Spiral Optimization Algorithm for solving Combined Economic and Emission Dispatch

Electrical Power and Energy Systems 62 (2014) 163–174

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Spiral Optimization Algorithm for solving Combined Economic and Emission Dispatch Lahouaria Benasla a,⇑, Abderrahim Belmadani b, Mostefa Rahli a a b

Department of Electrical Engineering, USTO, Oran, Algeria Department of Computer Science, USTO, Oran, Algeria

a r t i c l e

i n f o

Article history: Received 27 April 2013 Received in revised form 28 March 2014 Accepted 21 April 2014

Keywords: Combined Economic and Emission Dispatch Economic dispatch Emission dispatch Valve-point loading Spiral Optimization Algorithm

a b s t r a c t The Spiral Optimization Algorithm (SOA) is an optimization technique developed recently (2011) by K. Tamura and K. Yasuda at Tokyo Metropolitan University-Japan. SOA is a metaheuristic based on an analogy of spiral phenomena in nature and it is simple in concept, few in parameters and easy in implementation. In this paper, SOA is proposed for solving the Combined Economic and Emission Dispatch (CEED) problem. It is aimed, in the CEED problem, that scheduling of generators should operate with both minimum fuel costs and emission levels, simultaneously, while satisfying the load demand and operational constraints. The CEED problem is formulated as a multi-objective problem by considering the fuel cost and emission objectives of generating units. The bi-objective optimization problem is converted into a single objective function using a price penalty factor. The proposed algorithm has been implemented on three test systems with 3, 6, and 40 generating units, with different constraints and various cost curve nature. In order to see the effectiveness of the proposed algorithm, its results are compared to those reported in the recent literature. Those results are quite encouraging showing the good applicability of SOA for CEED problem. Ó 2014 Elsevier Ltd. All rights reserved.

Introduction Economic dispatch problem is one of the most important optimization problems in power system operation and forms the basis of many application programs. The main objective of economic load dispatch of electric power generation is to schedule the committed generating unit outputs to meet the load demand at minimum operating cost while satisfying all unit and system constraints. One of those constraints which are always taken into account is the environmental constraints. That is minimization of pollution emission (NOx, CO2, SOx, toxic metals, etc.) in case of power plants [1,2]. Thus, we are facing with a bi-objective optimization problem to deal with. Traditionally, electric utilities dispatch generation using minimum fuel cost as the main criterion. However the best economic dispatch does not lead to minimum emission and vice versa. The goal of emission dispatch is to determine the generation schedule that has the minimum emission cost. The two criteria are contradictory to each other and are in trade-off relationship. It therefore ⇑ Corresponding author. Tel.: +213 774753515. E-mail addresses: [email protected] (L. Benasla), abderrahim.belmadani@gmail. com (A. Belmadani), [email protected] (M. Rahli). http://dx.doi.org/10.1016/j.ijepes.2014.04.037 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

makes it difficult to handle such problem by conventional approaches that optimize a single objective function. One feasible approach to solve this kind of problem using conventional optimization method is to convert the bi-objective into a single objective function by giving relative weighting values. In this case the emission dispatch is added as a second objective to the economic dispatch problem which leads to Combined Economic Emission Dispatch (CEED) [3]. Environmental issues add complexity to the solution of the economic dispatch problem due to the nonlinear characteristics of the mathematical models used to represent emissions. In addition, the Economic Emission Dispatch (EED) problem can be complicated even further if nonsmooth and nonconvex fuel cost functions are used to model generators, such as valve-point loading effects. All these considerations make the EED problem a highly nonlinear and a multimodal optimization problem [4]. Several Economic Emission Dispatch (EED) strategies have appeared in the literature over the years. Lagrange relaxation method, weighted sum method, e-constrained algorithm, Linear programming method, Goal programming technique are used to solve the EED problem [5]. But unfortunately these methods are not able to find a solution with a significant computational time for medium or large-scale ED.

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Recently, with the development of computer science and technology, evolutionary algorithms like Genetic Algorithm (GA) [6], Particle Swarm Optimization (PSO) [7], Ant Colony Optimization (ACO) [8], Firefly Optimization Technique [9] and Harmony Search (HS) [10] are used to eliminate many difficulties in the classical methods and to solve non-linear CEED problem without modifying the shape of its fuel and emission cost curves. Those evolutionary algorithms are general-purpose stochastic search methods simulating natural selection and biological evolution. They differ from other optimization methods in the fact maintaining a population of potential solutions to a problem, and not just one solution. Generally, these algorithms work as follows: a population of individuals is randomly initialized where each individual represents a potential solution to the problem. The quality of each solution is evaluated using a fitness function. A selection process is applied during each iteration in order to form a new solution population. This procedure is repeated until convergence is reached. The best solution found is expected to be a near-optimum solution [11]. Two dimensional SOA that was recently proposed by Tamura and Yasuda [12] is a multipoint metaheuristics search method for two-dimensional continuous optimization problems based on the analogy of spiral phenomena in nature. Focused spiral phenomena are approximated to logarithmic spirals which often appear in nature and the universe, such as tropical cyclones, tornadoes and galaxies (Fig. 1). Then Tamura and Yasuda [13] proposed n-dimensional SOA using a design philosophy of 2-dimensional optimization. The SOA has several advantages including its few control variables, local searching capability, fast results, easy using process, simple structure and introduction of both phases of diversification and intensification in the same process. In this paper, n-dimensional SOA is applied to solve CEED problem which it is converted into mono-objective optimization problem by introducing price penalty factor. In order to investigate the effectiveness of the n-dimensional SOA, the algorithm is simulated for the systems with 3, 6, and 40 generating units, with different constraints and various cost curve nature. Numerical results obtained by the proposed algorithm were compared with other optimization results reported in the recent literature. CEED formulation

These objectives and constraints, and the formulation of the CEED problem are expressed as follows: Classical economic dispatch problem In power stations, every generator has its input/output curve. It has the fuel input as a function of the power output. But if the ordinates are multiplied by the cost of $/Btu, the result gives the fuel cost per hour as a function of power output [15]. The fuel cost of generator i may be represented as a polynomial function of real power generation:

F i ðPGi Þ ¼ ai P2Gi þ bi P Gi þ ci ð=hÞ i ¼ 1; 2; . . . ; ng

ð1Þ

where PGi is real power output, nG is the number of generators including the slack bus, ai, bi and ci are the cost coefficients of the ith unit. The Economic Dispatch Problem can be mathematically represented as:

(

0

ng X Min F ¼ F i ðP Gi Þ

) ð2Þ

i¼1

F ($/h) is the total fuel cost function for the entire power system. It is written as the sum of the fuel cost model for each generator. Economic dispatch problem with valve point effect Large steam turbine generators will have a number of steam admission valves that are opened in sequence to control the power output of the unit. As the unit loading increases the input to the unit increases and the incremental heat rate decreases between the opening points for any two valves. However, when a valve is first opened, the throttling losses increase rapidly and the incremental heat rate rises suddenly. This is called valve-point effect that leads to non-smooth, non-convex input–output characteristics as shown in Fig. 2 [16]. Usually, valve-point effect is modeled by adding a recurring rectified sinusoid to the basic quadratic cost curve [4,16]. Therefore, (1) can be modified as:

F i ðPGi Þ ¼ ai P2Gi þ bi P Gi þ ci þ jdi sin ðei ðPGi min  PGi ÞÞjð=hÞ

In the solution of the CEED problem, the point at issue is to minimize both fuel cost and emission, simultaneously, while satisfying equality and inequality constraints. Cost and emission functions, which are independent of each other, make the CEED problem bi-objective. Bi-objective problem solving can be done by two objective functions turned into a single objective function [14]. In this paper, this operation is enhanced using a price penalty factor and the CEED problem is converted into a single-objective function.

i ¼ 1; 2; . . . ; ng

ð3Þ

where di and ei are the coefficients of generator i reflecting valvepoint effects and PGimin is the minimum generation limit of unit i. Emission dispatch The emission function can be expressed as the sum of all types of emission considered, such as NOx, SO2, CO2, particles and

Fig. 1. Natural spiral phenomena.

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PL ¼

ng X ng ng X X PGi Bij PGj þ Boi PGi þ Boo i¼1 j¼1

ð9Þ

i¼1

where Bij, Boi and Boo are the transmission loss coefficients. Inequality constraints According to those constraints, the power output of each generator is restricted by minimum PGimin and maximum PGimax power limits.

PGi min  PGi  PGi max

i ¼ 1; . . . ; ng

ð10Þ

Combined Economic and Emission Dispatch The objective function of CEED problem can be formulated as:

Min ½F; E Fig. 2. Input–output curve with and without valve-point loading.

thermal emissions, with suitable pricing of weighting on each pollutant emitted [17]. Different mathematical models were proposed to represent the emission function of thermal generating units [18]. In this study, the following emission function will be considered to model the total emissions of all generating units [19–21]:

Ei ðPGi Þ ¼ 102  ðai P2Gi þ bi PGi þ di ÞðTon=hÞ

ð4Þ

Emissions from pollutants can be also summed up as a single equation in quadratic and exponential for ith generator [22,23].

Ei ðPGi Þ ¼ 102  ðai P2Gi þ bi PGi þ di Þ þ gi exp ðci PGi ÞðTon=hÞ

ð5Þ

ai, bi, di, gi and ci are the emission coefficients of the ith unit. The Emission Dispatch Problem can be mathematically represented as:

(

) ng X Min E ¼ Ei ðPGi Þ

During the minimization process, some equality and inequality constraints must be satisfied. In this process, an equality constraint is called a power balance and an inequality constraint is called a generation capacity constraint. Equality constraints The total power generation must cover the total demand Pd and the total power transmission losses in the network PL. Hence,

ð7Þ

i¼1

To determine economic scheduling of generating plants, the effect of transmission losses must be included. The total system losses are represented by loss coefficients (Bij), normally referred to as B-coefficients. The B-coefficients approximate the system losses as a quadratic function of the generator real powers. The simplest form of loss equation is George’s formula [24], which is:

PL ¼

ng X ng X PGi Bij PGj

Min½C T ¼ h  F þ ð1  hÞ  Pf  E

ð8Þ

i¼1 j¼1

The most popular approach for finding an approximate value of the losses is by Kron’s loss formula as given in (9), which represents the losses as a function of the output level of the system generators [25].

ð12Þ

where CT is the total operating cost in $/h, h is the weighting factor that can be varied between 0 and 1 and Pf is the price penalty factor ($/kg or $/ton) which blends the emission cost with the normal fuel costs. The values of h indicate the relative significance between the two objectives. By varying the value of h, the trade-off between the fuel cost and the environmental degradation cost can be determined over the range of h. If h = 1.0, the solution is that of minimum cost, and if h = 0.0 the solution is minimum emissions. The price penalty factor is the ratio between the maximum fuel cost and maximum emission of corresponding generator [21,26]:

ð6Þ

Constraints

ng X PGi  Pd  PL ¼ 0

The bi-objective Combined Economic Emission Dispatch CEED problem is converted into single optimization problem as the following equation [4,23]:

Pfi ¼

i¼1

ð11Þ

FðPGi max Þ =ton i ¼ 1; . . . ; ng EðPGi maxÞ

ð13Þ

The steps to determine the price penalty factor for a particular load demand are [26,27]: 1. Find the ratio between maximum fuel cost and maximum emission of each generator. 2. Arrange the values of price penalty factor in ascending order. 3. Add the maximum capacity of each unit one at a time, starting P from the smallest Pfi until i PGimax P Pd. 4. In this stage, Pfi associated with the last unit in the process is the price penalty factor of the given load. Spiral optimization algorithm The logarithmic spiral discrete model which satisfies the requirement that the center can be located in an arbitrary point is formulated as follows:

xðkþ1Þ ¼ rMðhÞ  xðkÞ  ðr  MðhÞ  In Þ  x

ð14Þ

Tamura and Yasuda [12,13] adopt not a one-point search model but a multipoint search model because of the following reason. In case of the one-point search model, the search based on (14) with such x⁄ does not work completely because the initial point becomes the best solution and the center x⁄ when evaluating the initial point. From the above considerations, Tamura and Yasuda propose the Spiral Optimization that is a multipoint search based on: ðkþ1Þ

xi

ðkÞ

¼ rMðhÞ  xi  ðr  MðhÞ  In Þ  x ; ⁄

i ¼ 1; 2; . . . ; m

ð15Þ

With the common center x as a best solution obtained in the search process. Namely x⁄ becomes an interaction.

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Adopting the multipoint and adding the interaction have the following meanings:  Interaction: contribution to realize the intensification that should be done around a good solution.  Multipoint: contribution to enhance both intensification and diversification of the spiral model.

Tamura and Yasuda [13] propose the using of composition rotation matrix M(n)(h) which consists of rotation matrices Eq. (16) based on all combination (n(n  1)/2 combinations) of 2 axes. M(n)(h) is defined as follows:

MðnÞ ðhÞ ¼

Y

 ðnÞ  M i;j hi;j

ð17Þ

i j

Algorithm of n-dimensional Spiral Optimization [13]: 0 6 h < 2p is the rotation angle around the origin at each k and 0 < r < 1 is a convergence rate of distance between a point and the origin at each k. M(h) is the rotation matrix. The rotation matrix for two-dimensional Spiral Optimization is defined as follows:

ð2Þ

M 1;2 ðhÞ ¼



cosðhÞ  sinðhÞ sinðhÞ



cosðhÞ

Fig. 3 for example, show trajectories of 40 points when simulating Eq. (14) until 50 steps with the following parameters: r ¼ 0:95; h ¼ p4 ; xð0Þ ¼ ð10; 10Þ and x ¼ ð4; 4Þ. This figure shows that the behavior of spiral model Eq. (14) has diversification in the early phase and intensification at the center in the late phase. Each 2-dimentional rotation matrix in n-dimensional space is defined as follows:

ð16Þ Whose blank elements mean 0. From this definition, many rotation matrices exist according to the way selecting 2 axes which consist of each rotation plane through their permutations or combinations.

Step 0: Preparation: Select the number of search points m P 2, the parameters 0 6 h < 2p, 0 < r < 1 of rM(n)(h) and the maximum iteration number: kmax. Set k = 0. Step1: Initialization: ð0Þ Set initials points xi 2 IRn ; i ¼ 1; 2; . . . ; m in the feasible ð0Þ ð0Þ region at random and center x* as x ¼ xig ; ig ¼ arg mini f ðxi Þ; i ¼ 1; 2; . . . ; m. ðkÞ ðkþ1Þ ðkÞ Step 2: Updating xi : xi ¼ rMðnÞ ðhÞ  xi  ðr  MðnÞ ðhÞ  In Þ  x . ðkþ1Þ ðkþ1Þ *  Step 3: Updating x: x ¼ xig ; ig ¼ arg mini f ðxi Þ; i ¼ 1; 2; . . . ; m. Step 4: if k = kmax then terminate. Otherwise, set k = k + 1 and return to step 2. Numerical examples and simulation results In order to evaluate the performance of the proposed algorithm, three test systems are considered. The first case study compromises a three-unit test system with a simple smooth fuel cost quadratic functions. The second test case considers a six-unit system. The third case study consists of a large test system with 40 thermal units presenting valve-point effects. These test systems are widely used as benchmarks in the power system field for solving the CEED problem and have been used by many other research groups around the world for similar purposes. In each case study, the problem is solved as pure economic dispatch problem and also as pure emission dispatch problem by considering weight factor h = 1 and h = 0, respectively. Simulations were carried out using DELPHI computational environment, on an Acer Intel Core 2 i5-3210M (2.5 GHz) with 4 GB total memory. For all simulations cases, the Spiral Optimization Algorithm is allowed to run 100 searches (Kmax = 100) for a population of 20 search points (m = 20). Test system 1 A system with three units considering NOx and SOx emission is used to examine the applicability of the proposed algorithm with a total demand of 850 MW. The operating limits, fuel cost coefficients and emission coefficients for this system aretaken from [28] and illustrated in Table 1. The expression for transmission line losses is given by:

PL ¼

Fig. 3. Illustration of equation 26.

0:00003P2G1 þ 0:00009P2G2 þ 0:00012P2G3

ðMWÞ:

After careful testing, the following SOA parameters have been used: Rotation angle around the origin h = p/2, spiral radius r = 0.99. For this system, the fuel cost given by (1) and the NOx and SOx emission given by (4) are individually, optimized and these individual results are reported in Tables 2–4 respectively. In order to demonstrate the performance of the SOA, the results obtained by the proposed algorithm are compared to those obtained using Tabu Search approach (T-S) [29], Differential Evolution (DE) combined with Biogeography-based Optimization (BBO) (DE/BBO) [29], Biogeography-based Optimization (BBO) [30], Non-dominated Sorting Genetic Algorithm-II (NSGA-II) [28] and Multi-Objective Evolutionary Programming (MOEP) [20]. From the comparison, it

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L. Benasla et al. / Electrical Power and Energy Systems 62 (2014) 163–174 Table 1 Generation limits, fuel cost and emission coefficients of three-generator system. Unit

PGimin

PGimax

ai

bi

ci

aiNOx

biNOx

ciNOx

aiSOx

biSOx

ciSOx

1 2 3

150 100 50

600 400 200

0.001562 0.00194 0.00482

7.92 7.85 7.97

561 310 78

1.4721848e-7 3.0207577e-7 1.9338531e-7

9.4868099e-5 9.7252878e-5 3.5373734e-5

0.04373254 0.055821713 0.027731524

1.6103e-6 2.1999e-6 5.4658e-6

0.00816466 0.00891174 0.00903782

0.5783298 0.3515338 0.0884504

Table 2 Comparison of best solutions for fuel cost minimization offered by the different algorithms for the three-unit test system. Method

SOA

T-S [29]

DE/BBO [29]

BBO [30]

NSGA-II [28]

MOEP [20]

PG1 (MW) PG2 (MW) PG3 (MW) Total generation (MW) Losses (MW) Fuel cost ($/h) NOx emission (ton/h) SOx emission (ton/h) Total CPU time (s)

445.404 289.445 130.427 865.27 15.277 8342.658 0.09816 9.00988 0.016

435.69 298.828 131.28 865.798 15.798 8344.598 0.09863 9.02146 NR

435.1978 299.9696 130.6604 865.8289 15.8289 8344.58319 0.098686 9.02194 NR

435.1966 299.9723 130.6600 865.8289 15.8289 8344.5927 0.098686 9.02195 NR

435.885 299.989 129.951 865.825 15.826 8344.598 0.09860 9.02129 NR

437.87 299.45 128.49 865.81 15.81 8344.69 0.0987 9.01951 NR

Bold means optimum values of the objective function for different optimization methods.

Table 3 Comparison of best solutions for NOx emission minimization offered by the different algorithms for the three-unit test system. Method

SOA

T-S [29]

DE/BBO [29]

BBO [30]

NSGA-II [28]

MOEP [20]

PG1 (MW) PG2 (MW) PG3 (MW) Total generation (MW) Losses (MW) Fuel cost ($/h) NOx emission (ton/h) SOx emission (ton/h) Total CPU time (s)

511.223 246.439 105.751 863.413 13.414 8355.227 0.09586 8.95982 0.016

502.914 254.294 108.592 865.8 15.8 8371.143 0.0958 8.9860 NR

508.5813 250.4433 105.7212 864.7459 14.7459 8365.11464 0.095923 8.973667 NR

508.5813 250.4433 105.7212 864.7459 14.7459 8365.11464 0.095923 8.973667 NR

505.810 252.951 106.023 864.784 14.784 8363.627 0.09593 8.97472 NR

507.26 253.13 104.41 864.80 14.8 8364.7 0.09593 8.97428 NR

Bold means optimum values of the objective function for different optimization methods. Table 4 Comparison of best solutions for SOx emission minimization offered by the different algorithms for the three-unit test system. Method

SOA

T-S [14]

DE/BBO [29]

BBO [30]

NSGA-II [28]

PG1 (MW) PG2 (MW) PG3 (MW) Total generation (MW) Losses (MW) Fuel cost ($/h) NOx emission (ton/h) SOx emission (ton/h) Total CPU time (s)

599.251 177.771 83.629 860.651 10.651 8413.031 0.09950 8.9371 0.016

549.247 234.582 81.893 865.722 15.722 8403.485 0.0974 8.974 NR

552.11106 219.44402 92.95969 864.51579 14.51579 8396.4573 0.096817 8.965927 NR

552.1111 219.4441 92.96053 864.5158 14.5158 8396.4665 0.096817 8.865937 NR

541.308 223.249 99.919 864.476 14.476 8387.518 0.09638 8.96655 NR

NR means not reported in the referred literature. Bold means optimum values of the objective function for different optimization methods.

is noticed that the proposed algorithm gives reduction in fuel cost (Table 2), in NOx emission (Table 3) and in SOx emission (Table 4) as compared to the reported ones. We can say that the proposed method is performing well in the solution of best fuel cost and the best NOx and SOx emission regarding the remarkable difference between the results of SOA, T-S, DE/BBO, BBO, NSGA-II and MOEP. From the results, it is inferred that, the fuel cost and emission are conflicting objectives. Emission has maximum value when cost is minimized. Convergence characteristics of the 3 generators system for minimum fuel cost, minimum NOx and minimum SOx emission in case of SOA are shown in Figs. 4–6, respectively. These graphs clearly indicate that Spiral Optimization converges rapidly to the optimal solution. Table 5 shows the best compromise solution between fuel cost, NOx emission and SOx emission for the 3 generator system obtained from proposed SOA, DE/BBO [29], BBO [30] and NSGA-II

[28]. Here, at the time of comparing best compromising solutions, emission level is converted into equivalent cost after multiplying emission level with corresponding price penalty factor. After adding fuel cost with emission cost overall cost is calculated. Results show that best compromise costs obtained by SOA is least compared with other methods mentioned. This emphasizes ability of SOA to give better solution quality. Convergence characteristics of the 3 generators system for total cost in case of SOA are shown in Fig. 7. Test system 2 The standard IEEE 30-bus six-generator system is considered as test system 2. This power system is interconnected by 41 transmission lines and the total system demand for the 21 load buses is 283.40 MW. The fuel and the emission coefficients including the limits of generation for the generators are illustrated in Table 6 [23].

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Fig. 4. Convergence characteristic for fuel cost minimization (3-generator system).

Fig. 5. Convergence characteristic for NOx emission minimization (3-generator system).

For the purpose of comparison with the reported results, the system is considered as lossless. Therefore, the problem constraints are the power balance constraint without PL and the generation capacity constraints only. For this test system, the fuel cost given by (2) and the emission given by (5) are, individually, optimized. To verify the parameter r effects on the optimization process, best fuel and best emission are solved under h = p/2 and r = 0.9, 0.95 and 0.99. The rotation matrix for h = p/2 is:

2

M ð6Þ

0

0

60 0 6 6 60 0 ¼6 60 0 2 6 6 4 0 1 1 0

p

0

0

0

0

0

1

0 1 0 1

0

0

0 0

0 0

0 0

1

3

0 7 7 7 0 7 7 0 7 7 7 0 5 0

The results obtained by adopting the SOA are presented in Table 7. The proposed SOA-based convergence profiles of the best solution for the fuel cost and emission objectives, as presented in Figs. 8 and 9, show better convergence characteristics yielded by the proposed algorithm. Convergence of fuel cost and NOx emission objective functions are shown in Figs. 7 and 8 respectively. To verify the rotation angle h properties, best fuel, best emission and CEED are solved under r = 0.99 and h = p/2 and p/4. The rotation matrix for h = p/4 is:

2

0:176777 0:176777 0:250000 0:353553 6 0:323223 0:676777 0:353553 0:250000 6 p 6 6 0:191942 0:588388 0:551777 0:353553 M ð6Þ ¼6 6 0:213388 4 0:066942 0:676777 0:551777 6 6 4 0:338388 0:213388 0:066942 0:588388 0:816942

0:338388

0:213388

0:191942

Table 8 presents the most appropriate results of the fuel cost and emission functions when optimized individually using SOA. The convergence profiles of the best solution for the fuel cost and the emission objectives are shown in Figs. 10 and 11, respectively.

Fig. 6. Convergence characteristic for SOx emission minimization (3-generator system).

The results for the best compromising solution with SOA for different values of r and h are tabulated in Table 9. Convergence curves of SOA for this case study are shown in Fig. 12. It should be noted that the total cost of the compromise solution was determined on the basis of the price penalty factor Pf of

0:500000 0:000000 0:250000 0:353553 0:676777 0:323223

0:707107

3

0:500000 7 7 7 0:353553 7 7 0:250000 7 7 7 0:176777 5 0:176777

6399.93 $/ton determined by applying the procedure described in Section ‘Combined Economic and Emission Dispatch’. The best compromising cost (or total cost) achieved by SOA reflects a value of 624.604 $/h for fuel cost and 0.18708 ton/h for emission level. The average CPU time for this test case is 0.015 s.

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L. Benasla et al. / Electrical Power and Energy Systems 62 (2014) 163–174 Table 5 Best compromise solution of fuel cost and SOx, NOx emission for 3 generator system (Pd = 850 MW). Power output

SOA

DE/BBO [29]

BBO [30]

NSGA-II [28]

PG1 (MW) PG2 (MW) PG3 (MW) Total generation (MW) Losses (MW) Fuel cost ($/h) NOx Emission (ton/h) NOx price penalty factor ($/ton) Equivalent cost of NOx emission ($/h) SOx Emission (ton/h) SOx price penalty factor ($/t) Equivalent cost of SOx emission ($/h) Total cost ($/h) Total CPU time (s)

513.82835 238.96754 110.40421 863.200 13.32001 8356.27194 0.09594 147,582.78814 14,158.52277 8.95838 970.031569 8689.90946 31204.70417 0.093

507.11941 251.64185 106.00028 864.76255 14.76255 8364.3019 0.095924 147,582.78814 14,156.84686 8.974190 970.031570 8705.24820 31,226.39702 NR

507.11954 251.64262 106.00042 864.76258 14.76258 8364.31126 0.0959248 147,582.78814 14,156.84943 8.974201 970.031570 8705.25828 31,226.41898 NR

496.328 260.426 108.144 864.898 14.898 8358.896 0.09599 147,582.78814 14,166.47183 8.97870 970.03157 8709.62245 31,234.99029 NR

NR means not reported in the referred literature. Bold means optimum values of the objective function for different optimization methods.

Fig. 7. Convergence characteristic for total cost minimization (3-generator system).

Fig. 8. Convergence characteristic for fuel cost minimization (6-generator system).

Table 6 Generation limits, fuel cost and emission coefficients of six-generator system. Unit

ai ($/MW2 h)

bi ($/MW h)

ci ($/h)

ai (ton/MW2 h)

bi (ton/MW h)

di (ton/h)

gi (ton/h)

ci (1/MW)

1 2 3 4 5 6

100 120 40 60 40 100

200 150 180 100 180 150

10 10 20 10 20 10

6.490 5.638 4.586 3.380 4.586 5.151

5.554 6.047 5.094 3.550 5.094 5.555

4.091 2.543 4.258 5.326 4.258 6.131

2.0e-4 5.0e-4 1.0e-6 2.0e-3 1.0e-6 1.0ev5

2.857 3.333 8.000 2.000 8.000 6.667

Table 7 Best solutions for fuel cost and emission minimization offered by the SOA for the IEEE 30-bus system for different values of r. Case

Best fuel cost

r

0.9

0.95

0.99

Best NOx emission 0.9

0.95

0.99

PG1 (pu) PG2 (pu) PG3 (pu) PG4 (pu) PG5 (pu) PG6 (pu) Total generation (pu) Fuel cost ($/h)

0.05051 0.23738 0.51928 0.90113 0.81066 0.31503 2.834 605.204

0.06267 0.33947 0.51656 1.10993 0.51291 0.29245 2.834 601.509

0.05169 0.31359 0.46012 1.04331 0.61133 0.35397 2.834

0.47812 0.59328 0.34949 0.46345 0.56023 0.38943 2.834 643.715

0.47452 0.52948 0.53952 0.43685 0.42591 0.42772 2.834 640.749

Emission (ton/h)

0.21010

0.21120

600.986 0.20877

0.32151 0.29858 0.54351 0.70120 0.39608 0.57313 2.834 615.777 0.19167

0.18904

Total CPU time (s)

0.031

0.031

0.032

0.016

0.015

0.18729 0.016

Bold means optimum values of the objective function for different optimization methods. Bolditalics, underline : best solution found by the SOA for a specific value of the actual parameter.

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Fig. 9. Convergence characteristic for NOx emission minimization (6-generator system). Table 8 Best solutions for fuel cost and emission minimization offered by the SOA for the IEEE 30-bus system for different values of h. Case

Best fuel cost

h

h = p/2

PG1 (pu) PG2 (pu) PG3 (pu) PG4 (pu) PG5 (pu) PG6 (pu) Total generation (pu) Fuel cost ($/h)

0.05169 0.31359 0.46012 1.04331 0.61133 0.35397 2.834

Emission (ton/h)

600.986 0.20889

Total CPU time (s)

0.031

Best NOx emission h = p/4

h = p/2

h = p/4

0.07360 0.33030 0.55534 0.95201 0.49709 0.42566 2.834 601.104

0.47452 0.52948 0.53952 0.43685 0.42591 0.42772 2.834 640.749

0.39039 0.58495 0.45158 0.57157 0.42814 0.40737 2.834 630.418

0.20301

0.18729 0.016

0.18790

0.047

0.031

Bold means optimum values of the objective function for different optimization methods. Bolditalics, underline : best solution found by the SOA for a specific value of the actual parameter.

Fig. 11. Convergence characteristic for NOx emission minimization (6-generator system).

Table 9 Best solutions for Combined Economic and Emission Dispatch minimization offered by the SOA for the IEEE 30-bus system. Generation (pu)

PG1 PG2 PG3 PG4 PG5 PG6 Total power generation (pu) Fuel cost ($/h) Emission (ton/h) Emission cost ($/h) Total cost ($/h) Average CPU time (s) Price penalty factor ($/ton)

r = 0.99

r = 0.90

h = p/2

h = p/4

h = p/2

h = p/4

0.39595 0.41611 0.55805 0.53119 0.50243 0.43027 2.834 624.604 0.18708 1197.310

0.38345 0.44067 0.49435 0.58440 0.35069 0.58045 2.834 627.287 0.18841 1205.796 1833.083

0.18951 0.45583 0.57035 0.70932 0.39608 0.51290 2.834 612.410 0.19188 1228.047 1840.457

0.21034 0.48066 0.70669 0.47351 0.66721 0.29559 2.834 625.280 0.19347 1238.210 1863.490

0.015

0.016

0.016 6399.93

1821.914 0.015

Bold means optimum values of the objective function for different optimization methods. Bolditalics, underline : best solution found by the SOA for a specific value of the actual parameter.

Fig. 10. Convergence characteristic for fuel cost minimization (6-generator system).

From the results, it is inferred that, h = p/2 is superior to h = p/4 and that r should be selected as nearer to 1 to improve performance. In order to demonstrate the performance of the SOA, the results are compared to those reported in the literature like linear programming (LP) [52], multi-objective stochastic search technique (MOSST) [53], NSGA [36], SPEA [37], Niched Pareto genetic

Fig. 12. Convergence characteristic for total cost minimization (6-generator system).

algorithm (NPGA) [37], Modified Bacterial Foraging Algorithm (MBFA) [46], Fuzzy Clustering based Particle Swarm Optimization (FCPSO) [32], and Differential Evolution (DE) [32].

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L. Benasla et al. / Electrical Power and Energy Systems 62 (2014) 163–174 Table 10 Comparison of best solutions for fuel cost minimization offered by the different algorithms for the IEEE 30-bus system. Generation (pu)

SOA

LP [31]

MOSST [32]

NSGA [33]

NPGA [33]

SPEA [33]

MBFA [34]

OHS [23]

PG1 PG2 PG3 PG4 PG5 PG6 Total generation Fuel cost ($/h)

0.05169 0.31359 0.46012 1.04331 0.61133 0.35397 2.834

0.150 0.300 0.550 1.050 0.460 0.350 2.86 606.310

0.113 0.302 0.531 1.021 0.531 0.363 2.861 605.890

0.1038 0.3228 0.5123 1.0387 0.5324 0.3241 2.8341 600.34

0.1116 0.3153 0.5419 1.0415 0.4726 0.3512 2.8341 600.31

0.1009 0.3186 0.5400 0.9903 0.5336 0.3507 2.8341 600.22

0.1133 0.3005 0.5202 0.9882 0.5409 0.3709 2.834 600.17

0.1086 0.2995 0.5315 1.0121 0.5230 0.3591 2.8338 600.00

0.223 NR

0.222 NR

0.2241 NR

0.2238 NR

0.2206 NR

0.2200 NR

0.2219 1.7135

Emission (ton/h) Total CPU time (s)

600.986 0.20889 0.031

NR means not reported in the referred literature. Bold means optimum values of the objective function for different optimization methods. Bolditalics, underline : best solution found by the SOA for a specific value of the actual parameter. Table 11 Comparison of best solutions for emission minimization offered by the different algorithms for the IEEE 30-bus system. Generation (pu)

SOA

LP [31]

MOSST [53]

NSGA [33]

NPGA [33]

SPEA [33]

MBFA [34]

OHS [23]

PG1 PG2 PG3 PG4 PG5 PG6 Total generation Fuel cost ($/h) Emission (ton/h)

0.47452 0.52948 0.53952 0.43685 0.42591 0.42772 2.834 640.749

0.150 0.300 0.550 1.050 0.460 0.350 2.86 606.310 0.2230

0.4146 0.463 0.543 0.388 0.543 0.514 2.861 644.11 0.1942

0.4072 0.4536 0.4888 0.4302 0.5836 0.4707 2.8341 633.83 0.1946

0.41 0.4419 0.5411 0.4067 0.5318 0.4979 2.834 636.04 0.1943

0.4240 0.4577 0.5301 0.3721 0.5311 0.5190 2.834 640.42 0.1942

0.3943 0.4627 0.5423 0.3946 0.5346 0.5056 2.8341 636.73 0.1942

0.412116 0.466949 0.547454 0.371771 0.534693 0.501016 2.833999 639.5493 0.1942

NR

NR

NR

NR

NR

NR

1.7125

Total CPU time (s)

0.18729 0.016

NR means not reported in the referred literature. Bold means optimum values of the objective function for different optimization methods. Bolditalics, underline : best solution found by the SOA for a specific value of the actual parameter. Table 12 Fuel cost and emission coefficients of forty-generator system. Unit

ci ($/h)

bi ($/MWh)

ai ($/(MW2) h)

di ($/h)

ei (rd/MW)

ai (ton/h)

bi (ton/MW h)

ci (ton/(MW2) h)

gi (ton/h)

di (1/MW)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

94.705 94.705 309.540 369.030 148.890 222.330 287.710 391.980 455.760 722.820 635.200 654.690 913.400 1760.40 1760.40 1760.40 647.850 649.690 647.830 647.810 785.960 785.960 794.530 794.530 801.320 801.320 1055.10 1055.10 1055.10 148.890 222.920 222.920 222.920 107.870 116.580 116.580 307.450 307.450 307.450 647.830

6.73 6.73 7.07 8.18 5.35 8.05 8.03 6.99 6.60 12.9 12.9 12.8 12.5 8.84 8.84 8.84 7.97 7.95 7.97 7.97 6.63 6.63 6.66 6.66 7.10 7.10 3.33 3.33 3.33 5.35 6.43 6.43 6.43 8.95 8.62 8.62 5.88 5.88 5.88 7.97

0.00690 0.00690 0.02028 0.00942 0.01140 0.01142 0.00357 0.00492 0.00573 0.00605 0.00515 0.00569 0.00421 0.00752 0.00752 0.00752 0.00313 0.00313 0.00313 0.00313 0.00298 0.00298 0.00284 0.00284 0.00277 0.00277 0.52124 0.52124 0.52124 0.01140 0.00160 0.00160 0.00160 0.00010 0.00010 0.00010 0.01610 0.01610 0.01610 0.00313

100 100 100 150 120 100 200 200 200 200 200 200 300 300 300 300 300 300 300 300 300 300 300 300 300 300 120 120 120 120 150 150 150 200 200 200 80 80 80 300

0.084 0.084 0.084 0.063 0.077 0.084 0.042 0.042 0.042 0.042 0.042 0.042 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.035 0.077 0.077 0.077 0.077 0.063 0.063 0.063 0.042 0.042 0.042 0.098 0.098 0.098 0.035

60 60 100 120 50 80 100 130 150 280 220 225 300 520 510 510 220 222 220 220 285 285 295 295 310 310 360 360 360 50 80 80 80 65 70 70 100 100 100 220

2.22 2.22 2.36 3.14 1.89 3.08 3.06 2.32 2.11 4.34 4.34 4.28 4.18 3.34 3.55 3.55 2.68 2.66 2.68 2.68 2.22 2.22 2.26 2.26 2.42 2.42 1.11 1.11 1.11 1.89 2.08 2.08 2.08 3.48 3.24 3.24 1.98 1.98 1.98 2.68

0.0480 0.0480 0.0762 0.0540 0.0850 0.0854 0.0242 0.0310 0.0335 0.4250 0.0322 0.0338 0.0296 0.0512 0.0496 0.0496 0.0151 0.0151 0.0151 0.0151 0.0145 0.0145 0.0138 0.0138 0.0132 0.0132 1.8420 1.8420 1.8420 0.0850 0.0121 0.0121 0.0121 0.0012 0.0012 0.0012 0.0950 0.0950 0.0950 0.0151

1.3100 1.3100 1.3100 0.9142 0.9936 1.3100 0.6550 0.6550 0.6550 0.6550 0.6550 0.6550 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.5035 0.9936 0.9936 0.9936 0.9936 0.9142 0.9142 0.9142 0.6550 0.6550 0.6550 1.4200 1.4200 1.4200 0.5035

0.05690 0.05690 0.05690 0.04540 0.04060 0.05690 0.02846 0.02846 0.02846 0.02846 0.02846 0.02846 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.02075 0.04060 0.04060 0.04060 0.04060 0.04540 0.04540 0.04540 0.02846 0.02846 0.02846 0.06770 0.06770 0.06770 0.02075

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Table 13 Comparison of best solutions for cost and emission offered by the different algorithms for the 40-generating units test system. Unit

PGimin (MW)

PGimax (MW)

Generation (MW)

Generation (MW)

Fuel cost minimization

1 36 2 36 3 60 4 80 5 47 6 68 7 110 8 135 9 135 10 130 11 94 12 94 13 125 14 125 15 125 16 125 17 220 18 220 19 242 20 242 21 254 22 254 23 254 24 254 25 254 26 254 27 10 28 10 29 10 30 47 31 60 32 60 33 60 34 90 35 90 36 90 37 25 38 25 39 25 40 242 Total power generation Fuel cost ($/h) Emission (ton/h) Average CPU time (s)

114 114 120 190 97 140 300 300 300 300 375 375 500 500 500 500 500 500 550 550 550 550 550 550 550 550 150 150 150 97 190 190 190 200 200 200 110 110 110 550 (MW)

Emission minimization

SOA

MBFA [34]

OHS [23]

DE [4]

SOA

MBFA [34]

OHS [23]

DE [4]

93,298 93,298 99,298 169,298 76,298 119,298 279,298 279,298 279,298 279,298 94 354,298 125 302,286 479,298 264,182 479,298 479,298 529,298 529,298 529,298 529,298 529,298 529,298 529,298 529,298 10 10 10 47 169,298 169,298 169,298 179,298 179,298 179,298 89,298 89,298 89,298 529,298 10,500 125,248.114 334,880.054 0.14

114.0000 110.8035 97.4002 179.7333 87.8072 140.0000 259.6004 284.6002 284.6006 130.0000 168.7999 168.7998 214.7598 304.5195 394.2794 394.2794 489.2794 489.2794 511.2795 511.2795 523.2794 523.2794 523.2796 523.2794 523.2795 523.2796 10.0001 10.0002 10.0002 89.5070 190.0000 190.0000 190.0000 164.8026 164.8035 164.8292 110.0000 110.0000 110.0000 511.2795 10,500 121,415.653 356,424.497 NR

103.9422 112.2758 97.2549 179.8187 95.0966 139.5940 263.8120 293.0950 299.5109 130.8497 102.1717 95.9356 129.7004 384.2587 301.7727 301.7773 496.6855 490.4517 502.7192 510.7183 523.3236 524.7491 523.4055 522.6936 522.6783 538.1215 149.7369 131.3113 130.7194 92.7962 185.4022 174.0935 168.3946 176.0944 104.4641 167.3644 89.9918 102.7658 108.1541 532.2987 10,500 120,240.0 312,500.00 21.1036

110.8256 111.1008 97.3996 179.7336 92.2835 140.0000 259.6004 284.6014 284.6000 130.0000 168.7999 168.8004 214.7597 394.2796 394.2793 304.5195 489.2796 489.2793 511.2798 511.2796 523.2793 523.2795 523.2800 523.2796 523.2795 523.2797 10.0000 10.0000 10.0000 87.8823 190.0000 190.0000 190.0000 164.8000 164.8422 164.8171 110.0000 110.0000 110.0000 511.2794 10,500 121,414.9372 356432.902 25.80

84,971 92,705 98,705 168,705 75,705 118,705 240,058 254,173 256,325 257,545 244,86 266,32 351,094 329,081 474,338 441,45 409,975 413,73 449,937 470,073 475,645 457,038 453,762 449,661 444,009 436,705 128,705 128,705 128,705 75,705 168,273 164,126 162,689 175,576 178,705 178,705 88,705 88,705 88,705 528,705 10,500 152,262.549 120,590.839 0.14

114.0000 114.0000 120.0000 169.3671 97.0000 124.2630 299.6931 297.9093 297.2578 130.0007 298.4210 298.0264 433.5590 421.7360 422.7884 422.7841 439.4078 439.4132 439.4111 439.4155 439.4421 439.4587 439.7822 439.7697 440.1191 440.1219 28.9738 29.0007 28.9828 97.0000 172.3348 172.3327 172.3262 200.0000 200.0000 200.0000 100.8441 100.8346 100.8362 439.3868 10,500 129,995.00 176,682.27 NR

105.5679 88.2574 105.9739 150.3464 82.0595 119.5704 248.5154 276.3936 244.2866 282.1424 293.2579 294.5149 432.2395 391.8179 422.8119 414.8810 428.5659 432.1613 442.9423 435.9092 451.8724 456.1765 437.2970 442.0350 445.9564 429.3785 124.0932 117.8668 94.5359 89.6485 153.1318 159.0102 148.7814 176.4061 170.0710 181.6662 96.8108 94.3094 82.4816 456.2560 10,500 128,140.00 102,190.00 21.0001

114.0000 114.0000 120.0000 169.3731 97.0000 124.3216 299.7804 297.8574 297.2253 130.0000 298.5781 298.1021 433.3265 421.5685 422.8753 422.8293 439.2832 439.4235 439.4559 439.4793 439.3454 439.3786 439.6003 440.0332 440.0050 440.2423 29.0481 28.9881 29.0570 96.9999 172.4231 172.3000 172.4462 200.0000 200.0000 200.0000 100.7115 100.8505 100.7753 439.2871 10,500 129994.2488 176679.6837 27.88

Fig. 13. Convergence characteristic for fuel cost minimization (40-generator system).

Fig. 14. Convergence characteristic for NOx emission minimization (40-generator system).

L. Benasla et al. / Electrical Power and Energy Systems 62 (2014) 163–174 Table 14 Best solutions for Combined Economic and Emission Dispatch minimization offered by the SOA for the 40-generating units test system (PD = 10500 MW). Generation (MW) Unit

SOA

1 114 2 114 3 120 4 190 5 97 6 112,985 7 266,512 8 205,54 9 222,193 10 292,266 11 354,778 12 375 13 390,494 14 348,717 15 407,008 16 431,418 17 477,519 18 478,196 19 434,99 20 409,734 Total power generation (MW) Fuel cost ($/h) Emission (ton/h) Emission cost ($/h) Total cost ($/h) Price penalty factor ($/ton) Average CPU time (s)

Generation (MW) DE [04]

Unit

SOA

DE [04]

113.9442 113.9888 119.9907 178.2275 97.0000 129.2120 299.9832 299.8143 297.9646 130.0158 308.5642 307.1322 434.9219 408.3507 410.4238 410.8373 451.6820 452.9208 438.3952 436.2184 10500.000

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

306,24 415,941 477,55 388,988 355,981 504,901 63,89 73,316 106,149 97 183,017 189,8 144,264 200 170,576 200 110 78,706 110 481,332

437.5263 438.4409 437.3739 437.4427 437.9143 438.2851 18.7573 20.3499 20.5208 96.9996 175.3641 175.4229 175.5867 200.0000 200.0000 199.9991 104.6397 104.0792 104.9804 436.7296 10500.000

140044.6584 154291.1405 54307.3956 194352,0540 0.35198 0.016

128713.8868 178634.0971 62875.6295 191589.5164 0.35198 24.59

The SOA parameters are taken as follows: h ¼ p2 , r = 0.99. From the comparison, it is noticed that the proposed algorithm gives the same fuel cost (Table 10) but a reduction in NOx emission (Table 11) as compared to the reported ones.

Test system 3 This last case is considered for the aim to further demonstrate the effectiveness of the SOA for large scale EED problems in power systems. This test system consists of 40-generating units with valve point loading effects and NOx emission. The total system demand is 10500 MW and no transmission losses are considered. The input data for this test system are taken from [14,35] and presented in Table 12. The operating limits are also adopted from [14,35] and presented in Table 13. In this case study, the control parameters of SOA used are variable values for the spiral radius and a constant rotation angle. While individually considering the fuel cost minimization and the emission minimization for this test system, the generation schedule along with the values of the fuel cost and that of the emission are also presented in Table 13. The results obtained by adopting the proposed SOA algorithm are compared to those appeared in a recent literature. While considering the fuel cost minimization the values of minimum fuel cost and emission yielded by the SOA algorithm are noted as 125,248.114 $/h and 334,880.054 ton/h, respectively. On the other hand, the values of fuel cost and minimum emission yielded by the SOA algorithm are recorded as 152,262.549 $/h, and 120,590.839 ton/h, respectively, for emission minimization problem. For both the individual minimization problems, the minimum values of the fuel cost and the emission as obtained by using the proposed algorithm are found to be the worst ones as compared to the counter parts yielded by MBFA [34], OHS [23] and DE [4],

173

however the CPU time for SOA is insignificant when compared with the other methods. Convergence curves of SOA for this case study are shown in Figs. 13 and 14 respectively. The proposed SOA-based convergence profiles of both the objective values for this test system are found to be very promising. Form these figures it is observed that SOA consistently produces solutions near the optimum at very early iterations. For the sake of comparison best compromising solutions obtained from DE [04] are included in Table 14. The best compromising cost obtained by SOA is very close to that found with DE [4] method (1.4% difference). Moreover, SOA outperforms DE in the minimization of NOx. These comparisons with the previous ones reveal the potential of the proposed Spiral Optimization Algorithm to produce good solutions within a highly nonlinear and nonconvex solution space having multiple local minima. Conclusion In this paper, a newly metaheuristic Spiral Optimization Algorithm (SOA) has been introduced and applied to solve the multiobjective environmental/economic dispatch problem in the presence of generators with nonsmooth and nonconvex fuel cost functions. SOA witch is one of the recent heuristic algorithms improved by Tamura and Yasuda for solving optimization problems, has several advantages including its few control variables, local searching capability, fast results, easy using process, simple structure and introduction of both phases of diversification and intensification in the same process. In order to illustrate the application of the proposed algorithm, it has been tested and examined with three different test systems. Firstly, SOA is tested on three and six generators, with a quadratic cost function for combined economic emission load dispatch problems. Secondly, the proposed algorithm is applied to forty generators, with a nonsmooth cost function. The results of the implementation of this proposed algorithm clearly showed the efficiency of the SOA for solving the CEED problem under various test systems. Moreover, the results of the proposed algorithm have been compared to those techniques published in the literature which shows that the proposed method confirms the effective high-quality solution for CEED problems. However, from the simulation results, it seems that the proper selection of the optimal control parameters is of paramount importance for the convergence of the algorithm. Since SOA is relatively new, further extensions of this algorithm should be explored to include more objective functions or constraints with regard to more realistic problems, as well as other data sets and standard test problems. Furthermore, future comparison studies with other methods are necessary so as to identify the strengths and weaknesses of the current metaheuristic algorithm. References [1] Phanthuna N, Phupha V, Rugthaicharoencheep N, Lerdwanittip S. Economic load dispatch with daily load patterns and generator constraints by particle swarm optimization. World Acad Sci Eng Technol 2012;71:1232–6. [2] Hamedi H. Solving the combined economic load and emission dispatch problems using new heuristic algorithm. Int J Electr Power Energy Syst 2013;46:10–6. [3] Rani C, Kothari DP, Busawon K. Combined economic emission dispatch problem using chaotic self Adapt3ive PSO. In: International conference on power, energy and control (ICPEC); 2013. p. 399–404. [4] Sayah S, Hamouda A, Bekrar A. Efficient hybrid optimization approach for emission constrained economic dispatch with nonsmooth cost curves. Electric Power Energy Syst 2014;56:127–39.

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[22]

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[28]

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[31]

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