Rabih2002 a Primal-dual Interior Point Method for Optimal Power Flow Dispatching

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654 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 3, AUGUST 2002

A PrimalDual Interior Point Methodfor Optimal Power Flow Dispatching

Rabih A. Jabr, Alun H. Coonick, and Brian J. Cory

AbstractIn this paper, the solution of the optimal powerflow dispatching (OPFD) problem by a primaldual interiorpoint method is considered. Several primaldual methods foroptimal power flow (OPF) have been suggested, all of whichare essentially direct extensions of primaldual methods forlinear programming. The aim of the present work is to enhanceconvergence through two modifications: a filter technique to guidethe choice of the step length and an altered search direction inorder to avoid convergence to a nonminimizing stationary point.A reduction in computational time is also gained through solvinga positive definite matrix for the search direction. Numerical testson standard IEEE systems and on a realistic network are veryencouraging and show that the new algorithm converges where

other algorithms fail. Index TermsOptimization methods, power generation dis-

patching, second-order condition, step length control.

I. INTRODUCTION

OPTIMAL POWER FLOW DISPATCHING (OPFD) is an

optimization problem which minimizes the total genera-

tion dispatch cost while satisfying physical and technical con-

straints on the network. Primaldual interior point methods for

optimal power flow (OPF) have recently been discussed [1][6].

All these methods have been motivated by the success of interior

point methods for linear programming. Wu et al. [1], [2] estab-

lished that primaldual interior point methods offer an attractivesolution to the OPF problem. Their method was based on polar

coordinates and investigated two different ways of updating the

barrier parameter. At the same time, Granville [3] independently

proposed a similar method for optimal reactive power dispatch.

Through the concept of centering directions, Wei et al. [4] uni-

fied the OPF, the approximate OPF, and the classical power flow

into a single optimization problem. These authors also presented

a novel data structure to reduce fill-in during factorization, and

reduce computational time when using rectangular coordinates.

Torres and Quintana [5] compared the polar and rectangular co-

ordinates version and concluded that both perform equally well.

Very recently, Castronuovo et al. [6] presented an OPF solution

with high-performance computational techniques. Here, the useof vector techniques in order to enhance computational speed

is proposed. The solution of linear equations is done through

sparse LU factorization and a modification of the Tinney II

(minimum degree ordering) heuristic.

Manuscript received October 12, 1999; revised October 1, 2001.R. A. Jabr is with the Department of Electrical, Computer and Communica-

tion Engineering, Notre Dame University, 72 Zouk Mikayel, Lebanon.A. H. Coonick and B. J. Cory are with the Department of Electrical and Elec-

tronic Engineering, Imperial College, London SW7 2BT, U.K.Publisher Item Identifier 10.1109/TPWRS.2002.800870.

Although these methods [1][6] proved to be efficient for

solving OPFD problems, they lack a technique which induces

convergence and neglect the second-order sufficiency condi-

tions [7], [8] which are needed to prove solution optimality.

An algorithm for nonlinear nonconvex programming which

does not check for the second-order conditions can leave

the user unsure about the outcome of the optimization. This

has motivated Almeida et al. [9] to propose the parametric

OPF that tracks the trajectory of the solution using Newtons

method while satisfying the second-order KuhnTucker (KT)

conditions. In this paper, we discuss two modifications used

to convert the linear programming interior point method tononconvex nonlinear OPFD problems. These modifications are

aimed at overcoming the above-mentioned disadvantages while

keeping the OPFD algorithm efficient. Moreover, an increase

in efficiency was achieved by formulating the OPFD through

an inequality form which leads to a sparse symmetric positive

definite matrix. Symmetric positive definiteness is the highest

state a matrix can aspire to be [10], since it permits economy

and numerical stability in the solution of linear systems. The

proposed algorithm is compared with a similar version of the

interior point OPF algorithm presented in [6].

This paper is organized as follows. Section II introduces the

problem and Section III presents the optimality conditions. The

primaldual interior point method theory and implementationare given in Sections IV and V. Special emphasis is given to

the treatment of indefiniteness in Section VI. The filter tech-

nique, which guides the choice of the step length, is introduced

in Section VII. Section VIII explains the choice of the barrier

parameter followed by a pseudocode summary of the algorithm

in Section IX. Section X highlights the main features of the

method used as a benchmark for comparison. Numerical results

are given in Section XI. The paper is concluded in Section XII.

II. OPFD PROBLEM FORMULATION

We are interested in the nonlinear nonconvex optimizationproblem with inequality constraints

minimize subject to and (1a)

where , , , and

. The functions and are continuous and smooth

(i.e., with first and second continuous derivatives).

The objective function is the total generation dispatch cost.

The cost functions are assumed to be convex and piecewise

linear. They are modeled in the optimization problem using

0885-8950/02$17.00 2002 IEEE
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TABLE INOMENCLATURE USED IN (1b)

separable programming [11]. The inequality constraints in (1a)

represent

(1b)

The symbols used in (1b) are explained in Table I.

Section II-A presents the formulation of the regulating trans-

former model. The following points concerning the OPFD

formulation need to be observed.

1) The active power balance equations are traditionally ex-

pressed as equality constraints [1][6]. Our formulation

in (1b) uses inequality constraints for reasons which will

become clear in the following sections. Note that since the

cost curves are positive, the optimization process of min-

imizing the cost will reduce the generation level to meetthe total load. If the optimal solution leads to over satis-

faction of the load, then either the OPFD problem is in-

feasible or the system has a lower cost if it is dispatched

at a slightly higher load (which is not practical). Irving

and Sterling [12] have used a similar inequality form in

the context of economic dispatching.

2) To enforce line flow constraints properly, the real line

flow power is checked at both the sending and receiving

end of every line. The positive flow on a line has the

higher magnitude (since the sum of flows is equal to the

real losses which is nonnegative) and can therefore be

limited by the network capacity.

Fig. 1. Transformer model.

Fig. 2. Equivalent transformer model.

3) The generator active limits are not explicitly expressed

but are taken into account through the separable program-

ming approach [13].

A. Regulating Transformer Model

A regulating transformer has the capacity to regulate both

voltage magnitude and phase angle [14]. Fig. 1 shows the one

line diagram of the transformer where is a complex

ratio. A direct consequence of this is that the bus admittance

matrix ( bus) becomes unsymmetrical [14].

Our approach for modeling the regulating transformer allows

its easy incorporation into an OPF-based program without al-

tering the functions that evaluate the Jacobian and Hessian ma-

trices corresponding to the power flow equations. For a trans-

former branch, we introduce a voltage controlled bus (slave),

as shown in Fig. 2. The control is done through the master bus

according to the following constraints:

(1c)

where ( and ) are the lower and upper tap limits, re-

spectively, and ( and ) are the lower and upper phase

shift limits, respectively. Moreover, we ensure in the optimiza-

tion process that the transformer power extracted from is

injected into .

III. OPTIMALITY CONDITIONS

The Lagrangian function for (1a) is

(2a)

where and . The gradient of is denoted

by and the Jacobian of by . The Hessian

of the Lagrangian [with respect to ] is

(2b)

Let denote a point where the following conditions

hold.
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Together, (3o) and (3p) specify a linear

system which can be used to solve for the Newton steps

.

V. SOLVING THE PRIMALDUAL SYSTEM

As noted in Section IV, the solution of the linear systems (3o)

and (3p) does not take into account the positive definiteness con-

ditions in (3k). A robust second derivative method to (1a) must

not only be able to check the second-order KT conditions (con-

dition 3 in Section III), but also to move away from nonmini-

mizing stationary points [17], [18]. The particular choice of the

standard form (1a) makes it straightforward to check positive

definiteness of the reduced Hessian. We describe our approach

here.

We start by solving for and from (3p)

(4a)

and substitute (4a) in (3o) to get the augmented system

(4b)

where

We further solve for from the second equation in (4b)

(4c)

and substitute back into the first equation of (4b) to get the

normal system

(4d)

where

Our algorithm solves for the search direction

first by solving the normal system (4d)

for and then by evaluating the other components of

the search direction from (4c) and (4a).

VI. SECOND-ORDER KT CONDITIONS

A minimizing solution of (1a) is distinguished from a non-

minimizing stationary point by positive definiteness of the re-

duced Hessian. Since there is no identification of the binding

constraints during an interior point iteration, it is not obvious

how this condition can be checked in practice. However, Gay

et al. [17] show that as we approach the optimal solution,

[in (4d)] can be used to provide guidance about the eigenvalues

of the unknown matrix (condition 3 in Section III)

without any need to predict the active set. In other words, the

algorithm can make decisions about possible indefiniteness of

the reduced Lagrangian Hessian near the solution by checking

the positive definiteness of . The same test is also used by

Vanderbei et al. [18].

A. Treatment of Indefiniteness

In Section VI, we have seen that matrix can be used tocheck positive definiteness of the reduced Lagrangian Hessian.

If is not positive definite, the iterates are likely to converge to

a nonminimizing stationary point. Consider the simple example

from [18] of minimizing a concave function subject

to the bound constraints . The algorithm presented in

[1][6] when applied to solve this problem starting from

, converges to the global maximum at 0.5.

In our algorithm, whenever is indefinite, we replace it by

its 2-norm positive approximant [10]

(5a)

where is the identity matrix and

is positive definite . A warning associated with this perturba-

tion is that dual infeasibility may fail to decrease even with arbi-

trarily small steps [18]. This is the price to pay in order to obtain

positive definiteness of and descent of the barrier function.

However, empirical evidence suggests that is zero most

of the time and that the dual infeasibility does decrease to zero

close to a minimizer since ultimately becomes positive defi-

nite [15], [17], [18].

The value of is obtained through the bisection method

[10]. The idea is to find an interval containing

. If is positive definite where , ac-

cept containing , otherwise accept . This process

is repeated until the desired accuracy is reached as

(5b)

where is a relative error tolerance, typically 5 10 .

To initialize this procedure, we set to the computer tolerance,

and to the Frobenius norm of . The Frobenius norm is a

weak upper bound, but it is used to avoid the calculation of the

minimum eigenvalue of . Testing for positive definiteness is

done through the Cholesky decomposition of . The matrix

is declared positive definite if the process succeeds. Sparsity

techniques and minimum degree ordering are employed in com-

puting the Cholesky factorization [19].

VII. CHOICE OF STEP LENGTH

If started far from a solution, primaldual methods the iter-

ates of which are updated based on the ratio test [1][6] may

fail to converge to a solution as illustrated by the minimization

of s.t. using an initial esti-

mate . For this reason, primaldual methods usually use

a merit function in order to induce convergence [18]. There are,

however, problems associated with the merit function, particu-

larly with the choice of the penalty parameter [20]. This has led

to the use of methods such as the watch-dog strategy [21] in

which the merit function is allowed to increase a limited number

of iterations.
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On the other hand, empirical evidence [20], [18] suggests that

Newtons method should be hindered as little as possible. Moti-

vated by this evidence, we adapt the filter method to the choice

of the step length in the primaldual method. This method was

originally proposed by Fletcher and Leyffer [20] for setting the

trust-region radius in sequential quadratic programming. The

idea is to interfere as little as possible with Newtons method

but to do enough in order to give bias toward convergence.

A. Filter Technique

There are two competing aims in the primaldual solution of

(1a). The first aim is to minimize the objective, and the second

is the satisfaction of the constraints. Keeping in mind that the

positivity of iterates is easily maintained using the ratio test,

these two conflicting aims can be written as

minimize (6a)

and

minimize (6b)

A merit function usually combines (6a) and (6b) into a single

objective. Instead we see (6a) and (6b) as two separate objec-

tives, similar to multiobjective optimization. However, the situ-

ation here is different since it is essential to find a point where

if possible. In this sense, (6b) has priority. Nevertheless,

we will make use of the principle of domination from multi-

objective programming in order to introduce the concept of the

filter.

Definition 1 [20]: A pair is said to dominate an-

other pair if and only if and .

In the context of the primaldual method, this implies that

the th iterate is at least as good as the th iterate with respectto (6a) and (6b). Next, we define the filter which will be used in

the line search to accept or reject a step.

Definition 2 [20]: A filter is a list of pairs such that

no pair dominates any other. A point is said to be ac-

cepted for inclusion in the filter if it is not dominated by any

point in the filter.

The filter therefore accepts any point that either improves op-

timality or infeasibility. Fig. 3 shows the filter graphically in

the plane. Each point defines a block of nonacceptable

points. The union of these blocks represents the set of points not

acceptable to the filter.

We follow most primaldual methods in allowing separate

step lengths for the primal and dual variables [22]. A standardratio test is used to ensure that nonnegative variables remain

nonnegative

(6c)

where

if

(6d)

Fig. 3. Graphical representation of the filter.

The step lengths (6c) are successively halved until the iterate

(6e) becomes acceptable to the filter

(6e)

VIII. CHOICE OF THE BARRIER PARAMETER

An important issue in the primaldual method is the choice

of the barrier parameter. Many methods are based on approx-

imate complementarity where the centering parameter is fixed

a priori [17]. Mehrotra [23] suggested a scheme for linear pro-

gramming in which the barrier parameter is estimated dynam-

ically during the iteration. Owing to its success [24], in our

algorithm we follow the heuristic originally proposed in [23].

First, the Newton equations system (see Section V) is solved

with the barrier set to zero. The direction obtained in this

case is called the affine-scaling

direction. The barrier parameter is estimated dynamically from

the estimated reduction in the complementarity gap along the

affine-scaling direction

(7a)

where

The step lengths in the affine-scaling direction are obtained

using (6c) and (6d). To avoid numerical instability, we define

by (7a) when the absolute complementarity gap ,

but when , we define

(7b)

We also make use of second-order correction for the comple-

mentarity condition [1], [2], [4], [5], [24].
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IX. PSEUDOCODE SUMMARY OF THE ALGORITHM: MPCIP

The algorithm presented will be referred to as the modified

predictorcorrector interior point (MPCIP). The quantities

itmax and lsmax are limits on the maximum number of itera-

tions in the main loop and maximum number of steps in the

filter search loop, respectively. In this implementation, itmax

100 and lsmax 25. The quantity is the exit tolerance and

is set to 10 .

Algorithm MPCIP:

initialize ;

given a starting solution , set

, and

;

initialize the filter;

iter 0;

repeat

iter iter 1;

if iter 1 then

perform minimum degree ordering of

end

compute the Cholesky decomposition of

;

obtain by modifying if necessary;

set and solve for the affine-

scaling direction:

from (4d), from (4c) and

from (4a);

obtain the primal and dual steps in theaffine scaling direction from

(6c) and (6d);

obtain using (7a) and (7b);

obtain the actual search direction

taking into account the second-order

correction terms: from (4d),

from (4c) and from (4a);

obtain the primal and dual steps in the

actual direction from (6c) and

(6d);

lsteps 0

repeat

compute a trial pointif the trial point is dominated by

the filter

and

else update the filter

break

end

lsteps lsteps 1

until lsteps lsmax;

update the step using (6e);

until(pfeas tol and dfeas tol and

opt tol) or iter > itmax or lsteps

lsmax.

TABLE IITEST PROBLEM STATISTICS

Needless to say, the same decomposition of is used in com-

puting the affine-scaling and actual directions. Note that in the

above algorithm, the relative primal infeasibility, dual infeasi-

bility, and optimality used to set up the stopping criteria are de-

fined as

(8a)

(8b)

(8c)

where

X. OTHER INTERIOR-POINT METHODS

Primaldual interior point methods have been applied before

to the solution of the OPF [1][6]. We have chosen to compare

our work with the most recently presented algorithm [6], which

mainly differs from the others by the choice of the linear

equations solver. In [6], Castronuovo et al. propose storage of

the augmented system by sparse lists as opposed to compact

form by blocks. This type of storage reduces the number of

floating operations, because only nonzeros are considered,

while in block storage, explicit inversion of the blocks with

zero element is required. The solution to the augmented systemis obtained through LU factorization with minimum degree

ordering modified to dynamically evaluate the condition of the

pivot. MATLAB [19] offers a similar function which allows

column minimum degree ordering to be performed at the

beginning of the solution together with dynamic partial pivoting

for numerical stability. To make the comparison fair, we in-

clude using the method presented in [6], the same heuristic for

evaluating the barrier parameter and second-order correction

step as explained in Section VIII. Different primal and dual

steps are used [1][5]. The resulting method is implemented in

MATLAB and referred to as predictor corrector interior point

(PCIP) method.
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TABLE IIIOPFD COMPUTATIONAL RESULTSPEAK DEMAND (NUMBER IN BRACKETS IS THE

STARTING POINT FOR CONSTRAINED OPFD WITH REGULATING TRANSFORMERS)

TABLE IVOPFD COMPUTATIONAL BEHAVIORTEN LOAD LEVELS

XI. NUMERICAL RESULTS

This section presents some numerical results obtained with an

implementation of the MPCIP algorithm. The algorithm is tested

on eight different networks each dispatched at ten different load

levels. The MPCIP is compared to the PCIP referred to in Sec-

tionX.AllroutinesarewritteninMATLABandrunonaPentium

II 400 MHz PC with128 Mb RAM. Table IIshows a summary of

thetestproblems.AllthesystemsarefromtheIEEEstandardsex-

cept the 175 Bus and the 175 Bus E (with an embedded network)

systems whichare derivedfrom an actual network.

The performance of the algorithms is tested starting from two

different values of the interpolatory variables but always with

a flat voltage profile and zero bus angles. Results on two dif-

ferent types of problems are reported:

1) constrained OPFD i.e., with the enforcement of line flow

constraints;

2) constrained OPFD with regulating transformers and line

flow constraints.

The results of theOPFD at peak demandare given in Table III,

which shows that with the MPCIP, we could solve all of the

test systems. For the PCIP, convergence was not attained forthe 57 Bus and 300 Bus systems even when starting from dif-

ferent points. The number of binding constraints obtained from

the OPFD solutions in Table III were monitored throughout the

simulations. The 300 Bus system in particular showed that the

MPCIP is able to detect a high number of active constraints.

To assess the relative performance of both methods, each of

the eight systems was dispatched at the ten load levels obtained

from a load duration curve and starting from two different ini-

tial points. When convergence was attained, the algorithms al-

ways reached the same schedule solution, within the tolerance

error. Table IV shows the percentage of success for each of the

methods together with number of iterations in which the filter


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Fig. 4. CPU times of MPCIP as a proportion of PCIP times.

Fig. 5. Iterations of MPCIP as a proportion of PCIP iterations.

technique (nF) and the estimation of diagonal perturbation (nE)was invoked. A problem number is also assigned for ease of

identification. Failures in the PCIP occurred for cases 8, 15, 16,

22, 29, and 30. Reasons for this can be attributed to the indefi-

niteness of the reduced Lagrangian Hessian, which this method

ignores. For the constrained OPFD cases, with or without regu-

lating transformers, the MPCIP algorithm showed a higher de-

gree of robustness. Failures in this case occurred only in the 57

bus system starting from the same point. This was due to a high

diagonal perturbation at the beginning of the iterations, which

prevented the dual infeasibility from decreasing to zero. How-

ever, starting from a different point resulted in 100% success.

Table IV also shows that diagonal perturbation was mostly re-

quired for the constrained OPFD with regulating transformers.Generally, the use of diagonal perturbation was rare since

close to the solution becomes positive definite, as expected.

Moreover, the number of iterations in which the filter enforced

a reduction in step length was very small. While this may seem

surprising, one should remember that the search direction is a

Newton step, which generally moves toward a minimizer and a

feasible point. A similar conclusion was reached by Vanderbei

[18] when using a merit function. Nevertheless, the filter is

needed in some cases to enhance convergence.

The ratio (MPCIP to PCIP) of average iteration count and

CPU time is shown in Figs. 4 and 5, respectively. The figures

indicate that in most cases, the PCIP requires a lower number of

iterations because it solves the OPF with both equality and in-

equality constraints i.e., a part of the active set is known a priori

which means that the complementarity condition (3l) has to be

checked for a lesser number of variables. However, the MPCIP

requires less computation time since it solves a symmetric pos-

itive definite system using Cholesky decomposition, which is

bound to be much faster than the solution of symmetric indef-

inite systems using LU decomposition. From the point of viewof the user, a reduction of computational time is more important

than a reduction in number of iterations.

XII. CONCLUSIONS

This paper describes a modified primaldual method applied

to the OPFD problem. Two modifications are included as com-

pared to previous published solutions:

1) a filter to ensure step length control;

2) diagonal perturbation of the normal equations matrix in

order to guide convergence toward minimizing stationary

points.

The method presented also provides a means to detect

and treat indefiniteness in the reduced Hessian (second-order

sufficiency condition for optimality) without predicting the

active set. Moreover, the formulation of the OPFD through

inequality constraints requires less CPU time as compared to

previous methods since it solves a positive definite matrix the

size of which is equal to the number of primal variables. The

paper also presents a model for the inclusion of regulating

transformers in the OPF solution. This model overcomes the

difficulty of modeling the phase shift, which has precluded its

appearance in previous interior point OPF solutions. Numerical

testing on eight different networks shows that the method is

promising.

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Rabih A. Jabr received the B.E. degree in electrical engineering (with highdistinction) from the American University of Beirut, Lebanon, in 1997, and thePh.D. degree in electrical engineeringfrom the Imperial College, London, U.K.,in 2000.

Currently, he is an Assistant Professor in the Department of Electrical,Computer, and Communication Engineering at Notre Dame University, ZoukMikayel, Lebanon. His research interests are in operations research and powersystem optimization.

Alun H. Coonick received the M.Sc. degree from the University ofSouthampton, Southampton, U.K., in 1980, and the Ph.D. degree from theImperial College, London, U.K., in 1991.

Currently, he is a Lecturer in the Department of Electrical and Electronic En-gineering, Imperial College, London, U.K. His research interests include powersystem stability, control using FACTS devices, and artificial intelligence.

Brian J. Cory lectured at Imperial College, London, U.K., from 1956 until hisretirement in 1993. He is now a Visiting Professor and Senior Research Fellow.His main research interests are in all aspects of electrical energy supply. Theseinclude planning, pricing, operating modern systems, and coping with deregu-lation, privatization, and competition.

Rabih2002 a Primal-dual Interior Point Method for Optimal Power Flow Dispatching

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