Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844
Vol. V (2010), No. 3, pp. 375-384

A Swarm Intelligence Approach to the Power Dispatch Problem

D.C. Secui, I. Felea, S. Dzitac, L. Popper

Dinu Călin Secui, Ioan Felea, Simona Dzitac, Laurenţiu Popper
University of Oradea, Faculty of Energy Engineering, Romania
E-mail: csecui@uoradea.ro, ifelea@uoradea.ro, simona.dzitac@gmail.com,
director@perfect-service.ro

Abstract: This paper examines how two techniques of the Particle Swarm Optimiza-
tion method (PSO) can be used to solve the Economic Power Dispatch (EPD) prob-
lem. The mathematical model of the EPD is a nonlinear one, PSO algorithms being
considered efficient in solving this kind of models. Also, PSO has been successfully
applied in many complex optimization problems in power systems. The PSO tech-
niques presented here are applied to three case studies, which analyze power systems
having four, six, respectively twenty generating units.
Keywords: economic dispatch problem, constrained optimization, particle swarm
optimization.

1 Introduction

An important issue in optimizing the power systems is the economic power dispatch. This problem
consists in determining the power generated by the plant units of a system in order to minimize the total
generation cost of units, taking into account the active power balance and the constraints imposed to the
capability of the units.

The mathematical optimization model is nonlinear, where both the objective function and the restric-
tions imposed by equations describing the system functionality are nonlinear.

The mathematical model can be solved using conventional optimization techniques such as the
lambda iteration method, the gradient method and others [1, 2], if constraints are considered as linear,
the objective function is continuous and the domain of the values is convex. Some disadvantages that
arise in these situations - long solving time, objective function discontinuity etc. - may be overcome by
applying the artificial intelligence techniques. The most common optimization techniques based upon ar-
tificial intelligence used for solving economic power dispatch problems are: the genetic algorithm [3–6],
the Hopfield neural networks [2, 7], the differential algorithm [8], the evolutionary programming [9, 10],
fuzzy-optimization [12, 13], tabu search [14], particle swarm optimization [15, 16, 27, 28]. Also, the EPD
can be formulated as a multi-objective optimization problem [11, 13, 17].

TWe mention that the particle swarm optimization method was successfully applied to other opti-
mization problems, such as optimal power flow [18–20], reactive power optimization and voltage con-
trol [23], power loss reduction in distribution systems [24], network reconfiguration [25], unit commit-
ment problem [26], due to its good convergence, low computational time and good quality solutions.

In this paper there are proposed two versions of applying PSO method for solving EPD, comparing
the results obtained for three systems consisting of four, six and respectively, twenty generating units.

The paper is structured as follows: section 2 exposes the EPD problem, section 3 and 4 describe the
proposal for solving this problem using PSO algorithm, section 5 presents the results obtained through
the application of PSO algorithm for three power systems, and section 6 outlines the conclusions drawn.

2 Formulation of the economic power dispatch problem

We consider a power system containing n generating units, each unit having its own generated power
Pj, j = 1...n. The total load required in the system is considered to be known and equal to PD. The fuel

Copyright c© 2006-2010 by CCC Publications



376 D.C. Secui, I. Felea, S. Dzitac, L. Popper

cost (Fj(Pj)) for each generator j is represented by a quadratic function:

Fj(Pj) = a j ·Pj + b j ·Pj + c j (1)
where: a j, b j and c j are fuel cost coefficients of generator j;
Pj represent the power of generator j.

EPD solution consists in determining the Pj powers of generating units, so that the total fuel cost of
the entire system to be minimal, respecting the restriction of power balance on the overall system and the
inequality restrictions for each unit j.

The objective function is:

min F =
n∑

j=

Fj(Pj) (2)

The problem constraints are given by relations (3) and (4):

Fp =
n∑

j=

Pj − PD − ∆ P =  (3)

Pminj ≤ Pj ≤ Pmaxj , j = , , ..., n (4)
Where: Pminj and P

max
j represent the maximum and the minimum operation limits of a generator j;

The power loss at the level of the entire system is a quadratic function in relation to variables Pj and
it is calculated by using constant B coefficient formula:

∆ P =
n∑

i=

n∑

j=

Pi ·Bi j ·Pj +
n∑

i=

Bi ·Pi + B (5)

Where Bi j is an element of the loss coefficient matrix, Bi is the element i of the loss coefficient
vector, and B is the loss coefficient constant.

3 Presentation of different Particle Swarm Optimization techniques

PSO is a heuristic algorithm, used for solving nonlinear and noncontinuous optimization problems,
being introduced by Kennedy and Eberhart [22], in 1995. Since then several techniques in applying the
PSO method have been developed, but in the current paper only two techniques are presented, namely the
classical PSO (PSO Classical) and PSO with time varying acceleration coefficients (PSO Accelerated).

Classical PSO: To search for the optimal solution in a space with the dimension n, PSO uses a
population of NP particles. For a given particle i within NP population, vector solutions at a certain
iteration k are represented by X ki =(x

k
i, x

k
i,..., x

k
i j,..., x

k
in). In any optimization process, switching from one

solution (xki j) to another solution (x
k+
i j ) is accomplished by using the velocity of particles, represented

by the vector V ki =(v
k
i, v

k
i,..., v

k
i j,..., v

k
in), according to the relation:

X k+i = X
k
i + V

k+
i , i = , , ..., NP (6)

The updated velocity of the particle in the next iteration (k+1) is given by the relation:

X k+i = ω ·V ki + c ·r ·(Pbest ki − X ki ) + c ·r ·(Gbest k − X ki ) (7)
Where: V ki , V

k+
i represent the velocity vector of particle i at iteration k, respectively k+1;

X ki , X
k+
i represent the solution vector of particle i at iteration k, respectively k+1;



A Swarm Intelligence Approach to the Power Dispatch Problem 377

Pbest ki represent the best solution vector of particle i, until iteration k;
Gbest k represent the vector corresponding to the best solution of the group, until iteration k;
c and c are coefficients corresponding to cognitive and social behavior;
r and r are random numbers between 0 and 1, and w is the inertia weight factor determined using

the relation:

ω = ωmax −
ωmax − ωmin

kmax
·k (8)

Where ωmax and ωmin represent the initial, respectively final weights, kmax is maximum iteration
number and k is current iteration number for the algorithm.

PSO with time varying acceleration coefficients tries to improve the global search in the early stages
of the optimization process and to accelerate the convergence of the particles to the global optimum in
the final part of the process. In this case, the calculation of the velocity (V k+i ) and of the solution (X

k+
i )

for the next iteration is done with relations (6) and (7), and according to [21], the coefficients c and c
are determined by the relations:

c = (c f − ci)·
k

kmax
+ ci (9)

c = (c f − ci)·
k

kmax
+ ci (10)

Where ci, ci, c f and c f are initial and final weights for cognitive and social acceleration coeffi-
cients.

4 The methodology based on PSO for solving the EPD problem

The implementation of PSO techniques for solving the EPD problem involves the following steps:
Step1. Initialization of the parameters and of PSO solution. The PSO algorithm parameters are set
in reference to ωmax, ωmin, kmax, number of particles (NP), coefficients c and c (for PSO Classical),
respectively, ci,ci, c f and c f (PSO Accelerated). Initially a population of NP particles is randomly
formed. Each particle defines a possible solution to the problem, which should respect the constraints
given by relations (3) and (4).

Step 2. Evaluation of the objective function F and of the auxiliary function f. The problem contains
an equality restriction shown by relation (3). Thus, the auxiliary function f is formed, using the relation:

f = F + α ·F p (11)
Where α is the penalty factor.
For each particle and each iteration the values of function f will be calculated, and by comparing

them solutions Pbesti and Gbest are selected. At the end of the optimization process, functions f and F
will have approximately equal values, according to the calculation error admitted by choosing factor α .

Step 3. Update velocity and solution. The minimum (V minj ) and the maximum limits (V
max
j ) of the

velocity for each generating unit j are calculated:

V maxj = β ·(Pmaxj − Pminj ) and V minj = −V maxj (12)
Where factor β was considered between 0.05 and 0.1.
The update of the particle position and velocity is done with relations (6) and (7). For each solution

(Xi) ) it is verified if the components xi j satisfy the constrain (4). If the constrain is satisfied, then the



378 D.C. Secui, I. Felea, S. Dzitac, L. Popper

Unit Pmin[MW] Pmax[MW] a[$/MW ] b[$/MW] c[$]
1 30 120 0.00875 18.24 750
2 50 160 0.00754 18.87 680
3 50 200 0.00310 19.05 650
4 100 300 0.00423 17.90 900

Table 1: Cost coefficients and limits of generated powers for a thermal power plant with four units (CS4)

Unit Pmin[MW] Pmax[MW] a[$/MW ] b[$/MW] c[$]
1 100 600 0.001562 7.92 561
2 100 400 0.00194 7.85 310
3 50 200 0.00482 7.97 78
4 140 590 0.00139 7.06 500
5 110 440 0.00184 7.46 295
6 110 440 0.00184 7.46 295

Table 2: Cost coefficients and limits of generated powers for a thermal power plant with six units (CS6)

calculated value for xi j is kept. Otherwise xi j is set with the value nearest to the limit of the domain (Pmaxj
or Pminj ).

The vectors Pbest and Gbest are obtained based on the evaluation of the auxiliary function f and
on the comparison of the f values calculated in two consecutive steps. If the new value of function f is
better than the previous value of f for previous Gbest, then Gbest is set at the new value. Similarly, Pbest
vector is updated.

Step 4. Stopping process. In this paper the criterion of stopping the calculation process is given
through achieving the maximum number of iterations set.

5 Numerical examples and simulation results

In this section three case studies on how EPD solving by applying the two PSO techniques (PSO
Classical and PSO Accelerated) are presented. The objective function is given by the relation (2) and
restrictions (3) and (4). All case studies were implemented in Mathcad, on a personal computer having a
1.58 GHz processor and 896 MB of RAM.

5.1 Description of tested systems

Case study 1 - four unit system (CS4). The first system (CS4) is a thermal power plant having four
generating units, where the total power losses (∆ P) are considered zero. The data for the four generators
(cost coefficients and limits of generated powers) are presented in Table 1. The total power demanded in
the system is PD=520 MW.

Case study 2 - six unit system (CS6). The second system (CS6) is a thermal power plant with six gen-
erating units, where the total (∆ P) is considered zero. The data for the six generators (cost coefficients
and limits of generated powers) are presented in Table 2 [5]. The total power demanded in the system is
PD=1800 MW.

Case study 3 - twenty unit system (CS20). The third system contains twenty units, and the demanded
power in the system is PD=2500MW. The data for the generators and the values of the coefficients Bi j
are available in [9].



A Swarm Intelligence Approach to the Power Dispatch Problem 379

Scenarios ci c f Best F Worst F Average F Standard deviation F
[$/hour] [$/hour] [$/hour] [$/hour]

1 1.5 1.5 12919.76 12920.77 12919.84 0.026
2 1.5 2 12919.76 12920.36 1291981 0.015
3 1.5 2.5 12919.76 12921.01 12919.84 0.035
4 2 1.5 12919.77 12921.00 12919.83 0.028
5 2 2 12919.76 12920.04 12919.79 0.007
6 2 2.5 12919.76 12920.42 12919.80 0.016
7 2.5 1.5 12919.76 12922.35 12919.84 0.052
8 2.5 2 12919.76 12920.17 12919.80 0.009
9 2.5 2.5 12919.76 12920.27 12919.80 0.013

Table 3: The influence of the coefficients c1i,c2f upon the results obtained through PSO Accelerated, for
one hundred trials (CS4)

5.2 PSO parameters and PSO convergence

PSO parameters involved in the calculation process may affect the algorithm performances and qual-
ity of the solutions. For the system having four generators (CS4) the parameters were set to values:
Wmin = ., Wmax = , c = ., c = ., NP=6, kmax =  (PSO Classical) and respectively Wmin = .,
Wmax = , ci = , ci = ., c f = ., c f = , NP=6, kmax =  (PSO Accelerated).

In case of PSO Accelerated algorithm, in order to assess the influence of the coefficients on EPD
problem solving, 100 distinct trials were performed, noting the best value for F (Best F), the worst
value for F (worst F), the average value for F (Average F) and the standard deviation of cost F. Table
3 shows the values Best F, Worst F, Average F and standard deviation F, considering that the values of
coefficients (ci, c f ) vary between limits ci, c f ∈ [., .], and the values of coefficient (c f ) and (ci)
are constant and equal to c f = ci = ..

Regarding the four unit system (CS4), in Table 3 it can be noticed that coefficient changes do not
affect the solutions, but the best results are obtained for scenario 5, where ci = , c f = .

For the six generator system (CS6) in both PSO Classical and PSO Accelerated algorithm, the co-
efficients have little influence upon the outcomes, considering that they vary within the limits c1,c2 [1,
4], respective c1i,c2f [1.5, 2.5] and c1f, c2i [0.1, 0.4]. The results are shown in Table 4, the parameters
being set to values: wmin=0.4, wmax=1, c1=2, c2=2, NP=15, kmax=30 (PSO Classical), and wmin=0.4,
wmax=0.9, c1i=2.5, c2i=0.2, c1f=0.4, c2f=1.6, NP=15, kmax=30 (PSO Accelerated).

In case of the twenty unit system (CS20), PSO Classical was applied taking into account the follow-
ing settings: w1=0.4, w2=1.1, c1=2, c2=2, NP=500, kmax=200 and =0.1; for PSO Accelerated it was
considered: w1=0.4, w2=1.1, c1i=2.5, c2f=2, c1f=0.4, c2i=0.2, NP=500, kmax=200 and =0.1.

The number of particles that constitute the population is another important factor in the PSO al-
gorithm. In case of the four unit system, for both methods (PSO Classical and PSO Accelerated), the
changes in cost F in relation to particles number (NP) were graphically represented, considering the
number of iterations set at kmax=15 (Fig. 1). Also, in Fig. 2 the variation of cost F in relation to the
number of iterations (kmax) is represented, considering the number of particles set to NP=10.

Analyzing the diagram shown in Fig. 1 it is found that for both algorithms (PSO Classical and PSO
Accelerated) the best solution (Best F) is obtained considering a population consisting of 5-6 particles.
Fig. 2 shows that the converging process towards the best value of F is obtained after fifteen iterations
for both algorithms, the initial solution being different.

In the six unit system, increasing the number of variables involves a higher number of particles
(NP=15) and a higher number of iterations to (kmax=30) in order to obtain the same solution, presented
in Table 4.



380 D.C. Secui, I. Felea, S. Dzitac, L. Popper

Figure 1: F convergence with particles number (NP)/CS4; Figure 2. F convergence with kmax for CS4

Algorithms P1 [MW] P2 [MW] P3 [MW] P4 [MW] P5 [MW] P6 [MW] Cost F [$/hour]
FLCGA [3] 250.49 215.43 109.92 572.84 325.66 325.66 16585.85

PGA [5] 248.14 217.74 75.20 587.80 335.56 335.56 16579.33
AECGA [6] 248.07 217.73 75.30 587.70 335.60 335.60 16579.33

IHN [7] 248.08 217.74 75.18 587.90 335.55 335.55 16579.33
PSO Classical 247.95 218.44 75.16 587.58 335.43 335.44 16579.33

PSO Accelerated 248.00 217.71 75.16 588.02 335.52 335.59 16579.33

Table 4: The comparison of the results obtained for the system with six generating units (CS6)

To assess the efficiency of PSO algorithms, they are compared to other four algorithms previously
presented using the same data, available in [5]. It can be seen that both PSO Classical, and PSO Accel-
erated algorithm reach the same cost F as the algorithms presented in [3], [5], [6] and [7]. The resulting
solution and the cost value are comprised in Table 4.

5.3 The assessment of the solutions

The quality of the solutions was assessed by determining the values Best F, Worst F, Average F and
standard deviation F considering 100 trials.

FFor the system with four generating units the results are presented in Table 5. In order to assess the
convergence process and quality of the solutions the average of cost values F (Average F - Fig.3) and its
standard deviation (Standard deviation F - Fig.4) for each iteration are determined, considering only one
trial. The system considered in Fig.3 and Fig.4 proves that both algorithms converge quickly toward the
best solution for F, the curves presenting a continuous decrease in relation to the number of iterations.

The best solution obtained in 100 trials, using algorithms PSO Classical and PSO Accelerated, is
presented in Table 5, together with the results of the gradient method.

In case of the six units system (CS6), the values Best F, Worst F, Average F and standard deviation
F considering 100 trials are presented in Table 6.

Algorithm robustness was tested starting from different initial solutions, randomly obtained, and
retaining the best value of F for a number of algorithm trials within the [1-50] interval.

For the system having twenty units (CS20) the following values were obtained, considering 100 trials
(Table 7). In Table 8 is presented the solution for the twenty units system, and the comparison of the
results with those obtained in [2].



A Swarm Intelligence Approach to the Power Dispatch Problem 381

Figure 3. Average variation of F with kmax (CS4); Figure 4. Variation of standard dev. F with
kmax(CS4)

Unit power output [MW] PSO Classical PSO Accelerated Gradient method
P1 88.554 92.536 92.493
P2 65.340 65.539 65.559
P3 134.662 130.293 130.431
P4 231.444 231.632 231.517

Total power [MW] 520.00 520.00 520.00
Total generated cost (Best F)[$/hour] 12919.96 12919.76 12919.76

CPU time (s) <  <  < 

Table 5: The best solutions (Best F) obtained for the system with four generating units (CS4)

Algorithm Best F Worst F Average F Standard deviation F
[$/hour] [$/hour] [$/hour] [$/hour]

PSO Classical 16579.33 16582.64 16579.51 0.0650
PSO Accelerated 16579.33 16581.93 16579.49 0.0362

Table 6: The values Best F, Worst F, Average F and standard deviation F for CS6

Algorithm Best F Worst F Average F Standard deviation F
[$/hour] [$/hour] [$/hour] [$/hour]

PSO Classical 62457.1805 62469.64 62462.45 0.4346
PSO Accelerated 62456.4380 62466.63 62461.05 0.3340

Table 7: The values Best F, Worst F, Average F and standard deviation F for CS20



382 D.C. Secui, I. Felea, S. Dzitac, L. Popper

The value of cost F obtained in Table 7 presents a smaller dispersion for both PSO algorithms,
showing a high quality of the solutions. The calculation of power losses in the transmission lines was
achieved with an error of 0.03 MW. It is also seen that PSO Accelerated algorithm gets a solution as
good as the algorithms presented in [2], but PSO Classical leads to a farther solution, comparing to the
solutions presented in Table 8.

Unit power output [MW] PSO PSO Lambda-iteration Hopfield neural
Accelerated Classical method [2] network [2]

P1 511.1808 514.2219 512.7805 512.7804
P2 171.9583 163.3928 169.1033 169.1035
P3 125.9410 125.7172 126.8898 126.8897
P4 99.6666 99.4738 102.8657 102.8656
P5 114.9377 112.8359 113.6836 113.6836
P6 75.1378 76.8715 73.5710 73.5709
P7 113.2613 109.2679 115.2878 115.2876
P8 116.2341 116.8953 116.3994 106.3994
P9 101.5174 100.5633 100.4062 100.4063
P10 102.5556 113.8036 106.0267 106.0267
P11 150.6753 154.2216 150.2394 150.2395
P12 292.5836 289.6717 292.7648 292.7647
P13 120.2476 118.0356 119.1154 119.1155
P14 34.9866 38.2194 30.8340 30.8342
P15 117.3186 115.2359 115.8057 115.8056
P16 36.1563 36.4369 36.2545 36.2545
P17 68.7329 69.8707 66.8590 66.8590
P18 83.7616 81.9408 87.9720 87.9720
P19 98.9061 103.8655 100.8033 100.8033
P20 56.4137 51.6055 54.3050 54.3050

Total power [MW] 2592.1728 2592.1468 2591.9670 2591.9669
Total generation cost [$/hour] 62456.4380 62457.1805 62456.6391 62456.6341

Total lines losses [MW] 92.1728 92.1468 91.9670 91.9669

Table 8: The best solutions (Best F) obtained for the system with twenty generating units (CS20)

6 Conclusions

In this paper, the economic power dispatch problem is solved using two PSO techniques, namely,
PSO Classical and PSO Accelerated. Both techniques are effective in solving this problem, but PSO
Accelerated leads to a better quality of solutions and a lower computing time.

For EPD problems with small number of variables and linear restrictions, the classical solving tech-
niques (gradient method, lambda iteration method) are also applicable, obtaining the same results as
PSO. For EPD with nonlinear restrictions, PSO techniques are more effective, having a better conver-
gence, robustness and stability, indicated by low values of standard deviation. The number of particles
and the number of iterations required to obtain stable solutions are related to a reduced computing time.

PSO techniques are also compared with other techniques, such as Hopfield neural network, the results
being almost identical for our applications.



A Swarm Intelligence Approach to the Power Dispatch Problem 383

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