An accurate analysis method for transient characteristics of DC line faults in voltage source converter-based DC systems

Voltage source converter-based DC systems face severe overcurrent problem under DC line faults, which causes signiﬁcant inﬂuence on security, reliability and stability of the power system. As the theoretical basis for relay coordination and protection, the transient fault analysis of voltage source converter-based DC systems needs an in-depth study. Based on the conventional fault analysis, an accurate transient fault analysis method for the DC pole-to-ground fault and pole-to-pole fault through freewheeling diode switch state signals is proposed. In the capacitor discharge stage, fault current calculation error is reduced by taking into consideration the current fed from the grid side. In addition, through detailed analysis of the conduction condition and state of the freewheeling diode, clear calculation equations of transient fault currents are obtained. With the prior knowledge of fault response characteristics, deﬁnitions of freewheeling diode switch state signals assist in unifying calculation processes of different fault stages and simplifying transient fault calculation. Finally, a typical simulation model of the voltage source converter-based DC system was built in PSCAD/EMTDC software. The simulation results veriﬁed the conciseness and correctness of the proposed fault analysis method compared with the conventional fault analysis.

the PPF, capacitor discharge stage, diode freewheel stage and grid-side current feeding stage are included in the transient process. In [17,18], the PPF is studied and similar results are shown, while the fault response is described as a four-stage process. Another stage is introduced between capacitor discharge and diode freewheel stage when the DC-link voltage dips below the line-to-line voltage of the AC generator. Besides, short-circuit fault characteristics are investigated in [19] for VSC-based DC distribution networks connected with different distributed generators. Current-limiting capability of the VSC-based DC distribution system is considered in [20]. And it is found that, for the PPF, diode freewheel stage exists only when fault resistance is small enough and DC-link voltage is underdamped. On the contrary, when fault resistance is quite large, DC-link voltage is over damped and diode freewheel stage does not occur. In addition, literature [21] provides short-circuit fault analysis of multiterminal AC/DC hybrid distribution system with different network topologies, which is considerably valuable for topology determination. A new dq frame modelling of VSCs for DC fault study is proposed in [22], and it achieves good results while only the PPF is considered. In sum, there have been much research work involving fault characteristics of VSC-based DC systems, and the adopted analysis methods have shown reliability and effectiveness [23][24][25]. However, these analyses still expose some deficiencies, one of which lies in the thought that the capacitor discharge current dominates the capacitor discharge stage and thus the current fed from grid side is ignored. When the current fed from grid side is much smaller than capacitor discharge current, neglecting grid-side current can reasonably simplify fault circuits and fault characteristics can still be well expressed. But when the grid-side current is large enough to cause influence on fault calculation accuracy, it should be taken into account to obtain more accurate results. However, there have been few studies that provide a good solution to the aforementioned problem. Moreover, another deficiency may result from the grid-side current feeding stage. The derived fault calculation equations of grid-side current feeding stage are usually aimed at a certain fault circuit. It means that, with changing freewheeling diodes' states, the fault response of grid-side current feeding stage cannot be solved continuously. As for lack of in-depth research on freewheeling diodes' switch on-off states, there have been few methods that manage to obtain detailed calculation equations of the grid-side current feeding stage.
This paper proposes an accurate and simple transient analysis method for both pole-to-ground and PPF in VSC-based DC systems. The proposed method makes a thorough investigation into switch on-off conditions and states of freewheeling diodes. And freewheeling diode switch state signals (FDSSSs) are defined to integrate fault calculation between AC and DC sides of the VSC. In capacitor discharge stage, current fed from grid side is considered and fault calculation accuracy is improved. Furthermore, fault circuits in different stages are represented by the changing FDSSSs. Transient fault responses of pole-to-ground and PPF are solved continuously. Finally, simulation results verified the superiority of the proposed method compared with the conventional method. The proposed fault The rest of this paper is organised as follows. Section 2 presents the structure of VSC-based DC systems, and introduces the conventional fault analysis of DC line faults. Sections 3 and 4 propose the improved fault analysis method for the pole-to-ground and PPF, respectively. In section 5, the proposed fault analysis method is verified on the basis of simulation results. Finally, conclusions are drawn in Section 6.

Structure of voltage source converter-based DC systems
A typical structure of the VSC-based DC system is shown in Figure 1. The AC utility grid is represented by a three-phase AC source with equivalent resistance and inductance. A twolevel VSC is connected to the grid to achieve energy conversion between AC and DC power. The DC-link midpoint is selected as the grounding point, which regulates the DC output voltage at ±U dc /2. Pulse width modulation (PWM) is employed for the VSC to generate trigger signals, which control turnon and turn-off of power electronic devices [9]. It means that trigger signals realize the energy conversion between AC and DC power by the control of power electronic devices. Besides, direct current control is selected as the control strategy for the VSC to fulfil active/reactive power control. In Figure 1, F 1 means the PGF, namely, the positive pole is grounded; F 2 means the PPF.
Considering the severity of DC line faults in VSC-based DC systems, rapidly rising fault currents will cause serious damage to the insulated gate bipolar transistors (IGBTs) in the VSC. In the engineering practice, the IGBTs will be blocked for self-protection when suffered from large fault current [14][15][16]. Given the self-protection of IGBTs, the VSC is assumed to be FIGURE 2 Different stages of the pole-to-ground fault in the conventional method. (a) Capacitor discharge stage, (b) grid-side current feeding stage blocked immediately after the fault occurrence. It is worth to mention that, although the converter is blocked, the AC grid can still feed fault current to DC lines through freewheeling diodes.

2.2
Conventional fault analysis method

Pole-to-ground fault
As mentioned above, the transient process of PGF is divided into capacitor discharge stage and grid-side current feeding stage [15,16]. Brief introduction of these two stages is as follows, and detailed fault calculation is provided in Appendix. Figure 2 illustrates the equivalent circuits of different stages in the PGF. The AC grid is represented by a three-phase AC source named u sa,b,c ; R s and L s mean its equivalent resistance and inductance respectively. The pole-to-ground DC-link capacitor is 2C. R and L mean π-model equivalent resistance and inductance of DC lines from the VSC to the fault point. Owning to the large DC-link capacitor, the DC line grounding capacitor is omitted here.
1. Capacitor discharge stage: When the fault occurs, the positive pole-to-ground DC-link capacitor discharges and the voltage of positive pole drops rapidly. The capacitive discharge current is large, and thus the current fed from AC grid is ignored. The equivalent circuit of this stage is shown in Figure 2(a) 2. Grid-side current feeding stage: When positive DC voltage drops below any phase voltage of the AC grid, the grid feeds current to faulted DC lines through freewheeling diodes. Generally, inductance L s is much larger than resistance R s and therefore inductance is considered only. The equivalent circuit of this stage is shown in Figure 2(b). It needs to be mentioned that the fault calculation equations in Appendix are aimed at a certain fault circuit, while clear explanation of changing fault circuits is not given

Pole-to-pole fault
In general, the transient response process of PPF is divided into capacitor discharge stage, diode freewheel stage and grid-side current feeding stage [15,16]. Relevant calculation can be found in Appendix. Figure 3 shows equivalent circuits of different stages in the PPF. The equivalent pole-to-pole DC-link capacitor is C. 2R and 2L mean π-model equivalent resistance and inductance of DC lines from the VSC to the fault point.
1. Capacitor discharge stage: Similar to the PGF, equivalent DC-link capacitor C discharges rapidly after the occurrence of PPF. Without the consideration of grid-side current, the equivalent circuit of this stage is represented by Figure 3(a)  Figure 3(b) 3. Grid-side current feeding stage: Similar to PGF, both AC grid and DC-link capacitor provide fault current to DC lines in this stage, whose equivalent circuit is shown in Figure 3(c).
It should be noted that the calculation results in Appendix are obtained by the approximation to a three-phase shortcircuit fault

IMPROVED ANALYSIS METHOD FOR TRANSIENT CHARACTERISTICS OF POLE-TO-GROUND FAULT
When the PGF occurs in the VSC-based DC system, the equivalent circuit is illustrated in Figure 4. The positive pole is short circuited to ground, and negative pole is omitted for negligible fault current. IGBTs in the VSC are indicated by dotted lines to show that they are all blocked.
As seen in Figure 2(a), the conventional method only considers the capacitive discharge current in capacitor discharge stage, thus ignoring the current fed from AC grid. To accurately obtain transient characteristics of the PGF, the current fed from AC grid is considered immediately after the fault occurs. According to Figure 4, fault calculation equations can be established based on Kirchhoff 's voltage law and Kirchhoff 's current law for AC side of the VSC: Where the functions G p (i φ ) and G n (i φ ) are defined to judge switch on-off states of freewheeling diodes by the positive or negative value of phase current i φ (φ = a,b,c). i' φ means the derivative of i φ , and other variables are denoted similarly.
Definitions of G p (i φ ) and G n (i φ ) are: G p (i φ ) = [1 + sgn (i φ )]/2 and G n (i φ ) = [1 -sgn(i φ )]/2, in which sgn(⋅) means the signum function. When i φ > 0, it is known that the upper phase-leg freewheeling diode is conducted and the lower is blocked. Based on the conduction of diodes, the related phase is conducted to the positive pole. At this point, from definitions of G p (i φ ) and G n (i φ ), it can be deduced that G p (i φ ) = 1 and G n (i φ ) = 0. Then, it leads to the result that u p G p (i φ ) + u n G n (i φ ) = u p , which is consistent with the conduction of diodes. Contrariwise, When i φ <0, it is obtained that G p (i φ ) = 0, G n (i φ ) = 1 and u p G p (i φ )+u n G n (i φ ) = u n , which means the related phase is conducted to the negative pole. Therefore, these analyses demonstrate that definitions of G p (i φ ) and G n (i φ ) can reflect the conduction of diodes and help to establish fault calculation equations.
As for DC side of the VSC, fault calculation equations are given in Equation (2) according to Figure 4. ( Combine Equations (1) and (2), then the whole fault calculation equations are obtained as follows: There are 8 unknown quantities and 8 equations in Equation (3), and therefore the fault calculation equations can be solved. Considering the complexity of the derived differential equations, the numerical method can be utilized to solve Equation (3). From calculation results of Equation (3), how fault currents at AC and DC sides of the VSC change in transient process can be attained and analysed.
It is noteworthy that there exists a problem in Equation (3). The condition under which upper and lower diodes are both blocked is not considered in Equation (3). In this case, the phase current is always 0 and its derivative is also 0, which does not correspond to the equation of i' φ in Equation (3). To solve this problem, a FDSSS called S φ is defined to describe the conduction state of upper and lower diodes. When S φ = 1, it means that either the upper or lower phase-leg diode is conducted; when S φ = 0, it means that both of the two diodes are blocked. With the use of S φ , updated calculation equations are shown in Equation (4). Hence, the condition under which the two phase-leg diodes are both blocked can be well expressed in Equation (4). As for the calculation of S φ , it should change with the conduction state of the two phase-leg diodes.
By solving Equation (4), the transient process of PGF is calculated interval by interval, in which every interval corresponds to a fixed fault circuit. With the switch on and off of phase-leg diodes, the equivalent fault circuit will change accordingly and S φ should be updated as well. When the phase current drops to 0, S φ should be determined whether to update by the comparison of phase voltage u sφ to positive pole voltage u p and negative pole voltage u n . If u n < u sφ <u p , the two phase-leg diodes are not conducted according to the conduction condition of diodes, and S φ should be set as 0. If u sφ ≥ u p or u sφ ≤ u n , the upper or lower diode are conducted, and S φ should be set as 1. The update of S φ means the change of the equivalent circuit. Therefore, solving process of the former circuit should be ended and that of the latter circuit should be started. Meanwhile, the end values of the former solving process should be set as the initial values of the latter solving process. The flow chart of transient fault calculation for PGF is shown in Figure 5, where S φ0 = |sign(i φ0 )| and t set means the specified calculation time for the transient process.
With the consideration of grid-side current in capacitor discharge stage, the improved analysis method is more accurate. Moreover, the introduction of S φ solves the aforementioned problem that the fault calculation equations only apply to a certain fault circuit in grid-side current feeding stage. The changing equivalent fault circuit is described by the changing S φ . Besides, the definition of S φ assists in unifying calculation processes of capacitor discharge stage and grid-side current feeding stage. As a result, the transient fault response of PGF can be solved continuously using Equation (4) and the flow chart in Figure 5.

IMPROVED ANALYSIS METHOD FOR TRANSIENT CHARACTERISTICS OF POLE-TO-POLE FAULT
When the PPF occurs and all IGBTs are blocked, the equivalent circuit is shown in Figure 6. The system parameters are same with those of the PGF.
Like the analysis method of PGF, fault calculation equations for AC and DC sides of the VSC are studied and derived according to Figure 6. In addition, the FDSSS called S φ is added to describe the conduction state of upper and lower diodes. Finally,   When DC-link voltage drops to 0, the diode freewheel stage starts for the diode clamping effect. However, the diode freewheel stage is not taken into account in Equation (5). In this stage, the upper and lower diodes are all conducted, and the voltages of positive and negative poles remain constant. Consequently, it is deduced that u p´= 0 and u n´= 0, which are not consistent with the expressions of u pá nd u ní n Equation (5). Similar to the FDSSS called S φ , another FDSSS called S d is defined to describe the condition when the upper and lower diodes are all conducted. When S d = 0, it means that the upper and lower diodes are all conducted, and the diode freewheel stage starts. When S d = 1, it means that the diode freewheel stage ends and grid-side current feeding stage starts. The fault characteristics of different fault stages can be applied to the determination of S d . Thus, S d is added and Equation (5) is modified as Equation (6): S d should be adjusted according to the start and end of different stages. The DC-link voltage and relevant fault currents can reflect the fault characteristics of different stages. After the occurrence of PPF, S d is set as 1 because there does not exist the condition when the upper and lower diodes are all conducted in the capacitor discharge stage. When the DC-link voltage drops to 0 (u p −u n = 0), the diode freewheel stage starts and S d should be set as 0, which means that all freewheeling diodes are conducted. When the sum of upper diode currents fed from AC grid exceeds the DC fault current (i p > i f ), the diode freewheel stage ends and the grid-side current feeding stage starts, and S d should be set as 1. The flow chart of transient fault calculation for PPF is similar to that of PGF. With the consideration of the diode freewheel stage, the differential Equation (6), instead of Equation (4), are solved to achieve fault calculation. In addition, the part of state judgement and signal adjustment in Figure 5 needs to be updated with a new one, which is given in Figure 7. Besides, the initial values of FDSSSs are modified to S 0 = (S a0 , S b0 , S c0 , S d0 ), where S d0 = 1.
Similar to the analysis of PGF, the improved analysis method for PPF considers the grid-side current immediately after the fault occurrence. Therefore, the transient fault calculation accuracy is improved in the capacitor discharge stage. Meanwhile, instead of the approximation to three-phase short-circuit fault, the exact fault circuit in the grid-side current feeding stage is described by S φ . The changing S φ can represent the changing fault circuit, and thus the fault response is solved continuously. Besides, on the basis of S φ , the definition of S d further unifies the fault calculation processes of capacitor discharge stage, diode freewheel stage and grid-side current feeding stage. In brief, the transient fault calculation of the PPF is more accurate and more concise.

SIMULATION RESULTS AND ANALYSES
A typical VSC-based DC system model shown in Figure 1 has been built using PSCAD/EMTDC, whose detailed system parameters are presented in Table 1. Direct current control is selected for the VSC to fulfil active/reactive power control. This  [14][15][16].
Considerable simulations of pole-to-ground and PPF have been carried out to make the comparison between the conventional and improved methods. The simulation step is 25 μs, and the fault occurrence time is 4.0 s. The VSC is blocked immediately after the fault initiation. The superiority of the proposed transient fault analysis method is verified by the simulation results.

5.1
Simulations of pole-to-ground fault In Figure 8(a), it can be seen that there are some errors between the calculation results i f,CM and the simulation results i f,SIM . The maximum error of the calculation results i f,CM is about −10% from Figure 8(b). It reveals that some calculation error will occur if using the conventional method. By contrast, using the improved method, the calculation results i f,IM are very close to the simulation results i f,SIM after taking the current fed from grid side into account. At the end of the capacitor discharge stage, the DC fault current calculation error of the conventional method is −9.86%, and that of the improved method is −0.42%. The superiority of the improved method to the conventional method is verified. It needs to be noted that the fault calculation of capacitor discharge stage is an extremely important issue in the protection design of DC systems. The fault detection and isolation requires to be accomplished in several milliseconds, which corresponds to the capacitor discharge stage. Thus, an accurate fault calculation method within the capacitor discharge stage is the necessary prerequisite for system protection.

Comparison between capacitor discharge current and grid-side current
As for the capacitor discharge stage shown in Figure 8, the current fed from AC grid i p,SIM and the capacitor discharge current i C,SIM are given together in Figure 9 to make a further comparison. It can be found that, in the built VSC-based DC system, the current fed from AC grid is at a high value compared with the capacitor discharge current in a short period after the fault occurs, which lasts for 0.81 ms. The period during which i p,SIM is larger than i C,SIM is indicated by the red area in Figure 9. In such a situation, omitting the fault loop at AC side of the VSC in the conventional method may cause certain calculation error,

FIGURE 9
Comparison between the current fed from AC grid and capacitor discharge current in pole-to-ground fault which explains the reason for higher accuracy of the improved method.

Calculation results of phase currents and DC fault current in transient state
Based on Equation (4) and the flow chart in Figure 5, transient fault currents within 0.04 s (two periods) after PGF occurs are calculated, and these results together with simulation results are shown in Figure 10. Comparison between calculation and simulation results of phase currents is made in Figure 10(a). Besides, calculation and simulation results of DC fault current are plotted in Figure 10(b). In these two subfigures, simulation results are presented by the thin and solid curves; while calculation results are presented by the heavy and dashed curves. From Figure 10(a), the simulation results and calculation results of phase currents are nearly identical. The improved method shows high calculation accuracy of phase currents. In Figure 10(b), the error between calculated and simulated curves of DC fault current is quite small. The peak value of DC fault current in simulation is 20.04 kA and the calculated result is 20.01 kA, whose error is −0.15%. Therefore, the conclusion is that the improved method has high calculation accuracy of the DC fault current. Moreover, it needs to be mentioned that the improved method solves the transient process of the PGF continuously.

Changes of freewheeling diode switch state signals in transient state
As for the transient stage shown in Figure 10, the changes of FDSSSs S a , S b and S c involved in Figure 5 are displayed in Figure 11. Besides, with the consideration of the FDSSS S d involved in Figure 7, the change of S d is also calculated and shown in Figure 11. It can be seen that S d is constantly 1, which means that the diode freewheel stage does not exist in the transient state of the PGF. And it is consistent with the actual fault response. In addition, the period under which S a , S b or S c is 0 corresponds

5.2
Simulations of pole-to-pole fault

Calculation results of DC fault current in capacitor discharge stage
In Figure 12(a), the calculation and simulation results of DC fault current are compared. And calculation errors of the conventional and improved methods are presented in Figure 12(b). It indicates that the improved method has little calculation error; and the conventional method may cause larger error. The maximum error of the conventional method exceeds −10% as shown in Figure 12(b). The short vertical line in every curve in Figure 12(a) means the corresponding end point of the capacitor discharge stage. It shows that the conventional method may obtain an inaccurate end point of the capacitor discharge stage. As a result, faster protection speed is required for isolation devices in the conventional method, which hurts the system economy. By contrast, the improved method exhibits much better calculation performance, and has less error in calculating the end point of the capacitor discharge stage. Furthermore, using the improved method, the fault response of the capacitor discharge stage and diode freewheel stage is solved continuously. The calculation complexity of the PPF in transient state is greatly simplified.

FIGURE 13
Comparison between the current fed from AC grid and capacitor discharge current in pole-to-pole fault

Comparison between capacitor discharge current and grid-side current
As for the capacitor discharge stage shown in Figure 12, the current fed from AC grid i p,SIM and the capacitor discharge current i C,SIM are given together in Figure 13 for further analysis. It is found that, in the early period of capacitor discharge stage, the current fed from AC grid is at a high value compared with the capacitor discharge current. Based on the calculation results of the conventional method in Figure 12, it is indicated that ignoring the grid-side current in capacitor discharge stage may cause error accumulation problem, and affect the fault calculation accuracy.

Calculation results of phase currents and DC fault current in transient state
Based on Equation (6) and the flow chart in Figure 7, transient currents within 0.04 s after the PPF occurs are calculated and presented in Figure 14. Calculation and simulation results of phase currents are compared in Figure 14(a). Calculated and simulated curves of DC fault current are plotted together in Figure 14(b). It can be seen from Figure 14(a) that the calculation results of phase currents are sufficiently close to the simulation results. The improved method exhibits high accuracy in phase current calculation. From Figure 14(b), the DC fault current in simulation peaks at 17.20 kA and the calculated result is 17.24 kA, whose error is 0.23%. Thus, it comes the conclusion that the improved method has high calculation accuracy of the DC fault current.

Changes of freewheeling diode switch state signals in transient state
Furthermore, the changes of FDSSSs S a , S b , S c and S d are plotted in Figure 15. It shows that S b is constantly 1 which means that there is no such a period when the phase B current is  Figure 14(a). Moreover, it is observed that there exist two periods when S d is constantly 0 which means the diode freewheel stage occurs twice. And the two occurrences of the diode freewheel stage is verified by the simulation results of the DC-link voltage shown in Figure 15(b). This figure also compares the simulation and calculation results of DC-link voltage, and effectiveness of the improved method is validated. From Figure 15, it is concluded that the improved method using FDSSSs can grasp firmly the transient characteristics of the PPF. In addition, the diode freewheel stage may happen more than once, which is not mentioned in the conventional method.

Comparison analysis under different parameters
To analyse the influence of different parameters, considerable simulation cases under different parameters have been carried out. The peak of DC fault current and the peak time are chosen to compare the calculation accuracy of the conventional and improved methods.
Icm and Tcm mean peak current and peak time of the conventional method; Iim and Tim mean those of the improved

Different DC-link capacitors
The simulation and calculation results under different DC-link capacitors are shown in Figure 16. In PGFPGF with DClink capacitors between 500-6000 μF, and in PPF with these between 500-2000 μF, the peak currents occur in the grid-side current feeding stage. Due to small DC-link capacitor, the capacitor discharge current is small while the grid-side current is relatively large. The conventional method ignores the large grid-side current, and so it gets incorrect peak times Tcm, which are in the capacitor discharge stage. These cases are shown by "incorrect Tcm cases" in Figure 16. For other cases, the conventional method still shows more error than the improved method, especially for the peak current. With consideration of grid-side current, the improved method shows high accuracy for both peak current and peak time under different DC-link capacitors.

Different fault resistances
Simulation cases under different fault resistances have been investigated. These results are given in Figure 17. For PGF, DC fault current reaches its peak in the grid-side current feeding stage when the fault resistance is relatively small (R f = 0.005, 0.01 and 0.05 Ω). In these cases, the conventional method provides incorrect peak times Tcm for omitting grid-side current. When the fault resistance is relatively high (R f = 5 and 20 Ω), the RLC circuit in the conventional method is overdamped. The calculated peak current Icm is the initial value, and the calculated peak time Tcm is 0. It is clearly inconsistent with simulation results. From Figure 17, it is indicated that the improved method shows much less error, and has high accuracy for both pole-to-ground and PPF.

FIGURE 18
Comparison analysis under different source parameters

Different source parameters
This part considers different source parameters (source resistance R s and source inductance L s ) in simulation. Figure 18 shows the simulation and calculation results. It needs to be noted that changing source parameters has no effect on calculation results of the conventional method. From simulation results, when source impedance is relatively small, peak currents occur in the grid-side current feeding stage. The reason for this is that, with small impedance of AC source, the grid-side current feeding effect is obvious. In this situation, the conventional method will underestimate the peak current and peak time.
With consideration of grid-side current from AC source, accurate peak current and peak time are obtained in the improved method for both pole-to-ground and PPF.

Different line parameters
In this part, simulation cases under different line parameters have been discussed. From Figure 19, for all the cases of PGF, the peak currents happen in the grid-side current feeding stage.
For PPF, DC fault current experiences its peak in the gridside current feeding stage when the line resistance is relatively small and impedance relatively large. With small resistance R and large impedance L in grid-side current loop, time constant L/R is large, and grid-side current attenuates slowly. It means that grid-side current feeding effect is obvious under small line resistance and large line impedance. The conventional method cannot cope with these cases. It is observed that, under different line parameters, the improved method keeps high calculation accuracy. And it shows much better performance than the conventional method.

FIGURE 19
Comparison analysis under different line parameters

CONCLUSIONS
An accurate analysis method for transient characteristics of pole-to-ground and PPF using FDSSSs is proposed in this paper. The FDSSS is defined and utilized to describe fault characteristics of the VSC. The current fed from grid side is considered in the capacitor discharge stage. Besides, the fault calculation processes of different stages are unified with the definitions of FDSSSs. By the simulation validation, the improved method shows high calculation accuracy for phase currents and DC fault current. Moreover, comparison analysis under different parameters indicates that the improved method has much better calculation performance than the conventional method. In this study, simulation results show that the diode freewheel stage may occur more than once in the PPF. And it is found that the peak of DC fault current may happen in the grid-side current feeding stage. With consideration of grid-side current feeding effect, the proposed method can accurately describe the fault response characteristics and obtain good results.

APPENDIX
The pole-to-ground fault: 1. Capacitor discharge stage ) . (A1) 2. Grid-side current feeding stage The pole-to-pole fault: 1. Capacitor discharge stage . (A3) 2. Diode freewheel stage 3. Grid-side current feeding stage In the conventional method, transient process of the poleto-ground fault is divided into capacitor discharge stage and grid-side current feeding stage; besides, transient process of the pole-to-pole fault is divided into capacitor discharge stage, diode freewheel stage and grid-side current feeding stage. The faulted pole in pole-to-ground fault is selected as the positive pole here and the analysis method can also be applied to the scenario when the negative pole is faulted.
Fault current expressions or calculation methods in different stages are given in (A1-A5). Where u p means the voltage of positive pole, i f means DC fault current, i a,b,c means phase currents, u dc means DC-link voltage, V 0 means the initial value of DC-link voltage, I 0 means the initial value of DC fault current, I f0 means the value of DC fault current at the end of capacitor discharge stage, I a0 means the current of phase A at the end of diode freewheel stage, α means the voltage angle of phase A at the end of diode freewheel stage, φ 0 means load angle, I g means steady-state current of threephase short-circuit fault, ω s means angular frequency of AC grid.