Optimal Economic Load Dispatch of the Nigerian Thermal Power Stations Using Particle Swarm Optimization (PSO)

The International Journal Of Engineering And Science (IJES)
|| Volume || 6 || Issue || 1 || Pages || PP 17-23 || 2017 ||
ISSN (e): 2319 – 1813 ISSN (p): 2319 – 1805
www.theijes.com The IJES Page 17
Optimal Economic Load Dispatch of the Nigerian Thermal Power
Stations Using Particle Swarm Optimization (PSO)
Y. S. Haruna1
, Y. A. Yisah2
, G. A. Bakare1
, M. S. Haruna3
and S. O. Oodo3
1
Department of Electrical and Electronics Engineering, A. T. B. U Bauchi-Nigeria,
2
Power Equipment and Electrical Machinery Development Institute (PEEMADI), P.M.B. 1029, Okene, Kogi State.
3
Department of Electrical and Electronics Engineering, Nile University of Nigeria, Abuja.
--------------------------------------------------------ABSTRACT-------------------------------------------------------------
This paper deals with the optimization of economic load dispatch (ELD) problem; this is to find the optimal
combination of generators in order to minimize the operating costs of the system. This is done by using the
particle swarm optimization (PSO) algorithm. PSO is applied to search for the optimal schedule of all the
generator units that can supply the required demand at minimum fuel cost while satisfying all system
constraints. The PSO algorithm has been implemented using MATLAB optimization toolbox and was applied to
solve the ELD problem of the Nigeria thermal power stations. The results were compared with published results
obtained via micro-GA, conventional-GA and differential evolution (DE) techniques.
Keywords: Economic Load Dispatch, Particle Swarm Optimization and Nigeria Thermal Power Stations.
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Date of Submission: 28 December 2016 Date of Accepted: 20 January 2017
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I. INTRODUCTION
The modern power system around the world has grown in complexity of interconnection and power demand.
The focus has shifted towards the enhanced performance, increased customer focus, low cost, reliable and clean
power. In this changed perspective, scarcity of energy resources, increasing power generation cost,
environmental concern necessitates optimal economic load dispatch (ELD). In reality power stations neither are
at equal distances from load nor have similar fuel cost functions. Hence for providing cheaper power, load has
to be distributed among the various power stations in a way which results in lowest cost of generation [1].
The main aim of electric power utilities is to provide high-quality, reliable power supply to the consumers at the
lowest possible cost while operating to meet the limits and constraints imposed on the generating units. This
formulates the economic load dispatch (ELD) problem for finding the optimal combination of the output power
of all the online generating units that minimizes the total fuel cost, while satisfying an equality constraint and a
set of inequality constraints. As the cost of power generation is exorbitant, an optimum dispatch results in
economy [2].
Traditional algorithms like lambda iteration, base point participation factor, gradient method, and Newton
method can solve this ELD problems effectively if and only if the fuel-cost curves of the generating units are
piece-wise linear and monotonically increasing. Practically the input-output characteristics of the generating
units are highly non-linear, non-smooth and discrete in nature owing to prohibited operating zones, ramp rate
limits and multi fuel effects. Thus the resultant ELD becomes a challenging non-convex optimization problem,
which is difficult to solve using the traditional methods [3, 4].
Several heuristic approaches like evolutionary programming (EP), genetic algorithm (GA), ant colony search
(ACS), tabu search (TS), artificial neural network (ANN), bio-geography based optimization (BBO), differential
evolution (DE) and simulated annealing (SA) have been developed for solving both linear and non-linear ED
problems [5 - 8].
In this paper, particle swarm optimization (PSO) algorithm is proposed to solve the ELD problems in power
systems. The viability of the method is analyzed for its accuracy and rate of convergence on the Nigerian power
network (1999 model) and results were compared with other heuristics methods.
1.1 Nigerian Power System
The electricity demand in Nigeria far outstrips the supply and the supply is epileptic in nature. This is hindering
its development, notwithstanding the availability of vast natural resources in the country. Constant power supply
is the hallmark of a developed economy. Any nation whose energy need is epileptic in supply prolongs her
development and risks losing potential investors.
Reliable Power production is critical to the profitability of electricity utilities. This can only be realized when
the power generators are scheduled efficiently to meet electricity demand. Economic load dispatch have been
applied to obtain optimal fuel cost while satisfying systems system constraints and generator scheduling for

Optimal Economic Load Dispatch Of The Nigerian Thermal Power Stations Using Particle Swarm…
hourly anticipated load within a period of 24 hours. The efficiency of generating unit, the transmission looses
and the operating costs are important factors to be considered for the economic operation of the system.
In recent years, the Power Holding Company of Nigeria (PHCN) has been experiencing serious problem in
generation, transmission, distribution, maintenance, financial constraints and increase in power demand [9],
considering the generation/power demand problems, several units were on emergency/forced outages, which led
to system disturbance such as; partial and total system collapse. These problems were attributed to over stressing
the units to generate outside their normal operating conditions. This will thus lead to generating electric power
at loss.
In view of the above problem, it becomes necessary for one to study the cost functions of the available thermal
units, their power limits and the maximum power demand of the whole country at a particular time so as to carry
out the ELD problem.
The aim of this research is to apply a particle swarm optimization technique to solve the economic load dispatch
(ELD) problem; for the purpose of optimal allocation of the total power demand among the available generating
units that minimizes the total generation cost subject to specified system constraints.
1.2 Statement of the Problem
Consider a system consisting of N thermal units connected to a transmission network as shown in Figure1
below.
Figure 1: Interconnected Power System Network
Where, i
F  fuel cost of unit i,
iG
P  power delivered by unit i,
D
P  total power demand and
L
P  total power loss.
The quadratic cost function of unit i, is given by;
2
( )
i i Gi
i G i i G i
F P P P     …(1)
Where, i
 - Constant cost coefficient of unit i,
i
 -Linear constant cost coefficient of unit i
i
 -A quadratic constant cost coefficient of unit i.
Operating the system subject to generation constraints, this objective function can be expressed mathematically
as:
1
( )
i
N
T i G
i
F F P

  i  1,2…….N … (2)
Where, T
F is the total cost of power generation and
( )
ii G
F P is the generation cost of unit i.
The ELD problem seeks to find the optimal combination of thermal power generations that minimized the total
cost while satisfying the total power demand and the systems constraints. The ELD problem is formulated as
minimization of Eqn. (2) subject to the following constraints:
1.2.1 Equality constraints
1
i
N
G D L
i
P P P

  …(3)
Using the B-coefficient method, network losses are expressed as:

 
T
Li G i G i
P P B P …(4)
Where,  B is the B matrix coefficient.
1.2.2 Inequality constraints
m in m a x
G i G i G i
P P P  …(5)
Where,
m in
G i
P is the minimum power limits of the unit i
m a x
G i
P is the maximum power limits of unit i.
Finally, PSO was applied to the coordination of the Nigerian thermal power plants. This research on economic
load dispatch is modeled using the quadratic cost function. The issues of environmental constraints, generation
ramp-rate limits, valve-point effect and piecewise linear cost function are out of scope of this paper.
II. LITERATURE REVIEW
The ELD problem has been widely studied and reported by different authors. The techniques used in the
literature ranges from the classical optimization techniques to the recent meta- heuristic optimization techniques
such as evolutionary algorithms (EA), genetic algorithm (GA), simulated annealing (SA) and particle swarm
optimization (PSO).
[3] developed a Pareto frontier Differential Evolution (PDE) technique to solve MOED problem. The proposed
method was implemented on the standard IEEE-30 bus system having six generating units including valve point
effects to evaluate its performance and applicability. From the results obtained, the proposed method
demonstrated its effectiveness by solving the Multi Objective economic dispatch problem considering security
constraints.
[4] proposed and developed evolutionary algorithms to solve the economic dispatch problem for the optimum
performance of a nonlinear and complex systems. They considered all the constraints of power dispatch for
economic operation of the power system.
[5] presented the economic power dispatch problems using ant colony optimization (ACO) technique which is a
meta-heuristic approach for solving hard combinatorial optimization problems. This technique was tested
using the standard IEEE 26-Bus RTS and the results revealed that the proposed technique has the merit in
achieving optimal solution for addressing the problems. Comparative studies with artificial immune system
(AIS) were also conducted in order to highlight the strength of the proposed technique.
In the year 2012, [6] presented an effective and reliable particle swarm optimization (PSO) technique for the
economic load dispatch problem using the standard 3-generator and 6-generator systems with and without
consideration of transmission losses. The final results obtained using PSO are compared with conventional
quadratic programming and found to be encouraging.
[7] proposed a method for solving economic dispatch problem using Particle Swarm Optimization (PSO)
Algorithm and Simulated Annealing (SA) for the three generating units as a case study. PSO and SA were
applied to find out the minimum cost for different power demand. They compared their results with the
traditional technique, where PSO displayed better result and better convergence characteristic.
[8] presented an improved exponential harmony search algorithm for improving the HS algorithm to solve the
SELD problem considering the valve-point effect. The numerical results show that the proposed method has
better convergence and also lower production costs than the conventional HS and particle swarm optimization
methods.
[10] presented the overview of different methods for solving economic load dispatch (ELD) problem using
MATLAB. They concluded that lambda iteration method converges rapidly but complexities increases as
system size increase. Gradient and newton methods can only be applied where cost function is much more
complex while for the non convex input-output curves, dynamic programming method can be used to solve the
economic load dispatch problem.
[11] presented lambda iteration method to solve the ELD problem using MATLAB for the three and six
generating units with and without transmission losses.
[12] presented an application of the GAMS method to power economic dispatch (PED) problem with Power
loss for 3 and 6 generator test case systems. The simulation results show that the proposed GAMS Method
outperforms previous optimization methods.
2.1 Heuristic Optimization Techniques
As an alternative to the conventional mathematical approaches, the heuristic optimization techniques such as
genetic algorithms, Tabu search, simulated annealing, and recently introduced particle swarm optimization

(PSO) are considered as realistic and powerful solution schemes to obtain the global optimums in power system
optimization problems.
2.2 Particle Swarm Optimization
Particle swarm optimization was first introduced by Kennedy and Eberhart in the year 1995. It is an exciting
new methodology in evolutionary computation and a population-based optimization tool. PSO is motivated from
the simulation of the behaviour of social systems such as fish schooling and birds flocking. The PSO algorithm
requires less computation time and less memory because of its inherent simplicity [7].
The basic assumption behind the PSO algorithm is that birds find food by flocking and not individually. This
leads to the assumption that information is owned jointly in the flocking. The swarm initially has a population of
random solutions. Each potential solution, called a particle (agent), is given a random velocity and is flown
through the problem space.
All the particles have memory and each particle keeps track of its previous best position (pbest) and the
corresponding fitness value. The swarm has another value called gbest, which is the best value of all the pbest.
Particle swarm optimization has been found to be extremely effective in solving a wide range of engineering
problems and solves them very quickly.
III. APPLICATION OF PSO IN ELD
The aim of this research is to distribute the total power demand among the available thermal generating stations
to minimizing the total fuel cost subject to both equality and inequality constraints as earlier stated in Eqns. 1-3.
Particle Swarm Optimization was used to achieve this desired goal.
The solution of the ELD problem using the classical approach presents some limitation in its implementation.
One of such limitation is that the Lamda-iteration method assumes the cost coefficient to be a continuous
function. The method breaks down when it is applied to a discontinuous function with prohibited zones or larger
steam turbine generating units [13]. Also, there is a large tendency for this approach to converge at a local
minimum when the power system operating status is far outside the normal situation, for instance during and
after large disturbances. For this purpose, the Particle Swarm Optimization technique was applied in this paper
to solve an ELD problem in order to eliminate the limitation of the Lamda-iterations enumerated above.
In a PSO system, population of particles exists in the n-dimensional search space. Each particle has certain
amount of knowledge and will move about the search space on the basis of this knowledge. The particle has
some inertia attributed to it and hence will continue to have a component of motion in the direction it is moving.
The Particle knows its location in the search space and will encounter with the best solution. The particle will
then modify its direction such that it has additional components towards its own best position, pbest and towards
the overall best position, gbest. The particle updates its velocity and position using the following Eqn. (6) and
(7).
𝑉𝑖
𝑘+1
= 𝑤𝑉𝑖
𝑘
+ 𝑐1 𝑅𝑎𝑛𝑑1() × 𝑝𝑏𝑒𝑠𝑡𝑖 − 𝑆𝑖
𝑘
+ 𝑐2 𝑅𝑎𝑛𝑑2() × 𝑔𝑏𝑒𝑠𝑡𝑖 − 𝑆𝑖
𝑘
… (6)
( 1 ) ( 1 )k K k
i i i
S S V
 
  …(7)
Where, k
i
V is the velocity of individual i at iteration k,
k is pointer of iterations,
W is the weighing factor,
C1,C2 are the acceleration coefficients,
Rand1( ), Rand2( ) are the random numbers between and 1,
Sk
is the current position of individual i at iteration k,
pbesti is the best position of individual i and
gbest is the best position of the group.
The coefficients C1 and C2 pull each particle towards pbest and gbest positions. Low values of acceleration
coefficients allow particles to roam far from the target regions, before being tugged back. Hence, the
acceleration coefficients C1 and C2 are often set to be 2 according to past experiences. The term C1Rand1 () x
(pbesti -Si
k
) is called particle memory influence or cognition part which represents the private thinking of the
itself and the term C2 Rand2( ) x (gbest - Si
k
) is called swarm influence or the social part which represents the
collaboration among the particles.
In the procedure of the particle swarm paradigm, the value of maximum allowed particle velocity Vmax
determines the resolution, or fitness, with which regions are to be searched between the present position and the
target position. If Vmax
is too high, particles may fly past good solutions. If Vmax
is too small, particles may not
explore sufficiently beyond local solutions. Thus, the system parameter Vmax
has the beneficial effect of

preventing explosion and scales the exploration of the particle search. The choice of a value for Vmax
is often set
at 10-20% of the dynamic range of the variable for each problem.
Suitable selection of inertia weight W provides a balance between global and local explorations, thus requiring
less iteration on an average to find a sufficiently optimal solution. Since W decreases linearly from about 0.9 to
0.4 quite often during a run, the following weighing function of Eqn. (8) is used in Eqn. (6):
m a x m in
m a x
m a x
W W
W W ite r
ite r

   …(8)
Where, Wmax is the initial weight,
Wmin is the final weight,
Itermax is the maximum iteration number,
iter is the current iteration number.
Eqn. (6) is used to calculate the particle's new velocity according to its previous velocity and the distances of its
current position from its own best experience (position) and the group's best experience. Then the particle flies
towards a new position according to Eqn. (7). The performance of each particle is measured according to a
predefined fitness function, which is related to the problem to be solved.
3.1 Implementation of PSO for ELD
The main objective of ELD is to obtain the amount of real power to be generated by each committed generator,
while achieving a minimum generation cost within the constraints. The evaluation function for evaluating the
minimum generation cost of each individual in the population is adopted as follows:
Minimize
1
( )
d
T i i
i
F F P

  …(9)
The search procedure for calculating the optimal generation quantity of each unit is summarized as follows:
i) In the ELD problems the number of online generating units is the 'dimension' of this problem. The particles
are randomly generated between the maximum and the minimum operating limits of the generators and
represented using equation (9).
ii) To each individual of the population calculate the dependent unit output Pdu from the power balance
equation and employ the B-coefficient loss formula to calculate the transmission loss PL using constraint
satisfaction technique.
iii) Calculate the evaluation value of each individual Pgi in the population using the evaluation function f, given
by equation (10).
iv) Compare each individual's evaluation value with its pbest. The best evaluation value among the pbest is
identified as gbest.
v) Modify the member velocity V of each individual Pgi according to the following equation:
𝑉𝑖𝑑
𝑡+1
= 𝑤𝑉𝑖𝑑
𝑡
+ 𝑐1 𝑅𝑎𝑛𝑑1() × 𝑝𝑏𝑒𝑠𝑡𝑖𝑑 − 𝑃𝑔𝑖𝑑
𝑡
+𝑐2 𝑅𝑎𝑛𝑑2() × 𝑔𝑏𝑒𝑠𝑡 𝑑 − 𝑃𝑔𝑖𝑑
𝑡
i =1,2 ,. n, d =1,2, .. m … (10)
Where, n is the population size, m is the generator units.
vi) Check the velocity constraints of the members of each individual from the following conditions:
( 1) m ax ( 1) m ax
( 1) m ax ( 1) m in
m in m in m ax m ax
, ,
, ,
0.5 , 0.5
t t
id d id d
t t
id d id d
d d d d
ifV V thenV V
ifV V thenV V
w hereV P V P
 
 


   

 …(11)
vii) Modify the member position of each individual Pg1 according to Eqn.(12):
( 1) ( ) ( 1)
id
t t t
g gid id
P P V
 
  …(12)
( 1)
id
t
g
P

must satisfy the constraints, namely the generating limits, described by Eqn. (5). If
( 1)
id
t
g
P

violates the
constraints, then
( 1)
id
t
g
P

must be modified towards the nearest margin of the feasible solution.
viii)If the evaluation value of each individual is better than previous pbest, the current value is set to be pbest. If
the best pbest is better than gbest, the best pbest is set to be gbest.
ix) If the number of iterations reaches the maximum, then go to step (x). Otherwise, go to step (ii).
x) The individual that generates the latest gbest is the optimal generation power of each unit with the
minimum total generation cost.

IV. RESULTS AND DISCUSSION
The procedure described in chapter three for the Particle Swarm Optimization based for solving ELD problems
for the thermal power plant has been implemented using the developed PSO software on Matlab 7.1 for
windows. The feasibility and the effectiveness of the method have been tested on the Nigerian thermal power
plant. This was executed on hp laptop computer with the specification as follows, Processor: Intel® Celeron®
CPU N2830 @ 2.16GHz, installed Memory (RAM): 2GB, system type: 64-bit operating system, hard disc:
500GB and operating system: windows 8.1. The results of the stimulation studies are presented.
[1] used the two approaches to solve this problem; micro genetic algorithm (MGA) and conventional genetic
algorithm (CGA). [2] used differential evolution (DE) to solve the same problem. They all used three sets of
power demand PD: 340MW, 850MW and 1150MW.
PSO was applied to the above system for obtaining economic load dispatch of similar load requirements. PSO
was implemented according to the flow chart shown. For each sample load, under the same objective function
and individual definition, 20 trials were performed to observe the evolutionary process and to compare their
solution quality and convergence characteristics.
4.1 Simulation Results of the Nigerian Power System
The developed PSO software for ELD problem was applied to the Nigerian power system whose single line
diagram is shown in Figure 2. The Nigerian power system grid is essentially a 31-bus, 330-kV network
interconnecting four thermal generating stations and three hydro stations to the various load points. The
network data (Bus data, Generator data and Branch) were obtained from [1, 2]. Table 1 presents the cost
coefficients of the four Nigerian thermal power stations and their minimum and maximum loading limits.
Figure 2: Nigerian 330 kV, 31- Bus Grid System
Table 1: Nigerian Thermal Power Plants Characteristics [1]
Station Α Α Α PG
min
(MW) PG
max
(MW)
Sapele 6929.0 7.84 0.13 137.5 550.0
Delta 525.74 -6.13 1.20 75.0 300.0
Afam 1998.0 56.0 0.092 135.0 540.0
Egbin 12787.0 13.1 0.031 275.0 1100.0

Table 2: Results Comparison between MGA. CGA, DE and the Proposed PSO Techniques
Power Station MGA CGA DE Proposed Method
Sapele PG1(MW) 365.35 482.79 390.931 525.0564
Delta PG2(MW) 78.48 79.51 80.008 76.5315
Afam PG3(MW) 340.46 255.82 343.363 281.1126
Egbin PG4(MW) 878.39 844.56 848.078 839.5335
Kainji PG5(MW) 350 350 350 350
Shiroro PG6(MW) 490 490 490 490
Jebba PG7(MW) 450 450 450 450
Total Power Generated(MW) 2952.80 2952.68 2952.38 3012.242
Total Power Demanded(MW) 2823.10 2823.10 2823.10 2975.203
Total Power Loss (MW) 130.28 130.28 129.28 37.04
Total Cost(N/hr) 114,521.33 116,946.55 107,430.00 97,321.00
Figure 3: Convergence Characteristics of PSO
V. CONCLUSION
PSO method was successfully employed to solve the ELD problem. The comparison of results for the 31-bus
Nigerian grid system clearly shows that the proposed PSO method was indeed capable of obtaining high quality
solution efficiently for ELD problems. Figure 3 shows the convergence characteristics of the proposed method
at the normal demand. The convergence is good since the algorithm takes few numbers of iterations to converge
hence less computation time. From the results obtained, the proposed PSO technique minimizes the total
production cost and transmission losses better than MGA and CGA, except in some cases where the DE also
performed equally good.
REFERENCES
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