, it accurately depicts the transient characteristics of the fault current.
Te computational principles proposed in this paper are also applicable to wind turbine control systems employing alternative positive-negative sequence decompositions.It is only necessary to substitute the equations corresponding to the DSOGI decomposition method used in this paper with those associated with other positive-negative sequence decomposition methods during the computation process.+ when u PCC drops to 30%, (b) i q + when u PCC drops to 30%, (c) i d + when u PCC drops to 60%, and (d) i q + when u PCC drops to 60%.

Figure 1 :
Figure 1: Schematic diagram of a permanent magnet direct-drive wind power system.

Figure 4 :
Figure 4: Te reference value of dq-axis current.

Figure 5 :
Figure 5: GSC's output current of the positive sequence under external three-phase fault.(a) d-axis output current of positive sequence and (b) q-axis output current of positive sequence.

Figure 7 :
Figure 7: Te voltages at the PCC point of dq-axis.(a) d-axis voltage and (b) q-axis voltage.

Figure 8 :
Figure 8: Output voltage of GSC in the dq-axis coordinate frame of positive sequence.(a) d-axis voltage and (b) q-axis voltage.

Figure 9 :
Figure 9: Te diference between the calculated output voltage of GSC and the actual output voltage of GSC in the dq-axis coordinate frame of positive sequence.
) and 11(b): It can be seen from Figures 11(a) and 11(b) that the errors between the dq-axis output currents of the positive sequence solved by (25) and the simulated currents are basically small.Te simulation results show that the proposed method can efectively improve the accuracy of the research results of fault current transient characteristics by considering the positive and negative sequence decomposition.Trough comparing the transient currents calculated by the existing research methods in Figures 5(a) and 5(b) with Figures 11(a) and 11(b), it can be learned that the positive and negative sequence decomposition link makes the transient currents have an obvious overshoot characteristic and a long duration time, which means the link has a great infuence on the solution accuracy of the transient fault currents.

Figure 11 :
Figure 11: GSC output current of positive sequence when u PCC drops to about 7%.(a) d-axis output current of positive sequence and (b) q-axis output current of positive sequence.

Figure 12 :Figure 13
Figure 12: Te diference between the calculated output voltage of GSC and the actual output voltage of GSC in the dq-axis coordinate frame of positive sequence.(a) Te diference of the d-axis output voltage of the positive sequence and (b) the diference of the q-axis output voltage of the positive sequence.

Figure 13 :
Figure 13: GSC output current of positive sequence.(a) i d+ when u PCC drops to 30%, (b) i q + when u PCC drops to 30%, (c) i d + when u PCC drops to 60%, and (d) i q + when u PCC drops to 60%.

14( 1 )
International Transactions on Electrical Energy Systems Te main results obtained are as follows: Te transient change process of the electrical volume obtained after the positive and negative sequence decomposition is studied, and it is shown that the transient response of the positive and negative sequence decomposition leads to signifcant fuctuation and a long duration of the positive sequence component transient process.Te reason why the coordinate transformation process of positive and negative sequence decomposition should be taken into account when establishing the calculation model of short-circuit current in the time domain is revealed.(2) Based on the change characteristics of the positive and negative sequence decomposition, the dq-axis positive sequence circuit equation after the positive and negative sequence decomposition transformation is derived, and the characteristics of the circuit equation are analyzed.At the same time, the relationship between the output voltage of the converter in the circuit equation and the modulation signal of the converter output voltage in the control equation is derived.Te analysis results show that the transient diference between the two makes it impossible to eliminate the converter output voltage as the intermediate variable when the control equation and the circuit equation are combined.(3) A time-domain short-circuit current calculation model is formed by combining the dq-axis positive sequence circuit equation, the positive sequence control equation, the converter output voltage difference relation, and a short-circuit current transient characteristic analysis method for AC side faults of permanent magnet direct drive wind farm is proposed.Te proposed method takes into account the infuence of positive and negative sequence decomposition on the time domain short circuit current calculation model and is suitable for situations where the voltage of the PCC point has diferent sags.Te calculated results accurately represent the characteristics of the dq-axis positive sequence current with an insurge current and a long transient duration.
Figure 6: Input signal and output signal of SOGI link.