2004 IEEE International Conference o n Electric Utility Deregulation, Restructuring a n d Power Technologies (DRPT2004) April 2004 Hong Kong

Parallel Evolutionary Programming for Optimal

Power Flow
C . H. Lo, Member, IEEE, C. Y. Chung, Member, IEEE, D. H. M. Nguyen, Member, IEEE, and K. P.
Wong, Fellow, IEEE

Absrruct--This paper proposes a parallel evolutionary OPF problem for searching the global optimum solution.
programming (EP) approach for solving the optimal power The EP technique is a stochastic optimization method
flow (OPF) problem. The parallel EP-OPF approach is less which uses the mechanics of evolution to search for
sensitive to the choice of starting points and types of optimal solution to a given problem. A population of
generator cost curves. The developed algorithm is
candidate solutions is evolved toward the global optimum
implemented on a Beowulf cluster with 31 Intel Pentium IV
2.66GHz processors which are arranged in master-slave through the use of a mutation operator and competition
structure. The proposed approach has been tested on the scheme. The EP technique is capable of incorporating
IEEE 30- and 118-bus systems. Computational speedup and new constraints arising from open access, non-convex
performance of the master-slave topology is then compared solution surfaces and FACTS devices. It is also
to those of the sequential EP approach. particularly suitable to non-monotonic solution surfaces
where many local minima may exist.
Index Terms-Evolutionary Programming, Optimization,
Optimal Power Flow. In our previous paper [6], an EP-based OPF solution
algorithm (EP-OPF) has been developed, which is capable
I. INTRODUCTION of determining the global optimal solution to the OPF
problem for a range of constraints and objective functions.
PTIMAL power flow (OPF) provides a means for
0 minimizing the cost of power system operation so as
to meet the load demand under various operational
The problems of the starting-point sensitivities and
linearisation of non-convex generator cost curves are
overcome by the developed serial EP-OPF algorithm in
constraints. Solving the OPF problem is of paramount
[6]. To improve the speed of the computation, this paper
importance in power system operation under the de-
reports work on the development of a parallel form of the
regulated environment of the electricity industry. It is a
previous EP-OPF algorithm and its implementation on a
highly constrained and large dimensional nonlinear
Beowulf computer cluster with 31 Intel Pentium IV
optimization problem, which is difficult to solve.
2.66GHz processors. The computational speedup and
Previously, various approaches such as, linear
performance of the parallel algorithm is then investigated
programming [ 11, nonlinear programming [2] and interior
using the IEEE 30- and 118-bus systems under the
point method [3], have been employed to solve the OPF
master-slave structure.
problem. These methods rely on convex and continuous
The organization of the paper is as follows: A brief
input/output generator curves to obtain the global
review of the OPF problem is given in Section 11. Section
optimum solution, and as such, these curves must be
111 addresses the general scheme of the EP-OPF for
approximated by continuous and monotonic functions.
solving the OPF problem. Implementation of the parallel
However, the OPF problem is in general non-convex, and
EP-OPF on the Beowulf cluster is stated in Section IV.
hence, these simplifying assumptions will lead to sub-
Section V presents the performance of the proposed
optimal solutions. The degree of non-convexity is further
parallel algorithm under the IEEE 30- and 118-bus
increased with the inclusion of FACTS devices on the
systems. Finally, Section VI concludes the paper.
network or the consideration of valve-point loading
effects of thermal generators [4, 51.
FLOWPROBLEM
POWER
11. OPTIMAL
Besides, classical optimization methods are highly
sensitive to starting points and frequently converge to The OPF problem was introduced in the early 1960s
local optimum solutions or diverge completely due to the and has grown into a powerful tool for power system
non-monotonic solution surface. Hence, evolutionary operation and planning. The OPF seeks to determine the
programming (EP) technique [6] has been applied to the optimal control parameter settings to minimize a desired
objective function J; while satisfying numerous
operational constraints [7]. For optimal active- and
This work was supported by the Faculty of Enginecring and reactive-power dispatch, the objective function J; can be
Department of Electrical Engineering, The Hong Kong Polytechnic the total generation cost. Voltage profile optimization or
University.
The authors are with the Computational Intelligence Applications minimization of active power loss may also be the
Research Laboratory (CIARLab), Department of Electrical Enginecring, objective. Mathematically, the objective is a function of
The Hong Kong Polytechnic Univcrsity, Hung Horn, Kowloon, Hong system control and dependent variables and can be written
Kong. (e-mail: eechlo@polyu.edu.hk; eecychun@polyu.edu.hk;
eenguyen@polyu.edu.hk; cekpwong@polyu.cdu.hk). as:

0-7803-8237-4/04/$17.0002004IEEE
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2 0 0 4 IEEE Intemational Conference o n Electric Utility Deregulation, Restructuring and P o w e r Technologies (DRPT2004) April 2004 H o n g K o n g

Minimize: A x , U) (1) solutions for subsequent use in initializing the loadflow

subject to: on the next iteration to reduce computational time within
g(x, U) = 0 (2) the loadflow algorithm.
h(x, U) I O (3)
where x is the vector of dependent variables (e.g. load
(PQ) node voltages, reactive power generation); U is the
I Initialize Population I
vector of control variables (e.g. transformer tap settings, J.
Evaluate Loadflow and
generator activeholtage magnitudes); AX, U) is the Assign Fitness
objective to be optimized; g(x, U) represents the equality
constraints which are the nodal load-flow equations; h(x, c
U) represents the inequality constraints on dependent and
control variables.

PROGRAMMING
111. EVOLUTIONARY BASEDOPF
w Mutate Population

Evaluate Loadflow and

Assign Fitness
Evolutionary programming is a powerful optimization
technique that is well suited to problems where analytical
methods have difficulties. It searches for the optimal Compete Populations
solution by evolving a population of candidate solutions
(individuals) over a number of generations. During each
Terminate
generation, a new population is formed from an existing
population through the use of the mutation operator. This
operator creates new individuals from existing individuals
by perturbing their variables by a random amount. After
I Finish I
mutation, the mutated population undergoes an evaluation Fig. 1. Flowchart of the EP-OPF algorithm
process where a fitness value is assigned to each
individual according to a pre-defined objective function. B. Initialization
Individuals from the original and mutated populations A uniform random number generator initializes each
are competed with each other through the use of a control variables of an individual within its feasible range.
competition scheme. The winning individuals then form a
For example, the voltage magnitude for a PV node 6,
resultant population which is regarded as the next
with upper and lower limits of yimin and 6 "
generation. As analogy to the survival of the fittest law,
individuals with higher fitness will have a greater chance respectively, we have
to survive during the competition than those with lower 5 = U[~::",Vi""] (4)
fitness. Since the degree of optimality of an individual is
measured by its fitness, the competition scheme ensures where .[vjmin,vimm1 is a uniform random number
that the population will evolve towards the global optimal generator between cmin and 5". Beside to the random
solution. initialization, one individual (candidate solution) will
Since the EP technique is an iterative process, a have its specified active power generation for all PV
stopping rule is usually required to terminate the process. nodes except slack node initialized using the economic
Two stopping rules are widely used: (i) stop after a pre- dispatch solution supplying the system load and
defined number of generations is reached or (ii) stop transmission losses. This economic dispatch solution is
when there is no appreciable change in the best individual obtained using the EP-based method in [8].
for a certain number of generations. Rule (i) is adopted in
this paper. C. Fitness Function Evaluation
Based on the EP methodology, an algorithm for Each individual is assigned a fitness to measure its
solving the OPF problem has been established in [6]. Fig. optimality with respect to the objective function being
1 shows the basic flowchart of the EP-OPF algorithm. optimized. Before fitness assignment, it is necessary to
solve the loadflow problem as shown in Fig. 1. The
A. Solution Coding
loadflow is solved using an EP-based loadflow algorithm
An individual in the population represents a candidate developed in [9]. Once the nodal active and reactive
solution to the OPF problem and consists of control and powers are known, the fitness of an individualJ will be,
dependent variables. Typically, control variables are the
M (5)
tap positions for variable tap transformers, the specified
f;=Ci+Z:VPj+SQ
voltage magnitude at all generator (PV) nodes and the
j
specified power generation at all PV nodes except slack
node. Dependent variables are the most recent loadflow

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2004 IEEE Intemational Conference on Electric Utility Deregulation, Restructuring and Power Technologies (DRPT2004) April 2004 Hong Kong

VPj = K,<v~
-1.0)~
if vj > vj"" or vi < vyin(6) Assuming the upper limit of an individual's slack node
active power generation is exceeded, and that the slack
!U otherwise
unit is unit 1. Then, the total available capacity of

1. K, (!&io& - QsT$k ) 2 i f Qsiuck > Q Z z k generators 2 to N in that individual is given by

N
SQ = Kq (Qsiock - Q:gk 1' i f Qsiack
min (7)
Qslack c2 = c (Pj""
j=2
-Pj) (10)
otherwise
where Pi is the jth generator's active power generation
In ( 5 ) , M is the maximum possible cost of generation, and N is total number of generators. The excessive
Ci is the total generation cost of individual i; VPjdenotes generation of the slack node is
a penalty term on PQ or switched PV node j that violates N
E2=D,5-(C P j + P Y M ) (1 1)
voltage limits ( p i n , pa);and SQ represents a penalty
J J j=2
on the slack node for violating its reactive power limits. where DL is the sum of the active power demand and the
K, and Kq in (6) and (7) are penalty weighting constants. transmission loss from the previous loadflow solution of
It is not necessary to detennine a penalty to the active that individual. The loading of generator 2 is then
power generation violation at the slack node since the modified according to
constrained mutation helps to satisfy this constraint. The
Pi = P2 Ez(PTM
i- - P I ) / Cz (12)
higher is the fitness, then the better is the individual.
If the modified loading exceeds the maximum loading of
D. Mutation
generator 2 ( PT'), it is set to its limiting value. Then,
Mutation is a process of producing a new population
from the existing (parent) population. A new individual p < the amount of excessive generation of the slack node left
to be shared is
is mutated from each parent pi, where thejth OPF variable
<
in the new individual p is computed as, E3 =E2 -(Pi -Pz) (13)
+ N(O,oij)2
(8) The above procedure is repeated to modify the loading of
! =I x..
x!. v generators 3 to N. The slack node active power will be on
where x!. represents the mutated variablej i n p {, x..is the its upper limit after the constrained mutation. Similar
!I !
I
value of variablej in the parent individual pi, and ~ ( 0G,$ process is used once the lower active power limit of the
slack node is violated.
is a Gaussian random number with zero mean and a
standard deviation of qj.The value of q, is calculated by F. Competition Scheme
(91, Selection of fitter individuals from parent and mutated
populations is performed by a competition scheme, which
! =(x""
(5.. I J -X;nin)(f" -fi +ar) (9)
is known as tournament scheme. All individuals within
fmm
where f;: is the fitness of the individual i; fmm is the the two (parent and mutated) populations will undergo a
maximum fitness within the parent population; xmnr min series of NI tournaments with randomly selected
J ,x.i opponents. Each individual i is then assigned a score si,
denote the upper and lower limits of the variablej; a is a according to (14),
positive number constant slightly less than unity; and r is N,
the generation counter. The term ay provides a decaying si = 2 n,
mutation offset with its rate depends of the value of a [6].
E. Constrained Mutation
The loadings of all generators other than the slack unit
nj= 'r
j=l

U iotherwise
ffi' f r
where is the fitness of individual i; fr is the fitness of
of an individual are assigned previously using the
opponent r; and nj is the result of a tournament between
standard mutation as stated in (8). The sum of these
individual i and the randomly chosen opponent r. If the
assigned loadings is then compared to the total generation
population size is k, the k highest scoring individuals are
found from the previous loadflow of that individual. If the
selected to form the resultant population in the next
difference between them is within the operating limits of
generation.
the slack unit, the mutated individual is accepted.
Otherwise, the process is repeated for another four
IV. PARALLEL
EP-OPF ALGORITHM
ANDITS
attempts. If within these attempts a feasible mutated IMPLEMENTATION
individual is not obtained, the constrained mutation is
employed to share the excessive generation of the slack The population size is one of the factors that will affect
unit among the remaining generators. The constrained the performance of the EP-OPF algorithm for seeking the
mutation forces the satisfaction of the slack node's active optimal solution. If a large population size is used, the
power generation constraint as follows. chance of obtaining the optimal solution by mutation and
competition is high. It is obvious that more computing

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2004 IEEE Intemational Conference on Electric Utility Deregulation, Restructuring and Power Technologies (DRPT2004) April 2004 Hong Kong

time is then required. To improve the computation speed be controllable. For both test cases, quadratic generation
while maintaining the same quality of solution, the cost curves have the form, cj = ai + biq + ciq2 . The
parallel EP-OPF algorithm is implemented.
performance of the parallel EP-OPF was compared with
The basis of the proposed parallel EP-OPF algorithm is
the sequential EP-OPF proposed in [6].
to divide the initial population into several sub-
The parallel EP-OPF algorithm was executed on a
populations. Since the size of sub-populations should be
Beowulf cluster consisting of 31 Intel-Pentium IV 2.66
smaller than the initial population, the computation time
GHz processors, which are connected through a Gigabit
will be reduced. A Beowulf cluster is arranged in master-
Ethernet switch. In the two test cases, one processor was
slave structure and the message passing interface (MPI)
assigned as the master node and the remaining thirty
protocol in C language is used [lo] to implement the
processors as slaves. Hence a total of 31 processors were
parallel EP-OPF program.
used in the computation. The specific settings for both the
Fig. 2 depicts the Beowulf cluster configuration for the
sequential and parallel EP-OPF algorithms and system
master-slave parallel EP-OPF topology. In this topology,
data are summarized in the Appendix.
a single master processor is employed to coordinate m
slave processors. The master processor performs the A. IEEE 30-bus Test Case
tournament competition for selecting the fittest The standard IEEE 30-bus loading was used and the
individuals among the sub-populations from the slave quadratic generation cost curve from [ 121 were
processors, which perform mutation and fitness evaluation. sumniarized in Table I. For the sequential EP-OPF, the
The parallel EP-OPF procedure can be described as average cost of solution obtained was $803.45 with the
follows. minimum being $802.91 and maximum of $804.03. For
Master processor randomly initializes the whole the parallel EP-OPF, the average cost of solution obtained
population of individuals. was $803.33 with the minimum being $802.51 and
Master processor divides the whole population maximum of $804.28. The average execution times for
among m slaves and then sends the sub- sequential and parallel EP-OPF were 55.14s and 5.02s
populations to each slave processor. respectively, with a computational speedup of 10.98 for
After receiving the sub-populations, the slave the parallel EP-OPF. The solution details for the
processors execute mutation and fitness evaluation minimum cost were provided in Table 11. The voltage
independently. Then, each slave processor sends profile of the minimum cost solution for the parallel EP-
its result back to the master. OPF is shown in Fig. 3.
After gathering the sub-populations from each
slave, the tournament competition scheme is TABLE I
conducted by the master so as to select the highest GENERATOR
DATAAND COST COEFFICIENTS FOR THE IEEE 3o-BUS
scoring individuals from the parent (master) and
mutated (slaves) populations to form a resultant
population in the next generation.
The stopping rule is checked. If it is not satisfied, 1.75 0.01750
return to step 2. -15 0.00 1.00 0.06250
IO 35 -15 60 0.00 3.25 0.00834
11 10 30 -10 50 0.00 3.00 0.02500
Master I 13 12 40 -15 60 0.00 3.00 0.025001

TABLE I1
MINIMUMSOLUTION FOUND BY EP-OPF IN THE IEEE 3o-BlJS TESTCASE
Sequential EP-OPF I Parallel EP-OPF I

Slave 1 Slave 2

Fig. 2. Configuration of master-slavc parallel EP-OPF algorithm. 1.0815

V. CASESTUDIES
The proposed parallel EP-OPF algorithm has been In this case, solutions from the sequential and parallel
applied to the OPF problem in the IEEE 30-bus system EP-OPF were similar but the execution time of the
and then the IEEE 118-bus system [ 111. The objective parallel algorithm was much faster since the work was
function to be minimized is the total system active power divided among 30 slaves. From [12], a solution of
generation cost, permitting all generators' active power $802.40 was reported which violated the slack node lower
generations, voltage magnitudes and transformer taps to Q-limit slightly by approximately 1.7MVAr. However,

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2004 IEEE International Conference on Electric Utility Deregulation, Restructuring and Power Technologies (DRPT2004) April 2004 Hong Kong

the proposed parallel EP-OPF returned a solution without

PV nodes violation or switching. The average statistics of
the population over the 100 trials are plotted in Fig. 4 to
illustrate the relative fast convergence of the parallel EP-
OPF algorithms. The nodal voltages and generator
reactive powers at minimum cost solution obtained by the
parallel EP-OPF are presented in the Appendix.

"7 ................

Scquential EP-OPF Parallel EP-OPF

PIO 543.412 V i 0 1.0562 Pi0 550.000 VIO 1.0072
5 15 20 25 30
P~s 128.449 V25 0.9582 pzs 84.042 V ~ S 1.0823
Node 391.377 VZ6 402.965 v36 1.026I
Fig. 3. IEEE 30-bus system voltage profile of the parallel EP-OPF. 51.136 V 4 9 53.195 V 4 9 0.9754
490.501 v 6 3 489.953 V 6 5 1.0293
980, I I , , . I I , . , 492.000 v 6 6 489.623 v66 0.9674
787.702 Vas 795.025 v 6 9 1.0466
577.000 V a 0 576.679 V,(j 0.9976
707.000 V 8 9 701.756 V 8 9 1.0635
277.068 V/oo 292.951 Vi00 1.0592
0.98 t64.61 0.90 164.61 1.05
tZ6 25 1.04 t65.66 0.97 165.66 0.90
tJ0-17 0.94 168.69 0.92 t6&59 1.02
t38.37 1.00 t8/.80 0.96 t8/.80 0.94
163.59 0.90 0.93

TABLE IV
RESULTS R\I THE IEEE 30- AND 1 18-BUS SYSTEMS
Generation
Processor Run cost ($)
Method Number Time Speedup Min. Max. Ave.
Fig. 4. Convergence of the parallel EP-OPF in the IEEE 30-bus system.
IEEE 30-bus System
B. IEEE II8-bzrs Test Case Sequential
EP-OPF I 55.14 1 802.91 804.03 803.45
The proposed parallel EP-OPF algorithm was further Parallel
tested on the lEEE 118-bus system with 54 generators. EP-OPF 31 5.02 10.98 802.51 804.28 803.33
Although a solution could be obtained, the speed of
IEEE 118-bus Svstcm
convergence was very slow and at most of the time, the Sequential
solution was unable to converge. It was due to the large EP-OPF 1 4692.5 1 1125.00 1131.38 1129.06
search space where most of the solutions were infeasible. Parallel
The infeasibility was due to a large power flow along the EP-OPF 31 394.87 11.88 1122.92 1132.53 1128.98
corridor of the power system, which surpassed generators
reactive power and PQ node voltage limits. VI. CONCLUSIONS
The search space can be reduced by reducing the A parallel evolutionary programming-based optimal
number of generators. In this case, 10 generators were power flow (EP-OPF) algorithm has been established
selected. The selected generators were allowed to mutate based on the sequential EP-OPF reported in [6]. The
during the EP searching process. Since the search space parallel EP-OPF algorithm has been successhlly and
was significantly reduced, the convergence for optimal effectively implemented on the Beowulf cluster arranged
solution was greatly improved. in master-slave structure. The performance of the
For the sequential EP-OPF, the average cost of proposed parallel EP-OPF algorithm has been
solution obtained was $1 129.06 with the minimum being demonstrated by its application to the IEEE 30- and 118-
$1 125.00 and maximum of $1 131.38. For the parallel EP- bus systems with quadratic generation cost curve. The
OPF, the average cost of solution obtained was $1 128.98 algorithm has accurately and reliably converged to the
with the minimum being $1122.92 and maximum of global optimum solution in each case, and the quality of
$1 132.53. The average execution times for sequential and the solution is comparable to the sequential counterpart. A
parallel EP-OPF were 4692.5s and 394.87s respectively, faster execution time is yielded for the parallel EP-OPF

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2004 I E E E Intemational Conference on Electric Utility Deregulation, Restructuring and P o w e r Technologies (DRPT2004) April 2004 Hong Kong

algorithm as computation loads have been divided among K. P. Wong, and Y. W. Wong, “Genetic and genetickimulated-
annealing approaches to economic dispatch”, IEEE Proc., Gen.
the slaves. Hence, a large search space can be handled Trans. BDistrib., vol. 141,110. 5 , pp. 507-513, 1994.
effectively by the parallel EP-OPF algorithm. A method IEEE Committee Report: “Present practices in the economic
has also been developed to effectively initializing the operation of power systems”, IEEE Trans., PAS-90, pp. 1768-1775,
population of the large 118-bus test case. While further 1986.
J. Yuryevich, and K. P. Wong, “Evolutionary programming based
work can be performed to enhance the computational optimal power flow algorithm”, IEEE Trans. on Power Systems,
speed more, the proposed algorithm can also be extended vol. 14, no. 4, pp. 1245-1250, Nov. 1999.
to include environmental and fuel constraints. H. W. Dommel, and W. F. Tinney, “Optimal power flow solutions”,
IEEE Trans. (Power App.& Syst.), vol. PAS-87, pp. 1866-1876,
Oct. 1968.
VII. APPENDIX K. P. Wong, and J. Yuryevich, “Evolutionary-programming-based
algorithm for environmentally constrained economic dispatch”,
System data: For both the IEEE 30- and IEEE 118-bus IEEE Truns. on Power Systems, vol. 13, pp. 301-306, Jan. 1998.
systems, the lower voltage magnitude limits for all buses K. P. Wong, J. Yuryevich, and An Li, “Evolutionary programming
is 0.95 p.u. while the upper limit is 1.05 p.u. for slack based method for evaluation of power flow”, In Proc. Genetic and
node and all load nodes, and 1.1 p.u. for all generation , .
Evolutionarv Comoutation Conference. DD. 1756-1761. Jul. 1999.
I .

[IO] W. Gropp, E. Lusk, and A. Skjcllum, Using MI: Portable Parallel

nodes. In phase tap-changing transformers are with Programming with the Message-Passing Interface. London: MIT
allowable tapping ranges of +lo% with a step size of 1%. Press, 1994.
Sequential and parallel algorithms parameter settings: [ I l l Power System Test Case Archive. Available:
The population size was set at 150 and the total number of http://www.ee.washington.cdu/research/pstea/
[I21 0. Alsae, and B. Stott, “Optimal loadflow with steady-state
generations was 50. N, in (14) was set at 100 and the security”, IEEE Trans., PAS-93, pp. 745-75 1, 1974.
value of a in (9) was 0.9. K, = 1,000 in (6) and Kq =
10,000 in (7). Both algorithms were run 50 times for each C. H. Lo (M’03) obtained his B.Eng and Ph.D. degrees from The Hong
test system. Kong Polytechnic University, Department of Electrical Engineering, in
1999 and 2003, respectively.
He is currently a research associate in the Department of Electrical
Engineering, The Hong Kong Polytechnic University. His current
Node Magnitude Phase (deg.) Node Magnitude Phase (deg.) research interests include application of evolutionary computation to
optimal power flow problem, parallel processing applications to power
(P.U.) (P.U.)
1 1.0498 0.000 16 1.0387 -11.853 system optimization, fuzzy control and multi-objective optimization.
2 1.0367 -3.704 17 1.0358 -12.203
3 1.0281 -5.782 18 1.0229 -12.884 C. Y. Chung (M’OI) received the B.Eng. degree (with First Class
4 1.0226 -6.946 19 1.0211 -13.065 Honors) and the Ph.D. degree in electrical engineering from The Hong
5 1.0072 -10.494 20 1.0256 -12.870 Kong Polytechnic University, Hong Kong, China.
6 1.0179 -8.073 21 1.0304 -12.509 After his Ph.D. graduation, he worked in the Electrical Engineering
7 1.0057 -9.595 22 1.0311 -12.499 Department at the University of Alberta, Edmonton, AB, Canada, and
8 1.0136 -8.230 23 1.0237 -12.695 Powertech Labs, Inc., Surrey, BC, Canada. Currently, he is a Lecturer in
9 1.0354 -10.334 24 1.0219 -12.936 the Electrical Engineering Department of The Hong Kong Polytechnic
10 1.0422 -12.052 25 1.027 -12.722 University. His research interests include power system stability/control,
11 1.0994 -9.228 26 1.0095 -13.134 computational intelligence applications and power markets.
12 1.0488 -11.220 27 1.0386 -12.327
13 1.0815 -10.371 28 1.0122 -8.546 D. H. M. Nguyen received his bachelor degree and the Ph.D. degree in
14 1.0349 -12.130 29 1.0191 -13.520 electrical engineering, at the University of Western Australia in 1994 and
15 1.0313 -12.252 30 1.0078 -14.376 2002 respectively. He is currently a postdoctoral fellow in the
Department of Electrical Engineering, The Hong Kong Polytechnic
University.
GENERATOR REACTIVE POWER AT MINIMUM COST SOLUTION FOR THE
IEEE 3o-BUS TESTCASE(PARALLELEP-OPF) K. P. Wong (M’87, SM’90, F’02) He obtained M.Sc and Ph.D. degrees
Node Rcactive Power (MVAR) Status from the University of Manchestcr, Institute of Science and Technology,
1 -16.013 Slack Node in 1972 and 1974 respectively. Prof. Wong was awarded the higher
2 23.035 Un-switched doctoral DEng degree by UMIST in 2001. Prof. Wong is currently a
5 25.057 Un-switched Chair Professor and is the Hcad of Department of Electrical Engineering,
8 19.459 Un-switched The Hong Kong Polytechnic University.
11 33.929 Un-switched Prof. Wong received three Sir John Madsen Medals (1981, 1982 and
13 25.384 Un-switched 1988) from thc Institute of Engineers Australia, the 1990 Outstanding
Engineer Award from IEEE Powcr Chapter Westem Australia and the
REFERENCES 2000 IEEE Third Millennium Award. He has published numerous
research papers in power systems and in the applications of artificial
R. Ristanovic, “Successivc linear programming based OPF intelligence and evolutionary computation to power system planning and
solution”, Optimal Power Flow: Solution Techniques, operations. His current rcscarch interests include cvolutionary
Requirements and Challenges, IEEE Power Engineering Society, optimization in power, power market analysis, power system planning
pp. 1-9, 1996. and operation in deregulated environment, power quality. He is a Fellow
S. M. Shahidehpour, and V. C. Ramesh, “Nonlinear programming of IEEE, IEE, H U E and IEAust.
algorithms and decomposition stratcgics for OPF”, Optimal Power
Flow: Solution Techniques, Requirements and Challenges, IEEE
Power Engineering Soeicty, pp. 10-24, 1996.
J. A. Momoh, S. X. Guo, E. C. Ogbuobiri, and R. Adapa, “The
quadratic intcrior point mcthod solving power system optimization
problems”, IEEE Trans. on Power Svstems, vol. 9, pp. 1327-1336,
Aug. 1994.

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