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 Mean-Variance Mapping Optimization (MVMOS). Global J Technol Optim 6:184. doi:10.4172/2229-8711.1000184
Copyright: © 2015 Khoa TH, et al. This is an open-access 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 Mean-variance mapping optimization (MVMOS) for solving the economic dispatch. The proposed optimization algorithm is the extension of the original single particle mean-variance mapping optimization (MVMO). The novel feature is the special mapping function applied for the mutation base on the mean and variance of n-best 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 large-scale 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; MVMOS
N Total number of generator
FT Total operationcost
Fi Fuel cost function of generator i
ai, bi, ci Fuel cost coefficients of generator i
Bij,B0i,B00 B-matrix coefficients for transmission power loss
PD Total system load demand
Pi Power output of generatori
Pi,max Maximum power output of generator i
Pi,min Minimum power output of generator i
PL Total transmission loss
itermax Maximum number of iterations
n_var Numberof variable (generators)
n_par Number of particles
mode Variable selection strategy for offspring creation
archive zize n-best individuals to be stored in the table
di Initial smoothing factor
Initial smoothing factor increment
Final smoothing factor increment
Initial shape scaling factor
Final shape scaling factor
Dmin 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 calculus-based techniques), methods based on artificial intelligence and hybrid methods.
For mathematical convenience, the objective cost function of ED problem is the quadratic function approximations , was solved by methods based on mathematical programming such as lambda iteration method, Newton’s method, gradient search, dynamic programming , linear programming , non-linear programming  and quadratic programming . 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) , Differential Evolution (DE) , Genetic Algorithm (GA) , Ant Colony Optimization (ACO) , Particle Swarm Optimization (PSO) [13,14] Bacterial Foraging (BF) , and Artificial Bee Colony (ABC) algorithm . 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 (EP-SQP) , combining of chaotic differential evolution and quadratic programming (DEC-SQP)  hybrid technique integrating the uniform design with the genetic algorithm (UHGA) , self-tuning hybrid differential evolution (selftuning HDE) , and fuzzy adaptive particle swarm optimization algorithm with Nelder–Mead simplex search (FAPSO-NM) . 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 meta-heuristic which can deal with multiform optimization problems . Recently, Prof. István Erlich has been conceived and developed a novel optimization technique which is named Mean-variance mapping optimization (MVMO) . This algorithm is so-called “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) , 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 MVMO-MVMOS 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 Fi , togenerates a power out Pi. The total fuel costof the system,FT,is sum of fuel cost of each unit.
The optimization problem of the ED is to minimize the total fuel cost FT which be written as:
Generally, the fuel cost curve of a thermal generating unit is presented as quadratic function as:
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:
The power loss PLis calculated by the below formulation :
Generator capacity limits
The active power output of generating units must be within the allowed limits:
Review of MVMO
Mean-variance mapping optimization (MVMO) is a novel optimization algorithm falls into the category of the so-called “population-based 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 mean-variance of the solutions stored in the archive.
The mean and variance vi are calculated as follows:
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 si1 and si2 as follows:
The scaling factor fs is a MVMO parameter which allows for controlling the search process during iteration. si 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 single-agent search algorithm because it uses a single parent-offspring in each iteration. Therefore, the number of fitness evaluations is identical to the number of iterations.
Extension of MVMO-MVMOS
Recently,theswarm version of MVMO has been developed. This version is abbreviated as MVMOS. 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 MVMO-algorithm extends two important parameters. These two parameters are used for calculation and assignment of si1 and si2 as follows:
Variable FS factor: In (10), the factor fs allows the modification of the shape factor calculated from the variance.
The extension of fS 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:
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 MVMOS algorithm uses the factor Δd as presented below:
if si> 0 then
= (1+)+ 2 .+ (rand() – 0.5)
if rand() ≥ 0.5 then
si1=si ; si2=di
si1=si ; si2=di
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 1-5) are set for all variables. At every iteration, if si>di , di will be multiplied by Δd leads to increased di. In case si<di, the current diis divided byΔd which is always greater than 1.0 and thus resulting in reduced value of di. Therefore, di 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.
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 MVMOS 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 xlbest,i to the global best xgbest is calculated by:
where, n denotes the number of optimization variables.
The i-th particle is discarded from the optimization process if the distance Di is less than a certain user defined threshold Dmin.
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*np+ np) 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 xgbest, instead of xlbest,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 MVMOS 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 N-1 generating units in the system. The method for calculation of power output for the slack unit is given in . The power output of the slack unit is as follows:
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
Substituting (17) into (13), a quadratic equation is abtained as follows:
where A, B and C are given by:
The power output of the slack generator is the posstive root of (18) between the two ones abtained as follows:
Based on the slack variable method, the fitness function for the proposed MVMOS will include the objective function (2) and penalty terms for the slack unit if inequality (6) is violated. The fitness function is as follows:
Implemention of MVMOS to ED: The steps of procedure of MVMOS for the ED problem are described as follows:
Step 1: Setting the parameters for MVMOS including itermax, n_var, n_par, mode, di, , archive zize, , n_randomly, n_randomly_min, indep.runs(m), Dmin
Set i=1, i donates the function evaluation
Step 2: Normalize initial variables to the range [0,1] (i.e. swarm of particles).
Step 3: Set k=1, kdonate particle counters.
Step 4: Using de-normalized variables to evaluate fitness function, store fbestand xbestin 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 i-th particle is discarded from the optimization process if the distance Di is less than a certain user defined threshold Dmin.
Step 7: Create offspring generation through three evolutionary operators: selection, mutation and crossover.
Step 8: if k<np ,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.
De-nomalization 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. De-nomalization of optimization variables is carried by using (24):
Pi=Pi,min + Scaling.x_normalized(ι,:)
Termination criteria:The algorithm of the proposed MVMOS is terminated when the maximum number of iterations itermax 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 large-scale systemwith 140 units are carried out. For each case, the algorithm of MVMOS is run 50 independent trialsona Intel Core i5-3470 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 . Afer all simulations, stragy 3 (mode=4) is suporior to the other stragy.
, The range ofin (14) is [0.01-0.4]. 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 . 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.
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 . 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 Pi (MW)|
|PD= 450MW PD= 850MW|
|Min Cost ($/h)||4652.4735||8194.3561|
|Average CPU time (s)||0.87||0.88|
Table 1: Power output of 3-unit system for load demand of 450 MW and 850 MW by MVMOS.
The parameters for MVMOS are set as follows: itermax=1000, n_var(generators)=3, np=5, archive size=4, indep.runs (m)=100, n_randomly=2, n_randomly_min=2, , , ,, ,Dmin=0
The total cost comparison between MVMOS and the other methods are presented in Table 2. In case ofthe 450MW load demand, the results and computational time of MVMOS are less than PSO and ABC. In case ofthe 850MW load demand, the results ofMVMOS is less than IEP, HS, GA, BGA, and the same as NM, PSO. The proposed MVMOS is faster than HS, GA and BGA. There is no computer processor reported for PSO, ABC, HS, GA and BGA and no computational time for the other methods. Table 1 shows that the power output obtained by the MVMOS is always satisfy the constraints.
|Method||450 (MW)||850 (MW)|
|Cost ($/h)||CPU (s)||Cost ($/h)||CPU (s)|
Table 2: Comparison of results and CPU time by MVMO and other techniques for 3-unit system.
Case 2: 13 unit - system: The data of 13 generating unit test system is from . In this case, the power transmission loss is neglected. The obtained results by the MVMOS corresponding to the twoload demand of 1800MW and 2520MW are shown in Table 3.
|Unit||Power outputs Pi (MW)
PD= 1800MW PD= 2520MW
|Total power (MW)||1800.0000||2520.0000|
|Min Cost ($/h)||17932.4741||24050.1408|
|Average CPU time (s)||2.97||16.29|
Table 3: Power output of each generating unit in 13-unit 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 than-Iteration, GA and SQP and same result as ALHN. The computational time of MVMOS is slower than these methods. The computational times for-Iteration, 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)|
Table 4: Comparison of results and CPU time by MVMO and other techniques for 13-unit system.
Case 3: 20 unit-system: The test system includes 20 generators with the system load demand of 2500MW. The data of this system is from . The power transmission loss is ignored in this case. The obtained results by the MVMOS is shown in Table 5.
|Unit||Power outputs Pi (MW)||Unit||Power outputs Pi (MW)|
|Total power (MW)||2500.0000|
|Min Cost ($/h)||60152.5509|
|Average CPU time (s)||31.45|
Table 5: Power output of 20-unit system for load demand of 2500MW by MVMOS
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 less-Iteration, 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 for-Iteration, 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 13-unit test system.
|Method||Total CostPD= 2500MW||CPU(s)|
Table 6: Comparison of results and CPU time by MVMO and other techniques for 20-unit system.
Case 4: large-scale 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 . 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)|
Table 7: Power output of 140-unit 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 2.0-GHz computer.
|Method||Best Total Cost||CPU|
Table 8: Comparison of results and CPU time by MVMOS and other techniques for 140-unit system.
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|
|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 ($/h)||0||0||0||0.0309|
Table 9: Robustness analysis of the proposed MVMOS by 50 independent trials for Case 1 and Case 2.
|Case 3: 20 unit||Case 4 : 140 unit|
|Min Cost ($/h)||60152.5509||1557461.803|
|Average Cost ($/h)||60153.2215||1557481.743|
|Max Cost ($/h)||60152.7388||1557481.743|
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 large-scale system problem. The computation time of technique on the large-scale 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 large-scale 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 swarm-based 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 large-scale 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 non-convex 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 lagre-scale system. The proposed method has merits as follows: easy implementation, good solutions, robustness of algorithm; applicable to large-scale 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.