Decaupled Power Flow Solution Method for Well-conditiones and Ill-contditiones Power Systems

8/10/2019 Decaupled Power Flow Solution Method for Well-conditiones and Ill-contditiones Power Systems

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Decaupled power f low so lu t ion m ethod forwell-conditioned and i l l-co nd itioned power systems

M.M.M. El -Ar in i

Indexing terms: Power qnfem.5. Loud flow , Simulation

Abstract: A decoupled power flow solutionmethod for well- and ill-conditioned powersystems is derived. The proposed method issimple, has no mathematical a pproxim ations andrequires less storage and com putationa l t ime tha neither the Newton-Raphson (NR) metho dfor ill-conditioned systems,or the Fast Decoupled (FD)method for well-conditioned systems. In thederived me thod, the power flow is decoupled intoP a n d Q power models without any approx-imations and the P model is decomposed into agenerator active power model and a load activepower model. The difference between the load

active power model m atrix and the reactive powermodel is combined with the mismatch power andthe second-order vectors, which enables the samematrix trian gulation to be used to solve the loadactive and load reactive power flow models. Theproposed method is adjusted with the optimalmultiplier to improve the convergence anddecrease the computational time of the ill-conditioned systems. To examine their effec-tiveness, two ill-conditioned systems, i.e.,11- and43-bus systems and a well-conditioned 90-nodeGerm an util i ty network ar e studied using the pro-posed method and compared with the FD andNR methods. The results show that the proposedmethod has greater comp utational speed, smallermemory requirements and better convergencethan the FD method for well-conditioned systemsand gives a convergent solution for ill-conditionedsystems while theFD gives a divergent solution.

1 I n t r o d u c t i o n

In the early history of load flow comp utat ion, the digitalcom puter solution presented by Ward an d Hale [ l ]needed many iterative calculations. The Newton-Raphson method [2] was then applied to obtain goodconvergence in few iterations, from a flat star t. The Fas tDecoupled (FD ) load flow method by Stott an d Alsac 131then appeared using polar co-ordinates, and fast loadflow retaining nonlinearity using the rectangular co-ordinates was proposed [4]. Th ese methods can solvewell-conditioned systems.

The following factors have been taken to characteriseill-conditioned load flows[SI

Pa pe r 895K W ) ,eceived 29th April 1992The author is with the Department of Electrical Power Engineering,Faculty of Engineering, Zagazig University, Zagaug, Egypt

I E E P R O C E E D I N G S - C ,Vo l . 140, No. I J A N U A R Y 993

(i) Netw ork configuration (longitudinal network and

(ii) Operating conditions (heavy load conditions).(iii) Loa d flow spec ifications (high ratio of numb erof

( P, Q ) buses/number of ( P, V )buses, position of the slackbus, inadequacy of the calculation and inadequacy of theinitial values.

The degree of ill-conditioning is evaluated by one of thethree scalar indices: the number of iterations or theoptimal multiplier p [ 6 ] or the value of m a ? (J- l ) , themaximum eigenvalues of the inverse matrix of theJacobian.

It is well known that load flow calculations can be

regarded as a nonlinear programming problem [7-9]which determines the direction and magnitude of thesolution so that a certain cost function can be minimised.

network with high ratio ofR / X .

2 Proposed load f l o w m e t h o d

The well known rectangular form of the load flow equa-tions

p , 1 dGLJeJ B, , f , ) +f,(G,, L B I j e j 1)J = l

are expanded using the Taylor series to obtain thecompact form

where n is the total number of system buses, ng is thenumber of generating buses, G and B are the real andimaginary parts of the bus ad mittance matrix,A P andAQ are the vectors of the active and reactive power mis-matches at all buses except the slack bus,f a n d e a r evectors of the imaginary an d real com ponents of the busvoltage and H P and H Q are the second-order termvectors, respectively.

For a PV -bus system, the load flow equation s to besolved are the P equation s at all buses except the slackbus and the Q equa t ions a t the load buses alone. The Qequation at the generating buses are then omitted andreplaced by A e of the generating buses, in the oth er equa-tions, by its equivalent from AeL ( % / e , ) AL. i = 2, . . .,

ng. Also, the P model in eqn. 3 is decomposed into thegenerator active power, P, , and the load active power,P, , equations. So that th e load flow equ ation (eqn. 3) canbe rewritten as

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where A P c i A P i ,i = 2, . . . n y, A P L i= APi and A Q L iA Q , , i = n y + , . . . , n,

H P, A j ; 1 GijA h Bij A e jj = 2

+h e , 1 Gij h e j + Bij AA)j = Z

H Q , = AA G,j Aei+ Bij AJj)J = 2

Aei1 G L I J, Bij A e j )j = 2

Due to the weak coupling betweenP a n d e and Q a n d f ,eqn . 4 can be rewritten in decoupled form as

J d A e , ) = A Q Z )

where

A P ; = A P c H P, , Ae,

A P, = A P I . H P, , he,

A Q L= AQ L HQ L AJG n AJL

The na tu re of the Jacobian submatrices shows that

J9 i , ) = J 5 i , j ) for i # j

Jg i , ) = J 5 i , j ) Si for i =

J8 i , = , i, j )

Jn i , j ) = J i , ) + 2 R ,for # j

for i

where

R ; G t j e j BiJfi)J = 1

SE c GI j + B,,A )J - 1

Now , the proposed load flow equation (eqn.5 ) becomes

Where AQ, = A Q , + ( 2 S I )Ae, a n d I is a unit matrix.The result of triangularisation of the models (eqns.6 and7) are

where

Now, the decomposition active power flow model (eqn.9)and the reactive power flow model (eqn. 10) can berewritten, as follows, for forward a nd b ackward process

CAP21 [ ] C X f L 1

3 De t e rm i na t i on of step s i z e length ( o p t i m a lm u l t i p l i e r )

In the propos ed me thod, the iterative value of the correc-tion vector A f k a n d Aek is modified by a p a n d aq duringthe solution process to improve the convergence. Thevalues of a, and aq can be obtained from the followingprocedure.

The active power, eqn.6, can be rewritten as

3::

;;) 3::

t)H b l ; HP,,, + H P C f J )

(12)H P + HP,,, + HP,,,

where

HP,,,, de, 1 ij Aej

HPC. ,~ Ae, 1 i JAJ, AJ 2 Bij A e jf f P ~ f , i= AA ij AI;

HI,,,: Aec1 j AeI

j = 2

2, . . ng

J = 2

I E E P R O C E E D I N G S - C , Vo l . 140, N o . I J A N U A R Y 1993


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To adjust the length of the correction vectorA f ( A f f t ,Afi , in cqn. 3 multiply the scalar quantity a p by Af toobtain the equation a t , b +a: c = 0 where a, b and care vectors with

A P c , AeL HP, , ,A P, , AeL HP, , ,

To determine the value ofap in a least squared sense, thecost function

is minimised. The optimal solutiona f can be obtained bysolving 2Wp/aa, 0 analytically using the Cardanformula

(14)o + apY1 + a;Yz a;Y3 0where

- 1

g , bz + 2a ic i )L = 1

n - l

- ,. .g 3 2 c:

i = 1

Also the reactive power equations (eqn.5b can be rewrit-ten as

AQL 7 L ? f 8 AfL+ 2 W AeL H Q L ~ ~HQLe/ s AeL HQLJJ = 0 (15)

where

HQLeel= ei

HQL,/i = A h G i jA e j + Ae, G i jAfjH Q L f f i A h x B i j A&

Eqn. 15 can be rewritten as

Bi jA e j

f o r i = n y + 1 , _ _ _

y + r q z + a,2h 0where

I = AQL i Afc AfL ~ HQL / /

z = 2 S I )Ae, + HQLeJ , A e ,h = - H Q Lee

The cost function of the Q equations to be minimisedusing the least squared metho d is

W = C Y ,+ a q z + a,Zh,Y

and th e optimal solutionaq* is obtained from the solutionof the cubic equation obtained from

1 n - n g

= 1

dW /?a = 0 (17)as in the active pow er flow equatio ns (eqn. 14).

I E E P R O C E E D I N G S - C , Vo l . 140, No. I J A N U A R Y 1993

4 Proposed m e m o d algor ithm

The simplest iteration schemeof the proposed metho d isdefined by the following steps for ill-conditioned system s

St e p 1 : Set the iteration indexK 0.Step 2: C o m p u t e A , , A , , A , , A , , A ,a n d A , and set

k = k l .S t e p 3: C o m p u t e A P c a n d A P, . If max APcl a n d

max J A P, are less than c p , go to s tep 5 , otherwise go tostep 4.

Step 4 : Solve eqns. l la- ll d to find A f k , obta in a*from eqn. 14 and update f according to f' = pS t e p 5 : C o m p u t e A Q L If max A P c 1 m a x A P L < c p

and max IA Q , 1< e g o t o s t e p7, otherwise, go to step6 .Step 6 : Solve eqns. l le , and llf to calculate Be',

obta in a: from the solution of eqn. 17 and u pdateeaccording to e k + = e + a: Be' if K < K go to s tep 2,otherwise, go to s tep 7.

+ a f Afk then find eG from eqn . l lg .

Step 7: Print the results and stop.

Num e r i c a l i l l u s t r a t i on

5.1 Ill cond itioned systemsTo illustrate the proposed method numerically the11-and 43-bus systems [ 6 ] which have difficulty convergingare tested. The results show that the proposed methodgives the same solution, as that obtained by the m ethod

suggested in Reference6 . In the case of the 11-bus systemthe solution does not exist. For the 43-bus system thesolution exists from the initial flat start using the doubleprecision at the same number of iterations but decreasesthe computa t iona l t ime by abou t 30 as compared tothe method proposed in Reference 6 while the FDmethod gives a divergent solution for both systems.

5 2 Well condi tioned systemsHere, the values of a p and a4 in the proposed method areset equal to one, the Jacob ian matrix is only determinedat th e starting point and the process of tr iangularisationoccurs only once. The load flow of an example systemwith 90 nodes, (62 load buses,28 generator buses and 177lines) a German utility network (Bayernewerk 90-nodesystem) is determined. The results, show that the pro-posed method gives the solution after 3 iterations whilethe FD method needs 6 i terations. The comp utation timerequired by the proposed method is abou t 75 of thetime required by the FD method and abou t 87 of thatneeded by the method proposed in Reference10. If thenumbers of the generating buses are treated asP Q buses,the number of generating buses in the system is relativelysmall, the computational speedof the proposed methodis increased and the saving in computation time isincreased comp ared to the F D method time (e.g. forn y 10, the proposed method takes abou t 65 of thetime required by theFD method).

Conclusions

A decoupled power flow solution method for well condi-tioned and ill-conditioned systems has been considered.For well-conditioned power systems with relatively fewgenerators the proposed method has greater computa-tional speed, smaller memory requirements and at leastthe same or better convergence than theFD method.

For ill-conditioned power systems the proposedmetho d is very simple, there are no approximations andit is quite practical. When the solution continues to oscil-

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late with the normal N R method, if the values of theoptimal multipliers m p and mq approach zero as the iter-ation count increases, the solution does not exist accord-ing to the initial estimate, whileif the values of up and m qremain at 1.0, regardless of the num ber of iterations, thesolution exists but does not converge due to the lackofprecision of the computer. The proposed methoddecreases the com putational t ime by a bout 30 com-pared to the time required for the method proposed inReference 6 .

7 References

1 WARD, J.B., and HALE, H.W.: Digital computer solutionof power

2 VAN NESS, J.E.: Iteration method for digital load flow studies,flow problems,A lEE Tr an s . ,1956,PAS-75, pp. 398-404

A lEE Tr an s . ,1959,PAS-78, pp. 583-589

3 STOTT, B. nd A L S A C , 0 Fast Decoupled load fiow, I b L 6Trans., 1974,PAS-93, pp. 859-869

4 IWAMOTO, S., and TAMURA, Y.: fast load flow methodretaining nonlinearity,IEEE Trans., 1978,PA S W, pp. 1586-1599

5 TAMURA, Y. , YORINO, N. . IGODA,K., NAKAZAWA, T. , andHASHIMOTO,J : Factor analysisof load flow ill-conditions. Pro-ceeding of the Ninth Power Systems Computation Conference,Cascais. Portugal, 1987,pp. 920-926

6 IWAMOTO, S., and TAMURA, Y.: A load flow calculationmethod for ill-condition power systems,IEEE Trans. , 1981, PAS-100, pp. 1736-1743

7 WALLACH, Y Grad ien t meth od sfor load flow problems, f E E ETrans. , 1968,PAS-87, pp. 1314-1318

8 SASSON, A.M.: Nonlinear programming solutionsfor the loadflow, minimum-loss, and economic dispatching problems,I E E ETrans. , 1969,PAS-88, pp. 399-409

9 SASSON A.M., TREVINO, C., and ABOYTES,F.: ImprovedNewtons load flow through a minimization technique, I EEETrans., 1971,PAS-W. pp. 1974-1981

10 JOVANOVIC, S M., and BABIC, B.S.: Decouled and decomposedpower flow solution m ethod,EIectr. Power & Energy Syst., 1987,9

10 I EE P RO CEED I N G S - C, Vo l .140, N o . I J A N U A RY 993

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Decaupled Power Flow Solution Method for Well-conditiones and Ill-contditiones Power Systems