Fast method for computing power system securitymargins to voltage collapse

L.A.Ll. Zarate and C.A. Castro

Abstract: The paper presents a simple, fast and reliable method for determining the distance inparameter (load) space from the system current operating point to the critical point for whichvoltage instability and even voltage collapse may occur. Sensitivity analysis is used to drive thesystem from the base case operating point to the vicinity of the critical point along a predefinedload increase direction. In case the sensitivity analysis drives the system to the infeasible operatingregion, a special load flow method with step size optimisation is used so that relevant informationcan be obtained to restore the feasibility. Finally, the critical point is estimated from a certainnumber of conventional load flow calculations for different operating points in the vicinity of thecritical point, by increasing the load along the predefined direction. The security margin to voltagecollapse is obtained at a low computational cost. Simulation results for small and large powersystems are shown to validate the effectiveness of the proposed method.

1 Introduction

Voltage stability can be defined as the ability of the systemto maintain acceptable voltage levels under normal operat-ing conditions and after the occurrence of disturbances [1].Massive systems interconnections, demand increase, insuffi-cient generation and transmission expansion, economicaland environmental factors have led power systems tooperate with its equipment very close to their limits.Consequently, voltage instability and even voltage collapsesituations became very likely to occur, imposing importantlimitations on power systems operation. The voltageinstability process is characterised by a monotonic voltagedrop, which is slow at first and becomes abrupt after sometime. Voltage collapse occurs when the system is unable tomeet the demand, and it is usually related to a poor reactivepower support.Even though voltage stability is essentially a dynamic

phenomenon, analyses based on static approaches havebeen widely used, since they provide results with anacceptable accuracy and little computational effort. Thesefeatures are desirable in restrictive environments from thecomputational effort standpoint, such as in a real timeoperation environment.Many voltage stability margin calculation methods can

be found in the literature. Continuation methods are widelyknown as very powerful, though slow, methods to estimatethe system maximum loading [2, 3]. An iterative methodthat drives the system from the base case condition to themaximum loading situation through sensitivity analysis wasproposed in [4, 5]. Gradual load increases are defined alonga predefined direction based on the sensitivity of the loadswith respect to the reactive power injections at generationbuses. Zeng et al. [6] proposed a very interesting method to

estimate the maximum loading conditions by using thevoltages from a set of operating points, obtained byconventional load flow calculations along a predefined loadincrease direction. The method is based on the analysis ofJacobian matrix behaviour near the maximum loadingpoint, where it becomes singular [7]. Recently, the utilisationof interior point methods to obtain the critical point wasproposed in [8, 9]. Also, many voltage collapse proximityindices were proposed, such as the one based on theJacobian matrix minimum singular values [10].The main goal of this paper is to propose a simple, fast

and reliable method for the estimation of the maximumloading point and, consequently, the security margin tovoltage collapse. The method is based on the improvementand convenient integration of existing methods, in such away to make effective use of their main features, over-coming their drawbacks.

2 Maximum loading point calculation methodbased on sensitivity analysis [4, 5]

The operating condition of a power system is obtained bysolving the following set of nonlinear algebraic equations:

g x; u; pð Þ ¼ 0 ð1Þ

where x is the system state (voltage magnitudes and phaseangles) vector, u is the vector of control variables and prepresents the system parameters (loads).Given a predefined load increase direction, a sensitivity

analysis based method was proposed in [4, 5] for thecalculation of the maximum loading point and the securitymargin to voltage colapse. It can be shown that changes inthe reactive powers at generating units DQG can beexpressed as a function of changes in control variablesand parameters by

DQG ¼ SwuDu þ SwpDp ð2Þ

where Swu and Swp are sensitivity matrices. Suppose that theload at bus i is to be increased up to a point where agenerator reaches its reactive power generation limit. For asystem with NG generating units available, the load increase

The authors are with the Power Systems Department, School of Electrical andComputer Engineering, State University of Campinas-Brazil, DSEE/FEEC/UNICAMP, CP 6101, Campinas SP 13081-970, Brazil

r IEE, 2004

IEE Proceedings online no. 20040058

doi:10.1049/ip-gtd:20040058

Paper first received 20th May 2003 and in revised form 17th September 2003

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 1, January 2004 19

is defined by

DSi ¼ minj

QlimGj � QGj

� �s

24

35; j ¼ 1; . . . ;NG ð3Þ

where DSi may represent an increase in real power, reactivepower or both, andQGj andQGj

lim are respectively the currentreactive power generated at bus j and its reactive powerlimit. s is the sensitivity factor, obtained from Swp. Thebasic idea is to compute load increases iteratively till themaximum loading point is reached.The method proposed in [4] presents problems when the

load increase defined by (3) drives the system to aninfeasible operating region, that is, the specified load level islarger than the maximum acceptable load. This situationmay occur due to errors caused by the linearisationperformed in the sensitivity analysis. An attempt to solvethis problem was proposed in [5], through the utilisation ofa special load flow method with step size optimisation(LFSSO), which will be briefly described in Section 3. Incase of infeasible situations, LFSSO provides usefulinformation about the system condition, from which loadadjustments can be set to pull the system back to thefeasible operating region. Even though the methodproposed in [5] presented good results in general, somesituations led to accuracy problems or to a large number ofiterations.

3 Load flow method with step size optimisation(LFSSO)

LFSSO was first developed to be used for solving the loadflow equations of ill-conditioned systems. For those theconventional load flow methods exhibit poorer perfor-mance, or simply diverge, although the system is indeed inthe feasible region. The iterative calculation process ofLFSSO consists of updating state vector x at an iteration kby

xkþ1 ¼ xk þ m J xk� ��1

DSk ð4Þ

where DSk is the vector of power mismatches, J (xk) is theJacobian matrix and m is the optimal multiplier, computedso as to minimise a quadratic function based on the powermismatches. This idea was first presented in [11]. In thispaper the method proposed in [12] will be used. The latter isbased on the work of [11]; however, its formulation includesvoltages in polar co-ordinates.The main features of LFSSO are: (a) the performance for

well conditioned systems is identical to the conventionalload flow (mE1); (b) the performance for ill-conditionedsystems is significantly improved; (c) the results (systemstate variables and power mismatches) obtained for aninfeasible operating condition provide useful informationabout the system condition, and load adjustments can bedefined to pull the system back into the feasible operatingregion. In this case the optimal multiplier m tends to zeroalong the iterative process, which converges to an operatingpoint on the feasibility boundary [13].In particular, feature (c) will be used extensively in this

paper, playing a very important role in the security margincalculation method proposed in this paper.

4 Maximum loading point estimation methodproposed in [6]

Zeng and others [6] proposed a method to estimate themaximum loading point based on the behaviour of the

Jacobian matrix near the singularity. Starting from an initialpoint s0 in parameter (load) space, and following apredefined load increase direction Ds, the critical (maximumloading) point is given by

scr ¼ F xcrð Þ ¼ s0 þ lcr Ds ð5Þwhere xcr is the voltage (magnitudes and phase angles)vector at the critical point, F is the set of load flowequations and lcr the critical loading factor. The distancebetween the initial point s0 and the singularity point scr isgiven by k lcrDs k, as shown in Fig. 1.

A sequence of load levels is defined by

sk ¼ s0 þ kDs k ¼ 0; 1; 2; . . . ; n ð6Þ

The corresponding state vectors xk are obtained byperforming (n+1) load flow calculations. From themathematical analysis described in detail in [6], thefollowing relationship between each xk and xcr holds:

Dxk ¼ xk � xcr ¼ v l� kð Þ1=2 þ wðl� kÞ ð7Þwhere vector v represents the Jacobian matrix null spaceand w is a vector orthogonal to v and represents theJacobian matrix range space. The following vectors are alsodefined:

Djxk ¼ Dj�1xk � Dj�1xk�1 ð8Þwhere D0xk¼ xk. The following scalar can be computed:

bn ¼k Dn�1xn kk Dn�1xn�1 k

ð9Þ

For instance, if n¼ 4 (five load flow calculations and fivestate variables vectors), critical loading factor lcr can befound by solving

f lð Þ ¼ l� 4ð Þ1=2 � 3þ b4ð Þ l� 3ð Þ1=2

þ 3 1þ b4ð Þ l� 2ð Þ1=2

� 1þ 3b4ð Þ l� 1ð Þ1=2 þ b4l1=2 ¼ 0 ð10Þ

Although a larger value of n could be used, the gain inprecision is not worth the increase in computational effort[6]. Thus, n¼ 4 is usually chosen.

5 Evaluation and improvement of the methods

In this Section the methods described in Sections 2 and4 will be evaluated. Their important features as well astheir drawbacks will be highlighted. Some improvementswill also be proposed, to overcome their drawbacks. Theidea is to integrate their important features and overcometheir drawbacks, to obtain a simple, fast and robust securitymargin calculation method.

feasibilityboundary

Σ

scr

sk

λcr∆s

k∆s∆s

s0

Fig. 1 Critical point estimation in parameter (load) space

20 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 1, January 2004

5.1 Evaluation of the method proposedin [5]The method proposed in [5] and briefly described in Section2 presents a good overall performance. However, it is basedon linearisation of the system equations. The errorsprovided by the sensitivity analysis increase with systemloading. In other words, heavily loaded systems exhibit amore pronounced nonlinear behaviour, and a linearisedanalysis may result in larger errors. These errors are suchthat the load increase computed through sensitivity analysismay lead the system to the infeasible operating region. SinceLFSSO (Section 3) is used, the final power mismatches ofnonconverging load flow calculations provide an indicationof load adjustments that should be done to pull the systemback into the feasible operating region. In some cases, theprocess of searching for a feasible operating point can takea large number of iterations. This constitutes the maindrawback of the method.Table 1 shows some results obtained with the method

proposed in [5] for a 904 bus, 1283 branch system,corresponding to a reduced version of the USA South-western system. Load increases for each bus were doneindividually.

Even though the results are good for some situations, twokinds of problems were identified: (a) difficulties in thefeasibility restoration procedure led to an infeasible finalsolution, and (b) lack of precision, since the informationgiven by the power mismatches are not sufficient forobtaining a good estimate. The expected values shown inTable 1 correspond to the maximum loading level for whichLFSSO converges.One of the main drawbacks of the method proposed in

[5] is the way information from LFSSO is handled once aninfeasible point is reached. For a load increase in a certainbus k, only the power mismatch at bus k is used to define anappropriate load decrease to pull the system back into thefeasible region. The load decrease is defined as equal to thepower mismatch. Even though the load increase occurs inone bus only, other power mismatches may presentsignificant values, especially in the vicinity of bus k.

5.2 Improvement of the method proposedin [5]In this paper a more efficient way to define the loaddecrease mentioned in Section 5.1 is proposed. The idea isto pull the system into the feasible region with the leastnumber of load flow calculations.

Once a load in the infeasible region is defined throughsensitivity analysis, LFSSO does not converge, but providesthe best solution possible. This solution is on the feasibilityboundary. Consequently, there are nonzero real andreactive power mismatches. Since an inaccurate loadincrease led to the infeasible region, a load decrease(curtailment) must be determined to pull the system backto the feasibility region.Suppose that the load increase direction consists of

increasing the real power load at bus k only. The followingoptimisation problem is formulated to define the amount ofreal power load to be curtailed at bus k:

min f x; Pkð Þ ¼ 1

2k Ds k22¼

1

2DsTDs ð11Þ

subject to g x; Pkð Þ ¼ 0 ð12Þwhere f is the objective function defined in terms of powermismatches, x is the vector of bus voltages (magnitudes andphase angles), Pk is the scheduled power injection at bus kand g corresponds to the load flow equations. The followingLagrangean function is defined:

L x; Pk; lð Þ ¼ f x; Pkð Þ þ lT g x; Pkð Þ ð13Þwhere l is the vector of Lagrangean multipliers. By apply-ing the Karush–Kuhn–Tucker (KKT) optimality condi-tions [14] to (13), the load curtailment at bus k isgiven by

Pnewk ¼ Pcurrent

k � f x; Pkð ÞDPk

ð14Þ

where DPk is the component of Ds corresponding to the realpower mismatch at bus k.Simulations have shown that load curtailments according

to (14) result in a very slow process. Considering that thereal power mismatch at bus k is much larger than the othercomponents of Ds, we have

f x; Pkð ÞDPk

� 1

2DPk ð15Þ

which is indeed a small amount of load curtailment. Anacceleration factor a is proposed in this paper, so (14) isrewritten as

Pnewk ¼ Pcurrent

k � af x; Pkð ÞDPk

ð16Þ

where

a ¼ 2DPk

k Ds k2ð17Þ

From (16) and (17), the new load at bus k is given by

Pnewk ¼ Pcurrent

k � k Ds k2 ð18Þ

Considering again that the real power mismatch at bus kis much larger than the other components of Ds, we have

k Ds k2� DPk ð19Þ

In this case the load curtailment is the same as proposedin [5]. Since in practice other power mismatch componentsmay also be significant, the amount of real powercurtailment at bus k will be larger than DPk and theprocess will reach the feasible region in a smaller number ofiterations. In case the reactive power is also increased, say,by assuming constant power factor, the load curtailmentmust also follow the same pattern.

Table 1: Maximum loading point – 904 bus system

Bus Pmax, MW [5] Load flowcalculations

Pmax, MW(expected)

19 296.40 9 296.70

20 3046.83 6 (a) 2490.00

102 473.38 7 475.70

230 418.50 4 (a) 299.50

326 1230.37 7 1230.40

362 2747.71 4 (a) 2388.23

386 932.68 3 (a) 1250.81

505 1665.67 7 (b) 1686.83

654 308.76 4 (b) 278.50

722 916.68 4 (b) 921.50

823 1929.18 5 (a) 1843.50

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 1, January 2004 21

Table 2 shows the results obtained with this new loadcurtailment approach. By comparing Table 2 to Table 1,the improvement in the results is clear, as far as the numberof load flow calculations and maximum loading estimatesare concerned.

5.3 Evaluation of method proposed in [6]Regarding the method proposed in [6], simulations haveshown that its accuracy and efficiency are closely related to:(a) the distance from the set of operating points to thecritical point; (b) the load increase step size; (c) the value ofn; (d) the discontinuities resulting from reactive power limitsof generating units; and (e) the procedure for the calculationof the critical loading factor.Figure 2 shows the results obtained for the 14 bus, 20

branch system [15]. The real power load at bus 9 wasincreased only. DP represents the load increase step size,defined as a percentage of the base case real power.

The maximum loading estimates are more accurate whenthe set of operating points is closer to the critical point.Therefore, it is expected that an inaccurate maximumloading point is estimated for a lightly loaded system. Thisresult alone justifies the need for a procedure that drives thesystem from base case conditions to the vicinity of thecritical point first. Then, the method proposed in [6] wouldwork properly and provide an accurate estimate. The otherpoints that will be discussed in this Section will confirm thisneed.

Figure 3 shows how the accuracy of the estimates isaffected by the load increase step size. Results for differentvalues of n are shown. Larger step sizes provide betterestimates since the last point of the set of operating points iscloser to the maximun loading point.Discontinuities in the PV curve are observed whenever a

generator reaches its generated reactive power limit. Theconsideration of these limits plays an essential role inmethods for the calculation of the maximum loading point.Figure 4 shows a discontinuity that occurs for a certainloading level Pk

G. At this point a generator reaches itsreactive power generation limit. An immediate consequenceof this discontinuity is a decrease in the maximum loading(Pk

croPkcr0). It is possible to define three possible cases for

the maximum loading estimation, as shown in Fig. 4.

Each case comprises five operating points (n¼ 4). Nodiscontinuities occur for cases 1 and 3, that is, no reactivepower violation occurs within the respective load increaseranges. On the other hand, a discontinuity occurs for thecase 2. The difference between cases 1 and 3 is that theoperating points of case 3 are closer to the maximumloading point. As discussed before, the set of operatingpoints of case 3 provides a better estimate of the maximumloading point.Figure 5 shows the behaviour of the voltage magnitude at

bus 4 of the 14 bus system as a function of its real power.For this simulation, the synchronous condenser at bus 6had its upper reactive power limit set to 84MVAr. All otherloads and limits have been kept unchanged.

Table 2: Maximum loading point – 904 bus system

Bus Pmax, MW [5] Load flowcalculations

Pmax, MW(expected)

19 295.55 6 296.70

20 2481.76 5 (*) 2490.00

102 469.11 5 475.70

230 299.54 5 (*) 299.50

326 1215.85 4 1230.40

362 2356.24 5 (*) 2388.23

386 1246.77 5 (*) 1250.81

505 1678.12 6 1686.83

654 277.87 5 278.50

722 916.53 4 921.50

823 1843.67 6 (*) 1843.50

50 100 150 200 250

500

1000

1500

2000

2500

3000

3500

4000

max

. loa

ding

, MW

∆ P = 10 %∆ P = 30 %∆ P = 50 %

max. loading

0P9, MW

Fig. 2 Critical point estimate as a function of s0 for different loadincrease step sizes

0 50 100 150 200 250 3000

0.5

1.0

1.5

2.0

2.5

step size

max

. loa

ding

, MW

(×

104 )

n = 4n = 5n = 6

Fig. 3 Critical point estimation as a function of load increase stepsizes for different values of n

criticalpoint

case 1case 2

case 3

Vk

PkPkG

Vkcr

Pk Pkcr1 cr0

Fig. 4 Different cases for maximum loading estimation

22 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 1, January 2004

The system has four generators (plus the slack generator),each one with its lower and upper bounds. Therefore, thePV curve can be divided into sectors. In sector I allgenerators operate within their limits. In sector II, generator2 has reached its upper limit. In sector III, generators 2 and3 have reached their limits, and so on. It is possible toobtain a correspondence between the cases shown in Fig. 4and the sectors shown in Fig. 5. For instance, case 1 wouldcorrespond to the situation where all operating points usedare within one sector. Case 2 would correspond to operatingpoints in sectors II and III, among others. Case 3 is similarto case 1, except that the operating points are closer to thecritical point. Table 3 shows some simulation results fordifferent cases.

Five operating points (n¼ 4) equally spaced within thespecified range were used. Large errors are observed whenthe set of operating points fall into case 1. The same occursfor case 2, except that the estimates seem to be better whenthe last operating point gets closer from the critical point.However, since the latter is not known a priori, there is noguarantee that a good estimate will be obtained. As far ascase 3, for which all operating points fall into sector V, goodestimates were obtained, even for small ranges (small loadincrease step size). Figure 6 illustrates the results shownconcisely in Table 3.

It is important that the number of active generators(those that present generated reactive power within thelower-upper limit range) be the same for all operating pointsfrom the set used for the estimation. This will lead toaccurate estimates for the maximum loading point.

5.4 Improvement of method proposedin [6]The procedure for the calculation of the critical loadingfactor lcr by solving (10) also deserves special attention.Function f(l) is ill-conditioned, in the sense that conven-tional iterative methods, such as Newton’s method, couldexperience numerical difficulties, resulting in a large numberof iterations or even divergence. This ill-conditioningproblem is a direct consequence of the ill-conditioning ofthe load flow equations in the vicinity of the maximumloading point.This ill-conditioning problem was solved by: (a) in-

corporating step size optimisation in the calculation processand (b) using initialisation functions in order to initialise theiterative process very close to the final solution. Step sizeoptimisation in the Newton method had already been usedfor solving ill-conditioned power systems, as mentioned inSection 3 [11, 12]. Its basic idea is that at iteration z thevalue of l is updated by

lz ¼ lz�1 þ mDl ¼ lz�1 � mf lz�1ð Þf 0 lz�1ð Þ ð20Þ

where m is an optimal multiplier, calculated in such a way asto minimise the following quadratic function:

F mð Þ ¼ 1

2f 2 lz�1 þ mDlð Þ ð21Þ

Figure 7 shows that the calculation of lcr using theoptimal multiplier m results in a more stable iterativeprocess, if compared to the conventional Newton method.However, in both cases (with or without the use ofmultiplier m) the number of iterations is significantly large.A great reduction in the number of iterations was

obtained by using initialisation functions. Since function f (l)is always the same for any system, and the differentoperating points used contribute to the definition of thevalue of b only, the relationship bewteen l and b is alwaysknown, as shown in Fig. 8 for n¼ 4 (fb,l).Therefore, the iterative process for the calculation of lcr

can always be initialised with l0 very close to the solutionitself by defining initialisation functions. In this paper, threeinitialisation functions were defined for each value of n.Each initialisation function is chosen to fit f (l) within arange of b (see Fig. 8). For n¼ 4 (set of five operating

0P4 = 0.4780

crV4 = 0.721

crP4 = 3.7015 crP4

P4, pu

V4, pu

V (G2,G3,G8,G6)IV (G2,G3,G8)

III (G2,G3)I (0) II (G2)

Fig. 5 Bus 4 of the 14 bus system

Table 3: Critical point estimates – 14 bus system

Range, MW P4cr, MW

(estimate)Sectors Case

47.80–68.83 601.65 I 1

47.80–124.34 6009.33 I II 2

47.80–143.40 4159.07 I II III 2

47.80–239.00 8601.42 I II III IV 2

47.80–334.60 335.82 I II III IV V 2

70.50–126.90 133.47 II 1

70.50–143.82 143.82 II III 2

70.50–296.10 2370.23 II III IV 2

70.50–352.50 352.58 II III IV V 2

145.31–313.87 387.43 IV 1

145.31–360.37 360.36 IV V 2

319.30–362.72 370.93 V 3

360–365.76 369.03 V 3

0

1000

2000

3000

4000

5000

6000

7000

8000

max

. loa

ding

, MW

Pcrit = 370.15MW

P0 = 319.30P0 = 145.31P0 = 70.50P0 = 47.80

Fig. 6 Critical point estimates as a function of the different sectors

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 1, January 2004 23

points) the initialisation functions are:

fa bð Þ ¼ 4þ 9:61e�0:89b b 2fb bð Þ ¼ 4þ 45e�1:63b 1:5 bo2fc bð Þ ¼ 4þ 650e�3:40b bo1:5

ð22Þ

It is worth noting that there are also maximum allowedvalues for b. No real solution can be found for (10) in case bis greater than a maximum value. Such maximum valuesare approximately 10.5, 15.5 and 21.0 for 5, 6 and 7 loadflow calculations, respectively.

6 Proposed method

The main idea of the proposed method is to integrate themain features of the methods proposed in [5, 6] along withthe improvements discussed in Section 5 in order to comeup with a simple, fast and robust voltage collapse securitymargin estimation method.The method proposed in [5], taking into account the

improvement introduced and discussed in Section 5.2,showed itself to be appropriate to efficiently drive thesystem from the base case to the vicinity of the critical point.The method proposed in [6], along with the improvementproposed in Section 5.4, is able to efficiently estimate thecritical point, provided that the aspects discussed in Section5.3 are taken into consideration. The proposed method willbe described in detail with the help of Fig. 9. It should be

noted that load increases (real power, reactive power orboth, keeping a constant power factor) in one bus only areconsidered in this paper, even though different load increasepatterns can be accommodated:(a) From an initial load level (base case) s0, obtain new

load levels along a predefined load increase direction: s0-s1-y-si. Load increases are determined through sensi-tivity analysis, as described in Section 2 and in [5].Suppose that at iteration (i+1) the new load level si+1

lies outside the feasible operating region defined byboundary S, that is, it represents a load beyond the systemloading capacity. This is the usual situation found forrealistic systems, even though it is theoretically possible thatall generators reach their limits before the maximumloading point is reached. In this case, the procedure basedon (3) could not be used.(b) Given a scheduled load level si+1 in the infeasible

operating region, LFSSO converges to a certain point s* onthe boundary S (m tends to zero), and the powermismatches provide information on the approximatedistance from s* to scr. Using the ideas presented in Section5.2, a load adjustment is made, in an attempt to drive thesystem to a load level sj within the feasible operating region.The load adjustment consists of load curtailment, and theamount is defined by (18). If the load curtailment is notsufficient and the system is still in the infeasible region,repeat step (b) until an operating point in the feasible regionis obtained. It is not necessary to search for a point on S,but for a feasible point in its vicinity.(c) Obtain load level s0i such that the number of active

generators is the same as for sj. In general it is expected thats0i ¼ si. However, sensitivity errors resulting from thelinearisation process can result in s0i4si. This is more likelyto occur for buses with large security margins. In this papers0i is obtained by

s0i ¼ 1� xcþ 1ð Þn2cþ 1

sj c ¼ 0; 1; 2; . . . ð23Þ

where c is a counter, updated whenever a new load level s0iis computed by (23), and x¼ 0.01 is a constant empiricallyobtained to avoid the convergence problems discussed inSection 5.(d) Find equally spaced intermediate load level points

within the interval ½s0i; sj�. The number of intermediatepoints depends on the value of n. Estimate the maximumloading point scr using the method proposed in [6] with the

1 2 3 4 50

5

10

15

20

25

30

35

40

45

50

�

num

ber

of it

erat

ions

without �

with �init. fct.

n = 4

Fig. 7 Number of iterations for calculation of l

1 2 3 4 5 60

5

10

15

20

25

30

f�,�

fa

fb

fc

fc

fb

fa

f�,�

λ

�

Fig. 8 Initialisation functions for computation of lcr

V

P

Σ

s0 s1 si si+1sj scr

Fig. 9 Idea of proposed method

24 IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 1, January 2004

modifications proposed in Section 5.4. Compute the securitymargin to voltage collapse.

7 Simulation results

The results of simulations carried out for the 14 bus and 904bus systems using the procedure proposed in Section 6 areshown in this Section.Table 4 shows the security margins for a number of buses

from the 14 bus system. These margins are compared tothose obtained by the continuation method proposedin [16].

The results obtained by the proposed method are veryclose to the expected ones. Moreover, the number of loadflow calculations is small, as shown in Fig. 10. This is a veryimportant feature of the proposed method, indicating itspotential to be used in restrictive environments as far ascomputational time is concerned. The average CPU timefor the simulations shown in Table 4 was 0.08 s, obtained byrunning a Fortran program in a Sun workstation Ultra 1.Input/output processing times were not included.

Table 5 shows the maximum loading point estimates fora number of buses from the 904 bus system. These resultscan be compared to those shown in Tables 1 (methodproposed in [5]) and 2 (method proposed in [5] with themodifications of Section 5.2). The proposed methodpresented a good performance and provided more accurateestimates.Table 6 shows the security margins obtained by the

proposed method for a number of buses of the 904 buses.The expected values correspond to the maximum load levelsfor which LFSSO converges to a feasible solution. It can beseen that both values are very close.The average CPU time for the simulations shown in

Tables 5 and 6 was 11.5 s, obtained by running a Fortranprogram in a Sun workstation Ultra 1. Input/outputprocessing times were not included.

8 Conclusion

Security analysis studies require the use of accurate, efficientand robust methods. In this paper a very simple, efficientand accurate method was proposed for the estimation ofsecurity margins to voltage collapse. It was the result of acombination of existing methods which were analysed andimproved. The proposed improvements aimed to deal witha number of important aspects, such as discontinuities inthe PV curve due to limitations in the reactive power atgenerating units, feasibility restoration and ill-conditioning,among others.The load margins estimated with the proposed method

were accurate, and the procedure required both a smallnumber of load flow calculations and short CPU times,indicating that it has potential to be used in more restrictiveenvronments, such as real time operation.The estimation of security margins using the proposed

method for any load increase direction is under investiga-tion. Even though this additional step does not pose anydifficulty from the conceptual standpoint, its utilisation forlarge systems requires further analysis and development ofnew procedures as far as computational efficiency isconcerned.

9 Acknowledgments

The authors would like to acknowledge the financialsupport provided by FAPESP, Brazil.

10 References

1 Kundur, P.: ‘Power system stability and control’ (McGraw-Hill,New York, 1994)

2 Mansour, Y. (ed.): ‘Suggested techniques for voltage stability analysis’(IEEE, New York, 1993)

3 Ajjarapu, V., and Christy, C.: ‘The continuation power flow: a tool forsteady state voltage stability analysis’, IEEE Trans. Power Syst., 1992,7, (1), pp. 1441–1450

Table 4: Security margins – 14 bus system

Bus Margin, MW [16] Margin, MW (proposed)

4 339.17 322.59

7 196.42 194.41

9 176.86 176.57

14 109.38 111.21

.

.

.

bus 4bus 7bus 14

load

pow

er, M

W

load flow calculations

0

50

100

150

200

250

300

350

400

450

2 4 6 8 10 12

370.39

194.41

126.10

Fig. 10 Iterative process evolution – 14 bus system

Table 5: Maximum loading points – 904 bus system

Bus Proposed, MW Load flow calculations

20 2490.45 9

326 1228.14 8

362 2385.31 9

386 1251.82 9

505 1679.35 10

654 278.69 9

722 927.47 8

Table 6: Security margins – 904 bus system

Bus Proposed, MW Expected, MW

38 672.18 672.00

92 58.72 58.74

449 418.20 418.81

555 594.58 594.66

654 249.01 248.82

759 584.90 583.98

772 315.47 315.90

IEE Proc.-Gener. Transm. Distrib., Vol. 151, No. 1, January 2004 25

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