Bi-objective Optimization Apply to Environment a land Economic Dispatch Problem Using the Method of Corridor Observations

ISSN (e): 2250 – 3005 || Vol, 04 || Issue, 10 || October – 2014 ||
International Journal of Computational Engineering Research (IJCER)
www.ijceronline.com Open Access Journal Page 16
Bi-objective Optimization Apply to Environmental and Economic Dispatch Problem Using the Method of Corridor Observations Salomé Ndjakomo Essiane1, Stève Perabi Ngoffe2, Adolphe Moukengue Imano3, Grégoire Abessolo Ondoa4 1Lecturer, Electronic, Electrotechnic, Automatic and Telecommunications Laboratory, ERSEE, University of Douala - Cameroon 2 Doctorate Student, Electronic, Electrotechnic, Automatic and Telecommunications Laboratory, ERSEE, University of Douala - Cameroon 3 Associate Professor, Electronic, Electrotechnic, Automatic and Telecommunications Laboratory, ERSEE, University of Douala, BP. 8698 Douala - Cameroon 4 Master of Sciences with thesis student, Electronic, Electrotechnic, Automatic and Telecommunications Laboratory, ERSEE , University of Douala - Cameroon
I. INTRODUCTION
The economic dispatching (ED) is one of the key problems in power system operation and planning. The basic objective of economic dispatch is to schedule the committed generating unit outputs so as to meet the load demand at minimum operating cost, while satisfying all equality and inequality constraints. This makes the ED problem a large – scale highly constrained non linear optimization problem. In addition, the increasing public awareness of the environmental protection and the passage of the Clean Air Act Amendments of 1990 have forced the utilities to modify their design or operational strategies to reduce pollution and atmospheric emissions of the thermal power plant. Several strategies to reduce the atmospheric emissions have been proposed and discussed. These include: installation of pollutant cleaning equipment, switching to low emission fuels, replacement of the aged fuel-burners with cleaner ones, and emission dispatching. The first three options require installation of new equipment and/or modification of the existing ones that involve considerable capital outlay and, hence, they can be considered as long-term options. The emission dispatching option is an attractive short-term alternative in which the emission in addition to the fuel cost objective is to be minimized. Thus, the ED problem can be handled as a multi-objective optimization problem with non-commensurable and contradictory objectives. In recent years, this option has received much attention [1–5] since it requires only small modification of the basic ED to include emissions.
ABSTRACT:
This paper presents a proposed and applicable optimization algorithm based on the evolutionary algorithm to solve an environmental and economic Dispatch (EED) problem. This problem is seen like a bi-objective optimization problem where fuel cost and gas emission are objectives .in this method, the optimal Pareto front is found using the concept of corridor observation and the best compromised solution is obtained by fuzzy logic. The feasibility of this method is demonstrated using power systems with three generation units, and it is compared with the other evolutionary methods of Optimization Toolbox of Matlab in terms of solution quality and CPU time. The simulated results showed that the proposed method is indeed capable of obtaining higher quality solutions in shorter computational time compared to other evolutionary methods. In addition the other advantage is that our approach solves in the same time a unit commitment problem and it can use with n-generation units.
KEYWORDS: bi-objective, corridor observations, Evolutionary algorithm, fuel cost, gas emission, optimal PARETO front, optimization

Bi-objective Optimization Apply to…
In the literature concerning environmental/economic dispatch (EED) problem, different technics have
been applied to solve EED problem. In [1, 2] the problem was reduced to a single objective problem by treating
the emission as a constraint. This formulation, however, has a severe difficulty in getting the trade-off relations
between cost and emission. Alternatively, minimizing the emission has been handled as another objective in
addition to the cost [5]. However, many mathematical assumptions have to be given to simplify the problem.
Furthermore, this approach does not give any information regarding the trade-offs involved. In other research
direction, the multi-objective EED problem was converted to a single objective problem by linear combination
of different objectives as a weighted sum [3], [6]. The important aspect of this weighted sum method is that a set
of non-inferior (or Pareto-optimal) solutions can be obtained by varying the weights. Unfortunately, this
requires multiple runs as many times as the number of desired Pareto-optimal solutions. Furthermore, this
method cannot be used in problems having a non-convex Pareto optimal front. To overcome it, certain method
optimizes the most preferred objective and considers the other objectives as constraints bounded by some
allowable levels [5]. The most obvious weaknesses of this approach are that, then are time-consuming and tend
to observed weakly non-dominated solutions [5].
The other direction is to consider both objectives simultaneously as competing objectives. The recent
review to the Unit Commitment and Methods for Solving [7] showed that evolutionary algorithms are the most
used in this case; certainly because they can efficiently eliminate most of the difficulties of classical methods
[5]. The major problems of these algorithms, is to find the pareto optimal front and to conserve the non-dominated
solutions during the search. In this paper we perform and apply one optimization method proposed in
[9] to solve the EED problem. The particularity of this method is based to the fact that the search of front and
optimal Pareto is found by the concept of corridor, and the archives is dynamic. This dynamism reduce the loses
of the non-dominated solution during the different generations. In the second part of this paper, we present
materials and methods to solve the problem, and the third part, present simulation and results obtained.
II. MATERIALS AND METHODS
In this part, we formulate the EED problem and present our approach to solve it.
2.1. Problem formulation
The EED problem is to minimize two competing objective functions, fuel cost and emission, while satisfying
several equality and inequality constraints. Generally the problem is formulated as follows :
2.1.1. Problem objectives
 Minimization of fuel cost
The generator cost curves are represented generally by quadratic functions. The total fuel cost ($/h) in terms of
period T, can be expressed as:
(1)
Where
2
, , , ( ) fi i t i i i t i i t c p  a  b p  c p (2)
and ( ) fi i , t c p is the generator fuel cost function; ai, bi and ci are the cost coefficients of ith generator; Pi,t, is the
electrical output of ith generator; Ng is the number of generators committed to the operating system ; Ii,tthestatut
of differents generators ; STi the start-up cost .
 Minimization of gas emission
The atmospheric pollutants such as sulpher dioxides (SO2) and nitrogen oxides (NOX) caused by fossil-fueled
thermal units can be modeled separately. However, for comparison purposes, the total emission (ton/h) in one
period T of these pollutants can be expressed as:
  i t i t
T
t
Ng
i
i t fi E p e p I , ,
1 1
,   ( )
 
 (3)
where
( ) ( )
2
fi i , t i i i , t i i , t e P     p   p (4)
and αi, βi, δi are the emission coefficients of the ith generator.
  i t i t i t i t i t
T
t
Ng
i
i t fi F p c p I ST I I , , , , 1 ,
1 1
, ( ) (1 ) 
 
    

2.1.2. Objective constraints
 Power balance constraint


 
Ng
i
load t i t i t p p I
1
, , , 0
(5)
 Spinning reserve constraint :


  
Ng
i
load t t i i t p R p I
1
, max, , 0
(6)
 Generation limit constraints :
i i t i t i i t p I p p I min, , , max, ,   , i  1 ...... Ng
(7)
 Minimum up and down time constraint :






or otherwise
if T T
if T T
I
down
i
off
i
up
i
on
i
i t
0 1
0
1
, 

(8)
Where Ti
up represent the minimum up time of unit i; Ti
down the minimum down time of unit iTi
off is the
continuously off time of unit i and Ti
on the continuously on time of unit- i .
 Start-up cost




  

down
i
cold
i
off
i i
cold
i
down
i
off
i
down
i i
i
CST if T T T
HST if T T T T
ST

2.2. The proposed approach
2.2.1. Algorithm of corridors observations method
The different steps of the proposed approach to solve EED problem is summarize in the follow figure
Figure 1: Different steps of the algorithms
 Step 1
In the first step, we start with to the status of different unit generation, were we create randomly the initial
population. Each individual is a combination of each power generation unit
 Step 2
In the second, using equations (1) and (3) we evaluate the objective functions of this population
 Step 3
Generate initial
population
· Evaluate function cost
· Evaluate emission cost
Segment objectives space function
in corridor
Search the best persons in
corridors
Archive the best persons
Stopping criteria
determine the pareto front
determine the best compromize
solution
End
Apply cross and mutation operator
Form a new population

Using the minimum of the different objective functions of individuals who respect the constraints (5) to (8), we
define the space solution and segment it to the corridors observation following the different axes which are
specify by each function.
 Step 4
In each corridor, we search the best individual who have the minimum objectives functions, and the non feasible
solutions are classified using the number and the rate of violation constraints. Those solutions will be used to
increase the number of feasible solutions.
 Step 5
We keep in the archives those best individuals
 Step 6
We verify the stoping criteria define as [9] :
  ln( d )
(9)
where
 
 




















F N
j
cl
i
t
j i
t
j i
F F
F F
Cl
d
1 1 max min
1
1 , ,
explain the metric progression of the best individuals in each corridor. NF is the number of objectives functions ;
Cl the number of corridor ; Ft
j,i , Ft-1
j,i the jth objective function of the best individual in ith corridor ; Fmin and Fmax
the minimum and maximum of the j function ; t is the present generation, t-1 the anterior generation. At times
the maximum number of generation can be the alternative stopping criteria
 step 7
If the stopping criteria is not verified, we construct the new population using the selection ,cross and mutation
operators apply to the archive population and we return to step 2.
 Step 8
If the stopping criteria is verified we find the best compromise solution among the individuals of the Pareto
front. Due to imprecise nature of the decision maker’s judgment, each objective function of the i-th solution is
represented by a membership i  function defined as



 



 




max
min max
max min
max
min
0
1
i i
i i i
i i
i i
i i
i
if F F
F F F
F F
F F
if F F

For each non-dominated solution, the normalized membership function is k
 calculated as:
 

 
 
M
k
N
i
k
i
N
i
k
i
k
F
F
1 1
1



where M is the number of non-dominated solutions. The best compromise solution is the one having the
maximum of k
 .
2.2.2. Implementation and strategy to search solution
To find the problem solution algorithm, we have to :
 Explore space solutions
During this step, the algorithm explore space solutions with using 0.9 percentage of random mutation operator
and 0.1 percentage of uniform cross operator, it corresponding to the inequality   8 .
 Exploit space solutions
To exploit space solutions, the algorithm applied to the population random mutation a percentage of 0.9 and
0.1 percentage of arithmetic cross operator. it corresponding to the inequality  18     12 .
 Apply hybrid mode
It’s a transition between exploration and exploitation which correspond to 0.5 percentage of cross probability
and 0.5 percentage of mutation probability. The range of stopping criteria is:  12     8
(11)
(12)
(10)

III. SIMULATION AND RESULTS
In order to validate the proposed procedure and to verify the feasibility of the corridor method to solve
the EED, a 3-units generation system is tested [10] and extent to 6, 10 and 15 units génération. The proposed
method is implemented with Matlab 2010.b on a core i3 2.1 GHz and the results are compared with other
evolutionary algorithms (GA and GA multi-obj) of optimization tool. The data concerning the unit generation is
given in table 1 to table 2 [10].
Table 1. The 3-units system data
Unit Pi
max
(MW)
Pi
min
(MW)
a
($ /h)
b
($ /MWh)
c
($ /MW2h)
Ri
up
($ /MW)
Ri
down
($ /MWh)
1 600 150 561 7.29 0.00156 100 100
2 400 100 310 7.85 0.00194 80 80
3 200 50 78 7.97 0.00482 50 50
Table 2. SO2 and NOx coefficients emission gas data of 3-units
Units
so 2  (tons/h)
Nox 
(tons/h) so 2  (tons/MWh)
Nox  (tons/MWh)
so 2  (tons/MW2h)
Nox 
(tons/MW2h)
1 0,5783298 0,04373254 0,00816466 -9,4868099 e-6 1,6103e-6 1,4721848 e-7
2 0,3515338 0,055821713 0,00891174 -9,7252878 e-5 5,4658 e-6 3,0207577 e-7
3 0,0884504 0,027731524 0,00903782 -3,5373734 e-4 5,4658 e-6 1,9338531 e-6
In the implementation, we add the different coefficients of each gas per groups to have the coefficient of the
whole gas.
3.1 Achievement of Pareto front
The Pareto front is obtained, keeping the best individuals (individuals who have the minimum fuel cost are
conserved in relation to the emission axes and vice versa) that respect all the contraints in each corridor during
the evolution. To simulate the evolution of pareto front we have initialize the population size at 300 , the
maximum generation at 1000 ,the number of corridor at 50 and load demand is 1000MW.
9260 9270 9280 9290 9300 9310 9320
10.5
10.55
10.6
10.65
10.7
10.75
10.8
10.85
10.9
Cost
Emission
generation 764
individual of population
feasible individual
Best compromise solution
Figure 2 : Pareto optimal front at the end of process (1000MW load demand).
In the search of the best solution in each corridor, at the beginning the algorithm could not find the best feasible
solutions but in the process of evolution the number of these solutions increases up to the Pareto optimal front.
3.1. Study of convergence
The curve convergence show that until one number of generations the algorithm finds the best solution,
from this times gas emission and fuel cost is uniform. It expresses the performance of stopping criteria.

0 100 200 300 400 500 600 700 800 900 1000
9255
9260
9265
9270
9275
9280
cost convergence with generation
generation
cost($)
0 100 200 300 400 500 600 700 800 900 1000
10.7
10.75
10.8
10.85
10.9
10.95
emission convergence with generation
generation
emission globale(ton/h)
Figure 3: Convergence of fuel cost($) and gas emissions(ton/h) objective functions (1000MW).
3.2. Apply of method to 24 hours
To present the effectiveness of our approach to unit commitment and EED, we have apply it to plan the
production of 3-units during 24 hours.
Table 3. Unit commitment and EED during 24 hours
Hours
(H)
Demands
(MW)
P1
(MW)
P2
(MW)
P3
(MW)
Fuel cost
X103($)
Emission
gas
(ton/h)
Starting
cost
($/MWh)
Total cost
($)
1 550 0 0 549.9 5.0415 5.6380
2 600 0 0 599.9 5.4957 6.1468
3 650 108.0382 0 541.8742 5.9646 6.6979 50
4 700 114.5484 0 585.351 6.4169 7.2059
5 750 150.228 0 599.67 6.8777 7.7323
6 800 0 224.27 575.62 7.4424 8.5731 80
7 850 0 250.314 599.58 7.8893 9.1188
8 900 0 300.541 599.3621 8.3352 9.7188
9 950 114.9563 239.4409 595.5072 8.8142 10.1626 50
10 1000 128.8152 275.9956 595.0968 9.2607 10.7347
11 1050 141.965 308.5501 599.3873 9.7144 11.3159
12 900 0 300.03 599.8287 8.3342 9.7188
13 850 0 250.0713 599.8287 7.8894 9.1184
14 800 0 225.58 574.3207 7.4419 8.5747
15 750 150.7585 0 599.1479 6.8779 7.7327 50
16 700 114.5434 0 585.355 6.4116 7.2059
17 650 108.0326 0 541.8732 5.9646 6.6979
18 600 0 0 599.9 5.4957 6.1468
19 730 130.2853 0 599.6148 6.6913 7.5171 50
20 820 0 233.9595 585.9423 7.6209 8.7892 80
21 860 0 260.1659 599.7343 7.9778 9.2361
22 900 0 300.531 599.3721 8.3352 9.7188
23 950 114.9553 239.4419 595.5072 8.8142 10.1626 50
24 1000 128.8152 275.9956 595.0968 9.2607 10.7347
Total 178.368 410 178778
In function of demand we see that some unit could be on and other off, to minimize the fuel cost and emission
gas. The plan production of different units is represent in follow figure

Figure 3: Production plan of different units
3.3. Comparison of results
To show the effectiveness of the proposed method, we have compared it with other evolutionary algorithm of optimization tool of matlab. The same parameter is considered and the criteria for comparisons are cost and average CPU times. Tableau 4: Comparison methods with 1000MW of load demand
GA
GA multi-obj.
corridor P1 (MW) 400.0587 399,999 598.4477
P2 (MW)
399.9885
399,999
275.8265 P3(MW) 199.9518 199,999 125.6293
Fuel Cost($ /h)
9879.2053
9878,988
9260,6 Emission ton/h 11.755 11,644 10.5768
Average CPU times
244.9135
261.9536
23.29854
This table show that our method is best in term of fuel cost and emission gas
3.4. Impact of numbers of units to fuel cost, gas emission and average CPU times
To show this impact, we have multiplied the number of units of our 3-units. The results are presented to follow table. Tableau 5: Impact of numbers of units to fuel cost, gas emission and average CPU times
Number of units
3
6
10
15 Fuel cost($) 9.2606 9.1911 9.176 9.177
Gas emission(ton/h)
10.7347
10.2771
10.3679
10.3683 Average times(s) 23.29 31.92 49.40 104.61
This table show that when the number of units increases, the fuel cost, gas emission and average CPU decrease.
IV. CONCLUSION
In this paper, an approach based on the evolutionary algorithm has been presented and applied to environmental/economic power dispatch optimization problem. The problem has been formulated as a bi- objective optimization problem with competing fuel cost and environmental impact objectives. The proposed approach has a diversity-preserving mechanism to find Pareto-optimal solution. The optimal Pareto front is obtained from minimizing each objective function in each corridor and keeping the best individuals in dynamic achieves. Moreover, a fuzzy-based mechanism is employed to extract the best compromise solution over the trade-off curve. The results show that the proposed approach is efficient for solving bi-objective optimization

where multiple Pareto-optimal solutions can be found in one simulation run. In addition, the non-dominated solutions in the obtained Pareto-optimal set are well distributed and have satisfactory diversity characteristics. Comparatively to other approach, the most important aspect of the proposed approach is the reduce time to find optimal Pareto front, fuel cost, gas emission, the consideration of unit commitment problem and the possibility to manage system which have n-generation units and best compromise. REFERENCES [1] Brodesky S. F, Hahn R. W. Assessing the influence of power pools on emission constrained economic dispatch. IEEE Trans Power Syst 1986;1(1).pp:57–62. [2] Granelli G. P, Montagna M, Pasini G. L, Marannino P. Emission constrained dynamic dispatch. Electr Power Syst Res 1992;24.pp: 56–64. [3] Chang C. S, Wong K. P, Fan B. Security-constrained multiobjective generation dispatch using bicriterion global optimization. IEE Proc—GenerTransmDistrib 1995;142(4):406–14. [4] R Manoj Kumar. Bavisetti, T.KranthiKiran. Optimization Of Combined Economic and Emission Dispatch Problem – A Comparative Study for 30 Bus Systems. IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE) 2012 ;Vol 2 : PP 37-43 [5] M. A. Abido. Environmental/Economic Power Dispatch Using Multiobjective Evolutionary Algorithms. IEEE Transactions On Power Systems ; 2003 VOL. 18, NO. 4. Pp :1529-1537 [6] Alkhalil Firas. «Supervision, économie et impact sur l’environnement d’un système d'énergie électrique associé à une centrale photovoltaïque »thèse de Doctorat Paris tech. 2011 [7] Samani.h, Razmezani.M and Naseh.M.R. (2013). Unit Commitment and Methods for Solving; a Review .J. Basic. Appl. Sci. Res., 3(2s) 358-364 [8] Lubing Xie, Songling Wang & Zhiquan Wu. Study on Economic, Rapid and Environmental Power Dispatch. Modern applied sciences vol.3 N06. 2009. pp :38-44 [9] Jean Dipama. « Optimisation Multi-Objectif Des Systèmes Énergétiques ».thèse soutenue à l’université de Montréal. 2010 [10] Farid Benhamida, Rachid Belhachem. Dynamic Constrained Economic/Emission Dispatch Scheduling Using Neural Network .Power Engineering And Electrical Engineering.Volume: 11 N01 .2013.pp :1-9

Bi-objective Optimization Apply to Environment a land Economic Dispatch Problem Using the Method of Corridor Observations

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