A New Solution to the Optimal Power Flow Problem

7/28/2019 A New Solution to the Optimal Power Flow Problem

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Paper accepted for presentation at PPT 20012001 IEEE Porto Power Tech ConferenceIOth 13Ih September, Porto, Portugal

ANew Solution to the Optimal Power FlowProblem

EdmCa C. Baptista, Edmarcio A. Belati and Geraldo R.M. da Costa Member. ZEEE

Abstract-A new approach to solving the Optimal Power Flowproblem is described, making use of some recent findings,especially in the area of primal-dual methods for complexprogramming. In this approach, equality constraints are handledby Newtons method inequality constraints for voltage andtransformer taps by the logarithmic barrier method and the otherinequality constraints by the augmented Lagrangian method.Numerical test results are presented, showing the effectiveperformance of this algorithm.

Index Terms-Interior-point methods, optimization methods,optimal power flow, nonlinear programming.

I. INTRODUCTIONThe Optimal Power Flow (OPF) is nonconvex static

nonlinear programming problem, was proposed by Carpentierin the early 60 s based on the economic dispatch problem [l].His major contribution is in placing economic dispatch on afirmmathematical basis. Since then, many papers have beenwritten in the attempt to solve it. Recently, the performance ofInterior Point algorithms has been motivating your applicationin the resolution of the OPF problem. In [2]-[4] a primal-duallogarithmic barrier algorithm is applied directly to a nonlinearOPF problem and the Karush-Kuhn-Tucker (KKT) conditionsare solved by Newtons method. A crucial step in thealgorithm is the choice of the barrier parameter. Up to thepresent there is no single robust, reliable and fast approach thathlf ills the requirements of the EMS. It is evident that all thesesolutions have followed close behind advances in numericaloptimization.

In this paper, we consider the application of logarithmicbarrier approach to voltage magnitude and tap-changingtransformer variables. The other constraints are treat byaugmented Lagrangian method [5]. The interest in thisapproach is that verified that when the voltage magnitude andtap-changing transformer variables violate your limits theotimization process can be not effective.

The paper is organized as follows: First, the OPF problemis explained. The new approach is then discussed.Implementation of this approach is described. The results ofthe comparative tests on two systems (3 0 and 57) are reported.Finally, some concluding comments are made.

11.OPTIMAL POWER FLOWROBLEMThe optimal power flow problem can be expressed as

follows: Minimize f ( x )subject to g j ( x )=0 i = 1,2,....,m 0 j =1, ... , r

0-7803-7139-9/01/$10.00 02001 IEEE


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sl> 0

The minimum of La with respect to the slack variable, zj ,, scalculated from the following stationary conditions:

( 5 )aLaaz- = O , j = I ...,r .

where:zj :slack variable related to inequality constraints,hj(x);su and sl : slack variables related to inequality constraints,x

'V, f x)' +AtJ ( x )+ m*)'I+(d) ' +( p j +ch,(x))V ,h,(x) ' , ifh ,(X )> -- puic =o

j= l 0, i f h j ( x ) < - A.C

+nuk =0,k =1, ..,n6--Su k

Problem (2) can be modified to incorporate the strictlypositive variables into the objective function, the former viathe logarithrmc barrier function After modification, theproblem becomes:

Minimize f(x) - s 5 n suk + i: In sl , )subject to: gi(x) =0, i =I , ...,m,k= l k =l

hj(x) +zj =0, j =1, ..., (3 )x +SU =xmaxx - s l = Xm'n

Substituting (7 ) into (4) we obtain the logarithmic barrier-augmented Lagrangian function:

k= 1 k= lm n

i= I k=l

zj 2 0, j =1, ..., r +i;~(x);i-c~(x), f h j ( x ) 2 - 2.U '-_ f h j ( x ) I-JThe logarithmic barrier-augmented Lagrangian function ' = I 2c Cassociated with problem (3) is:

The process involves the determination of values forx,s,A,and K , such that the conditions of optimality aren nLa(x,s, / z , ~ , ~ , z ) = f ~ ) - s ( C ~ n s u kC l n s i k ) +

k=l k= l satisfied. Thus:m

2 j=1 i=l+- - c Z ( h j ( x ) + z j ) '' + c A i g i ( x ) + V, a( x , s , A , n , p ) = O

V L a ( x ,s,1, , )=0

Theses conditions are satisfied by

z . -ma x O , - l - h . ( x ) , = 1, . . .J - 1 t' J J

dk=0, k =I , ..,n'lkg i ( x )=0,i=1,...,mX f S U - x m a x=o

X - s l - x m ' n =oA s zj 20,we have: where:

J(x) = V , g ( x ) and Z is the identity matrix.


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This system of equations was solved using the Newtonmethod in which Ax ,A s , AA , and A n were calculated. Thus,the equations' (10) can be represented as:W.Ad=-bwher

W =

with

V L L a 0 0 J ( x ) ' I I0 S(Su) 0 0 I O0 0 S(S1) 0 0 - I

J ( x ) 0 0 0 0 0I I 0 0 0 0I 0 - I 0 0 0

Ad'=(Ax,Asu,AsI,A/z,A m , A n l ) ;and

b=

V x ( x ) ' + A ' J ( x ) + ( n u ) 'I + ( n l ) ' I +

+nu,, k = l , .., nsu L

--

ni,,k =I , ..,n6SI,

Using the search directions obtained from (1 l) , the vectorsof the variables, x, su and SI and of the Lagrange multipliers, A,m and ni are updated, as follows:

where the scalar step size, q, nd Q ,are chosen to preservethe nonnegativity conditions. It is not always possible to take afull Newton step, so a~ (OJ] is damping factor not only toimprove the convergence but also to keep the nonnegativevariables strictly positive instead of just nonnegative. Themaximum step length a in the Newton direction is determinedby:

where ~ 0 . 9 9 9 5s an empirical value which, according toWright [ 6 ] , can be derived from the formula 1-1 /9&,where m is the number of constraints in the problem.

A . Updating the Multipliers and Penalty FactorsThe gradient of the augmented Lagrangian at the minimum

point x* s:

The terms [ . can be seen as the Lagrange multipliers thatsatisfy the optimality condition of the problem (l), [7].Therefore, a good approximation to the unknown ,U is :

+c k h j ( x k + l ) , f h j ( x k i l ) 2 -&C k , = l , ..,rI1 = ,U0, if h j ( x k + ' ) 5 --C k

The penalty parameter v and barrier parameter 6 may alsobe adjusted during the process.C k + l =p c k


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(17) IV. TESTRESULTS

where: p > l and p >O.The ill conditioning usually associated with the penalty

function approach can be controlled. The method treats theequality and inequality constraints in a unified manner, andavoids a search for binding constraints.

B. AlgorithmThe algorithm proposed to solve the problem is an iterative

process consisting of the following steps:i. Make starting estimates for d = x , s, A, a) nd p

x :can be the same as the initial values for a power flowA =p =0 or any reasonable guessIT >O or 6 0 KT conditions.

ii. Evaluate b as a function of diii. If KKT conditions are satisfied the problem is solved,

otherwiseiv. Update p (eq.16), c and 6 (eq.17)v. Evaluate the matrix W as a function of dvi. Solve the system W M=-b for Mvii.Update d by Mviii. Return to step ii

Tests were done to verify the efficiency of the proposedapproach. The algorithm was implemented in FORTRAN,using double precision arithmetic on a 500 M HZmicroprocessor, in the Power Systems OptimizationLaboratory of EESC, USP. The cases studied were theminimization of active power losses in transmission in theIEEE 30-bus and IEEE 57-bus systems.5. 30-bus system

This test was accomplished with the following initialconditions: Vk =1.0pu (100 MVA base) and 0 k =0.0 for k =1 ...,30, and 6 = 1.0 for i =1...,4. The Lagrange multipliersrelated to the equality and inequality constraints are,respectively, A =0 , =0 and n 0.

The initial penalties were defined as c = 1. All penaltyincreasing factors were defined as p =1.2. The processconverged in 4 iterations. The amount of reactive powergeneration was 144.30 MVAr, with a total active power lossof 17.57 M W . The initial barriers were defined as S= ,0001.All barriers increasing factors were defined as p = 10. Theoptimization process for this case is summarized in table 1.

Iteration Active Mismatch Mismatchpower IDP( Max (Mw) I D Q ~ax (MVh)l nw IMW

1 17.827 0.0207 0.47062 17.686 0.0902 0.41983 17.571 0.0143 0.01894 17.572 0.0000 0.0000The initial values allocated to the variables x can be in the

infeasible region of the problem or can be a power flowsolution. In the evaluation of matrix W in step (v), the updatedvalues of p and c must be used. For sufficiently large c, W is

TABLE IOPTtMIZATION SUMMARY FOR 3o-BUS SYSTEM

positive definite, thus a problem can be convexifed (at leastlocally). 5 .2 57-bus system

C, Implementation

Most of the work in the algorithm is in the solution ofsystem (1 1). The Lagrangian matrix, W , hat results from thelinear approximation of the KKT conditions, has a structurethat facilitates the application of sparsity of techniques. Thlsmatrix is sparse and symmetric. It needs to compute and storeonly half of LU factorization due to symmetry. The matrixstructure is constant through iterations, the ordering andsymbolic analysis are done only once to create a static datastructure. Thus the numerical factorization is carried outefficiently at every iteration [8].

This test was accomplished with the following initialconditions: x = t , V ,e ) unconverged network solution. TheLagrange multipliers for the equality and inequality constraintswere, respectively, A = O p=O and n # O . The initialpenalties were defined as c = 1.0. All penalty increasingfactors were defined as p = 1.2. The process converged in 10iterations to a specified tolerance 5 = The amount ofreactive power generation was 363.60 MVAr, with a totalactive power loss of 27.55 MW. The load flow does not keepthe voltage magnitude within desirable limits on the loadbuses, with a total of 378.80 MVAr of reactive powergeneration and 28.9 MW of active power losses. The initialbarriers were defined as 6=0,0001. All barriers increasingfactors were defined as p= 10 .The optimization process forthis case is summarized in table 2


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TABLE I1OPTIMIZATION SUMMARYFOR 57-BUS SYSTEMIteration Active Mismatch Mismatch

power 1 ~ ~ 1 ~ ~Mw) l ~ ~ l ~ ~ ~MVAr)loss(MW)1 29.012 0.1646 0.18652345678910

28.39328.04727.66227.60227.57127.55827.55327.55127.550

0.36420.12180.01580.00230.00100.00040.00010.00000.0000

0.05710.04210.03190.02460.01580.00810.00230.00120.0001The results achieved in the tests show the viability of the use

of the logarithmic barrier in association with the augmentedLagrangian function.

V.CONCLUSIONSOptimal Power Flow is a large-scale nonconvex nonlinear

programming problem with nonlinear constraints. The paperpresents a new approach to the solution of this problem. Thelogarithmic barrier-augmented Lagrangian functioncorresponding to the original problem is solved by the Newtonmethod. The difficulty in identifylng the binding constraint setis removed by introduction of dual variables and quadraticpenalty terms into the augmented Lagrangian. The applicationof logarithmic barrier approach to voltage magnitude and tap-changing transformer variables improved the performance ofthe approach. The method can start from infeasible points andveri fj the increase of the penalty factors, thus avoiding the ill-conditioned Lagrangian matrix. The tests demonstrated theeffectiveness of the proposed approach. The hardest task inthis approach is to find the initial values of the barrierparameter for the voltage magnitude and tap-changingtransformer variables.

VI . REFERENCES[ I] J.L Carpentier, Contribution a 1Ctude du Dispatching Economique,Bull-Soc. Fr. Elec. Ser.B3, pp. 43 1-447, 1962.[2] S . Granville, Optimal Reactive Dispatch through Interior PointMethod, IEEE Transactions on Power Systems, vol. 9, no.1, pp.136-146, Feb. 1994.[3] Y.Wu, A.S.Debs, and R.E.Marsten, A direct nonlinear preditor-corrector primal-dual interior point algorithm for optimal power flow,IEEE Transactions on Power Systems, vol. 9, no.21, pp.876-883, May1994G.L.Torres and V.H.Quintana, An interior point method for nonlinearoptimal power flow using voltage rectangular coordinates, IEEETransactions on Power Systems, vol. 13, no.4, pp.1211-1218, Nov.1998G.R.M. da Costa, Optimal Reactive Dispatch through primal-dualmethod, IEEE Transactions on Power Systems, vol. 12, no.2, pp.669-674, May 1997.[6] M.H. Wright, Why a pure prinal newton bamer step may heinfeasible?, SIAM Journal on Optimization, vol. 5 , no. 1, pp. 1-12,1995.

[4]

[5]

[7][SI

M.R. Hestenes, Multiplier and Gradient Methods, JOTA, vol. 4,K. Zollenkopf, Bi-Factorization-Basic Computational Algorithm andProgramming Techniques, J.K. Reid ed.-Large Sparse Sets of LinearEquations, pp. 75-97, 1971.

pp.303-320, 1969.

VII. BIOGRAPHIESEdmCa C Baptista: Licentiate in Mathematics at the University of the Stateof SBo Paul0 (UNESP) at Bauru, SP, Brazil. Master in Applied andComputing Mathematics at the University of SBo Paul0 (USP) at SBo Carlos,SP, Brazil. EdmCa is currently an assistance professor in the Department ofMathematics, UNESP (Bauru campus), and studying for a Doctorate in theDepartment of Electrical Engineer ing, USP (SBo Carlos campus). Herresearch interests are in linear and nonlinear optimization.Edmarcio A. Belati received the electrical engineering degree fromFaculdade de Engenharia de Lins and M.S. degree in the Department ofElectrical Engineering-FEIS-UNESP. He is presently a Ph.D. student in theDepartment Electrical Engineering of SBo Carlos Engineering School ofUniversity of SBo Paulo. His research interests are power system operationand planning.Gerald0 R. M. da Costa received his B.S. and M.S. degrees in theDepartment of Electrical Engineering of SBo Carlos Engineering School ofUniversity of SHo Paul0 and Ph.D. degree at University of Campinas(UNICAMP). He is an associated professor of the Department ElectricalEngineering of SBo Carlos Engineering School of University of SBo Paulo.His research interests are power system operation and planning.

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A New Solution to the Optimal Power Flow Problem