Abstract

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.

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 populations 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

Keywords

Flower pollination algorithm ; Economic load dispatch ; Combined economic emission dispatch ; Emission constraints ; Valve point loading effect ; Swarm intelligence

Nomenclature

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 generator in $

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 generator in $/MW2 , $/MW and $ respectively

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 generator in MW

d- The number of generators connected in the network

PD- The total load of the system in MW

PL- The transmission losses of the system in MW

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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_j}

- The real power injections at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i^{th}} 

 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle j^{th}}
 buses respectively

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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_{0i}}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_{00}}

- The loss-coefficients of transmission loss formula

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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_i^{max}}

- The minimum and maximum values of real power allowed at generator i

ei , fi- The coefficients of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i^{th}} 

 generator due to valve point effect in $ and MW−1  respectively

F- The optimal cost of total generation and emission

Fi (Pi ),Ei (Pi ) - The total fuel cost and total emission of generators respectively

a , b , c- The emission coefficients of generators in Kg/MW2 , Kg/MW and Kg respectively

ηi , δi- The emission coefficients of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i^{th}} 

 generator in Ton and MW−1  respectively

h- The price penalty factor value in $/Kg

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g * - The current best solution found at the current generation

γ- The scaling factor controlling the step size

Γ(λ ) - The standard gamma function

p- Switch probability

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]  and [3] . Well known long-established techniques such as gradient method [4] , lambda iteration method [5]  and [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] , [12] , [13]  and [14] . However, it requests more fine tuning and simulation before operational. Another technique like Artificial Neural Network (ANN) 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] , [16]  and [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] , [26] , [27]  and [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.

2.1. Objective function of ELD

For thermal generating units, the cost of fuel per unit power output varies significantly with the output power of the unit. Fuel costs are usually represented as a quadratic function of output power [31] , as shown in equation (1) .

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( 1)

Minimize

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( 2)

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

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( 3)

The total transmission loss using Krons loss formula is given in equation (4)

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( 4)

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

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( 5)

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] .

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( 6)


Valve point effect.


Fig. 1.

Valve point effect.

2.3. Objective function of CEED

The atmospheric pollutants such as sulfur oxides, nitrogen oxides and carbon dioxide caused by fossil fuel fired generator can be modeled separately [34] , [35]  and [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]  and [38] :

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( 7)

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] .

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( 8)

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] .

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( 9)

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]  and [33] , blends the emission with fuel cost, then F is the total operating cost in $.

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( 10)

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

  • Find the ratio between maximum fuel cost and maximum emission of each generator.
  • Arrange the values of price penalty factor in ascending order.
  • Add the maximum capacity of each unit (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_i^{max}}
) one at a time, starting from the smallest hi                                            , until Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sum P_i^{max}\geq P_D}
.
  • At this point, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle h_i}
 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, inhomogeneity 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 flowering 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 Lévy 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.

Step 4: The interaction of local pollination and global pollination is controlled by a switch probability p  ∈ [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:

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( 11)

In fact, L (λ ) is the Lévy 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 Lévy flight can be used to mimic this behavior efficiently. That is, L  > 0 from a Lévy distribution.

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( 12)

This distribution is valid for large steps s  > 0.

For the local pollination, both Step 2 and Step 3 can be represented as

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( 13)

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_j^t} 

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 are pollen from different flowers of the same plant species mimicking the flower constancy in a limited neighborhood. For a local random walk, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_j^t}
 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle x_k^t}
 come from the same species, then ε  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 .


Flow chart of FPA.


Fig. 2.

Flow chart of FPA.

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 is taken into account to complete the analysis [45] , [46] , [47] , [48] , [49] , [50] , [51] , [52] , [53] , [54]  and [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, 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] , [48] , [49] , [50] , [51] , [52] , [53] , [54] , [55] , [56] , [57] , [58] , [59] , [60] , [61] , [62] , [63]  and [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.

Table 1. ELD Comparison for 40 generators at load of 10,500 MW.
Outputs PSO [[#bib0230|[45]]] PPSO [[#bib0230|[45]]] APPSO [[#bib0230|[45]]] MPSO [[#bib0235|[46]]] ARCGA [[#bib0240|[47]]] TSAGA [[#bib0245|[48]]] CCPSO [[#bib0250|[49]]] CDE_SQP [[#bib0255|[50]]] EDA/DE [[#bib0260|[51]]] SOMA [[#bib0265|[52]]] CSOMA [[#bib0265|[52]]] DE/BBO [[#bib0270|[53]]] DHS [[#bib0275|[54]]] ICA-PSO [[#bib0280|[55]]] Proposed FPA
P1 (MW) 113.116 111.601 112.579 113.9971 110.8252 114.0000 110.7998 111.7600 111.1110 112.8544 110.8016 110.7998 110.7998 110.8 72.4810
P2 (MW) 113.010 111.781 111.553 112.6517 113.9112 111.0400 110.7999 111.5600 110.8299 111.7795 110.8068 110.7998 110.7998 110.8 103.0314
P3 (MW) 119.702 118.613 98.751 119.4255 97.4000 97.3000 97.3999 97.3900 97.4122 97.4059 97.4007 97.3999 97.3999 97.41 83.2726
P4 (MW) 81.647 179.819 180.384 189.0000 179.7331 179.6000 179.7331 179.7300 179.7443 179.7274 179.7333 179.7331 179.7331 179.74 182.3106
P5 (MW) 95.062 92.443 94.389 96.8711 88.6454 90.7210 87.7999 91.6600 88.1510 87.9306 87.8180 87.9576 87.7999 88.52 76.1669
P6 (MW) 139.209 139.846 139.943 139.2798 140.0000 140.0000 140.0000 140.0000 139.9959 139.9880 139.9997 140.0000 140.0000 140.00 126.1346
P7 (MW) 299.127 296.703 298.937 223.5924 259.6000 260.0600 259.5997 300.0000 259.6065 259.7736 259.6010 259.5997 259.5997 259.60 258.8452
P8 (MW) 287.491 284.566 285.827 284.5803 284.6000 285.8700 284.5997 300.0000 284.6045 284.6280 284.6000 284.5997 284.5997 284.60 297.1636
P9 (MW) 292.316 285.164 298.381 216.4333 284.6000 284.7700 284.5997 284.5900 284.6149 284.7539 284.6005 284.5997 284.5997 284.60 290.8899
P10 (MW) 279.273 203.859 130.212 239.3357 130.0000 130.0000 130.0000 130.0000 130.0002 130.0291 130.0003 130.0000 130.0000 130.00 274.8232
P11 (MW) 169.766 94.283 94.385 314.8734 168.7985 94.0000 94.0000 168.7900 168.8029 168.7908 168.7999 168.7998 94.0000 168.80 356.9806
P12 (MW) 94.344 94.090 169.583 305.0565 168.7994 168.3800 94.0000 94.0000 94.0000 168.8084 168.7999 94.0000 94.0000 94.00 124.4054
P13 (MW) 214.871 304.830 214.617 365.5429 214.7600 214.4500 214.7598 214.7600 214.7591 214.7191 214.7599 214.7598 214.7598 214.76 493.3764
P14 (MW) 304.790 304.173 304.886 493.3729 394.2800 394.0100 394.2794 394.2800 394.2716 394.2888 394.2794 394.2794 394.2794 394.28 344.9029
P15 (MW) 304.563 304.467 304.547 280.4326 304.5200 394.2700 394.2794 304.5200 304.5206 304.5196 304.5196 394.2794 394.2794 394.28 372.3864
P16 (MW) 304.302 304.177 304.584 432.0717 394.2800 304.5700 394.2794 304.5200 394.2834 394.2952 394.2794 394.2794 394.2794 304.52 345.4624
P17 (MW) 489.173 489.544 498.452 435.2428 489.2798 489.2800 489.2794 489.2800 489.2912 489.2905 489.2796 489.2794 489.2794 489.28 422.6378
P18 (MW) 491.336 489.773 497.472 417.6958 489.2800 489.5600 489.2794 489.2800 489.2877 489.2779 489.2795 489.2794 489.2794 489.28 434.4065
P19 (MW) 510.880 511.280 512.816 532.1877 511.2806 511.2900 511.2794 511.2800 511.2977 511.2861 511.2794 511.2794 511.2794 511.28 461.3107
P20 (MW) 511.474 510.904 548.992 409.2053 511.2800 511.2700 511.2794 511.2800 511.2791 511.2792 511.2796 511.2794 511.2794 511.28 434.3828
P21 (MW) 524.814 524.092 524.652 534.0629 523.2803 523.2300 523.2794 523.2800 523.2958 523.2858 523.2797 523.2794 523.2794 523.28 545.2846
P22 (MW) 524.775 523.121 523.399 457.0962 523.2800 523.6300 523.2794 523.2900 523.2849 523.2899 523.2798 523.2794 523.2794 523.28 490.3572
P23 (MW) 525.563 523.242 548.895 441.3634 523.2800 523.8200 523.2794 523.2800 523.2856 523.2783 523.2801 523.2794 523.2794 523.28 506.0639
P24 (MW) 522.712 524.260 525.871 397.3617 523.2800 523.6200 523.2794 523.2800 523.2979 523.3199 523.2795 523.2794 523.2794 523.28 467.3109
P25 (MW) 503.211 523.283 523.814 446.4181 523.2800 523.3300 523.2794 523.2800 523.2799 523.2791 523.2797 523.2794 523.2794 523.28 488.1203
P26 (MW) 524.199 523.074 523.565 442.1164 523.2801 523.6800 523.2794 523.2800 523.2910 523.3076 523.2799 523.2794 523.2794 523.28 486.9019
P27 (MW) 10.082 10.800 10.575 74.8622 10.0000 10.0000 10.0000 10.0000 10.0064 10.0021 10.0004 10.0000 10.0000 10.00 16.8002
P28 (MW) 10.663 10.742 11.177 27.5430 10.0000 10.0000 10.0000 10.0000 10.0018 10.0054 10.0004 10.0000 10.0000 10.00 39.3475
P29 (MW) 10.418 10.799 11.210 76.8314 10.0000 10.1600 10.0000 10.0000 10.0000 10.0061 10.0003 10.0000 10.0000 10.00 23.6359
P30 (MW) 94.244 94.475 96.178 97.0000 88.7611 87.8700 87.8000 90.3300 96.2132 88.8932 92.7158 97.0000 87.7999 96.39 86.3295
P31 (MW) 189.377 189.245 189.999 118.3775 190.0000 190.0000 190.0000 190.0000 189.9996 189.9975 189.9998 190.0000 190.0000 190.00 165.9924
P32 (MW) 189.796 189.995 189.924 188.7517 190.0000 190.0000 190.0000 190.0000 189.9998 189.9919 189.9998 190.0000 190.0000 190.00 174.5707
P33 (MW) 189.813 188.081 189.714 190.0000 190.0000 190.0000 190.0000 190.0000 189.9981 189.9825 189.9998 190.0000 190.0000 190.00 184.0570
P34 (MW) 199.797 198.475 199.284 120.7029 164.8000 165.2300 164.7998 200.0000 164.9126 164.9291 164.8014 164.7998 164.7998 164.82 193.6668
P35 (MW) 199.284 197.528 199.599 170.2403 164.8000 200.0000 194.3976 200.0000 199.9941 164.8031 164.8015 200.0000 200.0000 200.00 191.6152
P36 (MW) 198.165 196.971 199.751 198.9897 164.8054 200.0000 200.0000 200.0000 200.0000 164.9387 164.8051 200.0000 194.3978 200.00 196.1763
P37 (MW) 109.291 109.161 109.973 110.0000 110.0000 110.0000 110.0000 110.0000 109.9988 109.9974 109.9998 100.0000 110.0000 110.00 90.0101
P38 (MW) 109.087 109.900 109.506 109.3405 110.0000 110.0000 110.0000 110.0000 109.9994 109.9856 109.9998 110.0000 110.0000 110.00 37.5421
P39 (MW) 109.909 109.855 109.363 109.9243 110.0000 110.0000 110.0000 110.0000 109.9974 109.9995 109.9996 110.0000 110.0000 110.00 89.4239
P40 (MW) 512.348 510.984 511.261 468.1694 511.2800 510.9800 511.2794 511.2800 511.2800 511.2813 511.2797 511.2794 511.2794 511.28 471.4405
Fuel cost * 105 $ 1.22323 1.21788 1.220446 1.216492 1.214101 1.214630 1.214035 1.217419 1.21412 1.214187 1.214147 1.214208 1.214035 1.214132 1.210745

Table 2. Statistical comparison between FPA and different algorithms.
Algorithm Best cost ($) Mean cost ($) Worst cost ($) Time (s)
ARCGA [[#bib0240|[47]]] 121410.1038 121462.1502 121536.8745 15.67
TSAGA [[#bib0245|[48]]] 121463.07 122928.31 124296.54 696.01
CCPSO [[#bib0250|[49]]] 121403.5362 121445.3269 121535.4934 19.3
CDE_SQP [[#bib0255|[50]]] 121741.9793 122295.1278 122839.2941 14.26
EDA/DE [[#bib0260|[51]]] 121412.50 121460.70 121517.80 NA
BBO [[#bib0265|[52]]] 121426.66 121508.03 121688.66 NA
SOMA [[#bib0265|[52]]] 121418.7856 121449.8796 121508.3757 NA
CSOMA [[#bib0265|[52]]] 121414.6978 121415.0479 121417.8045 NA
DE/BBO [[#bib0270|[53]]] 121420.89 121420.90 121420.90 60.00
DHS [[#bib0275|[54]]] 121403.5355 121410.5967 121417.2274 1.32
ICA-PSO [[#bib0280|[55]]] 121413.2 121428.14 121453.56 139.92
EP [[#bib0285|[56]]] 122624.35 123382.00 125740 1167.35
EP–SQP [[#bib0285|[56]]] 122323.97 122379.63 NA 997.73
PSO [[#bib0285|[56]]] 123930.45 124154.49 NA 933.39
PSO–SQP [[#bib0285|[56]]] 122094.67 122245.25 NA 733.97
GA–PS–SQP [[#bib0290|[57]]] 121458 122039 NA 46.98
CPSO [[#bib0295|[58]]] 121865.23 122100.87 NA 114.65
CPSO–SQP [[#bib0295|[58]]] 121458.54 122028.16 NA 98.49
NPSO_LRS [[#bib0300|[59]]] 121664.4308 122209.3185 122981.5913 16.81
APSO [[#bib0305|[60]]] 121663.5222 122153.6730 122912.3958 5.05
DE [[#bib0310|[61]]] 121416.29 121422.72 121431.47 NA
CDEMD [[#bib0315|[62]]] 121423.4013 121526.7330 121696.9868 44.3
HMAPSO [[#bib0320|[63]]] 121586.90 121586.90 21586.90 NA
FAPSO-NM [[#bib0325|[64]]] 121418.3 121418.803 121419.8 40
FPA 121074.5 121095.7 121196.3 0.89


Fuel cost for various algorithms for case 1.


Fig. 3.

Fuel cost for various algorithms for case 1.


Objective function for forty unit system.


Fig. 4.

Objective function for forty unit system.

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 .

Table 3. Results for the best simulations with 3-unit system considering emission.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_D} h Power outputs GA [[#bib0195|[38]]] PSO [[#bib0195|[38]]] FPA
400 (MW) 43.55981 P1 (MW) 102.617 102.612 102.4468
P2 (MW) 153.825 153.809 153.8341
P3 (MW) 151.011 150.991 151.1321
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P_L} 
 (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.


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


Fig. 5.

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


Total cost for various algorithms with demand = 400 MW.


Fig. 6.

Total cost for various algorithms with demand = 400 MW.

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] , [66] , [67]  and [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 .

Table 4. CEED comparison for ten unit system at demand of 2000 MW.
Outputs MODE [[#bib0330|[65]]] NSGAII [[#bib0330|[65]]] PDE [[#bib0330|[65]]] SPEA-2 [[#bib0330|[65]]] GSA [[#bib0335|[66]]] ABC_PSO [[#bib0340|[67]]] EMOCA [[#bib0345|[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


Change of objective function with iterations for ten units.


Fig. 7.

Change of objective function with iterations for ten units.


Total cost for various algorithms for case 3.


Fig. 8.

Total cost for various algorithms for case 3.

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 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 .

Table 5. CEED comparison for 40 generators at load of 10,500 MW.
Outputs MODE [[#bib0330|[65]]] PDE [[#bib0330|[65]]] NSGA-II [[#bib0330|[65]]] SPEA-2 [[#bib0330|[65]]] GSA [[#bib0335|[66]]] MABC/D/Cat [[#bib0350|[69]]] MABC/D/Log [[#bib0350|[69]]] Proposed FPA
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


Change of objective function with iterations for forty units.


Fig. 9.

Change of objective function with iterations for forty units.


Total cost for various algorithms for case 4.


Fig. 10.

Total cost for various algorithms for case 4.

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 Fig. 3 , Fig. 6 , Fig. 8  and Fig. 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 Fig. 4 , Fig. 5 , Fig. 7  and Fig. 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 numbers. It is confirmed to have a good convergence property. As seen in Table 1 , Table 2 , Table 3 , Table 4  and Table 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

  • Parameters of FPA for case 40 generators: Maximum number of iterations = 500, population size = 20, probability switch = 0.8.
  • Parameters of FPA for case 3, 10 generators: Maximum number of iterations = 500, population size = 25, probability switch = 0.75.

Appendix B

See Table B1 , Table B2  and Table B3 and the transmission line losses coefficient.

Table B1. Generator cost coefficients for the three unit system considering emission.
Unit γ $/MW2 h β $/MWh α $/h a (Kg/MW2 h) b (Kg/MWh) c (Kg/h) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^{min}} 
(MW)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^{max}} 
(MW)
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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_{ij}=0.0001{_\ast}\left[\begin{array}{ccc} 0.71 & 0.3 & 0.25\\ 0.3 & 0.69 & 0.32\\ 0.255 & 0.32 & 0.8 \end{array}\right],} 

.

Table B2. Ten unit generator characteristics.
Unit γ $/MW2 h β ($/MWh) α ($/h) e ($/h) f (rad/MW) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^{min}} 
(MW)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^{max}} 
(MW)
a (Ib/MW2 h) b (Ib/MWh) c (Ib/h) η (Ib/h) δ (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.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B_{ij}=0.0001{_\ast}\left[\begin{array}{cccccccccc} 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\mbox{ } & 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\mbox{ } & 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 \end{array}\right]} 

.

Table B3. Forty unit generator characteristics.
Unit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^{min}} 
(MW)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^{max}} 
(MW)
α $/h β $/MWh γ $/MW2 h e ($/h) f (rad/MW) c (Ib/h) b (Ib/MWh) a (Ib/MW2 h) η (Ib/h) δ (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

References

  1. [1] A. Chakrabarti, S. Halder; Power System Analysis: Operation and Control; Prentice-Hall of India, New Delhi (2006)
  2. [2] D.P. Kothari, J.S. Dhillon; Power System Optimization; (2nd ed.)PHI Learning Pvt. Ltd, New Delhi (2006)
  3. [3] G. Aydin; Modeling of energy consumption based on economic and demographic factors: the case of Turkey with projections; Renew. Sustain. Energy Rev, 35 (2014), pp. 382–389
  4. [4] J.C. Dodu, P. Martin, A. Merlin, J. Pouget; An optimal formulation and solution of short-range operating problems for a power system with flow constraints; Proc. IEEE, 60 (1) (1972), pp. 54–63
  5. [5] C.L. Chen, S.C. Wang; Branch and bound scheduling for thermal generating units; IEEE Trans. Energy Convers, 8 (2) (1993), pp. 184–189
  6. [6] P. Aravindhababu, K.R. Nayar; Economic dispatch based on optimal lambda using radial basis function network; Int. J. Electr. Power Energy Syst, 24 (7) (2002), pp. 551–556
  7. [7] J. Parikh, D. Chattopadhyay; A multi-area linear programming approach for analysis of economic operation of the Indian power system; IEEE Trans. Power Syst, 11 (1) (1996), pp. 52–58
  8. [8] J.Y. Fan, L. Zhang; Real-time economic dispatch with line flow and emission constraints using quadratic programming; IEEE Trans. Power Syst, 13 (2) (1998), pp. 320–325
  9. [9] J. Nanda, L. Hari, M.L. Kothari; Economic emission dispatch with line flow constraints using a classical technique; IEE Proc. Gener. Trans. Distrib, 141 (1) (1994), pp. 1–10
  10. [10] A.A. El-Keib, H. Ma, J.L. Hart; Environmentally constrained economic dispatch using the Lagrangian relaxation method; IEEE Trans. Power Syst, 9 (4) (1994), pp. 1723–1729
  11. [11] Y.S. Brar, J.S. Dhillon, D.P. Kothari; Multiobjective load dispatch by fuzzy logic searching weightage pattern; Electr. Power Syst. Res, 63 (2002), pp. 149–160
  12. [12] C.A. Roa-Sepulveda, M. Herrera, B. Pavez-Lazo, U.G. Knight, A.H. Coonick; Economic dispatch using fuzzy decision trees; Electr. Power Syst. Res, 66 (2) (2003), pp. 115–122
  13. [13] P. Attaviriyanupap, H. Kita, E. Tanaka, J. Hasegawa; A fuzzy-optimization approach to dynamic economic dispatch considering uncertainties; IEEE Trans. Power Syst, 19 (3) (2004), pp. 1299–1307
  14. [14] L. Singh, J.S. Dhillon; Fuzzy satisfying multiobjective thermal power dispatch based on surrogate worth trade-off method; Electr. Power Components Syst, 36 (1) (2008), pp. 93–108
  15. [15] C.T. Su, C.T. Lin; New approach with a Hopfield modeling framework to economic dispatch; IEEE Trans. Power Syst, 15 (2) (2000), pp. 541–545
  16. [16] A.Y. Abdelaziz, S.F. Mekhamer, M.A.L. Badr, M.Z. Kamh; Economic dispatch using an enhanced Hopfield neural network; Electr. Power Components Syst, 36 (7) (2008), pp. 719–732
  17. [17] P. Surekha, S. Sumathi; A self-adaptive fuzzy c-means based radial basis function network to solve economic load dispatch problems; Int. J. Comput. Appl, 25 (4) (2011), pp. 50–59
  18. [18] M.A. Abido; A novel multiobjective evolutionary algorithm for environmental economic power dispatch; Electr. Power Syst. Res, 65 (1) (2003), pp. 71–81
  19. [19] H. Bouzeboudja, A. Chaker, A. Alali, B. Naama; Economic dispatch solution using a real coded genetic algorithm; Acta Electrotechnica Informatica, 5 (4) (2005), pp. 1–5
  20. [20] K.P. Wong, Y.W. Wong; Genetic and genetic/simulated – annealing approaches to economic dispatch; IEE Proc. Gener. Transm. Distrib, 141 (5) (1994), pp. 507–513
  21. [21] K.P. Wong, C.C. Fong; Simulated annealing based economic dispatch algorithm; IEE Proc. C (Gener. Transm. Distrib.), 140 (6) (1993), pp. 509–515
  22. [22] N. Sinha, R. Chakrabarti, P.K. Chattopadhyay; Evolutionary programming techniques for economic load dispatch; IEEE Trans. Evol. Comput, 7 (1) (2003), pp. 83–94
  23. [23] W.M. Lin, F.S. Cheng, M.T. Tsay; An improved Tabu search for economic dispatch with multiple minima; IEEE Trans. Power Syst, 17 (1) (2002), pp. 108–112
  24. [24] J. Cai, X. Ma, L. Li, Y. Yang, H. Peng, X. Wang; Chaotic ant swarm optimization to economic dispatch; Electr. Power Syst. Res, 77 (10) (2007), pp. 1373–1380
  25. [25] J.B. Park, K. Lee, J. Shin, K.Y. Lee; A particle swarm optimization for economic dispatch with nonsmooth cost functions; IEEE Trans. Power Syst, 20 (1) (2005), pp. 34–42
  26. [26] M.R. Alrashidi, M.E. El-Hawary; A survey of particle swarm optimization applications in power system operations; Electr. Power Components Syst, 34 (12) (2006), pp. 1349–1357
  27. [27] L. Coelho, C. Lee; Solving economic load dispatch problems in power systems using chaotic and Gaussian particle swarm optimization approaches; Int. J. Electr. Power Energy Syst, 30 (5) (2008), pp. 297–307
  28. [28] D.C. Secui, I. Felea, S. Dzitac, L. Popper; A swarm intelligence approach to the power dispatch problem; Int. J. Comput. Commun. Contr, 5 (3) (2010), pp. 375–384
  29. [29] H.M. Dubey, M. Pandit, B.K. Panigrahi, M. Udgir; Economic load dispatch by hybrid swarm intelligence based gravitational search algorithm; Int. J. Intell. Syst. Appl, 5 (8) (2013), pp. 21–32
  30. [30] D.G. Prasad, D.H. Mohan, P. Manjree, B.K. Panigrahi; Economic load dispatch using artificial bee colony optimization; Int. J. Adv. Electron. Eng, 1 (1) (2011), pp. 119–124
  31. [31] A.J. Wood, B.F. Wollenberg; Power Generation Operation and Control; (2nd ed.)John Wiley and Sons, New York (1996)
  32. [32] D.C. Walters, G.B. Sheble; Genetic algorithm solution of economic dispatch with valve point loading; IEEE Trans. Power Syst, 8 (3) (1993), pp. 1325–1332
  33. [33] K. Senthil, K. Manikandan; Economic thermal power dispatch with emission constraint and valve point effect loading using improved Tabu search algorithm; Int. J. Comput. Appl, 3 (9) (2010), pp. 6–11
  34. [34] J.G. Vlachogiannis; On the profitability of load dispatch in power systems; Int. Rev. Electr. Eng, 8 (2) (2013), pp. 810–819
  35. [35] G. Aydin; The modeling of coal related CO2 emissions and projections into future planning; Energy Sources Part A Recovery Util. Environ. Eff, 36 (2) (2014), pp. 191–201
  36. [36] G. Aydin; The development and validation of regression models to predict energy-related CO2 emissions in Turkey; Energy Sources Part B Econ. Plann. Policy, 10 (2) (2015), pp. 176–182
  37. [37] M.A. Abido; Environmental/economic power dispatch using multiobjective evolutionary algorithms; IEEE Trans. Power Syst, 18 (4) (2003), pp. 1529–1537
  38. [38] A.L. Devi, O.V. Krishna; Combined economic and emission dispatch using evolutionary algorithms – a case study; ARPN J. Eng. Appl. Sci, 3 (6) (2008), pp. 28–35
  39. [39] P. Venkatesh, R. Gnanadass, N.P. Padhy; Comparison and application of evolutionary programming techniques to combined economic emission dispatch with line flow constraints; IEEE Trans. Power Syst, 18 (2) (2003), pp. 688–697
  40. [40] X.S. Yang; Flower pollination algorithm for global optimization, unconventional computation and natural computation; Lect. Notes Comput. Sci, 7445 (2012), pp. 240–249
  41. [41] X.S. Yang, M. Karamanoglu, X. He; Multi-objective flower algorithm for optimization; Procedia Comput. Sci, 18 (2013), pp. 61–68
  42. [42] A. Ochoa, S. Gonzalez, L. Margain, T. Padilla, O. Castillo, P. Melin; Implementing Flower Multi-Objective Algorithm for Selection of University Academic Credits; NaBIC  (2014), pp. 7–11
  43. [43] S. Łukasik, P.A. Kowalski; Study of Flower Pollination Algorithm for Continuous Optimization; Advances in Intelligent Systems and Computing, vol. 322, Springer International Publishing, Warsaw, Poland (2015), pp. 451–459
  44. [44] A.Y. Abdelaziz, E.S. Ali; Static VAR compensator damping controller design based on flower pollination algorithm for a multi-machine power system; Electr. Power Components Syst, 43 (11) (2015), pp. 1268–1277
  45. [45] C.H. Chen, S.N. Yeh; Particle Swarm Optimization for Economic Power Dispatch with Valve-Point Effects; IEEE PES Transmission and Distribution Conference and Exposition Latin America, Venezuela  (2006), pp. 1–5
  46. [46] Hardiansyah; A modified particle swarm optimization technique for economic load dispatch with valve-point effect; Int. J. Intell. Syst. Appl, 7 (2013), pp. 32–41
  47. [47] N. Amjady, H.N. Rad; Solution of nonconvex and nonsmooth economic dispatch by a new adaptive real coded genetic algorithm; Expert Syst. Appl, 37 (7) (2010), pp. 5239–5245
  48. [48] P. Subbaraj, R. Rengaraj, S. Salivahanan; Enhancement of self-adaptive real coded genetic algorithm using Taguchi method for economic dispatch problem; Appl. Soft Comput, 11 (1) (2011), pp. 83–92
  49. [49] J.B. Park, Y.W. Jeong, J.R. Shin, K.Y. Lee; An improved particle swarm optimization for nonconvex economic dispatch problems; IEEE Trans. Power Syst, 25 (1) (2010), pp. 156–166
  50. [50] L.D.S. Coelho, V.C. Mariani; Combining of chaotic differential evolution and quadratic programming for economic dispatch optimization with valve-point effect; IEEE Trans. Power Syst, 21 (2) (2006), pp. 989–996
  51. [51] Y. Wang, B. Li, T. Weise; Estimation of distribution and differential evolution cooperation for large scale economic load dispatch optimization of power systems; Inf. Sci. (Ny), 180 (12) (2010), pp. 2405–2420
  52. [52] L.D.S. Coelho, V.C. Mariani; An efficient cultural self-organizing migrating strategy for economic dispatch optimization with valve-point effect; Int. J. Energy Convers. Manag, 51 (12) (2010), pp. 2580–2587
  53. [53] A. Bhattacharya, P.K. Chattopadhyay; Hybrid differential evolution with biogeography-based optimization for solution of economic load dispatch; IEEE Trans. Power Syst, 25 (4) (2010), pp. 1955–1964
  54. [54] L. Wang, L.P. Li; An effective differential harmony search algorithm for the solving non-convex economic load dispatch problems; Int. J. Electr. Power Energy Syst, 44 (2013), pp. 832–843
  55. [55] J.G. Vlachogiannis, K.Y. Lee; Economic load dispatch – a comparative study on heuristic optimization techniques with an improved coordinated aggregation-based PSO; IEEE Trans. Power Syst, 24 (2) (2009), pp. 991–1001
  56. [56] T.A.A. Victoire, A.E. Jeyakumar; Hybrid PSO–SQP for economic dispatch with valve-point effect; Electr. Power Syst. Res, 71 (1) (2004), pp. 51–59
  57. [57] J.S. Alsumait, J.K. Sykulski, A.K. Al-Othman; A hybrid GA–PS–SQP method to solve power system valve-point economic dispatch problems; Appl. Energy, 87 (5) (2010), pp. 1773–1781
  58. [58] C. Jiejin, L. Qiong, L. Lixiang, P. Haipeng, Y. Yixian; A hybrid CPSO–SQP method for economic dispatch considering the valve-point effects; Int. J. Energy Convers. Manag, 53 (1) (2012), pp. 175–181
  59. [59] A.I. Selvakumar, K. Thanushkodi; A new particle swarm optimization solution to nonconvex economic dispatch problems; IEEE Trans. Power Syst, 22 (1) (2007), pp. 42–51
  60. [60] A.I. Selvakumar, K. Thanushkodi; Anti-predatory particle swarm optimization: solution to nonconvex economic dispatch problems; Electr. Power Syst. Res, 78 (1) (2008), pp. 2–10
  61. [61] N. Noman, H. Iba; Differential evolution for economic load dispatch problems; Electr. Power Syst. Res, 78 (8) (2008), pp. 1322–1331
  62. [62] L.D.S. Coelho, R.C.T. Souza, V.C. Mariani; Improved differential evolution approach based on cultural algorithm and diversity measure applied to solve economic load dispatch problems; Math. Comput. Simul, 79 (10) (2009), pp. 3136–3147
  63. [63] R. Kumar, D. Sharma, A. Sadu; A hybrid multi-agent based particle swarm optimization algorithm for economic power dispatch; Int. J. Electr. Power Energy Syst, 33 (1) (2011), pp. 115–123
  64. [64] T. Niknam; A new fuzzy adaptive hybrid particle swarm optimization algorithm for non-linear, non-smooth and non-convex economic dispatch problem; Appl. Energy, 87 (1) (2010), pp. 327–339
  65. [65] M. Basu; Economic environmental dispatch using multi-objective differential evolution; Int. J. Appl. Soft Comput, 11 (2011), pp. 2845–2853
  66. [66] U. Güvenç, Y. Sonmez, S. Duman, N. Yorükeren; Combined economic and emission dispatch solution using gravitational search algorithm; Sci. Iran. D Comput. Sci. Eng. Electr. Eng, 19 (6) (2012), pp. 1754–1762
  67. [67] E.D. Manteaw, N.A. Odero; Combined economic and emission dispatch solution using ABC_PSO hybrid algorithm with valve point loading effect; Int. J. Sci. Res. Publ, 2 (12) (2012), pp. 1–9
  68. [68] R. Zhang, J. Zhou, L. Mo, S. Ouyang, X. Liao; Economic environmental dispatch using an enhanced multi-objective cultural algorithm; Electr. Power Syst. Res, 99 (2013), pp. 18–29
  69. [69] D.C. Secui; A new modified artificial bee colony algorithm for the economic dispatch problem; Int. J. Energy Convers. Manag, 89 (2015), pp. 43–62
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