Electric PowefSystems Research, 10 (1986) 157 - 166 157

A Probabilistic Approach to the Voltage Stability Analysis of Interconnected Power Systems

MICHELE BRUCOLI, MASSIMO LA SCALA and FRANCESCO TORELLI

Dipartimento di Elettrotecnica ed Elettronica, Facolt~ di Ingegneria, Universit~ degli Studi di Bari, Via Re David 200, 70125 Bari (Italy)

(Received November 24, 1985)

SUMMARY

In planning a power system it is always necessary to assess whether a voltage collapse occurs during a pref ixed system operating condition. However, present approaches to the analysis o f voltage instability phenomena in interconnected power systems are deter- ministic and, consequently, they cannot take into account the unavoidable uncertainties associated with the bus load forecast. This is indeed an important limitation. In this case the application o f probabilistic techniques is the most feasible alternative. On the basis o f this observation, in this paper a probabilistic approach to the voltage stability analysis o f interconnected power systems is presented; it treats loads as random uncorrelated variables with normal distributions. The method proves suitable for determining systematically, for each expected system operating condition, the statistics o f all the node voltages which are critical from the voltage stability view- point. The capability and usefulness o f the suggested approach are illustrated by carrying ou t simulation studies on a sample power system.

1. INTRODUCTION

Voltage instability phenomena in electric power systems have been the object of con- siderable interest in recent years. Much work has been done to analyse the mechanism of the above phenomena and several criteria have been suggested to evaluate whether voltage instability occurs during a prefixed operating condition of a power system [1 - 9].

A review of the literature reveals, however, that these approaches, except for a rare

a t tempt [9], are deterministic, that is, they assume that the data provided are absolutely precise. Hence, they do not take into account the unavoidable uncertainties associated with the bus load forecasting and changes in gener- ator operating points [10, 11]. This is an important limitation which makes the results of present deterministic methods not very reliable, particularly for the planning purposes of electric power systems.

In order to overcome these difficulties when a deterministic method is used, it would be necessary to carry out the voltage stability analysis for all combinations of possible loads considered over their range of variability, with a consequent prohibitive amount of calcula- tion and great difficulty in synthesising and analysing the formidable number of results.

In effect, the only way to get a practical solution to the problem of assessing accu- rately the risk of voltage instability in a power system, when the input data are uncertain, is to consider the above uncertainties as random variables and to apply probabilistic techniques of analysis. In the present paper a proba- bilistic method for the voltage stability analysis of interconnected power systems is proposed; it takes into consideration the random behaviour of the load demands and, consequently, of generator operating condi- tions. As is well known [4, 8], a voltage col- lapse at a load node occurs when the active power fed from the node reaches a maximum value. Consequently, the critical voltage stability conditions for the system can be derived by evaluating those voltages at the nodes of the network which maximize the active power fed from a prefixed load node and which, at the same time, satisfy the net- work constraints and dispatch laws. This non-

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158

linear constrained optimization problem can be solved by using appropriate techniques [121.

Now, by imposing the necessary first-order conditions for the opt imum on the Lagrange function associated with the above optimiza- tion problem, a set of equations is obtained which constitutes the stochastic process capable of correlating the input random quantities (that is, the reactive and active power demands at the nodes) with the critical state of the network.

Then, the probability density functions (PDFs) of the maximum active power at the prefixed load node, and of all the node voltages that are critical from the voltage stability viewpoint, are computed on the basis of the linearization, about an expected oper- ating point, of the equations describing the stochastic process.

In addition, in order to facilitate the volt- age stability analysis, a suitable and practical factor is introduced which is capable of quan- tifying the risk of voltage instability at a load node for each expected operating condition for the system. The probabilistic approach presented in this paper assumes that the input loads are independent and normally distrib- uted random variables [13], whereas a linear model of the dispatch strategy is considered [11, 14].

The method allows a systematic individu- alization of all the causes which affect the voltage instability phenomena and can be used advantageously to achieve reliable pre- dictions of voltage collapses at the level of power system planning.

To validate the capability and usefulness of the approach, tests were carried out on a sample system and some selected results are presented.

2. F O R M U L A T I O N OF THE V O L T A G E

S T A B I L I T Y P R O B L E M

This section deals with the deterministic voltage stability analysis for interconnected power systems. For this purpose, consider an (N + 1)-node power system and let nodes 1 to N L be PQ nodes, nodes N L + 1 to N be P V nodes; finally, assume that node N + 1 is the slack.

In § 1 it has been pointed out that a voltage collapse at a load node occurs when the active

power demand at that node reaches its maxi- mum value. Consequently, by assuming that the controlled voltage at each P V node is constant, the critical voltage stability condi- tion for the generic kth PQ (load) node can be found, for a given operating condition, by solving the following non-linear optimization problem with equality constraints [12]:

m a x PLk

subject to

P G i - - P L i - - P i ( O, V) = 0

QGi - - QL~ - - Qi(O, V) = 0

(1)

i = 1, 2 , . . . , N (2)

i = 1, 2 , . . . , NL(3)

with

N Pai= ai + bi ~ PLr i = 1, 2, . . . , N (4)

r = l

where PGi, QGi is the generation at node i; PLi, QLi is the load at node i; and Pi, Qi is the net power injection at node i as a function of the set of phase angles 0 and magnitudes V of the system bus voltages.

The voltage phase angles 0 are referred to the slack node. It should be noted that eqns. (2) and (3) are the standard load flow equa- tions, whereas eqn. (4) represents the eco- nomic dispatch law of the generating units expressed as a linear function of the bus load demands. This law, which satisfies the mini- mum production cost criterion, can be derived by assuming that the incremental cost for every generator in the system is linear, power losses in the network are negli- gible, and no inequality constraints are in- cluded [11, 15].

It is important to point out that PLk for the kth PQ node, V and 0 for each PQ node and 0 for each PV node are the N + N L + 1 prob- lem unknowns. The remaining quantities determine the system operating point. Taking into account that maximizing PLk is equiva- lent to minimizing --PLk, to derive the neces- sary conditions for the opt imum the Lagrange function takes the following form:

L(PLk , ~, V, ~,, V)

N = --PLk - - ~ Xj[PGj - - P L j - - P j ( O, V ) ]

j=l

N -- ~ vj[QGI -- QLJ -- Qj(O, V)] (5)

i=1

159

where ~i and vj are Lagrange multipliers asso- ciated with the ] t h node active power and reactive power balance equations, respec- tively. The set of necessary first-order condi- tions for the opt imum are [12]

~}L N - - - l + k k - - E b i k i = 0 (6)

0PLk j = 1

OL N OPj(O, V) NL OQj(O, V)

00i j= ~ 00~ j=, OOi - 0

i = 1, 2 , . . . , N (7)

~L N OPj(O, V) NL OQj(O, V) - + - o

O Vi j= , ~ V~ j= , O V~

i = 1, 2 , . . . , N L (8)

OL NL O~'i a] - - bi r=lE P L r + PLj + Pi( O, V) = 0 (9)

0L

Ovi

j = 1, 2, . . . , N

QGj + QLy + Qi( O, V) = 0 (1o)

j = 1 , 2 . . . . . NL

Equations (6) - (10) can be written in the form

LPL k = - - 1 - - bkT~k = 0 (11)

Lx = jpT~ + jQT~, = 0 (12)

LX = --a --BkpL - - b k P L k + P ( x ) = 0 (13)

L~ = --QG + QL + Q ( x ) = 0 (14)

where T denotes transposition and

bk = [b,, b2 . . . . . bk-,, bk--1, bk+,, ..., bN] T

N X 1

PL = [PL1, PL2 . . . . , PLY- l , PLk+I , - . - , P/.~v] T

(N-- 1) × 1

QG = [QG1, QG2, - - ' , QGNL] T N L × 1

QL = [QL1, Qtn, ..., QUVL] T NL × 1

B k= [bl, b2 . . . . . bk-1, b~+l, ..., bN]

N X ( N - - 1)

P ( x ) = [P,(x), P~(x) . . . . , PN(X)] T

N X 1

Q(x) = [Q,(x), Q2(x) . . . . . QNL(X)] T

N~Xl

Jp and JQ are the submatrices of the (N + NL) × (N + NL) Jacobian matrix [16]

Equations ( 1 1 ) - ( 1 4 ) constitute the deter- ministic model of the voltage stability prob- lem in an interconnected power system. Using these equations and starting with a given system operating point, the critical voltage stability condition for the system can be derived efficiently [8]. It should be pointed out, however, that this technique, since it is inherently deterministic, is inadequate to account for the random nature of the un- certainties which unavoidably affect the input data and which can cause significant changes in the solution of the voltage instability prob- lem.

It is apparent that these difficulties can be overcome by developing a technique which recognizes that loads vary stochastically and which is capable of determining how the stochastic nature of the input data affects the output quantities of the problem.

= [~1, ~2, "" ", ~N] T N × 1

P = [btl, /22 . . . . . /)NL ]T N L × I

x = [0,, (9: . . . . . ON, Vl , V~, . . . , V N J r

(N+NL) × 1

a = [al, a2 . . . . . aN] T N × I

3. PROBABILISTIC FORMULATION

The purpose of this section is to provide a technique which permits the voltage stability problem for interconnected power systems to be analysed probabilistically. The approach is developed by considering loads as the only sources of uncertainty in the problem, where-

160

as the voltages at the P V nodes and all the other network parameters are kept fixed.

3.1. Load mode l To account for the random behaviour of

loads, due to forecasting and measurement errors, it is assumed that the generic bus load demand in terms of active (PLi) and reactive (QLi) powers can be modelled as

PLi = pO + APbi i = 1, 2 . . . . . N (16)

QLi = Q°i + z~QLi i = 1, 2 . . . . , g L (17)

where P° i (Q°i) is the forecasted best estimate of the ith bus active (reactive) power and APL~ (AQLI) is the random variation from pOi (Q°i). To simplify the analysis, the input fluc- tuat ions APLi and AQLi are assumed to be normally distributed with the following mean values and variances, respectively [ 17 ] :

E(APLi ) = 0 V(~kPLi , APLi ) = OpL.2 (18)

i = 1, 2, . . . , N

E(AQT,i) = 0 V(AQLi , AQL~) = OQLi 2

i = 1, 2, . . . ,NL

In addition, it is assumed that

(19)

cov(APL, AP,.j) = 0

cov(APLi , AQLI ) = 0

cov(AQLi, AQLj) = 0

i , j = 1, 2 . . . . , N (20) i ¢ ]

i = 1, ..., NL (21a)

i, j = 1, ..., NL (21b) i C j

That is, the random fluctuations of active and reactive powers at the same node as well as at different nodes are assumed to be indepen- dent .

3.2. Generat ion mode l By assuming, for simplicity, tha t the

parameters ai and bi (i = 1, 2 , . . . , N ) appear- ing in eqn. (4) are deterministic and depend on the best estimate of the incremental costs, the generation PGi (i = 1 . . . . . N) can be con- sidered as a random variable whose statistics can be evaluated from the statistics of the bus load demands only.

3.3. D e v e l o p m e n t o f the m e t h o d Taking into account the random nature of

loads, eqns. ( 1 1 ) - ( 1 4 ) now const i tute the stochastic model of the voltage instability

critical condit ion for the system. Consequent- ly, the problem of determining the PDFs of the critical ou tpu t quantities of the above stochastic process has to be solved.

The non-linearity of eqns. (11) - (14) makes the development of a probabilistic ap- proach to the analysis extremely complicated. To circumvent this difficulty, a simple and practical approximative technique is proposed which is based on the first-order Taylor series expansion of the stochastic process around the expected value of the probabilistic op- t imum solution. This technique is based on the assumption that the values of tolerances for the load fluctuations are small, and leads to the following linearized description of the problem [17, 18]:

--1 -- b~W/~ = 0 (22a)

JpTttk + JQTtt, = 0 (22b)

--a -- Bk/~pL -- bkPPL~ + P(#x) = 0 (22c)

--ttQG + /tQL + Q(/t~) = 0 (22d)

and

LorA'N J , o l A~k / J p T 0 0 - - b k

AV /JQ T 0 0 0

Z~DLk - - b k T 0 0

Bk A(QG -- QL) × (23)

where p~ = E0,) and tt. = E( r ) are column vectors formed with the expected values of the components of the k and r vectors; Px = E(x ) is the column vector of the expected values of the components of the x vector, and so on; Ax = x -- ttx, A), = ), -- / t~, and so on.

In addition, Hp and HQ are matrices of dimensions (N + NL) × (N + N L ) N and (N + NL) X (N + NL)NL, respectively, with the fol- lowing structure:

Hv=[Hp1T Hp2 TI ... Hvj ~ I . . . I Hpu ~]

HQ = [HQ1T HQ2 T i . . . HQj T [ . . . [ HQNL T]

where the generic (N +NL) × (N + NL) Hpi and HQj matrices are the Hessians of the func-

tions Pj(x) and Qj(x), respectively. Finally, the matrices A and N have dimensions (N + NL)N × (N + NL) and (N + NL)NL × (N + NL), respectively, with the following structure:

I I . . . I . o . A = [)~lI I )HI , I ~jI ', ', XNI] T I I

I N =[vII Ip v21 ' ' J ' I"" I viI ' " ' ' , II P N L I ] T

Note that I and I0 are identi ty matrices of dimensions (N + NL) × (N + NL) and NL × NL, respectively.

The main consequence of this approxima- tion, and of the assumptions on the distribu- tion of the input data, is that all the PDFs of the unknowns follow the trend of a normal distribution [18]. In addition, the statistics of the problem unknowns can be completely determined from eqns. (22) and (23). Thus, the mean values can be evaluated by solving eqns. (22) iteratively by adopting, for ex- ample, the Newton-Raphson method [16]. In this case a flat start for the voltages can be used, whereas the starting values of ), and can be derived from the equation

which can be obtained by solving eqns. (11) and (12) for ~, and ~. It should be noted that in eqn. (24) the superscript + denotes pseudo- inverse [19] and 0 is an Nwdimensional column vector whose elements are null. Finally, to derive the covariance matrix, eqn. (221) can be written in the form

eo = Ael (25)

where

e O = [ ~ , A ~ , A ~ , ~ D L k ] T

eI = [ ~ P L , A ( Q G - - Q L ) ] T

A = I HpA+ HQN Je JQ 0 ] -1

Je T 0 0 - - b a

QT 0 0 0

- -ba T 0 0

B ~ 0 ×

161

Then, the covariance matrix of Co, taking into account eqn. (25) and after appropriate ma- nipulations, is given by [18]

coy(co) = E(eo eo T) = ARA T (26)

where

R = diag[apb~2, a 2 2 PL2 ' "" "' aPl.~J '

2 fir QGNL _ QI . ,N/ ] OQG1 - Q L I ' " " "

4. I N S T A B I L I T Y R I S K F A C T O R

The risk of voltage instability at a PQ node k and for a given system operating condition can be quantified by defining the factor

Pk = prob(P~, - - P ~ < 0) (27)

where PLk is the actual load demand at node k and P~k is the maximum active power demand at node k corresponding to a critical voltage stability condition for the system, whose dis- tr ibution can be evaluated by the procedure illustrated in § 3. Thus, this risk factor is the probabil i ty that the actual active power demand exceeds the critical active power demand at node k. To evaluate Pk, let the fol- lowing stochastic process be defined:

w #$ Yk - Pu~ --Pro (28)

Thus, we have

0

pk = prob(y~ < O) = f f(yQ dYk (29) - - o o

where f(yQ is the PDF of Yk. Now, it should be noted that, as both P ~ and PLa are nor- mally distributed and statistically indepen- dent, the difference process given by (28) is normally distributed and its moments are given by [ 17 ]

IJy~, = ~eL~ -- #eu~ (30)

oy~ 2= oeL~ 2+ oem 2 (31)

The use of this factor simplifies the identifica- tion of the causes which affect the system voltage instability and provides a simple and practical tool to compare the real voltage in- stability risks associated with different oper- ating conditions for the system.

162

5. N U M E R I C A L E X A M P L E

The technique described in the previous section has been applied to the test system of Fig. 1 which is referred to as the CIGRE 225 kV test system and consists of 10 nodes and 13 lines. There are seven generating nodes and loads are located at seven of the nodes. The deterministic line data, specified by the CIGRE, can be found in ref. 20, whereas all the other system data are given in Table 1; all p.u. values have been referred to 1000 MVA base and simulation studies have been carried out by taking node i as the slack node (Vl = 1.0 p.u.). It should be noted that all the loads have been assumed Gaussian, with standard deviations a i expressed as [17]

ai = 0 . 1 IAPLi/3 (32)

Obviously, by eqn. (4), the generation PGi is

S

) 3

9

8

(~) G E N E R A f O R

- -~ L O A D

Fig. 1. T e n - b u s C I G R E 225 kV s y s t e m .

also normally distributed and its mean value and standard deviation can be easily obtained using well-known techniques of probability theory [17].

Now, assuming node 10 to be the study node and keeping the voltages at the generat- ing nodes constant, the mean value and the standard deviation of the maximum supply of active power P* and of the corresponding Ll0 critical voltage magnitude V~* 0 at node 10 have been found: PRimo = 0.583 p.u., Op~,, o =

0.0274 p.u. and p v ~ o = 0.523 p.u., Ov~o =

0.001 68 p.u. Figure 2 shows, as an example, the PDF of * PL10- In order to test the validity of the suggested approach based on a linear- ization of the equations which describe the optimization problem, a Monte Carlo simula- tion was carried out by using the exact (non- linear) model of the problem. In this simula- tion, pseudo-random numbers with Gaussian distribution generated by a computer were used as input data. For purposes of compari- son, the results of 3000 trials are shown in Fig. 2 by means of a histogram representing the Monte Carlo density function of P~lo. It can be seen from Fig. 2 that the curve for the exact model follows the trend of a normal distribution and that, except for some differ- ences exhibited by the tails, it is very close to the approximate density curve evaluated by using the suggested approach. Furthermore, following the technique described in §4, the voltage instability risk factor was found to be PlO = 0.48%, which is sufficiently small. In particular, this factor is given by the shaded area of Fig. 3, where the PDF of the actual active power demand at node 10 minus the

T A B L E 1

S y s t e m o p e r a t i n g c o n d i t i o n

Bus Vo l t age m a g n i t u d e

No. T y p e (p .u . )

L o a d act ive p o w e r Load reac t ive p o w e r I n c r e m e n t a l cos t p a r a m e t e r s

(p .u . ) 0 (p .u . ) ,(1 (p .u . ) 0 (p .u . ) ( $ /MW ) ~ ($(MW 2)

1 Slack 1 2 P V 1 3 P V 1 4 P V 1 5 P V 1 6 P V 1 7 P V 1 8 PQ - -

9 PQ -- 10 P Q - -

0 .25 0 . 0 0 8 3 - - - - 1.9 . . . . . 1.9 1 0 . 0333 - - - - 2.0 . . . . 2.0 0.1 0 . 0 3 3 3 - - - - 1.9 0,1 0 . 0 3 3 3 - - - - 1.9 0.3 0 . 0 1 0 0 0.5 0 . 0167 - - 0.3 0 , 0 1 0 0 0.5 0 . 0167 - - 0.5 0 , 0167 0.6 0 .020 - -

0 .010 0 .010 0 .010 0 .010 0 .015 0 .015

163

20. I

~ >15- / ~ a )

¢. Q "0 1 0 -

5 "

4 o'5 ' ' P~o • 0 .7 0.6 Fig. 2. Probability density curve o f maximum active power demand at node 10 under critical voltage stabil ity condit ions: (a) suggested approximated method; (b) Monte Carlo m e t h o d on exact non-linear model.

.20 ¸

15- > .

e-

1 0 -

ri sk of 5-

i n s t a b i l i t y

\ o.1 o'.z Yl0

Fig. 3. Difference density curve o f actual active power demand minus critical active power demand at node 10.

maximum permissible active power demand at the same node has been plotted.

The above investigation was also carried out for the other PQ buses 8 and 9 and all simulation results are shown in Table 2. From

this Table it can be seen that node 9 presents the greatest risk of voltage collapse for the operating condition given in Table 1.

It may now be of interest to show how the suggested approach can provide important

164

T A B L E 2

M a x i m u m act ive power , cr i t ical vol tage m a g n i t u d e and ins tab i l i ty risk fac to r at d i f f e ren t PQ nodes

Node No. M a x i m u m act ive power Critical vol tage m a g n i t u d e Ins tab i l i ty risk f ac to r p (%)

(p.u . ) a (p .u . ) p (p .u . ) o (p .u . )

8 0 .474 0 .0679 0 .588 0 . 0 0 3 6 9 0.57 9 0 .357 0 . 0 3 4 8 0 .496 0 . 0 0 2 6 6 5.82

10 0 .583 0 . 0 2 7 4 0 .523 0 .00168 0.48

T A B L E 3

Effec ts of vol tage m a g n i t u d e changes at P V nodes o n the ins tabi l i ty risk fac to r and t he s ta t is t ics of m a x i m u m act ive power and cr i t ical vol tage m a g n i t u d e at n o d e 10

Vol tage m a g n i t u d e at P V nodes (p .u . )

M a x i m u m act ive power Crit ical vol tage m a g n i t u d e

U (p .u . ) a (p .u . ) p (p .u . ) o (p.u.)

Ins tab i l i ty risk f ac to r p (%)

1.01 0 .615 0 . 0 2 6 4 0 .531 0 .00185 0.01 1.00 0 .583 0 . 0 2 7 4 0 .523 0 . 0 0 1 6 8 0 .48 ( n o m i n a l case) 0 .99 0 .551 0 . 0 2 8 4 0 .516 0 .00147 6.18 0 .98 0 .519 0 , 0 2 9 6 0 .508 0 . 0 0 1 3 6 29.11 0 .97 0 .485 0 .0310 0 .501 0 . 0 0 1 2 4 66 .50 0 .96 0 .452 0 .0327 0 .494 0 .00121 90 .44 0 .95 0 ,417 0 .0344 0 .487 0 .00119 98.51

01o ~.0 . . . . . . . . . . . . . . . . . . . .

0.8-

018-

0.4-

0.2-

0.0 I

0.4 o'.~ o'.6 0.7 ,u PL~o Fig. 4. Vol tage ins tab i l i ty risk f ac to r Pzo versus t he m e a n value of act ive power d e m a n d at node 10.

information about the risk of voltage collapse when different operating conditions for the system are considered. Table 3 shows the in- stability risk factor along with the mean value and standard deviation of the critical active power and voltage at node 10 corresponding

to voltage changes at the P V nodes from +1% to --5% with respect to the nominal values listed in Table 1. It is notewor thy that a prob- ability of instability of 98% corresponds to a decrease of 5% in nominal voltage level at all the P V nodes. In addition, Fig. 4 shows the

relation between the instability risk factor at bus 10 and the mean value of the active power demand at the same bus.

The proposed probabilistic technique can fruitfully be used to find the most effective location of shunt compensation in the system in order to decrease the risk of voltage in- stability. For this purpose, the mean value of the reactive power at a generic PQ node was reduced (owing to the compensative action) by 10% with respect to the nominal value and, accordingly, the instability risk factor was evaluated at each PQ node. The results of this investigation are listed in Table 4. This Table clearly shows that a shunt compensa- tion at node 9 has the best effect on the reduction of the risk of a voltage collapse for the system.

TABLE 4

Values of instability risk factor Pk (k = 8, 9, 10) fol- lowing a partial compensative action at PQ nodes

Node No. Instability risk factor

Ps (%) P9 (%) P,0 (%)

8 0.0038 1.16 0.47 9 0.0001 0.0002 0.47

10 0.32 4.65 0.0010

6. CONCLUSIONS

In this paper the non-deterministic nature of loads has been taken into a c c o u n t to analyse voltage stability in interconnected power systems. Loads were treated as uncor- related random variables with normal distribu- tion, whereas the probabil i ty distribution for the generated powers was derived by linear modelling of the dispatch activity as a func- tion of bus load demands. The probabilistic technique was based on the linearization, about an expected operating condition, of the equations of the voltage instability stochastic model which correlates the random input quantities with the critical state of the system.

The suggested approach has proved suitable for achieving reliable predictions of voltage collapses at the level of power system plan- ning. In addition, this approach provides the possibility of analysing probabilistically the

165

effects on voltage stability due to variations of the voltage levels at the generating nodes or to the adoption of different strategies of shunt compensation.

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