3.1 Fault at double-terminal feeder
3.1.1 Fault analysis for the fault double-terminal feeder
The analysis for fault occurring at the double-terminal feeder uses the F1 fault in line BC as an example. In normal operation, the power flow of feeder BC consists of the following two situations.
-
(a)
The load demand of Sub-microgrids 4 and 5 is greater than the generation capacity of DGs, and thus the power flow is from B terminal to C terminal.
-
(b)
The load demand of Sub-microgrids 4 and 5 is less than the generation capacity of DGs, and thus the power flow is from C terminal to B terminal.
Situation (a): \( {\dot{U}}_B \), \( {\dot{I}}_B \) and \( {\dot{U}}_{1B} \), \( {\dot{I}}_{1B} \) are the pre- and post-fault voltages and currents of B terminal in line BC, respectively. For B terminal, it is in the upstream of the fault point and is connected with the distribution network, so the voltage will largely remain unchanged when the fault occurs but the current will significantly increase. The variation of pre- and post-fault current and voltage are:
$$ \left|{\dot{U}}_{1B}\right|\le \left|{\dot{U}}_B\right| $$
(13)
$$ \left|{\dot{I}}_B\right|\le \left|{\dot{I}}_{1B}\right| $$
(14)
The pre- and post-fault measured admittances of B terminal are:
$$ {G}_B={\dot{I}}_B/{\dot{U}}_B $$
(15)
$$ {G}_{1B}={\dot{I}}_{1B}/{\dot{U}}_{1B} $$
(16)
Combining (13)–(16), it is known that the measured admittance amplitudes have the following relationship:
$$ \left|{G}_{1B}\right|>>\left|{G}_B\right| $$
(17)
Both the pre- and post-fault current directions of B terminal are in the forward direction. It is also known from Table 1 that the phase difference of measured admittances of B terminal is:
$$ -{90}^{\mathrm{o}}\le \Delta {\varphi}_B\le {90}^{\mathrm{o}} $$
After the fault occurrence, the DGs in the downstream of C terminal cannot provide enough short circuit capacity, so the voltage of C terminal will drop significantly. Additionally, the DGs in the sub-microgrids are inverter interfaced distributed generators (IIDG), which are usually under PQ control strategy with low voltage ride-through capability. When the voltage of the fault point drops significantly, the output current of the IIDG may reverse and decrease in amplitude [20]. Thus, the traditional current protection scheme is probably invalid in this case. In addition, if the DG penetration reaches a certain level, the protection cooperation will also be interfered. Meanwhile, the amplitude change of the measured pre- and post- fault admittances of C terminal cannot be estimated directly with (15) and (16).
The pre-fault current of C terminal is in reverse direction, and when the fault occurs, it changes to forward direction. Thus, the phase difference of measured admittances of C terminal is:
$$ {90}^{\mathrm{o}}\le \Delta {\varphi}_C\le {270}^{\mathrm{o}} $$
Situation (b): In this case, the change information of measured admittance amplitude of B terminal is the same as (17). Before the fault, the current of B terminal is in reverse direction and when the fault occurs, it changes to forward direction. Thus, the phase difference of measured admittances of B terminal is:
$$ -{90}^{\mathrm{o}}\le \Delta {\varphi}_B\le {90}^{\mathrm{o}} $$
The pre- and post-fault current of C terminal are both in forward direction. Thus, the phase difference of measured admittances of C terminal is:
$$ -{90}^{\mathrm{o}}\le \Delta {\varphi}_C\le {90}^{\mathrm{o}} $$
3.1.2 Fault analysis for the healthy double-terminal feeder
For fault occurrences in F1, the analysis for healthy double-terminal feeders uses line AB as an example. Before the fault, the power flow of line AB consists of the following two situations.
-
(a)
The load demand of Sub-microgrids 3, 4 and 5 is greater than the generation capacity of DGs, and thus the power flow is from A terminal to B terminal.
-
(b)
The load demand of Sub-microgrid 3, 4 and 5 is less than the generation capacity of DGs, and thus the power flow is from B terminal to A terminal.
Situation (a): GA, G1A and GB1, G1B1 are the pre- and post-fault measured admittances of A and B terminals, respectively. In this case, A and B terminals are in the upstream of the fault point, and are connect to the distribution network. Therefore, the voltage will remain unchanged when the fault occurs, but the current will increase significantly. Thus, there are:
$$ {\displaystyle \begin{array}{c}\left|{G}_{1A}\right|>>\left|{G}_A\right|\\ {}\left|{G}_{1B 1}\right|>>\left|{G}_{B 1}\right|\end{array}} $$
Both the pre- and post-fault current of A terminal are in forward direction, while for B terminal, they are both in reverse direction. Thus, the respective phase differences of measured admittances of A and B terminal are:
$$ {\displaystyle \begin{array}{c}-{90}^{\mathrm{o}}\le \Delta {\varphi}_A\le {90}^{\mathrm{o}}\\ {}-{90}^{\mathrm{o}}\le \Delta {\varphi}_B\le {90}^{\mathrm{o}}\end{array}} $$
Situation (b): A and B terminals are in the upstream of the fault point, so there are:
$$ {\displaystyle \begin{array}{c}\left|{G}_{1A}\right|>>\left|{G}_A\right|\\ {}\left|{G}_{1B 1}\right|>>\left|{G}_{B 1}\right|\end{array}} $$
The pre-fault current of A terminal is in reverse direction, whereas it is in forward direction for B terminal. When the fault occurs, the current of A terminal changes to forward direction and B terminal becomes reverse direction. Thus, the respective phase differences of measured admittance of A and B terminal are:
$$ {\displaystyle \begin{array}{c}{90}^{\mathrm{o}}\le \Delta {\varphi}_A\le 27{0}^{\mathrm{o}}\\ {}{90}^{\mathrm{o}}\le \Delta {\varphi}_B\le 27{0}^{\mathrm{o}}\end{array}} $$
The power flow analysis for other healthy double-terminal feeders is similar to feeder AB, and thus is not covered here.
3.1.3 Fault analysis for single-terminal feeder
In this case, the current and voltage remain unchanged, and thus the measured admittance amplitudes will not change after the fault occurrence.
For the DG feeders, the current directions are from the DGs to the upstream bus, and all are in the reverse direction. Thus, the phase difference of measured admittances is:
$$ -{90}^{\mathrm{o}}\le \Delta {\varphi}_{DG}\le {90}^{\mathrm{o}} $$
For the load feeders, their current directions are from the upstream buses to the loads, and are all in forward direction. So the phase difference of measured admittances is:
$$ -{90}^{\mathrm{o}}\le \Delta {\varphi}_{Load}\le {90}^{\mathrm{o}} $$
3.2 Fault occurs in single-terminal feeder
The analysis for fault occurrence in a single-terminal feeder uses the F2-fault of feeder L3 and F3-fault of feeder L2 as examples.
3.2.1 Fault at F2
For faults occurring in F2, (Fig. 2), DG2 is connected to feeder L3, and P terminal is the access point that L3 connects with the upstream bus. The pre-fault current direction of P terminal is from DG2 to the upstream bus, and is in reverse direction. After the fault occurrence, it changes to forward direction from the upstream bus to the fault point. Thus, the phase difference of measured admittances is:
$$ {90}^{\mathrm{o}}\le \Delta {\varphi}_P\le 27{0}^{\mathrm{o}} $$
The power flow analysis for the other healthy feeders is the same as the feeder in Section 3.1 so not repeated here.
3.2.2 Fault at F3
For fault at F3 (Fig. 2), load2 is connected to feeder L2, and K terminal is the access point that L2 connects with the upstream bus. The pre- and post-fault current directions of K terminal are all in forward direction from the upstream bus to the fault point. Thus, the phase difference of measured admittances is:
$$ -{90}^{\mathrm{o}}\le \Delta {\varphi}_K\le {90}^{\mathrm{o}} $$
K terminal is in the upstream of the fault point, so the post-fault measured admittance amplitude of K terminal will increase significantly. For the other feeders that connect with the loads, the current and voltage will remain unchanged, and thus the post-fault measured admittance amplitude will not change.
The power flow analysis for the other feeders is similar to the feeders in Section 3.1, so not repeated here.