θ
−
δ
))
2
≤
1
(19)
Similarly, the line stability index by considering only the effect of active power can be
written as [
83
]:
L
P
=
4R
sr
P
r
V
2
s
(
cos
(
θ
−
δ
))
2
≤
1
(20)
where θ is the impedance angle. For the stable operation, the values of L
Q
and L
p
should
be less than one. These indices are known as the line voltage stability indices [
84
]. The
index L
p
is also termed as the power stability index (PSI). For transmission systems, these
stability indices (i.e., FVSI
sr
, L
Q
, and L
p
) are calculated by using δ
=
0, but this is not done
for microgrids.
Another important voltage stability index is the voltage collapse proximity index
(VCPI) and several studies are carried out to determine this index, which can be found
in [
85
,
86
]. The VCPI in [
85
] is obtained by considering maximum loading conditions
in the transmission system while the power transmission paths are considered in [
86
].
A novel line stability index (NLSI) is calculated in [
87
] based on the formula derived
for the FVSI with some assumptions. Several other indices have been introduced for
transmission systems. A new voltage stability index was proposed in [
88
]. Other VSIs
include voltage reactive power index (VQI) [
89
], voltage stability load index (VSLI) [
90
],
voltage stability index [
91
] or indicator (VSI) [
92
], voltage stability margin (VSM) [
93
], VSM
index (VSMI) [
94
], stability index (SI) [
95
], line collapse proximity index (LCPI) [
95
], and
power transfer stability index (PSTI) [
96
]. However, these cannot be directly employed for
microgrids, and need to be modified. All these stability indices are usually determined
based on FVSI
sr
, L
Q
, and L
P
with different assumptions where these assumptions include
the following:
•
Zero resistance of the line, i.e., R
sr
=
0;
•
No angle difference between the sending and receiving end, i.e., δ
=
0;
•
No DERs at either sending or receiving end, i.e., P
G
=
0 and Q
G
=
0; and
•
No shunt admittance.
All these assumptions are not feasible for microgrids and hence, thus these indices
cannot be used directly. These stability indices are categorized as the line voltage stability
indices in [
76
].
Apart from line voltage stability indices, different bus voltage stability indices are also
reviewed in [
76
]. In [
97
], a voltage stability evaluation index called the voltage collapse
proximity index (VCPI) is defined as:
VCPI
i
=
1
−
∑
N
j
=
1
j
6=
i
V
0
j
V
i
(21)
where VCPI
i
is the VCPI for bus i, V
i
is the voltage of bus i, and V
0
j
=
Y
ij
∑
N
k
=
1
k
6=
i
Y
ik
V
j
=
V
0
j
∠(
δ
0
j
)
with Y
ij
as the admittance between bus i and bus j. In determining VCPI, the
voltage magnitudes and angles of all buses are used plus the network bus admittance
Electronics 2021, 10, 115
16 of 27
matrix information. A value of VCPI close to 0 indicates stability and a value close to 1
indicates approach towards voltage collapse. This VCPI is used in [
98
] along with two
other indices, namely a voltage security index (VSI) and a power transfer stability index
(PTSI), for the assessment of voltage stability in microgrids, and to particularly send
signals to the microgrid control center when the dynamic state of the power system is not
acceptable. The concept of power flow study is used to determine several bus voltage
stability indices for transmission systems and these methods are L-index [
99
], voltage
stability index using voltage and current deviations [
100
], simplified voltage stability
index [
101
], and S difference criteria [
102
]. In [
103
], the bus voltage stability index is
calculated based on the impedance matching, while in [
104
], the impedance ratio is used to
determine the similarity index. All these indices are developed with assumptions related
to the topology of the system, efficiency of the system, and eliminating the incremental
changes in the power at the receiving end.
3.1. Impact of Load Variations on Voltage Stability Indices
Voltage stability indices are sensitive to loading variations and must present a pre-
dictable behavior by allowing extrapolation of the additional power requirement by the
network before moving to the voltage instability point. It must be noted that the voltage
stability indices can be monitored with the changes in system parameters, and their calcu-
lation must be fast enough for online system supervision feasibility. P-V curves are either
calculated offline or estimated. Thevenin-based parameters offer an effective way to esti-
mate P-V curves. The P-V method based on offline statistics cannot reliably predict voltage
instability. Similarly, the Thevenin-based method relying on the measured parameters of
aggregated loads has low accuracy.
3.1.1. Improved Thevenin Estimates
The interaction between the transmission and distribution system can cause the overall
voltage stability margin to be different from the individual network. A 3ϕ voltage stability
indicator incorporating the unbalances and coupling between the phases is proposed
in [
105
]. The proposed voltage stability indicator is based on estimated Thevenin equivalent
parameters through microphase measuring units (µPMUs). The ratio of the magnitude of
the total apparent power loss and the total apparent load power is an indicator of voltage
stability. This concept is extended to the three-phase (3ϕ) networks for the 3ϕ VSI, as given
by Equation (16):
VSI
D−3ϕ
=
S
LossT−3ϕ
+
S
LossD−3ϕ
S
LD−3ϕ
(22)
where, S
LossT−3ϕ
is the apparent power loss at the transmission line side and S
LossD−3ϕ
is
the apparent power loss at the distribution side, while S
LD−3ϕ
is the apparent load. For
a balanced network and balanced load, the VSI
D−3ϕ
parameters are replaced by their
positive sequence impedances, as shown in Equation (17).
VSI
D−3ph−balanced
=
|
Z
TP
+
Z
DP
|
|
Z
LD
|
(23)
Another new real-time voltage stability index S
I
for a smart grid was proposed in [
106
].
The method utilizes the data of individual loads, which are obtained through the smart
meters for estimating the Thevenin equivalent parameters.
S
I
=
1
−
sin
πD
2
(24)
where
D
=
s
(
P
max
−
P
)
P
max
(25)
Electronics 2021, 10, 115
17 of 27
P
max
=
S
max
cos
(
θ
L
)
(26)
|
S
max
| =
3
|
V
th
|
2
|
Z
th
|(
2
+
2cos
(
|
