A probabilistic approach to the voltage stability analysis of interconnected power systems

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 p o w e r system it is always necessary to assess whether a voltage collapse occurs during a prefixed system operating condition. However, present approaches to the analysis o f voltage instability p h e n o m e n a in interconnected p o w e r systems are deterministic 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 p o w e r systems is presented; it treats loads as random uncorrelated variables with normal distributions. The m e t h o d 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 viewpoint. The capability and usefulness o f the suggested approach are illustrated by carrying o u t simulation studies on a sample p o w e r system. 1. INTRODUCTION Voltage instability phenomena in electric power systems have been the object o f considerable interest in recent years. Much work has been done to analyse the mechanism of the above phenomena and several criteria have b e e n 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 0378-7796/86/$3.50

a t t e m p t [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 generator 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 m e t h o d 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 a m o u n t of calculation 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 accurately 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 probabilistic m e t h o d 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 conditions. As is well known [4, 8], a voltage collapse at a load node occurs when the active power fed from the node reaches a m a x i m u m 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 network constraints and dispatch laws. This non© Elsevier Sequoia/Printed in The Netherlands

158

linear constrained optimization problem can be solved by using appropriate techniques [121. Now, by imposing the necessary first-order conditions for the o p t i m u m on the Lagrange function associated with the above optimization 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 m a x i m u m active power at the prefixed load node, and of all the node voltages that are critical from the voltage stability viewpoint, are c o m p u t e d on the basis of the linearization, about an expected operating point, of the equations describing the stochastic process. In addition, in order to facilitate the voltage stability analysis, a suitable and practical factor is introduced which is capable of quantifying 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 distributed random variables [13], whereas a linear model of the dispatch strategy is considered [11, 14]. The m e t h o d allows a systematic individualization of all the causes which affect the voltage instability phenomena and can be used advantageously to achieve reliable predictions 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 O F T H E V O L T A G E STABILITY PROBLEM

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 maxim u m value. Consequently, by assuming that the controlled voltage at each P V node is constant, the critical voltage stability condition 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]: (1)

m a x PLk

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

i = 1, 2 , . . . , N

QGi

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

-

-

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

(2)

with N

P a i = 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 equations, whereas eqn. (4) represents the economic dispatch law of the generating units expressed as a linear function of the bus load demands. This law, which satisfies the minim u m 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 negligible, and no inequality constraints are included [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 problem unknowns. The remaining quantities determine the system operating point. Taking into account that maximizing PLk is equivalent to minimizing --PLk, to derive the necessary conditions for the o p t i m u m 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)] i=1

(5)

159

where ~i and vj are Lagrange multipliers associated with the ] t h node active power and reactive power balance equations, respectively. The set of necessary first-order conditions for the optimum are [12] ~}L 0PLk

Ebiki =0

(6)

j =1

OL 00i

N

OPj(O, V)

NL

OQj(O, V)

j= ~

00~

j=,

OOi

N

OPj(O, V)

-

OVi

-0

NL

B k = [bl, b2 . . . . .

P ( x ) = [P,(x),

--',

...,

QGNL] T N L ×

1

QUVL]T NL × 1

bk-1, b~+l, ...,

bN]

P~(x) ....

, PN(X)] T NX1

Q(x) = [Q,(x),

OQj(O, V)

Q2(x) .....

QNL(X)] T

-o

~ V~

j= ,

N~Xl

OV~

i = 1, 2 , . . . , N L OL O~'i

QG2,

(7)

+

j= ,

QG = [QG1,

N X ( N - - 1)

i = 1, 2 , . . . , N ~L

(N-- 1) × 1

QL = [QL1, Qtn,

N

---l+kk--

PL = [PL1, PL2 . . . . , P L Y - l , P L k + I , -.-, P/.~v] T

(8)

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

NL a] -- bi r=lEP L r + P L j + Pi( O, V) = 0 (9) j = 1, 2, . . . , N

0L Ovi

(1o)

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

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

Equations ( 1 1 ) - ( 1 4 ) constitute the deterministic model of the voltage stability problem 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 uncertainties which unavoidably affect the input data and which can cause significant changes in the solution of the voltage instability problem. 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 o f the problem.

NX1 = [~1,

~2,

"" ", ~N] T

N ×1

3. PROBABILISTIC FORMULATION

NL×I

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-

P = [btl,

/22. . . . .

/)NL ]T

x = [0,,

(9:. . . . .

ON,

Vl,

V~,

...,

VNJ r

(N+NL) × 1 a = [al,

a2 . . . . .

aN] T

N ×I

160 as the voltages at the P V nodes and all the o t h er n e t w o r k parameters are kept fixed. 3.1. L o a d m o d e l To acco u n t f or the r a ndom behaviour of loads, due to forecasting and m eas ur em e nt errors, it is assumed t hat the generic bus load d eman d 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 . . . .

(17)

, g L

where P ° i (Q°i) is the forecasted best estimate o f the ith bus active (reactive) p o w e r and APL~ (AQLI) is the r a n d o m variation f r om pOi (Q°i). To simplify the analysis, the input fluctuations 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)

critical condition for the system. Consequently, the probl em of determining the PDFs of the critical o u t p u t quantities of the above stochastic process has to be solved. The non-linearity o f eqns. ( 1 1 ) - (14) makes the d e v e l o p m e n t of a probabilistic approach to the analysis e x t r e m e l y complicated. To circumvent this difficulty, a simple and practical approximative technique is proposed which is based on the first-order Taylor series expansion o f the stochastic process around the expect ed value of the probabilistic opt i m u m solution. This technique is based on the assumption t hat 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)

i = 1, 2, . . . , N E(AQT,i) = 0

V ( A Q L i , AQL~) = OQLi2

(19)

and

i = 1, 2, . . . , N L In addition, it is assumed t h a t c o v ( AP L, AP,.j) = 0

i , j = 1, 2 . . . . , N i¢]

cov(APLi , AQLI ) = 0

i = 1, ..., NL

(21a)

cov(AQLi, AQLj) = 0

i, j = 1, ..., NL iCj

(21b)

(20)

T hat is, the r a n d o m fluctuations of active and reactive powers at the same node as well as at different nodes are assumed to be independent. 3.2. Generation m o d e l By assuming, for simplicity, t h a t the parameters ai and bi (i = 1, 2 , . . . , N ) appearing in eqn. (4) are deterministic and depend on the best estimate o f t he incremental costs, the generation PGi (i = 1 . . . . . N) can be considered as a r a n d o m variable whose statistics can be evaluated f r om the statistics of t he 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 a c c o u n t the r a n d o m nature of loads, eqns. ( 1 1 ) - ( 1 4 ) now constitute the stochastic model o f the voltage instability

LorA'NJ, o l

A~k

/Jp T

0

0

--b k

AV

/JQ T

0

0

0

--bk T

0

0

Z~DLk

×

Bk

A(QG -- QL)

(23)

where p~ = E 0 , ) and tt. = E ( r ) are col um n vectors f o r m e d with the e x p e c t e d values of the c o m p o n e n t s of the k and r vectors; Px = E ( x ) is the col um n vector of the e x p e c t e d values of the c o m p o n e n t s 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 following structure: Hv=[Hp1T

Hp2 TI ...

Hvj~ I . . . I

Hpu ~]

HQ = [HQ1T

HQ2T i ...

HQjT [ . . . [ HQNLT]

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

161

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: A = [)~lI II )HI ,I

...

II ~jI ', . o . ', XNI] T I

I

R = diag[apb~2,

Note that I and I0 are identity matrices of dimensions (N + NL) × (N + NL) and NL × NL, respectively. The main consequence of this approximation, and of the assumptions on the distribution 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 example, the N e w t o n - R a p h s o n m e t h o d [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 pseudoinverse [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 = [~,

A~,

eI = [ ~ P L ,

A~,

~DLk ] T

A(QG -- QL)] T

HpA+ HQN

Je

JQ

Je T

0

0

--ba

QT

0

0

0

--ba T

0

0

I ×

2

OQG1 - Q L I '

a PL2 2' " ""

B~

0

0 ]

(26)

-1

"" "'

2

aPl.~J '

firQGNL _ Q I . , N / ]

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 distribution can be evaluated by the procedure illustrated in § 3. Thus, this risk factor is the probability that the actual active p o w e r demand exceeds the critical active power demand at node k. To evaluate Pk, let the following stochastic process be defined: #$ Yk -w Pu~ --Pro

(28)

Thus, we have 0

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

(25)

where

A=

coy(co) = E(eo eo T) = A R A T where

I

N =[vII Ipv21 'I"" I' viI J'" ' ' , 'II P N L I ] T

eo = Ael

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

(29)

--oo

where f(yQ is the PDF of Yk. Now, it should be noted that, as both P ~ and PLa are normally distributed and statistically independent, 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 o f this factor simplifies the identification of the causes which affect the system voltage instability and provides a simple and practical tool to compare the real voltage instability risks associated with different operating conditions for the system.

162

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 generating nodes constant, the mean value and the standard deviation of the m a x i m u m supply of active power P*Ll0 and of the corresponding 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 P L*1 0 - In order to test the validity of the suggested approach based on a linearization of the equations which describe the optimization problem, a Monte Carlo simulation was carried out by using the exact (nonlinear) model of the problem. In this simulation, pseudo-random numbers with Gaussian distribution generated by a computer were used as input data. For purposes of comparison, 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 differences 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

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 t h a t all the loads have been assumed Gaussian, with standard deviations a i expressed as [17] ai

=

(32)

0 . 1 IAPLi/3

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

) 3

9

8

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

LOAD

Fig. 1. T e n - b u s C I G R E 225 k V s y s t e m . TABLE 1 System operating condition Bus No.

Type

Voltage magnitude (p.u.)

Load active p o w e r (p.u.)

Load reactive p o w e r

0 (p.u.)

,(1 ( p . u . )

0 (p.u.)

Incremental cost parameters ($/MW)

1

2 3 4 5 6 7 8 9 10

Slack

1

PV PV PV PV PV PV PQ PQ

1 1 1 1 1 1

PQ

--

---

0.25 . 1 . 0.1 0,1 0.3 0.3 0.5

. .

0.0083 . 0.0333 . 0.0333 0.0333 0.0100 0,0100 0,0167

-.

-.

--

--

--0.5 0.5 0.6

--0.0167 0.0167 0.020

.

1.9 1.9 2.0 2.0 1.9 1.9 ----

~ ( $ ( M W 2)

0.010 0.010 0.010 0.010 0.015 0.015

163

20.

I

~>15-

/~a)

¢. "0Q 10-

5"

4

o'5

•

0.6 '

'

0.7

P~o

Fig. 2. Probability density curve o f maximum active power demand at node 10 under critical voltage stability conditions: (a) suggested approximated m e t h o d ; (b) Monte Carlo m e t h o d on exact non-linear model.

.20 ¸

15>.

e-

10-

ri sk

of

5-

instability

\

o.1 o'.z Yl0 Fig. 3. Difference density curve of 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 o f voltage collapse for the operating condition given in Table 1. It may n o w be of interest to show h o w the suggested approach can provide important

164 TABLE 2 M a x i m u m active p o w e r , critical voltage m a g n i t u d e a n d i n s t a b i l i t y risk f a c t o r at d i f f e r e n t PQ n o d e s N o d e No.

M a x i m u m active p o w e r (p.u.)

8 9 10

0.474 0.357 0.583

Critical voltage m a g n i t u d e

a (p.u.)

p (p.u.)

o (p.u.)

0.0679 0.0348 0.0274

0.588 0.496 0.523

0.00369 0.00266 0.00168

I n s t a b i l i t y risk f a c t o r p (%) 0.57 5.82 0.48

TABLE 3 E f f e c t s o f voltage m a g n i t u d e c h a n g e s at P V n o d e s o n t h e i n s t a b i l i t y risk f a c t o r a n d t h e statistics o f m a x i m u m active p o w e r a n d critical voltage m a g n i t u d e at n o d e 10 Voltage m a g n i t u d e at P V n o d e s (p.u.) 1.01 1.00 ( n o m i n a l case) 0.99 0.98 0.97 0.96 0.95

I n s t a b i l i t y risk f a c t o r p (%)

M a x i m u m active p o w e r

Critical voltage m a g n i t u d e

U (p.u.)

a (p.u.)

p (p.u.)

o (p.u.)

0.615 0.583

0.0264 0.0274

0.531 0.523

0.00185 0.00168

0.01 0.48

0.551 0.519 0.485 0.452 0,417

0.0284 0,0296 0.0310 0.0327 0.0344

0.516 0.508 0.501 0.494 0.487

0.00147 0.00136 0.00124 0.00121 0.00119

6.18 29.11 66.50 90.44 98.51

01o ~.0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0.8-

018-

0.4-

0.2-

0.0

0.4

o'.~

o'.6

I

0.7

,u PL~o

Fig. 4. V o l t a g e i n s t a b i l i t y risk f a c t o r Pzo versus t h e m e a n value of active p o w e r d e m a n d at n o d e 10.

information about the risk of voltage collapse when different operating conditions for the system are considered. Table 3 shows the instability 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 n o t e w o r t h y that a probability 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

165

relation between the instability risk factor at bus 10 and the mean value of the active p o w e r 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 instability. 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 compensation 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) following a partial compensative action at PQ nodes Node No.

8 9 10

Instability risk factor

Ps (%)

P9 (%)

P,0 (%)

0.0038 0.0001 0.32

1.16 0.0002 4.65

0.47 0.47 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 p o w e r systems. Loads were treated as uncorrelated random variables with normal distribution, whereas the probability distribution for the generated powers was derived by linear modelling of the dispatch activity as a function of bus load demands. The probabilistic technique was based on the linearization, a b o u t 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 planning. In addition, this approach provides the possibility of analysing probabilistically the

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.

REFERENCES

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