Voltage stability evaluation of real power transmission system using singular value decomposition technique

Abstract

Voltage stability is an important factor that needs to be taken into consideration during the planning and operation of power systems in order to avoid voltage collapse and subsequently partial or full system blackout. This paper presents singular value decomposition (SVD) technique to assess voltage stability of Qatar power transmission system. A MATLAB program has been developed for SVD technique. The voltage stability of the system has been analyzed during the peak load condition of summer 2007. The SVD results obtained by the developed MATLAB programs confirm that the Qatar power transmission system is "voltage stable". The critical load buses that could lead the system to voltage instability have been identified and the computer results have shown that there are critical load buses in the system. These critical load buses are located in an industrial area feeding from Qatar power system. The results of the critical load buses obtained by SVD have been verified by Q-V sensitivity.

Full text

Page 1

2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia
Voltage Stability Evaluation of Real Power
Transmission System Using Singular Value
Decomposition Technique

K. Ellithy*, M. Shaheen*, M. Al-Athba*
A. Al-Subaie**, S. Al-Mohannadi**, S. Al-Okkah**, S. Abu-Eidah**

* Qatar University/Electrical Engineering, Doha, Qatar. Email: k.ellithy@qu.edu.qa
** Qatar General Electricity and Water Cooperation, Doha, Qatar.



Abstract—Voltage stability is an important factor that needs
to be taken into consideration during the planning and
operation of power systems in order to avoid voltage
collapse and subsequently partial or full system blackout.
This paper presents singular value decomposition (SVD)
technique to assess voltage stability of Qatar power
transmission system. A MATLAB program has been
developed for SVD technique. The voltage stability of the
system has been analyzed during the peak load condition of
summer 2007. The SVD results obtained by the developed
MATLAB programs confirm that the Qatar power
transmission system is “voltage stable”. The critical load
buses that could lead the system to voltage instability have
been identified and the computer results have shown that
there are critical load buses in the system. These critical
load buses are located in an industrial area feeding from
Qatar power system. The results of the critical load buses
obtained by SVD have been verified by Q-V sensitivity.
Keywords ? Qatar Power Transmission System; Voltage
Stability; Singular Value Decomposition; Q-V Sensitivity

I.
INTRODUCTION
System failures and blackouts have already been
observed in Europe, Japan, and USA due to the voltage
instability [1, 6]. In addition, voltage stability has become
one of the biggest concerns in operating and planning
electric power systems. Voltage stability is defined as the
ability of power system to maintain acceptable voltages at
all buses in the system under normal operating conditions
and after being subjected to a disturbance [6]. A system
enters a state of voltage instability when a disturbance,
increase in load demand, or change in system condition
causes a progressive and uncontrollable drop in voltage.
The main factor causing voltage instability is the
inability of power system to meet the demand of reactive
power [1, 6].Thus; it is becoming increasingly important
for power system planning and operating engineers to be
capable of performing comprehensive voltage stability
analysis of the systems. This need is largely due to the
recent trends towards operating systems under stressed
conditions as a result of increasing system loads without
sufficient transmission or generation enhancements. In
recent years many researchers have suggested several
methods and techniques for voltage stability analysis such
as singular value decomposition [3, 4], sensitivity analysis
[7], modal analysis [2, 5], P-V and Q-V curves [1, 2] and
energy method [8] as well. Mainly, singular value
decomposition and sensitivity analysis techniques will be
discussed briefly in this paper and will be applied to Qatar
Power System.
II.
POWER FLOW SOLUTION
The power flow solution for the base case is obtained
using Newton Raphson (N-R) method which has been
proven to be the most reliable method to solve the power-
flow equation. By using N-R method [1]:

?
?
?

where:
?
?
?
??

i: is the number of iteration.

?P = Pspecified – Pcalculated
?Q = Qspecified – Qcalculated

Both ?P and ?Q specify the required tolerance and
therefore called the mismatch power vectors.

?V: is the unknown voltage magnitude correction vector.
??: is the unknown voltage angle correction vector.

J: is called the full Jacobian matrix.

?
=
Q
J
θ
where
?
?
?
?
?
?
Δ
Δ
=
?
?
?
Δ
Δ
)(
)(
)(
)(
)(
i
i
i
i
i
V
J
Q
P
θ
(1)
?
?
?
−
−
=
?
?
?
?
Δ
Δ
)(
)(
)(
)(
i
i
i
i
QQ
PP
Q
P

?
?
?
?
?
QV
PVP
J
JJ
J
θ
(2)
1-4244-2405-4/08/$20.00 ©2008 IEEE
1691
Authorized licensed use limited to: QATAR UNIVERSITY. Downloaded on December 28, 2009 at 02:47 from IEEE Xplore. Restrictions apply.

Page 2

2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia
V
Q
J
Q
θ
J
V
P
J
P
θ
J
QVQ
PVP
∂
∂
=
∂
∂
=
∂
∂
=
∂
∂
=
,
,
θ
θ


III.
REDUCED JACOBIAN MATRIX
It is practically known that when the real power P
changes then both voltage magnitude V and angle ? will
change. However, the voltage angle will change much
more than the voltage magnitude and when the reactive
power Q changes then also both voltage magnitude V and
angle ? will change. However, the voltage magnitude will
change much more than the voltage angle. By utilizing
this coupling between Q & V and P & ? and as
mentioned earlier that the shortage of the reactive power
Q mainly causes the voltage instability in the power
system, the reduced Jacobian can be obtained as follows
[1, 5]:

Let
0
=ΔP
, then by using Eq. (1), Eq. (1) becomes:

?
=
??
Δ
Q
θ
i.e.
J
P
−=Δ
θ
θ
JQ
Q
Δ=Δθ
θ
Substituting Eq. (4) into Eq. (5) gives:

JJQ
QQV
−=Δ
(

where JR is called the reduced Jacobian and equal to [4];
(
QVR
JJJ
−=
From Eq. (1);
QJV
=Δ
??
?
??
?
Δ
Δ
?
?
?
?
?
?
??
?
V
JJ
JJ
QVQ
PVP
θ
θ
0
(3)
V
Δ
J
+
PV
J
Δ
−1
(4)
V
(5)
QV
VJVJJ
RPVP
Δ=Δ
−
)
1
θθ
(6)
)
1
θ
PVPQ
JJ
−
θ
(7)
RΔ
−1
(8)

IV. SINGULAR VALUE DECOMPOSIYION TECHNIQUE
The Singular Value Decomposition (SVD) is a very
useful tool for analyzing matrices and related problems in
many fields. It has been applied to power systems for
voltage stability analysis to obtain decomposition of the
Jacobian matrix [3, 4]. For a real square matrix A of n x n
dimension, the singular value decomposition is given by:
T
iii
n
i
T
vuUSV
σ
?
=
==Α
1
(9)
Where U and V are n x n orthogonal matrices, ui and vi
are called the left and right singular vectors respectively,
and S is a diagonal matrix with:

[
)(
diagA
i
σ=
?

] )
A
(
i= 1,2,3,…,n (10)
where σi≥0 for all i. The diagonal elements in the matrix
S are usually ordered so that σ1≥ σ2 ≥σ3≥……≥σn≥0. If
the matrix A has rank r (r≤n), its singular values σ1, σ2,
σ3… σr are the only nonzero entries in the nxn diagonal
matrix S.

To use the above theory for voltage stability analysis, the
linear power-flow equation based on the Newton-
Raphson method [6] has to be found which is established
by the power-flow reduced Jacobian matrix JR

Now, if the singular value decomposition is applied to JR,
one has
?
=
i
==
n
T
iii
T
R
vuUSVJ
1
σ
(11)
The minimum singular value
close to singularity the Jacobian is. From equation (3) in
the case of a small disturbance point of view, let

P
??
Δ
where Un is the last column of U, and

?
n
σ(JR) is a measure of how
n u
Q
=
??
?
?
=Δ
0
(12)
nnv
V
1
−
=
??
?
??
Δ
Δ
σ
θ
(13)

where vn is the last column of V. From the above
equations it can be concluded that for the minimum
singular value and corresponding left and right singular
vectors [6], the following apply:
? The smallest singular value
proximity to the steady state stability limit.
? The right singular vector,
indicates sensitive voltage.
The left singular vector,
the most sensitive direction for changes of active and
reactive power injections [2, 3, 4].

n
σ is an indicator of the
nv corresponding to
n
σ
n ucorresponding to
n
σ indicates
V.
SENSITIVITY ANALYSIS TECHNIQUE
The power flow Jacobian matrix, J gives the sensitivity
between power flow and bus voltage changes. Both P and
Q are affecting the system voltage stability. However, at
each operating point, P can be kept constant and the
voltage stability analysis will consider the incremental
relation between Q and V. Based on this consideration
the sensitivity equation can be written as follows [1]:

V
Δ
or
QJ
RΔ=
−1
(14)
1
−
R
=
Δ
Δ
J
Q
V
(15)

The matrix
its ith diagonal element is the V-Q sensitivity at bus i. The
1
−
R
J
is the reduced V-Q inverse Jacobian and
1692
Authorized licensed use limited to: QATAR UNIVERSITY. Downloaded on December 28, 2009 at 02:47 from IEEE Xplore. Restrictions apply.

Page 3

2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia
V-Q sensitivity of a bus can also obtained from the slope
of the Q-V curve. By using this method one can know if
the system is voltage stable or not; i.e. if the V-Q
sensitivity is positive, then the system is stable, and it is
unstable if the V-Q sensitivity is negative. The system is
more stable if the positive sensitivity magnitude is
smaller. However, a small negative sensitivity magnitude
represents a very unstable system. Due to the nonlinear
nature of the V-Q relationships, the magnitudes of the V-
Q sensitivity for different system conditions do not
provide a direct measure of the relative degree of stability
[1, 7].
VI. RESULTS AND DISCUSSION
Mainly, the methods described above were applied to
one of the planning cases proposed for Qatar transmission
power system for summer peak load (base case) [9]. Fig.
9 shows single line diagram captured using PSS/E
software. The developed MATLAB programs for singular
value decomposition have been applied to Qatar
transmission power system to study its voltage stability.
The singular values of reduced Jacobian matrix JR of the
system have been calculated and the results showed that
the system is voltage stable because all singular values
are positive. Fig. 1 shows the most dominant singular
values of JR. Tables 1, 2, 3, 4 show the four smallest
singular values and the four largest bus singular vectors
corresponding to the smallest singular values. Also, the
singular vectors are represented as histograms in Fig. 2,
3, 4 and 5.

Dominant Singular Values
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
?1 ?2?3?4
Singular Value
Singular Value

Figure 1 Dominant singular values

Table 1 First singular value & corresponding singular vectors
First Singular Value ? = 0.551
Bus Singular Vector
0.133
0.085
0.039
0.022
Bus Number
439
216
89
565

Table 2 Second singular value & corresponding singular vectors
Second Singular Value ? = 0.682
Bus Singular Vector
0.126
0.068
0.028
0.017

Bus Number
46
628
627
76
Table 3 Third singular value & corresponding singular vectors
Third Singular Value ? = 0.817
Bus Singular Vector
0.067
0.0403
0.0402
0.024
Bus Number
82
81
603
366

Table 4 Fourth singular value & corresponding singular vectors
Fourth Singular Value ? = 0.939
Bus Singular Vector
0.059
0.03
0.0299
0.0101
Bus Number
215
498
662
655


Bus Singular Vector for First Singular Value = 0.551
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
439 216 89565
Bus Number
Singular Vector

Figure 2 Singular vectors for first singular value

Bus Singular Vector for Second Singular Value = 0.682
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
46 62862776
Bus Number
Singular Vector

Figure 3 Singular vectors for second singular value

Bus Singular Vector for Third Singular Value = 0.817
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
8281 603366
Bus Number
Singular Vector

Figure 4 Singular vectors for third singular value
1693
Authorized licensed use limited to: QATAR UNIVERSITY. Downloaded on December 28, 2009 at 02:47 from IEEE Xplore. Restrictions apply.

Page 4

2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia
Bus Singular Vector for Fourth Singular Value = 0.939
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
215498 662 655
Bus Number
Singular Vector

Figure 5 Singular vectors for fourth singular value

Form table 1 it can be seen that the smallest singular
value with the largest singular vector is related to load
bus number 439 which is located in the 33kV industrial
load area. According to this result, load bus number 439
is the weakest bus or the closet bus to the voltage
instability in the system. This clearly can be seen from
Fig. 6. In order to confirm that bus number 439 is the
weakest bus in the system, we have stressed this bus by
increasing the reactive power. Fig. 7 compares the
singular values for both cases: base case and stressed
case.


Figure 6 Singular vectors of the critical load buses

Base and Stressed Cases
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
?1?2?3?4
Singular Values
Base Case
Stressed Case
B

Figure 7 Singular values for base and stressed cases

The sensitivity analysis technique has been applied to
Qatar power system using the developed MATLAB
program and the results of the first six buses are shown in
Fig. 8. The results of the V-Q sensitivity analysis show
that the system is voltage stable, since all sensitivity
factors are positive. The sensitivity analysis results
indicate that bus number 439 is the weakest bus because
it has the highest sensitivity factor.
Sensitivity Factors
0
0.5
1
1.5
2
439 498662 655507 524
Bus Number

Figure 8 Sensitivity factors for the most six critical buses


The reactive power security margins of the critical load
buses have also been determined. Figure 10 show the
Mvar security margin of the weakest load bus 439. The
security margins are important information to the system
operator to avoid system voltage instability.



Figure 9 Single line diagram of Qatar system












Figure 10 Q-V curve of the weakest bus
1.023
1.024
1.025
1.026
1.027
1.028
1.029
1.03
0 0.51 1.52 2.53 3.5
Mvar Security Margin
1694
Authorized licensed use limited to: QATAR UNIVERSITY. Downloaded on December 28, 2009 at 02:47 from IEEE Xplore. Restrictions apply.

Page 5

2nd IEEE International Conference on Power and Energy (PECon 08), December 1-3, 2008, Johor Baharu, Malaysia
VII. CONCLUSIONS
The methods used for voltage stability study of power
systems, namely the singular value decomposition and Q-
V sensitivity have been applied to assess the voltage
stability of Qatar transmission power system. A
MATLAB computer programs have been developed for
the two methods. The results of singular value
decomposition have shown that Qatar transmission power
system under summer 2007 load condition was voltage
stable. The weakest buses which lead the system to
instability have been identified. The weakest load buses
were determined based on the values of the elements of
the right (or the left) singular vector corresponding to the
minimum singular value. The weakest buses are 493, 216,
46, 603 and 215. Furthermore, the sensitivity factors have
confirmed that the power system was voltage stable, since
all sensitivity factors were positive and greater than zero.
The results of the sensitivity analysis method have also
shown that bus number 439 is the most sensitive load bus
in the system. The results presented in this paper are
important information to the system operator to avoid
system voltage instability or voltage collapse.

ACKNOWLEDGMENT
The authors gratefully acknowledge Qatar University,
Qatar General Electricity and Water (Kahramaa) and
Qatar Foundation-UREP for the support in achieving this
work.
REFERENCES
[1] P. Kundur, Power system stability and control, Electric Power
Research Institute, McGraw Hill, 1994.

[2] K. Ellithy, A. Gastli, T. Alkuhsaibi and M. Irving "Voltage
Stability Margins Assessment for Muscat Power System"
International Journal of Electrical Power and Energy Systems,
Vol. 24, pp. 665-67, Aug. 2002.

[3] P.A. Lof,, G. Anderson and D.J. Hill, "Voltage Stability Indices
for Stressed Power System. IEEE Trans. on Power Systems", Vol.
8, pp. 326-335, 1993.
[4] P.A. Lof,, T. Smed, G. Anderson and D.J. Hill, "Fast Calculation
of a Voltage Stability Index. IEEE Trans. on Power Systems",
Vol. 7, pp. 54-64, 1992.

[5] B. Gao, G. Morison, and P. Kundur, "Voltage Stability Evaluation
Using Modal Analysis, IEEE Trans. on Power Systems", vol. 7,
pp. 1423-1543, 1992.

[6] C.W. Taylor, Power System Voltage Stability, McGraw-Hill, New
York, 1994.

[7] N. Flatabo and H. Dommel, "Voltage Stability Condition in a
Power Transmission System calculated by Sensitivity Methods",
IEEE Trans. on Power Systems, vol. 5, pp. 1286-1293, 1990.

[8] T. Overbye and C. DeMacro "Improved Power System stability
Assessment Using Energy Method. IEEE Trans. on Power
Systems", vol. 6, pp. 1890-1896, 1995.

[9] "System Single Line Diagram and System Data", Qatar General
Electricity and Water Cooperation (Kahramaa), 2008.

1695
Authorized licensed use limited to: QATAR UNIVERSITY. Downloaded on December 28, 2009 at 02:47 from IEEE Xplore. Restrictions apply.