Probabilistic load flow consideringnetwork outages

A.M. Leite da Silva, M.Sc, Ph.D., Mem.I.E.E.E., R.N. Allan, M.Sc.Tech.,Ph.D., C.Eng., Sen.Mem.I.E.E.E., M.I.E.E., S.M. Soares, M.Sc,

and V.L. Arienti, M.Sc.

Indexing terms: Power transmission and distribution, Power system protection, Transformers, Mathematicaltechniques

Abstract: Many techniques have been proposed to solve the load flow problem probabilistically. The greatmajority have only accounted for load and generation data uncertainties, and therefore, the network configu-ration has been considered fixed. So far, the effects of the configuration uncertainties due to the probabilisticnature of the network have not been deeply analysed. The paper presents a new method for obtaining aprobabilistic load flow solution when network outages are modelled as a random variable. The proposedtechnique is applied to a typical power system and the results discussed.

List of principal symbols

cg,hpQVXY

zueaPd

1

= network configuration= load flow functions= active power= reactive power= voltage magnitude= state random vector (voltages and angles)= input random vector (power injections)= output random vector (power flows)= unavailability= voltage angle= expected value= standard deviation= variance scaling factor= partial derivative

Introduction

Probabilistic load flow (PLF) is a subject of continuinginterest [1-13]. The usefulness of the PLF technique is dueto its ability to assess adequacy indices such as the prob-ability of a line flow being greater than its thermal rating,and the probability of a busbar voltage being outside itsoperational constraints. These indices are obtained whenthe probabilistic nature of load, generation and network isanalysed for power networks operating under steady-stateconditions. This analysis can be carried out by MonteCarlo simulation (MCS) techniques, analytical methods, orby a combination of both.

Theoretically, there is no constraint for the MCSmethod. For instance, the exact nonlinear power flowequations can be used, statistical dependence betweenloads and generation can be easily considered, probabilityof different network configurations can be taken intoaccount etc. On the other hand, the MCS method requiresa large number of trials to ensure results of reasonableaccuracy. This means that a huge amount of storage andcomputing time is necessary. To reduce the computationaleffort, linearised power flow equations can be used whenthe input uncertainty level is not very large and the degreeof nonlinearity in the load flow functions is reasonablysmall.

Paper 3800C (P9), first received 19th September 1984 and in revised form 21stJanuary 1985

Dr. Leite da Silva, Mr. Soares and Mr. Arienti are with the Department of Electri-cal Engineering, Catholic University of Rio de Janeiro, PUC-RJ, C. Postal 38063,Rio de Janeiro, Brazil, and Dr. Allan is with the Department of Electrical Engineer-ing & Electronics, University of Manchester Institute of Science & Technology,Sackville Street, Manchester M60 1QD, United Kingdom

Whereas MCS methods only use the law of largenumbers to justify their results, analytical methods usemuch more elaborate techniques and are based on prob-ability theory. Owing to the inherent complexity of theanalytical solution, the following assumptions have beenconsidered by most PLF algorithms:

(a) linear load flow equations(b) independence between input parameters(c) constant network configuration.

Therefore the accuracy of the analytical methods is alsolimited because of the assumptions used to overcome theinherent difficulties. The main analytical methods can besummarised as follows.

The use of linearised power flow equations about anexpected operating point [1-6] can be considered, in termsof accuracy, to be a reasonable assumption for a widerange of input uncertainties. As these uncertainties becomevery large, a more elaborate algorithm [7], or even anMCS technique using the exact load flow equations,should be used. Assumptions (a), (b) and (c) mean the PLFsolution becomes a sum of independent random variablesweighted by sensitivity coefficients. Consequently, the solu-tion is obtained by a convolution process which can beefficiently carried out using fast Fourier transform algo-rithms [14]. However, there are various reasons for correl-ations to exist between nodal powers [8-10]. For example,correlation between generation exists because of the needto balance the active system power. This balance isachieved according to the utility's operating policy whichincludes economic dispatch, redispatch and load shedding.A direct analytical solution to this was not realisableowing to the complexity of the problem, because even avery simple criterion of economic dispatch means that anextra nonlinearity must be included in the PLF analysis.This solution was efficiently achieved by combining theconvolution method with MCS techniques using the linearpower flow equations [10].

The network configuration is assumed to be a fixed par-ameter in the great majority of PLF methods. Consequent-ly, the probability of the basic configuration is assumed tobe unity, and therefore the probability of losing anynetwork element, such as transmission lines, transformersetc., is neglected. However, for a given operating point(load and generation strategy), there are several possiblenetwork configurations. Therefore assumption (c) may beconsidered unrealistic, particularly when the power uncer-tainties are small.

So far, only a few formulations [11-13] have consideredthe effects of network outages in the PLF analysis. Refer-

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985 139

ence 13 did not assign a probability to every contingency,and so it is not possible to obtain an overall uncertaintyinterval, for every variable of interest, owing to the com-bined uncertainty effects from nodal powers and network.It will be shown in this paper that the probabilistic modelused in References 11 and 12 have deficiencies. Also, theseReferences have used a DC model for power flow equa-tions, which is another restriction.

A new PLF algorithm is proposed in this paper thataccounts for the probabilistic contingency effects ofnetwork elements.

2 Problem formulation

Any change in the network of power systems will alter theset of functions relating inputs and outputs. Therefore eachprobability density function (PDF) from the outputrandom vector will change and so will the technical andeconomic decisions. Two basic sources of variation can beidentified. The first is the variation in the actual par-ameters defining lines, transformers or other components.For example, the inductance of a line changes with tem-perature which is a random variable (RV). These changesare normally assumed to be negligible so that the par-ameter values of the network have a probability equal tounity. The second source of variation is associated with theavailability or unavailability of components such as trans-mission lines, transformers, switchgear etc., as all aresubject to outages due to faults and maintenance. In termsof planning, such outages may be satisfactorily modelled asRVs. Consequently each network configuration has anassociated probability; i.e. it can be considered as an RVwhich follows a discrete distribution. Therefore, for a givenconfiguration c, the load flow equations are

Y = 9C(XC) (1)

Zc = K(Xe) (2)

whereY = input random vector

Xc = state random vector for configuration cZc = output random vector for configuration c

gc, hc = load flow equations for configuration c

Linearising eqns. 1 and 2 around the expected value region[5, 7], and considering configuration c, gives

ACY (3)

(4)

wheren yO A vO. c — A c — Ac IT' <70 D VO

3XC

Vectors A" and Z'c are deterministic linear conditions. Acand 5C are sensitivity coefficient matrices for configurationc. Vectors X® and Z° are obtained from a conventionalpower flow using as input the expected value of vector Y,i.e. E{Y) = Y°, and configuration c.

As the main objective of this work is to model probabil-istically the network contingencies, the components of theinput vector Y are assumed to be independent[assumption (&)]. Therefore, for a given configuration c,

the PDFs/^(x) and/^(z) for components x and z of vectorsX and Z, respectively, are evaluated by convolving theinput PDFs, weighted by sensitivity coefficients obtainedfrom matrices Ac and Bc .Thus

fx(x) = N(X'C,O) *

fz(z) = N(Z'C, 0) *

* f2(ac2Y2)

* f2(bc2 Y2)

(5)

*--*fJbcmYm) (6)

The vector X'c is interpreted as a normal random variableN( , ) with expected value X'c and standard deviationequal to zero. The same applies to Z'c.

Considering each possible network configuration to beassociated with a probability of occurrence, the problemconsists of combining statistically the PDFs obtained fromeqns. 5 and 6.

3 Methods of solution

First, the formulations proposed in Reference 11 and 12are briefly analysed to identify their main deficiencies. Thisanalysis is followed by the proposed formulation.

3.1 Formulation 7 [11]This formulation uses a DC model for the power flowequations. The simulation of different contingencies iscarried out through the compensation method. Thus, anychange in the transmission system, i.e. addition or removalof lines, can be simulated by suitable injections into thesystem, without the removal or addition of any lines. TheDC model is the first restriction of this formulationbecause only the variables related to the active power(voltage angles and active flows) are considered.

The major defect of this formulation is the probabilisticmodel used for the contingencies. Only first-order contin-gencies are considered and the final active power flow F ina given element e is a weighted summation of flows in thiselement for each contingency analysed; i.e.

F= 1 - (7)

where Fo = flow in the element e with the basic networkconfiguration (no outage case), F, = flow in the element ewhen element i is on outage, u, = unavailability of elementi and n = number of elements (transmission lines andtransformers).

The weights used in eqn. 7 are, in fact, approximationsfor the probabilities associated with the basic configu-ration p0 and with the configuration considering theoutage of element i, p,, because

Po = i= 1- « . ) * i - z

1 1 X J. \ }>

(8)

(9)

Using approximations 8 and 9, the space of all configu-rations considered by formulation 1 has a probabilityequal to unity which is very desirable. On the other hand,from the statistical point of view , the approximate valuesfor p0 and p{ used to define F in eqn. 7 can give incoherentresults under some conditions. To illustrate this point,Table 1 shows the comparison between the exact andapproximate probability values of the basic configuration,considering network elements with the same unavailability.Therefore, for a system with 1000 elements each having

140 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985

Table 1: Comparison between p0 (exact) and p (formulationDn

u

ExactFormu-

lation 1n x u

10

10-2

0.904

0.9000.100

10-3

0.990

0.9900.010

100

10-2

0.366

01

10-3

0.905

0.9000.100

1000

10-2

4.3 X 1 0 - 5

-910

10-3

0.368

01

u = 10 2, the probability of occurrence of the basic con-figuration, according to formulation 1, is equal to —9,which is not possible. Although not discussed in Reference11, one condition to validate the above probabilistic modelis that the product of the number of elements n and theunavailability u must be significantly smaller than 1.

Finally, formulation 1 also assumes that the PDFs forthe power flows are normally distributed, which has beenshown to be an unreliable assumption [5].

3.2 Formulation 2 [12]This formulation also uses a DC model for the power flowequations. The contingency evaluation is carried out in avery efficient way, similar to formulation 1, although theefficiency is facilitated by the simplicity of the DC model.

The probabilistic contingency model has the followingcharacteristics: (i) all first and selected second orderoutages are considered, and (ii) the exact probability valuefor each configuration is used, assuming independencebetween outages. Any type of distribution can be used tomodel load and generation uncertainties. The final PDFsfor the components of vectors X and Z are computedthrough a weighted summation of the density functionsobtained for each analysed configuration.

It should be noted that, because the exact probabilityvalues for the configurations are used, and because alimited number of contingencies are analysed, the subspaceS' of these configuration states summates to a probabilityless than unity, if no compensatory action is taken. Table 2

Table 2: Probability of {S'} - formulation 2

10 100 1000

0.1 0.73610 3.2169 x 10~4 1.9596 x 1 o~44

0.01 0.99573 0.73576 4.7924 x 1O"4

0.001 0.99996 0.99536 0.73576

shows the probability of S' for some pairs (u, n), assumingthe same unavailability for all lines and considering firstorder outages only. Therefore, for a system with 1000 ele-ments, each having u = 0.01, the area under the probabil-ity density curves, obtained through formulation 2, is equalto 4.79 x 10~4 due to the truncation of network states.Therefore, if only first and selected second order outagesare analysed, a number of possible configurations will beneglected. This information should be included in the con-tingency probabilistic model to allow the statistical resultsobtained from this analysis to be easily interpreted. In fact,it was suggested in Reference 12 that all second order con-tingencies plus higher orders should be included to mini-mise the truncation effects. It was also suggested thatcommon mode failures may have to be considered.

3.3 Proposed formulationThe proposed formulation is an extension of formulation2, in which the DC model is substituted by an AC linearmodel described by eqns. 3 and 4. Also, the probabilisticmodel of contingencies is structured in a way which com-pensates for the truncation effects.

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985

Consider S to be the space of all possible system con-figurations c, and pc the associated probability of each con-figuration Sc. Thus

S = S l U S 2 U ••• USCU ••• U S , (10)

It is not possible for two or more configurations to existsimultaneously; therefore the events Sx to St are mutuallyexclusive:

ps = probability{S} = £ pc = 1c = l

(11)

Eqns. 5 and 6 show how PDFs for components x and zfrom vectors X and Z, respectively, are evaluated condi-tioned by the network configuration c, i.e. by the event Sc.As all events Sc are mutually exclusive, then the final PDFfor each variable x or z is given by

/*(*) == t

(12)

(13)

Eqns. 12 and 13 take the network changes into accountand show that the final solution is obtained from aweighted sum of density curves. Appendix 8.1 shows howthe expected value and variance of x and z can be evalu-ated.

The crucial point is the evaluation of the probability pc

associated with each configuration. As proposed in formu-lation 2, pc can be calculated from the unavailability u, ofeach element i of the network. This is done by enumeratingall physical network states. Therefore, for a given networkc, the probability pc is given by

pc = ft (1 - «,) f l uj (14)

wherena = number of elements availablenu = number of elements unavailable

Clearly, the analysis of all possible network configurationsis impracticable for real power networks due to the hugeamount of computing involved. Therefore, some criteriahave to be used to reduce the number of analysed configu-rations. Usually, these criteria consider simultaneouslythose contingencies more likely to occur and their impacton the system operating point [15]. Although this pro-cedure minimises the truncation effects, the adequacyindices obtained from this analysis have to be regarded asoptimistic values. These indices will become more realisticby increasing a well selected list of contingencies.

As the number of networks to be considered is reduced,the probability ps. associated with subspace S' which con-tains all the analysed networks will be less than unity.Considering that S' is an adequate approximation forspace S according to the state truncation criteria, the pre-vious probability value pc associated with each configu-ration c can be approximated to p'c as follows:

Pc = Pc/Ps (15)

This ensures, for the t' network configurations analysed,that

(16)c = l

Finally, the contributions of the proposed formulation inrelation to formulations 1 and 2 are summarised asfollows:

141

(i) The DC power flow model used in formulations 1and 2 is extended to an AC linear power flow model, prop-erly defined.

(ii) The approximation used for the probabilistic contin-gency model in the proposed formulation is more consis-tent from the statistical point of view than the one used informulation 1.

(iii) Although the probabilistic contingency model usedin the proposed formulation is as precise as that used informulation 2, the statistical interpretation of the resultsobtained through the proposed formulation is more ade-quate.

4 Results and discussions

The probabilistic system data used to demonstrate the pro-posed method are shown in Appendix 8.2. Such data weresimulated keeping the original characteristics of the IEEE14 busbar system.

Two distinct analyses were carried out and the resultsare shown and discussed as follows. The probabilistic con-figuration data shown in Appendix 8.2 are only used inanalysis 2.

4.1 Analysis 1This is a sensitivity analysis for the random vectors X andZ in relation to load, generation and network uncer-tainties. The load uncertainties are increased by a multi-plication factor p [5, 7] which takes values 0(Y = deterministic), 1 (Appendix 8.2), 3 and 7. The unavail-ability u is assumed to be the same for all network ele-ments and changes from 0 (fixed network) up to 10"1.Only first order contingencies are considered, and conse-

quently a correction factor is used to compensate for thestate truncation. Table 3 shows the network probability forthe basic configuration (no outage case) and for the firstorder contingencies.

Table 4 shows the results (/i = expected value ando = standard deviation) for five typical variables, using dif-ferent values of p and u.

An important conclusion from this analysis is that theconfiguration uncertainties are practically absorbed by theinput random uncertainties, when these are large (p = 7).Therefore, the element outage effects of a network have tobe carefully analysed, particularly in operational planning,because at this stage the load uncertainties are generallysmall. Although this is not a general conclusion, it shouldbe valid for the great majority of existing power transmis-sion networks.

4.2 Analysis 2The network probabilistic data described in Appendix 8.2were used. Each configuration has an associated probabil-ity. It can be seen that most contingencies are of firstorder, but some of second and third orders are also con-sidered. A truncation factor has already been used. Thecriteria used to limit the number of network configurationsto be analysed are not discussed. Although this contin-gency list is relatively small compared with all possiblesystem configurations, it can be considered sufficient forillustrating some useful results and interesting effects.

Three cases are analysed and summarised in Table 5and in Figs. 1, 2 and 3. In the first case {AY), only theuncertainties in the input vector are considered and thenetwork is kept fixed with no outages. This is the conven-tional algorithm of probabilistic load flow. In the second

Table 3: Probability of configuration against u

Configuration

Basic

First order

Table 4: Sensitivity \x

Variable u

010-4

83, degrees 10"3

10-=10-1

010-4

l/5,p.u. 10-3

10-=10-1

010-4

P, 2 , MW 10-3

10-=10-1

010-4

Q, 2, MVAR 10-3

10-2

10-1

010-4

S5 6 , MVA 10-3

10-=10-1

U

0

1

0

, 0

10-4

0.9980

I .Ox iO - 4

10-3

0.9804

9 .8x10"

against u against p

fJ

-12.83-12.83-12.88-13.28-14.67

1.018671.018661.018601.018121.01641

160.16160.15160.08159.47157.34

-19.68-19.68-19.68-19.65-19.55

45.7645.7645.7645.7845.85

a

p = 0

00.341.022.945.74

00.000370.001090.003120.00601

02.055.82

16.8333.97

00.310.862.505.05

00.581.704.94

10.00

10-2

0.83204 8 .4x10-

P = 1

0.931.021.413.125.87

0.000780.000960.001410.003300.00619

11.6111.8213.0220.4735.92

2.692.722.843.675.72

0.881.051.915.02

10.05

10-1

0.310CI

3 3 .5x10" =

p = 3

2.722.772.964.176.73

0.002350.002410.002670.004220.00719

28.3128.4829.0033.0644.40

6.566.586.637.028.26

2.662.723.165.64

10.42

P = 7

6.346.376.497.449.95

0.005480.005570.005780.007290.01091

64.2764.3864.6264.6473.26

14.8914.9114.9315.0915.66

6.186.216.438.04

12.11

142 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985

TableV5.p.

Case

AYACAYC

5: Network outage effectsu.

1.018671.017971.01796

6 9 , degrees

Case

AYACAYC

Case

AYACAYC

' i2-1:

Case

AYACAYC

Qs-e-

Case

AYACAYC

S 2 _ 4 ,

Case

AYACAYC

S 5 _ 6 ,

Case

AYACAYC

fJ

-15.015-15.477-15.478

MW

/J

44.1344.3244.32

3, MW

fJ

1.621.641.64

MVAR

fJ

12.0711.7411.75

MVA

55.9355.5555.55

MVA

fJ

45.7645.8945.88

a

0.000780.003350.00349

a

0.5612.6112.690

a

0.964.594.69

a

0.160.530.56

a

0.321.611.66

a

2.339.439.72

a

0.884.574.67

in the PLF solution

p < 1.014

0.00000.05900.0573

P ^ - 1 7

0.00030.05400.0556

P^42

0.00180.02600.0327

pO-4

0.04390.01500.0791

p < 11.0

0.00000.05900.0585

p < 4 8

0.00010.02600.0261

p ^44

0.00590.04800.0512

p ^ 1.017

0.01860.06100.0830

P^ -15

0.51650.98800.5669

P<43

0.11430.04800.1625

P<1.6

0.49160.03700.5073

P<117

0.12510.06900.1974

p ^ 53

0.11140.02600.1333

p s$46

0.66150.95000.6654

p < 1.020

0.95900.99700.9636

P ^ - 1 4

0.96990.99700.9708

p ^45

0.80810.94800.7864

p^1.8

0.81260.93900.7877

p ^ 12.4

0.85250.98300.8606

p ^ 58

0.81950.96400.7974

P^48

0.99680.97000.9614

p ^ 1.022

1.00000.99700.9970

p < - 1 3

1.00001.00000.9992

P^47

0.99990.96100.9621

p^2.0

0.99910.97000.9667

p ^ 13.0

0.99930.99200.9898

p ^ 63

0.99960.96900.9675

P^49

1.00000.97000.9702

case (AC), only the configuration network uncertainties areconsidered and the expected value of the input randomvector is used. This is an algorithm of contingency analysiswhere each configuration has an associated probability.Finally, in the third case (AYC), both input and networkuncertainties are analysed simultaneously. Note that casesA7 and AC are, in fact, particular cases for the proposedmethod.

Table 5 shows seven typical state/output random vari-

:100i

S 7 5

ables in terms of probabilistic parameters (/i, a andprobabilities). It can be seen that case AFC is extremelyrelevant in assessing X and Z uncertainties compared withthe individual analysis A Y or AC. For example, considerthe random variable S5_6. The probability of this flowbeing less or equal to 48 MVA is 0.9968 considering onlythe uncertainties AY, 0.9700 considering only the uncer-tainties AC and 0.9614 as both uncertainties AYC are con-sidered at the same time.

0.20r

50-

1.40

(?2_1 3

1.60MW

1.80 2.00

Fig. 1 MW flow distribution Pl2_l32-1.

Fig. 2 MVA flow density S2

AYAYC

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985 143

The probability distribution function for variablesP12_13 and S5_6, and the probability density function forvariable S2-4 are shown in Figs. 1, 3 and 2, respectively.

100

75

50

.^25

S 5 -6 ' M V A

Fig. 3 MVA flow distribution S5_6

AYACAye

From these Figures, the differences between AY, AC andA7C are better visualised. Owing to scale limitations, thetails of some functions are not represented because theyare very long. For the probability density function S2_4,case AYC, two impulses of 0.0265 and 0.0321 are placed atpoints 48.64 MVA and 63.34 MVA, respectively, to com-pensate for the statistical information from the tails. As thelevel of uncertainties A Y is not very large, the tails of thedistribution functions are influenced more significantly bythe uncertainties AC.

The average computer time using a CDC-CYBER170/835 computer for evaluating each PDF, representedby 128 points, was 0.21 s for case AY, 1.40 s for case ACand 3.50 s for case AYC.

4.3 Further commentsIn practice, the evaluation of state and output uncer-tainties, in terms of only expected value and variance par-ameters, is easily implemented in any conventional powerflow algorithm. On the other hand, such probabilisticinformation is insufficient to determine which network ele-ments (transmission lines, transformers, busbars etc.) arelikely to operate inadequately. Therefore it is alwaysnecessary to evaluate the probability density or distribu-tion functions to obtain system adequacy indices such as:

(a) the probability of a transmission line and trans-former loading being greater than its thermal rating

(b) the probability of a busbar voltage going above orbelow its limits

(c) the probability of a generator violating its reactivepower limits

(d) the probability of generation deficiency in a particu-lar system.

For example, suppose that transformer 5-6 in analysis 2has a maximum capacity of 48 MVA. It can be seen fromTable 5 that there will be a probability of(1 - 0.9614) = 0.0386 (i.e. 3.86%) that an overload occurswith this transformer. If a risk of 1 % is considered accept-able, then some measure has to be taken in relation to thisequipment. This analysis can be extended for all networkequipments.

The above overload risk of 3.86% would not be thesame if practical operating policy criteria, including eco-nomic dispatch, load shedding and redispatch, were con-sidered. Also, in the previous analysis, loads were assumedto be statistically independent. Therefore all these con-

siderations have to be incorporated [10] into the proposedmethod to provide reliable adequacy indices.

Finally, the development of load and generation models[8] to be used for steady-state probabilistic analysis is avital step to ensure that PLF algorithms will process rea-listic data.

5 Conclusions

This paper has presented a new probabilistic load flowmethod which considers the network configuration as adiscrete random variable. The network uncertainties aremodelled to account for the availability of componentssuch as transmission lines, transformers, switchgear etc.,which are all subject to outages due to faults and main-tenance.

The proposed solution for the state and output prob-ability density functions is obtained from a weighted sumof density functions evaluated for each possible networkconfiguration. These weights are probabilities associatedwith the configurations. An AC linear power flow model isused.

It has been shown that the probabilistic nature of thenetwork is extremely relevant in the probabilistic load flowsolution. Moreover, the network uncertainties have moreinfluence in the solution when the load uncertainty level isnot very high, which usually occurs in operational plan-ning.

Finally, although the proposed method requires morecomputing effort in relation to the conventional one, theevaluated adequacy indices contain, undoubtedly, moreinformation.

6 Acknowledgments

The authors gratefully acknowledge the financial assist-ance granted by CNPq, CAPES and FINEP of Brazil insupport of this project. Mr. Arienti is also indebted to theFederal University of Minas Gerais for granting leave ofabsence and financial support.

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144 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985

13 PRADA, R.B., and CORY, B.J.: 'Stochastic security assessment',Proc. 2nd Int. Symposium on Security of Power Systems Operation,Wroclaw, Poland, June 1981

14 ALLAN, R.N., LEITE DA SILVA, A.M., ABU-NASSER, A.A., andBURCHETT, R.C.: 'Discrete convolution in power system reliability',IEEE Trans., 1981, R-30, pp. 452^56

15 MIKOLINAS, T.A., and WOLLENBERG, B.F.: 'An advanced con-tingency selection algorithm', ibid., 1981, PAS-100, pp. 608-617

8 Appendixes

8.1 Expected value and variance evaluationThe expected value of component Xt and Zk from vectorsX and Z, respectively, are evaluated, conditioned by thenetwork configuration c, i.e.

c = l

E{Zk}=

(17)

(18)

where £{.} represents the expected value operator, XCiand X°. are the ith components of vectors Xc and X°,respectively, ZCk and Z°k are the kth components of vectorsZc and Z°, respectively.

Representing the variance operator by VAR{.}, thenthe variance of Xt and Zk can be evaluated as follows:

VAR{Xt} = t Pc[VAR{XCi}

VAR{Zk] = £ Pc[VAR{ZCk}c = l

where

(19)

- £2{Zk} (20)

VAR{XCi}=

VAR{ZCk}=

where Afi. and BCk. are elements of matrices Ac and Bc,respectively.

Note that, owing to the linearisation of eqns. 1 and 2,eqns. 17-20 are only approximations for the expectedvalue and variance of the state and output random vari-ables, the difference being related to the degree of uncer-tainty or dispersion of the input quantities and the system.

8.2 Data for 14 busbar systemThe system studied is based on the IEEE/AEP 14 busbarsystem. The line and transformer data used in these studiesare identical to those used in Reference 4. Table 6 showsthe selected configurations and their associated probabil-ities after correction by the truncation factor. Table 7shows the nodal probabilistic data, where a in (ii) isexpressed as a percentage of the expected value pi.

Table 6: Configuration probabilistic data

Elements onoutage

none1-21-52-32-42-54-95-6

Configurationprobability

0.9000.0080.0120.0090.0110.0100.0030.003

Elements onoutage

9-1410-1113-142-4/2-52^/4-59-10/9-146-11/6-12/6-134-7/7-8/7-9

Configurationprobability

0.0090.0110.0100.0020.0050.0030.0020.002

Table 7: Nodal probabilistic data

(i) Binomial distribution

Busbar

Number Type

Voltage,p.u.

Unit rating,MW

FOR Number ofunits

PV 1.045 20.0 0.09

(ii) Normal distribution, p = 1

Busbar Voltage, Active power Reactive power

Number

23456789

1011121314

Type

PVPVPQPQPVPQPVPQPQPQPQPQPQ

p.u.

1.0451.010

1.070

1.090

/y, MW

-21.74-94.20-47.80

-7.60-11.20

0.000.00

-29.50-9.00-3.50-6.10

-13.50

a, %

9.0010.0011.00

5.001.000.000.001.00

10.009.501.001.00

H, MVAR

3.90-1.60

0.000.00

-16.60-5.80-1.80-1.60-5.80-5.00

a, %

9.705.00

0.000.005.00

10.009.508.609.508.60

(iii) Discrete distribution, p = 1

Busbar Voltage,

Number Type p.u.

14 PQ

Active

MW

-18.00-15.00-13.00

Power

Probability

0.200.450.35

Reactive power

MVAR Probability

1EE PROCEEDINGS, Vol. 132, Pt. C, No. 3, MAY 1985 145