Moth Swarm Algorithm for Solving Combined Economic and Emission Dispatch Problem

In this paper, the Moth Swarm Algorithm (MSA) is applied to the Combined Economic and Emission Dispatch (CEED) problem in thermal power plants. The analysis of behavior and the evaluation of performances of the algorithm are carried out on the standard test systems with 3 and 6 generators. The results of the MSA application to these test systems are compared with the results published in recent literature. The present paper shows that the proposed MSA gives an accurate and effective solution of the CEED problem. DOI: http://dx.doi.org/10.5755/j01.eie.23.5.19267


I. INTRODUCTION
Minimization of the fuel cost and toxic gases emission (SO2, CO2, NOx) in thermal power plants represents a key task in the planning and operation of a power system.This problem is solved in the framework of the Economic Load Dispatch (ELD) and Economic Emission Dispatch (EED) problems.The ELD and EED problems are defined as the processes in which the optimal distribution of generators' output power is determined to deliver the required energy with minimal fuel cost or with minimal emission under specified constraints for power system operating conditions.
The fuel cost can be specified as a quadratic function without or with a sinusoidal term modelling the turbine valve-point loading effect in thermal power plant.The functions describing the dependence of gas emission on generator output power have complex shapes as well.Reducing operating costs (including the fuel cost) and gas emission are mutually conflicting goals.Namely, the reduction of emission requires an increase in operating costs and vice versa.Therefore, the problem is solved as the CEED problem with the two following objectives: minimization of fuel cost and minimization of emission, simultaneously or separately.In case of the CEED problem with such complex functions of operating costs and emission, the classical deterministic algorithms do not give results because they usually converge to locally optimal solutions.reported results of solving the CEED problem.

II. MODEL OF THE CEED PROBLEM
Usually, the function of fuel cost for a generator in a thermal power plant is described by a quadratic function of output power Pg   2 , g g g g g g g

F P a b P c P
where g = 1, 2, …,G, Fg ($/h) is the fuel cost of generator g, Pg (MW) is the output power of generator g, ag, bg and cg are the fuel costs coefficients of generator g.The function   g g F P is more complex if one is to take into account the valve-point loading effect in a thermal power plant [13], i.e.
where dg and eg are the valve-point coefficients and min g P is the minimum power of generator g.The emission of a generator in a thermal power plant can be modelled by different functions [3].According to [21] and [31], these functions are quadratic functions or a sum of quadratic and exponential functions, i.e.
where Eg (ton/h) is the emission of generator g, Pg (MW) is the output power of generator g, and αg, βg, ηg, ξg and λg are the emission coefficients of generator g.
The CEED problem is formulated by combining the function (1) or (2) with the function (3) using the weighted sum method.Then, the solution of the CEED problem is obtained by minimizing the following objective function [31] under the constraints, where γ is the scaling factor, w is the weight factor and G is the total number of generators in the thermal power plant.In (4), the limit of the weight factor w = 1 corresponds only to the minimization of Fg(Pg), while w = 0 corresponds only to minimization of Eg(Pg).The scaling factor γ is introduced in (4) in order to solve the bi-objective CEED problem as a single-objective problem.In accordance with other similar studies, this minimization process uses the constraints on the generation capacity and the power balance of the transmission system.The generator capacity constraint is defined as where PD and Ploss are the total load demand and the power loss in the transmission system, respectively.The power loss Ploss is expressed as a quadratic function of the actual powers of generators.The coefficients of this function Bgj are defined by the B-loss matrix.Then, the power loss is 0 00 , where Bgj, B0g and B00 are the coefficients of the associated B-loss matrices.
During the optimization process, in order to satisfy the constraint (6), one of the generators g is selected to be a dependent generator (i.e. the slack generator).For this generator (e.g.generator G), the value of output power PG is calculated from the following equation In (8) and (10), the variable PG represents the dependent variable.Thereafter, a quadratic penalty term with the penalty factor λp is added to the objective function FE, giving the following extended objective function to be minimized.

III. SUMMARY OF THE MSA
The MSA is developed in 2017 [37] and represents one of the newest population-based algorithms.This paper gives the fundamental principle for understanding the application of the MSA.The MSA was developed based on simulation of moth swarms flying toward the moonlight.
In order to be on a straight-line trajectory, during the night, a swarm of moths uses celestial navigation, i.e. a technique in which the direction of flying lies at a constant angle to the parallel light rays of the moon as a remote light source.In applying this technique, the moths encounter the nearby light sources representing obstructions for them.The position of the moth swarm relative to the moon is taken as an optimal solution of the problem, while the quality of the solution is measured on the basis of the intensity of the moth's luminescence.
Each swarm consists of the three following groups of moths [37]: (i) pathfinders that have ability to select the best position as light sources to guide the swarm, i.e. light its path; (ii) prospectors that tend to wander into a spiral path nearby the light sources, which have been marked by the pathfinders; and (iii) onlookers that drift directly towards the moonlight, which represents the best global solution obtained by the prospectors.
One moth in a group is labelled with mj, while its luminescence intensity is defined by f(mj).In each iteration, the whole group of moths is divided into the three groups.The first group consists of pathfinders that have the highest luminescence intensity (they have the best position in the swarm).The second and third best fitnesses in the swarm, as well as the associated groups, are considered as the positions of the prospectors and onlookers, respectively.The MSA consists of the following phases: 1. Phase of initialization.The initial positions of moths are defined as follows where the problem, q is the population number, mk min is the lower limit and mk max is the upper limit.After initialization, the fitnesses of all the moths are calculated.In addition to this, the moths are classified into the groups based on the calculated fitnesses.
2. Phase of reconnaissance.The precocious convergence (i.e. a stagnation situation) may occur during the optimization process.Pathfinders have the task to prevent this phenomenon by updating its position in interaction with other moths (using lévy-mutation).Lévy-mutation is carried out in accordance with the procedure explained in [37], where the relative dispersion σk i of the pathfinders in the k th dimension and the variation coefficient μ i as a limit of relative dispersion need to be first calculated.Pathfinders that have a low degree of dispersion are inserted in the group cp of crossover points .
For crossover points qc cp, the following vectors are formed: sub-trail vector The sub-trail vector is expressed, using Lévy-mutation [37], as , , , , , where and Lp2 are the variables generated by a heavy tail Lévyflights [37], [38] using the expression where ⊕ is the entrywise multiplications; Levy (α) is the Lévy flight-random walk, in which the step sizes probability distribution is heavy-tailed [38], [39]; α is the stability index of Lévy α-stable distribution.The mutually indexes (r 1 , r 2 , r 3 , r 4 , r 5 , p) are selected from the pathfinder solutions.
The crossover operation is carried out so that the trail solution The selections are carried out according to the following: The fitness values of trail solution and host solution are compared and a better solution is selected to survive for the next generation, as follows The probability value of solutions is estimated as where fitp is the luminescence intensity, which is calculated from the objective function fp for the minimization problems as 1 , 0, 1 1 , 0.
3. Spiral movement of prospectors.The moths that have the lower best luminescence are called the prospectors and their number qf is The position of each prospector mj is updated according to the mathematical expression for the spiral flight path 1 cos 2 , where p , ,....,q ; j q ,q ,...,q ,      mp is chosen on the basis of the probability function Pp (18); θ ∈ onlookers that move according to Gaussian distribution of steps (their number is qG = round (q0/2)); and (ii) the remaining onlooker moths, with number qA = q0 -qG, that move according to the Associative Learning Mechanism (ALIM) with short-term memory [37].The updating equation for the first group of onlookers is created in accordance with the Gaussian walk of random steps , where   where   At the end of each iteration, the fitnesses of the whole swarm become available to redefine the role of each moth for the upcoming iteration.Fig. 1 shows the flowchart of the MSA.

IV. SIMULATION RESULTS
The proposed MSA algorithm is tested on two test systems, one with 6 generators and another with 3.These test systems are often used for solving the CEED problems by different optimization algorithms.For reasons of comparison, along with the MSA, the authors apply the Firefly Algorithm (FFA) [40] and the PSOGSA [41].Moreover, the simulated results are compared to the existing results.The MSA, FFA and PSOGSA algorithms have been implemented in MATLAB 2011b computational environment and run on 2.20 GHz, with 3.0 GB RAM.The parameters used for the simulations are presented in the Table I.The best results of the simulations are obtained after 30 runs.The standard IEEE 30-bus six generator system with total load demand of 283.4 MW, with NOx emission and without the valve point effect is taken as the test system.In order to compare the obtained results to the existing ones, two cases of the test system 1 are considered, one without Ploss (Case I) and another with Ploss (Case II).The error tolerance is δ = 10 -6 MW.The B-loss matrices are given in the Table A-I.The fuel cost coefficients and NOx emission coefficients appearing in ( 2)-( 4) are taken from [31].A scaling factor γNOx of 1000 ($/ton) is applied.Table II shows the best solutions for the power output, fuel cost and emission of the test system 1.Minimization is carried out for the cases: w = 1 (fuel cost minimization), w = 0 (emission minimization) and w = 0.5 (minimization of fuel cost and emission, simultaneously).The test system 2 consists of 3 generators with a load demand of 850 MW, as well as with NOx and SOx emissions.For this system the fuel cost coefficients and NOx and SOx emission coefficients are taken from [13].In this case, the scaling factors appearing in ( 4) are taken from [3] and they are as follows: γNOx = 147582.78814($/ton) -for minimization of the NOx emission and γSOx = 970.031569($/ton) -for minimization of the SOx emission.The test system 2 is considered as a lossless system.Table III shows the minimum, maximum and Standard Deviation values for the cases of applications of the MSA, FFA and PSOGSA to the test system 2.According to Table III, the minimum values of the fuel cost and emission are the same for all the three algorithms.However, the Standard Deviations related to the application of the MSA are by far the lowest of the Standard Deviations obtained for these three applications (i.e. the differences are between 3 and 4 orders of magnitude lower).
In addition, Table IV shows the best solutions for the power output, fuel cost and emission of the test system 2 obtained by means of the MSA for the cases where w = 0, w = 1 and w = 0.5.The results obtained by the MSA and FFA for the test system 1 along with corresponding data from the literature are summarized in the Table V.
As can be seen in Table V, the MSA and FFA provided better values for the minimum fuel cost in regard to the values obtained by the algorithms proposed in [7], [13], [15], [17] and [23], as well as ones that are the same or very close to the results obtained by the algorithms from [18] and [35].The minimum value of NOx emission calculated by the MSA and FFA are the same or better than the associated results reported in [7], [15], [17], [18], [23] and [35].The Table V also reveals that the effect of the power loss Ploss on the valus of the minimum fuel cost and emission is very small or negligible.Figure 2 shows the convergence behaviors of the MSA, FFA and PSOGSA in the cases of minimization of fuel cost and minimization of emission for the test systems 1 and 2. According to Fig. 2, the MSA converges to the minimum value in a number of iterations which is lower than the one for the PSOGSA.Compared to the FFA, the MSA converges in a number of iterations which is lower (Fig. 2(b)) or approximately the same (Fig. 1(a) and Fig. 1(c)).For all three algorithms, ascend speeds are high at the beginning.

V. CONCLUSIONS
The application of the MSA to the CEED problem has been proposed in this paper for the first time.The algorithm has been successfully tested on the two standard IEEE test systems with 3 and 6 generators.The comparative analysis of the results obtained by means of the MSA, FFA and PSOGSA showed that the Standard Deviations of the results are the lowest for the MSA (between 4•10 -12 -for minimization of the fuel cost and 8.73•10 -16 -for minimization of the SOx emission), indicating a larger degree of robustness for the MSA.
Moreover, the MSA generated the minimal values of the fuel cost and NOx emission that are the same as to the corresponding results of the FFA and PSOGSA.In addition, these minimal values are better than the associated ones that are obtained using the recently developed algorithms, thus resulting in the high quality solution.The convergence profiles of the objective functions used in the MSA, FFA and PSOGSA showed that the ascend speeds are high at the beginning for all three algorithms.Compared to the FFA and PSOGSA, the MSA can achieve the optimal solution much faster.Thus, the MSA was demonstrated to have a better convergence property.Finaly, comparisons between the convergence profiles, Standard Deviations and optimal values of the results presented in this paper and in the existing literature also showed the best effectiveness and robustness of the MSA for solving CEED problem.
The MSA is also suitable for solving other complex and non-smooth problems, of course, having in mind the "No Free Lunch" theorem [37] which states that it is not usually possible to find a single algorithm that can solve all optimization problems.Therefore, the MSA, which is currently one of the newest population-based optimization algorithms, should be tested and applied to other scientific and engineering problems.

[r, 1 ]
is a random number and r = −1−i/I.The fact that the new solution for each prospector is better than current pathfinder solutions comes from(21).4.Onlookers movement.The onlooker moths have the lowest luminescence and they fly to the moonlight.The applied search technique for onlooker solutions is more effectively than the previously illustrated search technique for prospector solutions.The number of onlookers is q0 = qqf -qp.They are classified in the two following groups: (i) /G and 2g/G are the cognitive and social factors, respectively; r1 and r2 are random numbers within the interval [0,1]; bestp is the pathfinder solution randomly chosen on the basis of its probability value.

Fig. 2 .
Comparative convergence curves of the MSA, PSOGSA and FFA: (a) in the case of minimization of NOx emission for the test system 1 without Ploss; (b) in the case of minimization of SOx emission for the test system 2; and (c) in the case of minimization of fuel cost for the test system 2.
I. THE B-LOSS MATRICES FOR THE TEST SYSTEM 1.

TABLE II .
THE BEST SOLUTIONS OBTAINED BY MEANS OF THE MSA FOR THE TEST SYSTEM 1.

TABLE III .
MIN, MAX AND SD VALUES OF THE RESULTS OBTAINED BY MEANS OF THE MSA, FFA AND PSOGSA FOR THE TEST SYSTEM 2.

TABLE V .
A COMPARISON OF THE BEST SOLUTIONS FOR THE FUEL COST AND NOX EMISSION OF THE TEST SYSTEM 1.

TABLE A -
II.THE FUEL COST COEFFICIENTS, NOX EMISSION COEFFICIENTS AND GENERATION LIMITS FOR THE TEST SYSTEM 1.

TABLE A -
III. THE FUEL COST COEFFICIENTS, NOX AND SOX EMISSION COEFFICIENTS AND GENERATION LIMITS FOR THE TEST SYSTEM 2.