1. A GENERALIZED QUADRATICBASED MODEL FOR
OPTIMAL POWER FLOW
James A. Momoh, Senior Member, IEEE
Dept. of Electrical Engineering
Howard University
Washington, DC 20059
ABSTRACT voltage deviation, cost, flows and their combinations. The
current approach for solving optimal power flows used by
A quadratic form of power systems has been used to researchers is based on the sensitivity method, generalized
formulate a generalized optimum power system flow problem. reduced gradient and, recently, the MINOS augmented method
The conditions for feasibility, convergence and optimality are [ 8 1. These methods are generally implemented on quite
included in the construction of the optimal power flow (OPF) large systems and have drawn considerable attention by
algorithm. It is also capable of using hierarchical structures researchers in the optimal power flow work.
to include multiple objective functions and selectable
constraints. The generalized algorithm using sensitivity of A literature review and experience of operators reveals
objective functions with optimal adjustments in the that these methods lack the capability of identifying which
constraints yields a global optimal solution. The generalized constraints or controls are necessary to achieve a given
algorithm includes special cases of the OPF that reduces objective. Moreover, the inclusion of various constraints
computational burden when the number of constraints is which are not adequately prioritized makes it burdensome on
equal to the number of variables and can select the constraint the operator to use existing OPF packages efficiently. Another
sets that are binding or nonbinding. The optimal power flow area of concern is the approximation of the models used in
package developed is flexible and can accommodate other OPF as demonstrated in the so-called quadratic sequential
models and solution methods. models.
The algorithm has been tested using different objective The power industry is seeking improved, flexible and easy
functions on actual power systems, and optimal solutions to use models for OPF packages with conditionality checks on
reached in relatively few iterations. The potential of the feasibility and optimality. The work reported here utilizes a
method for on-line application is being investigated with quadratic model of OPF with capability to handle generalized
several utility companies. sets of constraints, convert two-sided constraints into one to
reduce computational burden and criteria for checking
Kevwords: Optimal Power Flow, Quadratic Model, Multiple feasible conditions, necessary and sufficient conditions while
Objective Functions. converging to a global optimum. The algorithm incorporates
multiple objective functions such as losses, voltage deviation,
cost and flows, and selectable constraints on the generation,
I. INTRODUCTION loads, transformer taps, flows, etc.
The task of operating a power system economically and This paper is organized as follows: formulation of
securely is commonly referred to as the OPF problem. An quadratic model OPF, feasibility and necessary conditions for
OPF solution gives the optimal active and reactive power quadratic forms, development of generalized OPF algorithm,
dispatch, and the optimal setting of all controllable variables special cases of OPF solutions, implementation procedure,
for static power system loading conditions. Computationally, scenarios for selecting multiple objective functions, and test
this is a very demanding nonlinear programming problem due cases and conclusions.
to the large number of variables, in particular, to the large
number and types of limit constraints imposed on the power
System by engineering design limits. These constraints, II. FORMULATION OF QUADRATIC MODEL
which define the technical feasibility, can be equalities or OPTIMAL POWER FLOW
inequalities.
Interest in the OPF problem has been growing since the In general, the OPF problem in standard form is given as
early 1960s and is perhaps at its peak at present [ 1, 2, 3 1
because of its potential for real-time economic and secure Min F(x, U) (1)
power system dispatch. Several techniques of optimal power
flow models using linear and nonlinear programs have been subject to
proposed. Their capability to include complete sets of
constraints or controls is developed in-house to suit different g o , U) = 0 (2)
utilities. To date no generalized algorithm is available to the
user to accommodate various objectives such as losses, h(x, U) f; (3)
261
CH2809-2/89/0000-0261$1.00 '1989 IEEE

2. where TABLE 2 SUMMARY OF DESCRIPTIONS OF OBJECTIVE
FUNCTIONS
x is the state variable for voltage and U represent the
controllable generator or transformer taps. The function
F(x,u) denotes costs, voltage deviation, losses or maximum
power transfer. g(x,u) denotes the equality constraints and
h(x,u) the inequality constraints. The optimal power flow
formulation is expressed in hierarchical structure. The
objective function in scalar form is represented as:
Min Fo(X) = 1/2 XT R X + aTX (4)
where the constraints are denoted as
Ki = Fi (X) = 1/2 XT Hi X + biT X (5)
TABLE 3: Summary of Constraints
and
Ci < Ki < Di i = 1, 2, ..., m
The matrices R and Hi, vectors a and bi and scalars Ci
and Di are described by the hierarchical structure tables
1 -71. These hierarchical structure tables are described as
1
follows:
Table 1, gives the summary of objective functions and
constraints. The relationship between the different objective
functions and constraints is listed in this table. For example
for objective functions OF1, the corresponding constraints
are 1, 2, 3 and 4. The summary of description of objective
functions is listed in Table 2. The detail formulae of the
different objective functions are given in this table. In Table
3, the summary of contraints formulations is given in a list.
Table 4 contains the objective function variable description. TABLE 4: Objective Function Variable Description
Table 5 provides the objective function parameter
description, Table 6 gives the constraint variable ~
description, while Table 7 details the constraints parameter VariabkS DEScRlpTION
description. For any given objective function, the
formulation for the objective functions and its constraints Voltage vector, 2n x 1, real and imag;narY part of voltage.
are easily obtained from these tables. For example, when loss
minimization is desired, the objective function from Table 2 V Expected voltage valw vector. 2n x 1
is OF3 and the corresponding constraints are constraints1,
2, 3 and 5 in Table 1. Real Power of Generator. g x 1.
r Bus number of of a given system.
TABLE 1: SUMMARY OF OBJECTIVEFUNCTIONS g Generatornumber of a given system.
AND CONSllWNTS
I Objanve
PmcUm I Constraints I TABLE 5: Objective Function Parameter Description
262

3. TABLE 6: Constraint Variable Description This formulation of the OPF in quadratic form lends itself
to the following questions.
1. Can we derive the optimum solutions using the
VpriablcS DESCRIPTION properties of quadratic forms analytically?
1 ii 1 1
X The voltage vector, the same as objective function 2. What is required to guarantee an optimum solution
when numerical techniques are used for power
~ w t e BU Voltage vector
d system dispatch problems?
Realgmemtionpowervector 3. What is the scenario for selecting candidate
constraint objectives to attain a desired optimal
Reactive generationpower vector operation andlor schedule?
Real load power vector
P Demands power vector for generator
111. FEASIBILITY AND NECESSARY CONDITIONS
i Real power for the imBus FOR QUADRATIC FORMS
I 'ii I CurrentbetweenBusiandj I In this section we present some theorems which are
developed and proved by the Energy Systems Network
Lower limit of variable (*) Laboratory (ESNL). The adaptation of the development of
these theorems is to build up the basis for the further sound
Uppa limit of variable (*) development of optimal power flow methodologies, and to
search for the mathematical formulation of the OPF problem.
These theorems are only stated; the proofs in detail are
available in [ 91.
Proposition 1: The matrices H 2 ~ - 1have exactly 2n-4
TABLE 7: Constraint Parameter Description zeros, two positive and two negative eigenvalues. In other
words, they are of rank four unless the bus K is isolated from
other buses. As such, the quadratic forms of Pk and Qk can be
Variables convex or concave for arbitrary vector X.
H" Theorem 2: If X is a solution, for the nonlinear
programming problem, then it is necessary that Lx(x, h) = 0
and that the following Extended Kuhn-Tucker(EKT) conditions
H must be satisfied for all i > p.
I
(a) hi=OwhenCis fi(X)lDi
(b) hi 2 0 when fi (x) = Di
(c) hi 2 0 when fi (x) = Ci
H
2k-1
where Lx(X,h) is differentiation of the Lagrange equation:
and h are the Lagrange multipliers.
Lemma 3: The quadratic optimization has a minimum if and
only if it is impossible to find a X to make f(X) + M = 0 as M
approaches infinity.
Theorem 4: The quadratic optimization has a minimum if
there is a set of real numbers a i , ap,... am such that M is
__ ~ _ _
positive definite where M is given as
Linear term coefficiency,depend on the system
See objective function parameter d d p t i o n M=R+ .x aiHi
m
See objective function parameter description 1= 1
(7)
Theorem 5: If there exists a scalar number such that the
square matrix Sp + PFxT Fx is positive definite, then the X
263

4. obtained from the EKT condition is a minimum
Substituting equations (4) and (5),
L = [ F&x) + W X )I (10)
m
where S=R + z hi ~i (8) L, = [ FOx(x) + hFx(x) I = 0 (11)
i= 1
and FTx = [HI X +bl, H ~ +b2,
x ..., Hmx + bm] (9) Expanding equation (11) yields
m
Proposition 1 deals with the nonconvexity and
=+a+ .Z h i ( H i X+ b i )
I= 1
=o (12)
nonconcavity of power systems. Theorems 2 and 4 give the
results of necessary and existence condition of nonlinear solve for
programming problem in quadratic form, and Theorem 5 is
the sufficiency condition for the problem. x = - s-lw (13)
IV. DEVELOPMENT OF GENERALIZED OPTIMUM where m
POWER FLOW ALGORITHM S=R+ .x %Hi
1=1
A generalized sensitivity algorithm is developed to solve
m
the OPF problem. It satisfies the feasibility, necessary and W=a + , E hibi (15)
sufficiency conditions for quadratic model power systems. 1=1
Basically, the algorithm consists of three parts. The first
phase is the input data preparation with the capability to if
easily interface with additional data. The input data consists
of generator limits, voltage limits, load limits, transformer
taps and network conditions. In addition, the optimization F[ X ( h )I = k (16)
constants include the range of tolerance and criteria for
achieving convergence. then X is determined using Newton Raphson method
iteratively as :
The second phase of the algorithm includes the following:
1. Selection of constraints for:
a) cases where the number of variables, n, equals
the number of constraints, m; X and h are determined from equations (13) and (17), and
are used in satisfying feasibility and necessary conditions.
b) cases of constraints that are binding or non-
binding and The optimal adjustment scheme used for preventing
oscillations and overshoot are obtained using one of the
c) cases where all candidate constraints are
included.
a) For constraints with the number of variables equal to
2. Satisfaction of the EKT conditions and criteria for the number of constraints, AK is simply calculated
optimal adjustment. Two approaches are used in from
the optimal adjustment scheme, namely:
AK = LE (1 8)
a) sensitivity technique and
b) least square estimation of eigenvalues and where E is a predetermined number which can be
eigen-vectors. specified by users.
3. Criteria for convergence evaluation. b ) For the case where all candidate constraints are
involved, in the adjustment scheme, AK is obtained as:
The third phase of the algorithm involves:
AK = A(-Ah) (19)
a) criteria for evaluation of termination ana
convergence critieria for optimal solution and where A is a normal transformation in m spaces, it is
computed from
b) display of control variables objective
functions and other pertinent data.
264

5. where pi and its transpose piT are associated eigenvectors
with the corresponding eigenvalues, 81, 82, ..., pm.
READ IN DATA
1
I
m F’REPROCESS OF THE DATA
So that AK= .Z dipi
I= 1
L
SELECTION OF OBJECTIVEFUNCTION
where
di = P.c~
Because ti = 0 for i > q, we have
AK= Md
For cases of binding and nonbinding constraints, new
values of F(x, k)=k are obtained using equations (13) and
(17). The function values are set to either the upper or Fig. 1 Flow Chart o f the OPF Package
lower values by using the sign of the difference between the
new F(x, k) and the given constraint limits. Investigation of
the direction of h allows us to separate the constraints into
binding and nonbinding sets. The separation of constraints VI. SCENARIOS FOR SELECTING MULTIPLE
must satisfy the EKT conditions thus feasibility and necessity OBJECTIVE FUNCTIONS
conditions can be guaranteed.
The selection scenarios of objectives and constraints for total
economic operation of power systems are given as follows:
V. IMPLEMENTATION PROCEDURE
1. Voltage Deviation Objective
A program package has been developed for achieving
total economic and reliable operation of power systems. It is This criterion minimizes the deviation of overvoltage
capable of selecting different objectives and their associated and undervoltage conditions for a given power system. The
constraints to meet the desired economic objective. For the optimal adjustments of generators, transformer taps, loads,
case of optimal power flow problem with the number of etc. alleviate voltage deviation problems. The capability to
variables equal to the number of constraints. the package uses minimize voltage deviation will prevent power system
the sensitivity method (SM) which is a special case of the instability and improve economic systems operation.
generalized SM discussed in the paper.
2. Cost Objective
For the general case n # m, two possible approaches are
employed. Firstly, the simplex-like method is used for This criterion, in general, minimizes the production
determining which constraints are binding or nonbinding. costs of generating plants. The associated constraints include
The second approach includes all candidate constraints in the controllable devices such as generators, loads, transformer
algorithm and uses the least square estimate to obtain the taps, etc. It has the additional capability to reduce cost of
optimal adjustments during the optimization procedure. operation when coupled with the voltage deviation objectives.
The program module for the multiple objective optimization
is shown in Figure 1. It consists of network data needed for 3. Loss Objective
optimization, the description of selectable objective functions
and the various approximations of optimization techniques, While minimization of voltage objective prevents
and finally, the output of optimized objectives and status of system instability and possibly decreases the chance of
constraint limits. system voltage collapse, the loss minimization increases the
optimal power while guaranteeing minimum cost of operation.
The associated constraints are given in Table 3 and are
adjustable to obtain the desired optimal conditions.
265

6. 4. Flow Objective Table 8: Sample of Constraints Status
for 39 bus system
This objective represents the determination of
maximum mwer transfer capability of a given network. This
objective is needed in networks, such as the WSCC system in
the United States, where there exists excess power generation .owbound lalue of Cons upbwa
but with limited networks for power transfer. The 1.01 1.05 1.11
maximization of the power transfer objective is obtained by 0.918 0.961 1.01
satisfying the limits imposed on generator outputs, 5.07 5.59 6.19
transformer taps, load demands and power flow, etc. -0.656 -0.656 -0.537
4.57 4.57 5.59
1.43 1.58 1.75
The impact of these objectives, when coupled with 25.2 57.1 75.5
appropriate weighting functions, guarantees optimum
1.01 1.06 1.11
maintenance costs and increased system reliability. The
0.918 0.957 1.01
proposed OPF package includes these objectives and
5.07 6.05 6.19
constraints in the hierarchical form discussed earlier in the
-0.656 -0.558 -0.537
paper.
4.57 5.54 5.59
1.43 1.54 1.75
0.724 1.22 1.36
VII. TEST CASES AND DISCUSSION
4.50 5.06 5.50
A variety of sim,ulations using several medium-size and
practical systems have been performed using the generalized 1.01 1.06 1.11
OPF algorithm. The optimum solutions are obtained by 0.918 0.958 1.01
utilizing the algorithm to guarantee the feasibility, necessary 5.07 6.08 6.1 9
conditions and convergence criteria. Moreover, employing -0.656 -0.551 -0.537
optimal adjustments and optimization constants discussed in 4.57 5.22 5.59
the paper, the algorithm is prevented from wandering, 1.43 1.47 1.75
oscillation and overshoot. 25.2 56.7 75.5
A 39-bus example is presented to demonstrate the
detailed behavior of the algorithm for selected objective
functions. Other higher order systems were evaluated to
extend the capability of the algorithm. The operating
constraints limits for different control variables and their Table 9: Results for 39-bus System
status at optimal objective function values are presented in
Table 8.
Specifically, the tables include the corresponding values
of voltage, power and loads for different objectives as
determined by different optimization schemes discussed in the
paper. The computational tractability of the methods are also
evaluated. The CPU time and number of iterations required to
achieve convergence are displayed in Table 9.
The special forms of the generalized algorithm based on
simplex-like method (SLM), distinguishes between binding
and nonbinding constraints. The SLM reduces the
computational burden of the optimization process. Objective Function: Cost
Accordingly, the S reduces the CPU time for possible cases
M Method: SLM
where the number of constraints equals the number of
variables. The convergence criteria for various tolerances
were determined. The cost objective using SLM and
generalized S are displayed in Figures 2 and 3.
M
Other objective functions are evaluated using the
generalized algorithm. The potential of these objectives to
minimize cost of operation and improve reliability has been
verified. Adaptability of the method to power utility needs is
being undertaken.
1 2 3 4 Iteration
Fig. 2 Convergence of SLM for 39-bus System
266

7. [ 2,] Happ, H. H., Vierath, D.R., "The OPF for Operation
Objective Function: Voltage Planning and for Use On-line", P Q Z
B E- of the Snd
. .
Method: GSD e on Power -S
a d Durham, England, pp. 290-295, July
n ,
1986.
1 [ 3 .I Stott, B., Alsac, O., Marinho, J.L., "The Optimal Power
Flow Problem", Power P r o w : t h e
MathematicalChallenae, 1986.pp. 327-351, SIAM,
73
[ 4 .] Stott, B., Marinho, L., Alsac, O., "Review of Linear
Programming Applied to Power System Rescheduling",
the 1 1 th PICA Conference,Cleveland,
Ohio pp.142 - 154, May 1979.
Fig.3 Convergence of GSD for 39 bus System
[5.] Sun, D.I., Ashley, B., Brewer, B., Hughes, A., Tinney,
W.F., "Optimal Power Flow by Newton Approach",
JFFF Transactkm on Power -
,
Vol. PAS 103, pp. 2864-2880, 1984.
VIII. CONCLUSION
[6.] Burchett, R.C., Happ, H.H., Vierath, D.R.,
A general algorithm extending the basic Kuhn-Tucker "Quadratically Convergent Optimal Power Flow",
conditions and employing a quadratic model in formulating ns on P,-
oe
wr Vol.
OPF problem has been developed. The algorithm incorporates PAS-103, pp. 3267-3276, 1984.
the general multiple objective functions such as voltage
deviation, losses, production costs, flow and their [7.] Chieh, Hua T., Hsieh, W.C., Optimization Theory with
combinations. The associated constraints of these objectives w a t i v e P r o w, Mon Min Co., 1981.
are organized in a hierarchical structure. The proofs of the
feasibility, necessary and sufficiency conditions provide [8.] Gill, P., Murray and Wright, M.,
. . "m
insight in the development of the mathematical basis for the QDtlmlzatian" Academic Press, 1981.
algorithm. Special cases of the algorithm to reduce
computation time have been tested on several practical power L9.1 Momoh, J. "Corrective Control of Power System
systems. During An Emergency" NMI Ph.D Dissertation
Information Service, 1983.
Computational memory and execution time required have
been significantly reduced by exploiting the properties of [lo.] Aoki, K., and Satosh, T. "Economic Dispatch with
quadratic forms in the optimization process. The sign of Network Security Constraints Using Parametric
Lagragian multipliers have been used to convert two-sided Quadratic Programming", Paper No. 82, SM 426-5,
constraints into one. Nonbinding constraints using the SLM Presented at the PES Summer Power Meeting, San
for certain classes of problems are easily circumvented Francisco, California, July 1982.
which reduces the CPU time for the OPF problem.
[ 11 .I Sun, D.E., et. al., "Optimal Power Flow Solution By
Results of tests performed on other system sizes Newtonian Approach, JFFF T- rn on PAS, Vol.
demonstated the same characteristics using the algorithm. In PAS-103, No. 10, October 1984, pp. 2864-2880.
general, the algorithm converges in few iterations.
Presently, the algorithm is developed on a VAX 11/780.
Similar results would be achieved on smaller computers. The [12.] Ponrajah, R.A., and Galiana, F.D., "The Minimum Cost
potential of the algorithm to enhance flexibility of the OPF Optimal Power Flow Problem Solved Via the Restart
package and be suited to different utility needs is being Homotopy Continuation Method", Proceedinps of
investigated. JEFF/PFS 1988 Wn i- ' , New York, New
York, 1988.
IX. ACKNOWLEDGEMENTS
[13.] Aoki, K., Fan, M., and Nishikor, A., "Optimal Var
The research is supported by the U.S. Department of Planning by Approximation Method For Recursive
Energy and National Science Foundation ECS 8657559 PYI. Mixed-Integer Linear Programming" procee-
Special gratitude also goes to Dr. Yi Zhang and Dr. Arunsi
Chuku at the Energy Systems Network Laboratory, Howard Meetiag. New York, New York,
University, who provided valuable suggestions. 1988.
( 1 4 -1 Momoh, James A. and B.Wolienberg, "Quadratic
Module for Voltage Reduction During an
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26 7