Voltage Stability Analysis during Power Network
Node Type Conversion
TAO YI
College of Electrical Engineering,
Shanghai Dianji University,
300# Rd ShuiHua, Shanghai 200240,
CHINA
Abstract: - During the analysis of the power network, a node type conversion will occur under certain extreme
conditions. This brings the problem of stability of the power system voltage. This article extracts the
characteristic equation of the system voltage critical point by establishing a node voltage-branch current to the
power network equation. Therefore, the corresponding solution equation group was in the conversion process
of the PQ-PV node. Through this equation group, the difference in the difference between the voltage value of
the node and the critical point voltage is formed to determine the system stability after the node type conversion
is converted. In the process of calculating the voltage critical point, the improved Newton iteration method is
used to avoid Jacques bizarre. This method allows calculations to smoothly reach a state of convergence. The
simulation calculation shows that the method proposed in this article is correct and effective.
Key-Words: - static voltage; stability; branch current; characteristic equation; node type conversion.
1 Introduction
In the process of static voltage stability analysis of
the power system, due to the limit of the operating
parameters of certain power equipment, the
conversion of the node type was generated, which
brought more loss of voltage stability [1]. At
present, due to the power -free power limit of nodes,
the situation of PV nodes into PQ nodes has been
more studied. In the case where the system loses
stability, the analysis has undergone extreme
induced bifurcation or saddle-node bifurcation [2].
The research on the transition of PQ nodes to PV
nodes due to the increasing limit of the voltage of
the load node is relatively small.
In the process of node type conversion, the study
of node voltage loss of voltage is mainly aimed at
the solution of the critical point of voltage disability,
and then comparative analysis. The calculation
methods adopted are mainly indirect and direct
methods. The indirect method is performed by
continuously changing parameters to form a P-V
curve. However, near the critical point, as the
elegant matrix tends to be strange, the conventional
trend algorithm fails. Therefore, the calculation of
the critical point of the voltage is often combined
with the pathogenic trend algorithm [3]. The
continuous trend method [4,5] Tracks the balanced
and dislike of the trendy equation through prediction
and correction, and improves the pathological
phenomenon and convergence of conventional trend
algorithms. It is a more commonly used and
effective calculation method. The direct method
[6~8] According to the nature of the tide of the
critical point, it establishes a trendy equation, and is
iterated with Newton-Rafson method to find a more
accurate critical point. In addition, the non-linear
planning [9~11] method transforms the critical point
conditions into an optimized load problem, and uses
the optimal conditions of Kun- Tuk to solve it. The
above methods also have some shortcomings in the
process of practical application. When the power
injection is very close to the feasible border, it will
greatly increase the number of iterations of the
optimal multiplier algorithm leading to the difficulty
of solution. At the same time, obtaining sensitivity
information requires the left special vector of the
Jacques rather than the zero symbol of the matrix,
which increases a lot of additional calculations.
This article introduces the branch circuit current
representing the trend of the branch, and establishes
the equation group of the power network as a
variable with the node voltage. By analyzing the
trendy equation, the characteristic equation of the
node voltage stable boundary is extracted. Using
this feature equation can solve the critical point of
node voltage discharge. In the process of mutual
conversion of the PQ and PV nodes, the comparison
of the conversion point voltage and the critical point
Received: July 24, 2021. Revised: August 12, 2022. Accepted: September 9, 2022. Published: October 11, 2022.
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.23
Tao Yi
E-ISSN: 2224-2678
215
Volume 21, 2022
voltage is used to determine that the node is in the
position of the PV curve after the conversion, and
then determines the stability of the system after the
node type conversion
2 The Power Network Equation is
Represented by Node Voltage and
Branch Circuit Current as Variables
Under the right-angle coordinate system, when
ignoring the ground branch conductivity, the power
network can be described as a branch circuit
current-a mixed form of node voltage equation [12]:
0
0
jiij
r
lij
a
l
jiij
r
lij
a
l
ffRiXi
eeXiRi
(1)
For node
i
:
i
il lii
il
a
li
il
r
li
i
il
r
li
il
a
li
qBfeifie
pifie
)( 22
(2)
Wherein:
Ll ,,2,1
is branch collection,
is node collection, are the real
part and imaginary part of the current of branch
respectively, are the real part and imaginary
part of the voltage of node
i
respectively,
are the real part and imaginary part of the
impedance voltage of branch respectively, is the
1/2 susceptance to ground of branch , are the
active and reactive power injected into the node.
Assuming and respectively
represent the sum of the real and imaginary parts of
the node
i
injected current (excluding the branch
current to the ground), is the sum of the
ground susceptance of the branch connected to the
node .
The equation (1) and (2) form the power network
hybrid equation group with node voltage and branch
circuit current. This article analyzes the stability of
the power network based on this equation group.
3 Stable Critical Condition of Node
Voltage
In the process of converting the PQ-PV node, the
comparison of the voltage amplitude of the
conversion point needs to be relying on the analysis
method of judgment basis. Therefore, the obtaining
PQ and PV nodes under the given conditions has
become the key. Let's first study the critical
condition of stable node voltage.
3.1 PQ node
From the formula (2):
)(2
4)(4)(
)(2
)(2
4)(4)(
)(2
22
2222222
22
22
2222222
22
iiil
iiliiiiliii
iiiiiil
i
iiil
iiliiiiliii
iiiiiil
i
yxb
pbyxqbyxx
yxxypb
f
yxb
pbyxqbyxy
yxyxpb
e
(3)
That is to obtain a node voltage explicit
expression with a branch current as the parameter.
From the formula (3), it can be seen that only:
04)(4)( 2222222 iiliiiilii pbyxqbyx
(4)
which is:
2222 22-)( iiiliilii qpbqbyx
(5)
When the above conditions are met, the trendy
equation exists. And there are the following physical
meanings. 1) The solution is about the circle with 0
as the center of the circle and
)(2 22 iiiil qpqb
as the radius. 2) When the amplitude of the node
injection current is outside the circle, that is, only
">" is established in the equation (5), and there are
multiple solutions. The system can run stably. 3)
When the "<" is established, that is, when the square
of the node's injection current is in the interior of
this circle, there is no solution, and the system
cannot run stably. 4) When "=" was established, the
only solution existed, and the solution was on the
circle, that is, the system's voltage stable boundary.
To solve the operating status of the system at this
time, the system's voltage stable critical point was
found.
3.2 PV Node
Except for PQ nodes, generator nodes are usually
defined as PV nodes. For PV nodes, the reactive
power formula in the equation (2) are usually
replaced by the next formula:
222 iii Ufe
(6)
Wherein:
i
U
is the amplitude of the voltage of node
i
. The analysis expression of the PV node voltage
from the formula (2) is:
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22
2222
22
2222
)(
)(
ii
iiiiiii
i
ii
iiiiiii
i
yx
pVyxxyp
f
yx
pVyxyxp
e
(7)
For PV nodes, the condition of solution existence
is:
0)( 2222 iiii pUyx
(8)
Its physical significance is similar to PQ nodes.
The solution is a circle with 0 as the center and
ii Up /
as the radius. When the amplitude of the
node injection current is distributed on the circle or
out of the circle, the power network equation has a
solution.
4 Analysis of the Loss of Voltage
Stability during the Mutual
Conversion of the PQ-PV Node
4.1 Judging the Loss of Stability during the
Mutual Conversion of the PQ-PV Node
Due to the reactive power limit or voltage limit
value, the type of PV and PQ nodes will be
converted to each other. During the conversion
process, the system may lose stability. Take PV
converting PQ nodes as an example here to indicate
the loss of stable discrimination during the
conversion process. In Figure 1, point A represents
the stable critical point of the voltage of a node, and
B and C represent the conversion point of the PV
node to the PQ node. The horizontal axis indicates
the power of the node, and the vertical axis indicates
the voltage value of the node.
Fig. 1: The instability judgment from node PV to
PQ
So the voltage disability judgment can be
expressed as:
atriUUT
(9)
Wherein:
tr
U
is the voltage amplitude of node
i
at
the conversion point b or c.
a
U
is the voltage
amplitude of the critical point a. If
i
T
is greater than
0, it means that the conversion point is on the upper
half of the PV curve, and the system can still run
stably after the conversion. If
i
T
is less than 0, the
system will lose stability in the lower half of the PV
curve. Because whether the PV node is converted to
a PQ node or the PQ node converted to PV nodes,
the voltage amplitude on the conversion point b and
c is known as
b
U
and
c
U
. Therefore, the voltage
amplitude of the voltage stable critical point a is the
most critical. It can be used to determine which
node that is on the conversion time is on which
solution curve. Then judge whether the system is
stable.
In the same way, when the node is converted
from PQ to PV, there are similar conclusions.
4.2 Analysis of the Loss of Stability during
the PV Node Converted to the PQ Node
When the equation is established, in the equation
(5), the system's stable boundary is reached, and the
characteristic equation of the extraction system is:
)(2)( 2222 iiiilii qpqbyx
(10)
So the node voltage becomes:
)(2
)(2
)(2
)(2
22
22
22
22
iiil
iiiiiil
i
iiil
iiiiiil
i
yxb
yxxypb
f
yxb
yxyxpb
e
(11)
Because when the node voltage approaches the
stable critical point, the corresponding Jache
compare matrix is strange. It makes the trend
calculation based on Newton's method unable to
converge. In order to calculate the critical point of
the voltage stability, this node can be removed in the
node voltage equation, that is, the method of
reducing the dimension of the node voltage
equation. In the branch circuit current equation, the
corresponding node voltage equation is replaced by
the equation (10) and (11). As shown in the equation
(12), it is assumed that the
m
node will be
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DOI: 10.37394/23202.2022.21.23
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Volume 21, 2022
converted.
)(2)(
)(2
)(2
)(2
)(2
0
0
)(
2222
22
22
22
22
22
mmmmlmm
mmml
mmmmmml
m
mmml
mmmmmml
m
jiij
r
lij
a
l
jiij
r
lij
a
l
iilii
il
a
li
il
r
li
i
il
r
li
il
a
li
qpqbyx
yxb
yxxypb
f
yxb
yxyxpb
e
ffRiXi
eeXiRi
qbfeifie
pifie
(12)
Wherein:
mi
is established in node power
equation (2). In the branch circuit current equation
(1), the voltage of node
i
is solved by
m
e
and
m
f
instead. After solving node
m
voltage, the voltage
difference between the voltage of the PV node and
the critical point after the conversion is calculated
by the equation (9). Then you can make system
stability judgments.
4.3 Analysis of the Loss of Stability during
the PQ Node Converted to the PV Node
When the "=" in the equation (8) is established, the
feature equation is:
2
22 )(
i
i
ii U
p
yx
(13)
Then the node voltage becomes:
22
22
ii
ii
i
ii
ii
i
yx
yp
f
yx
xp
e
(14)
When PQ nodes is converted to PV nodes, the
voltage amplitude V of the PV node is known. The
equation group to be solved is:
2
22
22
22
222
)(
0
0
m
m
mm
mm
mm
m
mm
mm
m
jiij
r
lij
a
l
jiij
r
lij
a
l
iii
i
il
r
li
il
a
li
U
p
yx
yx
yp
f
yx
xp
e
ffRiXi
eeXiRi
Vfe
pifie
(15)
The solution process of this equation group is
similar to (12).
5 Calculation Steps
During the conversion of PQ and PV nodes, the
calculation steps adopted by static voltage
calculation are as follows:
1) The initial value of node voltage
V
and
branch circuit current
I
is given.
2) PV nodes or PQ nodes
m
to be solved, use
(12) or (15) to form the Jacques matrix with node
voltage and branch current as variables, and use
Newtonian method to iterate to solve.
3) Calculate the critical point of the PQ or PV
node voltage amplitude
a
U
.
4) Use formula (9) to calculate the voltage
amplitude difference
i
T
.
5) Determine the loss stability of the system
according to the difference
i
T
.
6 Calculation Example
Take the IEEE118 node system as an example.
Among them, the 69th balance nodes and node
118 are confrontation, and the power factor of the
stable critical point of the voltage is 0.9. Table 1 is
the analysis of the stable situation of the system
when the PV node reaches the upper limit of the
reactive power to the upper limit of the PQ node,
where
it
Qlim
represents the limit of the reactive
power. Table 2 is the calculation of the PQ node
which the voltage reaches the lower limit to the PV
node. Table 1 and Table 2 both selected
representative calculation results.
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Table 1. The stability analysis of node PV
transforming to PQ
No.
tr
U
a
U
it
Qlim
i
T
18
0.9668
0.9520
0.500
0.0148
42
0.9881
0.9670
3.000
0.0211
49
0.9903
0.9611
2.100
0.0292
54
0.9692
0.9551
3.000
0.0141
55
0.9511
0.9624
0.230
-0.0113
56
0.9528
0.9598
0.150
-0.0070
59
0.9789
0.9619
1.800
0.0170
61
0.9812
0.9662
3.000
0.0150
72
0.9771
0.9572
1.000
0.0199
74
0.9562
0.9611
0.390
-0.0041
76
0.9505
0.9529
0.530
-0.0024
77
0.9871
0.9670
0.700
0.0201
105
0.9592
0.9638
0.300
-0.0046
107
0.9501
0.9529
2.000
-0.0019
110
0.9619
0.9555
0.430
0.0064
It can be seen from the calculation results of
Table 1 that the
i
T
value of 55, 56 and other nodes
is negative. This shows that when the node type is
converted from PV to PQ, the conversion point is on
the lower half of the PV curve, so the system is in
an unstable state after the conversion. The
parameters of the system can be seen that the
reactive power limit value of such nodes are
generally small and the adjustment capabilities are
poor. And the nodes are in the heavy load area, the
voltage stable critical point will be reached easily.
Therefore, the voltage amplitude (
a
U
) of the
critical point is large, so the conversion point is
easier to fall into the lower half of the PV curve,
resulting in the system's loss of stability.
Table 2 The stability analysis of node PQ
transforming to PV
No.
tr
U
a
U
i
T
20
0.9600
0.9533
0.0067
21
0.9600
0.9562
0.0038
22
0.9600
0.9711
-0.0111
43
0.9600
0.9641
-0.0041
44
0.9600
0.9508
0.0092
45
0.9600
0.9633
-0.0033
51
0.9600
0.9691
-0.0091
52
0.9600
0.9579
0.0021
53
0.9600
0.9638
-0.0038
82
0.9600
0.9666
-0.0066
83
0.9600
0.9613
-0.0013
In Table 2, the lower limit of the node's voltage
is 0.96, and
i
T
is the negative value indicates that
the system is in a state of losing stability after the
conversion. Observing the unstable node will find
that the cause of the disability is similar to Table 1.
Therefore, in the process of system stability
adjustment, first of all, nodes such as 45 and 51
should be adjusted to adjust their reactive power
reserves and reduce the load rate.
7 Conclusion
This article establishes the power network equation
based on node voltage and branch current as
variables. The voltage stability conditions
corresponding to the PQ and PV nodes are
proposed. Then analyze the system voltage stability
during the mutual conversion of the PQ-PV node.
The following conclusions are obtained through
simulation calculations:
1) The method proposed in this article can be
applied to the stability analysis of the node type
conversion of the power system, which is used to
determine the disability of the static voltage.
2) This article proposes a critical point feature
equation and an unstable judgment of voltage
stability. It provides a new way for the use of node
voltage amplitude values in the use of node
conversion as a method.
3) During the mutual conversion of the PV and
PQ nodes, nodes with low reactive power reserves
are more likely to occur type conversion. The
possibility of the lower half of the PV curve is great
during the conversion of the heavy load area. This
can provide a basis for system stable adjustment.
References:
[1] Bahram SHAKERIGHADI, Farrokh
AMINIFAR, Saeed AFSHARNIADai. Power
systems wide-area voltage stability
assessment considering dissimilar load
variations and credible contingencies, Power
Syst. Clean Energy, Vol.7, No.1, 2019, pp.
78-87.
[2] PABLO DANIEL PAZ SALAZAR,
YAVDAT ILYASOV, LUÍS FERNANDO
COSTA ALBERTO, etc. Saddle-Node
Bifurcations of Power Systems in the Context
of Variational Theory and Nonsmooth
Optimization, IEEE Access, Vol.8, 2020, pp.
110986-110993.
[3] Mengqi Yao, Daniel K. Molzahn, Johanna L.
Mathieu, etc. An Optimal Power-Flow
Approach to Improve Power System Voltage
Stability Using Demand Response, IEEE
TRANSACTIONS ON CONTROL OF
WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2022.21.23
Tao Yi
E-ISSN: 2224-2678
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Volume 21, 2022
NETWORK SYSTEMS, Vol.6, No.3, 2019,
pp. 1015-1025.
[4] Ke Chen, Anwar Hussein, Martin E. et al. A
Performance-Index Guided Continuation
Method for Fast Computation of Saddle-Node
Bifurcation in Power Systems[J]. IEEE
Transaction on Power Systems, 2003, 18(2):
753-761.
[5] Rafael J., Claudio A., Federico Milano, et al.
Equivalency of Cantinuation and
Optimization Methods to Determine Saddle-
Node and Limit-Induced Bifurcation in Power
Systems[J]. IEEE Transaction on Circuits and
Systems, 2009, 56(1): 210-222.
[6] Ajjarapu V, Christy C. The Continuation
Power Flow: A Tool for Steady State Voltage
stability Analysis[J]. IEEE transactions on
Power Systems, 2002, 7(10): 304-311.
[7] V. Ajjarapu and C. Christy. The Continuation
Power Flow: A Tool for Steady State Voltage
Stability Analysis[J]. IEEE Trans. Power
Syst., 1992, 7(1): 416-423.
[8] Z. Feng, V. Ajjarapu, and D. J. Maratukulam.
Identification of voltage collapse through
direct equilibrium tracing[J]. IEEE Trans.
Power Syst., 2000, 15(3): 342-349.
[9] Xiaoyuan Xu, Zheng Yan, Mohammad
Shahidehpour, etc. Power System Voltage
Stability Evaluation Considering Renewable
Energy With Correlated Variabilities, IEEE
TRANSACTIONS ON POWER SYSTEMS,
Vol.33, No.3, 2018, pp. 3236-3244.
[10] Shunjiang Lin, Yuerong Yang, Mingbo Liu,
etc. Static voltage stability margin calculation
of power systems with high wind power
penetration based on the interval optimization
method, IET Renewable Power Generation,
Vol.14, No.10, 2020, pp. 1728-1737.
[11] Syed Mohammad Ashraf, Ankur Gupta,
Dinesh Kumar, etc. Voltage stability
monitoring of power systems using reduced
network and artificial neural network,
Electrical Power and Energy Systems, Vol.87,
2017, pp. 43-51.
[12] Yi Tao, Wang Yanjie. The Closest Stability
Margin by Analyzing Full-Dimensional
Saddle-Node Bifurcation Point in Power
System, WSEAS TRANSACTIONS ON
POWER SYSTEMS, Vol.12, 2017, pp. 31-38.
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