Energy and Power Engineering, 2013, 5, 986-991
doi:10.4236/epe.2013.54B189 Published Online July 2013 (http://www.scirp.org/journal/epe)
Reactive Power Reserve Improvement Using Power
Systems Inherent Structural Characteristics
Tajudeen H. Sikiru1, Adisa A. Jimoh2, Yskandar Hamam3, John T. Agee2, Roger Ceschi4
1Department of Electrical Engineering, Tshwane University of Technology, South Africa and LISV of UVSQ, France
2Department of Electrical Engineering, Tshwane University of Technology, South Africa
3ESIEE Paris, France and FSATI at Tshwane University of Technology, South Africa
4LISV of UVSQ and ESME Sudria, France
Received January, 2013
ABSTRACT
This paper considers the use of the inherent structural characteristics of power system networks for improving the reac-
tive power reserve margins for both topologically weak and strong networks. The inherent structural characteristics of
the network are derived from the Schur complement of the partitioned Y-admittance matrix using circuit theory repre-
sentations. Results show that topologically strong networks, operating close to the upper voltage limit could be made to
increase their loadability margin by locating reactive power compensators close to generator sources, whereas topo-
logically weak (ill conditioned) networks could be made to operate within the feasible operating limits by locating reac-
tive power compensators on buses farther from generator sources.
Keywords: Power System Networks Inherent Characteristics; Reactive Power Reserve Margin; Loadability Margin;
Schur Complement
1. Introduction
Transmission network plays a critical role in power sys-
tems operations. Its role in ensuring reliable operations of
power systems was acknowledged after post-mortem
analysis of major blackouts in many advanced countries
[1]. However, the concept of using transmission net-
works inherent structural characteristics to resolve power
systems operational issues has not been fully considered.
Transmission networks traditionally serve the purpose of
transporting power from generating stations to load cen-
tres. The amount of power that could be supplied from
generating stations to load centres and the routes for the
power flows depend on the transmission network struc-
tural interconnections [2]. The structural interconnections
between the power system nodes define the power sys-
tem inherent structural characteristics [3]. These charac-
teristics are governed by the value of line impedances
and how they are interconnected. Line impedances con-
sist of resistive and reactive components. The reactive
components account for the reactive power presence on
the network, which majorly affect the network operating
voltages and the amount of transferable active power a
transmission network can support [4]. Transmission net-
works with excess reactive power are in general topo-
logically strong networks and have network bus voltages
that are very high beyond the nominal limit. On the other
hand, topologically weak (ill conditioned) networks,
which are in deficit of reactive power, have low voltages
below the nominal limit [5]. Therefore, there is the need
to balance the amount of reactive power in a network
against the desired voltage operating limits [6]. However,
due to scarce resources, the difficulty of securing rights
of way and environmental issues, power system networks
are forced to operate within tight technical constraints.
The effect in recent times is the total blackouts caused by
voltage collapsed experienced by many matured power
system networks around the world [1].
So far, the approaches used to address reactive power
reserve mainly range from the linear programming tech-
nique to nonlinear programming techniques [7-13].
However, the challenge for these optimisation techniques
is the nonlinear, non-convex nature of the problem for-
mulation [14]. Due to the non-convex nature of the problem,
many optimisation techniques could easily be trapped in
local minima [11]. Secondly, ill conditioned networks
could lead to suboptimal solutions because of the need to
locate fictitious reactive power compensators first, to
achieve convergence of load flow before the loadability
of the network can be properly addressed [15]. Thirdly,
the large sizes of practical networks could be a challenge
when the nonlinearity of the problem formulation is fully
considered, since large solution variables would need to

T. H. SIKIRU ET AL. 987
be generated and may present memory storage issues
[13]. Finally, different buses on the network affect the
network operations differently because of the nonlinear-
ity of the network parameters [16]. Because of these
challenges, there is therefore, the need to reconsider the
fundamental circuit theory properties of network, in or-
der to identify its inherent structural characteristics that
could be used to achieve better reactive power manage-
ment.
This paper focuses on the inherent structural charac-
teristics of power system networks as a solution guide to
the issue of reactive power reserve management. For the
remainder of the paper, section II presents a brief over-
view of reactive power reserve management tools, sec-
tion III discusses the inherent structural characteristics of
networks and section IV presents a case study and result
discussion. Finally, section V concludes the paper and
highlights the major findings.
2. Techniques for Assessing Reactive Power
Reserve Margin
The purpose of adequate reactive power in a network is
to ensure operation of the network at both normal and
stressed conditions. The stressed condition consists of
lost of major lines, transformers, generators or a situation
where load gradually increase until the network cannot
support such load demand corresponding to the nose
curve point D in Figure 1, referred to as voltage collapse
point [15]. A power system with operating voltage at
point A is ill conditioned, because, it is operating below
the nominal voltage limit. This is caused by the unavail-
ability of sufficient reactive power in the network, hence,
load flow may not converge for such networks [17]. It is
necessary to move the operating point to between points
B and C for which load flow will converge [15]. Figure
1 shows that as the network voltages move more towards
point C, the amount of extra power demand it can sup-
port increases (i.e. increased loadability margin) until
beyond point C where it is infeasible to operate the net-
work. The relationship between voltages and network
loadability is nonlinear. The amount of load demand the
network can support before voltage collapse is referred to
as its maximum loadability margin. The loadability mar-
gin is a function of reactive power reserve in the network
[18].
The techniques used for assessing the maximum load-
ability of networks are continuous power flow (CPF)
technique and optimal power flow-direct method (OPF-
DM) or mathematical optimisation techniques [14,18,19].
The difference between the two techniques is that in the
latter, to ensure adequate reactive power margin, system
security variables to be maintained within limits must be
defined, hence constituting an indirect approach to secu-
rity assessment of the network [19].
max
V
min
V
Figure 1. Power-voltage curve.
The CPF technique uses a modified power flow whose
loading margin can be expressed as
0c



 (1)
where c

is maximum loading at the critical point and
0

is the current or base loading margin. As

changes with load increase, it relationship with variation
in generation and load pattern are


0GG G
PP KP   S (2)
0
L
L
PP P
D


 (3)
0
L
LL
QQ KP
D


 (4)
where 0G is the generation base level, 0 P
L
P and 0
L
Q
are the base level of active and reactive respectively,
G
K
represents distributed slack bus and
L
K
represents
loads with constant power factor. S and P
D
P repre-
sent generation and load directions respectively [19].
In the case of optimisation techniques, the maximum
loadability can be expressed as [14]
,,
,,
max
LGG
VVQ



(5)
Subject to

,, ,
,0
GiLiPLGijij
PPGVVGB 


 (6)

,, ,
,0
GiLiqL Gijij
QQGVVGB 



(7)
min max
GiGi Gi
PPP (8)
min max
GiGi Gi
QQQ (9)
min max
iii
VVV (10)
where 1in


Q, Gi is the active power generation at bus , Gi is the reactive power generation at bus i,
G is the generator bus voltage magnitude,
P
i
V
L
V is the
load bus voltage magnitude,

is the bus voltage phase
angles,
L
i is the active power demand at bus i, P
L
i is
the reactive power demand at bus i, ij is the conduc-
tance of line and ij is the susceptance of line
[19]. Equations (6) and (7) are the power balance equa-
tions of the network, while Equations (8) - (10) are the
Q
G
ij Bij

T. H. SIKIRU ET AL.
988
inequalities that must be satisfied for the network to op-
erate between points B and C of Figure 1. For topologi-
cally weak (ill conditioned) networks, convergence of
load flow may not hold since the network is operating
around point A [17]. Other approaches besides those
presented in this section are necessary to identify suitable
locations for reactive compensators for such networks
[20]. On the other hand, topological strong networks
have voltages that are between points B and C of Figure
1; however, if the loadability of the network is to be in-
creased then suitable locations for reactive power com-
pensators are required. In order to satisfy these objectives,
the inherent structural characteristics of network which
may serve as a guide in selecting suitable reaction power
compensators locations is presented in the next section.
3. Inherent S tructural Characteristic s of
Networks
The fundamental circuit theorem law applicable to power
system networks can be written as
I
YV (11)
where I is current, Y is network admittance and V is
voltage.
Suppose that the Y-admittance matrix is partitioned as
GG GL
LG LL
YY
YYY




(12)
where GG is the generator-generator coupling in the
system admittance matrix with dimension G
YG

, GL
is the generator-load coupling in the system admittance
matrix with dimension ,
Y
GL 
L
G is the load-genera-
tor coupling in the system admittance matrix with di-
mension and YLL is the load-load coupling in the
system admittance matrix with dimension . G and
L are the number of generator and load buses in the net-
work respectively.
Y
LG LL
We can express (11) as
GG GLG
LG LL
G
L
L
YY
I
V
YY
I
V
  

 
  
(13)
where G
I
is injected generator bus currents,
L
I
is
injected load bus currents, G is generator bus complex
voltages and
V
L
V is load bus complex voltages.
Since GG , the leading submatrix in (12) is non-sin-
gular, the Schur complement [21,22] of in Y is
Y
GG
Y
1
L
L LGGGGL
YYY YY

 (14)
The determinant of the Y-admittance matrix based on
Schur complement formula [21] is

1
detdet det
GGLLLG GG GL
YY YYYY

 (15)
In compact form (15) is expressed as
detdetdet
GG LL
YYC

(16)
where 1
L
LLL LGGGGL
CYYYY

 and represents the
equivalent admittance of the network with all influences
associated to generators eliminated.
The importance of matrix
L
L in relation to power
system network is clearer from the algebraic manipula-
tion of (13) which gives
C


1
LLLLLGG
VCIWI

 (17)
where 1
L
GLGG
WYY
G


The right hand side of (17) shows that matrix
L
L is
inversely related to the network bus voltages. Combining
this fact with the determinant relationship of this matrix
with the entire network structure presented in equation
(16), it shows that matrix
C
L
L holds essential informa-
tion about the network structure. In order to identify
these inherent structural characteristics contained in ma-
trix
C
L
L, eigenvalue decomposition technique [23] is
applied as
C
1
n
L
L
i
CMRM mm 
iii




 (18)
where M is a orthonormal matrix with eigenvectors ,
while
i
m
i

are the eigenvalues.
Since the inverse of matrix
L
L exist due to the
non-singularity of , the generalized inverse of matrix
C
GG
Y
L
L
C is
11
1
nii
LL ii
mm
CMRM





 (19)
Substituting Equation (19) into Equation (17) gives

1
T
nii
LL
ii
vm
VIW





LGG
I (20)
The buses associated with the smallest eigenvalues in
matrix
L
L would have the most effect on the network
bus voltages as mathematically expressed in Equation
(20), due to the reciprocal relationship between eigen-
values and the load voltages. From power system per-
spective, the smallest eigenvalue
C
0
n

 based on a
predefined precision level, will occur when the network
buses are electrically far from one another, because of
the shortage of adequate reactive element within the
network structure. The corresponding left eigenvectors
(matrix M) in this case will have column vectors with
constant values, indicating non-participation between the
network buses. This indicates a topologically weak (ill
conditioned) network [5]. As previously discussed, topo-
logically weak network have low voltages [5]. Buses
associated with the smallest eigenvalues indicate where
reactive power support are required [20]. Hence, to im-
prove the overall voltage profile these buses are suitable
locations for reactive power compensators [24].

T. H. SIKIRU ET AL. 989
On the hand, when the smallest eigenvalue is 0
n


based on a predefined precision level, there is adequate
reactive element presence in the network structure. The
degree of sufficiency of the reactive elements dependent
on the participation between the network buses observ-
able from the eigenvectors of matrix M. Networks that
exhibit such characteristics are topologically strong net-
works [5]. As already mentioned, such networks have
adequate voltages. In order to improve the loadability
margin for such networks, generators should be pre-
vented from reaching their reactive power limits by add-
ing reactive power compensators close to the generators.
Buses associated with the largest eigenvalues are suitable
locations for achieving this objective, since they are the
ones closest to the generator buses. The next section il-
lustrates this concept with a case study.
4. Case Study and Discussion of Results
The test network is a 40 bus Southwest networks shown
in Figure 2. The voltage profile of this test network
without any reactive power compensator is shown in
Figure 3. The purpose of adding reactive power com-
pensators is mainly for increasing the loadability margin
of this test network [14].
The smallest eigenvalue for this network is 0.0045 (in
absolute value) from the application of equation (19). A
set of five suitable locations for installing reactive power
compensators associated with the largest eigenvalues for
improving the loadability margin of the test network are
presented in Table 1.
In order to ascertain the effectiveness of these loca-
tions on reactive power reserve margin, comparison with
locations obtained using multi start-Benders decomposi-
tion technique published in [14] for the same network is
used in this paper. Continuous power flow (CPF) reactive
power assessment technique implemented in Power System
Figure 2. Southwe st England 40 bus ne twork.
Analysis Toolbox (PSAT) with generation and load di-
rections set was used to determine the maximum load-
ability margin of the test network. The maximum load-
ability for both approaches are shown in Table 2 using a
Static Var Compensator (SVC) of .
0.04 pu
The power-voltage curves for the lowest voltage of
each approach are shown in Figure 4 for the maximum
loadability corresponding to installation of five SVCs.
The proposed approach improves the network load-
ability margin better compared to the multi start-Benders
decomposition technique as shown in Table 2 and Fig-
ure 4 respectively. This is because the propose approach
seeks to locate reactive power compensators close to
generators, in order to prevent the generators from
Figure 3. Voltage profile of Southwest 40 bus networ k.
Table 1. Suitable locations for reactive power compensa-
tors.
S/NLargest eigenvalues Bus number
1 16.4309 9
2 9.9136 11
3 9.3788 10
4 8.2647 12
5 7.3916 2
Table 2. Comparison of maximum loadability.
Proposed Approach Multi start-Benders decomposition
Number
of SVCsBus number

(p.u) Bus number

(p.u)
1 9 1.143429 1.1237
2 9,11 1.193929,30 1.174
3 9,11,10 1.244529,30,32 1.1978
4 9,11,10,121.293429,30,32,31 1.2045
5 9,11,10,12,21.340529,30,32,31,28 1.2119
Base (No SVC) Maximum loadability (λ) = 1.0909 p.u.

T. H. SIKIRU ET AL.
990
00.2 0.4 0.6 0.8 11.2 1.4
0.8 4
0.8 6
0.8 8
0.9
0.9 2
0.9 4
0.9 6
0.9 8
1
1.0 2
1.0 4
Loading Parameter

(p.u.)
Voltage (p.u.)
No SVC (Bus 30)
Pro po sed ap proa c h (B u s 34)
MS-B e nd er deco m po s it i o n (Bus 25)
Figure 4. Power-voltage curve of the test network.
reaching their reactive power limits. This allows the gen-
erators to be free to supply more active power as the load
demand increases in topologically strong networks. On
the other hand, for topologically weak networks, the
compensators should be located on nodes farthest from
the generators, (i.e. on buses associated with the smallest
eigenvalues) [20,24] to ensure that the networks would
be within the acceptable voltage limits.
5. Conclusions
This paper has demonstrated that the network inherent
characteristics derivable from the Schur complement of
the partitioned Y-admittance matrix could be used to
identify suitable locations for improving reactive power
reserve margins in power system networks. For the case
of topologically weak (ill conditioned) networks, buses
associated with the smallest eigenvalues are suitable lo-
cations for installing reactive power compensators to
ensure feasibility of the network operating voltages. On
the other hand, topologically strong networks, operating
well within the desired voltage limits could increase their
loadability margin by installing reactive power compen-
sators on buses associated with the largest eigenvalues.
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