Truong Khoa H^{1*}, Vasant P^{1}, Balbir Singh^{1} and Vo Dieu N^{2}
^{1}Department of Fundamental and Applied Sciences, Universiti Teknologi Petronas, Malaysia
^{2}Department of Power Systems, HCMC University of Technology, Vietnam
Received Date: April 11, 2015 Accepted Date: June 29, 2015 Published Date: July 06, 2015
Citation: Khoa TH, Vasant P, Singh B, Dieu VN (2015) Solving Economic Dispatch By Using Swarm Based MeanVariance Mapping Optimization (MVMOS). Global J Technol Optim 6:184. doi:10.4172/22298711.1000184
Copyright: © 2015 Khoa TH, et al. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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This paper proposes a novel optimization which named as Swarm based Meanvariance mapping optimization (MVMOS) for solving the economic dispatch. The proposed optimization algorithm is the extension of the original single particle meanvariance mapping optimization (MVMO). The novel feature is the special mapping function applied for the mutation base on the mean and variance of nbest population.The MVMOS outperforms the classical MVMO in global search ability due to the improvement of the mapping. The proposed MVMOS is investigated on four test power systems, including 3, 13 , 20 thermal generating units and largescale system 140 units with quadratic cost function and the obtained results are compared with many other known methods in the literature. Test results show that the proposed method can efficiently implement for solving economic dispatch.
Economic dispatch; Quadraticfuel cost function; MVMO; MVMO^{S}
N Total number of generator
F_{T} Total operationcost
F_{i} Fuel cost function of generator i
a_{i}, b_{i}, c_{i} Fuel cost coefficients of generator i
B_{ij},B_{0i},B_{00} Bmatrix coefficients for transmission power loss
P_{D} Total system load demand
P_{i} Power output of generatori
P_{i,max} Maximum power output of generator i
P_{i,min} Minimum power output of generator i
P_{L} Total transmission loss
iter_{max} Maximum number of iterations
n_var Numberof variable (generators)
n_par Number of particles
mode Variable selection strategy for offspring creation
archive zize nbest individuals to be stored in the table
d_{i} Initial smoothing factor
Initial smoothing factor increment
Final smoothing factor increment
Initial shape scaling factor
Final shape scaling factor
D_{min} Minimum distance threshold to the global best solution
n_randomly Initial number of variables selected for mutation
indep.runs m steps independently to collect a set of reliable individual solutions
The Economic Dispatch (ED) is an essential optimization task in the power generation system and its objective is to determine the economical real power output of the thermal generating units to supply required power load demand at the minimum fuel cost while satisfying all units and system constrains [1,2]. Since the concept of economic dispatch (ED) started in the 1950’s, there are a lot of various methods have been employed for solving ED problems, but in short there are three main categories: Methods based on mathematical programming (Classical calculusbased techniques), methods based on artificial intelligence and hybrid methods.
For mathematical convenience, the objective cost function of ED problem is the quadratic function approximations [3], was solved by methods based on mathematical programming such as lambda iteration method, Newton’s method, gradient search, dynamic programming [3], linear programming [4], nonlinear programming [5] and quadratic programming [6]. These methods are conventional techniques that were early employed. Over the past years, more advanced methods based on artificial intelligence have been developed and implemented outstandingly to ED problem such as Hopfield Neural Network (HNN) [7,8], Evolutionary Programming (EP) [9], Differential Evolution (DE) [10], Genetic Algorithm (GA) [11], Ant Colony Optimization (ACO) [12], Particle Swarm Optimization (PSO) [13,14] Bacterial Foraging (BF) [15], and Artificial Bee Colony (ABC) algorithm [16]. These methods do not always guarantee to find the global optimal solution in finite computational time but their ability often find near global optimal solution for optimization problems. Besides the single mentioned methods, hybrid methods have been also developed for solving the ED problems such as hybridization of evolutionary programming with Sequential Quadratic Programming (EPSQP) [17], combining of chaotic differential evolution and quadratic programming (DECSQP) [18] hybrid technique integrating the uniform design with the genetic algorithm (UHGA) [19], selftuning hybrid differential evolution (selftuning HDE) [20], and fuzzy adaptive particle swarm optimization algorithm with Nelder–Mead simplex search (FAPSONM) [21]. These hybrid methods become powerful search methods for obtaining higher solution quality due to using the advantages of each element method to improve their search ability for the complex problems. Nevertheless, they may be slower and more complicated than the element methods because of combination of several operations.
The above artificial intelligence methods are population based metaheuristic which can deal with multiform optimization problems [22]. Recently, Prof. István Erlich has been conceived and developed a novel optimization technique which is named Meanvariance mapping optimization (MVMO) [23]. This algorithm is socalled “populationbased stochastic optimization techniques”. MVMO has the capability to find the optimum solution quickly with minimum risk of premature convergence.
The extensions of MVMO, which named Swarm based Meanvariance mapping optimization (MVMOS) [24], has been developed to become more effective. In this paper, MVMOS is proposed for solving the economic dispatch problem with quadratic cost function.
Section II presents the formulation of the ED. The review of MVMO, extension of MVMOMVMOS and implementation of the proposed MVMOS to ED problem are addressed in Section III. The numerical results are showcased in Section IV. The discussion is followed in Section V. After all, the conclusion are given.
The power system consists of N thermal generating units. Each unit has a fuel cost function, shown as F_{i} , togenerates a power out P_{i}. The total fuel costof the system,F_{T},is sum of fuel cost of each unit.
(1)
The optimization problem of the ED is to minimize the total fuel cost F_{T} which be written as:
(2)
Generally, the fuel cost curve of a thermal generating unit is presented as quadratic function as:
(3)
The constraints of the ED problem must be satisfied during the optimization process are peresented as follows:
Real power balance equation
The total active power output of generating units must be equal to total active power load demand plus power loss:
(4)
The power loss P_{L}is calculated by the below formulation [3]:
(5)
Generator capacity limits
The active power output of generating units must be within the allowed limits:
(6)
Review of MVMO
Meanvariance mapping optimization (MVMO) is a novel optimization algorithm falls into the category of the socalled “populationbased stochastic optimization technique”. The similarities between MVMO and the other known stochastic algorithms are three evolutionary operators: selection, mutation and crossover. However, the major differences between MVMO and other existing techniques are as follows:
 The key feature of MVMO is a special mapping function which applied for mutating the offspring based on meanvariance of the solutions stored in the archive.
The mean and variance v_{i} are calculated as follows:
(7)
(8)
where, j= 1,2,..., n (n is population size).
The mapping function is depicted in Figure 1. The transformation mapping function, h, is calculated by the mean x and shape variables s_{i1} and s_{i2} as follows:
(9)
where,
(10)
The scaling factor f_{s} is a MVMO parameter which allows for controlling the search process during iteration. s_{i} is the shape variable.
• All variables are initialized within the limit range [0,1]. The output of mapping function is always inside [0,1]. However, the function evaluation is carried out always in the original scales.
• MVMO is a singleagent search algorithm because it uses a single parentoffspring in each iteration. Therefore, the number of fitness evaluations is identical to the number of iterations.
Interested readers can find the basics of algorithm and reference values for the algorithm’s settings in [23,24].
Extension of MVMOMVMO^{S}
Recently,theswarm version of MVMO has been developed. This version is abbreviated as MVMO^{S}. The new approach extends the ability of global searching of the classical MVMO by starting the search with a set of particles.
Modified version of MVMO: The MVMOalgorithm extends two important parameters. These two parameters are used for calculation and assignment of s_{i1} and s_{i2} as follows:
Variable F_{S} factor: In (10), the factor fs allows the modification of the shape factor calculated from the variance.
The extension of f_{S} factor is for the need of exploring the search space at the beginning more globally whereas, at the end of the iterations, the focus should be on the exploitation. It is determined by:
(11)
Where
(12)
r and () is a random number with the bounds [0, 1].
In (12) , the variable i represents the iterationnumber.
For the more accuracy of the optimization ,the initial and final values of it is recommended that <1 and >1 The suggested range of initial values of is from 0.9 to 1.0 and forfinal values of is from 1.0 to 3.0 .
When , which means that the option for controlling the fs factor is not used.
Variable increment Δd : The MVMO^{S} algorithm uses the factor Δd as presented below:
s^{i1}=s^{i2}=s^{i}
if s_{i}> 0 then
= (1+)+ 2 .+ (rand() – 0.5)
if s_{i}>d_{i}
d_{i}=d_{i} .Δd
else
d_{i}=d_{i} /Δd
end if
if rand() ≥ 0.5 then
s_{i1}=s_{i} ; s_{i2}=d_{i}
else
s_{i1}=s_{i} ; s_{i2}=d_{i}
end if
end if
The extension of variable increment Δd is used for the asymmetric characteristic of the mapping function.
At the start ofthe algorithm, the initial values of di(typically between 15) are set for all variables. At every iteration, if si>di , di will be multiplied by Δd leads to increased di. In case s_{i}<d_{i}, the current diis divided byΔd which is always greater than 1.0 and thus resulting in reduced value of di. Therefore, d_{i} will always oscillate around the current shape factor si. Furthermore, Δd is varied randomly around the value(1 + 0 Δd ) with the amplitude of 0 Δd adjusted in accordance to (14), where 0 Δd can be allowed to decrease from 0.4 to 0.01.
(14)
Swarm variant of MVMO: Compared with classical MVMO, the swarm variant explores the solution space more aggressively. The search process is started with a set ofparticles, each having its own memory defined in terms of the corresponding archive and mapping function. Initially, each particle performs m steps independently to collect a set of reliable individual solutions. Then, the particles start to communicate and to exchange information.
The scheme of MVMO^{S} is depicted in Figure 2.
i and k donate the function evaluation and particle counters, respectively. Whereas m and np stand for maximum number of independent runs and total number of particles, respectively.
It is not worth it to follow particles which are very close to each other since this would entail redundancy. To avoid closeness between particles (i.e. redundancy), the normalized distance of each particle’s local best solution x^{lbest,i} to the global best x^{gbest} is calculated by:
(15)
where, n denotes the number of optimization variables.
The ith particle is discarded from the optimization process if the distance D_{i} is less than a certain user defined threshold D_{min}.
A zero threshold means that all particles are considered throughout the whole process whereas a unit threshold will result in the dropping of all particles except the global best. In this case after (m*n_{p}+ n_{p}) fitness evaluations only one particle, the gbest, remains. Intermediate threshold values entail better adaptation to any optimization problem.
After independent evaluation, and if the particle is further considered, the global best solution guides the search by assigning x^{gbest}, instead of x^{lbest,i}, as parent for every particle’s offspring. The remaining steps are identical with those of the classical MVMO: A subset of dimensions in the parent vector is directly inherited whereas the remaining dimensions are selected and mutated, based on local statistics (mean and variance) of the particle, via mapping function.
Implemention of MVMO^{S} to ED
Handing of constraints: To guarantee that the equality constraint (4) is always satisfied, a slack generating unit is randomlyselected from N generating units and therefore its power output will be dependent on the power outputs of remaining N1 generating units in the system. The method for calculation of power output for the slack unit is given in [25]. The power output of the slack unit is as follows:
(16)
where,s is a random unit selected from N units
The power transmission loss in (5) is rewritten by considering PS as an unknown variable
(17)
Substituting (17) into (13), a quadratic equation is abtained as follows:
(18)
where A, B and C are given by:
(19)
(20)
(21)
The power output of the slack generator is the posstive root of (18) between the two ones abtained as follows:
(22)
Based on the slack variable method, the fitness function for the proposed MVMO^{S} will include the objective function (2) and penalty terms for the slack unit if inequality (6) is violated. The fitness function is as follows:
(22)
Implemention of MVMOS to ED: The steps of procedure of MVMO^{S} for the ED problem are described as follows:
Step 1: Setting the parameters for MVMOS including iter_{max}, n_var, n_par, mode, d_{i}, , archive zize, , n_randomly, n_randomly_min, indep.runs(m), D_{min}
Set i=1, i donates the function evaluation
Step 2: Normalize initial variables to the range [0,1] (i.e. swarm of particles).
x_normalized= rand(n_par,n_var)
Step 3: Set k=1, kdonate particle counters.
Step 4: Using denormalized variables to evaluate fitness function, store f_{best}and x_{best}in archive
Step 5: Increase i =i+1. If i<m ( independent steps), go to Step 5. Otherwise, go to Step 6.
Step 6: Check the particles for the global best, collect a set of reliable individual solutions. The ith particle is discarded from the optimization process if the distance D_{i} is less than a certain user defined threshold D_{min}.
Step 7: Create offspring generation through three evolutionary operators: selection, mutation and crossover.
Step 8: if k<n_{p} ,increasek=k+1 and go to step 4. Otherwise, go to step 9.
Step 9: Check termination criteria. If stoping criteria is satisfied, stop. Otherwise, go to step 3.
Denomalization of optimization variables: The output of mapping function is always inside [0,1]. However, the function evaluation is carried out always in the original scales. Denomalization of optimization variables is carried by using (24):
P_{i}=P_{i,min} + Scaling.x_normalized(ι,:)
where,
Scaling=P_{i,max} P_{i,min}
Termination criteria:The algorithm of the proposed MVMOS is terminated when the maximum number of iterations iter_{max} is reached.
The proposed MVMOS has been applied to the ED problem with the quadratic cost function.Four test cases including 3, 13, 20 thermal generating units and largescale systemwith 140 units are carried out. For each case, the algorithm of MVMOS is run 50 independent trialsona Intel Core i53470 CPU 3.2 GHz PC, Ram 4GB. The implematation of the proposed MVMOS was done in Matlab R2013a platform.
Selection of parameters
The parameters of MVMOSinclude itermax, n_var, n_par, mode, di,, ,archive zize, , n_randomly, indep.runs(m), Dmin. Since different parameters of the proposed method effect on the performance of MVMOS. Hence, it is important to determine an optimal set of parameters of the proposed methods for ED problem. For each problem, selection of parameters is carried out by varying only one parameter at a time and keeping the other. The parameter is first fixed at the low value and then increased. Multiple runs are carried out to choose the suitable set of parameters. The typical parameters are selected as follows:
• itermax : maximum number of iterations depend on the dimension of problems. The maximum number of iterations is selected in the range from 1000 to 50000 iterations for case 1, case 2 and case 3, and 80000 for case 4.
• n_var: number of variable (generators), dimension of problems. n_var is set to 3,13, 20, 140 for case 1, case 2, case 3and case 4, respectively.
• n_par: number of particles is varied from 5, 10, 20, 30, 40 and 50, respectively. By experiments, the good solution is obtained when number of particles isset to 5. Hence, number of particles is set for all cases.
• mode: There are four variable selection strategy for offspring creation [23]. Afer all simulations, stragy 3 (mode=4) is suporior to the other stragy.
, The range ofin (14) is [0.010.4]. By experiments,andis set to 0.4 and 0.02, respectively for all cases.
,Therange of values ofis from 0.9 to 1.0 and forvalues ofis from 1.0 to 3.0 [24]. For all cases,is set to 0.95 in the range [0.9, 1.0] andis set to 3 in the range 3 in the range [1.0, 3.0].
indep.runs(m) : The maximum number of independent runs can be selected in the range from 100 to 800.
D min is set to 0 for all cases.
Numerical results
Case 1: 3 unit system: The test system consists of 3 generating units without transmission loss. Here, the system load demand is 450MW and 850MW, respectively. The data of the system is taken from [25]. The power transmission loss is neglected in this case. The obtained results by the MVMOS corresponding to the two load demand are given in Table 1.
Unit  Power outputs P_{i} (MW)  

P_{D}= 450MW P_{D}= 850MW  
1  205.3077  393.1698 
2  183.3457  334.6038 
3  61.3466  122.2264 
Total power(MW)  450  850 
Min Cost ($/h)  4652.4735  8194.3561 
Average CPU time (s)  0.87  0.88 
Table 1: Power output of 3unit system for load demand of 450 MW and 850 MW by MVMOS.
The parameters for MVMOS are set as follows: itermax=1000, n_var(generators)=3, np=5, archive size=4, indep.runs (m)=100, n_randomly=2, n_randomly_min=2, , , ,, ,Dmin=0
The total cost comparison between MVMOS and the other methods are presented in Table 2. In case ofthe 450MW load demand, the results and computational time of MVMOS are less than PSO and ABC. In case ofthe 850MW load demand, the results ofMVMOS is less than IEP, HS, GA, BGA, and the same as NM, PSO. The proposed MVMOS is faster than HS, GA and BGA. There is no computer processor reported for PSO, ABC, HS, GA and BGA and no computational time for the other methods. Table 1 shows that the power output obtained by the MVMOS is always satisfy the constraints.
Method  450 (MW)  850 (MW)  

Cost ($/h)  CPU (s)  Cost ($/h)  CPU (s)  
PSO[26]  4653  7.69     
ABC[26]  4653  3.91     
NM[25]      8194.3561   
IEP[25]      8194.3561   
PSO[25]      8194.3561   
HS[27]      8194.5  27.62 
GA[27]      8194.3591  10.94 
BGA[27]      8194.357  3.66 
MVMO^{S}  4652.4735  0.87  8194.3561  0.88 
Table 2: Comparison of results and CPU time by MVMO and other techniques for 3unit system.
Case 2: 13 unit  system: The data of 13 generating unit test system is from [27]. In this case, the power transmission loss is neglected. The obtained results by the MVMOS corresponding to the twoload demand of 1800MW and 2520MW are shown in Table 3.
Unit  Power outputs Pi (MW) PD= 1800MW PD= 2520MW 


1  506.9117  679.9970 
2  253.4558  359.9957 
3  253.4560  360.0000 
4  99.3627  155.1418 
5  99.3627  155.1929 
6  99.3627  155.0396 
7  99.3628  156.0892 
8  99.3628  154.4661 
9  99.3627  154.0705 
10  40.0000  40.0015 
11  40.0000  40.0038 
12  55.0000  55.0022 
13  55.0000  55.0000 
Total power (MW)  1800.0000  2520.0000 
Min Cost ($/h)  17932.4741  24050.1408 
Average CPU time (s)  2.97  16.29 
Table 3: Power output of each generating unit in 13unit system for load demand of 1800 MW and 2520 MW by MVMOS.
For the load demand of 1800MW, the parameters for MVMOS are set as follows: itermax=5000, n_var(generators)=13, npnp=5, archive size=4, indep.runs (m)=300, n_randomly=5, n_randomly_min=4, , , ,, ,Dmin=0
For the load demand of 2520MW, the parameters for MVMOS are set as follows: itermax=30000, n_var(generators)=13, np=5, archive size=5, indep.runs (m)=300, n_randomly=8, n_randomly_min=4 , , ,, ,Dmin=0
The results of MVMOS for 1800 MW and 2520 MW load demands are compared to the other methods as presented in Table 4. In case of the 1800 MW load demand, the total cost obtained by MVMOS is less than HS, GA and BGA. The computational time of MVMOS is less than HS, GA and slower than BGA. There is no computer processor reported for HS, GA and BGA. In case of the 2520 MW load demand, the total cost obtained by MVMOS is less thanIteration, GA and SQP and same result as ALHN. The computational time of MVMOS is slower than these methods. The computational times forIteration, GA, SQP and ALHN methods were from a Petium M 1.5 GHz PC. Table 3 shows that the power output obtained by the MVMOS is always satisfy the constraints
Method  1800 (MW)  2520 (MW)  

Cost ($/h)  CPU(s)  Cost ($/h)  CPU(s)  
HS[27]  18274.0065  16.135     
GA[27]  18194.9507  5.8     
BGA[27]  17971.5503  1.98     
λIteration[28]      24058.27  0.85 
GA[28]      25087.45  1.76 
SQP[28]      24058.29  4.57 
ALHN[28]      24050.14  0.044 
MVMO^{S}  17932.4741  2.97  24050.14  16.29 
Table 4: Comparison of results and CPU time by MVMO and other techniques for 13unit system.
Case 3: 20 unitsystem: The test system includes 20 generators with the system load demand of 2500MW. The data of this system is from [28]. The power transmission loss is ignored in this case. The obtained results by the MVMOS is shown in Table 5.
Unit  Power outputs P_{i} (MW)  Unit  Power outputs P_{i} (MW) 

1  600.0002  11  286.9466 
2  131.1723  12  432.7209 
3  50.0000  13  124.1584 
4  50.0000  14  73.3046 
5  92.9882  15  94.8873 
6  20.0000  16  36.2083 
7  125.0000  17  30.0000 
8  50.0000  18  37.5181 
9  111.7012  19  77.8376 
10  45.5563  20  30.0000 
Total power (MW)  2500.0000  
Min Cost ($/h)  60152.5509  
Average CPU time (s)  31.45 
Table 5: Power output of 20unit system for load demand of 2500MW by MVMO^{S}
The MVMOS is run 50 independent trials. The parameters for MVMOS are set as follows: itermax=70000, n_var(generators)=5, np=5, archive size=4, indep. runs(m)=400, n_randomly=7, n_randomly_ min=6,, , ,, ,Dmin=0
Table 6 shows the comparison of results obtained and computational time by MVMOS and the other methods. In this case, the results of MVMOS is lessIteration, GA and SQP and the same as ALHN. The computation time of MVMOS is less than GA and slower than other methods. The computational times forIteration, GA, SQP and ALHN methods were from a Pentium M 1.5 GHz PC. Table 5 shows that the power output obtained by the MVMOS is always satisfy the constraints. Although the parameters for two load demands is different, the MVMOS guarantees the convergence to the global solution for the 13unit test system.
Method  Total CostP_{D}= 2500MW  CPU(s) 

λIteration [27]  60245.67  0.32 
GA[27]  61107  61.17 
SQP[27]  60693.14  1.28 
ALHN[27]  60152.55  0.076 
MVMO^{S}  60152.55  31.45 
Table 6: Comparison of results and CPU time by MVMO and other techniques for 20unit system.
Case 4: largescale system 140 unit: The Korean power system consists of 140 thermal generating units is the test system for this case. Here, the system load demand is 49342 MW. The data of the system is given in [31]. The power transmission loss is also ignored in this case. The parameters for MVMOS are set as follows: itermax=80000, n_var(generators)=140, np=5, archive size=4, indep. runs (m)=800, n_randomly=20, n_randomly_min=10,, , ,, ,Dmin=0
The obtained results and computational time by the MVMOS are given in Table 7. As seen in Table 7, the power output obtained by the MVMOS is always satisfy the constraints.
Unit  Pi (MW)  Unit  Pi (MW)  Unit  Pi (MW) 

1  113.6343  47  239.5319  94  984.0000 
2  189.0000  48  250.0000  95  978.0000 
3  190.0000  49  250.0000  96  682.0000 
4  190.0000  50  250.0000  98  720.0000 
5  170.4101  51  165.0000  99  718.0000 
6  190.0000  52  165.0000  100  720.0000 
7  490.0000  53  165.0000  101  964.0000 
8  490.0000  54  165.0000  102  958.0000 
9  496.0000  55  180.0000  103  1007.0000 
10  496.0000  56  180.0000  104  1006.0000 
12  496.0000  57  103.0000  105  1013.0000 
13  496.0000  58  198.0000  106  1020.0000 
14  506.0000  59  312.0000  107  954.0000 
15  509.0000  60  279.9162  108  952.0000 
16  506.0000  61  163.0000  109  1006.0000 
17  505.0000  62  95.0000  110  1013.0000 
18  506.0000  63  160.0000  111  1021.0000 
19  506.0000  64  160.0000  112  1015.0000 
20  505.0000  65  490.0000  113  94.0000 
21  505.0000  66  196.0000  114  94.0000 
22  505.0000  67  490.0000  115  94.0000 
23  505.0000  68  490.0000  116  244.0000 
24  505.0000  69  130.0000  117  244.0000 
25  505.0000  70  280.6907  118  244.0000 
26  537.0000  71  137.0000  119  95.0000 
27  537.0000  72  334.031  120  95.0000 
28  549.0000  73  195.0000  121  116.0000 
29  549.0000  74  175.0000  122  2.0000 
30  501.0000  75  175.0000  123  4.0000 
31  501.0000  76  175.0000  124  15.0000 
32  506.0000  77  175.0000  125  9.0000 
33  506.0000  78  330.0000  126  12.0000 
34  506.0000  79  531.0000  127  10.0000 
35  506.0000  80  531.0000  128  112.0000 
36  500.0000  81  350.0494  129  4.0000 
37  500.0000  82  56.0000  130  5.0000 
38  241.0000  83  115.0000  131  5.0000 
39  241.0000  84  115.0000  132  50.0000 
40  774.0000  85  115.0000  133  5.0000 
Table 7: Power output of 140unit system for load demand of 49342MW by MVMOS.
Table 8 shows the comparison of results and computational time obtained by MVMOS and the other methods. In this case, the results of MVMOS is less than CTPSO, CSPSO, COPSO, CCPSO and KVMO. The computation time of MVMOS is slower than these methods. The computational times for CTPSO, CSPSO, COPSO and CCPSO were from Pentium IV 2.0GHz computer.
Method  Best Total Cost  CPU 

P_{D}= 49342MW  (s)  
CTPSO [29]  1655685  50.1 
CSPSO[29]  1655685  9.6 
COPSO[29]  1655685  76.9 
CCPSO[29]  1655685  42.9 
KMVO [30]  1577607   
MVMO^{S}  1557461  105.39 
Table 8: Comparison of results and CPU time by MVMOS and other techniques for 140unit system.
Robustness analysis
The convergence of heuristic methods may not obtain exactly same solution because these methods initialize variables randomly at each run. Hence, their performances could not be judged by the results of a single run. Many trials should be carry out to reach a impartial conclusion about the performance of the algorithm. Therefore, in this study, 50 independent trials were carried out. The mean cost, max cost, average cost and standard deviation obtained by the proposed method to evaluate the robustness characteristic of the proposed method for ED problem. The robustness analysis of four cases test are presented in (Tables 9 and 10).
Case 1: 3 unit  Case2: 13 unit  

450(MW)  850(MW)  1800(MW)  2520(MW)  
Min Cost ($/h)  4652.4735  8194.3561  117932.4741  24050.1408 
Average Cost ($/h)  4652.4735  8194.3561  17932.4741  24050.277 
Max Cost ($/h)  4652.4735  8194.3561  17932.4741  24050.189 
Standard deviation ($/h)  0  0  0  0.0309 
Table 9: Robustness analysis of the proposed MVMOS by 50 independent trials for Case 1 and Case 2.
Case 3: 20 unit  Case 4 : 140 unit  

2500 (MW)  49342(MW)  
Min Cost ($/h)  60152.5509  1557461.803 
Average Cost ($/h)  60153.2215  1557481.743 
Max Cost ($/h)  60152.7388  1557481.743 
Standard deviation($/h)  0.1469  3.0107 
Table 10: Robustness analysis of the proposed MVMOS by 50 independent trials for Case 3 and Case 4.
Tables 9 and 10 clearly show that the performance the proposed MVMOS is very robust.
A solution of optimization techniques needs concern with two elements:
• Computation time: time to get the best solution is the shortest one.
• Quality of solution: the quality of solution need robustness, near or better the global solutions of the other techniques.
In addition, it takes note the largescale system problem. The computation time of technique on the largescale system may be take more time but the quality of solution need optimum.
Based on the numerical results and robustness analysis of the proposed method, it indicates that the MVMOS obtained the global solution with high probability, especially for largescale system due to the global search capability is enhanced. Besides its ability, the proposed MVMOS is also easy to be implemented for ED problem. Unlike other swarmbased optimization techniques, MVMOS does not strictly require many particles to progess. In this study, number of particles is set to 5 for all cases. The MVMOS showed the good performance. However, the computation time is relatively high for largescale system. Similar to original MVMO, the number of iterations in MVMOS is equivalent to the number of offspring fitness evaluations which is in practical applications usually comsume more time than the optimization algorithm itself.
In future, the MVMOS is proposed for solving the nonconvex ED problems with complicated objective function.
In this paper the proposed MVMOS has been tested for the ED problem with quadratic cost function efficiently and effectively.The numerical results show that the MVMOS exhibits a robust performance and also provides the good solutions for all test systems, expecially for lagrescale system. The proposed method has merits as follows: easy implementation, good solutions, robustness of algorithm; applicable to largescale system. Therefore, the proposed MVMOS could be favorable for solving other ED problems.
This research work is financially supported by Graduate Assistant Scheme (GAS) of Universiti Teknologi PETRONAS and with the help of Department of Fundamental and Applied Sciences, Faculty of Science and Information Technology, UTP.
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