Layout optimization of a wind farm to maximize the power output using enhanced teaching learning based optimization technique

Accepted Manuscript Layout optimization of a wind farm to maximize the power output using enhanced teaching learning based optimization technique Jaydeep Patel, Vimal Savsani, Vivek Patel, Rajesh Patel PII:

S0959-6526(17)30863-6

DOI:

10.1016/j.jclepro.2017.04.132

Reference:

JCLP 9496

To appear in:

Journal of Cleaner Production

Received Date: 6 December 2016 Revised Date:

23 March 2017

Accepted Date: 22 April 2017

Please cite this article as: Patel J, Savsani V, Patel V, Patel R, Layout optimization of a wind farm to maximize the power output using enhanced teaching learning based optimization technique, Journal of Cleaner Production (2017), doi: 10.1016/j.jclepro.2017.04.132. 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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Layout optimization of a wind farm to maximize the power output using enhanced Teaching Learning Based Optimization technique

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Jaydeep Patel Pandit Deendayal Petroleum University

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Gandhinagar, Gujarat, India-382007

Vimal Savsani*

Gandhinagar, Gujarat, India-382007

Vivek Patel Pandit Deendayal Petroleum University

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Gandhinagar, Gujarat, India-382007

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Pandit Deendayal Petroleum University

Rajesh Patel

Pandit Deendayal Petroleum University

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Gandhinagar, Gujarat, India-382007

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*Corresponding Author

Email: Vimal Savsani:- [email protected] Jaydeep Patel:- [email protected]

Phone no:- +91 98250 92139

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Abstract The global warming is a major concern in the present era that arises a need of cleaner production of energy. The wind energy is a major source of contribution for such clean energy demands.

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The Wind farm layout optimization (WFLO) problem deals with the optimum placement of wind turbines in a wind farm so as to maximize the total power output with minimum cost of energy. The placement of turbines is crucial for a wind farm because the power generation of wind

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turbine decreases if it is in the wake effect produced by the upstream turbines. So, WFLO problem is a challenging combinatorial optimization problem for which many direct search and

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local optimization method fails to attend the global optimum solution. The Meta-heuristic method often provides the effective solution for such problems in terms of convergence and the quality of the solution. In this work two, different metaheuristics algorithms are proposed to solve WFLO problem. These algorithms are developed by incorporating changes in the basic

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Teaching-learning based optimization (TLBO) algorithm. The proposed algorithms eliminate the limitations of basic TLBO algorithm to enhance its exploration and exploitation by incorporating effective search techniques. The implementations of the proposed algorithm are effective to

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optimize the position of wind turbines in a wind farm to maximize the expected power output of a wind farm with a minimum investment cost. The proposed algorithm is investigated for WFLO

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problem and a set of 10 challenging real life benchmark problems. The numerical results indicate that the proposed method is an effective technique to solve the WFLO problem compared to its basic algorithm and other state of the art methods. The optimum design of the wind farm results in the economical utilization of the wind resource and leads to clean energy production.

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Keywords Wind farm layout optimization, Wind energy, Teaching-learning based optimization, Metaheuristic

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Highlights

New methodology is proposed for wind farm layout optimization

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New enhanced TLBO is proposed

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Experimentation done for WFLO and Real-world engineering applications

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Proposed method is effective to others methods

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Nomenclatures Notations in formulation


 = Effective wind velocity at turbine k under single wake (m/s)

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 = Free stream wind velocity (m/s)

 = Turbine thrust coefficient

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 = Distance from the upstream turbine to downstream turbine following the wind direction (m)

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 = Rotor diameter (m)

ℎ = Hub height of wind turbine (m)

ℎ = Surface roughness of farm terrain (m) = Velocity deficit (m)

 = Power extracted by the ith wind turbine (kW)

ACCEPTED MANUSCRIPT  = Rotor swept area (m2 )

 = Number of wind turbine installed in a wind farm = Teacher solution (best solution)  = Teaching factor

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 = New solutions

 = ith solution

Greek symbol

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 = kth random solution  ! =Best solution

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 & ∅ = Random number

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   = Total power extracted by the wind farm (kW)

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" = Efficiency of a wind turbine

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# = Air density (kg⁄m3 ) Abbreviations

WFLO = Wind farm layout optimization TLBO = Teaching-learning based optimization TLBOe = Enhanced Teaching-learning based optimization

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AL= ABC inspired search mechanism along with the basic learner phase PAL= Combined PSO and ABC inspired search mechanism along with basic learner phase

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DED= Dynamic Economic Dispatch ELD= Static Economic Load Dispatch

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1. Introduction

There are several environmental issues such as emission of greenhouse gases and global

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warming that result from the use of fossil fuels. So, there is a need to move towards the renewable energy sources which have less environmental effects and can be considered as a source of cleaner energy production. Out of all the available renewable energy sources, wind energy ranks at the top in terms of CO2 equivalent, water utilization and social impacts (Evans et

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al., 2009). So, Wind energy can be considered as one of the best cleaner production technology. The characteristic of the wind is stochastic in nature, and so, the power output of a wind turbine also follows stochastic characteristics. This phenomenon of wind power generation often

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required the forecasting of wind power generation. Nowadays, the power distribution follows the smart grid technology to satisfy the demand location (Boroojeni et al., 2015). The wind energy

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contributes nearly 500 GW of the total power produced across the globe (WWEA, 2016). In the last few decades, wind energy has a remarkable growth for the electric power generation that uses the wind turbine to convert the wind energy in the form of electric energy. The wind turbines are installed in a cluster in the wind farm to effectively use the available land that can produce more power. The cluster placement of the turbines reduces the operation, maintenance and installation cost (Pookpunt and Ongsakul, 2013). When the turbine comes in contact with the wind, it utilizes the kinetic energy of the wind and produces rotational motion of the blades

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which can be used to produce the electrical energy. The rotation of the turbine acts as a rotational obstacle which produces the wake zone and reduces the velocity of the wind behind it (Turner et al., 2014). If any turbine is placed in this wake zone, then it will receive the wind with the lower

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velocity compared to its upstream turbine. So, it is obvious that both upstream turbine and wake affected turbine will produce different power in which the power produce by the later will be less. Moreover, the power produced by the wake affected turbine depends on the position in the

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wake zone and the relative distance to its upstream turbine. This phenomenon can be converted in the form of an optimization problem to find the best placement which can optimize the

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required objective such as maximum power generation and minimum cost of energy produced by the wind farm. Such type of wind turbine placement problem is classified as a ‘wind farm layout optimization (WFLO)’ problem. So, the placement of turbine in a cluster can be observed as a combinatorial optimization problem which consists of finding an optimum set (placement) from the available finite set (discretization of the wind farm). Suppose, the available placement is ‘n’

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and it is required to place ‘r’ number of turbines, it leads to %!/(! (% − )!) combinations. This indicates that the computational complexity of the problem increases with ‘n’ and so it may lead

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to the failure of many exhaustive search methods as well as local search method such as integer programming. So, the meta-heuristics method such as genetic algorithm (GA), differential

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evaluation (DE), particle swarm optimization (PSO), etc., can provide an effective solution with comparative computational efforts for WFLO problem. Many researchers have used these types of methods to address such problems. The optimum positioning of turbines in a wind farm to maximize the power extracted was first proposed by the Mosetti et al. (1994). He employed a GA to a discrete formulation of the wind farm layout optimization and had used the Jenson’s wake decay model (Jensen, 1983) to

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calculate the influence of wake on the downstream turbine. Grady et al. (2005), Wu et al. (2014) and Gao et al. (2014) used more individuals (population size) in the genetic algorithm to enhance the exploration of the solutions in a search space to obtain the better output. In addition to this,

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advanced genetic algorithm approaches, i.e., a hybrid GA (Réthoré et al., 2014), a quadratic assignment problem-genetic algorithm(QA-GA) (Rahbari et al., 2014), a distributed genetic algorithm (DGA) (Huang, 2007) and hybrid DGA (Huang, 2009) had been used to solve the

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WFLO problems. Apart from single objective problems, Şişbot et al. (2010) proposed a multiobjective GA (MOGA) to obtained optimal placement of wind turbines in a farm on Gokceada

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Island-Turkey. Chen et al. (2015) had also applied the MOGA to obtained optimal placement of wind turbines but used the micro-sitting of the turbine instead of gird-based placement. Many other meta-heuristic methods such as ant colony optimization algorithm (ACO) (Eroğlu and Seçkiner, 2012), greedy randomized adaptive search procedure (GRASP) (Yin and Wang, 2012),

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random search (RS) algorithm (Feng and Shen, 2015), teaching-learning based optimization (Patel et al., 2015), differential evolution (Rašuo et al., 2010), quadratic optimization (OQ) with mixed integer linear (MIL) method (Turner et al., 2014), simulated annealing (SA) (Bilbao and

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Alba, 2009) and particle swarm optimization (PSO) (Chowdhury et al., 2013) algorithm had also been investigated to address the WFLO problems. Apart from the basic algorithms, many

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researchers also modify the basic algorithms to obtain the better layout of a wind farm, i.e., binary particle swarm optimization (BPSO) (Pookpunt and Ongsakul, 2016), the hybrid EPS/GA (DuPont and Cagan, 2016), etc. As observed from the above references that several research is reported to solve the WFLO problems by using different meta-heuristics technique. Many modifications are imposed on the basic algorithm such as GA, PSO to enhance its search technique which can solve WFLO

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problem effectively. To overcome the limitation of a genetic algorithm such as slow convergence rate and high computational cost, other methods such as PSO, DE and TLBO are implemented to solve WFLO problem. The latest method to solve WFLO problem is teaching-learning based

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optimization (TLBO) method which has produces effective results over the other methods (Patel et al., 2015). The search technique of TLBO can be enhanced by incorporating some modification in its basic search technique. The modified version of TLBO has proved its

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effectiveness on several engineering applications (Rao, 2016). No modified version of TLBO are reported to address WFLO problem and so, considering this as a motivation factor; the efforts are

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put to modify the basic version of TLBO so that it can solve WFLO problem effectively. In the present study, the authors have proposed a new method called enhanced TLBO (TLBOe) for solving WFLO problem. To different variant of TLBOe, namely AL (ABC inspired search mechanism with learner phase) and PAL (combined PSO and ABC inspired search mechanism

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with learner phase) are proposed in the present work. The TLBOe can overcome the limitation of the basic TLBO to maintain the proper trade-off between the exploration and exploitation capability of the algorithm. Moreover, earlier the TLBO algorithm had been applied to solve the

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WFLO problem considering the basic cases such as uni-directional and multi-directional uniform

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wind velocity (Patel et al., 2015). In the present work, the authors have considered the more realistic approach of wind characteristics by considering multi-directional variable wind velocities. The incorporation of this practical approach makes WFLO problem more challenging for an optimization algorithm to solve. So, there are two major contributions of this work, firstly the basic version of TLBO is modified by different search techniques to address the WFLO problem and Secondly, to investigate the applicability of the proposed enhanced TLBO method in the design of wind farm layout optimization by considering the practical case of wind

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characteristics with multi-directional variable wind velocity. The optimize layout of turbines in a wind farm will produce more power with high efficiency that reduce the CO2 emission (g/kWh) and surety towards cleaner energy. that Furthermore, the potential of the proposed enhanced

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algorithms is evaluated by applying them to a set of ten real world benchmark problems from CEC2011(Das and Suganthan, 2010).

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The rest of the paper is organized as follows; Section 2 describes the wake modeling of a wind farm followed by the description and the structure of enhanced TLBO (TLBOe) in Section 3.

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Section 4 presents the implementation and validation of the algorithm for wind farm layout optimization problem and ten real world benchmark problems. Finally, the conclusions are drawn in section 5. 2. Mathematical Wake model

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As a first step in the direction of solving the wind farm layout optimization problem, it is necessary to calculate the velocity deficit at each turbine to obtain the power output of a wind farm. The present study uses a Jenson’s wake decay model for calculating the velocity deficit at

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each turbine (Katic et al., 1986). This model works on the conservation of momentum within the wake region which states that when steady approaching wind smacks a turbine, a turbulent wake

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is developed behind the turbine (Schepers, 2003). This turbulent wakes expands linearly depending upon the downwind distance and increases velocity deficits at the downstream turbines in a wake region. Hence, the wind velocity at the downstream turbine is affected and reduced. A schematic illustration of Jensen’s linearly expanding wake decay model is pictured in Fig. 1.

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Fig.1. Schematic of the wind wake model It is observed from the Fig. 1, that the turbine K is affected by the wake region created by the

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turbine I only. Hence, the effective wind velocity
 at any single wake affected downstream

turbine K is computed by the Eq. (1).
 =
 +1 − -

./0./12

= 5 < 67(8⁄89 ):;

3.4

>?

(1)

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 is the free stream velocity,  is turbine thrust coefficient,  is a distance from the

upstream turbine to downstream turbine following the wind direction,  is a rotor diameter, ℎ is

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the hub height of wind turbine and ℎ represents the surface roughness of farm terrain.

The wake effect becomes more tricky when the turbine is located under many wake regions created by the upstream turbines. This phenomenon can be observed for turbine L where, the turbine L is encountering multiple wake regions created by the turbine I and J. Hence, the

F = @∑G. 31 −

E

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the kinetic energy deficits of individual wakes given as: BC D B9

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resulting velocity deficit of a turbine encounter  wakes which can be computed by summing <

(2)

Hence, the effective wind velocity
 at any turbine i affected by the many wake regions is


 =
 (1 − )

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computed by the Eq. (3).

(3)

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The effective wind velocity at any turbine given by the Eq. (3) is used to compute the power extracted by the wind turbine. The power extracted by the wind turbine is a cubic function of a wind velocity and given by the Eq. (4).

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 = " 3 #
H < . D

(4)

where, " is an efficiency of a wind turbine, # is an air density and  is rotor swept area. The total power extracted by the wind farm is cumulative of power extracted by the individual turbine placed in a wind farm and given by the Eq. (5).

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E

(5)

The objective of the wind farm layout optimization is to minimize the cost of energy per unit

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produced. This can be possible when the energy extracted by the wind farm is maximum. The present study considers the non-linear cost model proposed by Mosetti et al. (1994) for calculating the cost of energy.

NFOP5 QR;S

=

= = U T T Z2 U ∑O^U [3 \]BOT < =

E2 3 4  V9.99UXYZ2 <

Where,



(6)

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IJ%JKJLM

1 !

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Hence, the wind farm layout optimization problem is formalized as:


G
, (`ℎM% abcJ%M Jd b%M %e `fgMd) ./0./12

= 5 < 67(8⁄89 ):;

3.4

>?, (`ℎM% abcJ%M Jd b%M dJ%hiM `fgM )

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G
 +1 − -

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EF Du l

G
 k1 − mn o1 − p qr t , (`ℎM% abcJ%M Jd b%M Kf%v `fgMd)
 k t G. j s

Subjected to:

The area of the wind farm is required to be discretized by small square cells and the placement of each turbine is restricted to be placed at the center of each square cell.

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3. Variants of enhanced Teaching Learning based optimization (TLBOe) In this section, the concepts of basic teaching-learning based optimization (TLBO) and different

3.1 Teaching-Learning Based Optimization Algorithm

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variants of enhanced teaching-learning based optimization (TLBOe) are explained.

The genetic algorithm requires algorithm parameter, such as crossover probability, mutation

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probability and selection methods whereas, PSO needs the value of inertia weight, social and cognitive parameters for its working. ABC also requires parameter such as a number of

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employed and onlooker’s bees, as well as the value of limit for its working. Similarly, other population-based method need algorithm parameters for its working. These algorithm parameters affect the effectiveness of an algorithm, and so, there is a direct influence of these parameter on the obtained solutions. Hence, the effort of a designer is consumed to determine appropriately algorithm parameter for the specified problem. These efforts are reduced if a designer prefers an

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algorithm which does not require any algorithm-specific parameter. TLBO does not need any algorithm-specific parameter except the population size (set of solution) and the termination criteria. So, this algorithm does not require any efforts to tune the algorithm parameters as

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2011).

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required by many other population-based method such as GA, PSO, ABC ,etc., (Rao et al.,

The TLBO algorithm is a nature-inspired population-based stochastic global search metaheuristic algorithm developed by Rao et al. (2011) inspired by an idea of teaching and learning. The TLBO works on the strength of a teacher to teach a group of learners. TLBO algorithm works in two phases; the first phase consists of the ‘Teacher phase’ that simulates the influence of a teacher on learners, and the second phase is a ‘Learner Phase’ that models the learning through the interaction between learners. In TLBO, the population represents the group of

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learners whereas different design variables can be represented by various subjects (courses) offered to learners. Fitness value of the optimization problem is analogous to a learner’s grade or results (Rao and Patel, 2012). In TLBO, the teacher is considered as the best solution, i.e., the

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best value of the objective function so far (Rao et al., 2011). The basic steps of TLBO are

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summarizing as below.

Fig. 2. Vectorial representation of TLBO algorithm

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Step 1 initialization;

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Set of solutions are created randomly between the bounds of design variables. This set of solutions is referred as a population for the meta-heuristics method. The number of solutions required to be created depends on the population size which is to be supplied by a designer. In TLBO the population refers to a group of students.

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Step 2 Teacher phase; In this phase, students or learner updates their knowledge level by the teaching of a teacher. So,

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teaching methodology or the knowledge contents of a teacher influences the learning outcome of students. So, the solutions in the teacher phase are updated by using following mathematical expression.

(7)

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E =  +  ( −  I),

Where, T represents the teacher which is the best solution in the population, M represents the

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mean of different design variables in the population,  is the current solution, and TF is a

teaching factor which can attain the values of 1 or 2.

The vectorial representation of Eq. (7) is given in Fig. 2(a). It is observed from the Fig. 2 (a) that

best solution  ‘2’ will try to attract new solutions  ‘4’or ‘4ꞌ’ towards it. The equation to

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update the solution by teacher phase is shown in Fig. 2 (a). Here, I ‘3’ is the mean of learners

(population) and a good teacher (best solution) always try to bring the level of learners to his or her level in terms of knowledge,  is a teaching factor either 1 or 2 and  is a random number

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lies in the range of [0, 1]. The new solution  depends on the value of the  and the  . It is

also observed that for the value of  =1, the value of  may lies between ‘1’ and ‘4’

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depending on the value of  . Also one can see that for  =2, the value of  may lies between ‘1’ and ‘4ꞌ’ depending on the value of  . If  is near to zero, the point can be near to

current solutions ‘1’. The current solution ‘i’ updated by  if it has better function value

compared to  . The solution after teacher phase is passed to the learner phase.

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Step 3 Learner phase; In this phase, learners update their skill level by interacting with different learners and so the

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knowledge level of different learners enhance the present skill level of interacting learner. In this phase, the solutions are updated using the following condition based mathematical expression.

E =  +  ( −  ), `ℎMM J ≠ g

~idM

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E =  +  ( −  ), `ℎMM J ≠ g

(8)

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xy zy( ) < y( )|,

Accept E if it gives better function value.

The vectorial representation of Eq. (8) is given in Fig. 2 (b). The vectorial representation and the equation to update the solution by learner phase are described in Fig. 2 (b). Here, in the learner phase, the solution is modified by two ways. The two random solutions are picked up from the

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set of solutions available. If the random solution  ‘1’is better than another random

solution  ‘2’, the algorithm explores the new solution around the  ‘1’ and generates  ‘3’.

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If the solution  ‘2’ is better than solution  ‘1’, the algorithm try to move towards  ‘2’ and will explore the new region in a solution space and generates the new solution  ‘3ꞌ’. Here,

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the value of new solution  is influenced by the random number  and it may attain any value that lies between ‘1’ and ‘3’ or ‘1’ and ‘3ꞌ’ depending on the value of an objective function

of the randomly selected solution. The current solution ‘i’ updated by  if it has better

function value compared to  .

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Step 4 Termination; Step 2 and step 3 are repeated till termination criteria specified by the designer is reached. The

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termination criteria can be a number of iteration (generation) or function evaluations. Since, the development of TLBO, it is applied to many engineering applications due to its several features like free from the algorithm parameters, efficiency, ease of programming etc.,

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and effectiveness in handling wide variety of optimization problems like multi-objective placement of the automatic regulators in the distribution system (Hosseinpour et al., 2011),

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optimization of design parameter of planar steel frames (Toğan, 2012), image processing (Jani et al., 2013), heat exchanger optimization (Patel and Savsani, 2014a), capacitor placement in radial distribution systems (Sultana and Roy, 2014), machining parameter of modern machining processes (Venkata Rao and Kalyankar, 2013), optimization of vehicle suspension parameter

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(Gadhvi and Savsani, 2014), and many more. Since, the development of basic TLBO, researchers have tried to modify the existing version of TLBO to make it more effective and problem specific. To list few of them; Patel and Savsani (2014b) proposed a multi-objective improved

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Teaching-learning based optimization algorithm (MO-ITLBO) by introducing the concept of Number of teachers, Adaptive teaching factor and Self-motivated learning to improve the

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performance. Satapathy and Naik (2014) proposed the mTLBO by adding the extra term in the learner phase based on the concept of tutorial class. Xie et al. (2014) modified TLBO algorithm to address the permutation flow shop scheduling problem (PFSP). Huang et al. (2015) presented modified teaching-learning-based cuckoo search (TLCS) for continuous optimization problems. Tejani et al. (2016) presented a modified sub-population TLBO (MS-TLBO) for structural optimization. Rahiminejad et al. (2016) proposed an MTLBO by modifying the learner phase based on the philosophy that if the students of the class participate in groups of several students

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and enhance their knowledge by interaction in these groups. This philosophy helps to get better convergence of the algorithm. So, it can be summarized that TLBO is an effective method to solve engineering optimization problems and also researchers are taking interest to modify the

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existing version of TLBO to suit the specific application. However, it can also be observed that modifications by using the effective search technique of ABC and PSO along with TLBO are not investigated and proposed in the literature so far. Hence, this work is focused on developing an

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effective version of TLBO by incorporating the search technique of ABC and PSO to solve the challenging combinatorial WFLO problem. The details explanation for the development of

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enhanced TLBO is given in the next section.

3.2 Enhanced Teaching-Learning based optimization (TLBOe)

In any optimization algorithms, exploitation refers to the ability to find the better solution in the

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neighbourhood of a good solution while exploration refers to the ability to find the various unknown region in the solution space to discover the global optimum that is encouraged by high diversity (Gao and Liu, 2012). As exploration and exploitation are the opposing forces of each

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other, it’s balance is required to achieve the better performance on optimization problems for searching the global optimum solution. It can also be noted that better exploration and

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exploitation guarantee high convergence rate (Rao and Savsani, 2012). The TLBO has two phases, a teacher phase and a learner phase, and it can be seen from Fig. 2 (a) that the teacher phase is better in exploitation compared to exploration, whereas learner phase is better at exploration compared to exploitation. Though TLBO possesses both exploitation and exploration capabilities, it is not effective for many multi-modal and constrained real life problems (Patel and Savsani, 2015). This can be argued that there can be a scope of improvement in the basic TLBO to enhance a proper balance between exploitation and exploration. This limitation of

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TLBO motivated us to modify TLBO to achieve the proper trade-off between the exploration and exploitation capability of the TLBO algorithm. It is observed from the teaching phase that it possess better exploitation compared to exploration and so, the search mechanism of teaching

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phase can be replaced by some other search mechanism which may have the better balance of exploration and exploitation.

The present study considers ABC and PSO inspired search mechanisms to enhance the search

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technique of TLBO. An ABC algorithm is good at exploration, but the search technique of ABC would generate a candidate solution by moving the prior one towards or away from another

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randomly selected solution from the population that may lead to poor exploitation (local search capability). Hence, the exploitation capability was modified by Zhu and Kwong (2010) by using the concept of best solution inspired from PSO algorithm. The PSO algorithm uses the global best solution of the population in each generation for searching the new candidate solution;

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hence, it can have better exploitation capability compared to the exploration capability (Roy and Jadhav, 2015). Enhancement in the basic TLBO is carried out to incorporate the benefits of an ABC and the PSO inspired search mechanisms. The use of the effective component of ABC and

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PSO algorithm in search technique of TLBO may be assumed to provide the proper balance of exploration and exploitation in the search tendency of teaching phase. As explained earlier,

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enhancement in TLBO is carried out in the teacher phase where the search technique of the teacher phase is enhanced by the search technique of ABC and PSO inspired search mechanisms. So, two different variants of TLBO are proposed in this work; (1) enhancing teacher phase with ABC inspired search mechanism along with the basic learner phase. This variant of enhanced TLBO (TLBOe) will be refereed as AL in this work. (2) Combined PSO and ABC inspired search mechanism along with basic learner phase. This variant of enhanced TLBO (TLBOe) will

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be refereed as PAL in this work. These enhancements in TLBO are assumed to take advantage of different search mechanisms of TLBO, ABC and PSO algorithms. The vectorial representation of AL and PAL algorithms are described in Fig. 3 and Fig. 4

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respectively. In the first part of AL algorithm (Fig. 3), the new solution  (‘3’ or ‘3ꞌ’) is produced by using the ABC inspired search mechanism. The new solution  (‘3’ or ‘3ꞌ’) is generated by vector difference between the current solution  ‘1’ and other random solution 

‘2’ as obtained from the mathematical expression shown in Fig. 3 (a). The new solution 

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(‘3’ or ‘3ꞌ’) is generated around the  ‘1’. The value and direction of new solution  (‘3’ or

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‘3ꞌ’) is influenced by the random number ∅ which varies between -1 to 1 and it may attain any value that lies between ‘1’ and ‘3’ or ‘1’ and ‘3ꞌ’. Now, the greedy selection is applied between

the new solution  (‘3’ or ‘3ꞌ’), and current solution  ‘1’ and better solution is preserved.

All the solutions obtained after using the ABC inspired search mechanism is passed further to the

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learner phase. The vectorial representation of the learner phase is given in Fig. 3 (b) which is

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explained in the previous section.

Fig. 3. Vectorial representation of AL algorithm

Fig. 4. Vectorial representation of PAL algorithm

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In the first part of PAL algorithm (Fig. 4), the new solution  (4 or 4ꞌ) is produced by using

the ABC and PSO inspired search mechanism. The new solution  (4 or 4ꞌ) is generated by

the sum of the vector difference between the current solution  ‘1’ and the random solution  !

‘3’ and current solution  ‘1’ as given by the mathematical

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‘2’& the best solution 

expression shown in Fig. 4 (a). In PAL algorithm, the search mechanism is inspired by the PSO; hence it utilizes the information of the best solution 

!

‘3’ to guide the search direction.

Hence, new solution  (‘4’ or ‘4ꞌ’) tries to move towards the best solution  ! ‘3’ which is

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visualized in the Fig. 4 (a). The search direction of new solution  (‘4’ or ‘4ꞌ’) depends on

the value of random number ∅ which varies between -1 to 1 and  varies between 0 to 1.25.

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Now, the greedy selection is applied between  (‘4’ or ‘4'’) and  ‘1’, and better solution is preserved. All the solutions obtained after using the ABC inspired search mechanism is passed further to the learner phase. The vectorial representation of the learner phase is given in Fig. 4 (b) which is explained in the previous section. The stepwise procedure along with the counter for fitness evaluations and termination condition for AL and PAL algorithms is presented in Fig. 5.

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Pseudo code of AL and PAL algorithm Set the algorithm parameter:- NL= number of learners (population size), MNFE=Maximum number of function (fitness) evaluations begin // Initialization Phase num_eva
0; for i=1 to NL do Cij
random solution(); evaluate (Cij); num_eva ++; if num_eva = =MNFE then return best (Cij); end repeat for i=1to NL do // Enhanced teaching phase Cijnew
From ABC inspired search mechanism or PSO inspired search mechanism; evaluate (Cijnew); num_eva ++; if Cijnew is better than Cij then Cij
Cijnew; if num_eva = =MNFE then return best (Cij); //Learner Phase Ckj
randomly select another learners from population such that k ≠ i; if Cij is better than Ckj then; Cijnew = Cij +rij(Cij - Ckj); else Cijnew = Cij + rij (Ckj - Cij); end evaluate (Cijnew) num_eva ++; if Cijnew is better than Cij then Cij
Cijnew; if num_eva = =MNFE then return best (Cij); end // Removing similar solutions for each similar solutions (Cijd) do Cijd
random solution (); evaluate (Cijd); num_eva ++; if num_eva = =MNFE then return best (Cij); end until num_eva==MNFE; return best (Cij) end Fig. 5. Pseudo code of AL and PAL algorithm

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4. Computational results and discussions Experimental studies were carried out to demonstrate the robustness and effectiveness of the

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proposed algorithms. The algorithms were coded using Matlab by using a computer with Intel Core i3 2.1 GHz processor and 4 GB RAM memory. Experiments were conducted on the wind farm layout optimization (WFLO) problems and a set of real-world numerical optimization

results is presented in following subsections.

Table 1 Wind turbine characteristics. Thrust Coefficient ( )

0.88 0.3

Hub Height (ℎ)

60 m

Rotor Diameter ()

Turbine efficiency (")

40 m 40%

1.225 (gh⁄KH )

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Air density (#)

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Ground Surface Roughness (ℎ )

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4.1 Wind farm layout optimization (WFLO) problem

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benchmark problems from CEC2011 (Das and Suganthan, 2010). The detail discussion about the

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The objective of the present problem was to minimize the cost of power produced per unit by minimizing the influence of wake on turbines installed in a wind farm that leads to more power output of the farm. The present study considers the wind farm is spreading across the area of 2000K × 2000K, which was further divided into a 10 × 10 square grid. The turbines were

considered to be placed at the center of the square cell and five rotor diameters apart from each

other in a crosswind and downwind directions. In this work, a specific type of wind turbines with characteristics shown in Table 1 (Mosetti et al., 1994) was considered. So, it is required to place

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39 turbines in the available placement of the 100 cells. Hence, it gives 9e27 feasible combinations out of which optimum placement is required to be obtained. The placement of turbines was obtained with two distinct cases of wind velocities and its directional distribution

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across the wind farm site. The first case (case (a)) was assumed to have multi-directional uniform wind velocity and the second case (case (b)) was considered with multi-directional variable wind velocities. The results were obtained for ten different runs with a population size of 30 and the

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function evaluations of 9,000 (Patel et al., 2015) for case (a) and population size of 50 and the function evaluations of 50,000 for case (b). The best value obtained out of ten runs was presented

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as the optimum solutions for the WFLO problem.

The methodology followed in this work for solving WFLO problem with the proposed algorithm (TLBOe) is given in Fig. 6 in the form of a flow chart. As observed from the flow chart (Fig. 6) the procedure start by supplying the wind farm data in the form of a number of wind turbines to

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be placed, characteristics of the wind turbine, and the area of a wind farm. In this study, two different scenarios for the wind characteristics are considered that includes (a) Multi-directional identical wind velocity and (b) Variable wind speed with variable wind direction. Selection of

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the wind characteristics is required to be supplied by the user. The method progresses by the

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initialization of different parameters required by the proposed algorithm. These parameters include population size, termination criteria and bounds on design variables. The proposed algorithms are population-based methods which start the search with a set of solutions. These set of solutions are indicated by the population size. Each solution gives one layout for WFLO problems. A termination criterion is a condition to stop the algorithm, and for the proposed approach, the number of function evaluations is set as a termination criterion. The number of design variable equals the number of turbines to be placed in a wind farm so, the bounds

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imposed on the design variable is in terms of the available feasible location. In the first iteration, the proposed algorithm will give feasible positions of turbines for which, the desired quantities like power, the minimum cost of energy and efficiency are calculated by considering the wake

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effect in the wind farm. Proposed algorithm updates the position of the turbine in each iteration and accepts the better placement. This procedure is repeated till the termination criterion is satisfied. The placement obtained at the termination criteria is considered as the optimized

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the cases are discussed in the section to be followed below.

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placement of wind turbines that can produce maximum power with less cost. The results for both

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Start Input: Number of wind turbines to be placed, wind turbine characteristics, and wind farm boundary Variable wind speed with variable wind direction

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Case (a) Multi-directional identical wind velocity

Case (b) Wind characteristics

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Initialize TLBOe parameters: population size, algorithm termination criterion, design variable (number of turbine to be placed), limit the design variable (one to number of feasible placements) Update the position of turbines using TLBOe algorithm, Assign the X and Y coordinates to turbine position, check wake effect of each wind turbine and compute the local wind velocity of each wind turbine

No

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Consider the existing layout

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Evaluate the objective functions

Is new layout better than existing?

Yes

Is termination criteria satisfied?

Consider the new layout

No

Yes

Output: Optimized placements of turbines

Fig. 6. The schematic overview of the WFLO problem

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4.1.1 Comparative results for multi-directional identical wind velocity (case (a)) Layout by Turner et al.

Proposed Layout (TLBO)

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Layout by Grady et al.

Proposed Layout (AL)

Proposed Layout (PAL)

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Proposed Layout (ABC)

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Fig. 7. Wind farm layout for case (a) with 39 wind turbines

This case considers the uniform wind velocity of 12 m/s flowing across the 36 rotational directions with equal likelihood of occurring in each direction. The result obtained for each direction are equally weighted and combined. The WFLO problem is solved by assuming 39 turbines to be placed in the available farm area. The visual representation of the layouts obtained by the proposed methods (AL and PAL), basic algorithms (ABC and TLBO) and from the literature (Grady et al. (2005) and Turner et al. (2014)) are shown in Fig. 7. The comparative

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performances of the results are summarized in Table 2 where the statistical results are presented in the form of total power, mean of total power and standard deviation of total power. As compared to Grady et al. (2005), the method adopted by Turner et al. (2014) had shown the

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better distribution of turbines within the wind farm. The distribution given by Grady et al. (2005) produces a total power of 17,220 kW with an efficiency of 85.23%, while the distribution provided by Turner et al. (2014) produces a total power of 18,336 kW with an efficiency of

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90.7%. The distribution obtained by using the PAL algorithm finds the better layout which has resulted in an increase of power by 0.74%, 2.1%, 0.22%, 7.09% and 0.57% compared to AL,

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basic algorithms (ABC and TLBO), and the methods used by Grady et al. (2005) and Turner et al. (2014) respectively. The distribution obtained by using TLBO, ABC, AL, and PAL optimization technique had produced the power output of 18,401 kW, 18,062 kW, 18,305 kW and 18,441 kW with an efficiency of 91.01, 89.34, 90.53 and 91.21 percentages respectively. The

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proposed approach used in this work is a heuristics method, and it may produce different results in different runs. So, the performance of the proposed approach can be judged based on the best solution produced in different runs, the mean of the solutions produced in different runs and the

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standard deviation associated with the output of different runs. These results are highlighted in Table 2. The performance of the different algorithm can be finely evaluated based on the mean

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of the solutions obtained in different runs as this parameter reflects the consistency of the algorithm. The algorithm is effective if it has a better consistency which directly reflects that the algorithm should have a better mean value. Also, the algorithm with less standard deviation indicates the capability of the algorithm to produce less variable results in different runs. The comparison of mean solutions of total power obtained by ABC, AL and PAL algorithms is shown in Fig.8 in the form of a bar chart. As observed from the bar chart, it is visually clear that

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the power produced by a wind farm using PAL is better compared to the power produced by using AL and ABC. However, the power produced by a wind farm by using AL is better

using PAL is least compared to AL and ABC algorithm. 18,500

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18,300 18,200 18,100 18,000 17,900 17,800 17,700 ABC

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Total Power(kW)-Mean

18,400

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compared to ABC. It can also be observed from Table 2 that the standard deviation produced by

AL

PAL

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Algorithms

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Fig. 8. Comparison of mean solution of total power obtained by ABC, AL and PAL algorithm

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Table 2 Comparative results for multi-directional identical wind velocity (case (a))

Grady

Turner

et al.

et al.

No. of wind turbines

39

Total power (kW)

Proposed configuration

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Configuration by

(ABC)

(AL)

(PAL)

39

39

39

39

39

17,220

18,336

18,401

18,062

18,305

18,441

Total power (kW)-Mean

-

-

-

17,998

Total power (kW)-SD

-

-

-

62.136

Objective function

0.00166

0.00147

Efficiency (%)

85.23

90.7

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(TLBO)

18,417

28.956

24.685

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18,273

0.001463

0.001490

0.001466

0.001459

91.01

89.34

90.53

91.21

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4.1.2 Comparative results for variable wind speed and variable wind direction (case (b)) This case is a more realistic approach of wind environment. This case considers the multi-

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directional wind with variable velocities of 8 m/s, 12 m/s and 17 m/s. The wind is flowing from the 36 rotational directions with an unequal likelihood of occurrence at each wind velocity in

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each direction. The likelihood of occurrence for each wind direction at each wind velocity is shown in Fig. 9. The visual representation of the layout obtained by the proposed methods (AL and PAL), basic algorithms (ABC and TLBO) and from the literature (Grady et al. (2005) and Turner et al. (2014)) are shown in Fig. 10. The comparative performances of the results are summarized in Table 3 where the statistical results are presented in the form of total power, mean of total power and standard deviation of total power.

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17 m/s

0.06

12 m/s 8 m/s

0.05

0.03 0.02

330

310

290

270

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Wind Angle (o)

230

210

190

170

150

130

110

90

70

50

30

10

0

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0.01

350

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0.04

250

Wind Fraction of Occurrence

0.07

Fig. 9. Bar Graph of weighting fractions for variable direction and variable wind speed

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It can be observed from the visual representation of the layout obtained by PAL, AL and ABC algorithms that the turbines are allotted towards the outer boundaries of the wind farm, and majority of turbines are placed at the outer edge. These had happened because these algorithms

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had produced the layouts that had tried to maximize the downwind distance between turbines, as this distance directly affect the velocity that reaches to the wind turbines and also reduces the

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influences of the wake. The layout obtained by the PAL algorithm gave the better result concerning power output and energy extraction efficiency compared to other state of the art algorithms considered in the present work. The wind turbine positioning obtained using the PAL algorithm gives the power output of 33,810 kW with an energy extracted efficiency of 91.40%. The distribution obtained by using the PAL algorithm finds the better layout which has resulted in an increase of power by 0.23%, 0.47%, 2.03%, 5.37% and 4.81% compared to Al, basic algorithm (ABC and TLBO), and the algorithms used by Grady et al. (2005) and Turner et al.

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(2014). As compared to Grady et al. (2005), the method adopted by Turner et al. (2014) had shown the better distribution of turbine within the wind farm. The distribution given by Turner et al. (2014) produces a total power output of 32,453 kW with an efficiency of 87.53%, while the

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distribution provided by Grady et al. (2005) produces a power output 32,086 kW with an efficiency of 86.74 %. The distribution obtained by using the TLBO, ABC, AL and PAL optimization technique had produced the power output of 33,137 kW, 33,652 kW, 33,732 kW

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and 33,810 kW with an efficiency of 89.58, 90.97, 91.18 and 91.40 percentages respectively. The comparison of mean solutions of total power obtained by TLBO, ABC, AL and PAL algorithms

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is shown in Fig. 11 in the form of a bar chart. As observed from the bar chart, it is visually clear that the power produced by a wind farm using PAL is better compared to the power produced by using AL TLBO, and ABC. However, the power produced by a wind farm by using AL is better compared to TLBO and ABC. It can also be observed from Table 3 that the standard deviation

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produced by using PAL is least compared to AL, TLBO and ABC algorithm. The convergence plot to highlight computational cost is plotted in Fig. 12 to show the variation of the objective function value with function evaluations. Each curve represents the average objective function

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value obtained for ten independent runs for each algorithm. It is observed from the convergence plot that the PAL algorithm converges faster compared to TLBO, ABC, and AL algorithm. This

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shows the capability of enhanced TLBO algorithm in terms of solving the challenging combinatorial WFLO problems.

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Layout by Turner et al.

Proposed Layout (ABC)

Proposed Layout (AL)

Proposed Layout (TLBO)

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Layout by Grady et al.

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Fig. 10. Wind farm layout for case (b) with 39 wind turbines

Proposed Layout (PAL)

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Table 3 Comparative results for multi-directional variable wind velocity (case (b)). Configuration by Turner

et al.

et al.

No. of wind turbines

39

Total power (kW) Total power (kW) -Mean

(TLBO)

(ABC)

39

39

39

32,086

32,453

33,137

33,652

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Grady

-

-

33,018

39

33,732

33,810

33,435

33,654

33,758

100.763

72.096

66.339

53.282

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39

0.000812

0.0008

0.000798

0.000796

86.74

87.73

89.58

90.97

91.18

91.40

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0.00082

EP

33,800 33,700 33,600 33,500 33,400 33,300 33,200 33,100 33,000 32,900 32,800

(PAL)

0.00083

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Total Power(kW)-Mean

Efficiency (%)

(AL)

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Total power (kW) -SD Objective function

Proposed configuration

TLBO

ABC

AL

PAL

Algorithms Fig. 11. Comparison of mean solution of total power obtained by TLBO, ABC, AL and PAL algorithms

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Fig. 12. Convergence plot for case (b).

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4.2 Real world application problem

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It will be misleading to draw a conclusion of the proposed algorithm by solving only one problem as done in the previous section. Though the challenging combinatorial WFLO problem

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can be treated as a benchmark problem for any optimization algorithm, but still experimentation on similar challenging benchmark problem is required to build confidence in the decision. So, we have considered the challenging real life benchmark problems such as Dynamic Economic Dispatch (DED), Static Economic Load Dispatch (ELD) and Hydrothermal Scheduling problems which can be considered as combinatorial optimization problems that reassemble to the WFLO up to some extent. The major problem which is associated with the wind power as to find a way to deal with it fluctuation and intermittent to achieve specific reliability and power system

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security. So DED problem for wind penetrated power system is of interest to many researchers. For the effective and safe use of wind power, researchers’ forecast the wind speed or wind generation over a span of time in advances and then the statistical distribution of the wind speed

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with respect to wind power generation is obtained. The dispatch scheme is determine based on the wind distribution function and estimation load (Makarov et al., 2011). So, the DED problem is more concern for with the wind power that has to address the fluctuation and intermittent of

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the power generation and the demand. The DED problem is based on the objectives to minimize the operating cost of power for producing a given profile of electricity demand subjected to the

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constraints such as ramp rate limits, valve-point loading effects, etc. Because of the valve-point loading effect, the solution may be trapped at the local optimum (Das and Suganthan, 2010). The ELD problem is based on the minimization of the fuel cost of the generating unit for a definite time of operation. The Hydrothermal Scheduling problem is the complex problem that involves

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the nonlinear relationship between the decision variables and the operation planning of hydrothermal systems. The aim of this problem is to minimize the fuel cost of the thermal system.

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To demonstrate the performance of proposed enhanced algorithms on set of real-world benchmark optimization problems, several experiments were conducted to compare the results

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with other state of the art algorithms. The real world benchmark optimization problems were considered from a special session on numerical function optimization competition of CEC2011 (Das and Suganthan, 2010). This type of problems offers a greater challenge to the optimization algorithms and so it can be used to verify the effectiveness of the proposed algorithms. The details of considered problem are summarized in Table 4 where all the problems are mentioned with a code from T11.1 to T11.10. To maintain the consistency in the comparison of competitive

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algorithms, a common experimental platform was kept. The maximum number of function evaluations was kept as the termination criteria for each run of the algorithm. The average of results was computed for different runs on each problem, and it was used to check the

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performance of the different algorithm. The maximum number of function evaluation for considered test problem were kept as 1.5e5 with the population size of 50, and each benchmark problem was executed run for 25 runs. Also, it was claimed that the basic code of TLBO uses the

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duplicate solution removal step, and duplicate removals steps also consume some number of function evaluations (Rao and Patel, 2012); hence the function evaluations considered in this

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study incorporates the function evaluations consumed in the duplicate removal step of the TLBO. So, total function evaluations consist of function evaluation consumed in teacher phase, learner phase and the duplicate solution removal step.

In the current study, for each experiment, the fitness values of solutions of the algorithm in 25

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runs were sorted from minimum to maximum value. The comparative statistical results obtained for T11.1 to T11.10 benchmark problems were presented in Table 5 in the form of best, mean, and standard deviation (SD). The results are compared with other state of the art algorithm such

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as; the GA with multi-parent crossover (GA-MPC) (Elsayed et al., 2011), self-adaptive convergent DE algorithm with a hidden adaptation selection (SaCDEhaS), self-adaptive

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differential evolution (SaDE) (Hu et al., 2014), differential evolution algorithm with constraint sequencing (DE-CS), differential evolution with a random hill climber (DE-RHC), and estimation of distribution and differential evolution cooperation (ED-DE) (Asafuddoula et al., 2015).

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Table 4 Summary of the problems presented. Problem Name

No. of Dimensions

Constraints

T 11.1

DED* instance 1

120

Inequality constraints

T 11.2

DED instance 2

216

Inequality constraints

T 11.3

ELD* Instance 1

6

Inequality constraints

T 11.4

ELD Instance 2

13

Inequality constraints

T 11.5

ELD Instance 3

15

Inequality constraints

T 11.6

ELD Instance 4

T 11.7

ELD Instance 5

T 11.8

Hydrothermal Scheduling Instance 1

96

Inequality constraints

T 11.9

Hydrothermal Scheduling Instance 2

96

Inequality constraints

T 11.10

Hydrothermal Scheduling Instance 3

96

Inequality constraints

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Problem No.

Inequality constraints

140

Inequality constraints

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40

*DED=Dynamic Economic Dispatch and ELD=Static Economic Load Dispatch

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The DE-CS algorithm obtained the best mean value for problem T11.2 and T11.7 while PAL obtained the best mean value for T11.8 and T11.10. GA-MPC obtains the best mean value for problem T11.9. DE-CS and PAL obtained the best mean value for the problem T11.4 and T11.5

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while PAL, GA-MPC, ABC and ED-DE obtained the best mean value for problem T11.6. DECS, PAL, ED-DE and GA-MPC obtained the best mean values for problem T11.3 while DE-CS

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and GA-MPC obtained the best mean value for problem T11.1. The comparison of mean solutions of T11.1 to T11.10 benchmark problems obtained by DE-CS, DE-RHC, ED-DE, PAL, AL, TLBO, ABC and GA-MPC algorithms is shown in Fig. 13 (a)-(d) in the form of bar chart in which the data is plotted on logarithmic scale. As observed from the bar chart, the mean solution obtained by the PAL algorithm is comparative compared to others algorithms mentioning in the bar chart. The results for the benchmark problems indicate that no algorithm has produced better

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results compared to other algorithms for all the real life benchmark problems. So, it is difficult to judge the extent of the superiority of one algorithm compared to the others. Hence, Friedman rank test was performed to identify the best performing algorithm and its extent with respect to

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other algorithms. Friedman rank test was performed on the mean solutions obtained by different algorithms in 25 runs. The Friedman value, normalized value and rank for various algorithms are shown in Table 6. Friedman rank test had excluded problems T11.9 and T11.10, as results by

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using DE-CS, DE-RHC and ED-DE are not available in the literature for the same problem. The

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results show that PAL is the best performing algorithm followed by GA-MPC and ED-DE.

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Table 5 Best, mean, and Std. of function values obtained for test problem taken over 25 runs (T11.1-T11.10). T11.1

T11.2

T11.3

T11.4

T11.5

T11.6

T11.7

T11.8

T11.9

T11.10

DED-1

DED-2

ELD-1

ELD-2

ELD-3

ELD-4

ELD-5

HS-1

HS-2

HS-3

Best

1.10E+05

1.04E+06

1.54E+04

1.82E+04

3.27E+04

1.33E+05

Mean

3.02E+05

1.05E+06

1.54E+04

1.82E+04

3.27E+04

1.40E+05

SD

2.13E+05

2.68E+04

1.38E+00

1.40E+02

4.48E+01

4.59E+03

Best

5.18E+05

1.80E+07

1.54E+04

1.89E+04

3.29E+04

1.32E+05

Mean

5.31E+05

1.81E+07

1.55E+04

1.92E+04

3.31E+04

1.39E+05

SD

5.11E+05

4.00e+04

8.78E+04

1.54E+02

7.46E+01

3.19E+03

Best

5.21E+04

1.07E+06

1.54E+04

1.83E+04

3.28E+04

Mean

5.21E+04

1.08E+06

1.54E+04

1.83E+04

SD

3.65E+02

2.72E+04

9.10E+00

Best

5.16E+04

1.07E+06

Mean

5.20E+04

SD

1.68E+06

3.30E+06

-

-

1.73E+06

6.29E+06

-

-

6.30E+07

1.66E+06

-

-

1.91E+06

9.51E+05

-

-

2.06E+06

1.09E+06

-

-

2.38E+05

1.56E+05

-

-

1.33E+05

1.91E+06

9.41E+05

-

-

3.29E+04

1.33E+05

1.91E+06

9.41E+05

-

-

8.69E+01

5.95E+01

1.94E+03

9.09E+03

1.61E+03

-

-

1.54E+04

1.82E+04

3.27E+04

1.28E+05

1.89E+06

9.39E+05

1.01E+06

9.41E+05

1.07E+06

1.54E+04

1.82E+04

3.27E+04

1.33E+05

1.92E+06

9.40E+05

1.08E+06

9.46E+05

2.02E+02

1.22E+03

1.07E+00

4.91E+01

3.04E+01

2.32E+03

1.20E+04

1.66E+03

6.52E+04

2.44E+03

Best

5.19E+04

1.07E+06

1.54E+04

1.85E+04

3.27E+04

1.31E+05

1.93E+06

9.42E+05

1.09E+06

9.42E+05

Mean

5.25E+04

1.08E+06

1.55E+04

1.84E+04

3.29E+04

1.36E+05

1.12E+07

1.08E+06

1.66E+06

1.08E+06

SD

3.89E+02

8.67E+03

1.52E+01

1.21E+02

8.61E+01

3.69E+03

4.38E+07

2.75E+05

4.02E+05

2.75E+05

Best

5.26E+04

1.27E+06

1.54E+04

1.84E+04

3.29E+04

1.32E+05

1.89E+06

9.42E+05

1.01E+06

9.42E+05

Mean

7.79E+05

1.55E+06

1.55E+04

1.86E+04

3.31E+04

1.44E+05

8.63E+06

1.28E+06

1.54E+06

1.28E+06

SD

6.35E+05

1.78E+05

3.80E+01

1.03E+02

1.43E+02

7.42E+03

1.92E+07

8.49E+05

7.93E+05

8.49E+05

Best

5.20E+04

1.07E+06

1.54E+04

1.86E+04

3.28E+04

1.30E+05

1.91E+06

9.44E+05

1.14E+06

9.44E+05

Mean

5.31E+04

1.10E+06

1.55E+04

1.91E+04

3.28E+04

1.33E+05

2.18E+06

1.04E+06

1.53E+06

1.08E+06

SD

2.35E+03

3.26E+04

1.49E+01

1.92E+02

4.16E+01

1.63E+03

4.29E+05

3.00E+05

3.58E+05

4.01E+05

Best

5.09E+04

1.07E+06

1.54E+04

1.81E+04

3.27E+04

1.29E+05

1.92E+06

9.50E+05

9.72E+05

9.47E+05

Mean

5.21E+04

1.07E+06

1.54E+04

1.83E+04

3.28E+04

1.33E+05

1.95E+06

9.71E+05

1.06E+06

9.75E+05

SD

4.50E+02

1.62E+03

1.75E-07

6.98E+01

2.68E+01

1.88E+03

1.41E+04

1.04E+04

5.70E+04

1.18E+04

DE-

TLBO

ABC

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AL

EP

PAL

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ED-DE

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RHC

SC

DE-CS

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Alg.

GAMPC

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T11.1 DED-1

Algorithms

GA-MPC

AL

PAL

ED-DE

DE-CS

DE-RHC

1.00E+04

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1.00E+05

ABC

SC

1.00E+06

TLBO

Mean

1.00E+07

T11.2 DED-2

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1.00E+08

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(a)

T11.4 ELD-2

T11.5 ELD-3

EP

1.00E+04

T11.3 ELD-1

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(b)

ABC

TLBO

GA-MPC

Algorithms

AL

PAL

ED-DE

DE-CS

1.00E+03

DE-RHC

Mean

1.00E+05

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1.00E+08

T11.6 ELD-4

T11.7 ELD-5

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Algorithms

GA-MPC

SC

AL

PAL

ED-DE

DE-CS

DE-RHC

1.00E+04

ABC

1.00E+05

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1.00E+06

TLBO

Mean

1.00E+07

(c)

1.00E+07

T11.8 HS-1

Algorithms (d)

Fig. 13 (a)-(d). Comparison of mean solutions of different algorithms for T11.1 –T11.10 problems.

GA-MPC

ABC

TLBO

AL

PAL

DE-RHC

EP DE-CS

AC C

ED-DE

Mean 1.00E+05

1.00E+04

T11.10 HS-3

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1.00E+06

T11.9 HS-2

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Table 6 Friedman Rank test for the ‘Mean’ solution obtained by 8 algorithms for test problem taken over 25 runs (T11.1-

DE-CS

DE-RHC

ED-DE

PAL

AL

Friedman value

28.5

54

25

15.5

43.5

Normalized value

1.84

3.48

1.61

1.00

2.81

Rank

4

7

3

1

6

TLBO

ABC

GA-MPC

57

40.5

24

3.68

2.61

1.55

8

5

2

SC

Algorithm

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T11.10). Bold font means the best performance.

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5. Conclusion

Two different variants of basic TLBO algorithm is proposed in this work and investigated for the challenging WFLO problem and a set of real life benchmark problems. The results indicate that the solution produced by the proposed algorithm, i.e., AL and PAL, is better compared to the

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basic TLBO algorithm. So, the incorporation of ABC and PSO inspired search technique are indeed effective to improve the exploration and exploitation of the basic TLBO algorithm. It can also be summarized that PAL is a better variant of TLBO compared to AL. PAL has reported the

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improvement in the results by 0.22% and 2.03% compared to TLBO for the multi-directional

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identical wind velocity and multi-directional variable wind velocity cases respectively. The PAL algorithm has produced the results better by 7.09 % and 0.57 % compared to the literature approaches by using the genetic algorithm (GA) and the quadratic optimization (OQ) with mixed integer linear (MIL) method for the multi-directional identical wind velocity case. Also, The PAL algorithm has produced the results better by 5.37 % and 4.81% compared to the literature approaches by using the genetic algorithm (GA) and the quadratic optimization (OQ) with mixed integer linear (MIL) method for the multi-directional variable wind velocity case. The

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investigation on a set of real world benchmark problems indicates that PAL is nearly 3 times and 1.5 times better compared to the basic TLBO algorithm and the GA-MPC algorithm respectively. So, it can be outlined that enhancement in the basic TLBO is indeed effective and can produce

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better results on challenging problems. Moreover, it can be summarized that the proposed approach is effective for the WFLO problem that can produce maximum power with the

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minimum cost of energy and less CO2 emission required for the cleaner energy production. References

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