Optimal stochastic load flows

Electrical Power Systems Research, 2 (1979) 71 - 75

71

© Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands

Optimal Stochastic Load Flows MARIAN SOBIERAJSKI

Institute of Electric Power Engineering, The Technical University at Wroc}aw, Wroctaw (Poland) (Received October 27, 1978)

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SUMMARY

y = f(x)

This paper presents an original m e t h o d of calculation for the optimal stochastic AC load flow. The mathematical model has been presented and the equations of solution derived. To optimize the stochastic load flow the penalty m e t h o d was chosen. Finally, a numerical example on the four-bus test system is presented in which a minimum for the expected value of the losses is accomplished with the proposed method.

where y are the given values and x the calculated values. In a load flow study, the known quantities (y) are the injected active powers (P) at all busbars except the slack, the injected reactive powers (Q) at all load busbars, and the voltage magnitude (U) at all generator busbars. The calculated quantities ( x ) m a y be the magnitudes (U) and the angles (5) of the bus voltages, or the rectangular components of the bus voltages (U 1 and U2). From these known quantities and eqn. (1) the load flow can be solved by one of the deterministic load flow techniques. However, as was mentioned, the information available for load flow calculations is purely stochastic in nature. Hence, in general, the load flow problem can be formulated as: given

1. INTRODUCTION

The general problem of optimal deterministic load flow subject to inequality constraints has been solved completely. There are many practical, very fast methods of solution [1]. The formulation of the optimal deterministic load flow solution assumes that the data provided are absolutely precise. However, in practice, the input data can only be known with some finite precision. The input data values are random and the results obtained can be considered to define the range of variation of the results. There are various reasons for the randomness of the input data values: the randomness of load powers is the outcome of random switching on and off of a large number of small electric power receivers; the randomness of generation powers is mainly the result of the switching off of generators during failures and the technical conditions of the generator-system collaboration.

2. STOCHASTIC LOAD FLOW

The load flow problem involves the solution of a system of nonlinear equations

y = f(x) E(y) = E(y -- y)(y -- ~)W = My then E(x) = E(x -- ~)(x -- ~)T = M:

where y refers to the active and reactive powers at busbars, x to the rectangular components of the bus voltages, E is an expected value operator, T is the transposition symbol, and M a covariance matrix of y or x. To calculate the expected values and covariances of the variables x the relations between the expected values and covariances of the variables y and x will be first determined. From eqn. (1) one obtains

.~ = E [ f ( x ) ]

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72

My = E ( f ( x ) -- E [ f ( x ) ] } {f(x) -- E [ f ( x ) ] } T (3) T h e f u n c t i o n f(x) is a square f u n c t i o n , h e n c e , assuming t h a t the p r o b a b i l i t y d i s t r i b u t i o n o f x can be a p p r o x i m a t e d b y a n o r m a l distribution, one can o b t a i n [2] = f(~) + B)~/2

(4)

My = AMxA T + B(N x - - ) ~ T ) B T / 4

(5)

where A is a m a t r i x o f the first derivatives o f f u n c t i o n f ( x ) , B a m a t r i x o f t h e s e c o n d derivatives o f f u n c t i o n f ( x ) , )~ a covariance v e c t o r o f variables x o b t a i n e d f r o m the rows o f the covariance m a t r i x Mx w r i t t e n one a f t e r a n o t h e r , and N~ is a m a t r i x o f the f o u r t h - o r d e r mom e n t s o f the variables x. T h e a s s u m p t i o n o f n o r m a l d i s t r i b u t i o n is o f vital i m p o r t a n c e , since it allows o n e t o relate the f o u r t h - o r d e r m o m e n t s t o the secondo r d e r ones. In c o n s e q u e n c e , h o w e v e r , the p r o b a b i l i t y distributions o f each nodal p o w e r at bus or b r a n c h o n l y a p p r o x i m a t e the n o r m a l distribution. Owing t o the n o n l i n e a r i t y o f eqns. (4) and (5) an iterative s o l u t i o n m u s t be used. As the result of calculations, e x p e c t e d values (~), variances, and covariances (M~) o f the rectangular c o m p o n e n t s o f the n o d a l voltages are obtained. B e t w e e n bus voltages and b r a n c h p o w e r s t h e r e also exists a square d e p e n d e n c e : z = g(x)

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where z refers to the active and reactive b r a n c h p o w e r s (P and Q, respectively). Thus, the e x p e c t e d values are calculated straight f r o m the e q u a t i o n = g(~) + D~/2

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w h e r e D is a m a t r i x o f the s e c o n d derivative o f eqn. (6). T h e n the covariances o f the b r a n c h p o w e r s are calculated: M z = CMx Cw + D(N x --)~)~T)DT]4 w h e r e C is a m a t r i x eqn. (6).

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o f the derivatives of

buses, and additional constraints such as bus voltage limits recognized. T h e r e are t w o o p t i m a l load flow p r o b l e m s : the e c o n o m i c dispatching p r o b l e m , and the m i n i m u m loss p r o b l e m . This p a p e r presents the solution o f the m i n i m u m loss p r o b l e m . As in the load flow p r o b l e m , the variables will be the n o d a l voltages, which will be used in rectangular form. T h e bus voltages and the bus p o w e r s are stored as vectors with the real parts in positions I to w and the c o r r e s p o n d i n g image parts in positions w + 1 to 2w: P i = Y i is the active p o w e r at bus i Q i = Y i + w is the reactive p o w e r at bus i

U~ = (x~ + X2+w)1f2 is the voltage m a g n i t u d e at bus i xi is the real part o f the voltage at bus i x i + w is the image p a r t o f the voltage at bus i w is the n u m b e r of i n d e p e n d e n t buses. A voltage space is d e f i n e d b y giving (1) active p o w e r s at all g e n e r a t o r and load busbars, (2) reactive p o w e r s at all load busbars, (3) limits to the reactive p o w e r s at all g e n e r a t o r busbars, (4) limits to the voltage m a g n i t u d e at all busbars. H e n c e the v e c t o r y can be p a r t i t i o n e d into a v e c t o r u o f k n o w n p o w e r s and a v e c t o r v o f u n k n o w n reactive powers: y = (u, v) as

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T h e m i n i m u m loss p r o b l e m can be defined the o p t i m i z a t i o n o f the loss f u n c t i o n w

s(x)=

Z yi(x)

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i=1

w h e r e y i ( x ) is the active bus p o w e r in t e r m s o f the system voltages x. T h e f u n c t i o n s(x) is subject t o system constraints which include (1) the n e t w o r k e q u a t i o n s

y = f(x) (2) physical limits o n g e n e r a t o r reactive powers

3. O P T I M A L S T O C H A S T I C L O A D F L O W

An o p t i m a l load flow is a load flow in which some q u a n t i t y is m i n i m i z e d , with t h e o r d i n a r y load flow c o n s t r a i n t s a r o u n d all

Vlower ~ V(X) ~< Vup~r

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(3) physical limits o n voltage m a g n i t u d e s Glowe r ~ U ( x ) ~ Uuppe r

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73 In ref. 1 are presented general methods that solve such a deterministic optimal load flow problem. Some of these methods are based on the K u h n - T u c k e r theorem and the others on nonlinear programming. However, it was stated in the previous section that the load flow problem is really a stochastic one. The powers at all buses should be treated as random values. Inequality constraints should also be treated as random values. They are seldom rigid limits in the strict mathematical sense but they are rather soft limits. For instance, Uiup~r <~ Ui <~ U/lower really means Ui should not exceed U i u ~ or Uilowe~ by too much, i.e., Ug = 1.01 U i u ~ may still be permissible. It means that inequality constraints must be satisfied with a required probability. Hence there are the following known quantities: (1) the expected values of the active powers at all busbars and the reactive powers at all load busbars = E(u)

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(2) the variances and covariances of bus active and reactive powers at all busbars Mr = E ( y y T) (3) the probabilistic limits to the u n k n o w n reactive powers P{Vlowe r • V(X) < Vupper} ~> a v

(14)

(4) the probabflistic limits to the voltage magnitudes P{Ulower < U(x) ~< Vuppet} ~ 0/u

(15)

where p is the probability of violation of the constraints and a is the required probability of satisfaction of the inequality constraints. The minimum loss problem is defined in such a way that the value of the losses is minimized by variations in reactive generation. Hence the problem of the optimal stochastic load flow c a n be formulated as follows. Determine the expected value of the unknown reactive powers u(x) that minimizes the expected value of the system losses s(x*) = min E[s(x)]

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subject to l equality constraints g(x) = 0

and m inequality constraints

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p(h(x) ~> O} ~> a

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where g(x) refers to the stochastic load flow equations (4) and (5), and h(x) to the constraints on the reactive powers (14) and the voltage magnitudes (15). Because the inequality constraints are soft limits the penalty m e t h o d of minimization was chosen. In the penalty function the objective function is augmented by penalties for constraint violations [3]. The choice for the penalty function is as follows: F(x, ri) = s(x) + ri ~ gi(x) 27i + i= 1

ti(x)2/3i i= 1

(19) where ti(x) = hi(x) -- koi >1 0

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7i = 0 ifgi(x) = 0 1 if gi(x) ¢ 0

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/3i = 0 if ti(x) t> 0 1 if t~(x) < 0

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rj. > r~_ 1

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and k is a coefficient apertaining to the required probability a, o i is the standard deviation of hi(x), and rj is a vector of controlling parameters. This choice for F(x, r/) satisfies the convergence properties which indicate that in solving the sequence of unconstrained minimization problems, the values of both the penalized cost function F and the constrained problem function s(x) will be monotonically increasing. It appears that any limit point of the sequence xj* is a solution of the problem.

4. SOLVING THE OPTIMIZATION PROBLEM

The process of solution has been reduced to the initial satisfaction of equality constraints by the m e t h o d of ref. 2 to obtain a feasible nonoptimal starting point which is followed by a minimization process. The solving procedure of the optimization problem by penalty function is as follows. 0. Establish initial values: (a) choose expected values of u n k n o w n reactive powers

74 ~j,,

j = 0, m = 0

M * = M*j

if n o t , establish a new value

(b) c o m p u t e initial values o f bus voltages (x) f r o m the stochastic load flow e q u a t i o n s

D+I > r j

xi-, f

J = 0, m = 0

and r e t u r n to step 1.

Mxj mf

j = O, m = 0

(c) c h o o s e the initial value o f r/ rj>O

j--0

where j is the i t e r a t i o n n u m b e r o f the minim i z a t i o n process, m is the i t e r a t i o n n u m b e r o f the load flow calculation, and f refers t o the load flow solution. 1. C h e c k the i n e q u a l i t y constraints. Because the initial values xjmf and M~ymf are c o m p u t e d f r o m the stochastic load flow e q u a t i o n s , 7 = 0 and the e q u a l i t y constraints have n o t to be checked. 2. C o m p u t e the gradient o f F ( x , D) at a p o i n t "~j,nf and Mxjmf grad Fj,,f = grad F ( x , r/)]i/m f,Mxjm f 3. Find a new set o f xj,. = xjm~-- grad Fjm f

4. Compute the new expected values of the unknown reactive powers Vj,,, = V(X)Ixj,,, Mxym 5. Solve the stochastic load flow at a p o i n t ~y~. One obtains Xj(rn +l ) f , Mxj(m+ l ) f

6. If IF(xy(~

+ 1)fry)

--

F(xjmf, rj) l 5. EXAMPLE CALCULATIONS

and ]Xj~,rn + 1)f

--

ijmt

{

are sufficiently small t h e n X? = Xj(m + 1)f

M*j = Mx/(m + 1)f and go to step 7. If not, r e t u r n t o step 1. 7. If { F ( x * , r y ) - - F(X* l , r 1 1){

is sufficiently small t h e n

.~* =,~*

As the above m e t h o d works f r o m b o t h the i n t e r i o r and the e x t e r i o r o f the c o n s t r a i n t space, there is n o p r o b l e m o f violated constraints during the m i n i m i z a t i o n . All constraints t h a t are satisfied at any one time are r e m o v e d f r o m the calculations and are n o t c o n s i d e r e d again until t h e y are violated. F o r all b u t the last value o f r) an e x a c t m i n i m u m is n o t necessary and h e n c e the crit e r i o n for a m i n i m u m can be m a d e stringent o n l y t o w a r d the end o f the c o m p u t a t i o n s . It m u s t be r e m e m b e r e d t h a t r) is a very sensitive f a c t o r which can govern the whole o u t c o m e o f the process. L o w e r values for ri m a y lead to m o r e c o m p u t a t i o n s and higher values m a y result in diverging solutions. When discussing the c o m p u t e r t i m e one should keep in m i n d the fact t h a t the stochastic load flow c o r r e s p o n d s in practice to several h u n d r e d d e t e r m i n i s t i c flows. I n d e e d , the c o m p u t e r time for calculation o f the o p t i m a l stochastic load flow is m u c h longer than t h a t f o r the calculation o f one deterministic flow. This is because the m a t r i x o f covariance o f the nodal voltages m u s t be calculated at each iterative step. It was s h o w n in the previous section t h a t the load flow p r o b l e m is n o t deterministic b u t stochastic. T h e o p t i m a l load flow problem is also stochastic.

A c c o r d i n g to the m e t h o d described, a p r o g r a m in F o r t r a n and the e x a m p l e calculations have b e e n made. Because the matrices A, B, C, and D are sparse, the t e c h n i q u e o f sparse matrices has b e e n used. slaok nods U 4 = 220 kV

4~

l

2/'10/100

R/X/B/2

1

.~ •

50/60/I00

f~/J]./uS 20/80/100%

21

/

\

y

Fig. 1. The network with branch data.

/

/ .

gensrat or nods

20/60/1 O0

75 TABLE 1 The results of the solution Solution

Non-optimal

Reactive power at bus 2 (MW) System losses (MW)

Optimal

Deterministic D.V.

Stochastic E.V. ± C.L.

Deterministic D.V.

Stochastic E.V. ± C.L.

455.0 68.5

455.0 ± 141.0 75.3 ± 66.9

318.8 61.2

326.9 ± 141.0 70.0 ± 51.3

D.V. = deterministic variable, E.V. = expected value, C.L. = confidence limits.

To compare the deterministic and stochastic evaluations the four-busbar system was analysed. Expected values Expected values of bus powers (in MVA) are as follows: $1 = 260 + j 1 9 5

$2 = 480 + j453

Ss = 165 + j145

The results of the calculation are given in Table 1. It can be seen that the deterministic optimization in comparison with the stochastic one is far too big a simplification of what can be found in the future network. The optimal system losses are n o t equal to 61.2 MW but m a y range from 20 MW to 120 MW, owing to the considerable random changes of the bus powers.

Cons train ts p (900 ~< Q2 ~< 200)/> 0.997 p ( 2 5 0 ~< U 1 ~< 200} i> 0.997 p ( 2 5 0 ~< U 2 ~< 200) ~> 0.997 p ( 2 5 0 ~< U 3 ~< 200} ~> 0.997 Variances Variances of each nodal power were calculated assuming 20% power deviations from the expected values:

( 0.2yi 12

var(yi) = \--~/

where Yi is the expected value of the nodal power, either active or reactive. Covariances It was assumed that all the nodal powers were correlated in agreement with the following principle: active and reactive powers in the same nodes, ppQ

-~ 1

active and reactive powers in different nodes, ppp

-~ p p o

= pQO

~-

0.5

6. CONCLUSION

This paper has described a new m e t h o d of calculating stochastic load flow subject to probabilistic inequality constraints. To optimize the problem, the penalty function m e t h o d was chosen, because this m e t h o d produces soft limits. A gradient technique was tised for the minimization of the penalty function. Only the inequality constraints that had been violated were penalized. The equality constraints, i.e. stochastic load flow equations, were eliminated from the minimizing function because they had been satisfied earlier. Further improvements are expected.

REFERENCES 1 H. H. Happ, Optimal power dispatch -- a comprehensive survey. IEEE Trans., PAS-96 (1977) 841. 2 M. Sobierajski, A method of stochastic load flow calculations. Arch. Elektrotech. (Berlin), 60 (1978) 37. 3 W. T. Zangwill, Nonlinear Programming, McGrawHill, New York, 1969, p. 255.