Analysis Method of Single-pole Grounding Transient Fault Current in Symmetrical Bipolar Flexible DC System

Abstract

The present invention discloses a method for analyzing the transient fault current of a symmetrical bipolar flexible DC system under single-pole grounding, and relates to the technical field of power fault analysis. The key points of the technical solution are: establishing a model domain fault equivalent network and a phase domain fault equivalent network under a single-pole grounding fault; obtaining line phase domain equivalent parameters taking into account the coupling effect of positive and negative lines according to the equivalent relationship analysis of electrical quantities between the model domain fault equivalent network and the phase domain fault equivalent network; obtaining a phase domain low-order equivalent calculation circuit after reducing the order of the phase domain fault equivalent network; solving the phase domain fault current analytical formula according to the line phase domain equivalent parameters and the phase domain low-order equivalent calculation circuit. The present invention takes into account the line phase domain equivalent parameters of the coupling effect, and can use the phase domain equivalent network to perform analytical calculation on the fault transient current. Since the order of the phase domain equivalent network is lower than that of the model domain equivalent network, the system order and calculation complexity can be greatly reduced.

Description

Technical Field

[0001] The invention relates to the technical field of power fault analysis, and more specifically, to a method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system. Background Art

[0002] Flexible DC systems based on modular multilevel converters (MMC) are of great significance for achieving large-scale access and consumption of new energy. Among them, the flexible DC system with a symmetrical bipolar structure has high flexibility and reliability, and is the construction trend of future power grids. Fault analysis and short-circuit calculation are important links in the design and operation of flexible DC systems. DC faults in flexible DC systems mainly include three types: bipolar short-circuit faults, unipolar grounding faults, and line break faults. Among them, the probability of unipolar grounding faults is the highest, and the calculation and analysis of transient currents for unipolar grounding faults are of great significance.

[0003] At present, there are two main methods for calculating single-pole grounding transient fault current: numerical method and analytical method. The numerical method can obtain a relatively accurate fault waveform, but it is difficult to traverse all fault scenarios, so it has certain limitations in revealing the DC fault mechanism. The analytical method can give an expression of the fault current with respect to the system parameters, but there are great challenges in its implementation. On the one hand, due to the fact that multiple converters feed current to the fault point at the same time during a fault, the fault equivalent network has a high order and is difficult to analyze. On the other hand, since the single-pole grounding fault is an asymmetric fault, the electrical quantity of the fault line will be affected by the coupling effect from the sound pole line. The fault zero-mode network and the one-mode network are connected in series at the fault point, and the network order and calculation complexity increase exponentially, making analysis difficult.

[0004] In the prior art, the use of phase domain equivalent networks can reduce the system order and thus reduce the computational complexity, but the parameters in the phase domain equivalent network are unknown. Some studies have ignored the coupling effect of the positive and negative lines, and used the line first mode parameters or line zero mode parameters to approximate the phase domain equivalent parameters of the line. However, due to the large difference in impedance between the zero mode network and the first mode network under a single-pole grounding fault, there is also a large difference in the contribution to the fault current. The above approximation method will introduce a large calculation error. In addition, there are studies that propose to use data fitting optimization algorithms to calculate the equivalent parameters of phase domain DC lines, but they need to rely on existing fault data. In general, fault data needs to be obtained through simulation, so this method is difficult to apply when only the system design parameters are given. It can be seen that the current calculation of single-pole grounding transient fault current in flexible DC systems has the following deficiencies: the numerical method cannot obtain an analytical expression of the fault current, the existing analytical method does not fully consider the coupling effect between the positive and negative lines, the calculation error is large, and it is difficult to implement when only the system design parameters are given. Summary of the invention

[0005] In order to solve the deficiencies in the prior art, the purpose of the present invention is to provide a method for analyzing the transient fault current of a single-pole grounding system in a symmetrical bipolar flexible DC system. Taking into account the phase domain equivalent parameters of the line with coupling effects, the phase domain equivalent network can be used to analyze and calculate the fault transient current. Since the order of the phase domain equivalent network is lower than that of the mode domain equivalent network, the system order and calculation complexity can be greatly reduced.

[0006] The above technical objectives of the present invention are achieved through the following technical solutions: A method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system, comprising the following steps:

[0007] Establish the mode domain fault equivalent network and phase domain fault equivalent network under single-pole grounding fault;

[0008] According to the equivalent relationship between the electrical quantities of the mode domain fault equivalent network and the phase domain fault equivalent network, the line phase domain equivalent parameters taking into account the coupling effect of the positive and negative lines are obtained;

[0009] After the phase domain fault equivalent network is reduced in order, a phase domain low-order equivalent calculation circuit is obtained;

[0010] The analytical formula of phase domain fault current is solved based on the line phase domain equivalent parameters and the phase domain low-order equivalent calculation circuit.

[0011] Furthermore, the process of establishing the module domain fault equivalent network is specifically as follows:

[0012] Establish a zero-mode fault equivalent network and a one-mode fault equivalent network;

[0013] By converting the boundary conditions at the fault point in the phase domain to the mode domain, the connection mode of the zero-mode fault equivalent network and the one-mode fault equivalent network is determined;

[0014] The zero-mode fault equivalent network and the one-mode fault equivalent network are connected at the fault point according to the connection method to obtain the mode domain fault equivalent network.

[0015] Furthermore, the electrical quantities include resistance, inductance and capacitance of the line branches.

[0016] Furthermore, the line phase domain equivalent parameter is a phase domain equivalent impedance that takes into account the coupling effect of the positive and negative lines.

[0017] Furthermore, the phase domain equivalent impedance is obtained by weighting the zero mode impedance and the first mode impedance according to the weighting coefficient.

[0018] Furthermore, the calculation formula of the phase domain equivalent impedance is specifically:

[0019] Z L =λZ L0 +(1-λ)ZL1

[0020] Among them, Z L represents the phase domain equivalent impedance of a single branch; λ represents the weighting coefficient; Z L0 Z represents the zero-mode impedance of the corresponding branch in the zero-mode fault equivalent network; L1 Represents the first-mode impedance of the corresponding branch in the first-mode fault equivalent network.

[0021] Furthermore, the weighting coefficient is determined by the Thevenin equivalent impedance of the zero-mode fault equivalent network and the one-mode fault equivalent network starting from the fault point.

[0022] Furthermore, the calculation formula of the weighted coefficient is specifically:

[0023]

[0024] Among them, λ represents the weighting coefficient; L line_1 represents the first-mode inductance of the DC line to be determined; L line_0 represents the zero-mode inductance of the DC line to be determined; L eq1 L represents the Thevenin equivalent inductance of the next-mode fault equivalent network starting from the fault point under a single-pole grounding fault; eq0 It represents the Thevenin equivalent inductance of the zero-mode fault equivalent network starting from the fault point under a single-pole grounding fault.

[0025] Furthermore, the phase domain fault current analytical expression is represented by a time domain transient fault current analytical expression.

[0026] Furthermore, the solution process of the phase domain fault current analytical formula is specifically as follows:

[0027] Convert the phase domain low-order equivalent calculation circuit to the complex frequency domain;

[0028] The circuit theory method is used to analyze and obtain the complex frequency domain transient fault current analytical formula of the low-order equivalent calculation circuit in the phase domain after conversion in the fault circuit;

[0029] The transient fault current analytical expression in the complex frequency domain is converted to the time domain using the inverse Laplace transform, and the transient fault current analytical expression in the time domain is obtained.

[0030] Compared with the prior art, the present invention has the following beneficial effects:

[0031] 1. The method for analyzing the transient fault current of a symmetrical bipolar flexible DC system with single-pole grounding provided by the present invention takes into account the line phase domain equivalent parameters of the coupling effect, and can use the phase domain equivalent network to analyze and calculate the fault transient current. Since the order of the phase domain equivalent network is lower than that of the module domain equivalent network, the system order and calculation complexity can be greatly reduced;

[0032] 2. The present invention takes into account the coupling effect of the sound line on the faulty line in the case of a single-pole grounding fault. The calculation accuracy is higher than that of the method that does not consider the coupling effect. The approximate analytical expression of the phase-domain transient current of the faulty line can be calculated when only the system design parameters are given, which can provide mechanism support for the system DC line protection design and setting. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] The drawings described herein are used to provide a further understanding of the embodiments of the present invention, constitute a part of this application, and do not constitute a limitation of the embodiments of the present invention. In the drawings:

[0034] Figure 1 is a schematic diagram of a two-terminal flexible DC system in an embodiment of the present invention;

[0035] Figure 2 is a schematic diagram of a zero-mode fault equivalent network in an embodiment of the present invention;

[0036] Figure 3 is a schematic diagram of a fault equivalent network in an embodiment of the present invention;

[0037] Figure 4 is a schematic diagram of a module domain fault equivalent network in an embodiment of the present invention;

[0038] Figure 5 is a schematic diagram of a phase domain fault equivalent network in an embodiment of the present invention;

[0039] Figure 6 is a simplified schematic diagram of a module domain equivalent circuit in an embodiment of the present invention;

[0040] Figure 7 is a simplified phase domain equivalent circuit schematic diagram in an embodiment of the present invention;

[0041] Figure 8 is a schematic diagram of a phase domain low-order equivalent value calculation circuit in an embodiment of the present invention;

[0042] Fig. 9 is a schematic diagram of transient fault current calculation results under different fault distances in an embodiment of the present invention;

[0043] Fig.10 1 is a schematic diagram of transient fault current calculation results under different fault resistances in an embodiment of the present invention. DETAILED DESCRIPTION

[0044] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with embodiments and drawings. The exemplary embodiments of the present invention and their description are only used to explain the present invention and are not intended to limit the present invention.

[0045] Embodiment: Method for analyzing single-pole grounding transient fault current in symmetrical bipolar flexible DC system

[0046] The simulation model is as follows Figure 1 The two-terminal flexible DC system shown in the figure is used as an example. The system adopts a bipolar symmetrical earth return line connection method, with a symmetrical structure of positive and negative poles and an equivalent inductance of the grounding branch L g Inverters MMC1 and MMC2 are connected to busbars B1 and B2 respectively, and current limiting reactors are installed at both ends of the DC line, and their inductance values ​​are L s .

[0047] Step 1: Establish the mode domain fault equivalent network and phase domain fault equivalent network under single-pole grounding fault.

[0048] 1. Modular domain fault equivalent network under single-pole grounding fault

[0049] In the case of a single-pole grounding fault, the fault current only flows between the fault circuit formed by the positive and negative lines and the earth, and the fault current contains both zero-mode components and single-mode components. Figure 1 In the system, when a positive ground fault occurs in the line, the zero-mode fault equivalent network and the one-mode fault equivalent network are established as follows: Figure 2 , Figure 3 shown.

[0050] Figure 2 , 3 In the example, the fault point is defined as bus B0. i , L i , C i Indicates the inverter MMC i The fault equivalent parameter (i=1,2). The line between busbars B0 and B1 is the fault branch B. 01 .L 01 _0, R 01 _0 represents the fault branch B 01 The zero-mode equivalent inductance and resistance, L 01_1 , R 01_1 Respectively represent the fault branch B 01 The equivalent inductance and resistance of a module. 01 _0, i 01 _1 represents the fault branch B 01 The line between busbars B0 and B2 is the fault branch B. 02 .L 02 _0, R 02 _01 indicates fault branch B 02 The zero-mode equivalent inductance and resistance, L 02_1 , R 02_1 Respectively represent the fault branch B 02The equivalent inductance and resistance of a module. 02 _0, i 02 _1 represents the fault branch B 02 The zero-mode and first-mode currents. L g Indicates the equivalent inductance of the grounding branch. When a single-pole grounding fault occurs, the zero-mode electrical quantity forms a loop with the earth through the positive and negative lines and the grounding branch, while the single-mode electrical quantity does not pass through the grounding branch. Therefore, the zero-mode impedance of the grounding branch is 2L g U0 and U1 represent the port voltages of the zero-mode network and the one-mode network at the fault point, respectively.

[0051] get Figure 2 , Figure 3 After the zero-mode fault equivalent network (referred to as zero-mode network) and the one-mode fault equivalent network (referred to as one-mode network) are shown, the boundary conditions at the fault point in the phase domain are converted to the mode domain to determine the connection mode of the zero-mode network and the one-mode network.

[0052] In the case of a positive grounding fault, the phase voltage and current of the positive and negative lines at the fault point meet the following requirements:

[0053]

[0054] Where U dc , U p are the initial voltage and fault voltage at the fault point of the positive line respectively; i p 、i n is the current at the fault point of the positive and negative lines; R f is the fault resistance.

[0055] By performing phase transformation on equation (1), we can obtain:

[0056]

[0057] Where U0 and U1 are the voltages at the fault ports of the zero-mode network and the single-mode network, respectively; i0 and i1 are the currents at the fault ports of the zero-mode network and the single-mode network, respectively.

[0058] From formula (2), we can see that the zero-mode and one-mode fault equivalent network currents are equal at the fault point. Therefore, it is determined that the zero-mode network and the one-mode network are connected in series at the fault point. After the series connection, the fault port voltage is That is to say Therefore, the additional network for single-pole grounding mode fault can be obtained as follows: Figure 4 shown.

[0059] 2. Phase domain fault equivalent network under single-pole grounding fault

[0060] in accordance with Figure 1A phase domain equivalent network can be constructed when a positive pole grounding fault occurs in a DC line, such as Figure 5 shown.

[0061] Figure 5 In, L 01 , R 01 Respectively represent the fault branch B 01 Phase domain equivalent inductance and resistance after considering coupling. 01 Indicates fault branch B 01 The phase fault current. 02 , R 02 Respectively represent the fault branch B 02 Phase domain equivalent inductance and resistance after considering coupling. 02 Indicates fault branch B 02 Phase domain fault current.

[0062] In the above-mentioned mode domain fault equivalent network and phase domain fault equivalent network, the fault equivalent parameters of the converter, the zero mode and first mode parameters of the line, and the grounding branch parameters can be obtained through the design parameters, and they are all known quantities. Due to the coupling between the positive and negative pole lines, the fault pole line in the phase domain equivalent network has a coupling effect from the sound pole line, so the phase domain equivalent inductance and resistance of the line are unknown quantities to be determined.

[0063] Step 2: Based on the equivalent relationship between the electrical quantities of the mode domain fault equivalent network and the phase domain fault equivalent network, the line phase domain equivalent parameters taking into account the coupling effect of the positive and negative lines are obtained.

[0064] for Figure 4 The network shown, with Z L0 , Z R0 Respectively represent the impedance of the left and right branches of the zero-mode network. L1 , Z R1 Respectively represent the impedance of the left and right branches of a single-mode network. Figure 5 The network shown, with Z L , Z R Represent the impedance of the left and right branches of the phase domain network respectively. Then we can get Figure 6 , Figure 7 The simplified mode domain and phase domain equivalent circuits are shown.

[0065] Solve based on circuit analysis Figure 6 For the network in , the zero-mode and first-mode currents of the left branch are:

[0066]

[0067]

[0068] In the formula, Z eq0 and Zeq1 for Figure 6 The Thevenin equivalent impedance of the zero-mode network and the one-mode network as seen from the fault point.

[0069] Figure 7 is the phase domain equivalent network that is expected to be solved considering the coupling effect of positive and negative lines. Figure 7 It can be seen that when the positive pole is grounded, the terminal voltage at the fault point is U p , the phase domain fault current of the branch on the left side of the fault point is:

[0070]

[0071] but Figure 7 Phase domain equivalent impedance Z of the left branch L It can be expressed as:

[0072]

[0073] Substituting the zero-mode and first-mode current expressions in equation (3) into equation (6), the equivalent impedance of the left branch phase domain can be obtained as:

[0074]

[0075] In the above formula, the weighting coefficient λ is defined as:

[0076]

[0077] but Figure 7 The phase domain equivalent impedance of the left branch can be further expressed as:

[0078] Z L =λZ L0 +(1-λ)Z L1 (9)

[0079] Because Z L0 The fault branch B 01 The impedance of the fault branch B is connected in series with the impedance of the converter and the grounding branch. 01 The phase domain equivalent impedance also satisfies the relationship shown in formula (9). It can be seen that the phase domain equivalent impedance considering the coupling effect of the positive and negative lines is the weighted zero-mode and one-mode impedances of the line. Among them, the weighting coefficient is related to the Thevenin equivalent impedance of the system zero-mode network and the one-mode network from the fault point. When only the current in the initial stage after the fault is considered, the high-frequency part in the system plays a dominant role in the fault current, then λ is:

[0080]

[0081] Where, L line_0 and L line_1They represent the zero-mode and first-mode inductances of the DC line to be determined, respectively, L eq0 and L eq1 They are the Thevenin equivalent inductances of the zero-mode network and the single-mode network for a single-pole grounding fault as seen from the fault point.

[0082] Step 3: Reduce the order of the phase domain fault equivalent network to obtain the phase domain low-order equivalent calculation circuit.

[0083] for Figure 7 The circuit shown can be simplified to Figure 8 Two separate circuits are shown.

[0084] Figure 8 In, R f1 and R f2 The value of and the current ratio of the fault branches B01 and B02 i 01 / i 02 Since in the initial stage of the fault, the fault current is mainly dominated by the equivalent inductance of the fault branch and its connected converter station. Therefore, take i 01 / i 02 The current ratio is the ratio of the inductance of the corresponding fault branch and its connected converter station, that is:

[0085] i 01 / i 02 =(L 02 +L2+L g ) / (L 01 +L1+L g ) (11)

[0086] but Figure 8 The fault resistance value can be expressed as:

[0087] R f1 =(1+(L 01 +L1+L g ) / (L 02 +L2+L g ))R f

[0088] R f2 =(1+(L 02 +L2+L g ) / (L 01 +L1+L g ))R f (12).

[0089] Step 4: Solve the phase domain fault current analytical formula based on the line phase domain equivalent parameters and the phase domain low-order equivalent calculation circuit.

[0090] Will Figure 8The circuit shown is converted to the complex frequency domain and analyzed using circuit theory. The complex frequency domain transient fault current analytical formula of the fault branches B01 and B02 can be obtained as follows:

[0091]

[0092]

[0093] Where s is the Laplace operator.

[0094] In addition, by using the inverse Laplace transform to transform equation (13) into the time domain, the corresponding time domain transient fault current analytical equation i can be obtained: 01 and i 02 .

[0095] Step 5: Validation

[0096] In order to verify the accuracy of the method proposed in the present invention, the following Figure 1 The two-terminal flexible DC system is shown. The converter adopts a detailed equivalent circuit model (Detailed Equivalent Circuit Model), and the parameters of MMC1 and MMC2 are the same. In order to eliminate the influence of the frequency-variable distributed parameter characteristics of the line, the DC line adopts a coupled lumped parameter model. The impedance equivalent parameters of the line are given by the line constant program in PSCAD. The system parameters are shown in Table 1.

[0097] Table 1 Test system parameters

[0098]

[0099] For the DC line positive pole grounding fault, different fault distances and fault resistances are set respectively, and the transient fault current i at the MMC1 outlet is calculated by the method proposed in the present invention. 01 The fault components are analyzed and compared with the simulation values ​​and the calculated values ​​without considering the coupling.

[0100] Fig. 9 The calculation results at different fault distances are as follows: the fault distances are 10km, 70km and 140km respectively, and the fault resistance is 0.01Ω. Fig. 9 It can be seen that the existing method does not consider the coupling effect between lines, and the fault current calculated by line zero mode impedance will be lower than the simulation value. However, since the method proposed in the present invention fully considers the coupling effect between lines, the transient fault current obtained is basically consistent with the simulation value, with only a small error, which verifies the effectiveness of the present invention.

[0101] Fig.10The calculation results under different fault resistances are as follows: the fault resistances are 0.01Ω, 50Ω and 100Ω respectively, and the fault resistance distance is 50km. Fig.10 It can be seen that under different fault resistances, the method proposed by the present invention still has high accuracy, and the transient fault current calculation value and the simulation value basically coincide. However, the existing method has a large error when the fault resistance is small, and the error decreases with the increase of the fault resistance. This is because as the fault resistance increases, the resistance in the system will gradually replace the inductance and play a leading role in the fault current. Since there is no mutual induction and coupling between the resistors, the coupling effect between the lines is not considered, which has little effect on the fault current calculation.

[0102] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system, characterized in that: The following steps are involved: Establish the mode domain fault equivalent network and phase domain fault equivalent network under single-pole grounding fault; According to the equivalent relationship between the electrical quantities of the mode domain fault equivalent network and the phase domain fault equivalent network, the line phase domain equivalent parameters taking into account the coupling effect of the positive and negative lines are obtained; After the phase domain fault equivalent network is reduced in order, a phase domain low-order equivalent calculation circuit is obtained; The analytical formula of phase domain fault current is solved based on the line phase domain equivalent parameters and the phase domain low-order equivalent calculation circuit.

2. The method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system according to claim 1 is characterized in that: The process of establishing the model domain fault equivalent network is specifically as follows: Establish a zero-mode fault equivalent network and a one-mode fault equivalent network; By converting the boundary conditions at the fault point in the phase domain to the mode domain, the connection mode of the zero-mode fault equivalent network and the one-mode fault equivalent network is determined; The zero-mode fault equivalent network and the one-mode fault equivalent network are connected at the fault point according to the connection method to obtain the mode domain fault equivalent network.

3. The method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system according to claim 1 is characterized in that: The electrical quantities include resistance, inductance and capacitance of the line branches.

4. The method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system according to claim 1 is characterized in that: The line phase domain equivalent parameter is the phase domain equivalent impedance that takes into account the coupling effect of the positive and negative pole lines.

5. The method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system according to claim 4 is characterized in that: The phase domain equivalent impedance is obtained by weighting the zero mode impedance and the first mode impedance according to the weighting coefficient.

6. The method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system according to claim 5 is characterized in that: The calculation formula of the phase domain equivalent impedance is specifically: WITH L =λZ L0 +(1-λ)Z L1 Among them, Z L represents the phase domain equivalent impedance of a single branch; λ represents the weighting coefficient; Z L0 Z represents the zero-mode impedance of the corresponding branch in the zero-mode fault equivalent network; L1 Represents the first-mode impedance of the corresponding branch in the first-mode fault equivalent network.

7. The method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system according to claim 5 is characterized in that: The weighting coefficient is determined by the Thevenin equivalent impedance of the zero-mode fault equivalent network and the one-mode fault equivalent network starting from the fault point.

8. The method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system according to claim 5 is characterized in that: The calculation formula of the weighted coefficient is specifically: Among them, λ represents the weighting coefficient; L line_1 represents the first-mode inductance of the DC line to be determined; L line_0 represents the zero-mode inductance of the DC line to be determined; L eq1 L represents the Thevenin equivalent inductance of the next-mode fault equivalent network starting from the fault point under a single-pole grounding fault; eq0 It represents the Thevenin equivalent inductance of the zero-mode fault equivalent network starting from the fault point under a single-pole grounding fault.

9. The method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system according to any one of claims 1 to 8, characterized in that: The phase domain fault current analytical expression is represented by a time domain transient fault current analytical expression.

10. The method for analyzing single-pole grounding transient fault current in a symmetrical bipolar flexible DC system according to any one of claims 1 to 8, characterized in that: The specific solution process of the phase domain fault current analytical formula is as follows: Convert the phase domain low-order equivalent calculation circuit to the complex frequency domain; The circuit theory method is used to analyze and obtain the complex frequency domain transient fault current analytical formula of the low-order equivalent calculation circuit in the phase domain after conversion in the fault circuit; The transient fault current analytical expression in the complex frequency domain is converted to the time domain using the inverse Laplace transform, and the transient fault current analytical expression in the time domain is obtained.

Images