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876 IEEE Transactions on Power S yste m. Vol. 9. No. 2.May 1994A Direct Nonlinear Predictor-Corrector Primal-Dual Interior Point Algorithmfor Optimal Power Flows

Yu-chi w u Atif S . Debs Roy E. MarstenStudent Member Senior Member School of Industrial andSchool of Electrical Engineering Systems EngineeringGeorgia Institute of TechnologyAtlanta, GA 30332, U.S.A.

Abstract -A new algorithm using the primal-dual interiorpoint method with th e predictor-corrector for solvingnonlinear optimal power flow (OPF)problems is presented.The formulation and the solution technique are new. Bothequalities and inequalities in the OPF are considered andsimultaneously solved in a nonlinear manner based on theKarush-Kuhn-Tucker (KKT) condit ions. The majorcomputational effort of the algorithm is solving asymmetrical system of equations, whose sparsity structureis fixed. Therefore only one optimal ordering and onesymbolic factorization are involved. Numerical results ofseveral test systems ranging in size from 9 to 2423 buses arepresented and comparisons are made with the pureprimal-dual interior point algorithm. The results show thatthe predictor-corrector primal-dual interior point algorithmfor OPF is computationally more attractive than the pureprimal-dual interior point algorithm in terms of speed anditeration count.Keywords: Optimal power flow, Interior point method,Nonlinear programming, Sparsity techniques, Predictor-corrector method.

1. IntroductioqThe optimal power flow procedure consists of methods

of determining the optimal steady state operation of anelectrical power generation-transmission system, whichsimultaneously minimizes the value of a chosen objectivefunction and satisfies certain physical and operatingconstraints. It is a typical nonlinear programming (NLP)problem and can be mathematically expressed as

This paper was presented at the 1993 IEEE Power IndustryComputer Application Conference held in Scottsdale, Arizona,May 4 - 7. 1993.

whereX : the set of state variables including power

generations (Pg nd Q,),bus voltages (V andti),tap ratios of transformers(t and $), etc.,

: a scalar function representing the operationperformance of a power system,fix)

g(x) : power flow equations,h(x) : functional inequalities of the power system,hU,hi : the upper and lower bounds of inequalities,xu,xl : the upper and lower bounds on variables, x.In the past three decades, various optimization

techniques were proposed to solve the nonlinear OPFproblem expressed as (1).They range from improvedmathematical techniques to more efficient problemformulations [ll.Several references 11-31 provide overviewsof solution metho ds in existence. These availablemathematical techniques can be categorized as (a)gradient techniques, (b) successive quadraticprogramming (SQP) techniques, (c) Kuhn-Tuckernonlinear progr amming (NLP) techniques, and (d )successive linear programming (SLP) techniques.

The gradient techniques [8,21] were the firstapproaches proposed to solve OPFproblems. Despite theirmathematical rigor, these approaches exhibit slowconvergence, especially zigzagging near the optimum. TheSQP approaches [9-111, roposed in the 1980's, use thesecond-order derivatives to improve the convergence rateof the gradient approaches. Their modeling is based on theQuasi-Newton process, in which the approximation of theHessian matrix of the Lagrangian function is used toovercome the difficulties encountered in QP problems.However, as in all Quasi-Newton methods, the reducedHessian, built iteratively, is dense, which may make thesemethods too slow as the number of control variablesbecomes very large. The NLP approaches, based on theKuhn-Tucker techniques, attempt to solve OPF by meetingthe Kuhn-Tucker optimality conditions directly. Althoughthey were proposed as early as 1962, few are either reliableor fast. Until Sun et al. [5] roposed a Newton approachcombining with Kuhn-Tucker, quadratic, and advancedsparsity techniques, people began to re-evaluate thisfamily of methods. The attraction of Sun's algorithm is thatwith special techniques the sparsity structure of the system

0885-8950/941$04.000 993 IEEE

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877is not affected by the penalty function which is used toenforce the binding constraints, and the problem can bedecomposed into an active-power subproblem and areactive-power subp robl em. The challenge in thi salgorithm development is to identify binding inequalities,therefore some heuristics are involved . The SLPapproaches [4,6,7] are based on the linearization of OPFconstraints and the objective function. These techniquesare mostly applied to the active-power reschedulingproblems, in which the controllable quantities a re to bescheduled to comply with transmission-system line-loadinglimits, and perhaps other transfer restrictions. The centralmodeling issue is the linear P- 0 representation of networkequations. An incremental linear model is adopted, andthe dual simplex algorithm is used. The reasons forchoosing the dual simplex algorithm are that the dualmethod has a simpler and less time-consuminginitialization, and that the storage required is less than thatof primal problem. Due to the fact that LP approachesprovide solutions jumping from one vertex to another,during the OPF procedure a successive refinement of costcurves into smaller segments is needed to overcome thediscontinuities.

Recently, du e to the efficiency of the newly developedinterior point methods for solving large-scale linearprograms (LP), they became candidates for manyapplications. Although the earliest literature of interiorpoint methods appeared in the early 1950's and they havebeen formally studied in detail by Fiacco and McCormick[19] since then, the big breakthrough of interior pointmethods was accomplished in 1984 when Karmarker [l8lannounced his polynomial-time algorithm for LP. Thetheoretical foundation for interior point methods consistsof three crucial building blocks 1141: Newton's method forsolving nonlinear equations and hence for unconstrainedoptimization, Lagrange's method for optimization withequalities, and Fiacco & McCormick's barrier method foroptimization with inequalities. Among the many variants ofinterior point methods, the primal-dual interior pointmethod [14-19,221 pro ves to be the most eleganttheoretically and the most successful computationally forLP. In [20], a different formulation based on the primal-dual interior point method is proposed for power systemstate estimation problems. In this paper we present a newdifferent algorithm using the primal-dual interior pointmethod and the predictor-corrector to directly solve OPFproblems in a nonlinear manner based on KKT conditions.Also, numerical comparison results are provided todemonstrate the superiority of this new presentedalgorithm.

In the next section, the pure primal-dual interior pointalgorithm (PDIPA) for OP F and the predictor-correctorprimal-dual interior point algorithm (PCPDIPA) for OPFare described in detail . The implementation an dcomputational issues are discussed in section 3. Section 4

presents numerical comparison results using the newpresented algorithm and PDIPA. Conclusions and futureresearch are made in section 5.2 Direct Nonlinear Interior Point Aleonthm s for OPF

Although several papers 114-17,221 were published inthe past few years to address the formulation and theperformance of the primal-dual interior point method forLP problems, the use of the primal-dual interior pointmethod to directly solve large-scale nonlinear systems isstill in the development stage. In this section, two differentformulations based on the primal-dual interior pointmethod for solving the nonlinear OPF problems arederived and discussed in detail. They are the pure primal-dual interior point algorithm for OPF, and the predictor-corrector primal-dual interior point algorithm for OPF.

Consider the general form of the OPF problem (11, andthe transformed OPF problem- (2 ) by introducing slackvariables sh, ssh, and sx.

The nonnegativity conditions on (x-xl) and those slackvariables in (2) can be treated by appending thelogarithmic barrier functions to the objective,

(3)where n and rn are the numbers of state variables, x, andinequality constraints respectively, and the barrierparameter p is a positive number that is enforced todecrease towards zero iteratively. Based on the Fiacco &McCormick's theorem [19], as jt tends towards zero, thesolution of the subproblem, x(p), approaches x*, thesolution of (2). The resultant Lagrangian function of thesubproblem with barrier functions, therefore, is

(4)wherey, yh,ysh and yx are Lagrangian multipliers (dualvariables) for constraints (2.a), (2.b), (2x1, and (2.d)respectively. The stationary point of (4) is, thus, the optimal

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878solution of the subproblem, which satisfies Karush-Kuhn-Tucker (KKT) irst-order conditions:v x p =vf(X)- g ( X f y+Vh(X)Tyh+Y x -P(x- i>-'e

= O (a)VshLp=Yh+Ysh-&e = o (b)vs, L p = Ysh - @ z e

vy , = g (x >v L p = x + s x - x u

= o (c)= O (d)s xL, = Y x -&e= O (e) (5)= o 0y s , , L p = h ( x ) + s h - h

y x = o (g)vyh, = Sh +Ssh -hu +h i = O (h)where e=[l, .., lT , X=diag(xl ,...Xn), Sh =diag(shl,..., b ) ,Ssh=diag(Ssh1 .. ,S s h m1 SX =diag(sx1 .. ,Sx n1. Thesenonlinear equations are then solved by the Newton'smethod. The new approximation to the new minimizer, (x,Sx l Sh, SshlZ,YIYxIYhlJ'sh) is determined by (61,

xk+l =x k + a hs y s,"+ aAs,kk+l kssh = ssh ahsh

s h + a A s h

Z k + l = k +aA 2where the scalar step size a is chosen only to preserve thenonnegativity conditions on (x-xi),sx, sh, sSh,z,yx, ysh, and

Instead of taking several Newton steps to converge to theoptimal point of the subproblem with fixed p, at everyiterationp is then reduced and the problem is relinearized.This is the main fea ture of the presented algorithmdifferent from conventional SLP/SQP-based algorithms.

Depending on the approximations used in the Newton'smethod, two variants of the primal-dual interior pointalgorithms for OPF can be derived. They are discussed inthe following subsections.

(YhVsh).

Pure Primal-Dual Interior Point Alp o h PDIPA)By proper transformation and adding one extra equation

z=~O(-XI e, (7)the nonlinear equations (5.a)-(5.d) and (7) can be re-writtenas

( a > = ; , V f ( X ) - V g ( x ) T y + V h ( x > T y h + y x - z = O (8.a)( b )* h (yh+ ysh >e = pe (8.b)(cl* shyshe = pe (8.c)(4;,SxYxe = pe (8.d)

= pe (8.e)where equations (8.b)-(8.e) are the approximations of thecomplementary slackness conditions and equation (8.a)represents the set of dual constraints. If p=O, then (8.bH8.e)correspond to the ordinary complementary slacknessconditions.

By taking the first derivatives of (5.e145.h) and (81, thefollowing symmetric system of equations is obtained:

where H is the Hessian matrix of the Lagrangian function.Because in (9 ) the barrier parameter p appears in the

right hand side of equations, the factorization of the matrixon the left hand side of (9) is not affected by p. This featuremakes th e predictor-corrector method easil y beincorporated into the algorithm, deriving the predictor-corrector primal-dual interior point algorithm which will bediscussed in next subsection.

The outline of the PDIPA is as the following:Step0 : Initialization

Choose a proper starting point such that thenonnegativity conditions are satisfied.Step1 Compute the barrier parameter, p.

Step2 : Solve the system of equations (9).Step3: Determine the step size, a, nd update the

solution.Step4 : Convergence tes tIf the solution meets the convergence criterion,

optimal solution is found, otherwise go back tostep 1.

f i e Predictor-Corrector Primal-Dual Interio r Po&Algorithm for OPF (PCPDIPA)

In PDIPA, only the first-order linear terms aremodeled. However, rather than applying the Newton'smethod to KKT conditions to generate correction terms tothe current estimate, we can substitute the new point intoKKT conditions directly, yielding

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879&?,A.?,, A.?), ,GSh,Ai, Ajx,Ajh,Aj sh are then used toapproximate the nonlinear terms in the right hand side of(ll), nd to dynamically estimate p. Once the estimates ofnonlinear terms in the right hand side of (11) and p aredetermined, the actual new search direction (Ax, Asx, Ash,ASsh, Az,Ay, Ayx, Ayh, AYsh) are obtained by (11)

The outline of this algorithm can be stated as thefollowing:

Step0 : InitializationChoose a proper starting point such that thenonnegativity conditions are satisfied.

Step1 Solve the system of equations (12).Step2: Compute the barrier parameter, p, and the

estimated nonlinear terms.Step3 : Solve the system of equations (11).Step4: Determine the step size, a, and update the

solution.Step 5 : Convergence test

If the solution meets the convergence criterion,optimal solution is found, otherwise go back tostep 1.

where A x , Asx,AY,, AYh, A Y & , Ash, Assh and Az arediagonal matrices having elements of Ax, As,, Ay,, Ayh,Aysh, As h, As, h, and Az, respectively. The followingsymmetrical system is obtained:. -z-'(X-x,)

000000-I0

000

s-,'y,0I000

0 0 00 1 II O 00 1 00 0 00 0 00 0 0I 0 -VhT0 0 0

The major difference between (9) and (11)is the presenceof the nonlinear terms AXAZe, ASh(AYh+AYsh)e, ASxAYxe,and ASshAYshe in th e right hand side of (11).Thesenonlinear terms cannot be determined in advance,therefore(11)only can be solved approximately. In order toestimate these nonlinear terms, Mehrotra [15] suggestsfirst solving the defining equations for the primal-dualaffine directions:

In this section, several implementation issuesassociated with the algorithms presented in section 2 arediscussed in detail: the adjustment of the barrierparameter and the determination of the Newton step size,the stopping criterion, accuracy, the choice of the startingpoint, and sparse matrix techniques.

The Adjustment of the Barrier Parameter and theDetermination of the Newton's Step SizeBased on the Fiacco and McCormick's theorem, the

barrier parameter p must approach zero as the iterationsprogress. The primal-dual method itself suggests how pshould be reduced from step to step. For linearprogramming problems [14-17,221, the value of p s madeproportional to the duality gap, the difference between theprimal and dual objective functions. The duality gap of thenonlinear problem (2) defined as

(13)ap=YT&)+ YhT [hu @)I +YL(h, -4)T+y, (xu- x)+ zT (x -XI)is a positive quantity if the primal and dual variables meetall the primal and dual constraints and is zero at theoptimum point. However, due to the fact that the primaland dual variables are not feasible, the value of gup maynot be positive. We, therefore, use the complementary gapto approximate the duality gap

gap*= (Yh +ysh)T h +y$$& + + T (x-XI) (14)and by following Lustig [161 we chooseh + =h, - l -Sh - sh

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880

for the pure primal-dual interior point algorithm. For thepredictor-corrector prima ldual interior point algorithm,

(16)where

And the Newtons step size, a, s determined asa min I0.9995a*, .0) (19)where

a*=-(---- (x-x!>j (%)j -- (ssh)j -- (sh)j& j (b x) j (&sh)j (.bsh)j (20)-----Yx)j (Ysh)j (Ysh+Yh)j -5)

(AYx)j (AYYsh)j (AYsh+AYh)j A z jfor those A X

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881of the largest power mismatch (in pu.) is less than q= 103,and the absolute value of the largest dua l constraintmismatch is less than &3=104. The results of iterationcounts required to solve 9-bus to 244-bus systems arereported in Table 2 and Table 3 reports CPU times of allOPF runs, in which the numbers inside the parenthesesare, respectively, the iteration counts and CPU times byPDIPA. Table 4 lists the numbers of binding constraints forthose test cases with line flow constraints.

CaseName9-buS

workstations. By exploiting the memory hierarchy, itprovides better performance than conventional ones.

4. Test ResultsIn our OPF model, all voltage magnitudes and all

voltage angles except for swing bus voltage angle aremodeled as bounded variables. Both real and reactivepower flow equations are included. The optimal orderingscheme used in our implementation is the multipleminimum external degree algorithm. The algorithms wereimplemented in FORTRAN and compiled with - 03 option.Several different systems ranging in size from 9 to 2423buses are tested to evaluate the performance of th ealgorithm. Two basic types of OPF problems are tested:cost minimization and loss minimization. The costminimization OPF problem is to minimize the total cost ofpower generations, while the loss minimization OPF is tominimize the transmission losses. Quadratic cost curvesare used in cost minimization problems. Also, for 9-bus to244-bus test systems two sets of constraints are considered:the one consisting of only power flow equations and simple2-sided bounds on variables, and the one consisting of bothpower flow equations and line flow constraints and simple2-sided bounds on variables. All testing is done on the SunSparc 1workstation. Table 1 ists the specifications of thesetest cases.

without withmin min min mincost losses cost losses

line flow constraints line flow constraints

0.53 (0.70) 0.53 (0.64) 0.60 (0.73) 0.65 (0.75)

Table 1:Specifications of Test cases

30-bus39-bus118-bus

Case Name puses ILines D-Controls ICurve9-bUS I 9 1 9 1 3 1 3

. . . . . ,1.33 (1.43) 1.29(1.40) 2.20 (2.46) 1.651.68 (2.24) 1.83 (1.89) 2.23 (2.77) 2.165.05 (6.85) 5.23 (6.86) 7.27 (9.68) 6.38

462423-bus I 2423 I 3069 I 221 1 221

Table 2: Number of Iterations -- PCPDIPA vs. PDIPA

constraints constraints

. . . . . .30-bus 9(11) 8(10) 12(16) 8(12)39-bus 9(15) 10 (12) lO(16) 10 (14) I11 118-bus i 10(17) i IO (16) i 13(22) i 11 (17) 11244-bus I 12f23) I 13f241 I 13f231 I 131231

There are total 24 cases for testing the performance ofPCPDIPA. Among these test cases, two are based on a2423-bus system. Each case is then solved by PCPDIPA andPDIPA for comparison purpose. The iterations in thealgorithm a re terminated when the relativecomplementary gap is less than q=104 , the absolute value

Table 3: CPU Times -PCPDIPA vs. PDIPA (seconds)

. .44-bus 116.82 (24.80)117.74 (27.261117.87 (26.29)119.08 (27.9

Table 4 Numbers of Binding Constraints for Cases withLine Flow Constraints

IICase I min cost I min loss II

In order to see how the PCPDIPA responds to smallchanges in the constraints, two runs were performed basedon the 244-bus system, one for cost minimization and theother for loss minimization. The changes made in the firstrun are 1%decrease and increase respectively on voltageupper and lower bounds and 5MW decrease and increaserespectively on line flow upper and lower bounds. In thesecond run, 2% changes are made on voltage limits andlOMW changes on line flow bounds. The requirediterations and CPU times and th e numbers of bindingconstraints are reported in Table 5.

Table 5: Iterations, CPU times, and Numbers of BindingConstraints for 244-bus Systemwith Changes on Bound Limits

min cost I min lossiterations: 13. CPU: 17.39sec literations: 14 .CPU 1 9 . 3 1 ~ ~

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882

0.600E+01-0.133E+050.504E+01-0).129E+04o.494E+01-0.392E+020.488E+01-0.790E+010.466E+010.165E+010.452E+OI0.301E+010.446E+01

The results of Table 5 show that the PCPDIPA solvesthe changed cases as effectively as it solves the originalcases. Table 6 lists the convergence process of the lossminimization OPF run for the 244-bus system with line flowconstraints. After 13 iterations, the duality gap is less thanlo4 , he relative duality gap is less than the norms ofprimal and dual constraints are less than l o 5 .

0.134E+05 0.677E+Ol 0.178E+030.100E+01 0.125E+01 0.100E+010.130E+04 0.763E+00 0.187E+020.100E+01 0.146E+00 0.253E-030.442E+02 0.257E-01 0.900E+000.110E+01 0.502E-02 0.800E-020.128E+02 0.708E-01 0.139E+000.143E+01 0.414E-02 0.386E-030.299E+01 0.165E+00 0.107E+oO0.113E+01 0.595E-01 0.320E-030.150E+00 0.148E+00 0.468E-010.374E+oO 0.435E-01 0.383E-030.658E+00 0.524E-01 0.103E-01

Table 6: Convergence Process of Loss Minimization for244-bus System

1112

134**

~ 7

0.428E+01 0.286E-01 0.283E-02 0.994E-070.442E+Ol 0.371E-01 0.170E-01 0.895E-030.439E+01 0.688E-02 0.811E-03 0.244E-070.442E+Ol 0.455E-02 0.117E-01 0.1%E-030.442E+Ol 0.839E-03 0.113E-03 0.300E-080.442E+Ol 0.222E-04 0.178E-03 0.174E-040.442E+01 0.155E-06 0.962E-06 0.159E-050.442E+Ol 0.287E-07 0.346E-050.442E+01 0.409E-05 0.108E-05 0.146E-10

Iterations 26 43 18ICPU Time 279.85sec 370.15sec 221.87sec k65.31sec

Two other cases based on a 2423-bus system weretested for performance comparison between the PCPDIPAand PDIPA. The iterations and CPU times are reported inTable 7.

Table 7: Iterations & CPU Times for 2423-bus System

1 min cost I min lossPCPDIPA I PDIPA IPCPDIPA I PDIPA

The results in Tables 2, 3, 7 show that the PCPDIPAprovides considerably better performance in solvingnonlinear OPF problems than PDIPA. This is due to thefact that in PCPDIPA the second-order terms areconsidered with only little additional computational effortinvolved (one additional forward/backward substitution atevery iteration).

5. Conclusions an d Future ResearchA new algorithm and a new formulation using the

primal-dual interior point method with the predictor-corrector for the nonlinear optimal power flow problemshave been presented in detail. The computationalefficiency is du e to the spars ity of the Hessian of theLagrangian with respect to both primal and dual variables.The functional inequality constraints are directly treated inthis algorithm. Only one optimal ordering and onesymbolic factorization are needed, and one numericalfactorization and two forward/backward substitutions areinvolved at every iteration. Since the second-order termsare considered, the convergence of PCPDIPA is faster thanthat of PDIPA. The computa tional experience reportedhere shows that the algorithm is attractive.

Further work on the current algorithm falls under twocategories. First, there is the need to address such keyrefinements as infeasibility detection, hot starting,decomposition, constraint relaxation, all of which arerelated to the nature of this approach - being inherently abarrier function optimization approach. Second, thealgorithm will have to be extended to the challengingproblems of security constrained OPF problems, OPF-based external equivalencing, discrete controls, hydro-thermal coordination, etc..

Due to the structure of system (ll), t is possible toembed the decoupling schemes into the presentedalgorithm, like the conventional power flow or the Newton'sOPF. Moreover, partial refactorization or compensationmethod is also applicable to improve the performance.Since the primal-dual interior po int method convergesquickly in the early stages of iterations and slowly nearoptimal solution, the changes in the values of variables inthe latter stages are small. Therefore, partial refactorizationor compensation method can be applied in the latter stageof iterations to reduce the computational effort.

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BiographiesYu-Chi Wu (Student Member, IEEE) was born in Taiwan,the Republic of China. H e received the ElectricalEngineering Diploma from the National KaohsiungInstitute of Technology, Taiwan in 1984. Since 1988, he hasbeen a graduate student in the School of ElectricalEngineering, Georgia Institute of Technology, workingtowards the Ph.D. degree. He was a research assistant atNational Sun Yat-Sen University, Taiwan from 1986-1988and worked for Pacific Gas and Electric Company duringthe summers of 1989, 1990, and 1991, and for EnergyManagement Associates, Inc. during the summer of 1992.Currently, he is a research assistant at Georgia Institute ofTechnology. His primary research interests are in the areasof optimization and planning of electrical power systems,including optimal power flow, security analysis, unitcommitment, maint enance scheduling, productionsimulation, load forecasting.Atif S. Debs (Senior Member, IEEE) is a professor ofElectrical Engineering at the Georgia Institute ofTechnology, Atlanta, GA since 1972 where he became theco-founder of the Georgia Tech Electric Power Program. H eobtained his S.B., S.M., and Ph.D. degrees in 1964,1965, and1969 respectively, at MIT, Cambridge, Massachusetts. Hisfields of interes t are in the a reas of control centerapplications and power system planning including stateestimation, static and dynamic security assessment,production simulation, artificial neural networkapplications, and others. He is the author of a 1988textbook entitled, Modern Power System Control andOperation: A Study of Real-Time Power U tili ty ControlCenters as well as many technical papers in the abovefields. He is an active officer in both the Power Engineeringand Control Society.Ro y E. Marsten received his Ph.D. in OperationsResearch from UCLA in 1971. He taught at NorthwesternUniversity ,MIT, and the University of Arizona beforemoving to Georgia Institute of Technology. His expertise isin computational optimization. He shared, with DavidShanno and Irvin Lustig, the 1991 Beale/Orchard-HaysPrize for Excellence in Computational MathematicalProgramming awarded by the Mathematical ProgrammingSociety.

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