Voltage Stability Analysis Based on Probabilistic Power Flow and Maximum Entropy

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Published in IET Generation, Transmission & DistributionReceived on 12th February 2009Revised on 5th November 2009doi: 10.1049/iet-gtd.2009.0071

ISSN 1751-8687

Voltage stability analysis based on probabilisticpower flow and maximum entropyJ.F. Zhang1 C.T. Tse1 W. Wang2 C.Y. Chung1

1Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kong, People’s Republic of China2Department of Electrical Engineering, Zhengzhou University, Zhengzhou, Henan, People’s Republic of ChinaE-mail: [email protected]

Abstract: Methods of determination of voltage stability margin index had been well established. This study adoptsa new method to determine the probabilistic distribution of margin index taking into account the randomvariations of bus loads. First, the probabilistic technique and the Jacobian method are combined to determinethe probabilistic characteristics of stability margins and nodal voltages at the maximum load points. Then,according to these probabilistic characteristics, maximum entropy method is adopted to determine theprobabilistic distribution of stability margin. Last, the proposed method is investigated on two test systemswith random active and reactive loads. Monte Carlo simulations are used as a reference solution to evaluatethe accuracy of the proposed method.

1 IntroductionThere is an increasing concern about modern power systemvoltage stability particularly after some blackouts because ofvoltage instability or collapse in different countries [1, 2].Several methods based on different models for voltagestability analysis have been presented. Static voltage stabilityanalysis based on power flow attracts researchers for itssimple and fast calculation. It regards the maximum loadpoint as the critical point, or the so-called saddle-nodebifurcation. Static voltage stability analysis can providemargin index for power system operators. Stability marginis the distance between the current operating point and themaximum load point. Stability margin is a useful indexbecause it indicates how close the power system is tovoltage collapse.

Deterministic voltage stability analysis requires specificvalue for loads. In practice, the loads vary from time totime. It is impossible to carry out conventional voltagestability study for every possible or probable combination ofbus loads because of the extremely large computationaleffort required. Probabilistic methodology may be mostsuitable to analyse the system uncertainties. Probabilisticanalysis considering the uncertainties in loads, generationor network has been performed in literatures. The method

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of combined cumulants and Gram–Charlier series has beenused in probabilistic power flow to determine the majorinvestment on improving transmission system inadequacy[3]. Probabilistic eigenvalue analysis considering multi-operating conditions has been used to analyse and designrobust power system stabiliser (PSS) to improve the angularstability of power system [4, 5]. The results of theprobabilistic load flow have been used to assess voltageinstability [6]. A new nodal loading model, called thehyper-cone model whose thickness represents theuncertainty of future loading, has been proposed for voltagestability assessment of electric power system, and theformulation of worst cases based on this model, as well asrelated numerical methods, has been described in [7].Sobierajski used the P–Q curve to estimate the probabilityof the critical voltage violation under the assumption thatactive and reactive power at a given load bus are uniformlydistributed [8]. Schellenberg proposed a cumulant-basedmethod to solve a maximum loading problem incorporatinga constraint on the maximum variance of the loadingparameter [9]. This method takes advantage of someproperties regarding saddle-node bifurcations to create alinear mapping relationship between random bus loadingvariables and all other system variables so that the cumulantsof system variables are evaluated and the probabilistic densityfunction (PDF) of stability margin is obtained.

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This paper combines the probabilistic technique and theJacobian approach to determine the probabilisticcharacteristics of stability margin and nodal voltages at themaximum load points, and then applies the maximumentropy technique to study the distribution of stabilitymargin considering the load variations. Entropy, aconception in information theory, is a measure ofuncertainty. An entropy application is to deal with problemsinvolving the determination of unknown distributions butwith available information such as expected values or otherstatistical functions [10, 11]. Like the cumulant method,maximum entropy method is a general method and is notrestricted to a special distribution. With the moments ofrandom variable of stability margin obtained in terms ofprobabilistic Jacobian approach, the distribution can bedetermined with maximum entropy.

This paper is arranged as follows. In Section 2, threeprevailed methods to obtain static voltage stability marginare first revealed. Probabilistic voltage stability margin andits calculation are proposed in Section 3. The method ofmaximum entropy is introduced in Section 4. Theproposed probabilistic method is investigated on two testsystems and the results are compared with Monte Carlo inSection 5. Section 6 gives conclusions.

2 Static voltage stability analysesStatic voltage stability analysis still attracts researchers’ attentionfor its simple calculation and intuition. This analysis regardsthe maximum load point as the critical point. The distancebetween the current operating point and the maximum loadpoint is called voltage stability margin. Methods to determinethe maximum load, such as continuation power flow [12,13], non-linear programming method [14, 15] and Jacobianmethod [16, 17], have been used to obtain the saddle-nodebifurcation and stability margin. The pros and cons of thesemethods are discussed as below.

2.1 Continuation power flow

The model is described as follows

f (V )� lB � S0 ¼ 0

l0 � l � lcritical

(1)

where V is the vector of nodal voltages, B describes loadincreases and S0 is the normal injection power vector(including active and reactive power). l is the load factor.l ¼ 0 corresponds to normal load level and l ¼ lcritical

corresponds to critical load level.

To access the critical point and stability margin, the iterativecalculation starts from a known solution, and the predictor andcorrector technique is used to obtain the subsequent solution atdifferent load levels. As the Jacobian matrix of power flowequations is singular at the maximum load point, it is difficult

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to obtain the solution of power flow near the critical point.The difficulty is overcome by introducing other parameterand equation [12, 13]. This makes the power flow Jacobiannon-singular at the voltage collapse point.

Although the continuation power flow method can obtainthe critical point and stability margin without the calculationdifficulty near the critical point, this method only obtains anapproximate solution and is time consuming.

2.2 Non-linear programming method

The voltage collapse point is the maximum load point.Calculating the maximum load is represented as a static non-linear programming problem. The objective function is to findthe maximum load factor with the constraints of power flowequations and other inequalities. The problem is presented as

max l

s:t: f (V )� lB � S0 ¼ 0

g � h(V ) � l

(2)

The equalities are the ac power flow equations at different loadlevel. The inequality constrains contain any variable limitationsas well as reactive power limits of generators. The merit of thismethod is that some constrains can be considered easily.

2.3 Jacobian method at critical point

A salient characteristic of the critical point is that theJacobian matrix J of power flow equations is singular; J hasa zero eigenvalue but the corresponding eigenvectors (leftand right) are non-zero.

Based on this property, the exact critical point of powersystem is directly obtained by solving the extended powerflow equations as follows

f (V )� lB � S0 ¼ 0 (3a)

J TV w ¼ 0 (3b)

wTw� 1 ¼ 0 (3c)

where w is the left eigenvector of JV with respect to zeroeigenvalue and T stands for transpose. Equation (3a)describes the power flow and (3b) and (3c) ensure the JV issingular at the critical point. This method, also called pointof collapse method, is a particular case of non-linearprogramming method.

3 Stability margin assessment byprobabilistic approachThere are always uncertainties in power systems, such as thevariations of loads and the generators outputs. In this paper,loads are regarded as random variables. Based on the Jacobianapproach mentioned in Section 2.3, the nodal voltages, the

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left eigenvectors and hence stability margin are also randomvariables.

For the sake of convenience, the Jacobian approach methodis combined with probabilistic method to determine thecharacteristics of stability margin. If the voltages in (3) areexpressed with rectangular coordinates, every expression isthe second-order function of nodal voltage and lefteigenvector. Equation (3) can be represented as

F 1(V1V1, . . . , V1Vn, . . . , VnV1, . . . , VnVn)�lB�S0¼ 0 (4a)

F2(V1w1, . . . , V1wn, . . . , Vnw1, . . . , Vnwn)¼ 0 (4b)

F 3(w1w1, . . . , wnwn)�1¼ 0 (4c)

where Vi and wi, for i ¼ 1 to n, respectively, stand for theelements of voltage vector and left eigenvector. S0 iscomposed of initial injections of active and reactive powerPGj0 2 PLj0 and QGj0 2 QLj0 for j ¼ 1 to N. In this study,PLj0 and QLj0 are active and reactive power of load at bus jand are random variables. PGj0 and QGj0 are active andreactive power of generator at bus j and are not randomvariables. S0 is random vector. B describes load increase. Inthis paper, B is expectation of S0, that is B is composedby injections of active and reactive powers PGj0�PLj0 andQGj0�QLj0 for j ¼ 1 to N. l stands for stability margin. Atthe critical point, the load at bus j can be represented byPLj ¼PLj0þlPLj0 and QLj ¼QLj0þlQLj0.

If (4) is expanded at the means of nodal voltages, stabilitymargin, the left eigenvector and nodal injection

F1(V 1V 1, . . . , V 1V n, . . . , V nV 1, . . . , V nV n)

þF 1(DV1V 1, . . . ,DV1V n, . . . ,DVnV 1,. . . ,DVnV n)

þF 1(V 1DV1, . . . , V nDV1, . . . , V 1DVn,. . . , V nDVn)

þF 1(DV1DV1, . . . ,DV1DVn,. . . ,DVnDV1, . . . ,DVnDVn)

�lB�DlB�S0�DS0¼0 (5a)

F2( �V 1 �w1,. . . , �V 1 �wn, . . . , �V n �w1, . . . , �V n �wn)

þF2(DV1 �w1, . . . ,DV1 �wn, . . . ,DVn �w1, . . . ,DVn �wn)

þF2( �V 1Dw1, . . . , �V 1Dwn,. . . , �V nDw1, . . . , �V nDwn)

þF2(DV1Dw1, . . . ,DV1Dwn, . . . ,DVnDw1, . . . ,DVnDwn)

¼0 (5b)

F 3( �w1 �w1, . . . , �wn �wn)

þF3( �w1Dw1, . . . , �wnDwn)

þF3(Dw1 �w1, . . . ,Dwn �wn)

þF3(Dw1Dw1, . . . ,DwnDwn)�1¼0 (5c)


The expectation of (5) is expressed as (6)

F1( �V 1�V 1, . . . , �V 1

�V n, . . . , �V n�V 1, . . . , �V n

�V n)

þF 1(DV1DV1, . . . ,DV1DVn,. . . ,DVnDV 1, . . . ,DVnDV n)

� �lB� �S0¼0 (6a)

F2( �V 1 �w1,. . . , �V 1 �wn, . . . , �V n �w1, . . . , �V n �wn)

þF2(DV1Dw1, . . . ,DV1Dwn, . . . ,DVnDw1, . . . ,DVnDwn)

¼0 (6b)

F3( �w1 �w1, . . . , �wn �wn)þF 3(Dw1Dw1, . . . ,DwnDwn)�1

¼0 (6c)

As DViDVj¼CViVj, DViDwj¼CViwj

, DwiDwj¼Cwiwj, (6)

can also be represented with covariance of randomvariances of nodal voltages and left eigenvector. With thecovariances, the means of nodal voltages, stability marginand left eigenvector at critical point can be obtained bysolving (6) with Newton–Raphson method.

Omitting the second-order terms of (5) and (6), thelinearised relationships among nodal voltage V, stabilitymargin l, left eigenvector w and the nodal load S0 areshown as

JDVDwDl

24

35 ¼ DS0

00

24

35 (7)

where

J ¼

@F1

@V

��V¼ �V

0 �B

@F2

@V

��V¼ �V

@F 2

@w

��w¼ �w

0

0@F 3

@w

��w¼ �w

0

266666664

377777775

The deviations from the means of random variables nodalvoltage, left eigenvector and stability margin is obtained byrearranging (7) as

DVDwDl

24

35 ¼ J�1

DS0

00

24

35 (8)

The covariances of random variables nodal voltage, left


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eigenvector and stability margin are obtained as

C ¼

CVV CVw CVl

CwV Cww Cwl

ClV Clw Cll

264

375

¼ J�1E

DS0

0

0

264

375 DST

0 0 0� �0

B@1CA(J�1)T

¼ J�1

CS0S00 0

0 0 0

0 0 0

264

375(J�1)T (9)

The nth moment for [DV Dw Dl]T is computedfrom (8). As a result, the nth moment for[ V w l]T can be obtained using following equation

E(X n) ¼ ( �X )nþ (DX )n

þXn�1

r¼1

Crn

�Xr(DX )n�r (10)

The probabilistic characteristics of critical points can bedescribed by probabilistic characteristics of stability marginsand nodal voltages. The procedure of probabilistic criticalpoints calculation is as follows.

Step 1: Calculate the probabilistic power flow at a load level,and obtain the initial voltages and covariances of voltages CV.

Step 2: Calculate the eigenvalues and eigenvectors of Jacobianmatrix of power flow equation. The left eigenvectorcorresponding to minimum real eigenvalue is the initial lefteigenvector.

Step 3: Compute the mismatches in (6); if the mismatchesmeet the error requirement, go to step 6, otherwise continue.

Step 4: Form the Jacobian matrix J of extended power flow (6)with respect to means of nodal voltage, stability margin andleft eigenvector.

Step 5: Calculate the correction of means of nodal voltage,stability margin and left eigenvector and correct them; goto step 3.

Step 6: Calculate the covariances of nodal voltage, lefteigenvector and stability margin according to (9). Substitutethe covariances in (6), and calculate the mismatches. If themismatches meet the error requirement, stop, otherwise goto step 4.

Consequently, the means and covariances of both nodalvoltages and stability margin can be determined from theprobabilistic computation. The other moments of stabilitymargin can be obtained in terms of (10).

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4 Maximum entropy methodEntropy is a measure to tackle uncertainty. The entropy of aPDF p(x) is defined as [11]

H ¼ �

ðp(x) ln p(x) dx (11)

An application of entropy is to determine an unknowndistribution based on the principle of maximum entropy,provided that expected values and its other statisticalfunctions are given. The solution by maximum entropy, inits general form [18], is

max H ¼ �

ðp(x) ln p(x) dx

s:t: E{wn(x)} ¼

ðwn(x)p(x) dx ¼ mn, n ¼ 0, . . . , N

(12)

where m0 ¼ 1,w0(x) ¼ 1 and wn(x),n ¼ 1, . . . , N are Nknown functions, and mn, n ¼ 1, . . . , N are N givenexpectation data. The classical solution of this problem isgiven by

p(x) ¼ exp �XN

n¼0

lnwn(x)

" #(13)

The Nþ 1 Lagrange parameters l ¼ [l0, . . . , lN ] areobtained by solving the following Nþ 1 non-linear equations

ðwn(x)exp �

XN

n¼0

lnwn(x)

" #dx¼mn, n¼ 0, . . . , N (14)

For illustration, the maximum entropy is applied to a simpleexponential distribution with a PDF of

f (x)¼le�lx (15)

where x � 0, l . 0. If l ¼ 4, 0 � x � 4.

With the PDF, expectations of random variable functions,for example the geometrical moments, can be calculated aslisted in Table 1. The objective is to determine thedistribution of random variable only with known functionwn(x), n ¼ 0, . . . , 4 and expectations of random variablefunctions Efwn(x)g ¼ mn in Table 1. By solving the fivenon-linear equations in (14), the five Lagrange parametersl ¼ [l0, l1, l2, l3, l4] in (13) are obtained. The PDF of

Table 1 Functions and their expectation of random variable

n 0 1 2 3 4

wn(x) 1 x x2 x3 x4

mn 1 0.25 0.125 0.09374 0.09371

533


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random variable x is (13) with five known Lagrangeparameters l ¼ [l0, l1, l2, l3, l4 ].

The PDF obtained by maximum entropy is then comparedwith known PDF of exponential distribution in Fig. 1. Notethat for this simple case, the derived curve and the actual

Figure 1 PDF of exponential distribution


exponential curve almost coincide, and accurate result canbe obtained when N ¼ 4. This shows that maximumentropy method can determine any unknown distribution ifenough available information of random variables is known.

In this paper, according to (10), the moments ofstability margin and nodal voltage can be computed.Therefore wn(x) ¼ xn, and mn for n ¼ 1, . . . , 4 are availableinformation to determine the distribution stability margin.

5 Applications of probabilisticapproach and maximum entropyThe proposed method is investigated on two test systems: the 39-bus system and the 57-bus system. Normal distribution is one ofthe most common random distributions. All real and reactiveloads in this study are treated as Gaussian random variables.The loads are independent random variables with means at the

Table 2 Stability margin for the 39-bus system

sL Probabilistic method Monte Carlo Difference (%)

Mean Variance (1023) ISM index of 99% Mean Variance (1023) Mean Variance

(a) 0.025m 1.3513 0.2248 1.3163 1.3510 0.2252 0.0222 0.1776

(b) 0.05m 1.3500 0.8993 1.2801 1.3495 0.9036 0.0371 0.4759

(c) 0.075m 1.3480 2.0238 1.2431 1.3472 2.0428 0.0594 0.9301

(d) 0.1m 1.3450 3.5989 1.2052 1.3436 3.7263 0.1042 3.4189

Figure 2 Probabilistic distributions of stability margin with different sL of the 39-bus system

a sL ¼ 0.025mb sL ¼ 0.05mc sL ¼ 0.075md sL ¼ 0.1m


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normal bus loading, denoted bym, from the original system. Thevariance of each load has been chosen such that 99% confidencein within +10% from the normal loading value [9]. The loadstandard deviation sL ¼ 0.04m has been used for probabilistictransient stability analysis in [19]. In order to study the effectof uncertainty of load on stability margin, different standarddeviations of loads from 0.025m to 0.1m are, respectively,analysed. For simplicity, there are no limitations for reactivepower of generation or bus voltage levels considered. All theloads are constant power models. The probabilistic stabilitymargin and its distribution are determined by applyingprobabilistic Jacobian method (Section 3) and maximumentropy (Section 4). Monte Carlo simulations, consisting of1000 samples, are used to validate the proposed method.

5.1 Case study on the 39-bus system [20]

With different variance of loads, the means and variances ofstability margins are calculated from (6) and (9) using

probabilistic Jacobian method as listed in Table 2. FromTable 2, it is observed that although the mean of stabilitymargin has insignificant changes, the variance of stabilitymargin increases with load standard deviation sL (from0.025m to 0.1m) as expected. Based on the moments, thePDFs obtained by the maximum entropy approach areplotted in Fig. 2. In Table 2, a stability margin index ISM

of, say, 99% confidence is introduced. For example,ISM ¼ 1.3163 in (a) implies that if the system load isincreased by 1.3163B (B is the load expectation in (4)),99% of scenarios will not experience voltage collapse (i.e.99% of area under the PDF in Fig. 2a has a margin greaterthan 1.3163) for a system load standard deviationsL ¼ 0.025m. Of course, if the load variance increases, ISM

will be reduced.

To validate the probabilistic point of collapse method, themean and variance are compared with those obtained by

Table 3 Stability margin for the 57-bus system

sL Probabilistic method Monte Carlo Difference (%)

Mean Variance (1023) ISM index of 99% Mean Variance (1023) Mean Variance

(a) 0.025m 0.80157 0.3769 0.7872 0.80158 0.3769 0.0012 0.0000

(b) 0.05m 0.80140 1.5074 0.7727 0.80142 1.5074 0.0025 0.0000

(c) 0.075m 0.80112 3.3916 0.7582 0.80116 3.3914 0.0050 0.0059

(d) 0.1m 0.80073 6.0293 0.7435 0.80078 6.0053 0.0061 0.3996

Figure 3 Probabilistic distributions of stability margin with different sL of the 57-bus system

a sL ¼ 0.025mb sL ¼ 0.05mc sL ¼ 0.075md sL ¼ 0.1m

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Monte Carlo method based on 1000 deterministic loadflows(Section 2.3) as listed in Table 2. Results show that theirdifferences in the mean computation are very small, and thedifference in variance is only noticeable with high loaduncertainty.

To validate the maximum entropy method, the kernelsmoothing density estimation is used to obtain the actualdistribution of stability margin (in Fig. 2) from 1000stability margin samples. From the curve obtained MonteCarlo, it can be seen that the stability margin is not normaldistribution. The probabilistic distributions of the stabilitymargins obtained by the two approaches are compared withdifferent sL shown in Fig. 2 and the results show that theirdifferences are acceptably small.

5.2 Case study on the 57-bus system [21]

The 57-bus system includes four generating units and threesynchronous condensers. The means and variances ofstability margin with different sL are listed in Table 3 andthe probabilistic distributions are shown in Fig. 3. Asexpected, the so-called stability margin index ISM will bereduced with increase of load standard deviation sL (i.e.with increase of system uncertainty). The computationresults are then compared with Monte Carlo simulationand the difference between two methods are quite small.

6 ConclusionMethods of voltage stability margin index calculation had beenwell established, but these methods are based on deterministicsystem load flows. This paper proposes a new probabilisticapproach to evaluate this important margin index by takinginto account the uncertainty of system loads. With theassumption that all loads are independent Gauss distributions,the prevailed Jacobian method to evaluate stability margin ismodified such that the moments of stability margin and criticalnodal voltage can be obtained. Maximum entropy technique isthen applied to determine the distribution of the stabilitymargin. As expected, the stability margin will be reduced withhigher load uncertainty, implying that the stability marginsobtained based on traditional deterministic loadflows are too‘optimistic’. The proposed approach is tested on two testsystems. To validate the proposed probabilistic algorithm, thedistributions of the stability margin are compared with MonteCarlo technique (by 1000 computation simulations usingdeterministic method) and the computation results are foundto be quite close. As a conclusion, the proposed probabilisticapproach (based on single computer run) is very effective andpragmatic to compute the voltage stability margin undersystem uncertainties.

7 AcknowledgmentsThis work is supported by Research Grants Council of HongKong under project PolyU 5212/03E and the Hong KongPolytechnic University.


8 References

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

[2] TAYLOR C.W.: ‘Power system voltage stability’ (McGraw-Hill, New York, 1994)

[3] ZHANG P., LEE S.T.: ‘Probabilistic load flow computationusing the method of combined cumulants and Gram –Charlier expansion’, IEEE Trans. Power Syst., 2004, 19, (1),pp. 676–682

[4] WANG K.W., CHUNG C.Y., TSE C.T., TSANG K.M.: ‘Improvedprobabilistic method for power system dynamic stabilitystudies’, IEE Proc. Gener. Transm. Distrib., 2000, 147, (1),pp. 37–43

[5] TSE C.T., WANG K.W., CHUNG C.Y., TSANG K.M.: ‘Robust PSS designby probabilistic eigenvalue sensitivity analysis’, Electr. PowerSyst. Res., 2001, 59, (1), pp. 47–54

[6] HATZIARGYRIOU N.D., KARAKATSANIS T.S.: ‘Probabilistic loadflow for assessment of voltage instability’, IEE Proc. Gener.Transm. Distrib., 1998, 145, (2), pp. 196–202

[7] KATAOKA Y.: ‘A probabilistic nodal loading model andworst case solutions for electric power system voltagestability assessment’, IEEE Trans. Power Syst., 2003, 18,(4), pp. 1507–1514

[8] SOBIERAJSKI M., FULCZYK M.: ‘Voltage stability study by p–qcurve with rectangular probability distribution of bus load’.Eighth Int. Conf. on Probabilistic Methods Applied to PowerSystems, Iowa State University, Ames, Iowa, September2004, pp. 894–899

[9] SCHELLENBERG A., ROSEHART W., AGUADO J.A.: ‘Cumulant-basedstochastic nonlinear programming for variance constrainedvoltage stability analysis of power systems’, IEEE Trans.Power Syst., 2006, 21, (2), pp. 579–585

[10] PAPOULIS A.: ‘Probability, random variables, and stochasticprocesses’ (McGraw-Hill, New York, Boston, 2002, 4th edn.)

[11] ZELLNER A., HIGHFIELD R.A.: ‘Calculation of maximum entropydistributions and approximation of marginalposteriordistributions’, J. Econ., 1988, 37, (2), pp. 195–209

[12] AJJARAPU V., CHRISTY C.: ‘The continuation power flow:a tool for steady state voltage stability analysis’, IEEETrans. Power Syst., 1992, 7, (1), pp. 416–423

[13] IBA K., SUZUKI H., EGAWA M., WATANABE T.: ‘Calculation ofcritical loading condition with nose curve using homotopycontinuation method’, IEEE Trans. Power Syst., 1991, 6,(2), pp. 584–593


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[14] IRISARRI G.D., WANG X., TONG J., MOKHTARI S.: ‘Maximumloadability of power systems using interior pointnonlinear optimization method’, IEEE Trans. Power Syst.,1997, 12, (1), pp. 162–172

[15] PARKER C.J., MORRISON I.F., SUTANTO D.: ‘Application of anoptimisation method for determining the reactive marginfrom voltage collapse in reactive power planning’, IEEETrans. Power Syst., 1996, 11, (3), pp. 1473–1481

[16] CANIZARES C.A., ALVARADO F.L.: ‘Point of collapse andcontinuation methods for large AC/DC systems’, IEEETrans. Power Syst., 1993, 8, (1), pp. 1–8

[17] CHIANG H.-D., JEAN-JUMEAU R.: ‘A more efficient formulation forcomputation of the maximum loading points in electric powersystems’, IEEE Trans. Power Syst., 1995, 10, (2), pp. 635–646

Gener. Transm. Distrib., 2010, Vol. 4, Iss. 4, pp. 530–537i: 10.1049/iet-gtd.2009.0071

[18] MOHAMMAD-DJAFARI A.: ‘A matlab program to calculatethe maximum entropy distributions’. Maximum entropyand Bayesian methods: proc. of the eleventhinternational workshop on maximum entropy andbayesian methods of statistical analysis, Seattle,Washington, 1991, pp. 221–233

[19] ABORESHAID S., BILLINTON R., FOTUHI-FIRUZABAD M.:‘Probabilistic transient stability studies using the methodof bisection [power systems]’, IEEE Trans. Power Syst.,1996, 11, (4), pp. 1990–1995

[20] PAI M.A.: ‘Energy function analysis for power systemstability’ (Kluwer Academic Publishers, Boston, 1989)

[21] http://www.ee.washington.edu/research/pstca/pf57/pg_tca57bus.htm, accessed February 2009

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