2.1. Fault Point Is Located between the Bus and a DG
According to the positional relationship between the fault point and the point of common coupling (PCC), single-phase short-circuit faults of active distribution networks can be divided into three cases: the fault is located between the bus and a DG, between DGs, and between a DG and the load. The capacitive charging and discharging process to the ground is short under distribution network faults, and the influence of the fault transition process is minimal, so the capacitive reactance to the ground of the AC line can be neglected. Therefore, we refer to the existing studies on fault analysis and utilize the resistance and reactance series model to equate the AC line. The resistance grounding model is used as the fault model. According to the positive sequence equivalent rule, the composite sequence network of single-phase short-circuit faults incorporating all three cases can be established as shown in
Figure 1. In
Figure 1,
is the equivalent potential vector of the superior grid;
is the equivalent impedance of the AC line from the bus to the PCC for DG
1 (PCC
1);
it is the ratio of the distance from the fault point f
1 to the bus to the distance from the bus to PCC
1;
is the equivalent impedance of the AC line between PCC
1 and the PCC for DG
2 (PCC
2);
is the ratio of the distance from fault point f
2 to PCC
1 to the distance from PCC
1 to PCC
2;
is the equivalent impedance of the AC line from PCC
2 to the end of the feeder;
is the ratio of the distance from fault point f
3 to PCC
2 to the distance from PCC
2 to the end of the feeder;
is the load equivalent impedance;
is the additional impedance,
and
are the equivalent impedances of negative sequence network and zero sequence network of the single-phase short-circuit fault, respectively;
is the fault transition resistance; and
represents a single-phase short-circuit fault located between the bus and a DG, between DGs, and between a DG and the load, respectively.
In
Figure 1, when the single-phase short-circuit fault occurs between the bus and a DG, the current flowing head of the feeder is the short-circuit current provided by the superior grid. The current flowing through the upstream of PCC
1 is the sum of the short-circuit current vectors of DG
1 and DG
2. Because the capacity of the superior power grid is much larger than that of the distribution network, the bus voltage remains constant during the fault. With the bus to the load as the reference direction, the current vector at the head of the feeder can be expressed as follows:
where
is the current vector at the head of the feeder when a single-phase short circuit fault occurs between the bus and a DG;
is the equivalent potential magnitude of the superior grid;
and
are the voltage magnitude and phase angle at the additional impedance, respectively; and
and
are the magnitude and phase angle of the equivalent impedance of the AC line from the bus to PCC
1, respectively.
In the case of a single-phase short-circuit fault, the active distribution network is asymmetric, and there are negative- and zero-sequence components of the voltage of the PCC for DG. However, the connected transformer for the DG is usually not grounded, so there is no zero-sequence current on the DG side, but there is a negative-sequence current path [
28]. The superposition of positive- and negative-sequence currents on the DG side may exceed the converter capacity and threaten DG safety, so the DG usually needs to suppress its negative-sequence current under the single-phase short-circuit fault and provide reactive current to support the grid voltage. The short-circuit current of DG under grid fault usually reaches the maximum limit, so according to the control equation of the low-voltage ride-through for photovoltaic [
29], the short-circuit current vectors of DG
1 and DG
2 can be expressed, respectively, as:
where
and
are the short-circuit current vectors of DG
1 and DG
2, respectively;
and
are the maximum allowable current magnitude of DG
1 and DG
2, respectively;
and
are the voltage phase angles of PCC
1 and PCC
2, respectively; and
and
are the angles at which the short-circuit current vectors of DG
1 and DG
2 lag behind the voltage vectors of PCC
1 and PCC
2, respectively, calculated according to the following:
where
is the reactive current coefficient of DG, which is determined according to the grid code;
and
is the voltage magnitude of PCC
1 and PCC
2, respectively;
is the rated voltage of the distribution network; and
and
are the rated currents of DG
1 and DG
2, respectively.
According to Kirchhoff’s law, the relationship between the voltage vectors of PCC
1 and PCC
2 and the voltage vector at the additional impedance can be established as follows:
where
and
are the voltage vectors of PCC
1 and PCC
2, respectively; and
are the voltage vectors at the additional impedance.
Combining Equations (2)–(4), full-rank nonhomogeneous linear equations can be constructed for the voltage magnitudes and phase angles of PCC
1 and PCC
2 based on the equal vector magnitudes and phase angles. Therefore, the voltage magnitudes and phase angles of PCC
1 and PCC
2 can be uniquely represented by the voltage magnitude and phase angle at the additional impedance. The current vector upstream of PCC
1 can be expressed as a function of the voltage magnitude and phase angle at the additional impedance as follows:
where
is the current vector measured upstream of PCC
1 when a single-phase short-circuit fault occurs between the bus and a DG.
2.2. Fault Point Is Located between DGs
When the fault point is located between DGs, the current vector flowing downstream of PCC
1 is the sum of the short-circuit current of the superior grid and DG
1, and the current flowing upstream of PCC
2 is the opposite short-circuit current vector of DG
2. Referring to Equations (2) and (3), the current vector upstream of PCC
2 can be expressed as:
where
is the current vector measured upstream of PCC
2 when a single-phase short-circuit fault occurs between DGs.
Since the phase angle of the short-circuit current of DG
2 depends on the voltage magnitude and phase angle of PCC
2, according to Kirchhoff’s law, the voltage vector of PCC
2 can also be expressed by the voltage vector at the additional impedance as:
where
and
are the voltage magnitude and phase angle at the additional impedance; and
is the magnitude of the equivalent impedance of the AC line between PCC
1 and PCC
2.
Combining Equations (6) and (7), full-rank nonhomogeneous linear equations can be constructed for the voltage magnitude and phase angle of PCC
2. Therefore, when the fault point is located between DGs, the current magnitude measured upstream of PCC
2, i.e., at the tail of the fault section, is constant. At the same time, the phase angle depends on the voltage magnitude and phase angle at the additional impedance. The current measured at the head of the fault section can be expressed as:
where
is the current vector flowing downstream of PCC
1 when a single-phase short-circuit fault occurs between DGs. Its magnitude
and phase angle
are calculated according to the following:
where the parameters
p and
q are calculated according to the following:
From Equation (8), it can be seen that the current magnitude and phase angle flowing downstream of PCC
1 are related to the voltage magnitude and phase angle of PCC
1. The relationship between the voltage vector of PCC
1 and the voltage vector at the additional impedance can be expressed as:
where
is the voltage vector at the additional impedance.
Combining Equations (8) and (11) yields full-rank nonhomogeneous linear equations for the voltage magnitude and phase angle of PCC1. Therefore, the voltage magnitude and phase angle of PCC1 can be uniquely represented by the voltage magnitude and phase angle at the additional impedance. The current magnitude and phase angle flowing downstream of PCC1 depend on the voltage magnitude and phase angle at the additional impedance.