Energy Procedia 61 ( 2014 ) 2016 – 2021
ScienceDirect
1876-6102 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Peer-review under responsibility of the Organizing Committee of ICAE2014 doi: 10.1016/j.egypro.2014.12.065
The 6
thInternational Conference on Applied Energy – ICAE2014
Multi-objective optimization for economic emission dispatch
using an improved multi-objective binary differential
evolution algorithm
Yijuan Di
a,*, Minrui Fei
a, Ling Wang
aand Wei Wu
baShanghai Key Laboratory of Power Station Automation Technology, School of Mechatronic Engineering and Automation,
Shanghai University, Shanghai 200072, China
bDepartment of Chemical Engineering, National Cheng Kung University, Tainan 70101, Taiwan
Abstract
Economic emission dispatch (EED) problem is a non-linear constrained multi-objective optimization problem with two competing and conflicting objectives. In this paper, an improved multi-objective binary differential evolution (IMBDE) algorithm is proposed to solve the EED problem. In the proposed algorithm, a marginal analysis correction operator is presented to handle various system constraints. The simulation results demonstrate that the proposed algorithm is valid in solving the EED problem and its performance is superior to other approaches.
Key Words: Economic emission dispatch; Differential evolution algorithm; Marginal analysis
1. Introduction
Traditional economic load dispatch problem is one of the most important mathematical optimization issues in thermal power system. With the increasing concern over environmental pollution such as SOx,
NOx and COx caused by thermal power plants, minimum emissions have to be considered along with
economic load dispatch, which is called as economic emission dispatch (EED) problem. The EED problem aims to schedule optimal power outputs of all generators in thermal power system so as to minimize the fuel cost and emission simultaneously while satisfying various system constraints.
The EED problem is a complex non-linear multi-objective optimization problem. Compared with other evolutionary algorithms, Differential Evolution (DE) algorithm [1] is a simple, efficient and extremely robust direct search algorithm for global optimization. Recently, Pareto-based multi-objective DE algorithms [2] were studied to solve the EED problems. However, these algorithms have the premature convergence problem and lack a mechanism to effectively handle complicated constraints of the EED problems.
Therefore in this paper, an improved multi-objective binary differential evolution (IMBDE) algorithm
* Corresponding author. E-mail address: [email protected].
is proposed to solve the EED problem. In the proposed algorithm, a marginal analysis correction operator based on marginal analysis method [3], which is a quantity analysis method and widely used in the modern western economics, is presented for handling various constrains of the EED problem.
2. Problem formulation Objectives 2 1 1 [ ( )] [ ] min N N eco eco i i i i i i i i C P a P b P c F = = =
¦
=¦
+ + (1) 2 1 1 min [ ( )] [ ] N N emis emis i i i i i i i i F C Pα
Pβ
Pγ
= = =¦
=¦
+ + (2) where Feco and Femis are the total fuel cost and total emission of all generators, respectively. ai, bi and ci
are the cost coefficients of the ith generator. N is the number of generators. Pi is the active power output
of the ith generator. Ceco(P
i) is the fuel cost function of the ith generator. αi, βi and γi are the emission
coefficients of the ith generator. Cemis(P
i) is the emission function of the ith generator.
Constraints
(i) Power balance constraints
1 N i D L i P P P = = +
¦
(3)where PD is the load demand of power system, and PL is the transmission loss, which can be calculated as:
1 1 N N L i ij j i j P PB P = = =
¦¦
(4)(ii) Active power operating limits
min max i i i
P
≤
P
≤
P
i
∈
N
(5) where min i P , max iP are the minimum and maximum active power outputs of the ith generator.
3. Overview of differential evolution algorithm
The standard DE [1] algorithm contains three operations, i.e. mutation operator, crossover operator and selection operator.
Mutation: The mutation operation generates a new mutant individual
u
iG by three randomly selecteddifferent target individuals ( G1, r j
x , G2, r j
x and G3, r j
x ). The jth element of
u
iG can be calculated as follows:, 1, ( 2, 3, ) ,
G G G G
i j r j r j r j P
u =x +F x −x i∈N j∈N (7)
where F is the mutation factor which is a positive constant.
Crossover: The crossover operation generates trial individual
v
iG by mixing the target individual G ix
and its mutant individualu
iG. The jth element ofG i
v
can be obtained as follows: , , , , G i j G i j G P i j u if rand CR or j q v i N j N x otherwise ≤ = ° =® ∈ ∈ °¯ (8)Selection: The selection operation compares the fitness values of the trial individual
v
iG and thecorresponding target individual
x
Gi and the one with better fitness is chosen as new individual1
G i
x
+ of the next generation. The jth element ofx
iG1+
, , , 1 , , ( ) ( ) , G G G i j i j i j G i j G P i j v if f v f x x i N j N x otherwise + =°® ≤ ∈ ∈ °¯ (9)
4. Improved multi-objective binary differential evolution algorithm
4.1. Marginal analysis correction operator
When the new generated individual violates the constraints in optimization process, a marginal analysis correction operator is presented to transfer infeasible solutions into feasible solutions. The main steps of marginal analysis correction operator can be described as follows:
Step 1: If any element of new generated individual is outside the feasible boundaries, the following procedure will be applied.
min min max max i i i i i i i P if P P P i N P if P P < ° =® ∈ > °¯ (10)
Step 2: Calculate marginal fuel cost and marginal emission as shown in Eq. (11)-(12) by using the marginal analysis method [3]. Get the number, denoted as j, of the generator with minimum values.
( ) ( 1) 2
eco eco eco
i i i i i i i
M =C P −C P− = a P− +a b i∈N (11)
( ) ( 1) 2
emis emis emis
i i i i i i i
M =C P −C P− = αP− +α β i∈N (12) Step 3: Calculate the power output of the jth generator by Eq. (13) and generate a new individual.
' 1 1 1 ( ) N N N j j j D i ij j j i j P P P P PB P j N = = = = +
¦
− −¦¦
∈ (13)Step 4: If stopping criteria are met, the search process is stopped; otherwise go to step 1.
4.2. Procedures of IMBDE for the EED problem
The flow chart of IMBDE is shown in the Fig. 1.
Fig. 1. The flow chart of IMBDE
The test system consists of six generators, where the transmission loss is taken into account. The detailed data of generators are derived from [4]. The load demand of this system is 700MW.
To better demonstrate the proposed algorithm, the error is calculated as follows.
1 1 1 N N N i D i ij j i i j error P P PB P = = = =
¦
− −¦¦
(14) The population size NP and the maximum number of generations Gmax are selected as 20 and 100,respectively. F and CR are considered as 0.8 and 0.2, respectively. The detailed results of the best compromise solution are compared with those reported by using NR [5], FCGA [6], NSGA-II [4] and BBO [7] as shown in Tables 1 as well as the errors under these different algorithms.
Table 1 Best compromise solution under different algorithms
Algorithm P1(MW) P2(MW) P3(MW) P4(MW) P5(MW) P6(MW) Fuel cost ($/h) Emission (kg/h) error(MW)
NR 85.9240 60.9630 53.9090 107.1240 250.5030 176.5040 39070.7400 528.4470 3.4697 FCGA 80.1600 53.7100 40.9300 116.2300 251.2000 190.6200 38408.8200 527.4600 -0.0047 NSGA-II 86.2860 60.2880 73.0640 109.0360 223.4480 184.1110 38671.8130 484.9310 -0.0006 BBO 84.0180 61.9268 78.7284 110.9188 211.9852 187.8924 38828.2660 476.4080 -3.2520 IMBDE 76.6603 51.8044 51.3477 110.0785 255.3635 189.4812 38388.0363 523.4624 0.0001
Tables 1 show that the proposed algorithm can find better fuel costs and emissions than NR and FCGA. Compared with NSGA-II and BBO, IMBDE achieves better fuel costs but poorer emissions and these solutions obtained by these three algorithms are non-dominated solutions. From Table 1, it is also observed that the errors of IMBDE are least among all algorithms. Therefore, it is fair to claim that the proposed algorithm can effectively solve the EED problem and outperforms other approaches.
6. Conclusion
In this paper, an improved multi-objective binary differential evolution algorithm is proposed in this paper, where a marginal analysis correction operator is presented for constraints handling. Compared with other approaches reported in recent literatures, the simulation results and its errors demonstrate that the proposed IMBDE has better global searching ability in solving the EED problem and its performance is superior to other approaches.
References
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