Comparative study of the price penalty factors approaches for Bi-objective dispatch problem via PSO

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 10, No. 4, August 2020, pp. 3343~3349
ISSN: 2088-8708, DOI: 10.11591/ijece.v10i4.pp3343-3349  3343
Journal homepage: http://ijece.iaescore.com/index.php/IJECE
Comparative study of the price penalty factors approaches for
Bi-objective dispatch problem via PSO
Mohammed Amine Meziane1
, Youssef Mouloudi2
, Abdelghani Draoui3
1,2,3
Departement of Electrical Engineering, University of Tahri Mohammed, Bechar, Algeria
1,2,3
Smart Grids and Renewable Energies Laboratory (SGRE), Algeria
Article Info ABSTRACT
Article history:
Received Jul 28, 2018
Revised Jan 24, 2020
Accepted Feb 2, 2020
One of the main objectives of electricity dispatch centers is to schedule
the operation of available generating units to meet the required load demand
at minimum operating cost with minimum emission level caused by
fossil-based power plants. Finding the right balance between the fuel cost
the green gasemissionsis reffered as Combined Economic and Emission
Dispatch (CEED) problem which is one of the important optimization
problems related the operationmodern power systems. The Particle Swarm
Optimization algorithm (PSO) is a stochastic optimization technique which is
inspired from the social learning of birds or fishes. It is exploited to solve
CEED problem. This paper examines the impact of six penalty factors like
"Min-Max", "Max-Max", "Min-Min", "Max-Min", "Average" and
"Common" price penalty factors for solving CEED problem. The Price
Penalty Factor for the CEED is the ratio of fuel cost to emission value.
This bi-objective dispatch problem is investigated in the Real West Algeria
power network consisting of 22 buses with 7 generators. Results prove
capability of PSO in solving CEED problem with various penalty factors and
it proves that Min-Max price penalty factor provides the best compromise
solution in comparison to the other penalty factors.
Keywords:
Bi-objective dispatch problem
Combined economic emission
dispatch
Particle swarm optimization
Price penalty factor
Real West Algeria electrical
network
Copyright © 2020 Institute of Advanced Engineering and Science.
All rights reserved.
Corresponding Author:
Mohammed Amine Meziane,
Faculty of Science and Technology, Departement of Electrical Engineering,
Smart Grids and Renewable Energies Laboratory (SGRE),
University of Tahri Mohammed,
BP 417, 08000 Bechar, Algeria.
Email: aminemohmeziane@gmail.com
1. INTRODUCTION
Electric utility systems are interconnected to achieve high operating efficiency and to produce cheap
electricity with minimum production cost, maximum reliability, and better operating conditions [1].
The optimal power flow problem (OPF) is an important tool in operation and control of large modern power
systems, it was first discussed by Carpentier in 1962 [2], the main purpose of OPF is to find the optimal
output power of generators to minimize the total generation cost and satisfy the equality and inequality
constraints. Operating at absolute minimum cost can no longer be the only criterion for dispatching electric
power due to increasing concern over the environmental issues. The generation of electricity from fossil fuel
resources releases several contaminants, such as SOx, NOx and CO2 into the atmosphere [3]. In this paper
the used term Economic Dispatch Problem (ED) is the short-term which refers to the determination of
the optimal output of a number of electricity generation facilities.
The aim of every generating station is to produce electricity at the lowest possiblefuel consumption
and emission rates, but these two constraints cannot be metsimultaneously. Nowadays,the demand for energy
is increasing at a high pace, which makes it highly crucial to run generators at very minimal cost. This is
the main goal of an Economic Dispatch Problem. With the exceptional production of carbon emissions by

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 10, No. 4, August 2020 : 3343 - 3349
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thermal power plants [4], the environmental issues has become a big concern which has to be addressedto
mitigate the effects of pollution and hence rectify problem of global warming. Therefore, production of
electricity with an optimized costat a lower green gas emmissionsacts as two vital parts of economic dispatch
problem. Production at the minimum cost result in a relatively high amount of emissions. Similarly, ensuring
minimum gas emissions limits the production of utilities running on fossil fuels. In order to find a right
balance in the present tradeoff, this optimization problem can be modelled asa multi-objective function
(Economic/Emission) which involves minimization of the cost function of producing electrical energy and
minimization of the gas emission function, by satisfying the constraints of both functions.
In the modeling of the bi-objective economic dispatch problem, the presentcomparative study
examines different types of the constraints and various types of price penalty factors. The following
parameters are considered:
a. Fuel cost and emission functions are modelled as second order polynomial function for both.
b. The following types of price penalty factors are used for the multi-objective dispatch problem:
- Min-Max price penalty factor
- Max-Max price penalty factor
- Min-Min price penalty factor
- Max-Min price penalty factor
- Average price penalty factor
- Common price penalty factor
c. Type of constraints to be satisfied are:
- Load/supply balance
- Minimum/maximum limits of the energy produced by the generators
- Transmission line losses
In order to overcome the above illustrated drawbacks, heuristic methodologies have been under
research for solving CEED problem. In the past the traditional methods used to solve this economic load
dispatch problem are the Lambda iteration method, Gradient, Newton, linear programming and interior point
method. Recently, meta-heuristic techniques such as Simulated Annealing, Genetic Algorithm (GA), Particle
Swarm Optimization (PSO), and Tabu search algorithm are used to solve this problem [5]. In this paper,
the Particle Swarm Optimization based-approach is proposed to solve the CEED problem. In order to
facilitate the search for the optimized solution, the price penalty factor is used to convert the bi-objective
CEED problem into a single objective function. The proposed method has been examined and tested on a real
grid in west Algeria which consists of a 22-bus system of 220 Kvvoltage level. Satisfactory simulation results
show the effectiveness of the proposed algorithm.
2. MATHEMATICAAL FORMULATION OF CEED PROBLEM
The bi-objective function for CEED problem [6-12] is given as follows:
𝐶𝐸𝐸𝐷 = 𝑀𝑖𝑛(𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑡) + 𝑝𝑒𝑛𝑎𝑙𝑡𝑦 𝑓𝑎𝑐𝑡𝑜𝑟 × 𝑀𝑖𝑛(𝐸𝑚𝑖𝑠𝑠𝑖𝑜𝑛 𝑣𝑎𝑙𝑢𝑒)
𝐹𝑐 = 𝑀𝑖𝑛 ∑ 𝐹𝑖(𝑃𝐺𝑖)𝑛𝐺
𝑖=1 = 𝑀𝑖𝑛 ∑ (𝑎𝑖 𝑃𝐺𝑖
2
+ 𝑏𝑖 𝑃𝐺𝑖 + 𝑐𝑖)𝑛𝐺
𝑖=1 (1)
where Fc is the total fuel cost of the system is, ng is the number of generators, PGi is real power generation of
a generator unit i, and ai,biand ciare the cost coefficients of the ith
generating unit.
𝐸 𝑇 = 𝑀𝑖𝑛 ∑ (𝛼𝑖 𝑃𝐺𝑖
2
+ 𝛽𝑖 𝑃𝐺𝑖 + 𝛾𝑖)𝑛𝐺
𝑖=1 (2)
where, 𝐸 𝑇 is total emission; 𝛼𝑖, 𝛽𝑖,𝛾𝑖are emission coefficients of generating unit iin [kg/MW2
h], [kg/MWh]
and [kg/h] respectively. Price penalty factorℎ𝑖 is used to convert the bi-objective CEED optimization problem
into a single objective [6-13] problem:
𝐹𝑇 = ∑ [((𝑎𝑖 𝑃𝐺𝑖
2
+ 𝑏𝑖 𝑃𝐺𝑖 + 𝑐𝑖)) + ℎ𝑖((𝛼𝑖 𝑃𝐺𝑖
2
+ 𝛽𝑖 𝑃𝐺𝑖 + 𝛾𝑖))]𝐺𝑔
𝑖=1 (3)
where, FT is total CEEDfuel cost; hi is price penalty factor.
3. PRICE PENALTY FACTORS (PPF)
The PPF [6, 11, 13-23] for CEED problem is formulated taking the ratio fuel cost and emission
value of the corresponding generators as follows:

Int J Elec & Comp Eng ISSN: 2088-8708 
Comparative study of the price penalty factors approaches for Bi-objective dispatch … (Youssef Mouloudi)
3345
- Min-Max price penalty factor is described as:
ℎ𝑖 =
aiPGi,min
2
+biPGi,min+ci
αiPGi,max
2 +βiPGi,max+𝛾i
(4)
- Max-Max price penalty factor is described as:
ℎ𝑖 =
aiPGi,max
2
+biPGi,max+ci
diPGi,max
2 +eiPGi,max+𝛾i
(5)
- Min-Min price penalty factor is described as:
ℎ𝑖 =
aiPGi,min
2
+biPGi,min+ci
αiPGi,min
2 +βiPGi,min+𝛾i
(6)
- Max-Min price penalty factor is described as:
ℎ𝑖 =
aiPGi,max
2
+biPGi,max+ci
αiPGi,min
2 +βiPGi,min+𝛾i
(7)
- Average price penalty factor is formulated as:
ℎ 𝐴𝑉𝐸𝑅𝐴𝐺𝐸 𝑖 =
∑ ℎ 𝑖
4
1
4
(8)
- Common price penalty factor is formulated as:
ℎ 𝐶𝑂𝑀𝑀𝑂𝑁 𝑖 =
ℎ 𝐴𝑉𝐸𝑅𝐴𝐺𝐸 𝑖
4𝑛
(9)
where: n is operational generating unit.
4. CONSTRAINTS
4.1. Power balance constraints [24]
Where, PG, PDemand and PLossare the total generated power, load demand and transmission line loss
of the system respectively. Transmission line loss constraint can be given as, [25]:
PG = ∑ Pi = PDemand + PLoss
n
i=1 (10)
where, Pi, and Pj is the active power of unit iih
and jih
respectively. Bij, B0i and B00 is the transmission loss
coefficients.
PL = ∑ ∑ PiBijPj
n
i=j + ∑ B0iPi
n
i=1 + B00
n
i=1 (11)
4.2. Generator limits
The power output of each generator is restricted by minimum and maximum power limits,
is given as:
PGi min ≤ PGi ≤ PGi max (12)
5. PARTICAL SWARM OPTIMIZATION ALGORITHM
Particle swarm optimization PSO is a population-based optimization technique which was first
introduced by Kennedy and Eberhart in 1995 [26], inspired by social behavior of bird flocking or fish
schooling in search of food.The most important prominent features of PSO, compared to other existing
heuristic optimization strategies such as genetic algorithm, are its easy implementation, there are few
parameters to adjust and computation efficiency. In a PSO system, particles fly around in a multidimensional
search space. During flight, each particle adjusts its trajectory towards its own previous best position this
value is called (Pbest), and towards the best previous position attained by any member of its neighborhood or

 ISSN: 2088-8708
3346
globally, the whole swarm this value is called (Gbest) [27-32]. The two equations which are used in PSO are
velocity update equation (13) and position update equations (14). These are to be modified at each time step,
of PSO algorithm to converge the optimum solution.
Vi(t + 1) = ωVi(t) + c1r1[Pbesti(t) − Xi(t)] + c2r2[Gbesti(t) − Xi(t)] (13)
𝑋𝑖(𝑡 + 1)𝑋𝑖(𝑡) + 𝑉𝑖(𝑡 + 1) (14)
Where,i is the particle index;  is the inertia coefficient; are acceleration coefficients 22,10  cc ; rr 2,1 are
random values,  rr 2,10 regenerated every velocity cc 2,1 update; Vi is the particles velocity at time t ; X i
is the particles position at time t; Pbestis the particles individual best solution as of time t; Gbest is
the swarms best solution as of time t.
6. SIMULATION RESULTS AND ANALYSIS
The west algerian power network is a 22 bus system with 7production units. This latter is considered
in an attempt to solve the CEED problem using “Min-Max“, “Max-Max“, “Min-Min“, “Max-Min“,
”Average” and “Common” price penalty factors. The test system consists of 7 thermal units, 15 load buses
and 31 transmission lines, 03compensator VARSTATIC SVC [3* (+40Mvar and )10Mvar)]. The total system
demand is 856 MW. The data for the considering test system is shown in Table 1. The real power limits of
the generators, fuel cost coefficients are also given in the Table 1. Programming of the CEED using the PSO
method has been applied by using MATLAB software, tested on a CORE i5, personal computer with 2.20
GHz and 4 GO RAM. Table 2 show solution of CEED problem with different price penalty factors such as
“Min-Max”, “Max-Max”, “Min-Min”, “Max-Min”, Average and Common. Table 3 compares the results
obtained with all six penalty factors. As illustrated in Table 2 the results show an acceptable improvement in
the fuel cost, and total fuel cost CEED of the system when using the Min-Max price penalty factor compared
to other penalty factors. The emission value is less when using Max-Max price penalty factor in comparison
with the other penalty factors. The Max-Min price penalty factor is better in terms of the lowest transmission
loss compared to other penalty factors.
Table 1. 22 bus system data
Generator
Numbers
Generator limits [MW] Fuel cost coefficients
𝑃 𝑚𝑖𝑛 [𝑀𝑊] 𝑃𝑚𝑎𝑥 [𝑀𝑊] 𝑎𝑖[$/MW2
h] 𝑏𝑖[$/MWh] 𝑐𝑖 [$/h]
1 100 500 0.007 7.5 240
2 50 200 0.008 7 200
3 80 300 0.0085 7.5 220
4 50 150 0.009 7 200
5 50 200 0.009 9 220
6 50 120 0.0075 10 190
7 10 80 0.009 6.3 180
Table 2. Solution of CEED problem using PSO with various price penalty factors
Price Penalty Factors Data
From
SONELG
AZ [30]
Min-Max Max-Max Min-Min Max-Min Average Common
𝑃1 [MW] 200 100.0000 100.0000 100.0000 100.0000 100.0000 100.0000
𝑃2 [MW] 200 206.2277 186.6597 246.9342 210.3191 164.5518 185.9998
𝑃3 [MW] 300 188.3073 219.3888 224.8282 191.1794 257.3631 236.7597
𝑃4 [MW] 80 130.3337 96.3901 138.6912 141.6869 56.5279 60.8637
𝑃5 [MW] 100 124.7016 124.0204 65.8667 60.0516 135.4592 105.7232
𝑃6 [MW] 100 88.4415 86.7610 63.0696 109.6222 73.1928 104.4470
𝑃7 [MW] 10 19.5432 50.2276 27.0108 43.4842 73.6464 83.2097
Power Loss [MW] 21.4 20.882 20.175 20.087 17.409 21.550 19.049
Total output [MW] 990 857.5555 863.4476 866.4007 856.3434 860.7412 877.0031
Power demand [MW] 856 856 856 856 856 856 856
Generation cost[$/h] 9104.44 8892.0 8899.4 9089.8 8904.5 8909.5 9040
Emission [Kg/h] * 1096.1 1078.1 1228.9 1196.5 1101.4 1225.5
Total cost[$/h] * 10903 14406 18985 36863 32346 40640
Temps [S] * 0.095112 0.106198 0.080914 0.084423 0.096332 0.096956

3347
Table 3. Comparison of simulation results obtained from “Min-Max”, “Max-Max”, “Min-Min”, “Max-Min”,
“Average”,“Common” price penalty factors
Criterion Min-Max
price penalty
factor
Max-Max
price penalty
factor
Min-Min
price penalty
factor
Max-Min
price penalty
factor
Average
price penalty
factor
Common
price penalty
factor
Power Loss [MW] 100% 96.61% 96.19% 83.37% 103.19% 93.03%
Generation cost[$/h] 100% 100.10% 102.22% 100.14% 100.19% 100.23%
Emission[Kg/h] 100% 98.36% 112.12% 109.16% 100.48% 103.52%
Total cost[$/h] 100% 132.13% 174.12% 338.09 296.67% 652.79%
Figure 1 show clearly that the convergence profile obtained by PSO algorithm of functions such as
CEED total cost, generation cost, emission cost and transmission loss when using Min-Max, Max-Max,
Min-Min, Max-Min, average and common price penalty factors is faster and more effective, which proves
that the proposed algorithm has more ability to find the optimal points in a search space compared with data
provided by SONELGAZ, the company which is in charge of operating the above mentioned grid of west of
Algeria [30].
From Figure 1(a), the variation of CEED fuel cost values of the bi-objective dispatch problem using
Min-Max price penalty factor are the lowest compared to other penalty factors. Similarly, the variation of
fuel cost values of the bi-objective dispatch problem using Min-Max price penalty factor are the lowest
compared to other penalty factors, see Figure 1(b). Likewise, according to Figure 1(c) the variation of
emission values of the bi-objective dispatch problem using Max-Max price penalty factor has minimum
pollution control compared to other pnalty factors. Finally, From Figure 1(d) the variation of power
lossvalues of the bi-objective dispatch problem using Max-Min price penalty factor has lowest transmission
power loss compared to other penalty factors.
(a) (b)
(c) (d)
Figure 1. Convergence curve for functions such as, (a) CEED (comparison of CEED total cost using various
price penalty factors), (b) fuel cost (comparison of generation cost usingvarious price penalty facteur),
(c) emission value (comparison of emission value using various price penalty facteur), (d) power loss
(comparison of power loss using variousprice penalty facteur)
1 2 3 4 5 6 7 8 9 10
1
2
3
4
5
6
7
x 10
4
Iterations
CEEDTotalCost[$/h]
Min-MaxPrice penaltyfactor
Max-MaxPrice penaltyfactor
Min-MinPrice penaltyfactor
Max-MinPrice penaltyfactor
Average Price penaltyfactor
CommonPrice penaltyfactor
1 2 3 4 5 6 7 8 9 10
0.5
1
1.5
2
2.5
3
3.5
4
x 10
4
Iterations
GenerationCost[$/h]
1 2 3 4 5 6 7 8 9 10
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
Iterations
Emission[Kg/h]
1 2 3 4 5 6 7 8 9 10
16
18
20
22
24
26
28
30
32
34
36
Iterations
PowerLoss[MW]

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7. CONCLUSION
In this paper, the impact of price penalty factors on the solution of the bi-objective power system
economic dispatch optimization problem is examined on electric grid of west algeria which consists of
22-Bus system. The Particle Swarm Optimization algorithm is proposed for solving the combined economic
emission dispatch problem. On the basis of results obtained some conclusions are made: the simulation
results show that Min-Max price penalty factor yields a minimum generation cost for bi-objective power
dispatch problem. The results show that theminimum emission values are less in Max-Max price penalty
factor compared to other penalty factors. The Max-Min price penalty factor is better in terms of the lowest
transmission loss compared to other penalty factrors.
In Summary, it has been shown that the minimum overall cost for the bi-objective power system
dispatch optimization problem can be obtained using Min-Max Price penalty factor. From Table 2 the CEED
fuel cost values are significantly lower with Min-Max price penalty factor by 32.13% in comparison to
the solution using Max-Max price penalty factor. The results also show that the emission values are less in
Max-Max price penalty factor by 1.68% when compared to Min-Max price penalty factor.
ACKNOWLEDGEMENTS
Authors would like to thank the heads of Laboratory of Analysis, Control andOptimization of
Electro-Energetic Systems (CAOSEE) and Laboratory of Smart Grids & Renewable Energies
(ENERGARID) at the university TAHRI Mohammed, Béchar (Algeria).
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