## 1 Introduction

## 2 PV low-voltage ride-through characteristics

_{q.f}and I

_{d.f}are the reactive and active current generated during the fault, respectively; U

_{d.0}is the d-axis component of the voltage in normal operations; \(\mathop U\nolimits_{{{\text{d.f}}}}^{ + }\) is the positive-sequence component of the voltage at the time of the fault; I

_{max}is the maximum PV output current during the fault; I

_{amp.f}is the fault current amplitude; α is the fault current phase angle; and k is the reactive power compensation coefficient.

_{L}is the line equivalent impedance, Zs is the system equivalent positive-sequence impedance, and ΔI is the current fault component.

_{d}before the fault, where the output active current is I

_{d}, \({U}_{\mathrm{d}}^{^{\prime}}\) is the post-fault voltage, θ is the phase lag angle, \({I}_{1}^{^{\prime}}\) and \({I}_{2}^{^{\prime}}\) are the fault currents, and the fault current range is the sector area surrounded by the arc with radius I

_{max}. Additionally, \({\Delta I}_{\mathrm{f}1}\) and \({\Delta I}_{\mathrm{f}2}\) are the current fault components. If the voltage drop is large when a distribution network fault occurs (i.e., \({U}_{\mathrm{d}}^{^{\prime}}\) is small), the phase difference between ΔI and U

_{d}before the fault is > 90°. If the voltage drop is small (i.e., \({U}_{\mathrm{d}}^{^{\prime}}\) is large), the phase difference between ΔI and U

_{d}is < 90°. Thus, the voltage drop magnitude contributes to the phase difference between ΔI and U

_{d}. Furthermore, the post-fault voltage has a critical value, as indicated in Fig. 2b. \({I}_{\mathrm{d}}^{^{\prime}}\) and \({I}_{\mathrm{q}}^{^{\prime}}\) are the post-fault active and reactive current, respectively, and the phase angle between ΔI and U

_{d}is 90° [18].

_{d}is > 90°. When the post-fault voltage is greater than the critical voltage, the phase angle between ΔI and U

_{d}is < 90°. Analysis of the PV output fault current characteristics forms the basis of microgrid fault analysis.

## 3 Microgrid fault characteristics during PV low-voltage ride-through

_{1}, Z

_{2}, and Z

_{3}are the positive-sequence load impedances. ΔU

_{F}is the additional voltage source created by the fault point, and Z

_{F}is the fault impedance. Z

_{AF}, Z

_{BF}, Z

_{EM}, and Z

_{FG}are the equivalent positive-sequence impedances of the lines between the buses indicated by the subscripts; ΔI

_{1}and ΔI

_{2}are the current fault components output by IIDG1 and 2, respectively.

_{M}is the positive-sequence fault voltage component at bus M, and Z

_{2}is inductive; the phase angle between ΔI

_{M3}and ΔU

_{M}is < 90° and in the third quadrant. The phase relationship between the fault voltage and current components at bus M can be obtained through analysis, as shown in Fig. 4a.

_{E}is the positive-sequence fault voltage component at bus E, and Z

_{S}is inductive; the phase angle between ΔI

_{E1}and ΔU

_{E}is < 90° and in the third quadrant. From this analysis, we can extract the phase relationship between the fault voltage and current components for bus E, as shown in Fig. 4b.

_{G3}and ΔU

_{G}is < 90°, which is in the third quadrant. There is a large voltage drop when failure occurs at point f, because bus G is not supported by the utility grid.

_{1}and U

_{G}is > 90°; therefore, ΔI

_{1}is in the third quadrant. As shown in the fault component diagram, the fault components of each bus F feeder are \({\Delta I}_{\mathrm{F}3}={-\Delta I}_{\mathrm{G}1}\), \({\Delta I}_{\mathrm{F}2}={-\Delta U}_{\mathrm{F}}/{Z}_{3}\), and \({\Delta I}_{\mathrm{F}1}=-\left({\Delta I}_{\mathrm{F}2}+{\Delta I}_{\mathrm{F}3}\right)\). Therefore, the fault vector direction ΔI

_{G1}is known. As shown in Fig. 4c, the phasor diagram of each fault component for bus F can be obtained using this analysis approach.

_{1}, f

_{2}, f

_{3}, and f

_{4}) were set as shown in Fig. 5a, and b shows the positive-sequence fault additional network considering a failure at point f

_{4}. Z

_{F}is the resistance of the fault point, Z

_{S}is the equivalent positive-sequence system impedance, and Z

_{11}and Z

_{22}are the equivalent positive impedances of the lines. Z

_{EF}, Z

_{BF}, and Z

_{FG}are the equivalent positive-sequence impedances between the buses indicated by the subscripts.

_{1}–f

_{4}. Because failure at point f

_{1}is discussed above, the analysis is not repeated here.

_{2}, the microgrid fault network and the current phasor diagrams for buses E and M are similar to the network and phasor diagrams shown for point f

_{1}. However, analysis of bus G is necessary. When the bus G voltage drop is small, the phase angle between ΔI

_{1}and U

_{G}is < 90°; thus, ΔI

_{1}is in the fourth quadrant. Figure 6 shows the phasor diagram for the superimposed fault network.

_{3}, the relationships between the bus E and M positive-sequence fault phasors are similar to the relationships shown in Fig. 4a, b.

_{E3}is similar to the phasor shown in Fig. 7. The positive-sequence impedance of the load is inductive; the phase difference between ΔI

_{F2}and ΔU

_{F}is thus < 90°, which is in the third quadrant. The phase relationship between the bus F fault voltage and current components can be obtained as shown in Fig. 8.

_{4}is shown in Fig. 5b. In this case, the fault phasor components for buses E, M, and F are similar to the components in Fig. 4a, b and Fig. 8.

_{1}and U

_{G}is < 90°; accordingly, ΔI

_{1}is in the fourth quadrant. As shown in Fig. 5b, \({\Delta I}_{\mathrm{G}2}={-\Delta I}_{1}\), \({\Delta I}_{\mathrm{G}1}=-\Delta {I}_{F3}\), and \({\Delta I}_{\mathrm{G}3}=-\left({\Delta I}_{\mathrm{G}1}+{\Delta I}_{\mathrm{G}2}\right)\). Therefore, the fault voltage and current components can be obtained as shown in Fig. 9.

_{A1}, ΔI

_{A2}, and ΔI

_{A3}, respectively. In Fig. 10, Z

_{1}is the equivalent positive-sequence impedance of bus A between the front feeder and load, Z

_{2}is the equivalent impedance of branch 2, Z

_{3}is the impedance from the fault point to bus A in branch 3, Z

_{4}is the impedance from the fault point to the end of branch 3, and Z

_{F}is the additional impedance of the power supply at the fault point.

_{1}, Z

_{2}, Z

_{3}, and Z

_{F}are inductive. The phase angle of U

_{F}is θ

_{u}; the phase angles of Z

_{1}to Z

_{F}are θ

_{1}to θ

_{F}, respectively. Therefore, the limit:

_{x}is the phase angle that corresponds to the addition of vectors Z

_{1}to Z

_{F}. As shown in Fig. 10, the fault current of each branch feeder can be represented by an equivalent positive-sequence voltage and resistance at the fault point, where

## 4 Microgrid protection scheme

_{Y}is the target line, and ΔY

_{Yn}are the other lines on bus X. A branch is in fault if the difference is between 90° and 180°, and the fault state criterion output is –1. Otherwise, the branch feeder is healthy and the fault state is 1. When the detection process is initiated, all bus branch feeders are simultaneously inspected for faults; a trip signal is triggered if a faulty line is detected. A trip signal is sent to the opposite side. The system will trip if the signal is received simultaneously; otherwise, it will not trip.

## 5 Case study

^{–1}, and the positive-sequence inductive reactance of the line was 0.12 Ω km

^{–1}. The zero-sequence resistance of the line was 2.00 Ω km

^{–1}, the zero-sequence inductive reactance of the line was 0.4 Ω km

^{–1}, and the load was 0.4 MW. The test fault was a three-phase ground fault that occurred at 1 s.

System components | Parameters |
---|---|

Step-down transformer | 110 kV/10 kV |

System frequency | 50 Hz |

PV capacity | 0.4 MW |

Fault resistance | 0.01 Ω |

_{1}. The phase angle of feeder F

_{2}was opposite to the phase angles of feeders F

_{1}and F

_{3}. The fault lines were ΔI

_{E2}, ΔI

_{F1}, and ΔI

_{M1}; the healthy lines were ΔI

_{E1}, ΔI

_{E3}, ΔI

_{F2}, ΔI

_{F3}, ΔI

_{M2}, and ΔI

_{M3}. The phase angle between the faulty and healthy lines was between 90° and 180°; the phase angle between the healthy lines was in the range of 0° to 90° (Fig. 15).

_{2}. The phase angle of feeder F

_{2}was opposite to the phase angles of feeders F

_{1}and F

_{3}. The fault lines were ΔI

_{E2}, ΔI

_{F2}, ΔI

_{G1}, and ΔI

_{M1}; the healthy lines were ΔI

_{E1}, ΔI

_{E3}, ΔI

_{F1}, ΔI

_{F3}, ΔI

_{M2}, ΔI

_{M3}, ΔI

_{G2}, and ΔI

_{G3}. The phase angle between the faulty and healthy lines was between 90° and 180°; the phase angle between the healthy lines was in the range of 0° to 90°.

_{3}. The phase angle of feeder F

_{3}was opposite to the phase angles of feeders F

_{2}and F

_{1}. The fault lines were ΔI

_{E2}, ΔI

_{F3}, and ΔI

_{M1}; the healthy lines were ΔI

_{E1}, ΔI

_{E3}, ΔI

_{F2}, ΔI

_{F1}, ΔI

_{M2}, and ΔI

_{M3}. The phase angle between the faulty and healthy lines was between 90° and 180°; the phase angle between the healthy lines was in the range of 0° to 90°.

_{4}. The phase angle of feeder G

_{3}was opposite to the phase angles of feeders G

_{2}and G

_{1}. The fault lines were ΔI

_{E2}, ΔI

_{F3}, ΔI

_{G3}, and ΔI

_{M1}; the healthy lines were ΔI

_{E1}, ΔI

_{E3}, ΔI

_{F2}, ΔI

_{F1}, ΔI

_{M2}, ΔI

_{M3}, ΔI

_{G1}, and ΔI

_{G2}. The phase angle between the faulty and healthy lines was between 90° and 180°; the phase angle between the healthy lines was in the range of 0° to 90°.

Fault location | Fault resistance | Faulty feeders | Feeders with phase difference in range of 0° to 90° | Feeders with phase difference in range of 90° to 180° |
---|---|---|---|---|

f _{1} | 0.01 Ω | E2, F1, M1 | E2, E3 F2, F3 M2, M3 | E2, F1 M1 |

0.1 Ω | ||||

1 Ω | ||||

f _{2} | 0.01 Ω | E2, M1, F2, G1 | E1, E3 F2, F3 M2, M3 G2, G3 | E2, F2 M1, G1 |

0.1 Ω | ||||

1 Ω | ||||

f _{3} | 0.01 Ω | E2, M1, F3, G1 | E1, E3 F1, F2 M2, M3 G2, G3 | E2, F3 M1, G1 |

0.1 Ω | ||||

1 Ω | ||||

f _{4} | 0.01 Ω | E2, M1, F3, G3 | E1, E3 F1, F2 M2, M3 G1, G2 | E2, F3 M1, G3 |

0.1 Ω | ||||

1 Ω |

_{4}was set to 0.1 Ω. Differences between the positive-sequence currents of each bus F feeder were measured to demonstrate the efficacy of the proposed process. When a fault occurs at f

_{4}, the faulty line is ΔI

_{F3}, whereas the healthy lines are ΔI

_{F1}and ΔI

_{F2}. This fault condition is illustrated in Fig. 5b, and a simplified model of the microgrid is shown in Fig. 19.

_{4}is a single-phase ground fault are shown in Fig. 20.

_{F3}, and ΔI

_{F1}and ΔI

_{F2}is between 90° and 180° when a single-phase grounding fault occurs. The phase angle between the two healthy lines is in the range of 0° to 90°. The phase angle relationship is shown in Fig. 21.

_{4}is a two-phase short-circuit fault are shown in Fig. 22.

_{F3}, and ΔI

_{F1}and ΔI

_{F2}is between 90° and 180° when a two-phase short-circuit fault occurs. The phase angle between the two healthy lines is in the range of 0° to 90°. The phase angle relationship is shown in Fig. 23.

_{3}fault shown in Fig. 17 as an example, and the fault occurred at 3.5 s. When the fault occurs, the current waveforms at each branch feeder of bus F are shown in Fig. 25.