A modified crow search algorithm (MCSA) for solving economic load dispatch problem

Accepted Manuscript Title: A Modified Crow Search Algorithm (MCSA) for Solving Economic Load Dispatch Problem Authors: Farid Mohammadi, Hamdi Abdi PII: DOI: Reference:

S1568-4946(18)30374-0 https://doi.org/10.1016/j.asoc.2018.06.040 ASOC 4956

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

31-10-2017 22-5-2018 22-6-2018

Please cite this article as: Mohammadi F, Abdi H, A Modified Crow Search Algorithm (MCSA) for Solving Economic Load Dispatch Problem, Applied Soft Computing Journal (2018), https://doi.org/10.1016/j.asoc.2018.06.040 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Modified Crow Search Algorithm (MCSA) for Solving Economic Load Dispatch Problem Farid Mohammadi, Hamdi Abdi Department of Electrical Engineering, Faculty of Engineering, Razi University, Kermanshah, Iran. 

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Corresponding Author, [email protected]

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Highligths:  Applying a novel evolutionary optimization algorithm namely MCSA to ELD problem.  Proposing two modification methods for improving the CSA performance.  Applying the MCSA to five well-known ELD test systems.  Employing four well-known benchmark functions to verify the MCSA.  Addressing the MCSA as a highly competitive with some previous algorithms.

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Abstract This paper presents a novel evolutionary optimization algorithm namely the modified crow search algorithm (MCSA) for solving the non-convex economic load dispatch (ELD) problem which improves the crow search algorithm (CSA) by an innovative selection of the crows and adaptive adjustment of the flight length. MCSA is a population-based technique based on the intelligent behavior of the crows in finding food sources. In MCSA, each crow saves its food in hiding-places for the time it needs. Also, each crow searches environment to find the better foods by stealthily following other crows to discover their hiding-places. The proposed MCSA develops the search capability of crows in the original CSA and introduces a new way by which a destination is selected by a crow to follow. To indicate the applicability of MCSA in the ELD problem, it is applied on three different well-known test systems. The results are compared in terms of the solution quality, robustness, and computing time with other methods implying that the proposed method has a superior performance than the other techniques.

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Keywords: Economic Load Dispatch (ELD), Evolutionary Algorithms, Modified Crow Search Algorithm (MCSA), Optimization.

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Nomenclature cost curve coefficients of the ith unit 𝒂𝒊 , 𝒃𝒊 , 𝒄𝒊 valve-point effects coefficients of the ith unit 𝒆𝒊 , 𝒅𝒊 generating output power of the ith unit 𝑷𝒊 minimum and maximum generation capacities 𝑷𝒎𝒊𝒏 , 𝒊 𝒎𝒂𝒙 of the ith unit, respectively 𝑷𝒊

Nc itrmax 𝑋 𝑖,𝑖𝑡𝑟 d

number of crows in the flock maximum number of iteration of the algorithm position vector of ith crow at the iteration itr the number of decision variables

𝒂𝒊𝒋 , 𝒃𝒊𝒋 , 𝒄𝒊𝒋 , 𝒆𝒊𝒋 , 𝒅𝒊𝒋 NM

fuel cost curve coefficients of the ith unit for the jth fuel type

𝑀𝑖,𝑖𝑡𝑟

the memory of the crow i at the iteration itr

the number of fuel type

𝑟𝑖 , 𝑟𝑗

Nu

total number of the committed generating units

fl, AP

𝑷𝑫 , 𝑷𝑻𝑳

load demand and the total power loss of transmission lines, respectively power loss coefficients

K

uniform distributed random numbers in the interval (0,1) flight length and awareness probability, respectively number of selected best crows in the population minimum and maximum number of best selected crows, respectively

𝐷𝑡ℎ𝑟 𝑓𝑙𝑡ℎ𝑟 𝛾

distance threshold value flight length threshold value penalty coefficient

vector which contains the distances between the crow i and crow j

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𝒏𝒑𝒛 𝑷𝒊𝟎 𝑼𝑹𝒊 , 𝑫𝑹𝒊

the lower and upper boundaries of Prohibited Operating Zones (POZs) for each generating unit, respectively the number of POZs initial power output before scheduling the upper and down ramp limits of the ith unit, respectively

𝐾 𝑚𝑖𝑛 , 𝐾 𝑚𝑎𝑥 𝐷𝑖,𝑗

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𝑩𝒊,𝒋 , 𝑩𝒊𝟎 , 𝑩𝟎𝟎 𝑷𝒍𝒐𝒘 , 𝑷𝒖𝒑

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1. Introduction Economic load dispatch (ELD) is one of the important problems in the power system operation that has the objective of optimally scheduling the generating units’ outputs, so that the total generating costs are minimized while meeting some equality and inequality constraints [1-3]. Some of the main constraints are: meeting the power balance equality, generation limits, and ramp-rate limits [4, 5]. In the conventional ELD problem, the fuel cost function of the generating units is modeled by the quadratic functions which are smooth and convex. However, in the real power system, some practical constraints exist such as the ramp-rate limits, valve point loading effects (VPE), prohibited operating zones (POZs), etc. which should be taken into account. These limitations, convert the ELD problem into a non-linear optimization problem with a nonsmooth and non-convex objective function [6-9]. For solving the constrained ELD problem, many evolutionary algorithms (EAs) are proposed. EAs have many good features, such as no requirement for a differentiable or continuous objective function, and easy applicability.

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Some EAs used to solve the ELD problem are: the genetic algorithm (GA) [10], particle swarm optimization (PSO) [1], artificial bee colony algorithm (ABC) [11], evolutionary programming [10], differential evolution (DE) [11], teaching learning-based optimization (TLBO) [14], firefly algorithm (FA) [15], cuckoo search algorithm (CSA) [16], chaotic bat algorithm (CBA) [17], grey wolf optimization (GWO) [18,19], oppositional invasive weed optimization (OIWO) [20], simulated annealing (SA) [21], oppositional real coded chemical reaction optimization (ORCCRO) [22], crisscross optimization (CSO) [23], colonial competitive differential evolution (CCEDE) [24], etc. Some of these methods have flaws, for

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example, GA requires a large computational time and has a low quality optimization solution [25]. DE and PSO suffer from the premature convergence or may be caught in the local optimums, because of their individual shortcomings [26]. The main disadvantage of ABC is its poor performance in exploitation of the solution area [27]. Due to the above limitations, several modified algorithms and hybrid algorithms have been proposed to solve the ELD problem. These methods improve their overall performance by focusing on the modification or combination of the original versions of EAs. Some papers focusing on the combination of EAs to solve the ELD problem are detailed as follows. In [28], a combination of a special class of ant colony optimization and a real coded genetic algorithm (GA-API) is proposed. It combines the downhill behavior of API and a good spreading in the solution space of the GA search process. A hybrid cross-entropy (CE) method and sequential quadratic programming (SQP) is proposed in [29]. CE-SQP uses the CE exploration capability and the promising local search ability of SQP to search the solution space, simultaneously. A variant of the GRASP for continuous space (C-GRASP) and the differential evolution (DE) are combined in [30]. Each iteration of C-GRASP provides a series of construction, local improvement cycles as well as the discrete grid of the search space. However, it is highly dependent on the local search and internal parameters. The authors have overcome these drawbacks by proposing the C-GRASP approaches based on the DE local search (C-GRASP–DE). In [26], a hybrid algorithm namely DE-PSO-DE or DPD is proposed. The whole population in increasing the order of solution quality is divided into the inferior, mid, and superior groups. The DE optimizes the inferior and superior groups and the PSO is employed to minimize the mid group. The Elitism strategy and the Non-Redundant Search are proposed to retain the best solution values and improve the solution quality, respectively. In [31], a modified hybrid particle swarm optimization (PSO) and gravitational search algorithm (GSA) based on the fuzzy logic is introduced (FPSOGSA) which combines the social thinking ability (gbest) of PSO and the local search capability of GSA. Also, some of the researches which investigate modification on the original EAs for solving the non-convex ELD problem in the recent years are as following. In random drift particle swarm optimization (RDPSO), the main evaluation mechanism of the original PSO, is changed to a novel set of equations which inspired by the free electron model in the metal conductors placed in an external electric field. The results show that the proposed process improves the performance of the algorithm [7]. In [32], a modified artificial bee colony based on the chaos theory (CIABC) is proposed. This method leads to a better self-searching ability compares to the original ABC algorithm by using the chaotic local search. In [2], a self-tuning improved random drift particle swarm optimization (ST-IRDPSO) is proposed. The improvement is done by adding a crossover operation at the greedy selection process. Also, the self-tuning method, makes the selection of the algorithm parameters independent of characteristics of the considered optimization problem. An improved PSO (IPSO) is given in [33], in which the chaotic sequences and the crossover operation are employed to improve the original PSO algorithm. The chaotic sequences designed to enhance the global searching feature and the crossover operation is used to increase the diversity of the population. In [34], a modified PSO algorithm namely Ө-PSO is introduced. In this method, the velocity vector

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of the PSO algorithm is replaced with phase angle vector (Ө), and the new position of the particles are obtained based on mapping of the phase angles. Ref. [35] proposed a modified version of the krill herd algorithm (KHA) namely the opposition krill herd algorithm (OKHA), in which the opposition based learning (OBL) is employed to enhance the robustness and the convergence feature of the original KHA algorithm. Recently, a new evolutionary algorithm called the crow search algorithm (CSA) was proposed in [36]. CSA is a population-based optimization algorithm which works based on behavior of the crows in searching the environment for better food sources. CSA has a simple mechanism for optimization with two controlling parameters i.e. the flight length and awareness probability. So, CSA has less adjustable parameters than the other algorithms such as GA, PSO, and ICA [36]. Moreover, CSA is easier to be used which makes it very desirable for solving the complex optimization problems. In this paper, in order to solve the constrained ELD problem, a modified crow search algorithm (called MCSA) has been used. The difference between the proposed MCSA and the original CSA is in two parts. First, the modification introduces a priority based method that determines how each crow should choose one of the other crows of population as its destination to move toward which will be explained in the following Sections in details. On the other hand in the basic CSA, there is no criterion for choosing the destination and the selection is done randomly between all crows. The second modification is related to proposing an intelligent method for adjusting the efficient amount of flight length (fl) controlling parameter. In the basic CSA, the amount of fl is a constant value which may cause inappropriate searching by the crows in the solution space that results in trapping in the local optimum. Moreover, in this paper, the implementation of the CSA and MCSA algorithm to solve the ELD problem have been presented for the first time. To ensure the applicability of the proposed method, the simulation results have been carried out for the 6, 10, and 40 unit test systems and they have been compared in terms of solution quality, robustness, and computing time with other algorithms. Also, the comparison of the convergence characteristic of MCSA and CSA have been made for each test system separately.

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2. Problem formulation 2.1. Objective function In the traditional ELD model, the cost function of the generating unit is quadratic. However, in the practical power systems, the thermal units usually have boilers with the valve points for controlling their power outputs which imposes a wide variation in the input–output characteristics due to the wire drawing effect. The valve point effect (VPE) is usually modelled by adding a sinusoidal term to the quadratic cost function [1-5]. In this case, ELD with VPE is a non-smooth and non-convex problem with the multiple minima considering

ripples in the heat-rate curves of boilers [37]. The cost function of each unit with VPE is as (1).

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Fc  Pi   ai  bi Pi  ci  Pi   d i sin e i  Pi min  Pi  2

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(1)

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(2)

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 a  b P  c  P 2  d sin e  P min  P  , i1 i1 i i1 i i1 i1 i i   fuel 1, Pi min  Pi  Pi1   a  b P  c  P 2  d sin e  P min  P  , i2 i2 i i2 i i2 i2 i i   Fc  Pi    fuel 2, Pi1  Pi  Pi 2    2 min aiM  biM Pi  ciM  Pi   d iM sin eiM  Pi  Pi  ,  fuel NM, Pil 1  Pi  Pi max  

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Where 𝑎𝑖 , 𝑏𝑖 , and 𝑐𝑖 , are the cost curve coefficients, 𝑒𝑖 and 𝑑𝑖 are the valve-point effects coefficients; 𝑃𝑖 is the generating output power of the ith unit and 𝑃𝑖𝑚𝑖𝑛 is the minimum generation capacities of the ith unit. Many of the thermal generating units are supplied with the multiple fuel sources (MFs) such as the coal, natural gas, and oil. Therefore, their fuel costs functions may be segmented as piecewise quadratic polynomial cost functions for the different fuel types [4]. Therefore, the cost function is given as (2).

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Nu

i 1

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Minimize F  Fc  Pi

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where, 𝑎𝑖𝑗 , 𝑏𝑖𝑗 , 𝑐𝑖𝑗 , 𝑒𝑖𝑗 ,and 𝑑𝑖𝑗 are the fuel cost curve coefficients of the ith unit for jth fuel type; NM is the number of fuel type. The total generating cost relation (the objective function of ELD) that should be minimized is given in (3). (3)

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where, Nu is the total number of the committed generating units. The committed units, are the active units during the study time of the ELD problem. 2.2. Constraints Some of the equality and inequality constraints that should be met in the ELD optimization problem are described as follows.

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2.2.1. Equality constraint The sum of the generating power outputs should be equal to the sum of the total demanded power from the system and the total transmission power loss. Nu

 (P )  P i 1

i

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 PTL  0

(4)

where, 𝑃𝐷 and 𝑃𝑇𝐿 are the load demand and the total power loss of transmission lines. 𝑃𝑇𝐿 is calculated by the Kron’s loss formula which is expressed as (5).

Nu Nu

Nu

i 1 j 1

i 1

PTL  Pi B ij Pj  B i 0 Pi  B 00

(5)

where, 𝐵𝑖,𝑗 , 𝐵𝑖0 , and 𝐵00 are the power loss coefficients.

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2.2.2. Generation capacity constraint The ith generator output must be within its lower limit 𝑃𝑖𝑚𝑖𝑛 and upper limit 𝑃𝑖𝑚𝑎𝑥 as (6). Pi min  Pi  Pi max , i  1,, Nu (6) where, 𝑃𝑖𝑚𝑎𝑥 and 𝑃𝑖𝑚𝑖𝑛 are the maximum and the minimum capacity of the ith generating unit, respectively.

i  1,..., Nu Z  2,..., n PZ

(7)

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 Pi min  Pi  Pi low or ,1  up low or Pi ,z 1  Pi  Pi , z  P up  P  P max i , n pz i i 

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2.2.3. Prohibited operating zones (POZs) POZs of generating units denote that the units should not work in the specified regions because of the vibration of shaft bearing in the torsion frequencies. These regions produce discontinuities in the cost function curve, since the unit must operate under or over the certain specified limits [6, 9]. POZs of ith unit are as (7).

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where, 𝑃𝑙𝑜𝑤 and 𝑃𝑢𝑝 are the lower and upper boundaries of POZs for each generating unit, respectively; 𝑛𝑝𝑧 is the number of the POZs.

If the output power decreases If the output power increases

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Pi 0  Pi  DR i   Pi  Pi 0  UR i

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2.2.4. Ramp Rate limit In real-world systems, the power output of a thermal unit can’t suddenly change because of some physical limitation related to the turbine speed change, the steam pressure, and etc. The change of power output from an initial state to the next must not exceed a specified limit. This condition is known as the ramp rate constraint [9, 14]. The ramp rate constraints for the ith generation unit expressed in (8). (8)

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where, 𝑃𝑖0 is the power output of the previous hour, 𝑈𝑅𝑖 and 𝐷𝑅𝑖 are the upper and down ramp limits of the ith generating unit (MW/h), respectively. 3. Proposed method 3.1. Overview of crow search algorithm Crow search algorithm (CSA) is one of the newest population based evolutionary algorithms introduced in [36] based on crows behavior in finding their food. A crow checks the places while others birds hide their foods and the crow tries to steal them when the owner leaves its place [36].

i ,itr   ri  fl i ,itr   M X    a random position

j ,itr

X

i ,itr

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r j  AP j ,itr

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i ,itr 1

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In CSA, the position of each crow (its hiding places) represents a possible solution of the optimization problem. In fact, a crow has a similar role as a particle in the PSO algorithm or a country in the ICA algorithm. It is assumed that Nc is the number of crows in the flock (population size) and the itrmax is the maximum number of iteration of the algorithm. For a d-dimensional search space the position vector of ith crow is specified by 𝑋 𝑖,𝑖𝑡𝑟 = [𝑥1𝑖,𝑖𝑡𝑟 , 𝑥2𝑖,𝑖𝑡𝑟 , … , 𝑥𝑑𝑖,𝑖𝑡𝑟 ] where 𝑖𝑡𝑟 = 1,2, … , 𝑖𝑡𝑒𝑟𝑚𝑎𝑥 and 𝑖 = 1,2, … , 𝑁𝑐 and d is the number of decision variables. For each crow, there is a memory which the crow memorized the position of its best hiding place, actually the memory of the crow i which is shown by 𝑀𝑖,𝑖𝑡𝑟 = [𝑚1𝑖,𝑖𝑡𝑟 , 𝑚2𝑖,𝑖𝑡𝑟 , … , 𝑚𝑑𝑖,𝑖𝑡𝑟 ] represents the best experience of the crow i. Crows move in the environment and search for the better food sources (hiding places). For this purpose, it is assumed that in the iteration itr the crow i decides to follow the crow j (a random member in the flock) to access the best hiding place of crow j (𝑀 𝑗,𝑖𝑡𝑟 ). In this case, there are two possibilities. (i): the crow j does not know that the crow i is following it. So, the crow i will be able to reach the crow j in the hiding place. (ii): the crow j is aware that the crow i is following it. In this case, the crow j will try to deceive the crow i by moving to another random position in the search space in order to protect its food source. According to above possibilities (i, ii), the next position of the crow i is expressed by (9). (9)

otherwise

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where, 𝑟𝑖 and 𝑟𝑗 are the uniform distributed random numbers in the interval (0,1) and; 𝑓𝑙 𝑖,𝑖𝑡𝑟 and 𝐴𝑃𝑖,𝑖𝑡𝑟 are the flight length (fl) of the crow i and awareness probability (AP) of the crow j at the iteration itr, respectively. Setting small values for fl (fl < 1) causes the local search at the adjacency of 𝑋 𝑖,𝑖𝑡𝑟 and large values (fl > 1) results in the global search (far from 𝑋 𝑖,𝑖𝑡𝑟 ). The value of AP has an inverse relation to the rate of similarity of the crows position and has a direct relation to the rate of diversity of the crows position. By selecting a small AP, the algorithm performs a search in the area where the current good solution of the j-th crow is located. On the other hand, by selecting a large AP, the algorithm performs an exploration of the solution space - a random search - by decreasing the probability of searching the proximity of the current good solutions while increasing the diversity. [36]. The procedure of the CSA optimizing is as the following steps. Step1: Initialize the current position of each crow in the flock and setting the initial memory of crows to their initial position. Step2: For each crow, evaluate the position quality by inserting of the crow in the objective function. Step3: Generate the new position of crows. Step4: Check the feasibility of new positions. If the new position of each crow is not feasible, then the crow stays at its current position and does not move to the new position.

Step5: Evaluate the fitness function of the new positions for each crow. Step6: Update the memory of crows. The crow i uses (10) to update its memory. i ,itr 1  f X i ,itr 1 is better than f M i ,itr X i ,itr 1 M  (10) i ,itr otherwise  M In (10), f(.) represents the amount of objective function. Step7: For each iteration, steps 3-6 are repeated until the number of iteration reaches to the itrmax, then the best memory position is selected as the solution.

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Fig. 1. The crow i reposition process in CSA (a) fl < 1 and (b) fl > 1 (The position of the crow i is somewhere on the dash line).

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3.2. Modified crow search algorithm The proposed modified crow search algorithm (MCSA) is described in this section. General principles of the proposed MCSA is similar to CSA. Nevertheless, the proposed MCSA uses a different method for how each crow i chooses one of the crows in the flock as destination to follow it to discover the position of its hiding place. Moreover, the modified method for valuing the flight length (fl) are introduced. These terms are given in the following.

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3.2.1. Selection of the target crow As mentioned, when in the iteration itr, the crows want to generate new position in the search space, each crow (for example the crow i) should select one of the flock crows (for example the crow j) and follow it to discover the position of its hidden food place. In the basic CSA, choosing is randomly which means that the crow i selects a random crow (the crow j) in the flock. However, by random selection among population, there is a high possibility that the target crow (the crow j) may have a bad position with the large value of the objective function (in the minimization) which makes it an inappropriate destination to move. This

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selection should be made for all crows of the population at each iteration and this random selection process leads to that a large number of crows failed to improve their position by choosing inappropriate target crow to follow it. Actually, in this case a large number of crows are following the bad destinations which decreases the convergence. In this paper, a priority based selection is proposed. In this method, for the minimization problem at each iteration for the crow i, K of the best crows in the population are selected, then the crow i selects one of these top crows randomly to following it. The advantage of this selection is that by selecting the target crow among the number of top crows in the flock, the possibility of choosing inappropriate target crow is decreased. In this case, the crows can act better in improving their position by following the better target crows and it leads to increasing the convergence of the algorithm. However, choosing the value of K is very important. By choosing a very small value for K, the crow i should choose the crow j between a small number of best crows that may cause getting caught into the local optimum. On the other hand, by choosing the large value for K, the possibility of selecting the bad target crow will be problematic. To create balance in resolving these issues, in the beginning of iterations (for early iterations) for better exploration in the solution space the value of K starts from large number and its value is reduced according to (11) and at the last iterations where the exact searching around the best local optimums is important, the K has the small value.

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K  K min   K itr  round  K max  max  itr  (11) itrmax  

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3.2.2. The modification of the flight length (fl) According to (9), updating the position of crows is mainly up to the fl parameter as shown in Fig. 1. From Fig. 1(a), the new position of the crow i will be somewhere on the dash line between 𝑋 𝑖,𝑖𝑡𝑟 and 𝑀 𝑗,𝑖𝑡𝑟 if the value of fl is fixed less than 1. Otherwise, according to Fig. 1(b), the new position of the crow i will be on the dash line which might exceed 𝑀 𝑗,𝑖𝑡𝑟 . In (9), the term 𝑓𝑙 𝑖,𝑖𝑡𝑟 × (𝑀𝑗,𝑖𝑡𝑟 − 𝑋 𝑖,𝑖𝑡𝑟 ) determines a boundary where the next position of the crow i exists. As the main purpose of the movement is to reach 𝑀 𝑗,𝑖𝑡𝑟 and 𝑟𝑖 is the random number between 0 and 1, it can be argued that the best value for fl is 2 because 𝑓𝑙 = 2 specifies the boundaries that covers all proximity area around 𝑀 𝑗,𝑖𝑡𝑟 . In the reference article of CSA [36], the value of fl has been considered as 2 as well. However, selecting 𝑓𝑙 < 2 will determine the smaller boundaries which increases the possibility that the crow i stops before reaching the crow j and selecting 𝑓𝑙 > 2 will determines the larger boundaries which increases the possibility that the crow i exceeds the crow j. According to the fact that the crow i (𝑋 𝑖,𝑖𝑡𝑟 ) and the crow j ( 𝑀 𝑗,𝑖𝑡𝑟 ) can have different distances from each other, considering the value 2 for fl could not guaranteed the global search in the solution space. To understand this, if the position of the crow i be close to the position of memory of the crow j, the new position of the crow i would be very similar to the crow j

fl

i ,itr

2 if   flthr if

Di , j  Dthr Di , j  Dthr

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after updating its position. In this case, the movement of the crow i would be a focused (local) search around the crow j and the possibility that the movement can't help to exploration of the solution space will be increased. On the other hand, by selecting a larger value for fl (fl>2), the above issue could be handled and the larger fl can cause diversity between the new position of the crow i (𝑋 𝑖,𝑖𝑡𝑟+1 ) and 𝑀 𝑗,𝑖𝑡𝑟 . However, fl>2 will have good effect on those crows that have close position to their target crow and can't be used for all crows in the population, because in this case the chance that the crow i moves excessively and diverges from the crow j will increase. So, we need to define a threshold for the distance among 𝑋 𝑖,𝑖𝑡𝑟 and 𝑀 𝑗,𝑖𝑡𝑟 in order to using best value of fl. In this regard, equation (12) is proposed to the determination of flight length in MCSA which leads to select the suitable value of fl with respect to the situations of the crows. (12)

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where, 𝐷𝑖,𝑗 is the vector which contains the distances between the crow i and crow j (𝑀 𝑗,𝑖𝑡𝑟 − 𝑋 𝑖,𝑖𝑡𝑟 ), 𝐷𝑡ℎ𝑟 is the distance threshold value and 𝑓𝑙𝑡ℎ𝑟 is a number greater than 2.

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4. Implementation of MCSA for ELD problem In this part, the procedure for applying MCSA to solve the ELD problem is investigated. The ELD problem is an optimization problem in order to minimize the total generating cost (as the objective function). In the ELD problem, the power output of each generating units are the decision variables and the number of units (Nu) is equal to the number of dimensions of optimization problem (d). Each scheduled power outputs of generating units is a possible solution of the ELD problem (position of each crow for the ELD problem). Before presenting the solution method, it should be noted that in the ELD problem there are some equality and inequality constraints introduced in (4)-(8) that should be met. The most fundamental constraint is the equality of power production and the demanded load. The equality constraint in (4) can be handled by adding the penalty part to the objective function in (2) [5, 7, 15, 17, 23, 29, 34, 35]. The new objective function is the minimization of the following relation.

CC

Nu

FCN  FC  

(P )  P i

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 PTL

(13)

i 1

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where, 𝛾 is the penalty coefficient which is a positive real number. If the constraint (4) is nonzero, the amount of the second term in Eq. (13) will be nonzero too and by multiplying in 𝛾 will added to the calculated cost (Fc). In other words, a candidate which doesn’t meet the constraint (4), will have a large objective function and more likely will be discarded. On the other hand a candidate which meets the constraint (4) will have a relatively small objective function and consequently will be kept. If 𝛾 is fixed in very large value, may reduce the robustness of the algorithm and cause premature convergence. On the other hand by selecting a small value, the inequality constraint may not be met. The amount of 𝛾 is selected as 100

for simulations in this paper which have appropriate performance to meet the power equality constraint.

1   x 1   m 1   p11  p Nu  2  2  2 2   x    m    p1  p Nu          Nc   Nc   Nc Nc   p Nu  x   m   p1 

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The overall process to solve the ELD problem by MCSA is as follows. Also, the Pseudo code of the proposed MCSA for solving ELD is given in Fig. 2. Step1: Define the initial data including the characteristics of generation units, penalty coefficient, number of the crows (Nc); maximum iteration (itrmax); other parameters such as AP, fl, 𝐾𝑚𝑎𝑥 , 𝐾𝑚𝑖𝑛 , 𝐷𝑡ℎ𝑟 , and 𝑓𝑙𝑡ℎ𝑟 . Step2: Generate of the initial population by generating randomly position of each crows in the flock between the lower limit 𝑃𝑖𝑚𝑖𝑛 and upper limit 𝑃𝑖𝑚𝑎𝑥 capacity of the generating units. Also, because the crows in the initial iteration have no experience, it is assumed that the initial memory of crows is equal to their initial position. The initial population is Nc by Nu matrix as follows.

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(14)

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Step3: For each crow in the flock the power output of generating units should met the ramp rate limit and should not be placed in the prohibited operating zones. If a crow does not satisfy the mentioned conditions, modify the power outputs toward the near margin of the feasible solution. After generating the initial population, the main process of the MCSA algorithm runs as follows. Step4: Calculate the K factor as (11). Step5: Determine the flight length as (12). Step6: Select K of the best crows in the population except the current crow (the crow i), updating the position of the crow i by selecting one of these top crows randomly as target (crow j) and use (9) to generate the new position. Step7: Check this out for the crow i: a- The power outputs should not be placed in POZs (See (7)) or violated the generation capacity of the units (see (6)). b- Increase and decrease rates of each generating units from the initial state be in the acceptable ranges defined by (8). If the initial power outputs of generating units are not given, it is supposed that the initial power outputs of all generating units are in the acceptable ranges and there is no need to consider the ramp rate limits. If ramp rate limits and POZ limits are violated, modify the power outputs toward the near margin of the feasible solution. Step8: Calculate the total power loss of transmission lines for the crow i as (4). Step9: Calculate the quality of the crow i by inserting its output powers in the objective function as (14).

Step10: Update the memory of crows. The crow i uses (10) to update its memory. Step11: Repeat the steps 5-10 for all of the crows. Step12: Repeat the steps 4-11 until the maximum number of iterations is reached. Step13: the solution of ELD is the best memory of the flock in the last iteration.

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Generating the initial population: Define the initial data Randomly initialize the position of a flock of N crows in the search space Initialize the memory of each crow Check the ramp-rate and POZ constraints

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Main loop: while iter < itermax Calculate the K factor Determine the flight length Calculate the transmission loss Evaluate the position of the crows for i = 1 : NC (all NC crows of the flock) Select K of the best crows in the population Randomly choose one of the K crows to follow (for example j) if 𝒓𝒋 ≥ 𝑨𝑷𝒋,𝒊𝒕𝒓

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X i ,itr 1  X i ,itr  ri  fl i ,itr   M j ,itr  X i ,itr 

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else X i ,itr 1  a random position of search space end if end for Check the feasibility of new positions Evaluate the new position of the crows Update the memory of crows end while

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Fig. 2. Pseudo code of the proposed MCSA for solving ELD.

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5. Simulation and Results In this section, the MCSA algorithm has been used for solving the ELD problem by applying the proposed solving process on five known test systems. Also, comparison of MCSA with other optimization algorithms on the benchmark functions optimization has been done too. 5.1. Simulation setting The EAs have randomness nature in solving the optimization problem. It means that for each EAs, the quality of obtained solution in each run of algorithm is different and is affected from the initial population position and random movement of population in searching the

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optimum solution. So, judging about an evolutionary algorithm performance just based on one run of the algorithm is not proper comparison. Actually, for investigating the robustness of the various methods, the results of many runs must be compared to provide a reliable conclusion about the robustness of the algorithm. An algorithm has a good robustness feature, if it gives the consistent result during all runs. The numbers of independent runs used for obtaining the results for each test system adopted from previous literatures. We should take in mind that they selected a specific value for their relevant test system. For the fair comparison, we used the same values in this work, which are 50, 100, 50, 50, 50 independent runs for the systems 1 to 5, respectively. It should be noted that the population size and the number of iteration have different values in the previous literatures for the different optimization algorithms and different test systems, which depends on the nature of each algorithm and the size of the test system that causes different convergence power, may have large or small value. The fact is that the literatures set the values so that their proposed algorithm could have proper convergence and at the final iterations the variation of the cost of solutions be small. Like the previous literatures which tuned the controlling parameters of their algorithms for better performance, the amount of the controlling parameters of MCSA and the CSA have been selected at their optimum values for the ELD problem and the benchmark functions, so that the better solution can be achieved. The method of tuning the controlling parameters will be explained at section 5.7. These tuned values are as Table 1. Table 1. Parameter setting used in the test system and the benchmark functions.

TE

CSA

Parameter fl AP fl AP Kmin Kmax Dthr fthr

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MCSA

System1 2 0.1 2 0.1 5 25 0.8 3

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Algorithm

System2 2 0.1 2 0.1 5 25 0.8 3

System3 2 0.1 2 0.1 5 25 0.8 3

System4 1.9 0.1 1.9 0.1 7 27 0.8 2.8

System5 1.9 0.1 1.9 0.1 7 27 0.8 2.8

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5.2. Test system one This system consists of 6-thermal-units, 26 buses and, 46 transmission lines. The network load demand is 1263 MW. In this system, the ramp-rate limit and transmission losses are considered and all units have POZs. The characteristics of the system include the cost coefficients, B loss coefficients, and upper and down ramp limits, POZs, and initial output of units are as [1]. The number of crows (Nc) and maximum iteration (maxitr) are set to 100, respectively for both CSA and MCSA algorithms. MCSA with only first modification (MCSA-1), the MCSA with only second modification (MCSA-2), MCSA with both modification (MCSA), and the original CSA, are compared with the other optimization techniques include: GA [1], CBA [17], PSO [1], NPSO-LRS [5], MABC [38], MSSA [27], ST-IRDPSO [2], Differential

N

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evolution algorithm (DE) [13], and Decentralized approach algorithm (DE) [9]. The best scheduled power output of generating unit for each method after the 50 independent runs have been shown in Table 2. It is clear from Table 2 that both CSA and MCSA achieve solution with the lower cost (better quality) than other methods. Moreover, they could clearly handle the equality constraint of generation and demand. To understand this if the power inequality error defined as 𝑒𝑟𝑟𝑜𝑟 = |∑𝑁𝑢 𝑖=1(𝑃𝑖 ) − 𝑃𝐷 − 𝑃𝑇𝐿 |, using the relevant data form Table 2, the obtained inequality error will be equal to 0.05 MW and 0.06 MW, for CSA and MCSA, respectively which are acceptable. MCSA is actually an improvement of CSA. It has the better performance in the minimization of the total cost (has the better solution quality) subject to satisfying all constraint. Statistical result consist of the best, mean, standard deviation (Std. Div), and the worst generation cost during 50 independent runs shown in Table 3 which indicates that MCSA has competitive results compared to others. Moreover, it can be seen that the result of MCSA under each one of the proposed modification processes has better cost than the original CSA. Fig. 3 shows the convergence characteristic of CSA and MCSA in the ELD problem for the best run in the test system, which indicates that MCSA has better convergence than CSA.

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P3(MW) 262.2089 264.0759 263.4745 262.3436 263.4622 263.4630 263.4668 263.4110 263.4400 265.0000 264.3317 263.3939 265.0000

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Methods P1(MW) P2(MW) GA [1] 474.8066 178.6363 CBA [17] 447.4187 172.8255 PSO [1] 447.4970 173.3221 NPSO-LRS [5] 446.9600 173.3944 MABC [38] 447.5039 173.3187 MSSA [27] 447.5029 173.3186 ST-IRDPSO [2] 447.5131 173.2975 DE a [13] 447.7440 173.4070 DE b [9] 448.2700 172.9600 447.1090 175.7846 CSA MCSA-1c 446.6077 174.2183 MCSA-2d 447.6502 173.3722 444.6373 174.6410 MCSA a Differential evolution algorithm b Decentralized approach algorithm c The MCSA with only the first modification d The MCSA with only the second modification

A

Table 2. Best scheduled power achieved by different methods for the 6-unit system. P4(MW) 134.2826 139.2469 139.0594 139.5120 139.0650 139.0656 139.0360 139.0760 139.3000 136.7949 138.5819 138.8758 139.2251

P5(MW) 151.9039 165.6526 165.4761 164.7089 165.4731 165.4730 165.4843 165.3640 165.2800 161.0010 165.5090 165.4757 165.7121

P6(MW) 74.1812 86.7652 87.1280 89.0162 87.1350 87.1349 87.1605 86.9440 86.6800 90.2061 86.6686 87.1424 86.6807

Cost($/hr) 15459 15450.2381 15450 15450 15449.8995 15449.8995 15449.8945 15449.7660 15449.5826 15449.5736 15449.2473 15449.2230 15449.1672

Loss(MW) 13.0217 12.9840 12.9584 12.9361 12.9582 12.9580 12.9580 12.9570 12.9800 12.9506 12.9673 12.9604 12.9576

Table 3. Comparison of results for different methods for the 6-unit system. GA [1] CBA [17] PSO [1] NPSO-LRS [5] MABC [38] MSSA [27] ST-IRDPSO [2] DE b [13] DE c [9] CSA MCSA-1d MCSA-2e MCSA a

Best 15,459 15,450.2381 15,450 15,450 15,449.8995 15,449.899 15,449.8945 15,449.766 15,449.5826 15,449.5736 15,449.2473 15,449.2230 15,449.1672

Generation Cost ($/hr) Mean Worst 15,469 15,524 15,454.76 15,518.6588 15,454 15,492 15,452 15,454 15,449.8995 15,449.8995 15,449.937 15,453.545 15,450.70 NR 15,449.777 15,449.874 15,449.6171 15,449.6508 15,449.7623 15,449.8461 15,449.3432 15,449.6214 15,449.3163 15,449.5109 15,449.2358 15,449.3854

Not reported in the referred literature

b Differential

evolution algorithm Decentralized approach algorithm d The MCSA with only the first modification e The MCSA with only the second modification c

MCSA

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CSA

15565

N

15545 15525 15505

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Cost $/h

Std. Div NRa 2.965 NR NR 6.04×10-8 0.3647 1.416 NR NR 0.5702 0.4012 0.3803 0.2681

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Methods

15465 15445 0

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15485

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Number of Iteration

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Fig. 3. The cost convergence of the tested CSA and MCSA for the 6-unit test system.

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5.3. Test system two In this part, the ten generating units with multiple fuel sources under the valve point loading effects has been considered for testing the performance of the CSA and MCSA methods in solving the ELD problem. The input data of the system is available in [4] and the total load demanded is 2700 MW. The Nc and maxitr are set to 100, respectively. Nevertheless, the scheduled power outputs, and the fuel types (FT) obtained for MCSA and the CSA after 100 independent runs and are shown in Table 4 which indicates that MCSA has better solution quality. To judge about the solution quality and robustness of CSA and the proposed MCSA, a statistical analysis is done for 100 independent runs of MCSA and CSA and compared with CGA-MU [4], IGA-MU [4], NPSO [5], NPSO-LSR [5], cuckoo search algorithm (CSA) [16], one rank cuckoo search algorithm (ORCSA) [25], ST-IRDPSO [2], DEPSO [26], and CCEDE [24] in Table 5. As seen in Table 5, the solution by MCSA reaches to 623.8280 $/hr, which has the least cost of all compared methods. CCEDE, DEPSO and the original CSA have the best cost, respectively after MCSA. Also, for MCSA the standard deviation and the

difference between the best, mean, and worst solution is lower than all compared algorithms and lower than CSA which denotes the excellent robustness and high stability of MCSA. The convergence diagram of the best run for MCSA and CSA is shown in Fig. 4 that represents a faster convergence of MCSA. Table 4. Best scheduled power achieved by CSA and MCSA for the 10-unit system.

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1 2 3 4 5 6 7 8 9 10 Generation Cost ($/hr) Loss (MW) Error(MW) = |∑Nu i=1(Pi ) − PD |

MCSA 𝑃𝑖 (MW) FT 2 219.1319 1 212.4022 1 280.6570 3 239.2832 1 279.9351 3 239.7953 1 287.7274 3 239.4176 3 425.7811 1 275.8685 623.8280 0 0.00

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CSA FT 𝑃𝑖 (MW) 219.1817 2 211.6596 1 280.6571 1 239.9551 3 276.4164 1 239.7953 3 290.0985 1 239.8207 3 426.3626 3 276.0531 1 623.8361 0 0.00

Unit

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TE

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CGA-MU [4] IGA-MU [4] NPSO [5] NPSO-LRS [5] CSA b [16] ORCSA c [25] CSA ST-IRDPSO [2] DEPSO [26] MCSA-1d CCEDE [24] MCSA-2e MCSA

Best 624.7193 624.5178 624.1624 624.1273 623.8684 623.8608 623.8361 623.830 623.8300 623.8293 623.8288 623.8285 623.8280

a

Not reported in the referred literature Cuckoo search algorithm c One rank cuckoo search algorithm d

The MCSA with only the first modification The MCSA with only the second modification

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e

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b

Generation Cost ($/hr) Mean Worst Std. Div 627.6087 633.8652 NRa 625.8692 630.8705 NR 625.2180 627.4237 NR 624.9985 626.9981 NR 623.9495 626.3666 0.2438 623.8963 623.9353 0.0154 623.9626 624.8304 0.0116 623.838 NR 0.0052 623.9000 624.0800 0.49 623.8703 623.9113 0.0089 623.8574 623.8904 0.0076 623.8424 623.8751 0.0068 623.8311 623.8439 0.0050

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Table 5. Comparison of results for different methods for the 10-unit system.

820

CSA

800

MCSA

780 760

Cost $/h

740 720 700 680 660 640 0

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100

Number of Iterations

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620

Fig. 4. The cost convergence of the tested CSA and MCSA for the 10-unit test system

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5.4. Test system three In this part, the solution of the ELD problem has been presented for a test system consists of forty generating units incorporating the transmission loss and valve point effects. The results obtained in two cases for this system. In case1, the transmission losses neglected and in case 2, the transmission losses are considered. The total load demanded is 10500 MW and the system data is extracted from [20]. Both the population size and the maximum iteration are set to 100, respectively. The best power outputs of the units by applying MCSA for cases 1, and 2 are given in the Tables 6 and 8, respectively. It is clear from these tables that MCSA is able to determine the optimal solution impressively while satisfying the equality constraint (inequality error is almost zero). The minimum, mean, standard deviation, and maximum values have been compared with MCSA-1, MCSA-2, and CSA and some of the most prominent papers for the 50 independent runs for case 1 and 2, as shown in Tables 7 and 9, respectively. These algorithms for case 1 include: PSO [7], RDPSO [7], SSOA [8], IFEP [12], ABC [11], Ө-PSO [34], FA [15],CGRASP-SADE [30], MSSA [27], CE-SQP [29], DE [9], CBA[17], PSOGSA [31], MABC [38], one rank cuckoo search algorithm (ORCSA) [25], ST-IRDPSO [2], EGSSOA [8], and CIABC [32]. The compared methods for case 2 include: GAAPI [28], SDE [39], TLBO [21], QOTLBO [21], KHA-IV [19], OKHA [35], and OIWO [20]. In case1, the best and worst generation costs are 121,412.14 $/hr and 130,887.084 $/hr achieved by MCSA and PSO, respectively. Also, for case 2 the best and worst generation costs are related to MCSA and the GAAPI with the total cost equal to 136,448.6295 and 139,864.96, respectively. Moreover, Tables 7 and 9 show that the difference of the best, worst, and mean generation costs and also the standard deviation achieved by MCSA, MCSA-1, and MCSA-2 have the lower values than all other compared techniques in solving ELD in both cases 1 and 2, which indicates the capability of MCSA in reaching more stable result during the program runs. The convergence characteristics of the best economical solution for CSA and MCSA in cases 1 and 2 are shown in Fig. 5 and Fig. 6, respectively. The convergence curves imply the better computational efficiency of MCSA.

Table 6. Best scheduled power achieved by MCSA for the 40-unit test system for case1 (without transmission losses). 1 2 3 4 5 6 7 8 9 10

𝑷𝒊 (MW) 110.7989 110.7993 97.3982 179.7324 87.7999 139.9998 259.5995 284.5992 284.5999 130

Unit 11 12 13 14 15 16 17 18 19 20

𝑷𝒊 (MW) 94 94.0004 214.7597 394.2791 394.2791 394.2783 489.2790 489.2796 511.2795 511.2795

Unit 21 22 23 24 25 26 27 28 29 30

𝑷𝒊 (MW) 523.2791 523.2789 523.2793 523.2791 523.2794 523.2790 10 10 10 87.7997

Generation Cost ($/hr) = 121,412.140 Loss (MW) = 0 𝐄𝐫𝐫𝐨𝐫 (𝐌𝐖) = |∑𝐍𝐮 𝐢=𝟏(𝐏𝐢 ) − 𝐏𝐃 |= 0.00

Unit

𝑷𝒊 (MW)

31 32 33 34 35 36 37 38 39 40

190 190 190 164.7985 194.3926 199.9622 110 110 110 511.2793

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Unit

Table 7. Comparison of results for different methods for the 40-unit system in case 1 (without transmission losses).

C-GRASP-SADE[30]

TE

MSSA [27] CE-SQP [29] DE [9] CBA [17] PSOGSA [31] MABC [38] CSA ORCSA c [25]

ST-IRDPSO [2]

a b c

d

Cuckoo search algorithm Not reported in the referred literature One rank cuckoo search algorithm

The MCSA with only the first modification The MCSA with only the second modification

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EGSSOA [8] CIABC [32] MCSA-1d MCSA-2e MCSA

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Generation Cost ($/h) Mean Worst 132,614.197 135,008.739 129,482.097 131,129.086 123,963.047 124,433.516 123,382.00 125,740.63 121,995.82 122,123.77 NRb NR 121,509.842 121,852.424 121,416.57 121,424.56 121,736.025 122,245.696 121,466.61 121,521.73 121,423.65 NR 121,439.89 121,479.63 121,418.982 121,436.15 121,413.56 121,414.983 121,431.779 121,503.755 121,416.618 121,420.165 121,472.4534 121596.1789 121,443.792 NR 121,487.765 NR 121,623.15 NR 121,413.5528 121414.5420 121,413.4419 121414.5057 121,413.280 121414.324

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PSO [7] RDPSO [7] SSOA [8] IFEP [12] ABC [11] CSA a [16] Ө-PSO [34] FA [15]

Best 130,887.084 128,864.452 123,650.155 122,624.35 121,441.03 121,425.61 121,420.902 121,415.05 121,414.621 121,413.46 121,412,88 121,412.68 121,412.546 121,412.542 121,412,540 121,412.540 121,412.5355 121,412.535 121,412.5355 121,412,40 121,412.3642 121,412.2773 121,412.140

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Std. Div 618.8210 568.9333 NR NR NR NR 92.3956 1.784 166.896 28.6932 NR NR 1.611 NR 19.16 1.6507 58.6005 33.44 NR NR 1.3404 1.2312 0.8761

137000

CSA

MCSA

135000

Cost $/h

133000 131000 129000 127000 125000 123000 121000 20

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100

Number of Iterations

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0

Fig. 5. The cost convergence of the tested CSA and MCSA for the 40-unit test system in case 1 (without transmission losses). Table 8. Best scheduled power achieved by MCSA for the 40-unit test system in case 2 (with transmission losses). 𝑷𝒊 (MW) 550 550 523.2794 523.2794 523.3088 523.2805 10 10 10 87.8001

31 32 33 34 35 36 37 38 39 40

N

𝑷𝒊 (MW) 190 190 190 200 200 164.7998 110 110 110 550

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Unit 𝑷𝒊 𝑷𝒊 Unit (MW) (MW) 114 11 168.7998 21 1 114 12 94 22 2 120 13 484.0392 23 3 182.2038 14 484.0392 24 4 88.1831 15 484.0392 25 5 140 16 484.0392 26 6 300 17 489.2794 27 7 293.5382 18 489.2794 28 8 299.9987 19 511.2794 29 9 280 20 511.2795 30 10 Generation cost ($/h) = 136448.6295 Loss (MW) = 957.4059 𝐄𝐫𝐫𝐨𝐫(𝐌𝐖) = |∑𝐍𝐮 𝐢=𝟏(𝐏𝐢 ) − 𝐏𝐃 − 𝐏𝐓𝐋 |= 0.06 Unit

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Table 9. Comparison of results for different methods for the 40-unit system in case 2 (with transmission losses). Methods

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CC

EP

GAAPI[28] SDE [39] TLBO[21] QOTLBO[21] KHA-IV [19] OKHA [35] OIWO [20] CSA MCSA-1b MCSA-2c MCSA

a

Not reported in the referred literature The MCSA with only the first modification c The MCSA with only the second modification b

Best 139,864.96 138,157.46 137,814.17 137,329.86 136,670.3701 136,575.968 136,452.677 136,452.4871 136,451.1487 136,450.8991 136,448.6295

Generation Cost ($/h) Mean Worst NRa NR NR NR NR NR NR NR 136,671.2293 136,671.8648 136,576.15 136,576.64 136,452.677 136,452.677 136,453.3637 136,453.6962 136,451.3822 136,453.6962 136,451.0034 136,453.6962 136,448.7158 136,448.9490

Std. Div NR NR NR NR NR NR NR 2.3465 1.2961 1.2149 1.1005

152000

CSA

150000

MCSA

Cost $/h

148000 146000 144000 142000 140000 138000 0

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80

100

Number of Iterations

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136000

Fig. 6. The cost convergence of the tested CSA and MCSA for the forty-unit test system in case 2 (with transmission losses).

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5.5. Test system four This system consist of the 110 generating unit with the quadratic cost characteristic. The system data are extracted from [20]. The total load demand is equal to 15,000 MW. The population size and the maximum iteration are set to 100 and 200, respectively. The best scheduled powers achieved by MCSA are given in Table 10. The best, mean, worst, and standard deviation obtained using MCSA, MCSA-1, MCSA-2, and CSA, SAF [21], SAB [21], SA [21], BBO [22], DE/BBO [22], ORCCRO [22], and OIWO [20] are shown in Table 11 for the 50 trials. The convergence characteristic curve of MCSA and CSA are depicted in Fig. 7.

A

CC

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

𝑷𝒊 (MW) 68.9000 68.9000 68.9000 350.0000 400.0000 400.0000 500.0000 500.0000 200.0000 100.0000 10.0000 20.0000 80.0000 250.0000 360.0000 400.0000 40.0000 70.0000 100.0000 120.0000

Unit 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

TE

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

𝑷𝒊 (MW) 2.4000 2.4000 2.4000 2.4000 2.4000 4.0000 4.0000 4.0000 4.0000 64.2850 62.3237 36.1585 56.6586 25.0000 25.0000 25.0000 155.0000 155.0000 155.0000 155.0000

EP

Unit

D

Table 10. Best scheduled powers achieved by MCSA for the 110-unit test system. 𝑷𝒊 (MW) 157.2856 220.0000 440.0000 560.0000 660.0000 616.3731 5.4000 5.4000 8.4000 8.4000 8.4002 12.0000 12.0000 12.0000 12.0000 25.2007 25.2000 35.0000 35.0000 45.0000

Unit 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

𝑷𝒊 (MW) 45.0000 45.0000 185.0000 185.0000 185.0000 185.0000 70.0000 70.0000 70.0000 360.0000 400.0000 399.9998 104.7188 191.6113 90.0000 50.0000 160.0000 294.9932 175.1186 98.0302

Unit

𝑷𝒊 (MW)

Unit

𝑷𝒊 (MW)

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

10.0000 12.0000 20.0000 200.0000 325.0000 440.0000 15.2615 24.2229 82.1279 89.3664 58.1238 100.0000 440.0000 500.0000 600.0000 471.2337 3.6000 3.6000 4.4000 4.4000

101 102 103 104 105 106 107 108 109 110

10.0000 10.0000 20.0000 20.0000 40.0000 40.0000 50.0000 30.0000 40.0000 20.0000

Generation cost ($/h) = 197988.0932 Loss (MW) = 0 Error(MW) = Nu (Pi ) − PD | =0.00 |∑i=1

Table11. Comparison of results for different methods for the 110-unit system Best 207,380.5164 206,912.9057 198,352.6413 198,241.166 198,231.06 198,016.29 198,004.6432 197,998.2331 197,992,5198 197,989.14 197,988.0932

SAF[21] SAB[21] SA [21] BBO [22] DE/BBO [22] ORCCRO [22] CSA MCSA1b MCSA-2c OIWO[20] MCSA

Generation Cost ($/h) Mean Worst 207,813.37 NAa 207,764.73 NA 201,595.19 NA 198,413.45 199,102.59 198,326.66 198,828.57 198,016.32 198,016.89 198,004.9361 198,005.1251 197,998.5408 197,999.1109 197,992.7703 197,993.1263 197,989.41 197,989.93 197,988.3604 197,988.7840

a

Not reported in the referred literature The MCSA with only the first modification c The MCSA with only the second modification b

217000 215000

CSA

213000

MCSA

U

211000 209000 207000

N

Cost $/h

Std. Div NA NA NA NA NA NA 2.7831 1.9233 1.7808 NA 1.2643

SC RI PT

Methods

205000 203000

A

201000

197000 0

50

100

Number of Iterations

M

199000 150

200

D

Fig. 7. The cost convergence of the tested CSA and MCSA for the 110-unit test system.

A

CC

EP

TE

5.6. Test system five A complex large system consist of the 160 thermal units with the multiple fuel sources under the valve point loading effects has been considered here. The system data adopted from [4] which is obtained by duplicating the 10-unit system (the mentioned test system 2) 16 times. The total load demand is 43,200 MW for this system. The population size and the maximum iteration are set to 100 and 250, respectively. With a larger scale system, the number of the local minima in the solution space increases. This fact indicates that the optimization algorithms should have more powerful search ability to avoid the premature convergence. The best scheduled powers achieved by MCSA are given in Table 12 and the statistical analysis obtained using MCSA, MCSA-1, MCSA-2, CSA, CGA_MU [4], IGA_MU [4], BBO [22], DE/BBO [22], ORCCRO [22], CBA [17], cuckoo search algorithm (CSA) [16], ORCSA [25], and CSO [23] are shown in Table 13 for the 50 trials. The results indicate that MCSA outperforms all other compared algorithms in reaching better results. The convergence characteristic curve of the MCSA and the CSA are given in Fig. 8.

Table 12. Best scheduled powers achieved by MCSA for the 160-unit test system. 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

𝑷𝒊 (MW) 240.3582 430.1366 278.9024 218.2682 212.4023 278.6406 239.8207 276.2271 240.4672 285.8031 241.0301 422.2540 274.8176 217.0781 211.4120 286.7077 239.8207 277.4450 239.9297 285.6469 240.7613 429.7990 279.3884 217.0903 211.6596 278.6531 240.0895

Unit 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81

𝑷𝒊 (MW) 277.4305 240.3328 285.3985 240.2239 429.9553 277.3006 216.9899 211.6596 278.6232 239.5520 275.9339 239.5266 285.8861 239.6864 430.3782 277.6723 218.0853 210.9169 279.6489 239.8207 276.0575 240.6016 286.2812 240.8957 432.0265 279.3327 218.0878

82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108

𝑷𝒊 (MW) 211.1644 278.9582 240.0895 277.0038 239.6610 283.8708 240.2239 428.9775 279.4825 218.3928 210.6693 280.1091 239.2833 276.6758 239.6610 285.2345 240.3582 426.7809 278.6139 216.8916 211.1643 285.6992 243.7175 278.6097 240.1984 285.3170 240.3582

Unit

𝑷𝒊 (MW)

Unit

𝑷𝒊 (MW)

109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135

429.3719 278.3662 221.3582 210.9169 279.6466 239.6864 276.5958 239.9297 285.6932 239.9551 431.1971 280.1821 217.0954 211.4120 278.6407 239.8207 276.4344 239.6610 285.3530 239.1489 429.4483 279.2720 217.4805 211.6596 279.6488 240.3582 276.4552

136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160

240.1984 285.1193 240.7613 430.2026 278.3173 218.1031 210.6693 279.6486 240.2239 276.9728 242.4827 285.4954 240.8957 429.7445 276.3896 217.7515 209.6791 279.6484 239.8207 276.2760 239.9297 285.3801 240.0895 429.9943 278.3333

M

A

Generation cost ($/h) = 9982.6846 Loss (MW) = 0 𝐍𝐮 (𝐏𝐢 ) − 𝐏𝐃 | =0.00 𝐄𝐫𝐫𝐨𝐫(𝐌𝐖) = |∑𝐢=𝟏

Unit

SC RI PT

Unit

U

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

𝑷𝒊 (MW) 217.0759 213.8877 280.6569 239.6864 279.1978 240.6015 285.2960 239.9551 430.2418 279.5090 218.2033 211.9071 279.6485 239.8207 277.1311 238.1829 285.0039 240.3582 430.0298 275.5086 220.1583 210.6693 279.6707 239.6864 273.5166 240.1984 287.2023

N

Unit

Table 13. Comparison of results for different methods for the 160-unit system.

a

CC

EP

TE

CGA_MU[4] IGA_MU[4] BBO [22] DE/BBO [22] ORCCRO [22] CBA [17] CSAb[16] CSA ORCSA [25] MCSA-1c CSO[23] MCSA-2d MCSA

Best 10,143.7236 10,042.4742 10,008.71 100,7.05 10,004.20 10,002.8596 9,996.6390 9,991.1561 9,989.9444 9,986.1947 9,984.2438 9,984.0537 9,982.6846

D

Methods

Not reported in the referred literature The cuckoo search algorithm c The MCSA with only the first modification d The MCSA with only the second modification

A

b

Generation Cost ($/h) Mean Worst 10,143.7236 NAa 10,042.4742 NA 10,009.16 10,010.59 10,007.56 10,010.26 10,004.21 10,004.45 10,006.3251 10,045.2265 9,996.6390 10,014.0183 9,992.8468 9,994.1343 9,992.0503 9,996.8317 9,986.7602 9,987.9804 9,984.9163 9,986.3640 9,984.4809 9,985.7749 9,982.9734 9,983.6212

Std. Div NA NA NA NA NA 3.2106 4.9268 1.1146 1.2038 0.4311 0.4032 0.3790 0.2136

11300

CSA

11100

MCSA

Cost $/h

10900 10700 10500 10300 10100 9900 50

100 150 Number of Iterations

200

250

SC RI PT

0

Fig. 8. The cost convergence of the tested CSA and MCSA for the 160-unit test system.

A

CC

EP

TE

D

M

A

N

U

5.7. Parameter setting At above Sections, the results obtained using the optimum setting of MCSA and CSA algorithms to achieve the best solution in each test system are compared with other algorithms. The proposed MCSA introduced some new parameters which make more complex to be used than the original CSA. But, this extra parameters make the MCSA more powerful to solve the optimization problem as seen in the reported results for the test systems 1 to 5 and will be seen in the next section for the benchmark functions. In this part, the method which used for achieving the optimum parameters setting of MCSA and CSA is introduced and the effect of changing of each parameter is given too. Parameters fl and AP are common for both CSA and MCSA. Parameters 𝑓𝑙𝑡ℎ𝑟 , 𝐷𝑡ℎ𝑟 , 𝐾𝑚𝑖𝑛 , and 𝐾𝑚𝑎𝑥 are devoted to MCSA. Each of mentioned parameter has effect on the final solution, so the effect of each parameter should be studied. In this paper, for investigating of each parameter, its value varies in a certain range with a definite step size, while the remaining parameters are fixed in the initial defined values. Then, the results obtained for each step. At the end of cycle, the best step with best cost is selected as the optimum setting value for each parameter. At the beginning of adjusting process an initial set of parameters that have been considered for MCSA include: fl=1.5, AP= 0.1 for both CSA and MCSA; 𝑓𝑙𝑡ℎ𝑟 =2, 𝐷𝑡ℎ𝑟 =0.5; 𝐾𝑚𝑖𝑛 and 𝐾𝑚𝑎𝑥 are set to 5 and 20 percent of number of crows (NC). The changing ranges of parameters are as follows: 𝑓𝑙 ∈ (0,4] with the step size 0.1; 𝐴𝑃 ∈ (0,1] with the step size 0.05; 𝑓𝑙𝑡ℎ𝑟 ∈ (0,4] with the step size 0.1; 𝐷𝑡ℎ𝑟 ∈ (0,4] with the step size 0.1; 0.01𝑁𝐶 ≤ 𝐾𝑚𝑖𝑛 ∈ [0.01𝑁𝐶, 0.2𝑁𝐶] with the step size 0.01NC; 0.2𝑁𝐶 ≤ 𝐾𝑚𝑎𝑥 ∈ [0.2𝑁𝐶, 0.5𝑁𝐶] with the step size 0.01NC. The initial setting, the changing range and the step size of fl and AP for the original CSA are similar to the values which used in MCSA. The number of answers obtained from the above process are large and due to paper size limitation, all of results can’t be presented. So, only some parts of the calculations are presented in this section for different parameters of MCSA in order to understanding the effect of changing each one.

U

SC RI PT

The best cost achieved for test systems 1 to 5 by changing the fl, AP, 𝐾𝑚𝑖𝑛 , 𝐾𝑚𝑎𝑥 , 𝐷𝑡ℎ𝑟 , and 𝑓𝑙𝑡ℎ𝑟 are shown in Tables 14, 15, 16, 17, 18, and 19, respectively. It should be noted that, the result of changing of each parameter given for the steps at the proximity of the optimum value. Obviously, by moving away from the optimum values of MCSA parameters, the results have larger cost and based on the above explanations the results of other steps are not shown. As it can be seen from results, for all systems, the most suitable values of AP and 𝐷𝑡ℎ𝑟 are 0.1 and 0.8, respectively. Also, it can be seen that the most suitable values of fl, 𝐾𝑚𝑖𝑛 , 𝐾𝑚𝑎𝑥 , and 𝑓𝑙𝑡ℎ𝑟 are 2, 5, 25, and 3 for systems 1 to 3, respectively and are 1.9, 7, 27, and 2.8 for systems 4 and 5, respectively. In CSA, with the similar procedure to MCSA, the most suitable values of AP is 0.1 for all systems and the most suitable values of fl is 2 for systems 1 to 3, respectively and is 1.9 for systems 4 and 5, respectively. The results show that the complexity of choosing the optimum values of parameters in MCSA will not be problematic because the optimum value of the controlling parameters for different type of test systems have almost close values. These parameters have almost similar values for different types of benchmark functions as given in next section.

1.7 1.8 1.9 2 2.1 2.2 2.3

15,457.8361 15,457.2795 15,456.2158 15,455.1439 15,456.2194 15,457.3193 15,458.4539

631.7310 628.5894 626.4586 625.3193 627.1188 630.6305 632.8022

system 3 case1 127,412.4353 127,389.3426 127,375.2418 127,324.1218 127,372.2711 127,380.3823 127,411.4528

system 3 case2 140,718.4236 140,589.8635 140,449.6733 140,361.1068 140,479.8932 140,612.1437 140,803.8673

A

system 2

M

system 1

system 4

system 5

200,124.4284 199,837.3248 199,573.3741 199,756.4395 199,897.7630 199,940.0185 200,241.1935

10,197.9684 10,176.5495 10,151.7366 10,178.7245 10,187.1439 10,195.9832 10,215.7630

D

fl

N

Table 14. Effect of the 𝑓𝑙 on the MCSA performance in various test systems.

TE

Table 15. Effect of the 𝐴𝑃 on the MCSA performance in various test systems. system 1

system 2

0.05 0.10 1.15 0.20 0.25 0.30 0.35

15,455.3568 15,455.1374 15,455.3638 15,456.4500 15,457.8963 15,461.9530 15,465.0561

626.8740 625.7289 627.7612 628.0050 630.6683 633.6395 636.1781

CC

EP

AP

system 3 case1 127,381.0045 127,373.4295 127,383.7284 127,394.1044 127,407.8076 127,324.1045 127,450.3869

system 3 case2 140.491.7304 140,476.3349 140.511.0165 140.689.8745 140.767.2354 140.890.9817 140.996.8065

system 4

system 5

199,710.1576 199,607.1629 199,731.9706 199,804.9572 199,945.4854 200,345,8003 200,670.1419

10,167.9157 10,155.2349 10,179.7922 10,208.9595 10,235.6557 10,297.0357 10,343.8491

A

Table 16. Effect of the 𝐾𝑚𝑖𝑛 on the MCSA performance in various test systems. 𝑲𝒎𝒊𝒏

system 1

system 2

3 4 5 6 7 8 9

15,466.7577 15,463.7432 15,456.9340 15,461.9502 15,459.6555 15,464.1712 15,468.6787

633.0318 630.2769 626.7060 630.0344 622.8235 634.6948 635.3171

system 3 case1 127,447.3816 127,423.7655 127,392.4387 127,429.7952 127,441.1869 127,467.4898 127,498.4456

system 3 case2 140,810.6463 140,589.7094 140,511.7547 140,655.2760 140,735.6797 141,008.6551 141,323.1626

system 4

system 5

201,166.1190 200,478.4984 200,002.9597 199,875.3404 199,630.5853 200,056.2238 200,433.7513

10,680.2551 10,531.5060 10,377.6991 10,297.8909 10,201.9593 10,356.5472 10,473.1386

Table 17. Effect of the 𝐾𝑚𝑎𝑥 on the MCSA performance in various test systems. system 1

system 2

23 24 25 26 27 28 29

15,463.1493 15,458.2543 15,455.8143 15,464.2435 15,468.9293 15,478.1966 15,489.3500

637.2511 632.6160 628.4733 631.3517 634.8308 641.5853 648.5497

system 3 case1 127,441.9172 127,420.2858 127,401.7572 127,423.7537 127,445.3804 127,468.5678 127,487.0759

system 3 case2 140,810.6463 140,589.7094 140,496.7547 140,655.2760 140,735.6797 141,008.6551 141,411.1626

system 4

system 5

201,089.0540 200,666.5308 200,383.7792 199,875.9340 199,488.1299 199,834.4694 200,389.0119

10,563.3371 10,498.1622 10,355.7943 10,301.3112 10,254.5285 10,390.1659 10,420.6020

SC RI PT

𝑲𝒎𝒂𝒙

Table 18. Effect of the 𝐷𝑡ℎ𝑟 on the MCSA performance in various test systems. 𝑫𝒕𝒉𝒓

system 1

system 2

0.5 0.6 0.7 0.8 0.9 1.1 1.2

15,481.2630 15,474.6541 15,462.6892 15,455.7482 15,463.4505 15,470.0838 15,476.2290

644.9133 636.1524 629.8258 627.5383 632.9961 638.0782 644.4427

system 3 case1 127,490.1067 127,456.9619 127,428.0046 127,380.7749 127,466.8173 127,487.8687 127,500.1844

system 3 case2 140,946.3998 140,611.2599 140,507.8001 140,443.4314 140,650.9106 141,121.1818 141,352.2638

system 4

system 5

200,558.1455 200,166.1361 199.835.8693 199,462.5797 199,723.1145 200,324.8530 200,705.6221

10,721.3520 10,443.5132 10,279.4018 10,180.0760 10,288.2399 10,350.1233 10,673.1839

15,487.2400 15,481.4173 15,475.0497 15,467.9027 15,456.9448 15,463.4909 15,473.4893

644.3377 638.9001 634.3692 629.1112 626.7803 630.3897 635.2417

system 3 case2 141,223.8003 140,958.5448 140,720.1443 140,335.7893 140,281.5254 140,359.6476 140,752.3972

N

2.6 2.7 2.8 2.9 3.0 3.1 3.2

system 3 case1 127,481.4039 127,459.0965 127,411.9235 127,366.3155 127,348.0905 127,395.8916 127,427.5061

A

system 2

M

system 1

D

𝒇𝒍𝒕𝒉𝒓

U

Table 19. Effect of the 𝑓𝑙𝑡ℎ𝑟 on the MCSA performance in various test systems. system 4

system 5

200,438.4330 199,963.9259 199,555.7612 199,789.3466 199,870.1055 200,248.5436 200,681.0916

10,527.4146 10,329.3323 10,210.3762 10,374.4837 10,508.7010 10,622.0960 10,803.5942

A

CC

EP

TE

5.8. Multi-period ELD problem In the previous test systems, the single-period ELD was investigated which all the calculations obtained for the single hour. However, to test the MCSA ability in solving the ELD problem under ramp-rate constraint, the multi-period ELD should be considered. In this regard, a 10 unit test system in a 24-hours period is solved using the proposed MCSA while the transmission loss and the valve point effect are considered too. The system data consist of the cost coefficients, the ramp-rate limit, the power generation upper and down limits, the B loss coefficient, and the load data are totally extracted from [40]. The load data for 24 hours is shown in Table 20. The population size and the maximum iteration both are set to be equal to 100. With the similar process to the section 5.7, in CSA and MCSA, the parameters AP and fl are set to be 0.1 and 1.8, respectively. Also, in the MCSA, the amounts of 𝑓𝑙𝑡ℎ𝑟 , 𝐷𝑡ℎ𝑟 , 𝐾𝑚𝑎𝑥 , and 𝐾𝑚𝑖𝑛 are set to 2.9, 0.8, 25 and 5, respectively. The best scheduled powers achieved by the MCSA for 24-hour time interval are given in Table 21. According to this Table, the power rate of the units in each time interval within its lower and upper capacity and the changing of the power from one hour to the next ones satisfies the ramp-rate limit. Also, the error parameter which shows the violation of equality constraint (𝑬𝒓𝒓𝒐𝒓 = |∑Nu i=1(Pi ) − PD − PTL |) has an acceptable value in each time interval.

In Table 22, the results obtained with MCSA-1, MCSA-2, original CSA, and other reported algorithms in literatures are presented for the 30 independent runs. These algorithms include: EP [40], EP-SQP [40], DGPSO [40], IPSO [41], AIS [42], SOS [43], GA [44], ABC [44], PSO [44], CDBCO [45], MHEP-SQP [46], CE [47], ECE [47], TVAC-IPSO [48], ICA [49], IPM [50], and MILP-IPM [50]. The total cost of the MCSA is equal to 1,040,435 ($/h) for 24-hour interval, which is a lower value than other reported algorithms. Table 20. The load data for a period of 24 hours. 1 1036 13 2072

2 1110 14 1924

3 1258 15 1776

4 1406 16 1554

5 1480 17 1480

6 1628 18 1628

7 1702 19 1776

8 1776 20 2072

9 1924 21 1924

10 2072 22 1628

11 2146 23 1332

SC RI PT

Hour Load(MW) Hour Load(MW)

12 2220 24 1184

Table 21. Best scheduled powers achieved by MCSA for the 10-unit test system during 24hour time interval.

TE

EP CC A

𝑷𝟖 (MW) 47.0000 47.0000 47.0000 47.0000 47.0000 47.0000 77.0000 47.0000 47.0000 77.0000 85.3121 115.3121 85.3121 85.3121 85.3121 85.3121 85.3121 85.3121 85.3121 85.3121 85.3121 55.3121 47.0000 47.0000

U

𝑷𝟕 (MW) 129.5904 129.5904 129.5904 99.5904 129.5904 129.5904 129.5904 129.5904 129.5904 129.5904 129.5904 129.5906 129.5906 129.5904 129.5904 129.5904 129.5904 129.5904 129.5904 129.5904 129.5904 129.5904 130.0000 129.5904

N

𝑷𝟔 (MW) 122.4498 122.4499 122.4498 122.4545 122.4498 122.4713 122.8996 122.4504 160.0000 160.0000 160.0000 160.0000 122.5226 160.0000 122.4641 122.4498 122.4707 122.4498 122.4498 160.0000 123.4766 122.4498 122.4346 122.4498

A

𝑷𝟓 (MW) 73.0000 122.8666 73.0000 122.8666 172.7331 172.7331 222.7331 172.7331 222.5997 222.5997 234.5443 229.4122 222.5999 172.5999 122.8666 73.0000 123.0000 172.7327 222.5996 222.5997 222.5997 172.5997 122.8211 73.0000

M

𝑷𝟒 (MW) 60.0000 60.0000 60.0000 60.0025 60.0000 110.0000 120.4155 120.4152 170.4152 180.8305 230.8305 280.8305 300.0000 250.0000 200.0000 179.5180 129.5180 179.4733 129.4733 179.4733 180.8305 130.8305 117.8911 120.1993

D

𝑷𝟏 𝑷𝟐 𝑷𝟑 (MW) (MW) (MW) 226.6242 135.0000 179.0548 1 226.6242 135.0000 204.7815 2 303.2484 215.0000 251.8231 3 379.8726 222.2665 300.1509 4 303.2484 302.2664 296.5396 5 379.8726 309.5329 314.6606 6 379.8726 309.5329 299.3960 7 456.4968 389.5329 305.8253 8 456.4968 396.7994 314.3504 9 456.4968 460.0000 336.3757 10 456.4969 460.0000 340.0000 11 456.4971 460.0000 340.0000 12 456.4969 396.7994 303.0083 13 379.8726 396.7994 316.4233 14 379.8726 396.7994 302.7317 15 303.2484 316.7994 296.4351 16 226.6242 309.5329 305.0062 17 303.2484 309.5329 280.5813 18 379.8726 389.5329 283.0136 19 456.4968 460.0000 329.1628 20 456.4968 396.7994 300.3460 21 379.8726 316.7994 278.4586 22 303.2268 236.7994 198.4586 23 226.6242 222.2665 184.8288 24 Total generation cost ($/h) = 1,040,434.6164 Total loss (MW) = 858.6393 Hour

𝑷𝟗 (MW) 20.0000 20.0000 20.0000 20.0000 20.0000 20.0000 20.0000 20.0000 20.0000 50.0000 52.0571 52.0571 52.0571 22.0571 20.0000 20.0000 20.0000 20.0000 20.0000 50.0000 20.0000 20.0000 20.0000 20.0000

𝑷𝟏𝟎 (MW) 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000 55.0000

Cost ($/h) 28480.9817 30218.0425 33788.2926 36936.2300 38470.8887 42007.1779 43713.6461 45420.1948 49084.0587 53424.2995 55306.0108 57366.9109 52594.9420 49290.9512 45609.9058 40414.1936 38656.7900 42081.3654 45652.2976 53207.8574 48756.2576 42481.0633 35615.8875 31856.3706

Loss (MW) 11.7193 13.3125 19.1118 23.2040 28.8278 32.8611 34.4402 43.0443 48.2519 55.8931 57.8312 58.6995 51.3867 43.6548 38.6369 27.3533 26.0546 29.9209 40.8444 55.6351 46.4516 32.9132 21.6317 16.9590

Error (MW) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Table 22. Comparison of the results from different methods for the 10-unit test system during the 24-hour time interval. Generation Cost ($/h) Mean Worst 1,057,323 NAa 1,053,771 NA 1,058,041 1,062,511 1,052,349 NA 1,051,725 NA 1,052,092 1,057,170 NA NA 1,048,145 NA 1,047,050 1,048,431 1,045,159 NA 1,044,470 NA 1,044,963 1,046,805 1,044,700 NA 1,042,900 1,042,945 1,041,633 1,042,325 1,042,118 1,043,626 1,041,393 1,041,901 1,041,665 1,043,174 1,041,026 1,041,473 NA NA 1,040,490 1,040,587

Std. Div NA NA NA NA NA NA NA NA NA NA NA NA NA NA 28.8514 NA 21.5375 NA 19.2439 NA 9.1459

U

EP [40] EP-SQP [40] GA [44] MHEP-SQP [46] DGPSO [40] PSO [44] IPM [50] IPSO [41] AIS [42] CE [47] ECE [47] ABC [44] CDBCO [45] SOS [43] CSA TVAC-IPSO [48] MCSA-1b ICA [49] MCSA-2c MILP-IPM [50] MCSA

Best 1,054,685 1,052,668 1,052,251 1,050,054 1,049,167 1,048,410 1,047,294 1,046,275 1,045,715 1,044,051 1,043,989 1,043,381 1,042,900 1,042,869 1,041,098 1,041,066 1,040,978 1,040,758 1,040,712 1,040,676 1,040,435

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Methods

a

Not reported in the referred literature The MCSA with only the first modification c The MCSA with only the second modification

A

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b

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5.9. Benchmark functions At the previous sections, the comparison of different algorithms has been done for solving the ELD optimization problem. However, it is appropriate to test the performance of MCSA in other problems. Benchmark instances would be appropriate to deeply understanding the effectiveness of the proposed MCSA in the optimization problems. In this regard, four wellknown benchmark functions are optimized. These functions are introduced in Table 23. The dimensional size of each function (DS) are assumed to be 50 in this paper and the results found by MCSA, CSA, ICA, PSO and GA over the 50 independent runs. The population size and the total number of iteration for all methods set to 100 and 2000, respectively. The controlling parameters of each compared algorithm are tuned so that the best result can be achieved. The process of tuning of each parameters for each algorithm is similar to the method that introduced in the previous section, these settings are as follows. In CSA and MCSA, AP and fl are set to 0.1 and 1.8, respectively. Also, for MCSA, the amounts of 𝑓𝑙𝑡ℎ𝑟 , 𝐷𝑡ℎ𝑟 , 𝐾𝑚𝑎𝑥 and 𝐾𝑚𝑖𝑛 set to 2.3, 0.8, 20 and 3, respectively. In ICA, the number of empire select as 10, the movement factor of colonies (𝛽), and 𝜉 parameter which indicates the impact of colonies on the total price of empire are set to 2 and 0.05, respectively. In PSO, the acceleration constant C1 and C2 both are set to 2, the inertia weight decreases linearly from 0.9 to 0.4 during the iterations. In GA, the crossover rate (with the coefficients of 0.25 and 0.75), the uniform mutation and tournament selection are used. The crossover and mutation probabilities are set to 0.9 and 0.005, respectively.

The statistical results consist of best solution of all runs, the optimized solution average of the all runs and the standard deviation during the runs shown in Table 24. Table 24 shows that MCSA outperforms the other algorithms which confirms high capability of MCSA in optimizing different type of benchmark functions. Table 23. Used benchmark functions in this study Function

Name of the function 𝒇𝟏(𝒙) = ∑

Rosenbrock function

𝑓2(𝑥) = ∑

Rastrigin’s multimodal function

𝑓3(𝑥) =

𝑫𝑺

𝒙𝟐𝒌

𝒌=𝟏 𝑫𝑺−𝟏 𝒌=𝟏 𝑫𝑺

(100(𝑥𝑘+1 − 𝑥𝑘2)2 + (𝑥𝑘 − 1)2 )

∑ (𝑥𝑘2 𝒌=𝟏

− 10 cos(2𝜋𝑥𝑘 ) + 10)

𝐷𝑆 𝐷𝑆 1 1 𝑓4(𝑥) = −20𝑒𝑥𝑝 (−0.2√ ∑ 𝑥𝑘2) − 𝑒𝑥𝑝 ( ∑ cos 2𝜋𝑥𝑘 ) + 20 + 𝑒 𝑛 𝑛 𝑘=1 𝑘=1

Ackley function

Global minimum

[-100,100]

0

[-30,30]

0

[-5.12,5.12]

0

[-32,32]

0

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Sphere function

Search boundaries

CSA

ICA

PSO

TE

GA

F2 2.8767 10.4655 21.0764 4.1645 15.4217 26.9157 5.6922 18.9594 28.6557 5.9743 20.8865 31.0357 18.8491 96.9339 118.6787

N

Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div.

A

MCSA

F1 4.7395e-85 3.8723e-81 5.1459e-80 3.6323e-71 7.0975e-66 2.6059e-62 3.2785e-57 3.8723e-50 2.0603e-45 5.4853e-41 4.1568e-36 4.8575e-29 3.2636e-20 5.4584e-16 5.9758e-08

M

Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div.

D

Methods

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Table 24. Comparison of MCSA, CSA, ICA, PSO and GA performance in minimization of the benchmark functions. Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div.

F3 62.5286 94.7577 128.7431 76.3922 125.6554 156.1711 107.7060 148.0318 188.2769 97.0461 145.8234 182.6948 156.3171 243.9502 311.4387

Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div. Best Mean Std.Div.

F4 1.5648e-15 1.3815e-09 2.7655e-13 3.7952e-12 6.1868e-05 5.4897e-07 1.5396e-05 4.2748e-02 7.5385e-04 2.4455e-04 4.7436e-02 9.8975e-04 1.5465e-01 0.6463 0.2142

A

CC

EP

5.10. Mann-Whitney test Since there is no statistical test for comparing the average of two optimization method with arbitrary distributions (for example "MCSA and PSO" or "MCSA and GA" or "MCSA and ICA" or etc.), the results of statistical test namely Mann-Whitney test is presented to improve the confidence of the results. The Mann-Whitney test is a non-parametric test which used to compare whether two population means are equal or not [51]. In this test, two types of hypothesis used to compare the methods. The null-hypothesis (denoted by H0) is a default position that specifies there is no significant relationship between two measured groups, or the groups are similar with no difference. The alternate-hypothesis (denoted by H1) is the opposite of the null-hypothesis. Actually, at the beginning it assumed the H0 to be true and the H1 is accepted by rejection of the H0. Suppose the goal is to compare two EAs methods A and B which have been run n a and nb times, respectively. Also, fa and fb represent obtained probability density of solution quality using the A and B, respectively. The null-hypothesis defined as 𝑓𝑎 = 𝑓𝑏 and if the probability that H0 is true would be lower than a selected significance level, then the null-hypothesis will be rejected, accordingly, the alternate-hypothesis will be accepted [52]. It means that A, and

The cuckoo search algorithm

0.0000 0.0000 0.0000 0.0003 0.0000 0.0002 0.0006 0.0000 0.0013 0.0010

M

A

p-value

9.5682 9.6541 8.6485 9.6241 8.8134 7.6582 8.4982 8.3236 7.5896 7.5646

EP

a

Z-score

172625 165745 172926 173149 171267 168627 153462 161308 148624 151389

TE

MCSA and CGA_MU MCSA and IGA_MU MCSA and BBO MCSA and DE/BBO MCSA and ORCCRO MCSA and CBA MCSA and CSAa MCSA and CSA MCSA and ORCSA MCSA and CSO

Test result U-value

D

Compared methods

N

U

Table 25. The Mann-Whitney test results of comparison MCSA with other methods for the 160-unit test system.

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B have different solution quality. In other words, the probability that the method A have similar solution quality to the method B is rejected and the probability that A generates a solution better than B is lower (or larger) than the probability that B generates a solution better than A. In this section, the Mann-Whitney test introduced in [52] is used for the fifth test system (160 unit test) while the significance level is 1% for two-tailed. The test results of comparison the proposed MCSA with other method are given in Table 25. In Table 25 for MCSA and CGA_MU, the U-value and Z-score are calculated as 172625 and 9.5682, respectively and the p-value is almost equal to zero. Therefore, the result is significant at 𝑝 ≤ 0.01. Accordingly, the null-hypothesis is rejected and also by consideration the results reported in Table 13, the probability that the MCSA generates a solution better than CGA_MU is larger than the probability that the CGA_MU generates a solution better than MCSA. For other rows of Table 25, with a same reason, the superior performance of MCSA than other methods can be concluded.

6. Conclusions

A

CC

In this paper, the modified crow search algorithm (MCSA) was proposed to solve the nonconvex economic load dispatch problem. The modification applied in the original crow search algorithm (CSA) consist of two process. First, a priority selection method for each crow in the population to choose a more efficient target crow as the destination which decreases the possibility of flopping crows in finding the better food sources. Second, determining the efficient amount of flight length based on the amount of proximity of crows which helps in a better exploration. MCSA has been tested on five test systems with 6, 10, 40, 110, and 160 generating units, respectively. For investigating the performance of the proposed method in solving the problem with more consistent and better quality rate, the statistical analysis is done for each case and the results are compared with the previous literatures. The superiority of MCSA over other algorithms was confirmed. The convergence curves for MCSA and CSA are given in each case too. The better convergence feature of MCSA over the original CSA

was confirmed too. Also, MCSA was tested for minimization of the various benchmark functions which results in the success of MCSA in obtaining the better solutions. In the future work, MCSA will be used for solving the economic dispatch problem in microgrids considering the voltage stability.

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