Reactive Power Reserve Improvement Using Power Systems Inherent Structural Characteristics

doi:10.4236/epe.2013.54B189

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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

Email: [email protected], [email protected], [email protected], [email protected], [email protected]

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

min

Figure 1. Power-voltage curve.

The CPF technique uses a modified power flow whose

loading margin can be expressed as







 (1)

where c



is maximum loading at the critical point and



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)

PP P





 (3)

QQ KP





 (4)

where 0G is the generation base level, 0

P and 0

are the base level of active and reactive respectively,

represents distributed slack bus and

represents

loads with constant power factor. S and P

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



,, ,

GiLiPLGijij

PPGVVGB







 (6)



,, ,

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,

V is the

load bus voltage magnitude,



is the bus voltage phase

angles,

i is the active power demand at bus i, P

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

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

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

YYY









(12)

where GG is the generator-generator coupling in the

system admittance matrix with dimension G



, GL

is the generator-load coupling in the system admittance

matrix with dimension ,



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.

LL

We can express (11) as

GG GLG

LG LL



 



 

 



(13)

where G

is injected generator bus currents,

injected load bus currents, G is generator bus complex

voltages and

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

L LGGGGL

YYY YY



 (14)

The determinant of the Y-admittance matrix based on

Schur complement formula [21] is



detdet det

GGLLLG GG GL

YY YYYY



 (15)

In compact form (15) is expressed as

detdetdet

GG LL

YYC



(16)

where 1

LLL LGGGGL

CYYYY



 and represents the

equivalent admittance of the network with all influences

associated to generators eliminated.

The importance of matrix

L in relation to power

system network is clearer from the algebraic manipula-

tion of (13) which gives





LLLLLGG

VCIWI



 (17)

where 1

GLGG

WYY





The right hand side of (17) shows that matrix

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

L holds essential informa-

tion about the network structure. In order to identify

these inherent structural characteristics contained in ma-

trix

L, eigenvalue decomposition technique [23] is

applied as

CMRM mm



iii







 (18)

where M is a orthonormal matrix with eigenvectors ,

while



are the eigenvalues.

Since the inverse of matrix

L exist due to the

non-singularity of , the generalized inverse of matrix

C is

nii

LL ii

CMRM











 (19)

Substituting Equation (19) into Equation (17) gives



nii

VIW











LGG

(20)

The buses associated with the smallest eigenvalues in

matrix

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



 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





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

1.0 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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