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Multiobjective Bacteria Foraging Algorithm for Electrical Load Dispatch Problem
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IET Generation, Transmission & Distribution GTD-2008-0407 Research Paper 20-Aug-2008 Panigrahi, Bijaya; IIT, New Delhi, Electrical Engineering; IIT, New Delhi, Electrical Engineering Pandi, Ravikumar; IIT, Electrical Engineering Biswas, Arijit; Jadavpur University, Electronics and Telecomm Engg Dasgupta, Sambarta Das, Swagatam; Jadavpur University, Electronics and Telecomm Engg Environmental/Economic dispatch, Pareto front,, multiobjective optimization, Bacterial Foraging, Non-dominated sorting.
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Multiobjective Bacteria Foraging Algorithm for Electrical Load Dispatch Problem B K Panigrahi*, V. Ravikumar Pandi*, Arijit Biswas**, Sambrat Dasgupta** Swagatam Das** *Department of Electrical Engineering, IIT, Delhi, INDIA **Department of Electronics and Communication Engineering, Jadavpur University, Kolkata, INDIA Abstract: In this paper we try to extend the Bacteria Foraging meta-heuristic into the domain of multiobjective optimization. In this multiobjective bacteria foraging optimization technique, during chemotaxis a set of intermediate bacteria positions are generated. Next, we use pareto non-dominance criterion to determine final set of bacteria positions, which constitute the superior solutions among current and intermediate solutions. To test the efficacy of our proposed algorithm we have chosen a highly constrained optimization problem namely economic / emission dispatch. Economic dispatch is a constrained optimization problem in power system to distribute the load demand among the committed generators economically. Now-a-days environmental concern that arises due to the operation of fossil fuel fired electric generators and global warming, transforms the classical economic load dispatch problem into multiobjective environmental/economic dispatch (EED). In the proposed work, we have considered the standard IEEE 30-bus six-generator test system on which several other multiobjective evolutionary algorithms are tested. We have also made a comparative study of the proposed algorithm with that of reported in literature. Results show that the proposed algorithm is a capable candidate in solving the multiobjective economic emission load dispatch problem. Index Terms: Environmental/Economic dispatch, Pareto front, multiobjective optimization, Bacterial Foraging, Non-dominated sorting.
1. Introduction The operations of electrical power systems are designed to meet the continuous variation of power demand. In essence, to ensure economic operation, power generation scheduling is performed based on two important tasks, unit commitment and economic dispatch, of which, later is the topic of present research. The purpose of traditional economic dispatch is to allocate generation levels to various generators in the system in order to meet the load demand in the most economic way. However, the optimum schedule obtained may not be the best, in case environmental criteria are also considered. The passage of the clean air act amendments in 1990 has forced the utilities to reduce their SO2 and NOX emissions [1]. Therefore, apart from cost, emission objective must also be taken into account. Environmental/Economic dispatch (EED) is a multiobjective problem having conflicting objectives, as the minimization of emission is contrary to the maintenance of cost economy [2]. There has been much research pertaining to EED problem. A linear programming based technique has been proposed in [3] which considers one objective at a time. But the approach failed to give any information regarding the trade-off front. Refs. [4], [5], [6], [7], linearly combined different objectives through the weighted sum method to convert the multiobjective EED problem in single objective optimization problem. These methods generate the non-dominated solution by varying the weights, thus requiring multiple runs to generate the desired Pareto set of solutions. Moreover, these methods are not efficient in solving problems having non-convex Pareto optimal fronts. Refs. [8], [9] treated emission as a constraint and reduced the problem to single objective. However, the tradeoff relation between cost and emission is difficult to obtain through the aforementioned approach. The -constraint method was presented [10], [11], [12] to overcome this
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difficulty. The method involves the optimization of most preferred objective while considering other objectives as constraints, which are bounded by some allowable level . However, the approach provided only weakly non-dominated solution and that too in considerably large time. Recently, the research focus has shifted towards handling both the objectives simultaneously. Over the past decade, this option has received much interest due to the development of a number of multiobjective evolutionary search strategies. Nondominated Sorting Genetic Algorithm (NSGA [13]), Niched Pareto Genetic Algorithm (NPGA [14]), Strength Pareto Evolutionary Algorithm [2], Multi-Objective Stochastic Search Technique (MOSST [15]), Tabu Search [16], NSGA II [17], [18], Multiobjective Particle Swarm Optimization (MOPSO [20]) etc. constitutes the pioneering multiobjective approaches that have earlier been applied to solve the multi-objective environmental/economic dispatch problem. These algorithms have been implemented on standard IEEE 30 bus 6-generator system in order to obtain the trade-off between the cost and emission. In this paper we have tried to develop a new multiobjective algorithm to obtain a pareto-optimal set of solutions for the above-mentioned problem. In 2002 K.M. Passino proposed a new optimization technique Bacteria Foraging Optimization Algorithm (BFOA) [22,23]. An individual E.coli bacterium in a foraging swarm takes necessary action to maximize the energy utilized per unit time spent for foraging, considering all the constraints presented by its own physiology such as sensing and cognitive capabilities, environment. This natural foraging strategy can lead to optimization and this forms the theoretical basis of BFOA. Based on this conception, Passino proposed an optimization technique known as the Bacterial Foraging Optimization Algorithm (BFOA) Until date, BFOA has successfully been applied to real world problems like optimal controller design [24], harmonic estimation [25], transmission loss reduction [26], and active power filter synthesis [27]. To the best of our knowledge, BFOA has been successfully applied mainly in the domain of single objective function optimization. Our objective is to modify the algorithm to tackle multiobjective problem. In this paper we have developed a Multiobjective Bacteria Foraging (MOBF) algorithm which is then applied to EED problem. The rest of the paper is organized as follows. In section 2 and 3, we outline the classical BFOA and multiobjective optimization respectively. Section 4 describes formulation of the novel multiobjective bacteria foraging (MOBF) algorithm. In section 5, EED problem is briefly stated. Section 6 describes the simulation strategy for implementing the EED problem. Section 7 provides detailed experimental results comprising of final pareto fronts obtained by MOBF as well as for competitive algorithms. Various numerical metrics are computed in order to evaluate performance of various algorithms. Best compromise solution is determined applying a fuzzy technique. Also comparisons are carried out with the results reported in standard literature.
2.
The Classical BFOA algorithm
The bacterial foraging system consists of four principal mechanisms, namely chemotaxis, swarming, reproduction and elimination-dispersal [1]. Below we briefly describe each of these processes and finally provide a pseudo-code of the complete algorithm. I. Chemotaxis: This process simulates the movement of an E.coli cell through swimming and tumbling via flagella. Biologically an E.coli bacterium can move in two different ways. It can swim for a period of time in the same direction or it may tumble, and alternate between these two modes of operation for the entire lifetime. Suppose i ( j , k , l ) represents i-th bacterium at j-th chemotactic, k-th reproductive and l-th elimination-dispersal step. C(i) is the size of the step taken in the random direction
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specified by the tumble (run length unit). Then in computational chemotaxis the movement of the bacterium may be represented by (i ) i (1) ( j + 1, k , l ) = i ( j , k , l ) + C (i ) T (i ) (i ) Where
indicates a vector in the random direction whose elements lie in [-1, 1].
II. Swarming: An interesting group behavior has been observed for several motile species of bacteria including E.coli and S. typhimurium, where stable spatio-temporal patterns (swarms) are formed in semisolid nutrient medium. A group of E.coli cells arrange themselves in a traveling ring by moving up the nutrient gradient when placed amidst a semisolid matrix with a single nutrient chemo-effecter. The cells when stimulated by a high level of succinate, release an attractant aspertate, which helps them to aggregate into groups and thus move as concentric patterns of swarms with high bacterial density. The cell-to-cell signaling in E. coli swarm may be represented by the following function.
J cc ( , P( j, k , l )) = =
S i =1
J cc ( , i ( j, k , l )) (2) p
S
[ dattractant exp( wattractant i =1
( m=1
m
) )] +
p
S
i 2 m
[hrepellant exp( wrepellant i =1
(
m
i 2 m
) )]
m=1
where J cc ( , P ( j , k , l )) is the objective function value to be added to the actual objective function (to be minimized) to present a time varying objective function, S is the total number of bacteria, p is the number of variables to be optimized, which are present in each bacterium and = [ 1, 2,..................., p ]T is a point in the p-dimensional search domain.
d aatractant , wattractant , hrepellant , wrepellant are different coefficients that should be chosen properly. III. Reproduction: The least healthy bacteria eventually die while each of the healthier bacteria (those yielding lower value of the objective function) asexually split into two bacteria, which are then placed in the same location. This keeps the swarm size constant. IV. Elimination and Dispersal: Gradual or sudden changes in the local environment where a bacterium population lives may occur due to various reasons e.g. a significant local rise of temperature may kill a group of bacteria that are currently in a region with a high concentration of nutrient gradients. Events can take place in such a fashion that all the bacteria in a region are killed or a group is dispersed into a new location. To simulate this phenomenon in BFOA some bacteria are liquidated at random with a very small probability while the new replacements are randomly initialized over the search space.
3. Multiobjective Optimization: A Brief Overview Multiobjective optimization involves the simultaneous optimization of several incommensurable and often competing objectives [23]. In the absence of any preference information, a non-dominated set of solutions is obtained, instead of a single optimal solution. These optimal solutions are termed as Pareto optimal solutions. Let us consider a minimization problem
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r
r
r
r
Minimize J ( ) = ( J1 ( ), J 2 ( ),...... J M ( )) Subjected to constraints
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(3)
r gi ( ) = 0, i = 1,2.....m r h j ( ) 0, j = 1,2,....n
(4)
(5)
r
where, is called the decision vector, M is the number of objectives and m and n are the number of equality and inequality constraints respectively. Any solution vector a dominates b, if and only if a is partially less than b (a
J i (a ) J i (b) (6) Those solutions, which are not dominated, by other solutions of a given set are considered nondominated, regarding that set. The front obtained by mapping these non-dominated particles into objective space is called Pareto optimal front, POF. r r r r r POF = {J = ( J1 ( ), J 2 ( ),.........J M ( )) |
S}
(7) Where, S is the set of obtained non-dominated particles. The determination of complete Pareto optimal front is a very difficult task owing to the computational complexity involved in its computation due to the presence of a large number of suboptimal Pareto fronts. Considering the existing memory constraints, the determination of the complete Pareto front becomes infeasible, and thus requires the solutions to be diverse covering maximum possible regions of it. During past decade, a variety of stochastic approaches like NSGA [13], SPEA [21], NPGA [14], NSGA II [17], [18], SPEA 2 [19], MOPSO [20], MODE [28 ] etc. have been proposed. NSGA II, formulated by Deb et al., alleviates various shortcomings of NSGA and is found to be very efficient for multiobjective optimization. Zitzler et al. developed SPEA 2, which is also another robust algorithm. C.A. Coello Coello came up with Multiobjective Particle Swarm Optimization, a powerful tool in multiobjective problem handling. Following the footsteps of these algorithms we have tried to develop a new multiobjective optimization algorithm using chemotactic operator of BFOA.
4. Multiobjective bacteria Foraging (MOBF): Description of Algorithm In classical BFOA algorithm individual bacterium tries to get ample amount of nutrient substance and tries to avoid noxious substrates. It is the only objective that governs the search process. But in course of foraging, it may have another constraint say favorable temperature. One possible approach can be incorporation of temparature constraint by adding a penalty function to the actual nutrient concentration. It is expected that bacterium should not move to a region of unfavorable temperature. This approach leads to a single objective constrained optimization. In stead of it, we may take nutrient concentration and favorable temperature as two separate objectives. Individual bacterium tries to optimize these two simultaneously and it can be applied to multiobjective optimization. Corresponding to each possible position of bacterium in decision variable space we have two objective functions. In classical BFOA, during chemotaxis bacterium changes its position if it finds a region with higher nutrient gradient. But the scenario changes for Multiobjective Bacteria Foraging. So instead of differentiating between two positions on the basis of the single objective function, we introduce the concept of pareto non-dominance [13, 17]. In MOBF the population is ranked according to Pareto dominance criterion. During chemotaxis bacterium probabilistically moves towards another bacterium, belonging to a front with lower rank, otherwise it tumbles in a random direction. In this way a new set of positions are generated for the population. Previous positions are stored in memory. A non-dominated sorting approach is used to obtain final locations of bacteria population.
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Pseudo-code of MOBF Algorithm: Parameters: i
[Step 1] Initialize parameters p, S, Nc, , Ned, Ped,Pg ,C(i)(i=1,2…S), M, . Set ranks of all bacterium to 1. Where, p: Dimension of the search space, S: Total number of bacteria in the population, Nc : The number of chemotactic steps, Ned : The number of elimination-dispersal events, Ped : Elimination-dispersal probability, Pg : Guide Probabilty, C (i): The size of the step taken in the random direction specified by the tumble. M=Number of objective functions [Step 2] Elimination-dispersal loop: k=k+1 [Step 3] Chemotaxis loop: j=j+1 [a] For i =1,2…S take a chemotactic step for bacterium i as follows. [b] Compute fitness function, J l (i, j , k ) , Where l varies from 1 to M.
[c] Tumble: Generate a random number p. If p
p
with each element
m
(i ), m = 1,2,..., p, a random
number on [-1, 1]. [d] Move: i
( j + 1, k , l ) =
i
( j , k , l ) + C (i )
(i ) T
(i ) (i )
This results in a step of size C (i ) in the direction of the tumble for bacterium i. [f] Go to next bacterium (i+1) if i S (i.e., go to [b] to process the next bacterium). [e] Store these new positions along with the old positions in the memory and all these positions are sorted on the basis of non-dominated sorting. [f] Only the better-ranked positions are selected from the sorted pool to get the population of size S for the next iteration j+1. [Step 4] If j < N c , go to step 3. In this case continue chemotaxis since the life of the bacteria is not over. [Step 5] Elimination-dispersal: For i = 1,2..., S with probability Ped , eliminate and disperse each bacterium (this keeps the number of bacteria in the population constant). To do this, if a bacterium is eliminated, simply disperse another one to a random location on the optimization domain. If l < N ed , then go to step 2; otherwise end.
5. EED Problem Formulation
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A. Objectives 1) Fuel Cost Objective: The objective of classical economic dispatch is the minimization of total generation cost while satisfying several constraints. Mathematically,
Minimize J ( PG ) =
NG
J i ( PGi )
(9)
i =1 i
In eqn. 1, J i ( PG ) is the generation cost function and is approximated as a quadratic function of the power output from the generating units, i.e.
J i ( PGi ) = ai ( PGi ) 2 + bi PGi + ci
(10)
Where, ai, bi, ci are the cost coefficients, NG represents the number of generating units in the system and PGi is the power output of the ith generator. 2) Emission Objective (NOX): The minimum emission dispatch is the simultaneous minimization of classical economic dispatch including NOX emission objective which is modeled as:
ENOx =
NG i =1
10 2 (
i
+
i i PG
+ ( PGi ) 2 +
exp( i PGi ))
i
(11)
Here, i , i , i , i and i are the emission coefficients of the ith generator. Although a number of other substances such as SO2, dust particles etc. are also emitted during the operation of electrical generators, the NOX objectives are considered in this study as they are representative of the emissions generated by the fossil fuel fired generators. B. Constraints 1) Power Balance Constraint or Demand Constraint: The total power generated must cover total demand PD and total transmission losses PLOSS. Therefore, NG i =1
PGi
PLoss = 0
PD
(12)
Transmission losses are calculated by solving the load flow problem given as [22]: NB
PGi
PDi
Vi
QGi
Q Di
Vi
j=1
Vj G ij cos
(
i
j
) + Bij sin (
i
j
Vj G ij sin
(
i
j
)
i
j
NB j=1
Bij cos
(
)
=0
(13)
)
=0
(14)
where,NB: Total number of buses QGi : Reactive power generated at the ith bus PDi : Real power at the ith bus QDi : Reactive Power at the ith bus Gij : Transfer conductance between buses i and j Bij : Transfer susceptance between buses i and j Vi & Vj : Voltage magnitude at bus i and j i & j : Voltage angle at bus i and j Equations 5 and 6 are nonlinear constraint equations, which are solved using NewtonRaphson method, and the load flow solution thus obtained gives all bus magnitudes and angles. Then the real power loss is calculated as
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PLoss =
NL k =1
g k Vi2 + Vj2
2Vi Vj cos
(
i
j
)
(15)
where NL is the total number of transmission lines and gk is the conductance of the kth line which connects buses i and j. 2) Generation Capacity Constraints: The real output power of each generator is constrained by lower and upper limits, i.e. min PGi
PGi
max PGi
(16)
min max and PGi are the minimum and maximum operating outputs of unit i respectively. where, PGi 3) Line flow constraints: The line flow constraint is used to avoid undesired line loadings due to power distribution. Thus transmission line loading Sl is restricted by its upper limit as:
Sli
Slmax , i
I =1,2,…,NL
(17)
C. Problem Formulation The Environmental/Economic dispatch problem is formulated as a constrained multiobjective optimization problem and is given as:
Minimize F ( PG ) , E NOX ( PG ) NG
Subjected to:
i =1
min PGi
PGi
PGi
PD
(18)
PLoss = 0
max PGi
(19)
Slmax , I =1,2,…,NL i
Sli
6. Simulation Stategy The approach presented in this study is simulated on the standard IEEE 30-bus 6generator test system [13] (figure 1). The power system is connected through 41 transmission lines and the total system demand amounts to 2.834 p.u. Fuel Cost and NOX emission coefficients are provided in Tables 1 and 2 and the detailed data could be obtained from [2]. A. Representing Individual Bacterium
Implementation of the proposed MOBF on the problem at hand begins with the parameter encoding. In the proposed approach, each unit’s output is taken as the encoded parameter. Thus each particle consists of a 6-bit real coded string. During iterations, the output of five generators was calculated while the output of the last generator was adjusted to satisfy the equality constraints. Parametric setup for MOBF is provided in Table 1. B. Parametric Setup
p 6
S 100
Nc 50
Ned 15
Ped 0.10
Pg 0.25
Table 1: Parametric set-up of MOBF.
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C (i) 0.10
M 2
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Generator
ai
bi
ci
PGi (min)
PGi (max)
PG1
10
200
100
0.05
0.50
PG2
10
150
120
0.05
0.60
PG3
20
180
40
0.05
1.00
PG4
10
100
60
0.05
1.20
PG5
20
180
40
0.05
1.00
PG6
10
150
100
0.05
0.60
Table 2: Fuel cost coefficient.
Fig1: IEEE 30-bus system.
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Generator
i
i
i
i
i
1 G
P
4.091
-5.554
6.490
2.0E-4
2.857
PG2
2.543
-6.047
5.638
5.0E-4
3.333
PG3
4.258
-5.094
4.586
1.0E-6
8.000
PG4
5.326
-3.550
3.380
2.0E-3
2.000
PG5
4.258
-5.094
4.586
1.0E-6
8.000
PG6
6.131
-5.555
5.151
1.0E-5
6.667
Table 3: NOX emission coefficient. The simulations were carried out for three different cases: Case 1: Lossless system considering only generation capacity constraint. Case 2: Transmission losses are also considered. Case 3: All the constraints are considered. The results obtained by the implementation of the proposed approach were compared against those reported by other peer approaches. C. Determining Best Compromise Solution
After getting pareto front we have tried to extract the best compromise solution from the pareto. A fuzzy-based mechanism, as proposed in [2], is applied to get the best compromise solution, which can be offered to Decision Maker later. Membership value of each individual lying in Paretooptimal set Fi is computed using the membership function defined in the following way:
1 µi =
Fi Fi
max
max
if -Fi
-Fi
min
0
Fi
Fimin
if
Fimax
if
max
Fi
Fi
(20)
where µi stands for the membership value of the ith function (Fi). For each nondominated solution k the normalized membership value ( µ[k] ) is calculated using M
µ[ k ] =
µi [ k ]
i =1 N pareto M j =1
(21)
µi [ j ]
i =1
M is the number of objectives and N pareto is the number of bacteria in pareto-optimal front. The best compromise solution is that for which µ[ k ] is maximum.
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7. Experimental Results A. Case I
Initially, the system is considered as lossless, and all the constraints except generation capacity constraint are released. For this case MOBF as well as two other peer algorithms NSGA II and MOPSO are implemented. Pareto-fronts obtained by these algorithms are also provided in figure 2. Stopping criterion for these algorithms are taken as 10000 objective function evaluations. Fifteen independent runs of three algorithms are carried out. The Pareto fronts obtained has been shown in figure 3. The graphical results clearly reveal that the solutions found were well distributed and covered the entire Pareto front of the problem. Table 4 and 5 contained respectively the best results for fuel cost and emission obtained by the MOBF as compared to those reported using linear programming (LP) [3], multiobjective stochastic search technique (MOSST) [15], non-dominated sorting genetic algorithm (NSGA) [13], niched Pareto genetic algorithm (NPGA) [14], strength Pareto evolutionary algorithm (SPEA) [2] and NSGA II [17], FCPSO [21]. From Table 4 and 5 it is evident that the proposed algorithm is capable of providing better solution in comparison to competitor algorithms for minimum fuel cost and it performs comparably to some of the peer algorithms for minimum emission case. Case - I
LP [3]
MOSST [15]
NSGA [13]
NPGA [14]
SPEA [2]
FCPSO [21]
MOPSO
MOBF
0.1567
NSGAII [17] 0.1059
PG1
0.1500
0.1125
0.1080
0.1062
0.1070
0.0605
0.1099
PG2
0.3000
0.3020
0.2870
0.3177
0.3284
0.2897
0.2897
0.2522
0.2999
PG3
0.5500
0.5311
0.4671
0.5216
0.5386
0.5289
0.5250
0.5553
0.5245
PG4
1.0500
1.0208
1.0467
1.0146
1.0067
1.0025
1.0150
0.9761
1.0155
PG5
0.4600
0.5311
0.5037
0.5159
0.4949
0.5402
0.5300
0.6431
0.5243
PG6
0.3500
0.3625
0.3729
0.3583
0.3574
0.3664
0.3673
0.3466
0.3599
Cost Emission
606.31 605.89 600.57 600.16 600.26 600.15 600.13 601.3416 0.2233 0.2222 0.2228 0.2219 0.2212 0.2215 0.2223 0.2237 Table 4: Comparison of MOBF with other algorithms for minimum fuel cost (CASE I).
Case - I
LP [3]
MOSST [15]
NSGA [13]
PG1
0.4000
0.4095
PG2
0.4500
PG3
600.11 0.2221
NPGA [14]
SPEA [2]
FCPSO [21]
MOPSO
MOBF
0.4394
NSGAII [17] 0.4074
0.4002
0.4116
0.4097
0.4178
0.3948
0.4626
0.4511
0.4577
0.4474
0.4532
0.4550
0.4650
0.4520
0.5500
0.5426
0.5105
0.5389
0.5166
0.5329
0.5363
0.5828
0.5356
PG4
0.4000
0.3884
0.3871
0.3837
0.3688
0.3832
0.3842
0.3718
0.4068
PG5
0.5500
0.5427
0.5553
0.5352
0.5751
0.5383
0.5348
0.5047
0.5446
PG6
0.5000
0.5152
0.4905
0.5110
0.5259
0.5148
0.5140
0.4918
0.4991
Cost Emission
639.60
644.112
638.27
638.36
0.1942
0.1942
639.6896 0.1944
635.27
0.1942
639.18 0.1943
638.51
0.1942
639.23 0.1944
0.1942
Table 5: Comparison of MOBF with other algorithms for minimum NOX emission (CASE I).
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0.1942
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0.225
0.225
MOBF
0.22
0.22
0.215
0.215
0.21
0.21
Emission
Emission
NSGA-II
0.205
0.205
0.2
0.2
0.195
0.195
0.19 600
605
610
615
620 Cost
625
630
635
0.19 600
640
605
610
615
(a)
620 Cost
625
630
0.225 MOPSO
0.22
0.22
0.215
0.215
0.21
0.21
Emission
Emission
640
(b)
0.225
0.205
0.2
0.195
0.195
605
610
615
620 625 Cost
630
635
640
645
NSGA-II MOBF MOPSO
0.205
0.2
0.19 600
635
0.19 600
605
610
615
(c)
620 625 Cost
630
635
640
645
(d)
Fig 2: Optimal pareto front obtained by (a) NSGA II, (b) MOBF, (c) MOPSO for case I. (d) All pareto fronts combined together.
B. Case II
In this case, transmission losses as well as power balance constraints are also considered. Transmission losses are calculated using the power flow algorithm. The Pareto front obtained is shown in figure 3. From the figure, it can be deduced that the proposed algorithm was able to maintain the diversity over the trade-off front. It is evident from Table 6 and 7 that the results obtained by the proposed approach are competitive against other approaches.
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Case - II
NSGA [13]
NSGA- II [17]
NPGA [14]
SPEA [2]
FCPSO [21]
MOPSO
MOBF
PG1
0.4113
0.4141
0.3923
0.4043
0.4063
0.4002
0.4105
PG2
0.4591
0.4602
0.4700
0.4525
0.4586
0.4011
0.4601
PG3
0.5117
0.5429
0.5565
0.5525
0.5510
0.6717
0.5491
PG4
0.3724
0.4011
0.3695
0.4079
0.4084
0.4124
0.3999
PG5
0.5810
0.5422
0.5599
0.5468
0.5432
0.4309
0.53492
PG6
0.5304
0.5045
0.5163
0.5005
0.4974
0.5455
0.5115
Cost 647.25 644.13 645.98 642.60 642.89 642.48 643.84 Emission 0.19432 0.19424 0.19422 0.1942 0.1959 0.1942 0.19419 Table 6: Comparison of MOBF with other algorithms for minimum fuel cost (CASE II).
Case - II
NSGA [13]
NSGA- II [17]
NPGA [14]
SPEA [2]
FCPSO [21]
MOPSO
MOBF
PG1
0.1168
0.1182
0.1245
0.1086
0.1130
0.1157
0.1154
PG2
0.3165
0.3148
0.2792
0.3056
0.3145
0.2964
0.3053
PG3
0.5441
0.5910
0.6284
0.5818
0.5826
0.6631
0.5967
PG4
0.9447
0.9710
1.0264
0.9846
0.9860
0.9739
0.9812
PG5
0.5498
0.5172
0.4693
0.5288
0.5264
0.4341
0.5138
PG6
0.3964
0.3548
0.3993
0.3584
0.3450
0.3810
0.3537
Cost Emission
608.25 0.2166
607.80 0.2189
608.15 0.2236
607.81 0.2201
607.79 0.2201
608.0770 0.2203
607.51 0.2199
Table 7: Comparison of MOBF with other algorithms for minimum NOX emission (CASE II).
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0.22 0.225
MOBF NSGA-II
0.215
0.22
0.21 Emission
Emission
0.215
0.21
0.2
0.205
0.195
0.2
0.195 605
0.205
610
615
620
(a)
625 Cost
630
635
640
645
0.19 605
610
615
620
625 Cost
630
635
640
645
(b)
0.225
0.225 MOPSO
NSGA-II MOBF MOPSO
0.22
0.22
0.215
Emission
Emission
0.215
0.21
0.21
0.205
0.205 0.2 0.2
0.195 605
0.195
610
615
620
625 630 Cost
635
640
645
650
0.19 605
610
615
620
625 630 Cost
635
640
(c) (d) Fig 3: Optimal pareto front obtained by (a) NSGA II, (b) MOBF, (c) MOPSO for case II. (d) All pareto fronts combined together. C. Case III In this case, all the equality and inequality constraints are considered simultaneously. For comparison with reported results, the maximum line flow capacities used are 115% of the standard values provided in [11]. We compare the results, for the best cost and emission, obtained by the proposed MOBF with NSGA, NSGA-II, NPGA, SPEA, FCPSO, MOPSO and report in table VIII and IX. The results show that the proposed approach performed competitive with respect to NSGA and NPGA while its performance is slightly poor than SPEA, which provided best solutions in this case for both the objectives. The Pareto front obtained by the proposed approach is given in figure 8. It can again be observed that the proposed approach is capable of finding the Pareto front by effectively solving the problem when all the constraints are considered. The efficiency of the proposed approach is also verified by the results obtained from the -constraint method. The particles in the Pareto front obtained by the utilization of the proposed approach overlap with those obtained by the -constraint method, proving that the approach succeeded in obtaining the optimal points all over the Pareto front. In addition, the minimum values of each objective confirmed that the obtained solutions were able to cover the entire trade-off surface.
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Case – III
NSGA [13]
NSGA-II
NPGA [14]
SPEA [2]
FCPSO [21]
MOPSO
MOBF
PG1
0.1358
0.1870
0.1127
0.1319
0.1596
0.1772
0.1763
PG2
0.3151
0.3720
0.3747
0.3654
0.3535
0.4032
0.3581
PG3
0.8418
0.7378
0.8057
0.7791
0.7974
0.7382
0.7429
PG4
1.0431
0.5984
0.9031
0.9282
0.9719
0.6015
0.5970
PG5
0.0631
0.5669
0.1347
0.1308
0.0862
0.5432
0.5977
PG6
0.4664
0.3959
0.5331
0.5292
0.4961
0.3948
0.3861
Cost 620.87 619.1922 620.46 619.60 620.18 619.4951 619.03 Emission 0.2368 0.2027 0.2243 0.2244 0.2283 0.2028 0.2177 Table 8: Comparison of MOBF with other algorithms for minimum fuel cost (CASE III).
Case - III
NSGA [ 13]
NSGA-II
NPGA [14]
SPEA [2]
FCPSO [21]
MOPSO
MOBF
PG1
0.4403
0.4101
0.4753
0.4419
0.4797
0.4110
0.3980
PG2
0.4940
0.4630
0.5162
0.4598
0.5287
0.4649
0.4521
PG3
0.7509
0.5433
0.6513
0.6944
0.6711
0.5470
0.5516
PG4
0.5060
0.3894
0.4363
0.4616
0.5318
0.3921
0.4190
PG5
0.1375
0.5434
0.1896
0.1952
0.1257
0.5356
0.5338
PG6
0.5364
0.5147
0.5988
0.6131
0.5299
0.5133
0.5091
Cost Emission
649.24 0.2048
644.9830 0.1942
657.59 0.2017
651.71 0.2019
651.62 0.2047
644.8551 0.1942
641.43 0.1942
Table 9: Comparison of MOBF with other algorithms for minimum NOX emission (CASE III)
0.204
0.206 NSGA II
MOBF
0.203 0.204
0.202 0.201
0.202 Emission
Emission
0.2 0.199
0.2
0.198 0.198
0.197 0.196
0.196 0.195 0.194 615
620
625
630 Cost
635
640
645
0.194 615
620
(a)
625
630 Cost
635
(b)
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0.204
0.206
MOPSO
NSGA-II MOBF MOPSO
0.203 0.204
0.202 0.201
0.202 Emission
Emission
0.2 0.199 0.198
0.2
0.198
0.197 0.196
0.196
0.195 0.194 615
620
625
630 Cost
635
640
0.194 615
645
620
625
630 Cost
(c)
635
640
645
(d)
Fig 4: Optimal pareto front obtained by (a) NSGA II, (b) MOBF, (c) MOPSO for case III. (d) all pareto fronts combined together.
D. Calculation of Convergence and Diversity metric
For testing efficacy of the proposed multiobjective bacterial foraging optimization we take two different performance metrics: 1) C-metric [29]: This metric can directly compare the quality of two non-dominated sets. For the calculation of this metric, we don’t require any optimal pareto front and it is very easy to compute. If A and B are two different sets of non-dominated solutions,
If C ( A, B ) = 1 , all candidate solutions in B are dominated by at least one solution in A. If C ( A, B ) = 0 , no candidate solutions in B is dominated by any solution in A. 2) Moment of inertia based diversity metric [30]: This metric does not require any optimal pareto front and has a relation with the Hamming and Euclidean distance between solutions. If there is S number of points on a pareto front and the space is N-dimensional then centroid C i for
i th dimension is given by, S
x ij
, For i = 1,2,......, N . S th th Where, xij denotes the i dimension of the j point. Ci =
j =1
Then the diversity measuring D-metric is given by, I
=
N
S
( xij
Ci ) 2
i =1 j =1
The higher the value of I higher the higher the diversity of S points on the pareto front.
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Table 10 gives the convergence metric values taking the MOBF algorithm as the reference and also the competitive algorithms as the references. Table 11 describes the diversity of the obtained set of solutions for all three algorithms by means of diversity metric as defined above.
Case considered
Statistical Measure
C (MOBF, NSGAII)
C (MOBF, MOPSO)
C (NSGAII, MOBF)
C (MOPSO, MOBF)
0.0000
0.6000
0.0000
0.0000
Mean
0.0320
0.7400
0.0360
0.0400
Standard Deviation
0.3030
0.1538
0.0416
0.0418
0.400
0.1010
0.0200
0.1500
Mean
0.2120
0.2301
0.1280
0.2580
Standard Deviation
0.2179
0.2573
0.1052
0.0965
0.1230
0.2133
0.0102
0.0320
Mean
0.2610
0.3102
0.0976
0.1970
Standard Deviation
0.0831
0.2601
0.1088
0.2210
Minimum Case I
Minimum Case II
Minimum Case III
Table 10: Convergence metric calculated for MOBF, MOPSO, and NSGA II.
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Different Algorithms
Statistical Measures
MOBF
NSGA-II
MOPSO
Case I
Maximum
14325.1314
14730.3697
12430.9544
Mean
13778.2254
13491.8455
9528.3043
Standard Deviation Maximum
706.3708
1050.7916
2729.7640
13255.5736
13328.3598
11098.2964
Mean
12555.3948
12912.1349
10131.6287
Standard Deviation
556.8273
379.9596
867.5516
Maximum
7241.6285
6812.5383
5001.4785
Mean
6775.7275
6585.1316
4868.2960
Standard Deviation
1041.7862
207.5929
182.3676
Case II
Case III
Table 11: Divergence metric calculated for MOBF, MOPSO, and NSGA II. E. Best Compromise Solution and Epsilon Constraint: In this section we determine the best compromise solution using equation 20 and 21and the result is reported in figure 5. Solution on the non-dominated front, which has the maximum membership value with the decision maker, is given the title of best compromise solution. Here we also try to get an overview of the pareto front with epsilon constraint method [31]. Case I 0.225 MOBF Best Compromise Solution Epsilon Constraint
0.22
Emission
0.215
0.21
0.205
0.2
0.195
0.19 600
605
610
615
620 Cost
625
630
(a)
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0.22
0.215
0.21
0.205
0.2
0.195
0.19 605
610
615
620
625
630
635
640
645
(b) 0.204 MOBF Best Compromise SOlution Epsilon Constraint
0.203 0.202 0.201 0.2 0.199 0.198 0.197 0.196 0.195 0.194 615
620
625
630
635
640
645
(c) Fig 5: Pareto fronts with the best compromise solution and epsilon constraint for all the three cases.
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7. Conclusion In this paper we develop a novel multiobjective optimization algorithm based on bacterial foraging, which is then used for constrained Economic Electrical Load Dispatch problem. In this minimization problem cost and emission are two objectives, which are to be optimized, subjected to some other constraints. IEEE 30 bus 6-generator system has been chosen for this purpose. We also determine best compromise solution from the final pareto front obtained using a fuzzy approach. To test convergence and diversity of the obtained pareto fronts, some performance metrics are evaluated. Comparing values of these metrics we come to the conclusion that the proposed approach is capable to identify a good pareto of solutions maintaining adequate diversity. Apart from these we find the pareto using epsilon constraint method. We also graphically plot these solutions on final pareto front to test efficiency of the proposed multiobjective approach.
8. References 1) IEEE Current Operating Problems Working Group, Potential impacts of clean air regulations on system operations, vol. 10, pp. 647-653, 1995. 2) M. A. Abido, “Environmental/Economic Power Dispatch using Multiobjective Evolutionary Algorithms,” IEEE Transactions on Power Systems, vol. 18, no. 4, pp. 1529-1537, 2003. 3) A. Farag, S. Al-Baiyat and T.C. Cheng, “Economic load dispatch multiobjective optimization procedures using linear programming techniques,” IEEE Trans Power Syst, vol. 10, no. 2, pp. 731-738, 1995. 4) J. Zahavi, L. Eisenberg, “Economic-environmental power dispatch,” IEEE Trans Syst, Man, Cybernet, vol. 5, no. 5, pp. 485-489, 1985. 5) J.X. Xu, C.S. Chang, and X.W. Wang, “Constrained multiobjective global optimization of longitudinal interconnected power system by genetic algorithm,” IEE Proc-Gener Transm Distrib, vol. 143, no. 5, pp. 435- 436, 1996. 6) C.S. Chang, K.P. Wong, and B. Fan, “Security-constrained multiobjective generation dispatch using bicriterion global optimization,” IEE Proc- Gener Transm Distrib, vol. 142, no. 4, pp. 406-414, 1995. 7) J.S. Dhillon, S.C. Parti, and D.P. Kothari, “Stochastic economic emission load dispatch,” Electr Power Syst Res, vol. 26, pp. 186-197, 1993. 8) S.F. Brodesky, and R.W. Hahn, “Assessing the influence of power pools on emission constrained economic dispatch,” IEEE Trans Power Syst, vol. 1, no. 1, pp. 57-62, 1986. 9) G.P. Granelli, M. Montagna, and G.L. Pasini, “Emission constrained dynamic dispatch,” Electr Power Syst Res, vol. 24, pp. 56-64, 1992. 10) A.A. Abou El-Ela, and M.A. Abido, “Optimal operation strategy for reactive power control,” Model, Simulation Control, Part A, ASME, vol. 41, no. 3, pp. 19-40, 1992. 11) R. Yokoyama, S.H. Bae, T. Morita, and H. Sasaki, “Multiobjective generation dispatch based on probability security criteria,” IEEE Trans Power Syst, vol. 3, no. 1, pp. 317-324, 1988. 12) Y.T. Hsiao, H.D. Chiang, C.C. Liu, and Y.L. Chen, “A computer package for optimal multi-objective VAR planning in large scale power systems,” IEEE Trans Power Syst, vol. 9, no. 2, pp. 668-676, 1994. 13) M.A.Abido, “A Novel Multiobjective Evolutionary Algorithm for Environmental/Economic Power Dispatch,” Electr Power Syst Res, vol. 65, pp. 71-91, 2003. 14) M. A. Abido, “A Niched Pareto Genetic Algorithm for Environmental/ Economic Power Dispatch,” Electr Power Syst Res, vol. 25, no. 2, pp. 97-105, 2003. 15) D. B. Das, and C. Patvardhan, “New Multi-Objective Stochastic Search Technique for Economic Load Dispatch,” IEE Proceedings-Generation, Trans. and Distribution, vol. 145, no. 6, pp. 747-752, 1998.
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16) G. L. Viviani, and G. T. Heydt, “Stochastic Optimal Energy Dispatch,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-100, no. 7, pp. 3221-3228, 1981. 17) T. F. Robert, A.H. King, C. S. Harry, Rughooputh, and K. Deb, “Evolutionary MultiObjective Environmental/Economic Dispatch: Stochastic vs. Deterministic Approaches,” KanGAL Report Number 2004019, pp. 1-15, 2004. 18) T. F. Robert, A.H. King, C. S. Harry, Rughooputh, and K. Deb, “Stochastic Evolutionary Multiobjective Environmental/Economic Dispatch,” 2006 IEEE Congress on Evolutionary Computation, Vancouver, BC, Canada, pp. 94-953, July 2006. 19) E. Zitzler, Laumanns, M., and L. Thiele, “SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization,” In: Evolutionary Methods for Design, Optimisation and Control, 19-26, Barcelona, Spain 2002. 20) B. Zhao, and Y. J. Cao, “Multiple objective particle swarm optimization technique for economic load dispatch,” Journal of Zhejiang University SCIENCE, vol. 6A, no. 5, pp. 420-427, 2005. 21) Shubham Agrawal, B. K. Panigrahi , and M. K. Tiwari, “Multiobjective Particle Swarm Algorithm withFuzzy Clustering for Electrical Power Dispatch”, IEEE Trans on Evolutionary Computation. (Accepted). 22) Passino, K. M.: Biomimicry of Bacterial Foraging for Distributed Optimization and Control, IEEE Control Systems Magazine, 52-67, (2002). 23) Liu, Y. and Passino, K. M.: Biomimicry of Social Foraging Bacteria for Distributed Optimization: Models, Principles, and Emergent Behaviors, Journal of Optimization Theory And Applications: Vol. 115, No. 3, pp. 603–628, December 2002 24) Kim, D. H., Abraham, A., Cho, J. H.: A hybrid genetic algorithm and bacterial foraging approach for global optimization, Information Sciences, Vol. 177 (18), 3918-3937, (2007). 25) Mishra, S.: A hybrid least square-fuzzy bacterial foraging strategy for harmonic estimation. IEEE Trans. on Evolutionary Computation, vol. 9(1): 61-73, (2005). 26) Tripathy, M., Mishra, S., Lai, L. L. and Zhang, Q. P.: Transmission Loss Reduction Based on FACTS and Bacteria Foraging Algorithm. PPSN, 222-231, (2006). 27) Mishra, S. and Bhende C. N.: Bacterial Foraging Technique-Based Optimized Active Power Filter for Load Compensation, IEEE Transactions on Power Delivery, Volume 22, Issue 1, Jan. 2007 Page(s): 457 – 465. 28) Zamuda, A.; Brest, J.; Boskovic, B.; Zumer, V, “Differential evolution for multiobjective optimization with self adaptation”, 2006 IEEE Congress on Evolutionary Computation, pp.3617-3624. 29) Zitler E., Deb K., Thiele L., “ Multiobjective Evolutionary Algorithms : A Comparative Study and the Strength Pareto Approach”, IEEE Transactions on Evolutionary Computation, 3(4) pp-257-271, (1999). 30) Morrison R.W., De Jong K.A., “ Measurement of Population Diversity , Artificial Evolution”, Lecture Notes in Computer Science, 2310, Springer, 31-41,(2001). 31) K.Deb, “Multiobjective Optimization Using Evolutionary Algorithms”, Chichester UK: Willey 2001.
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