Evolutionary algorithm solution for economic dispatch problems

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 12, No. 3, June 2022, pp. 2963~2970
ISSN: 2088-8708, DOI: 10.11591/ijece.v12i3.pp2963-2970  2963
Journal homepage: http://ijece.iaescore.com
Evolutionary algorithm solution for economic dispatch
problems
Balasim Mohammed Hussein
Department of Electrical Power and Machine Engineering, College of Engineering, University of Diyala, Diyala, Iraq
Article Info ABSTRACT
Article history:
Received Jul 5, 2020
Revised Jan 4, 2022
Accepted Jan 25, 2022
A modified firefly algorithm (FA) was presented in this paper for finding a
solution to the economic dispatch (ED) problem. ED is considered a difficult
topic in the field of power systems due to the complexity of calculating the
optimal generation schedule that will satisfy the demand for electric power
at the lowest fuel costs while satisfying all the other constraints.
Furthermore, the ED problems are associated with objective functions that
have both quality and inequality constraints, these include the practical
operation constraints of the generators (such as the forbidden working areas,
nonlinear limits, and generation limits) that makes the calculation of the
global optimal solutions of ED a difficult task. The proposed approach in
this study was evaluated in the IEEE 30-Bus test-bed, the evaluation showed
that the proposed FA-based approach performed optimally in comparison
with the performance of the other existing optimizers, such as the traditional
FA and particle swarm optimization. The results show the high performance
of the modified firefly algorithm compared to the other methods.
Keywords:
Economic dispatch
Firefly
Fuel cost functions
M-FA
PSO
This is an open access article under the CC BY-SA license.
Corresponding Author:
Balasim Mohammed Hussein
Department of Electrical Power and Machine Engineering, College of Engineering, University of Diyala
Diyala, Iraq
Email: balasim@inbox.ru
NOMENCLATURE
FCz : Fuel cost of plant (z) expressed by $/h.
Pz : Real power generation of plant z.
a, b, c : Cost function constant.
N : Number of power plants.
Pz min : Minimum limit for real power generator (z) expressed by MW.
Pz max : Maximum limit for real power generator (z) expressed by MW.
PL : Total power losses in MW.
PD : Real power load in GfW.
B : B coefficients.
1. INTRODUCTION
It is important that power systems should be of a high grade and economically feasible, meaning
that the cost of building the system must be optimized. In the power systems, the economic dispatch (ED)
issue portrays the expected level of load that must be partitioned between the generators to ensure minimal
operating cost. The concept of optimization demands minimization of the objective functions while
maintaining a reasonable and acceptable level of system performance [1], [2]. Typically, ED is considered an

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important area of the control and operation of power systems as its main objective of the ED plant is to
schedule the operations of the generating units to ensure maximum performance at the minimum operation
cost; this amounts to low-cost electricity on the side of the customers and possible gain on the side of the
service provider in the electricity market. Being that the generated power by the generating units cannot be
stored as other types of products, coupled with the need for transmission and distribution networks to get the
generated power to the consumers, it is becoming difficult to accurately determine the cost of electric power
generation and services [3], [4]. The study by Touma [5] proposed the whale optimization approach of
finding a solution to the ED problem. The proposed system was evaluated in a standard IEEE 30-Bus testbed
and the performance was compared with those of other optimization algorithms, such as elephant heart
optimization (EHO), Garra Rufa optimization (GRO), particle swarm optimization (PSO), and genetic
algorithm [6]. Another study by Victoire and Jeyakumar [7] reported the use of PSO to solve different forms
of ED, including the two-area ED with tie line limits, ED of generators with forbidden operation zones, and
ED with different fuel choices. The study relied on quadratic programming for the modification of the PSO
used to determine the optimal solution. In [8], this study use teaching and learning based optimization
(TLBO) to analysis the practical non-convex economic demand dispatch objective. The presented method is
applied to determine demand dispatch for six units and ten units standard network among other methods.
While in [9], Rajesh and Visali presented hybrid method, modified non-dominated sorted genetic algorithm
and modified population variant differential evolution for solving and optimization economic demand
dispatch ED problems. The cost of electrical power generation, which reduced by sharing load between
power generation plant is a main target of the problem.
Genetic algorithm (GA) has been proposed in [10] for solving the ED issue; the study incorporated a
novel heuristic mechanism for finding infeasible solutions to the feasible area. The behavior of the algorithm
was enhanced using the dynamic relaxation for equality constraints and diversity mechanism. A hybrid
system called fuzzy-based hybrid-PSO has been presented by [11], for finding a solution to the ED problem.
To make the obtained results more practicable, the study investigated the actual conditions, valve-point
action, and containing multi-fuels process during the evaluation of the proposed hybrid system. The study by
Daniel and Chaturvedi [12] presented an algorithm for addressing the ED problem; the algorithm was
developed for the reduction of the nonlinear behavior of the fuel cost of power generation subject to practical
constraints. The evaluation was done on the IEEE standard system testbed and from the results, the presented
solution was efficient in finding solutions to the ED problem. A genetic algorithm approach to the ED
problem of thermal generators has been presented by [13]; the study applied the proposed method in 3 and 6
generator test systems. The considered system in the system is a lossless system. Another study by Chopra
and Kaur [14] presented a modified GA approach to the ED problem with the aim of minimizing costs and
overworking constraints. In this approach, the iteration method requires a precise modification of the Lambda
even though it does not guarantee the global optimal solution.
A genetic algorithm-based system has been presented by [15] for solving the ED problem; the
system is for the determination of the global solution to ED considering the transmission line losses. A fuzzy
based GA for solving ED problems has been presented by [16]; the system relies on the control and
exploitation capabilities of the heuristic search theory (the basis of the GA) to explore and exploit the
solution space more efficiently. In the study in [17], [18], a new stochastic optimization technique was
presented for finding a solution to the ED problem; the techniques depend on the hybrid bacterial foraging-
differential evolution (BFOM) method. This method combined the chemotaxis calculation of bacterial
foraging optimization algorithm (BFOA) BFOA (considered a stochastic gradient search) with the crossover
and mutation parameters of GA. In this study, a modified FA was proposed for solving the ED problem; the
proposed system was evaluated using a standard IEEE 30-Bus testbed and the performance was compared
with that of other existing optimization techniques, such as the traditional FA and PSO.
2. ECONOMIC DISPATCH PROBLEM
The ED problem is mainly concerned with the determination of the optimal distribution of power in
a manner that minimizes the overall operation cost while considering the equality and inequality constraints.
The operation characteristics of a generator with different fuel choices can be described using a second-order
function. For instance, the (1) defines an ED problem with a piece-wise quadratic function.
𝐹𝐶𝑧 = 𝑎𝑧𝑃𝑧
2
+ 𝑏𝑧𝑃
𝑧 + 𝑐𝑧 (1)
The following are considered constraints to the cost function:
− The inequality conditions in the formularization of the ED problem are described by the generator limits.

Int J Elec & Comp Eng ISSN: 2088-8708 
Evolutionary algorithm solution for economic dispatch problems (Balasim Mohammed Hussein)
2965
Pz min ≤ Pz ≤ Pz max , i = 1, … … . . , n
− The losses from the transmission line have an impact on the optimal flow of power in the power system.
These losses can be mathematically written as in (2).
𝑃𝐿 = ∑ ∑ 𝑃𝑧
𝑛
𝑧=1
𝑛
𝑧=1 𝐵𝑧𝑗𝑃𝑗 + ∑ 𝐵0𝑗
𝑛
𝑗=1 𝑃𝑗 + 𝐵0 (2)
For a variable system demand, the B coefficients are to be determined. The electrical equality conditions for
ED are represented in (3).
𝑃𝐷 = ∑ 𝑃
𝑧
𝑛
𝑧=1 − 𝑃𝐿 (3)
The objective function of the economic dispatch is minimized as in (4).
(FC) = min ∑ FCz
n
z=1 (4)
3. CASE STUDIED
A modified firefly algorithm (M-FA) was proposed in this article; the technique was evaluated on a
standard IEEE 30-Bus testbed with six power plants and 41 interconnected lines. The electric power load of
the system is 0.3 GW. The simulation study was performed in the MATLAB R2017b platform. Figure 1
showed the grid for the simulation of the proposed system while Tables 1 and 2 described the datasheet of the
test system [1].
Figure 1. Interconnected grid of IEEE 30-Bus

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Table 1. Cost equation factors
Generator Number Fuel cost factors Capacity levels (MW)
a($/MWhr) b($/MWhr) c($/hr) Pmin Pmax
1 0.0037 2.0000 0 50 200
2 0.0175 1.7500 0 20 80
3 0.0625 1.0000 0 15 50
4 0.0083 3.2500 0 10 35
5 0.0250 3.0000 0 10 30
6 0.0250 3.0000 0 12 40
Table 2. Fuel cost function parameter
B=
0.0002180 0.0001030 0.0000090 -0.00010 0.0000020 0.0000270
0.0001030 0.0001810 0.0000040 -0.0000150 0.0000020 0.0000300
0.0000090 0.0000040 0.0004710 -0.0001310 -0.0001530 -0.0001070
-0.000100 -0.0000150 -0.0001310 0.0002210 0.0000940 0.0000500
0.0000020 0.0000020 -0.0001530 0.0000940 0.0002430 -0.000000
0.0000270 0.0000300 -0.0001070 0.0000500 -0.000000 0.0003580
B00=-0.0000030 0.0000201 -0.0000560 0.0000340 0.0000150 0.000078
B0=0.0000140
4. MODELLING OF PSO ALGORITHM
PSO was developed in 1995 as a swarm-based heuristic by [19], [20]. In the PSO, each particle is
considered a potential solution, and the best particle in the solution space, p, is defined by the character, g. At
each iteration step, the speed of the kth
particle which is given as vɉ=(vɉ1, vɉ2, . . ., vɉd), is updated over each
axis j using the formula:
𝑣ɉ(𝑡 + 1) = 𝜔𝑣ɉ(𝑡) + 𝑐1𝑟1(Ƥɉ(𝑡) − Үɉ(𝑡)) + 𝑐2𝑟2(𝐺𝑘(𝑡) − 𝑋𝑘(t)) (5)
where, c1 and c2 represent the acceleration coefficients, Ƥɉ=local solution and Gk is the global solution at
each iteration, and ω=the inertia weight. The range of velocity of the particle is fixed and can be updated at
iteration:
𝑣𝑘𝑗 = [−𝑉
𝑚𝑎𝑥, 𝑉
𝑚𝑎𝑥] (6)
The following function is applied to determine the new location of a particle:
Үɉ(𝑡 + 1) = Үɉ(𝑡) + 𝑣ɉ(𝑡 + 1) (7)
This new coordinates of the particle are updated by (8):
Ƥɉ(𝑡 + 1) = {
Ƥɉ𝑡𝑓 (Үɉ(𝑡 + 1)) < 𝑓(Ƥɉ(𝑡))
Үɉ(𝑡 + 1)𝑓 (Үɉ(𝑡 + 1)) < 𝑓(Ƥɉ(𝑡))
(8)
where the global best index is:
𝑔 = arg min 𝑓(Үɉ; 𝑡 + 1)), 1 ≤ ɉ ≤ 𝑁 (9)
5. FIREFLY ALGORITHM
Abdullah et al. [21] introduces the FA for the first time in 2008 as a nature-inspired optimization
technique. The FA is inspired by the flashing nature of fireflies which are unisexual insects and hence, they
moved to each other with no gender preferences [22]. However, the flash of the firefly is the indication to
pull other fireflies. The attractiveness among fireflies is associated with their brightness, and for every two
fireflies, the one with less bright flash moves to the shining ones and so on. The brightness decrease as the
fireflies are onward from each other. If particles are equal in their flashes’ brightness, the movement will be
random. Like other evolutionary algorithms, FA was employed for the setting of the control parameters; here,
each firefly is considered a potential solution and described based on the position.

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The current position of a firefly in the d dimensional vector space is given as
𝑋𝑘 = (𝑥𝑘1, … . . , 𝑥𝑘𝑛,… . . , 𝑋𝑘𝑑). However, the initialization of the random positions of n fireflies is done
within a specified range. Hence, the formulation of the changes in position k subject to attraction by a
brighter firefly j is given thus:
𝑋𝑘(𝑡 + 1) = 𝑋𝑘(𝑡) + 𝛽0 exp(−𝛾𝑟𝑘𝑗)(𝑥𝑗 − 𝑥𝑘) + 𝛼(𝑟𝑎𝑛𝑑 − 0.5) (10)
where the changes in the position of the brighter firefly are captured thus:
𝑥𝑏𝑒𝑠𝑡(𝑛𝑒𝑥𝑡) = 𝑥𝑏𝑒𝑠𝑡(𝑐𝑢𝑟𝑟𝑒𝑛𝑡) + 𝛼(𝑟𝑎𝑛𝑑 − 0.5) (11)
In (10) and (11), the first terms are the current positions of the attracted firefly and the brightest firefly,
respectively while the second term in (10) represents the attractiveness of the firefly to the brighter light
intensity. Assume that β0=the initial attractiveness at r = 0, γ= the absorption parameter (ranging from 0 to 1),
and r=the distance between any 2 fireflies (k and j) at positions 𝑥𝑘 and 𝑥𝑗 respectively, then, the Euclidean
distance can be formulated as:
𝑟𝑘𝑗 = √∑ (𝑥𝑘,𝑛
𝑑
𝑛=1 − 𝑥𝑗,𝑛)2
(12)
where 𝑥𝑘 and 𝑥𝑗= the location vectors for k and j, respectively, and 𝑥𝑘,𝑛= the location value of the dimension. To
decrease the randomness, the third terms in (10) and (11) are applied; this implies a gradual reduction of the
velocity of the fireflies through 𝛼 = 𝛼0𝜌𝑘, where α0 ranges from 0 to 1, δ is the randomness reduction factor,
where (0.0 < 𝜌 < 1.0), and k= the counter of iterations.
6. THE PROPOSED ALGORITHM
The FA has been previously applied in the optimization of the application of different power
systems. This work strives to decrease the problem of local optima entrapment of FA to improve its search
ability via some modification steps on the FA algorithm. The following enhancements were made to the FA
to achieve the proposed M-FA [23], for every 2 mutations, 3 crossover processes are considered. The overall
generations are to be pushed toward optimal (either local or global) [24], [25]. In each iteration, the
modification must be done to complete the following two steps:
Үµ = Үµ + 𝛿 × (Үµ+1 − Үµ+2) (13)
Үµ2 = Үµ1 + 𝛿 × (Ү𝐵𝑒𝑠𝑡
𝑖𝑡𝑒𝑟
− Ү𝑊𝑜𝑟𝑠𝑡
𝑖𝑡𝑒𝑟
) (14)
where, δ is a number between [0, 1], Where (Ү1, Ү2 𝑎𝑛𝑑 Ү3) are random elements of firefly,
Ү𝑏𝑒𝑡𝑡1 = [𝑦𝑏𝑒𝑡𝑡1,𝑦𝑏𝑒𝑡𝑡2, …… , 𝑦𝑏𝑒𝑡𝑡𝑛 ]
Ү𝐵𝑒𝑠𝑡1 = [𝑦𝐵𝑒𝑠𝑡1,𝑦𝐵𝑒𝑠𝑡2, … … , 𝑦𝐵𝑒𝑠𝑡𝑛 ] (15)
Ү𝑖𝑚𝑝𝑟𝑜𝑣𝑒1,ɉ = {
Үµ1,ɉ ,𝑖𝑓 ɉ1 ≤ ɉ2
Ү𝐵𝑒𝑠𝑡1,ɉ 𝑖𝑓 ɉ1 > ɉ2
} (16)
Үµ1,ɉ ,𝑖𝑓 ɉ2 ≤ ɉ3
Ү,ɉ 𝑖𝑓 ɉ2 > ɉ3
} (17)
Ү𝐵𝑒𝑠𝑡1,ɉ ,𝑖𝑓 ɉ3 ≤ ɉ4
Ү,ɉ 𝑖𝑓 ɉ3 > ɉ4
} (18)
where, Ү𝑏𝑒𝑡𝑡
𝑖𝑡𝑒𝑟
𝑎𝑛𝑑 Ү𝑏𝑎𝑑
𝑖𝑡𝑒𝑟
are the better and the bad population respectively in each iteration, ɉ1 − ɉ5 have
amount in range [0, 1]. However, fireflies have to locate the objectives in order to meet the best value for
their iteration. The flowchart of the proposed algorithm is illustrated in Figure 2.

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Figure 2. Flowchart of the modified firefly algorithm
7. RESULTS AND DISCUSSION
The proposed M-FA algorithm was compared with the traditional FA and PSO algorithms for
identifying the impact of the modification proposed in this paper. The evaluation was done on a power
system of 6 units using MATLAB 2017b. The actual power generation by each unit based on the three
employed optimization methods was presented in Table 3, this is an important aspect of the optimal ED
calculation. The overall cost of the system when using each of the considered algorithms is presented in
Table 4. The table showed that the proposed MFA performed better cost reduction than PSO (by >0.3%) and
original FA (by >0.24%) within 24 h. Hence, the proposed M-FA outperformed PSO and original FA. The
calculated overall power losses by the three algorithms were 8.9140 MW (for MFA), 8.9144 MW (for PSO),
and 8.9172 MW (for original FA). The error and run time for each algorithm are presented in Table 5.
Observably, FA required the shortest time to achieve results but with bigger error margin while M-FA took
longer time to achieve results with the least error margin. Hence, the proposed M-FA is more suitable for
practical applications.
Table 3. Real generation of six generators in system, (0.3 GW load)
Algorithm P1(GW) P2(GW) P3(GW) P4(GW) P5(GW) P6(GW)
PSO 0.177316 0.051200 0.020318 0.019637 0.011764 0.012000
FA 0.177919 0.05136 0 0.020363 0.019959 0.011878 0.012000
M-FA 0.176959 0.05110 0 0.020291 0.019446 0.011697 0.012000
Table 4. Economic dispatch in (0.3 GW)
Evolutionary Algorithm PL(MW) Total Cost (Fuel Cost) $/hr
PSO 8.9144 799.7964
FA 8.9172 800.0274
M-FA 8.9140 799.7636

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Table 5. Error and run time of each algorithm
Evolutionary Algorithm Error Run Time
PSO 0.0082 0.7970
FA 0.0653 0.0881
M-FA 0.0079 0.1273
Figure 3 presents the overall cost input of using each of the evaluated methods. Notably, the
proposed M-FA achieved the least cost function compared to PSO and FA within 24 h of operation. This
suggests that M-FA can produce better performance in practical terms compared to FA and PSO. The rate of
changes in error for each method per iteration is captured in Figure 4. The figure showed that the error is not
constant. Despite the error margin in each method, the M-FA exhibited the least and more stable error margin
as shown in Figure 4.
Figure 3. PSO, FA and M-FA algorithms cost
Figure 4. PSO, FA and M-FA algorithms error
8. CONCLUSION
This study presented a modified FA for the determination of the optimal solution for ED problems.
The evaluation of the proposed system was done on the IEEE 30-Bus testbed with 6 generating units. From
the results, it can be concluded that the proposed method achieved optimal performance in providing a
solution to ED problems. The performance of the modified FA in comparison to the PSO and FA showed the
impact of the modification propose in this study. In the future, efforts can be channelled on other real high
complex electric power system, for example the Iraqi power system.
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[25] B. M. Hussein and A. S. Jaber, “Unit commitment based on modified firefly algorithm,” Measurement and Control, vol. 53, no.
3–4, pp. 320–327, Mar. 2020, doi: 10.1177/0020294019890630.
BIOGRAPHIES OF AUTHORS
Balasim Mohammed Hussein obtained his BSc in Electrical Power Engineering
from Diyala University (Iraq) in 2004 and a Master’s in Electrical Power Engineering from the
Technical University (Iraq) in 2008. He acquired his PhD from the Russian South State
University (Russia) in 2015. He is currently a faculty member in College of Engineering,
University of Diyala, Baqubah, Iraq. He has several articles in many journals like Science
Direct, measurement and control, Electromechanical Journal. He can be contacted at email:
balasim@inbox.ru.

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