Performance evaluation of metaheuristic optimization ...

2017 Nineteenth International Middle East Power Systems Conference (MEPCON), Menoufia University, Egypt, 19-21 December 2017

Performance evaluation of metaheuristic optimization methods with mutation operators for combined economic and emission dispatch Eman Mostafa, Mohamed Abdel-Nasser, and Karar Mahmoud Electrical Engineering Department, Aswan university, 81542 Aswan, Egypt turning off for a specified time according to variety in demand load and the change in electricity prices, a process called unit commitment is responsible for taking such decisions [4], [5]. Complex optimization problems can be solved by inspirations from nature, since it is known that nature is a system of vast complexity and always reach an equilibrium that is optimal, thus a near-optimum solution could be obtained. Artificial neural networks, simulated annealing, genetic algorithms, swarm intelligence and quantum computing are examples for natural inspired optimization methods [6]. In the last decade, various evolutionary and meta-heuristics optimization algorithms have been developed mimicking natural phenomena and they have been introduced as optimizers for economic dispatch (ED) problem. Some of these algorithms are genetic algorithm (GA), which mimics the natural evolution process based on Darwin principle of natural evolution, it belongs to the larger class of evolutionary algorithms (EA) [7]. Particle swarm optimization (PSO) which is based on the social behaviors of birds flocking and fish schooling, it is a population based evolutionary algorithm [8]. Grey wolf optimization (GWO) is a recently developed powerful evolutionary algorithm belongs to swarm intelligence algorithms. It mimics the grey wolves’ cooperation when hunting a prey [9]. Ant lion optimization (ALO) that mimics the antlion hunting technique, it is a population-based algorithm [10]. In addition, ant colony optimization (ACO) [11], firefly algorithm (FFA) [12], Krill herd algorithm (KHA) [13], modified group search optimization (MGSO) algorithm [14], and multi verse optimization (MVO) [15] are applied to solve ED problem. Flower pollination algorithm (FPA) [16], gravitational search algorithm (GSA) [17], PSO in [18] and [19], the collective neurodynamic optimization (CNO) method combining heuristic approach and projection neural network (PNN) [20], multiobjective evolutionary algorithms (MOEAs) [21], a summation based multi-objective differential evolution (SMODE) algorithm [22], and backtracking search algorithm (BSA) [23] are applied to solve the ED problem taking in consideration the emission reduction in a process defined as combined economic and emission dispatch (CEED) problem. The aim of this paper is to reach an optimized solution for the CEED problem. To do so, six optimization methods are used and two different mutation operators are applied to them. The rest of this paper is organized as follows. Section II

Abstract—This paper solves the combined economic and emission dispatch (CEED) problem which aims to achieve minimum generating costs with emission reduction using different optimization methods. Six methods are discussed: moth-flame optimization (MFO), moth swarm algorithm (MSA), grey wolf optimization (GWO), antlion optimization (ALO), sine cosine algorithm (SCA), and multi-verse optimization (MVO). Different mutation operators are integrated to these methods to improve their performance. Two test systems are simulated, and the results are compared to see the effectiveness of applying mutation operators to the optimization methods.

Index Terms—Economic Dispatch (ED); Combined Economic and Emission Dispatch (CEED); System Constraints.

I. I NTRODUCTION Recent power system networks include various generation types such as thermal, hydro, and nuclear power, but mainly depending on fossil fuel. The total operating cost of the system depends on the number of generators. In a power generating plant, the main running cost is the fuel cost per unit of electrical energy generation, taking into account the cost of lubricating oil, maintenance and repairs. These charges are proportional to the number of generator units [1]. Other power generation units called renewable energy resources or nonconventional energy of power generation, such as solar power generation, wind power generation and tidal power generation. Several countries might reach a time that they would run out of their entire reserve for fossil fuels; these resources will be the main sources of energy; so they also known as the future energy [2]. Renewable resources can co-generate required power with the conventional resources in a smart grid. Smart grid is the evolution of the current electrical grid depending on advanced technology for optimizing the generation and power delivery. It can integrate renewable resources, control power flow, reduce losses and reduce carbon emission. Therefore, smart grids can achieve optimal dispatch in an economic manner taking into account the environmental impact [3]. In a power system, economic dispatch is a computating process aims to minimize the total generation cost while meeting system constraints by sharing generation for the required power among the available units. It requires scheduling the generating units to decide which one is turning on and which is

c 978-1-5386-0990-3/17/$31.00 2017 IEEE

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mathematically models the CEED problem and its bi-objective function. Section III presents the different optimization methods. Section IV illustrates the mutation operators that are applied to the optimization methods. Section V discusses comparative results that are obtained in the paper. Finally, the conclusion of the paper and the future work are presented in section VI.

To combine the total fuel cost and total emission in a single objective function, a price penalty factor is needed. The objective function is expresses as:

II. PROBLEM FORMULATION

Where h is the price penalty factor ($/h) can be obtained as follows [24] : F C(Pimax ) (8) hi = EM (Pimax )

F =

n X

((ai Pi2 + bi Pi + ci ) + h(αi Pi2 + βi Pi + γi )) (7)

i=1

For a specified load condition, each unit in the plant has its own fuel cost and released emission as in Fig. 1.

III. OPTIMIZATION METHODS A. Moth-flame optimization (MFO) algorithm MFO is a population-based algorithm mimics the death behavior of moths when they get attracted by a flame and moves towards it until they dead [25]. This algorithm mathematically models the moth’s navigation manner in nature when flying in night by keeping a fixed angle with respect to the moon or an artificial light in order to guarantee travelling in a straight line for far distances. In the MFO algorithm, moths represent feasible solutions and their positions in the space are variables of the problem. The set of moths can be represented as:   m1,1 m1,2 . . . m1,d  m2,1 m2,2 ... m2,d    M = . (9) .. .. ..   .. . . . 

Fig. 1 Power plant. The CEED optimization problem aims to minimize the total generation cost of the power system satisfying the system constraints, it is a multi-objective function and can be formulated as the following: M inimize

F = F C + EM

(1)

F C = ai Pi2 + bi Pi + ci

(2)

EM =

αi Pi2

+ β i Pi + γ i

mn,1

Pi = Pd + Ploss

(3)

mn,d

where n is the number of moths and d is the number of variables (dimension). The fitness values for the flames are sorted. Moths are actual search agents moving in the search space, and flames are the best positions of moths, they are the best solution obtained so far. So, each moth searches for a flame and updates it if it finds a better solution. As a result, a moth never loses its best solution. To implement MFO to a CEED problem; best positions for moths represent the best power generating scenarios, fitness values for the flames represent the minimum total generating cost. Population can be expressed as follows:   P1,1 P1,2 . . . P1,d  P2,1 P2,2 ... P2,d    P = . (11) .. .. ..   .. . . . 

(4)

i=1

Inequality constraints: Pimax ≤ Pi ≤ Pimin , i = 1, . . . , n.

...

where n is the number of moths and d is the number of variables (dimension). For all moths, the corresponding fitness values are sorted. The following similar matrix represents the flames which the moths are searching for.   F1,1 F1,2 . . . F1,d  F2,1 F2,2 ... F2,d    F = . (10) .. .. ..   .. . . .  Fn1,1 Fn1,2 . . . Fn1,d

where F is the total generating cost, F C, and EM are the fuel cost and the released emission for the ith unit subject to the equality and inequality constraints of the system. Equality constraints: n X

mn,2

(5)

where ai , bi and ci are cost coefficients, αi , βi and γi are emission coefficients, n is the number of generating units to be scheduled, Pi is the real power generated from the ith unit, Pd is the total power demand, Ploss is power transmission losses, Pimin and Pimax are the minimum and maximum limits of the output power from the ith unit. Ploss can be expressed as a function of generator powers through George’s formula using B coefficients as: n X n X Ploss = Bij Pj Pi (6)

Pn,1

i=1 j=1

2

Pn,2

...

Pn,d

where n is the number of feasible solutions and d is the number of generating units. The corresponding best values for population are stored in the following array:   P best1  P best2    Pbest =  .  (12) .  . 

components os ~a decrease linearly from 2 to 0, and r~1 and r~2 are random vectors have values between [0,1]. For hunting the prey, grey wolves update theire positions according to the first three best solutions. GWO is applied to solve ED problem in [9]. D. Antlion optimization (ALO) algorithm ALO is a natural inspired population-based algorithm which mimics the hunting scenario between antlions and ants. Ants move with random walks over the search space searching for food while antlions are trying to catch those ants by building traps. When catching the ants, antlions rebuild traps. This process is repeated for a number of iterations. In each iteration, antlions update their positions according to ants location [25]. Ants locations are represented as:   A1,1 A1,2 . . . A1,d  A2,1 A2,2 ... A2,d    (17) Mants =  . .. .. ..   .. . . . 

P bestn where n is the number of feasible solutions. These best solutions will be optimized by calculating the corresponding total cost for each solution and checking all constraints. This process resulted in the best solution obtained so far which has the minimum total cost as equation 7 and satisfy system constraints. In order to reach the optimal solution, search agents still updating their positions untill reaching the best position using a logarithmic spiral starts from the present solution as initial point and ends to the fitness value as a final point.

Am,1

B. Moth swarm algorithm (MSA)

Alm,1

(14)

(15)

~ = 2.r~2 C

(16)

...

Alm,d

F. Multi-verse optimization (MVO) algorithm The inspiration of MVO algorithm is mainly based on three concepts; white holes, black holes, and wormholes. The white and black holes are utilized to explore search spaces by MVO, while the wormhole is utilized for exploiting the search space [29]. At every iteration, universes are sorted according to their inflation rates as follows:   u1,1 u1,2 . . . u1,d u2,1 u2,2 ... u2,d    U = . (20) .. .. ..   .. . . . 

~p represents the where t represents the current position, X ~ represents the position position vector of the prey, and X vector of a grey wolf. A and C are coefficient vectors calculated as follows: ~ = 2~a.r~1 − ~a A

Alm,2

E. Sine cosine algorithm (SCA) In SCA, solutions are randomly initialized. These solutions are improved through number of steps. Updating solutions positions follows the equation below: ( Xit + r1 ∗ sin(r2) ∗ |r3 Pit − Xit | r4 < 0.5 t+1 Xi = Xit + r1 ∗ cos(r2) ∗ |r3 Pit − Xit | r4 ≥ 0.5 (19) where Xit is the position of the ith solution for the tth iteration, r1 ,r2 ,r3 are random numbers indicate movement in the search space, Pi is the position of fitness point in the ith dimension, and r4 is a random number lies between [0,1] [28].

GWO mimics the mechanism that grey wolves use for hunting a prey. The alpha wolves (α) are the leaders responsible for taking decisions like hunting and sleeping. Beta wolves (β) are the second level that support the alpha’s commands throughout the pack and drive feedback to the alpha. Delta (δ) and omega (ω) wolves follow alpha and beta wolves. For an optimization problem, alpha wolves represent the best solution, beta and delta wolves are the second and third best solutions while omega wolves represent all the other solutions [27]. Encircling behavior is modeled as follows:

~p (t) − A.( ~ D) ~ X(t + 1) = X

Am,d

where m represents the number of antlions, and d represents the number of variables. Fitness of each antlion is also recorded. ALO is applied to solve ED problem in [10].

C. Grey wolf optimization (GWO) algorithm

(13)

...

where m represents the number of ants, and d represents the number of variables. Fitness of each ant is recorded. The following matrix describes the antlions positions which are hiding somewhere in the search space:   Al1,1 Al1,2 . . . Al1,d  Al2,1 Al2,2 ... Al2,d    Mantlionss =  . (18) .. .. ..   .. . . . 

In MSA, the navigation system of moth swarms is mathematically modeled for solving optimization problems [26]. The groups of moths that have the next best lighting intensity are set to be the prospectors. Each prospector updates its position according to the spiral path used in MFO searching for the best light source intensity. When a prospector gets better position, it will become a pathfinder. For an ED problem, moth swarms represent search agents which search for the best fitness. Prospectors update their positions and new finders are generated. The fitness value is calculated after each update. This process is repeated until the global best solution is obtained.

~ = |C. ~ X ~p (t) − X(t)| ~ D

A2,m

uz,1

3

uz,2

...

uz,d

where z is the number of universes (fesible solutions) and d is the number of parameters.

V. R ESULTS AND DISCUSSION Two power systems are tested in this paper using six optimization methods for solving the CEED problem. The results are compared using MFO, MSA, GWO, ALO, SCA, and MVO.

IV. A PPLYING MUTATION OPERATORS TO THE OPTIMIZATION METHODS

Two different mutation operators are applied in this paper in order to improve the performance of the optimization methods.

A. Test system (1) This power system consists of three units, load demand is 400 MW. System data and transmission losses are obtained from [16]. 1) Case (1): Comparison among all tested optimizers: It is shown in Table I that the least emission results when using GWO, while minimum fuel cost is achieved when using SCA. However, using MFO obtains the minimum total cost combining emission and fuel cost. MFO, SCA, and GWO consume less time than other methods. MFO obtains the best solution in this case. 2) Case (2): Applying Mut1 to all tested optimizers: From Table II, after applying Mut1, SCA results in the least emission but the highest fuel cost and total cost. The least fuel cost is achieved when using GWO which consumes the least time. The minimum total cost is obtained by using MFO.

A. Mutation operator 1 (Mut1) In this mutation, mutant vector is belt by randomly chosen vectors from the population to search in the entire search space around the best solution [30]. i i i i Xmut = XgBest + K ∗ Xr1 − Xr2

(21)

i i whre Xr1 and Xr2 are andomly chosen vectors from the population. K is a coefficient lies between 0 and 1.

B. Mutation operator 2 (Mut2) This mutation operator uses L´evy flights which are more efficient random walks in exploring unknown large-scale space for a global optimization [31]. Random walks of L´evy flights are drawn from L´evy distribution. levy ∼ u = t−λ , 1 < λ ≤ 3

(22) TABLE III Comparative results for test system (1) when applying Mut2 to optimiztion methods

The step length is expressed as: s=

µ |v|1/β

(23) N (0, σµ2 ),

and v = where s is the step length, β = 1.5. µ = N (0, σv2 ) are drawn from the normal stochastic distribution: σµ = [

Γ(1 + β) sin(π ∗ β/2) ]1/β , σv = 1 Γ((1 + β)/2) ∗ β ∗ 2(β−1)/2

(24)

Emission (∗102 )(Kg)

Fuel cost (∗104 ) ($)

Power losses (MW)

Total cost (∗104 ) ($)

MFO MSA GWO ALO SCA MVO

2.0023062932 2.0023064782 2.0022892622 2.0023062967 2.0025176930 2.0023013689

2.08378569996 2.08378562509 2.08379283857 2.08378569854 2.08375420025 2.08378772257

7.421039 7.421036 7.421167 7.421039 7.427946 7.421088

2.942647310198887 2.942647310238683 2.942647507684604 2.942647310198912 2.942708035347108 2.942647330089973

Elapsed time (sec) 2.70 6.00 2.53 5.39 2.53 2.74

Method

Fuel cost (∗104 ) ($)

Power losses (MW)

Total cost (∗104 ) ($)

MFO MSA GWO ALO SCA MVO

2.0023062938 2.0023063463 2.0023094770 2.0023062912 2.0022869007 2.0023082801

2.08378569968 2.08378567806 2.08378436894 2.08378570080 2.08379379677 2.08378489536

7.421039 7.421039 7.421090 7.421039 7.420929 7.421018

2.942647310198886 2.942647310199656 2.942647312418791 2.942647310198897 2.942647380885776 2.942647313556953

Fuel cost (∗104 ) ($)

Power losses (MW)

Total cost (∗104 ) ($)

MFO MSA GWO ALO SCA MVO

2.00230 2.02306 2.00231 2.00231 2.00267 2.00233

2.0837847 2.08887190 2.08378406 2.08378570 2.08365156 2.08377574

7.4211 7.3194 7.4211 7.4210 7.4238 7.4209

2.94264731 2.95671093 2.94264731 2.94264731 2.94266438 2.94264771

Elapsed time (sec) 2.66 3.23 2.62 5.45 2.63 2.81

Method

Emission (∗103 ) (Ib)

Fuel cost (∗105 ) ($)

Power losses (MW)

Total cost (∗105 ) ($)

MFO MSA GWO ALO SCA MVO

3.97753 4.02920 3.95136 3.94287 4.04530 3.94411

1.13262578 1.12729204 1.13500425 1.13756766 1.13795287 1.13632955

84.8833 85.4080 84.0661 83.9457 82.8331 84.1343

1.67879222 1.68055322 1.67757682 1.67897456 1.69342485 1.67790672

Elapsed time (sec) 4.07 4.65 3.93 12.59 4.13 4.70

TABLE V Comparative results for test system (2) when applying Mut1 to optimiztion methods

TABLE II Comparative results for test system (1) when applying Mut1 to optimiztion methods Emission (∗102 )(Kg)

Emission (∗102 ) (Kg)

TABLE IV Comparative results for test system (2) (load demand = 2000 MW)

TABLE I Comparative results for test system (1) (load demand = 400 MW) Method

Method

Elapsed time (sec) 5.10 5.67 5.05 7.79 5.06 5.27

4

Method

Emission (∗103 ) (Ib)

Fuel cost (∗105 ) ($)

Power losses (MW)

Total cost (∗105 ) ($)

MFO MSA GWO ALO SCA MVO

3.94098 3.95479 3.95383 3.92937 3.94272 3.94450

1.13639299 1.13525588 1.13470839 1.13840977 1.13718299 1.13591838

84.0171 84.0146 84.1427 83.8289 84.2073 84.0914

1.67754015 1.67829943 1.67762039 1.67796319 1.67856983 1.67755003

Elapsed time (sec) 8.00 8.57 7.81 16.21 7.89 8.12

TABLE VI Comparative results for test system (2) when applying Mut2 to optimiztion methods Method

Emission (∗103 ) (Ib)

Fuel cost (∗105 ) ($)

Power losses (MW)

Total cost (∗105 ) ($)

MFO MSA GWO ALO SCA MVO

3.936168 4.444699 3.946829 3.958549 4.084902 3.950458

1.137216912 1.226810004 1.135770216 1.134344206 1.132156983 1.135293718

83.8622 93.3740 84.1078 84.3185 85.6292 84.1812

1.677703918 1.683965964 1.677721118 1.677904420 1.693067056 1.677742915

Elapsed time (sec) 4.16 4.79 4.28 12.63 4.05 4.44

(a) MFO.

TABLE VII Performance evaluation with mutation operators for test system (1) Method MFO MSA GWO ALO SCA Mut1 N N (+) N (+) Mut2 N (-) (+) N (+) (+): Better performance with this mutation operator. (-): Worst performance with this mutation operator. N: No effect appears whith this mutation operator.

MVO (+) (-)

(b) MSA.

TABLE VIII Performance evaluation with mutation operators for test system (2) Method MFO MSA GWO ALO SCA Mut1 (+) (+) (-) (+) (+) Mut2 (-) (-) (-) (+) (+) (+): Better performance with this mutation operator. (-): Worst performance with this mutation operator.

MVO (+) (+)

(c) GWO.

3) Case (3): Applying Mut2 to all tested optimizers: From Table III, when applying Mut2, MFO achieves the least emission. The minimum fuel cost is achieved by SCA. MFO, GWO, and ALO obtain the minimum total cost, MFO and GWO consume the least time while ALO consumes much time. MFO achieves the best efficient solution for this system. B. Test system (2)

(d) ALO.

This power system consists of ten units, load demand is 2000 MW. System data and transmission losses are obtained from [16]. 1) Case (1): Comparison among all tested optimizers: From Table IV, ALO results in the least emission and the largest time consumed. The minimum fuel cost is obtained by MSA while the minimum total cost is obtained by GWO which consumes the least time. 2) Case (2): Applying Mut1 to all tested optimizers: From Table V, the least emission is achieved by ALO. GWO achieves the least fuel cost consuming the least time. The least total cost is obtained by MFO. 3) Case (3): Applying Mut2 to all tested optimizers: Table VI shows that MFO achieves the least emission and the least total cost while SCA obtains the minimum fuel cost with the least consumed time. For test system (1), optimization methods are slightly affected by applying mutation operators to them (see Table VII). For test system (2), the effect of applying mutation operators to

(e) SCA.

(f) MVO.

Fig. 2 Convergence characteristics for test system (2). 5

the optimization methods is more clear (see Table VIII). The sign (+) indicates that performance gets better by applying the mutation operator, the sign (−) indicates that performance becomes worest by applying the mutation operator, and N indicates that the optimizer is not affected by applying the mutation operator. MFO obtains better solutions when applying mutation. Applying Mut1 improves its performance better than applying Mut2 as the obtained cost decreased. MSA achieves less total cost when applying Mut1 and higher total cost when applying Mut2. GWO has its best performance without applying mutation. ALO is improved by applying mutation, applying Mut2 achieves better total cost than applying Mut1. SCA achieves less total cost when applying mutation, it obtains the best solution when applying Mut1. MVO is also improved by applying mutation, when applying Mut1 it obtains a total cost less than that obtained when applying Mut2. Fig. 2 shows the convergence of solutions.

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