preventive control approach for voltage stability improvement using voltage stability constrained...

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Published in IET Generation, Transmission & DistributionReceived on 30th May 2013Accepted on 14th November 2013doi: 10.1049/iet-gtd.2013.0724

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ISSN 1751-8687

Preventive control approach for voltage stabilityimprovement using voltage stability constrainedoptimal power flow based on static line voltagestability indicesTarik Zabaiou1, Louis-A Dessaint1, Innocent Kamwa2

1Department of Electrical Engineering, École de Technologie Supérieure (ÉTS), Montréal, QC, H3C 1K3, Canada2Hydro-Québec/IREQ, Power Systems and Mathematics, Varennes, QC, J3X 1S1, Canada

E-mail: [email protected]

Abstract: Voltage stability improvement is a challenging issue in planning and security assessment of power systems. As modernsystems are being operated under heavily stressed conditions with reduced stability margins, incorporation of voltage stabilitycriteria in the operation of power systems began receiving great attention. This study presents a novel voltage stabilityconstrained optimal power flow (VSC-OPF) approach based on static line voltage stability indices to simultaneously improvevoltage stability and minimise power system losses under stressed and contingency conditions. The proposed methodologyuses a voltage collapse proximity indicator (VCPI) to provide important information about the proximity of the system tovoltage instability. The VCPI index is incorporated into the optimal power flow (OPF) formulation in two ways; first it can beadded as a new voltage stability constraint in the OPF constraints, or used as a voltage stability objective function. Theproposed approach has been evaluated on the standard IEEE 30-bus and 57-bus test systems under different cases andcompared with two well proved VSC-OPF approaches based on the bus voltage indicator L-index and the minimum singularvalue. The simulation results are promising and demonstrate the effectiveness of the proposed VSC-OPF based on the linevoltage stability index.

1 Introduction

During recent years, the planning and the operation of largeinterconnected power systems while improving systemstability and security have become important concerns inthe daily operation of modern power networks. Thereforethere is a renewed interest in developing optimal powerflow (OPF) models that incorporate additional constraintsand new objective functions to enhance the security ofelectricity markets. As several blackouts around the worldhave been related to voltage phenomena [1, 2], much moreinterest has been devoted by planning engineers to thevoltage stability constrained OPF (VSC-OPF) problem. Thepresent paper deals with including the voltage stability issuein the conventional OPF to effectively improve systemvoltage stability as well as to reduce power losses whensubject to unexpected contingencies such as generationoutages, tripping of a transmission line in a heavily loadedsystem or an unpredictable increase in load demand.In the literature, many methods and techniques have been

reported for voltage stability analysis and voltage collapseprediction. Some of these methods are based on PV–QVcurves [3], modal analysis [4], singular valuedecomposition [5], sensitivity method [6], energy function[7], continuous power flow [8] and bifurcation theory [9].

On the other hand, a number of static voltage stabilityindices have been widely used for evaluating and predictingthe proximity of the system to voltage instability. Theseindices have been grouped into two categories. Bus voltagestability indices have been applied to provide informationabout the stability condition of the buses and to identify theweakest bus in the system such as L-index [10], voltagecollapse prediction index [11], voltage stability index (VSI)[12] and an improved VSI (IVSI) [13] whereas, the lineindices-based techniques have been used to determine themost critical line in an interconnected system, for example,fast VSI (FVSI) [14], line stability index (Lmn) [15],voltage collapse proximity indicator (VCPI) [16] and theline voltage factor (LQP) [17].Several works have been investigating the most efficient

and accurate manner to incorporate these differentapproaches as voltage stability criteria into the OPFformulation. An OPF for maximising voltage securitythrough the use of the minimum singular value (MSV) ofthe power flow Jacobian was proposed in [18] and a voltagesecurity constrained OPF based on the loading levelparameter was formulated in [19]. In these works, theauthors tried to minimise the operating costs and the losseswhereas maximising the distance to the voltage collapse.The voltage stability margin (VSM) obtained by the

ommons AttributionIET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 5, pp. 924–934

doi: 10.1049/iet-gtd.2013.0724

Fig. 1 Single transmission line model

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continuation power flow (CPF) was adopted as a staticvoltage constraint in the multicontingency VSC-OPFmodel [20]. In [21], a VSC-OPF formulation incorporatinga linear margin enhancement constraint (MEC) is proposedto enhance the interface flow margin. In [22], an optimaldispatch with voltage stability constraint using thebifurcation technique was applied to improve the voltagestability in deregulated power systems.In addition, many researchers have introduced bus voltage
indices in the OPF problem. In [23, 24] Huang et al. and Kimet al. presented an OPF with voltage stability constraint basedon the L-index to investigate the voltage constraint effect onthe fuel cost and the system voltage stability. In [25], theL-index was adopted as the voltage stability constraint tocompute the load curtailment evaluation. The L-index hasalso been used as an objective function for voltage stabilityenhancement in both nominal situations [26–29] andcontingency conditions [30, 31]. In [13], the total sum of animproved bus voltage index (IVSI) for all the system busesis used as an objective function for optimising the settingsof the compensation devices.Even though the above works have made important

contributions and improvements in the VSC-OPF problem,the use of the line VSIs in the OPF model has not beeninvestigated extensively. Kargarian et al. [32] studied theminimisation of the market payment for the reactive powerand the system energy losses, and simultaneouslymaximising the voltage stability by reducing the largest linestability index (Lmn) of the network. The FVSI and theLQP are used as objective functions in the optimal locationof the FACTS devices for minimising the power systemlosses and the voltage stability improvement [33, 34].Moreover, the good characteristics of the line VSIs

motivated the purpose of a VSC-OPF based on the linevoltage indices. These indices involve both power andvoltage variation and hence provide reliable informationabout the proximity to voltage instability. In addition, thechoice of the VCPI index is inspired by its good properties(accuracy and robustness) in predicting voltage collapse.Comparison studies of the performance of several line VSIs[35, 36] have shown an agreement between the differentline stability indices which were found coherent with theirtheoretical background. However, the VCPI has the bestaccuracy and robustness in predicting voltage collapse [37].Also, the VCPI index has some strong points such assimplicity, fast numerical calculation, flexibility forsimulating any type of topological and load modificationsin the network and its application for real time simulation[38]. Furthermore, a VSC-OPF study has revealed thatincorporating the VCPI index in the OPF is more efficientfor voltage stability enhancement and power losses reduction.This work is focused on including the VCPI index in the

classical OPF to effectively improve the voltage stability aswell as minimise the power losses. The OPF problem isformulated as a non-linear optimisation problem withequality and inequality constraints. The objective functionsare the minimisation of the fuel cost and the improvementof the voltage stability in both stressed and contingencyconditions. The paper is organised as follows. Thedescription of the VCPI is given in Section 2. In Section 3,the mathematical formulation of the OPF problem withvoltage stability consideration is presented. Section 4explains the VSC-OPF approach for preventive control,whereas Section 5 details the concept implementation of theproposed approach. Section 6 presents and discusses thenumerical results. Section 7 summarises a comparative

IET Gener. Transm. Distrib., 2014, Vol. 8, Iss. 5, pp. 924–934doi: 10.1049/iet-gtd.2013.0724 This is an open access art

study. Finally, conclusions and future works are outlined inSection 8.

2 Line VSI

The VCPI was proposed by Moghavvemi and Faruque [16]that has rigorously demonstrated its accuracy and reliabilityon the standard IEEE test systems with different load nodes.This indicator is adopted in our study to investigate thestability of each line of the system by determination of thecritical line referred to a bus. The VCPI index is based onthe concept of maximum power transferred through thelines of the network as presented in Fig. 1 and it is definedas follows

VCPI (power) = Pr

Pr( max )= Qr

Qr( max )(1)

The numerator is the real or reactive power transferred to thereceiving end and it is obtained from the power flowcalculations. The denominator is the maximum active orreactive power that can be transferred through a line. It canbe calculated by the following equations

Pr( max ) =V 2S

ZS

cosf

4 cos2 (u− f)/2( ) (2)

Qr( max ) =V 2S

ZS

sinf

4 cos2 (u− f)/2( ) (3)

where VS is the sending end voltage; ZS is the line impedance;θ is the line impedance angle and f = tan−1(Qr/Pr) is thephase angle of the load impedance.The values of the VCPI index increase with the increasing

of the power flow transferred by the transmission lines andvary from 0 (no load condition) to 1 (voltage collapse).Hence, the critical line will be the line with the highestVCPI value and the load bus connected to the line will bethe vulnerable bus in the system. This indicator is simpleand fast to calculate compared with the bus voltage stabilityindicator L-index [10] which requires additional computingtime for the inversion of the complex bus admittancesubmatrices. The VCPI index considers the power variationin addition to the voltage variation, this allows us to takeinto account the power limits of the system equipments(such as generators) which are not considered in theL-index [39]. The index is flexible for simulating any typeof system topology and load modifications, and easilyonline implemented in a realistic system.

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From the above equations, it is clear that the rise in voltage
increases the values of Pr(max) and Qr(max) (as they areproportional to V 2

S ), resulting in lower VCPI indicatorvalues therefore improving the voltage stability limit.Hence, this principle is investigated in a VSC-OPF problemby limiting (constraint) and minimising (objective function)the VCPI index to enhance the system voltage stability. Asa result of the voltage stability improvement, the linecurrent magnitude is reduced as along with the power losses.

3 Problem formulation

The main purpose of the OPF problem is to determine theoptimal control variables for minimising an objectivefunction subject to several equality and inequalityconstraints. The problem is generally formulated as follows

min f (x, u) (4)

subject to

g(x, u) = 0 (5)

h(x, u) ≤ 0 (6)

where f is the objective function to be minimised; g is theequality constraints and h is the system operatingconstraints. x is the vector of the state variables and u is thevector of the control variables. The control variables aregenerator active power outputs and bus voltages. The statevariables are the voltages and the angles of the load buses.The objective functions, the conventional constraints and

the voltage stability constraint are described as follows.

3.1 Objective functions

In this paper, two different objective functions are considered.

3.1.1 Fuel cost: The first objective function is to minimisethe total fuel cost (FC) of the system. The generator costcurves are modelled by quadratic functions and can beexpressed as

FC =∑Ng

i=1

ai + biPGi + ciP2Gi

( )$/h (7)

where Ng is the number of generator buses; FC is the total fuelcost; ai, bi and ci are the fuel cost coefficients of the ithgenerator and PGi is the real power output of the ith generator.

3.1.2 Voltage stability improvement: Maintaining theacceptable voltage stability level under normal, stressed andcontingency operating conditions is an important concern inpower system planning and operation. For this aim, theminimisation of the total VSI (VCPIT) is proposed as anobjective function to enhance the overall voltage stability ofthe system.The VCPIT is the sum of the voltage stability indices for all

the lines of the system and it is mathematically evaluated as

VCPIT =∑Nl

i=1

VCPIi (8)

where VCPIi is the VCPI for line i and Nl is the number oftransmission lines in the system.

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3.2 Conventional constraints

The conventional constraints of the OPF problem arerepresented by two categories of constraints described by(9)–(13) as follows.

3.2.1 Equality constraints: represent the non-linearpower flow equations:

PGi − PDi = Vi

∑Nj=1

Vj(Gij cos uij + Bij sin uij),

i = 1, . . . , N

(9)

QGi − QDi = Vi

∑Nj=1

Vj(Gij sin uij − Bij cos uij),

i = 1, . . . , N

(10)

3.2.2 Inequality constraints: include the systemoperating and the security limits:

PminGi ≤ PGi ≤ Pmax

Gi , i = 1, . . . , Ng (11)

QminGi ≤ QGi ≤ Qmax

Gi , i = 1, . . . , Ng (12)

Vmini ≤ Vi ≤ Vmax

i , i = 1, . . . , N (13)

where N is the total number of buses in the system; PGi andQGi are the active and the reactive power generations atbus i; PDi and QDi are the active and the reactive powerloads of bus i; Gij and Bij are the transfer conductance andthe susceptance between buses i and j, respectively; θij isthe phase angle difference between the voltages at buses iand j.

3.3 Voltage stability constraint

Generally, the voltage magnitude limits for each bus are usedas the voltage constraints. However, the voltage limits aloneare not sufficient to guarantee an acceptable voltagestability level of the system under different operatingconditions. In this paper, a voltage stability constraint basedon the VCPI is added to the classical OPF. The aim of thisnew voltage security constraint is to limit the maximumvalue of the line index and then move the system far fromthe voltage collapse.The additional voltage stability constraint is formulated as

VCPImax ≤ VCPIlimit (14)

where VCPIlimit is a desired threshold value to ensure acertain system security level and VCPImax is the maximumvalue of the VCPI index defined as

VCPImax = max (VCPIi), i = 1, . . . , Nl (15)

and Nl is the total number of lines in the system.

4 VSC-OPF approach for preventive control

The purpose of the VSC-OPF based on the VCPI index ismainly to move the power system operation state far awayfrom the voltage collapse by increasing the system stabilitymargin. The proposed algorithm is incorporated into an


doi: 10.1049/iet-gtd.2013.0724

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automatic security monitoring and control system (ASMCS)as the preventive control scheme as illustrated in Fig. 2.Two approaches could be implemented by the blockVSC-OPF to design a preventive control system for thevoltage stability improvement and the power lossesminimisation. The VSI is embedded in the OPF formulationas the new voltage stability inequality constraint, or as anobjective function with the minimisation of the total VSI.Fig. 2 summarises the general view of an ASMCS. The
output of the state estimator is used to compute thevulnerability [40] and the line voltage stability [16] indicesin order to verify the system security. When the network isoverloaded and the voltage instable, the ASMCS is incorrective mode. In this situation, a first threshold(Threshold 1) for the voltage indices is designed to evaluatewhether the system is stable or not. According to thedefinition in the previous section, the VCPI values increasewith the increasing of the power flow transferred by thetransmission lines and vary from 0 (no load condition) to 1(voltage collapse). Therefore a threshold of 90% isconsidered appropriate in this work, if the maximum VCPIindex values exceed this threshold it means that the voltageof the critical line is very likely to collapse. Then, fastcontrol actions such as fuzzy-logic-based generationrescheduling approach [40], load shedding, flexible ACtransmission systems (FACTS) controllers, high voltagedirect current (HVDC) links and corrective switching (lineand bus switching) should be initially carried out to movethe system into a voltage secure operating point. On theother hand, if the contingency is already screened, asolution exists, and the preventive control is appliedautomatically to the system.After applying the corrective control actions, the system

restores the voltage stability and the values of the VCPIindex will definitely move below Threshold 1. In that case,a contingency analysis is performed to provide a list of themost severe contingencies, in terms of voltage stability

Fig. 2 General flowchart of an ASMCS


margins from a set of credible contingencies. If after acontingency study the voltage stability margin isinsufficient, then it is necessary to further performthe preventive control based on the VSC-OPF to move thesystem operating point away from the critical point (theVCPI max value is limited by Threshold 2) and thusobtaining adequate security margins.

5 Implementation of the VSC-OPF algorithm

The proposed VSC-OPF defined in the previous section issolved by using the fmincon function provided by thestandard optimisation toolbox of MATLAB [41]. In thisoptimisation algorithm, an approximation of the Hessianmatrix of the Lagrangian function is calculated at eachiteration and the problem is solved by using a line searchprocedure.The following steps describe the computational procedure

for solving the VSC-OPF presented in the diagram of Fig. 3.

(1) Read the system data.(2) Initialise the VSC-OPF by a conventional OPF thatconsists of minimising (7) subjected to power flow (9) and(10) and technical limits (11)–(13).(3) Calculate the VCPI for all the transmission lines byusing (1). Find the critical line with the highest VCPI indexvalue.(4) If the objective of the optimisation study is to restrict thevalue of the VCPI index within a range of 0 to VCPIlimit toachieve a required voltage stability level and to study theeffect of the security voltage on the generation cost, thenthe approach with the VCPI as the voltage constraint iscompleted as follows:

† Formulate the cost objective function (7).† Construct the conventional constraints given by (9)–(13).† Construct the voltage stability constraint given by (14).



Fig. 3 VSC-OPF computational algorithm

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† Calculate the first and the second derivatives of theconventional constraints.

† Calculate the first and the second derivatives of thevoltage stability constraint.

† Solve the optimisation problem by the line searchmethod.

(5) If the aim of the problem is the improvement of theoverall voltage stability of the system and the minimisationof the power losses, then the VCPI index is used as anobjective function and the OPF is executed as follows:

† Formulate the voltage stability objective function (8).† Calculate the conventional constraints given by (9)–(13).† Calculate the derivatives of the conventional constraints.† Solve the optimisation problem by the line search

method.(6) The voltage stability condition has been achieved whenthe objective function and the control variables converge. Atthat time, the execution of the algorithm stops, if not, it isrepeated from Step 3.

6 Simulation results and discussions

To verify and investigate the effectiveness and theperformance of the proposed approach, the standard IEEE30-bus and 57-bus depicted in Fig. 4 are used as the testsystems. The IEEE 30-bus system includes six generators atbuses 1, 2, 5, 8, 11 and 13, four transformers at lines 6–9,6–10, 4–12 and 28–27, 21 loads and 41 transmission lines.The total network loads are 283.40 MW and 126.20 MVAR,respectively. The IEEE 57-bus system consists of seven


generators connected at bus numbers 1, 2, 3, 6, 8, 9 and12, 80 transmission lines, 15 tap changing transformersand 42 loads totaling 1250.80 MW and 336.40 MVAR,respectively. In this system, the shunt reactive powersources are considered at buses 18, 25 and 53. Thetopology and the complete data of these networks are givenin [42]. The minimum and the maximum operating limitsfor the control variables are adapted from MATPOWER[43]. The lower and the upper voltage magnitudes are set to0.94 pu and 1.06 pu, respectively, from MATPOWER.The results are carried out by using MATLAB 7.11

environment with AMD Phenom(tm) II processor, 2.79GHz and 3 GB RAM.In this simulation study, the VCPI index is incorporated in

the optimisation problem in two ways. First, it can be added tothe OPF constraints as the new voltage stability constraint. Onthe other hand, the VCPI index can be minimised byformulating the index as the objective function of theoptimisation problem. The two approaches are applied forthe voltage stability enhancement and the power lossesminimisation under the stressed and the contingencyconditions in the system.Since the main purpose of this paper is the performance

evaluation of the proposed VSC-OPF approach based onthe line VSI (VCPI), the OPF problem has been solved fordifferent cases described as follows:

† Case 1: Cost function. In this first case, the minimisationof the fuel cost of generation is considered as anobjective function. The generator cost coefficients forthe IEEE 30-bus system [44] and the IEEE 57-bussystem [43] are defined in Tables 6 and 7, respectively.


doi: 10.1049/iet-gtd.2013.0724

Fig. 4 Single line diagram of the test systems

a IEEE 30-bus systemb IEEE 57-bus system

Table 1 System performance for the stressed conditions (theIEEE 30-bus system)

Objective functions

Case 1 Case 2 Case 3

Qgen, MVAR 177.18 158.76 142.58Ploss, MW 13.896 8.616 4.080VCPImax 0.3671 0.2600 0.2597VCPIT 4.6634 4.1663 3.6212MSV 0.4704 0.4790 0.4880FC, $/h 1260.53 1403.04 1868.60

Fig. 5 Variables comparison for the stressed conditions (the IEEE30-bus system)

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† Case 2: Cost function with the voltage stabilityconstraint. In this case, the VCPI is included as thevoltage stability constraint in the OPF model. Thiswill allow limiting the VCPI index of the most criticalline in the system and then increasing the voltagestability margin of the system. Consequently, theoptimal solution can satisfy the economic and thesystem security requirements simultaneously.

† Case 3: Total sum of the VSI. From the system securitypoint of view, an objective function which incorporatesan improvement of the system voltage stability is foundto be more efficient. As the VCPI index value indicatesthe proximity of the system to the voltage collapse, theminimisation of the sum of the voltage stabilityindices (VCPIT) is selected as an objective function,such that the overall system voltage stability isimproved. Thus, the lower the value of the VCPIT, thebetter the voltage stability.

In this study, the MSV of the reduced power flow Jacobianmatrix is used to validate the improvement of the systemvoltage stability. It is well known that the MSV of thepower flow Jacobian is a measure of the voltage stabilityand an accurate indicator of the system proximity to thevoltage collapse point [5]. Therefore the higher MSV value


(lower VCPIT value) indicates an improvement in thevoltage stability [26, 47].

6.1 IEEE 30-bus system: stressed conditions

To analyse the system under stressed conditions, the activeand the reactive loads of each bus are increased to 140% ofthe base load conditions. In this first situation, the VCPIindex value is below Threshold 1 indicating a systemvoltage secure operation point. However, the obtainedvoltage stability level is small and assumed not sufficient.Therefore the proposed preventive control based on theVSC-OPF is carried out to move the system away from thevoltage collapse and to obtain a new operation point withan adequate voltage stability margin.To assess the effectiveness of the proposed approach, a

comparison between three different cases is performed. Theresults of this comparison are given in Table 1.

6.1.1 VCPI as the voltage stability constraint: Thissection investigates the effect of the new voltage stabilityconstraint on the voltage stability enhancement and thepower losses minimisation. It is clear from Table 1 andFig. 5 that the reactive power generation, real power loss,VCPI max and the sum of the VCPI index values aresignificantly reduced with the voltage stability constraint(Case 2) compared with the case without the voltagestability constraint (Case 1).The results show that the real power loss (Ploss) is reduced

from 13.896 to 8.616 MW, with a percentage reduction of38%, and the reactive power generation (Qgen) is decreasedfrom 177.18 to 158.76 MVAR, with a percentage reductionof 10.4%. These positive performances are a goodindication about the system relieving from the stressedconditions to a more secure level.



Fig. 6 Voltage improvement of the system for the stressedconditions (the IEEE 30-bus system)

Fig. 7 Results for the heavily loaded system (IEEE 30-bus)

a VCPI index valuesb Line real power loss

Table 2 System performance for the line outage contingency(the IEEE 30-bus system)

Objective functions



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In addition, the highest value of the VCPI index (VCPImax)corresponding to the critical line is reduced to 0.2600(improved by 29.17%) in comparison of 0.3671 in case ofthe OPF without the voltage constraint. The sum of theVCPI index values (VCPIT) is minimised from 4.6634 to4.1663 which represents a gain in the voltage stabilitymargin of 10.66%. The MSV is also increased from 0.4704to 0.4790 indicating an improvement of the voltage stabilitylevel of the system as illustrated in Fig. 6.The voltage stability constrained OPF based on the VCPI

index for the stressed conditions results in a reduction of38% in real power loss and 29.17% reduction in VSI, butthe generation fuel cost has increased by 11.3% which isacceptable considering voltage stability enhancement andpower losses minimisation and the stressed conditions ofthe system. Note that the fuel cost decreases when thesystem is less stressed.

6.1.2 VCPI as an objective function: Here, the resultsobtained with the minimisation of the sum of the VCPIindex values (Case 3) are compared with those obtained bythe addition of the voltage stability constraint to the fuelcost function (Case 2). As indicated from Table 1 andFig. 5, the real power loss (Ploss) is reduced to 4.080 MWin comparison to 8.616 MW in Case 2, a reduction of about52.65%. The total VSI (VCPIT) is reduced from 4.1663 inCase 2 to 3.6212 in Case 3, a reduction of about 13.08%.Finally, the MSV is increased from 0.4790 to 0.4880demonstrating a good improvement of the system voltagestability as shown in Fig. 6.From the above results analysis, it is observed that Case 3

has the best performance in the voltage stability improvementand also has the minimum power system losses confirmingthe advantage of using the VCPI index as an objectivefunction in the VSC-OPF problem. However, this casecauses an increase in the fuel cost compared withCase 2. This increase represents the additional cost toimprove the voltage security of the system.

6.2 IEEE 30-bus system: line outage contingency

Line outage contingency generally causes undesirableoperating conditions and has a significant effect on alteringthe system security that could lead to the voltage collapse.Therefore to maintain the system security against thevoltage collapse, it is important to estimate the effect of thecontingency conditions on the voltage stability.In this case study, the line with the highest VCPI index is

identified as the most critical line, and therefore selected as


a candidate for outage. According to the index valuesobtained from the stressed conditions presented in Fig. 7, itis found that line 5 (connecting buses 2 and 5) has themaximum index value means that this line is the criticalline of the system. The results have also shown that line 5has the greatest real power loss 2.90 MW.To analyse the performance of the proposed VSC-OPF

based on the VCPI index under system disturbance, theoutage of line 5 (2–5) is considered at 1.20 times base loadconditions. This outage results in a reduced voltage stabilitylevel with a maximum value of the VSI VCPImax = 0.3291and highest value of the real power loss Ploss = 18.043 MW.The considered contingency is ranked among the five mostcritical contingencies for this test power system [45, 46]. Forthis specified system scenario, the system voltage stability isconcluded to be insufficient. Therefore the preventive controlbased on the VSC-OPF is executed to improve the voltagestability and to minimise the power losses.The comparative values of the various variables of the

system such as reactive power generation (Qgen), real powerloss (Ploss), the maximum value of the VCPI index(VCPImax) and the sum of the VCPI index values (VCPIT)for Cases 1–3 are summarised in Table 2.


doi: 10.1049/iet-gtd.2013.0724

Fig. 8 Variables comparison for the line outage contingency (theIEEE 30-bus system)

Table 3 System performance for the line outage contingency(the IEEE 57-bus system)

Objective functions



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6.2.1 VCPI as the voltage stability constraint: Theresults clearly show a considerable reduction of the variablevalues with the addition of the VCPI index as the voltagestability constraint (see Table 2 and Fig. 8). The real powerloss (Ploss) and the reactive power generation (Qgen) arereduced by 8.525 MW and 27.62 MVAR, respectively,corresponding to the 47.25 and 17.07% reduction.Furthermore, it can be observed from Table 2 that the

maximum value of the VCPI index (VCPImax) issignificantly reduced from 0.3291 to 0.2220 (32.54%reduction), the sum of the VCPI index values (VCPIT) isdecreased by 14.15% (from 4.3906 to 3.7693) and theMSV is increased from 0.4832 to 0.4898, thus indicating anenhancement in the overall voltage stability of the system asshown in Fig. 9.For the line outage contingency case, it is obvious that both

the voltage stability improvement and the real power lossminimisation are satisfied when adding the voltage stabilityconstraint, with 47.25% reduction in the real power loss and32.54% reduction in the VSI. However, this case hascaused an increase of the fuel cost by 10.03%.

6.2.2 VCPI as an objective function: To verify theeffectiveness of using the VCPI index as an objectivefunction (Case 3) under the contingency conditions, thecomputed results for this case are compared with those ofCase 2 (VCPI index as the voltage stability constraint).As shown in Table 2 and Fig. 8, it is clear that the

performance obtained in Case 3 is better than Case 2. Thereal power loss (Ploss) is 2.923 MW less by 69.29%compared with 9.518 MW obtained in Case 2. The total

Fig. 9 Voltage improvement of the system for the line outagecontingency (the IEEE 30-bus system)


VSI (VCPIT) is reduced from 3.7693 to 2.8975, a reductionof about 23.13% and the MSV is improved from 0.4898 to0.4970 resulting in the voltage stability improvement asshown in Fig. 9.

6.3 IEEE 57-bus system: line outage contingency

To evaluate the effectiveness of the proposed VSC-OPFapproach in the larger power system, the standard IEEE57-bus system is considered under the contingencyconditions. Based on the contingency analysis, the outageof the critical line (8–9) was identified as a severe case witha VCPImax value of 0.3980 and large real power loss Ploss

of 30.362 MW. Table 3 summarises the system performancefor the different case studies.

6.3.1 VCPI as the voltage stability constraint: Fromthe results given in Table 3 and Fig. 10, it is observed thatCase 2 (with the voltage stability constraint) obtains betterresults than Case 1 (without the voltage stability constraint).The real power loss (Ploss) is reduced from 30.362 to24.164 MW (20.41% reduction) and the reactive powergeneration (Qgen) is reduced from 316.67 to 294.74 MVAR(6.93% reduction). Moreover, the maximum value of theVCPI index (VCPImax) is significantly reduced from 0.3980to 0.2800 (29.65% reduction) and the total VSI (VCPIT) isdecreased from 9.4363 to 8.8148 (6.59% reduction).Finally, the MSV is improved from 0.2379 to 0.2393showing an enhancement of the system voltage stability. Onthe other hand, the fuel cost has increased by a smallmargin (0.44%).

6.3.2 VCPI as an objective function: The resultsobtained in Table 3 and Fig. 10 indicate that using theVCPI index as an objective function (Case 3) achieves thebest reactive power generation, the best power losses and

Fig. 10 Variables comparison for the line outage contingency (theIEEE 57-bus system)



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the best voltage stability margin. Consequently, the total realpower loss Ploss = 15.989 MW is on average 50% less than thelosses obtained in the other cases, the total VSI (VCPIT) isdecreased to a smallest value of 7.3521 and MSV of thereduced power flow Jacobian is improved to 0.2401, thusindicating an enhancement in the voltage stability.From all the previous cases, the VSC-OPF based on the
VCPI index approach gives good results regarding thevoltage stability improvement and the power lossesminimisation. However, using the VCPI index as anobjective function in the VSC-OPF problem has the bestperformance in all respects. In addition, the proposedmethod has shown an increase in the fuel cost. Thisincrease indicates the extra cost to enhance the voltagestability margin in the stressed and the contingencyconditions. From the above statement, it is clear that theproposed algorithm is able to simultaneously satisfy thevoltage stability and the power loss objectives.Nevertheless, for achieving the best cost it is necessary tocombine both the fuel cost objective function and thevoltage stability enhancement objective function by solvinga multi-objective OPF problem with an efficientoptimisation algorithm [28, 48].

7 Comparative study

In this section, the performance of the proposed VSC-OPFbased on the VCPI index (minVCPIT) is further validatedby comparing its results with the performance of aVSC-OPF using the L-index as an objective function. Asmentioned in the Introduction, the minimisation of themaximum L-index value (minLmax) has been widely used asan objective function for the voltage stability enhancementin both the normal and the contingency conditions. Inaddition, the MSV of the modified power flow Jacobianmatrix is used as a voltage stability indicator to verify theimprovement of the voltage stability margin. Therefore the

Table 4 Comparison results of the VSC-OPF based on the L-index an

O

IEEE 30-bus: stressedconditions

IEEE 3

minLmax minVCPIT minLm

Qgen, MVAR 148.25 142.58 120.7Ploss, MW 3.928 4.080 2.83MSV 0.4882 0.4880 0.500CPU time, s 4.64 1.86 4.92FC, $/h 1904.20 1868.60 1703.

Table 5 Comparison results for various VSC-OPF techniques on the

O

IEEE 30-bus: stressed conditions

minLmax maxMSV minVC

Qgen, MVAR 148.25 145.95 142.5Ploss, MW 3.928 3.963 4.08MSV 0.4882 0.4884 0.488CPU time, s 4.64 4.02 1.86FC, $/h 1904.20 1891.00 1868.


two VSIs namely the L-index and the MSV are consideredfor the comparison and the validation of the obtained results.On observing the results given in Table 4, the VSC-OPF

based on the VCPI index is shown to be more efficient inreducing the reactive power generation (Qgen) for all thecase studies. However, for the IEEE 30-bus system theVSC-OPF based on the L-index is little better inminimising the real power losses (Ploss) under the stressedand the contingency conditions. Nevertheless, for the largersystem (IEEE 57-bus), the VSC-OPF based on the VCPIindex achieves the best power loss minimisation. Inaddition, the two approaches give approximately the sameMSV with a difference of about 0.1%, which indicates asimilar voltage stability improvement margin. Obviously, inthe two operation conditions (three tests), the VSC-OPFbased on the VCPI index is the fastest algorithm since itconverges in 1.86, 2.14 and 6.82 s compared with 4.64,4.92 and 25.32 s, respectively, for the VSC-OPF with theL-index. As we know, the L-index calculation (see theAppendix) requires complex matrix inversion of the busadmittance submatrices, thus increasing the burden of thecalculation [39]. In terms of the fuel cost, it appears that theproposed VSC-OPF approach has the best generation fuelcost.Finally, the conclusion drawn from the comparison study is

that the two approaches realise nearly the same performancewith an advantage of the VSC-OPF using the VCPI indexwhich is computationally efficient and less expensive thanthe approach based on the L-index. The same conclusioncan also be drawn when the proposed approach has beencompared with a VSC-OPF based on the maximising of theMSV (or minimal eigenvalue) of the Jacobian matrix [49,50]. The results of the comparison of these differentVSC-OPF techniques (minLmax, maxMSV and minVCPIT)are presented in Table 5.These encouraging results make the proposed VSC-OPF

based on the VCPI index a promising candidate for thepower system optimisation problems considering the

d the VCPI index

bjective functions

0-bus: line (2–5) outagecontingency

IEEE 57-bus: line (8–9) outagecontingency

ax minVCPIT minLmax minVCPIT

5 115.35 290.11 269.173 2.923 20.837 15.9895 0.4970 0.2414 0.2401

2.14 25.32 6.8230 1628.70 47459.18 47241.94

IEEE 30-bus and 57-bus systems

bjective functions

IEEE 57-bus: line (8–9) outage contingency

PIT minLmax maxMSV minVCPIT

8 290.11 293.72 269.170 20.837 21.710 15.9890 0.2414 0.2413 0.2401

25.32 17.28 6.8260 47459.18 48292.51 47241.94


doi: 10.1049/iet-gtd.2013.0724

www.ietdl.org
voltage stability. Furthermore, the obtained results can beimproved by using an efficient optimisation technique tosolve the OPF problem instead of the standard optimisationtoolbox of MATLAB [41].
8 Conclusions

In this paper, a novel VSC-OPF approach for the voltagestability preventive control has been presented. Theproposed approach is addressed by incorporating the VCPIinto the classical OPF problem. The VCPI index is firstused as the voltage stability constraint, this requires onlyone additional voltage constraint added to the conventionalOPF constraints. Hence, the dimension of the resultingoptimisation problem is similar to that of a conventionalOPF, and this feature help in reducing the complexity andimproving the numerical solution feasibility of the problem.On the other hand, this index is formulated as the OPFobjective function and minimised to improve the voltagestability of the system.The effectiveness and the robustness of the proposed

VSC-OPF based on the VCPI index are tested anddemonstrated on both the IEEE 30-bus and the IEEE57-bus systems. The simulation results obtained under thestressed and the line outage contingency conditions arepromising and clearly show the potential of the proposedapproach to enhance the power system voltage security byimproving the system voltage stability and minimising thepower losses. It is also found that the approach based onthe VCPI index achieves a performance comparable withthe other reported approaches (minLmax and maxMSV) witha superiority in terms of the computational efficiency andthe solution cost.The proposed technique is based on a simple concept and

can be practically applicable for the online voltage securityassessment. In future works, this technique could be easilycombined with other stability constraints such as thetransient stability or the small signal stability and could beimplemented in multi-objective optimisation problems.

9 References

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Table 6 Generator cost coefficients for the IEEE 30-bus testsystem

Bus no Cost coefficients

a b c

1 0.00 2.00 0.0037502 0.00 1.75 0.0175005 0.00 1.00 0.0625008 0.00 3.25 0.00834011 0.00 3.00 0.02500013 0.00 3.00 0.025000

Table 7 Generator cost coefficients for the IEEE 57-bus testsystem

Bus no. Cost coefficients

a b c

1 0.00 20 0.07757952 0.00 40 0.01000003 0.00 20 0.25000006 0.00 40 0.01000008 0.00 20 0.02222229 0.00 40 0.010000012 0.00 20 0.0322581

www.ietdl.org
36 Cardet, C.D.: ‘Analysis on voltage stability indices’. Master
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10 Appendix

10.1 L-index calculation

The L-index [10] is a quantitative measure to evaluate thevoltage stability at each bus of the system and to estimatethe proximity to the voltage collapse. The value of theL-index of load bus j is calculated by

Lj = 1−∑Ng

i=1

FjiVi

Vj/(u ji + di − dj)

∣∣∣∣∣∣∣∣∣∣∣∣,

j = Ng + 1 . . .N

(16)

where Vi and Vj are the voltage magnitudes at generator bus i


and load bus j, respectively. δi and δj are the phase angles atbus i and bus j, respectively, and θji is the phase angle of termFji. Ng is the number of generators and N the total number ofbuses. The values of Fji are obtained from the submatrix FLG

calculated as follows

FLG

[ ] = − YLL[ ]−1

YLG[ ]

(17)

where YLL and YLG are the submatrices of the Y bus matrix in(18).

ILIG

[ ]= YLL YLG

YGL YGG

[ ]VLVG

[ ](18)

IL, IG and VL, VG represent the currents and the voltages at theload buses and the generator buses.The indicator value varies in the range between 0 (no load)

and 1 (voltage collapse). The bus with the highest L-indexvalue will be the most vulnerable bus in the system.

10.2 Generator cost coefficients

See Tables 6 and 7.


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