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Engineering Science and Technology, an International Journal

journal homepage: http://www.elsevier.com/locate/jestch

Full Length Article

Flower pollination algorithm to solve combined economic and

emission dispatch problems

A.Y. Abdelaziza, E.S. Alib*, S.M. Abd Elazimb

a Electric Power and Machine Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt b Electric Power and Machine Department, Faculty of Engineering, Zagazig University, Zagazig, Egypt

ARTICLE INFO ABSTRACT

Article history: Received 5 June 2015 Received in revised form 26 September 2015 Accepted 1 November 2015 Available online

Keywords:

Flower pollination algorithm

Economic load dispatch

Combined economic emission dispatch

Emission constraints

Valve point loading effect

Swarm intelligence

Economic Load Dispatch (ELD) is the process of allocating the required load between the available generation units such that the cost of operation is minimized. The ELD problem is formulated as a nonlinear constrained optimization problem with both equality and inequality constraints. The dual-objective Combined Economic Emission Dispatch (CEED) problem is considering the environmental impacts that accumulated from emission of gaseous pollutants of fossil-fuelled power plants. In this paper, an implementation of Flower Pollination Algorithm (FPA) to solve ELD and CEED problems in power systems is discussed. Results obtained by the proposed FPA are compared with other optimization algorithms for various power systems. The results introduced in this paper show that the proposed FPA outlasts other techniques even for large scale power system considering valve point effect in terms of total cost and computational time.

Copyright © 2015, The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

licenses/by-nc-nd/4.0/).

Abbreviations: ELD, economic load dispatch; CEED, combined economic emission dispatch; FPA, flower pollination algorithm; ED, economic dispatch; FLC, fuzzy logic control; ANN, artificial neural network; EA, evolutionary algorithm; GA, genetic algorithm; SA, simulated annealing; EP, evolutionary programming; TS, Tabu search; PSO, particle swarm optimization; GSA, gravitational search algorithm; ABC, artificial bee colony; QP, quadratic programming; DE, differential evolution; PPSO, personal best-oriented PSO; APPSO, adaptive personal-best oriented PSO; MPSO, modified particle swarm optimization; ARCGA, adaptive real coded GA; TSAGA, Taguchi self-adaptive real-coded genetic algorithm; CCPSO, PSO with both chaotic sequences and crossover operation; CDE_SQP, combining of chaotic DE and quadratic programming; EDA/DE, estimation of distribution and differential evolution cooperation; SOMA, self-organizing migrating strategy; CSOMA, cultural self-organizing migrating strategy; DE/BBO, combination of differential evolution and biogeography-based optimization; DHS, differential harmony search; BBO, biogeography based optimization; PSO-SQP, integrating PSO with the sequential quadratic programming; GA-PS-SQP, hybrid algorithm consisting of GA, pattern search (PS) and SQP; CPSO, chaotic particle swarm optimization; CPSO-SQP, hybrid algorithm consisting of CPSO and SQP; NPSO_LRS, new PSO with local random search; CDEMD, cultural DE based on measure of population's diversity; HMAPSO, hybrid multi agent based PSO; FAPSO-NM, fuzzy adaptive PSO algorithm with Nelder-Mead; ICA-PSO, improved coordinated aggregation-based PSO; MODE, multiobjective differential evolution; NSGA-II, non-dominated sorting genetic algorithm-II; PDE, Pareto differential evolution; SPEA-2, strength Pareto evolutionary algorithm 2; ABC_PSO, ABC and PSO; EMOCA, enhanced multi-objective cultural algorithm; MABC/D/Cat, modified artificial bee colony with disruptive cat map; MABC/D/Log, modified artificial bee colony with disruptive logistic map; CPU, computational time; NA, not available; PV, photovoltaic.

* Corresponding author. Tel.: (002) 0111-2669781, fax: (002) 055-2321407.

E-mail address: ehabsalimalisalama@yahoo.com (E.S. Ali).

Peer review under responsibility of Karabuk University.

1. Introduction

Economic Dispatch (ED) problem has become a crucial task in the operation and planning of power system [1]. It is very complex to solve because of a nonlinear objective function and a large number of constraints. ED in power system deals with the determination of optimum generation schedule of available generators so that the total cost of generation is minimized within the system constraints [2,3]. Well known long-established techniques such as gradient method [4], lambda iteration method [5,6], linear programming [7], quadratic programming [8], Lagrangian multiplier method [9], and classical technique based on co-ordination equations [10] are applied to solve ELD problems. These conventional methods cannot perform satisfactorily for solving such problems as they are sensitive to initial estimates and converge into local optimal solution in addition to its computational complexity.

During the last decades many researches and techniques had dealt with ELD problems. Fuzzy Logic Control (FLC) has attracted the attention in control applications. In contrast with the conventional techniques, FLC formulates the control action in terms of linguistic rules drawn from the behavior of a human operator rather than in terms of an algorithm synthesized from a model of the system [11-14]. However, it requests more fine tuning and simulation before operational. Another technique like Artificial Neural Network (ANN)

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has its own advantages and disadvantages. The characteristics of the system is enhanced by ANN, but the main problem of this technique is the long training time, the selecting number of layers and the number of neurons in each layer [6,15-17].

An alternative approach is to employ Evolutionary Algorithm (EA) techniques. Due to its ability to treat nonlinear objective functions, EA is believed to be very effective to deal with ELD problem. Among the EA techniques, Genetic Algorithm (GA) is introduced in References 18 and 19, but it requires a very long run time depending on the size of the system under study. Also, it gives rise to repeat revisiting of the same suboptimal solutions. Simulated Annealing (SA) is illustrated in References 20 and 21, but this technique might fail by getting trapped in one of the local optimal. Evolutionary Programming (EP) is discussed in Reference 22, but it has a slow convergence rate for large problem. Improved Tabu Search (TS) is introduced in Reference 23, but the efficiency of this algorithm is reduced by the use of highly epistatic objective functions and the large number of parameters to be optimized. Also, it is a time-consuming method. Ant swarm optimization is presented in Reference 24, but its theoretical analysis is difficult and probability distribution changes by iteration. Particle Swarm Optimization (PSO) is discussed in References 25-28, but it pains from the partial optimism. Moreover, the algorithm cannot work out the problems of scattering and optimization. Gravitational Search Algorithm (GSA) in illustrated in Reference 29. However, this algorithm appears to be effective for solving ELD problem, it has poor performance at the later search stage due to the lack of agents' diversity in GSA. Artificial Bee Colony (ABC) is developed in Reference 30 to solve the complex non-linear optimization problem, but it is slow to converge and the processes of the exploration and exploitation contradict with each other, so the two abilities should be well balanced for achieving good optimization performance. On the other hand, FPA has only one key parameter p (switch probability) which makes the algorithm easier to implement and faster to reach optimum solution. Moreover, this transferring switch between local and global pollination can guarantee escaping from local minimum solution. Thus, FPA is proposed in this paper to overcome the previous drawbacks. In addition, it is clear from the literature survey that the application of FPA to solve ELD and CEED problems has not been discussed. This encourages us to adopt FPA to deal with these problems.

In this paper, a new approach for solving ELD and CEED problems using FPA methodology is discussed considering the power limits of the generator. The purpose of CEED is to minimize both the operating fuel cost and emission level simultaneously while satisfying load demand and operational constraints. This multi-objective CEED problem is converted into a single objective function using a modified price penalty factor approach. FPA is investigated to determine the optimal loading of generators in power systems. Simulations results for small and large scale power system considering the valve loading effect are implemented to indicate the robustness of FPA.

The remainder of this paper is organized as follows: Section 2 provides a brief description and mathematical formulation of ELD and CEED problems. In section 3, the concept of FPA is discussed. Section 4 shows the result on three, ten and forty unit thermal test systems. Finally, the conclusion and future work of research are outlined in section 5.

2. Problem formulation

The CEED problem is to minimize two computing objective functions simultaneously, fuel cost and emission, while satisfying various equality and inequality constraints. Generally the problem is formulated as follows.

A : Primary Valve B : Secondary Valve C : Tertiary Valve D : Quaternary Valve E : Quinary Valve

Fig. 1. Valve point effect.

costs are usually represented as a quadratic function of output power [31], as shown in equation (1).

F (P ) = rP2 +ßP + a

Minimize

d d Ft = 2 Fi (Pi )=£( +AP + a) i=i i=i

The minimization is performed subject to the equality constraint that the total generation must equal to the demand plus the loss thus:

X Pi = Pd + Pl i=i

The total transmission loss using Kron's loss formula is given in equation (4)

d d d Pl =XX(PiBijPj ) + X BoiPi + Bo, i=i j=i i=i

It is assumed with little error that these coefficients are constant (as long as operation is near the value where these coefficients are computed).

Based on the maximum and minimum power limits of generators the inequality constraint is

Pmin < Pi < Pmax i = i, 2.......d

2.2. Effect of valve point on fuel cost objective

To be more practical, the valve point effect is taken into account in the cost function of generators. The sharp increase in losses due to the wire drawing effects which occur as each steam admission valve starts to open leads to the nonlinear rippled input output curve [32] as shown in Fig. 1. The obtained cost function based on the rippled curve is more accurate modeling. Thus, the fuel cost function of each fossil fuel generator is given as the sum of a quadratic and a sinusoidal function [33].

Ft = 2 Fi (Pi ) = £( +ßiP +ai + ei * sin (( *(min - Pi )))

2.1. Objective function of ELD

2.3. Objective function of CEED

For thermal generating units, the cost of fuel per unit power output varies significantly with the output power of the unit. Fuel

The atmospheric pollutants such as sulfur oxides, nitrogen oxides and carbon dioxide caused by fossil fuel fired generator can be

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modeled separately [34-36]. However, for comparison purposes, the total emission of these pollutants which is the sum of a quadratic and an exponential function can be expressed as [37,38]:

Et = £Ei (Pi ) = £ ( + biPi + d * exp (Si * Pi )) i=1 i=1

Optimization of generation cost has been formulated based on classical ELD with emission and line flow constraints. The detailed problem is given as follows [38].

Minimize F = £{F(Pi), Ei (Pi)}

The minimum value of the above objective function has to be found out subject to equality and inequality constraints given by equations (3) and (5). The dual-objective CEED problem is converted into single optimization problem by introducing a price penalty factor h as follows [39].

Minimize F = Ft + h x Et

Subject to constraints given by equations (3) and (5), the price penalty factor h, which is the ratio between the maximum fuel cost and maximum emission of corresponding generator in $/Kg [30,33], blends the emission with fuel cost, then F is the total operating cost in $.

Ft (Pm

Et (Pm

i = 1,2,

The following steps are used to find the price penalty factor for a particular load demand:

A. Find the ratio between maximum fuel cost and maximum emission of each generator.

B. Arrange the values of price penalty factor in ascending order.

C. Add the maximum capacity of each unit ( Pmax) one at a time, starting from the smallest hi, until E Pmax ^ PD.

D. At this point, hi which associated with the last unit in this process is the approximate price penalty factor value (h) for the given load.

Hence, a modified price penalty factor (h) is used to give the exact value for the particular load demand by interpolating the values of (h), corresponding to their load demand values.

3. Overview of flower pollination algorithm

FPA was developed by Yang in 2012 [40]. It is inspired by the pollination process of flowering plants. Real-world design problems in engineering and industry are usually multiobjective. These multiple objectives often conflict with one another. Also, they have additional challenging issues such as time complexity, inhomoge-neity and dimensionality [41]. They are usually more time-consuming. FPA has been adopted in this paper to solve ELD and CEED problems.

3.1. Characteristics of flower pollination

The main purpose of a flower is ultimately reproduction via pollination. Flower pollination is typically correlating with the transfer of pollen, which often associated with pollinators such as birds and insects. Indeed, some flowers and insects have a very specialized flower-pollinator partnership, as some flowers can only attract a specific species of insect or bird for effective pollination. Pollination appears in two major forms: abiotic and biotic. About 90% of flow-

Fig. 2. Flow chart of FPA.

ering plants depend on the biotic pollination process, in which the pollen is transferred by pollinators. About 10% of pollination follows abiotic form that does not require any pollinators [42]. Wind and diffusion help in the pollination process of such flowering plants [43].

Pollination can be achieved by self-pollination or cross-pollination. Self-pollination is the pollination of one flower from pollen of the same flower. Cross-pollination is the pollination from pollen of a flower of different plants. The objective of flower pollination is the survival of the fittest and the optimal reproduction of plants in terms of numbers as well as the fittest. This can be considered as an optimization process of plant species. All of these factors and processes of flower pollination created optimal reproduction of the flowering plants [43].

3.2. Flower pollination algorithm

For FPA, the following four steps are used:

Step 1: Global pollination represented in biotic and cross-pollination processes, as pollen-carrying pollinators fly following Levy flight [44].

Step 2: Local pollination represented in abiotic and self-pollination as the process does not require any pollinators.

Step 3: Flower constancy which can be developed by insects, which is on a par with a reproduction probability that is proportional to the similarity of two flowers involved.

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Table 2

Statistical comparison between FPA and different algorithms.

Algorithm Best cost ($) Mean cost ($) Worst cost ($) Time (s)

ARCGA [47] 121410.1038 121462.1502 121536.8745 15.67

TSAGA [48] 121463.07 122928.31 124296.54 696.01

CCPSO [49] 121403.5362 121445.3269 121535.4934 19.3

CDE_SQP [50] 121741.9793 122295.1278 122839.2941 14.26

EDA/DE [51] 121412.50 121460.70 121517.80 NA

BBO [52] 121426.66 121508.03 121688.66 NA

SOMA [52] 121418.7856 121449.8796 121508.3757 NA

CSOMA [52] 121414.6978 121415.0479 121417.8045 NA

DE/BBO [53] 121420.89 121420.90 121420.90 60.00

DHS [54] 121403.5355 121410.5967 121417.2274 1.32

ICA-PSO [55] 121413.2 121428.14 121453.56 139.92

EP [56] 122624.35 123382.00 125740 1167.35

EP-SQP [56] 122323.97 122379.63 NA 997.73

PSO [56] 123930.45 124154.49 NA 933.39

PSO-SQP [56] 122094.67 122245.25 NA 733.97

GA-PS-SQP [57] 121458 122039 NA 46.98

CPSO [58] 121865.23 122100.87 NA 114.65

CPSO-SQP [58] 121458.54 122028.16 NA 98.49

NPSO_LRS [59] 121664.4308 122209.3185 122981.5913 16.81

APSO [60] 121663.5222 122153.6730 122912.3958 5.05

DE [61] 121416.29 121422.72 121431.47 NA

CDEMD [62] 121423.4013 121526.7330 121696.9868 44.3

HMAPSO [63] 121586.90 121586.90 21586.90 NA

FAPSO-NM [64] 121418.3 121418.803 121419.8 40

FPA 121074.5 121095.7 121196.3 0.89

Step 4: The interaction of local pollination and global pollination is controlled by a switch probability p e [0, 1], lightly biased toward local pollination.

To generate the updating formulas, the above rules have to be converted into proper updating equations. For example at the global pollination step, the pollinators such as insects carry the flower pollen gametes, so the pollen can travel over a long distance because of the ability of these insects to fly and move in much longer ranges. Therefore, global pollination step and flower constancy step can be represented by:

xt+1 = xt + yLW(g, - xt )

In fact, L(A) is the Levy flights based step size that corresponds to the strength of the pollination. Since long distances can be covered by insects using various distance steps, a Levy flight can be used to mimic this behavior efficiently. That is, L > 0 from a Levy distribution.

1.7 1.65 1.6 1.55 1.5 1.45 1.4 1.35 1.3 1.25 1.2

40 50 60 Iterations

Fig. 4. Objective function for forty unit system.

L _ ХГЩфХ/2)Mj ( ^ S0 > 0)

This distribution is valid for large steps s > 0. For the local pollination, both Step 2 and Step 3 can be represented as

x,t+1 = xt + e(xtj - xk )

Table 3

Results for the best simulations with 3-unit system considering emission.

Pd h Power GA [38] PSO [38] FPA

outputs

400 43.55981 P1 (MW) 102.617 102.612 102.4468

(MW) P2 (MW) 153.825 153.809 153.8341

p3(mw) 151.011 150.991 151.1321

Pl (MW) 7.41324 7.41173 7.4126

Fuel Cost ($) 20840.1 20838.3 20838.1

Emission (Kg) 200.256 200.221 200.2238

Total Cost ($) 29563.2 29559.9 29559.81

CPU (Sec) 0.282 0.235 0.175

The bold values are obtained using the proposed FPA algorithm.

Algorithms

122600 122400 122200 122000 121800 121600 121400 121200 121000 120800 120600 120400

0 10 20 30 40 50 60 70 80 90 100 Iterations

Fig. 3. Fuel cost for various algorithms for case 1.

Fig. 5. Objective function for 3-unit system with demand = 400 MW.

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FPA PSO GA

Algorithms

Fig. 6. Total cost for various algorithms with demand = 400 MW.

0 10 20 30 40 50 60 70 80 90 100 Iterations

Fig. 7. Change of objective function with iterations for ten units.

where xt and x'k are pollen from different flowers of the same plant species mimicking the flower constancy in a limited neighborhood. For a local random walk, xt and xk come from the same species, then e is drawn from a uniform distribution as [0,1].

In principle, flower pollination activities can occur at all scales. But in reality, adjacent flower patches are more likely to be pollinated by a local flower pollen than those far away. In order to mimic this, one can effectively use a switch probability (Step 4) to switch between common global pollination to intensive local pollination. To start with, one can use a naive value of p = 0.5. A preliminary parametric showed that p = 0.8 might work better for most applications. The flow chart of FPA is given in Fig. 2. The data of FPA are shown in Appendix A.

4. Results and discussion

FPA is employed to solve ELD and CEED problems for different cases to assure its optimization efficiency, where the objective function is limited by the output limits of generation units and transmission losses. The performance of FPA is compared with various optimization algorithms. Simulations were done under the Matlab environment.

4.1. Case study 1

This case considers 40 generators as a large scale power system to confirm the superiority of FPA over other algorithms in reaching optimum solution. Moreover, the effect of valve loading point

113450 -a

113400 3

FPA EMOCA ABC_PSO GSA SPEA-2 PDE NSGAII MODE Algorithms

Fig. 8. Total cost for various algorithms for case 3.

is taken into account to complete the analysis [45-55]. The data of this system are given in Appendix B.

Table 1 outlines the outputs of each unit for 10,500 MW load demand and the cost for each algorithm. It can be noticed that the suggested FPA achieves lower cost compared with other algorithms while achieving the constraints of generations. Therefore, these algorithms have trapped in local minimum solutions. Thus,

113600

113550

113500

113300

113250

Table 4

CEED comparison for ten unit system at demand of 2000 MW.

Outputs MODE [65] NSGAII [65] PDE [65] SPEA-2[65] GSA [66] ABC_PSO [67] EMOCA [68] Proposed FPA

P1 (MW) 54.9487 51.9515 54.9853 52.9761 54.9992 55 55 53.188

P2 (MW) 74.5821 67.2584 79.3803 72.813 79.9586 80 80 79.975

P3 (MW) 79.4294 73.6879 83.9842 78.1128 79.4341 81.14 83.5594 78.105

P4 (MW) 80.6875 91.3554 86.5942 83.6088 85.0000 84.216 84.6031 97.119

P5 (MW) 136.8551 134.0522 144.4386 137.2432 142.1063 138.3377 146.5632 152.74

P6 (MW) 172.6393 174.9504 165.7756 172.9188 166.5670 167.5086 169.2481 163.08

P7 (MW) 283.8233 289.4350 283.2122 287.2023 292.8749 296.8338 300 258.61

P8 (MW) 316.3407 314.0556 312.7709 326.4023 313.2387 311.5824 317.3496 302.22

P9 (MW) 448.5923 455.6978 440.1135 448.8814 441.1775 420.3363 412.9183 433.21

P10(MW) 436.4287 431.8054 432.6783 423.9025 428.6306 449.1598 434.3133 466.07

Fuel cost * 105$ 1.13484 1.13539 1.1351 1.1352 1.1349 1.1342 1.13445 1.1337

Emission (Ib) 4124.9 4130.2 4111.4 4109.1 4111.4 4120.1 4113.98 3997.7

Losses (MW) 84.33 84.25 83.9 84.1 83.9869 84.1736 83.56 84.3

CPU (s) 3.82 6.02 4.23 7.53 NA NA 2.90 2.23

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FPA performs better than these algorithms in terms of fuel cost even for large scale power system with valve loading effect. Also, Table 2 lists the statistical comparison between FPA and different algorithms reported in [47-64] in terms of the best, mean, worst cost and computational (CPU) time through 50 trials. It is clear that the fuel cost obtained by the proposed FPA is better than other algorithms. Fig. 3 shows the total cost for each algorithm. On the other hand, a graph for convergence rate of the objective function is given in Fig. 4. It can be seen that the objective function is stabilized after 9 iterations. Also, the mean CPU time of FPA is the shortest one.

4.2. Case study 2

This case studies a 3-unit generating thermal system considering emission impact. The generator cost coefficients, emission coefficients, generation limits and the transmission loss coefficient matrix are given in Appendix B. Table 3 summarizes the results of solving CEED using the proposed FPA compared with GA and PSO [38]. As shown from Table 3, FPA donates superior result in terms of fuel cost, total cost and CPU compared with other algorithms. Moreover, the equality and inequality constraints are accomplished. The proposed FPA gives better results in terms of minimum total cost and smaller CPU time than other algorithms. Fig. 5 shows the total cost associated with FPA for 400 MW demand. The superiority of the proposed algorithm in decreasing the total cost can be verified as shown in Fig. 6.

4.3. Case study 3

This case involves a ten unit generating thermal system with valve point effects. The fuel cost coefficients, generators constraint, emission coefficients and transmission loss coefficient matrix are shown in Appendix B. Table 4 outlines the results of solving CEED for 2000 MW load demand using FPA and comparing with other algorithms [65-68]. The result of the suggested algorithm is highlighted here. The suggested FPA yields a lower cost than ABC_PSO, GSA, EMOCA, MODE, PDE, SPEA-2 and NSGA-II by 50$, 120$, 75$, 114$, 140$, 150$ and 169$ respectively while achieving the constraints of system. Also, its emission is also lower than SPEA-2, PDE, GSA, EMOCA, ABC_PSO, MODE and NSGA-II. Thus, FPA succeeds in achieving the global minimum solution. Moreover, the CPU time is smaller than other algorithm. Hence, FPA outperforms and outlasts other algorithms in reducing the net cost with minimum time. In addition, the cost convergence for this demand is given in Fig. 7. The objective function is convergent after 5 iterations. Finally, the total cost for every algorithm is given in Fig. 8.

4.4. Case study 4

This test system consists of forty generating units with non-smooth fuel cost and emission functions. Unit data and loss coefficients have been found in Appendix B. Table 5 summarizes the results of solving CEED for 10,500 MW load demand using FPA and

Table 5

CEED comparison for 40 generators at load of 10,500 MW.

Outputs MODE [65] PDE [65] NSGA-II [65] SPEA-2 [65] GSA [66] MABC/D/Cat [69] MABC/D/Log [69] Proposed

P1 (MW) 113.5295 112.1549 113.8685 113.9694 113.9989 110.7998 110.7998 43.405

P2 (mw) 114 113.9431 113.6381 114 113.9896 110.7998 110.7998 113.95

P3 (mw) 120 120 120 119.8719 119.9995 97.3999 97.3999 105.86

P4 (MW) 179.8015 180.2647 180.7887 179.9284 179.7857 174.5504 174.5486 169.65

P5 (MW) 96.7716 97 97 97 97 87.7999 97 96.659

P6 (mw) 139.2760 140 140 139.2721 139.0128 105.3999 105.3999 139.02

P7 (mw) 300 299.8829 300 300 299.9885 259.5996 259.5996 273.28

P8 (mw) 298.9193 300 299.0084 298.2706 300 284.5996 284.5996 285.17

P9 (mw) 290.7737 289.8915 288.8890 290.5228 296.2025 284.5996 284.5996 241.96

P10(MW) 130.9025 130.5725 131.6132 131.4832 130.3850 130 130 131.26

P11 (MW) 244.7349 244.1003 246.5128 244.6704 245.4775 318.1921 318.2129 312.13

P12 (mw) 317.8218 318.2840 318.8748 317.2003 318.2101 243.5996 243.5996 362.58

P13(mw) 395.3846 394.7833 395.7224 394.7357 394.6257 394.2793 394.2793 346.24

P14 (mw) 394.4692 394.2187 394.1369 394.6223 395.2016 394.2793 394.2793 306.06

P15 (mw) 305.8104 305.9616 305.5781 304.7271 306.0014 394.2793 394.2793 358.78

P16 (mw) 394.8229 394.1321 394.6968 394.7289 395.1005 394.2793 394.2793 260.68

P17 (mw) 487.9872 489.3040 489.4234 487.9857 489.2569 399.5195 399.5195 415.19

P18 (mw) 489.1751 489.6419 488.2701 488.5321 488.7598 399.5195 399.5195 423.94

P19 (mw) 500.5265 499.9835 500.8 501.1683 499.2320 506.1985 506.1716 549.12

P20 (mw) 457.0072 455.4160 455.2006 456.4324 455.2821 506.1985 506.2206 496.7

P21 (Mw) 434.6068 435.2845 434.6639 434.7887 433.4520 514.1472 514.1105 539.17

P22 (Mw) 434.5310 433.7311 434.15 434.3937 433.8125 514.1455 514.1472 546.46

P23 (Mw) 444.6732 446.2496 445.8385 445.0772 445.5136 514.5237 514.5664 540.06

P24 (Mw) 452.0332 451.8828 450.7509 451.8970 452.0547 514.5386 514.4868 514.5

P25 (Mw) 492.7831 493.2259 491.2745 492.3946 492.8864 433.5196 433.5195 453.46

P26 (Mw) 436.3347 434.7492 436.3418 436.9926 433.3695 433.5195 433.5196 517.31

P27 (Mw) 10 11.8064 11.2457 10.7784 10.0026 10 10 14.881

P28 (Mw) 10.3901 10.7536 10 10.2955 10.0246 10 10 18.79

P29 (Mw) 12.3149 10.3053 12.0714 13.7018 10.0125 10 10 26.611

P30 (Mw) 96.9050 97. 97 96.2431 96.9125 97 87.8042 59.581

P31 (Mw) 189.7727 190.0000 189.4826 190.0000 189.9689 159.733 159.733 183.48

P32 (Mw) 174.2324 175.3065 174.7971 174.2163 175 159.733 159.7331 183.39

P33 (Mw) 190 190 189.2845 190 189.0181 159.733 159.733 189.02

P34 (Mw) 199.6506 200 200 200 200 200 200 198.73

P35 (Mw) 199.8662 200 199.9138 200 200 200 200 198.77

P36 (Mw) 200 200 199.5066 200 199.9978 200 200 182.23

P37 (Mw) 110 109.9412 108.3061 110 109.9969 89.1141 89.1141 39.673

P38 (Mw) 109.9454 109.8823 110 109.6912 109.0126 89.1141 89.1141 81.596

P39 (Mw) 108.1786 108.9686 109.7899 108.5560 109.4560 89.1141 89.1141 42.96

P40 (Mw) 422.0628 421.3778 421.5609 421.8521 421.9987 506.1879 506.1951 537.17

Total cost * 105 $ 1.2579 1.2573 1.2583 1.2581 1.2578 1.24490903 1.24491161 1.23170

Emission * 105 ton 2.1119 2.1177 2.1095 2.1110 2.1093 2.56560267 2.56560267 2.0846

CPU (s) 5.39 6.15 7.32 8.57 NA NA NA 4.92

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M 1.29

'J 1.28

O H 1. 27

10 20 30 40 50 60 70 80 90 100 Iterations

Fig. 9. Change of objective function with iterations for forty units.

comparing with MODE, PDE, NSGA-II, SPEA-2 [65], GSA [66], MABC/ D/Cat [69] and MABC/D/Log [69]. The result of the suggested algorithm yields to a lower fuel cost than others as shown in Table 5. Therefore, these algorithms have trapped in local minimum solutions. On the other hand, the objective function representing the total cost decreases gradually and converges after 18 iterations as given in Fig. 9. Moreover, the average CPU time of the proposed FPA is the smallest one compared with other algorithms. The superiority of the proposed FPA in reaching the global minimum cost is detected by examining Fig. 10.

4.5. Comparison and discussion

The superiority of the proposed FPA is investigated here by comparison with other optimization algorithms in terms of economic effects and computation efficiency.

4.5.1. Economic effects

As seen in Figs. 3, 6, 8 and 10, the proposed FPA can get the best solution among other algorithms in the literatures. From Table 2, it is obvious that the mean cost value obtained by the proposed FPA is comparatively less compared with other algorithms. Therefore, the proposed FPA can result in better economic effects than other algorithms. Moreover, it leads to higher quality solution than other algorithms.

4.5.2. Convergence property and computation efficiency

From Figs. 4, 5, 7 and 9, one can get that the descending speeds at the beginning are high; this indicates the high convergence of the proposed algorithm based on evolution search. FPA can be convergent quickly and get the optimum results in very small iteration

Algorithms Fig. 10. Total cost for various algorithms for case 4.

numbers. It is confirmed to have a good convergence property. As seen in Tables 1-5, CPU times of the proposed FPA are smaller than other algorithms since FPA has only one key parameter. Thus, it can get better computation efficiency than other algorithms.

5. Conclusions

In this paper, FPA has been developed to solve ELD and CEED problems in power systems. The performance of the FPA was tested for various test cases and compared with the reported cases in recent literatures. The superiority of FPA over other algorithms for settling ELD and CEED problems even for large scale power system with valve point effect is confirmed. Moreover, the economic effect, computation efficiency and convergence property of FPA are demonstrated. Therefore FPA optimization is a promising technique for solving complicated problems in power systems. Applications of the proposed algorithm to multi-area power system integrated with wind farms and PV system are the future scope of this work.

Appendix A

[a] Parameters of FPA for case 40 generators: Maximum number of iterations = 500, population size = 20, probability switch = 0.8.

[b] Parameters of FPA for case 3, 10 generators: Maximum number of iterations = 500, population size = 25, probability switch = 0.75.

Appendix B

See Tables B1-B3 and the transmission line losses coefficient. Table B1

Generator cost coefficients for the three unit system considering emission.

Unit Y ß a a b c pmn (MW) pmax (MW)

$/MW2h $/MWh $/h (Kg/MW2h) (Kg/MWh) (Kg/h)

1 0.03546 38.30553 1243.5311 0.00683 -0.54551 40.2669 35 210

2 0.02111 36.32782 1658.5696 0.00461 -0.5116 42.89553 130 325

3 0.01799 38.27041 1356.6592 0.00461 -0.5116 42.89553 125 315

The transmission line losses coefficient of three units system. f 0.71 0.3 0.251

B„ = 0.0001»

0.3 0.255

0.69 0.32 0.32 0.8

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Table B2

Ten unit generator characteristics.

Unit Y ß a e f pmn (MW) Pmax a b c n S

$/MW2h ($/MWh) ($/h) ($/h) (rad/MW) (MW) (lb/MW2h) (lb/MWh) (lb/h) (lb/h) (1/MW)

P1 0.12951 40.5407 1000.403 33 0.0174 10 55 0.04702 -3.9864 360.0012 0.25475 0.01234

P2 0.10908 39.5804 950.606 25 0.0178 20 80 0.04652 -3.9524 350.0056 0.25475 0.01234

P3 0.12511 36.5104 900.705 32 0.0162 47 120 0.04652 -3.9023 330.0056 0.25163 0.01215

P4 0.12111 39.5104 800.705 30 0.0168 20 130 0.04652 -3.9023 330.0056 0.25163 0.01215

P5 0.15247 38.539 756.799 30 0.0148 50 160 0.0042 0.3277 13.8593 0.2497 0.012

P6 0.10587 46.1592 451.325 20 0.0163 70 240 0.0042 0.3277 13.8593 0.2497 0.012

P7 0.03546 38.3055 1243.531 20 0.0152 60 300 0.0068 -0.5455 40.2669 0.248 0.0129

P8 0.02803 40.3965 1049.998 30 0.0128 70 340 0.0068 -0.5455 40.2669 0.2499 0.01203

P9 0.02111 36.3278 1658.569 60 0.0136 135 470 0.0046 -0.5112 42.8955 0.2547 0.01234

P10 0.01799 38.2704 1356.659 40 0.0141 150 470 0.0046 -0.5112 42.8955 0.2547 0.01234

The transmission line losses coefficient of ten units system.

0.49 0.14 0.15 0.15 0.16 0.17 0.17 0.18 0.19 0.20

0.14 0.45 0.16 0.16 0.17 0.15 0.15 0.16 0.18 0.18

0.15 0.16 0.39 0.10 0.12 0.12 0.14 0.14 0.16 0.16

0.15 0.16 0.10 0.40 0.14 0.10 0.11 0.12 0.14 0.15

0.16 0.17 0.12 0.14 0.35 0.11 0.13 0.13 0.15 0.16

0.17 0.15 0.12 0.10 0.11 0.36 0.12 0.12 0.14 0.15

0.17 0.15 0.14 0.11 0.13 0.12 0.38 0.16 0.16 0.18

0.18 0.16 0.14 0.12 0.13 0.12 0.16 0.40 0.15 0.16

0.19 0.18 0.16 0.14 0.15 0.14 0.16 0.15 0.42 0.19

0.20 0.18 0.16 0.15 0.16 0.15 0.18 0.16 0.19 0.44

B, = 0.0001

Table B3

Forty unit generator characteristics.

Unit pmin (MW) pmax (MW) a $/h ß $/MWh Y $/MW2h e ($/h) f (rad/MW) c (lb/h) b (lb/MWh) a (lb/MW2h) n (lb/h) S (1/MW)

P1 36 114 94.705 6.73 0.00690 100 0.084 60 -2.22 0.0480 1.3100 0.05690

P2 36 114 94.705 6.73 0.00690 100 0.084 60 -2.22 0.0480 1.3100 0.05690

P3 60 120 309.540 7.07 0.02028 100 0.084 100 -2.36 0.0762 1.3100 0.05690

P4 80 190 369.030 8.18 0.00942 150 0.063 120 -3.14 0.0540 0.9142 0.04540

P5 47 97 148.890 5.35 0.01140 120 0.077 50 -1.89 0.0850 0.9936 0.04060

P6 68 140 222.330 8.05 0.01142 100 0.084 80 -3.08 0.0854 1.3100 0.05690

P7 110 300 287.710 8.03 0.00357 200 0.042 100 -3.06 0.0242 0.6550 0.02846

P8 135 300 391.980 6.99 0.00492 200 0.042 130 -2.32 0.0310 0.6550 0.02846

P9 135 300 455.760 6.60 0.00573 200 0.042 150 -2.11 0.0335 0.6550 0.02846

P10 130 300 722.820 12.9 0.00605 200 0.042 280 -4.34 0.4250 0.6550 0.02846

P11 94 375 635.200 12.9 0.00515 200 0.042 220 -4.34 0.0322 0.6550 0.02846

P12 94 375 654.690 12.8 0.00569 200 0.042 225 -4.28 0.0338 0.6550 0.02846

P13 125 500 913.400 12.5 0.00421 300 0.035 300 -4.18 0.0296 0.5035 0.02075

P14 125 500 1760.400 8.84 0.00752 300 0.035 520 -3.34 0.0512 0.5035 0.02075

P15 125 500 1760.400 8.84 0.00752 300 0.035 510 -3.55 0.0496 0.5035 0.02075

P16 125 500 1760.400 8.84 0.00752 300 0.035 510 -3.55 0.0496 0.5035 0.02075

P17 220 500 647.850 7.97 0.00313 300 0.035 220 -2.68 0.0151 0.5035 0.02075

P18 220 500 649.690 7.95 0.00313 300 0.035 222 -2.66 0.0151 0.5035 0.02075

P19 242 550 647.830 7.97 0.00313 300 0.035 220 -2.68 0.0151 0.5035 0.02075

P20 242 550 647.810 7.97 0.00313 300 0.035 220 -2.68 0.0151 0.5035 0.02075

P21 254 550 785.960 6.63 0.00298 300 0.035 290 -2.22 0.0145 0.5035 0.02075

P22 254 550 785.960 6.63 0.00298 300 0.035 285 -2.22 0.0145 0.5035 0.02075

P23 254 550 794.530 6.66 0.00284 300 0.035 295 -2.26 0.0138 0.5035 0.02075

P24 254 550 794.530 6.66 0.00284 300 0.035 295 -2.26 0.0138 0.5035 0.02075

P25 254 550 801.320 7.10 0.00277 300 0.035 310 -2.42 0.0132 0.5035 0.02075

P26 254 550 801.320 7.10 0.00277 300 0.035 310 -2.42 0.0132 0.5035 0.02075

P27 10 150 1055.100 3.33 0.52124 120 0.077 360 -1.11 1.8420 0.9936 0.04060

P28 10 150 1055.100 3.33 0.52124 120 0.077 360 -1.11 1.8420 0.9936 0.04060

P29 10 150 1055.100 3.33 0.52124 120 0.077 360 -1.11 1.8420 0.9936 0.04060

P30 47 97 148.890 5.35 0.01140 120 0.077 50 -1.89 0.0850 0.9936 0.04060

P31 60 190 222.920 6.43 0.00160 150 0.063 80 -2.08 0.0121 0.9142 0.04540

P32 60 190 222.920 6.43 0.00160 150 0.063 80 -2.08 0.0121 0.9142 0.04540

P33 60 190 222.920 6.43 0.00160 150 0.063 80 -2.08 0.0121 0.9142 0.04540

P34 90 200 107.870 8.95 0.00010 200 0.042 65 -3.48 0.0012 0.6550 0.02846

P35 90 200 116.580 8.62 0.00010 200 0.042 70 -3.24 0.0012 0.6550 0.02846

P36 90 200 116.580 8.62 0.00010 200 0.042 70 -3.24 0.0012 0.6550 0.02846

P37 25 110 307.450 5.88 0.01610 80 0.098 100 -1.98 0.0950 1.4200 0.06770

P38 25 110 307.450 5.88 0.01610 80 0.098 100 -1.98 0.0950 1.4200 0.06770

P39 25 110 307.450 5.88 0.01610 80 0.098 100 -1.98 0.0950 1.4200 0.06770

P40 242 550 647.830 7.97 0.00313 300 0.035 220 -2.68 0.0151 0.5035 0.02075

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The total fuel cost of generation in $ The fuel cost function of ith generator in $ The cost coefficients of ith generator in $/MW2, $/MW and $ respectively

The real power generation of ith generator in MW The number of generators connected in the network The total load of the system in MW The transmission losses of the system in MW The real power injections at ith and jth buses respectively

The loss-coefficients of transmission loss formula The minimum and maximum values of real power allowed at generator i

The coefficients of ith generator due to valve point effect in $ and MW-1 respectively The optimal cost of total generation and emission The total fuel cost and total emission of generators respectively

The emission coefficients of generators in Kg/ MW2, Kg/MW and Kg respectively The emission coefficients of ith generator in Ton and MW-1 respectively

The price penalty factor value in $/Kg The pollen i

The current best solution found at the current generation

The scaling factor controlling the step size The standard gamma function Switch probability

Nomenclature

Fi (Pi )

Yi, ßi, ai

Pi d Pd Pl

Pi, Pj

Bij , B0i, B00 pmin pmax

et, fi F

Fi(Pi),Ei(Pt) a, b, c П, St

Г(А) P

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