A New Hybrid Metaheuristic Algorithm for Wind Farm Micrositing

This work focuses on proposing a new algorithm, referred as HMA (Hybrid Metaheuristic Algorithm) for the solution of the WTO (Wind Turbine Optimization) problem. It is well documented that turbines located behind one another face a power loss due to the obstruction of the wind due to wake loss. It is required to reduce this wake loss by the effective placement of turbines using a new HMA. This HMA is derived from the two basic algorithms i.e. DEA (Differential Evolution Algorithm) and the FA (Firefly Algorithm). The function of optimization is undertaken on the N.O. Jensen model. The blending of DEA and FA into HMA are discussed and the new algorithm HMA is implemented maximize power and minimize the cost in a WTO problem. The results by HMA have been compared with GA (Genetic Algorithm) used in some previous studies. The successfully calculated total power produced and cost per unit turbine for a wind farm by using HMA and its comparison with past approaches using single algorithms have shown that there is a significant advantage of using the HMA as compared to the use of single algorithms. The first time implementation of a new algorithm by blending two single algorithms is a significant step towards learning the behavior of algorithms and their added advantages by using them together.

T he topic of wind turbine micrositing has gathered a lot of attention internationally due to the rapid increase in the installation of wind power. It has been discussed in earlier literature that the wind turbine positioning can be improved by the use of the FA [1].
The goal of this work is to find a new method of solving the WTO problem. This solution examines the feasibility of using a multi-Algorithmic approach for the multi objective optimization is achieved.
Wind turbine micrositing is a complex problem of optimization because of the fact that it cannot be solved by exact methods.
The problem space is non-linear and totally dependent on The main objective of this work is to propose a HMA for advancing the purpose of research in this field. This HMA is the combination of a mainstream algorithm the DEA and recent algorithm, namely the FA. Both algorithms are selected on the basis of their relative strengths. The DEA is known for escaping local optima and the FA is well known for its fast processing and efficient operation.
The method of nesting two algorithms is well known in literature and there are a number of publications available [4][5]. Other implementation are of HMAs are such as a combination of two or more algorithms to attain a common objective are also discussed in [6][7][8][9]. During the development of the metaheuristic algorithm the major limitations of both the algorithms in use were mapped and compared. The main objective was to get the maximum power at the lowest cost.
This study focuses on the recent interest during the midst of 2015 from the optimization community in the use of HMAs for the solution of engineering problems. The WTO problem has moved on to other avenues since then and has become a multi-dimensional topic by the application of the CFD (Computational Fluid Dynamics) approach [10][11][12].
The program code for optimization is developed in Matlab and run on a Core i3 computer with 8 GB RAM.

LITERATURE REVIEW
The area of wind farm optimization was pioneered by several authors including Mosetti, Beyer, Barthelmieand Grady [13][14][15][16]. In their approaches the solution space was encoded into a square grid of 2x2 km to facilitate the search string. They used the GA (Genetic Algorithm) primarily for achieving this purpose. There was added interest by the use of metaheuristic algorithms and the Jensen Model as deployed by Wan et. al. [17] and Marmidis et. al. 18].
Other authors such as Acero and Kusiak used other metaheuristic techniques such as Virtual Gene Algorithm, Markov Chain methods and Simulated Annealing [19][20].
Interestingly the work for WTO by using CFD domain was initiated very early by Palma et. al. [21]. It was also reported that the CFD is a more accurate method of micrositing during its infancy.
Then Wan et. al. [22] utilized the GA to arrive at systematic results for the simulation of the wind farm. His work was based on Grady et. al. [16] results.
This was followed by Emami and Nougreh [23] who advanced the work by considering the three scenarios, uniform and unidirectional wind, uniform and multidirectional wind and non-uniform and multidirectional wind. However his results were based on less strenuous parameters.
Moreover, Rasuo and Bengin [24] presented his work based on the GA. He was quick to note that the realistic wake effects can only be mapped by the use of a viscous flow model such as a CFD model. However, he realized that the computational cost of using both CFD and GA would be great. Then Gonzalez et. al. [27] summed up the work done in this field in a review paper. The results were compared with Grady et. al. [16] and were an extension of the work of Mosetti et. al. [13].
This present work has evolved from the work of Mittal [3]. The work of Rajper [2] builds on the work of Mittal [3]. Much work has been done in the field of metaheuristics since the term was coined by Glover [28].
Other researchers contributed by adopting different approaches. Karampelas et. al. [29], utilized the downhill simplex optimization method to reach at the optimal number of wind turbines and the least cost of installing these turbines.
Then, Villarreal and Espiritu [30], compared the results of Mosetti et. al. [13] by using a viral based algorithm. Addition to this field was made by Chowdhry et. al. [31] who used a miniature wind farm in conjunction with the PSO (Particle Swarm Optimization) Algorithm. A key aspect of this study was that the use of different rotor sizes were used in the same wind farm to obtain better simulation results.
Particle flow simulation or the use of CFD was added to this field by Song et. al. [32] who showed that the results of CFD were better than widely used linear model given by Jensen [33]. Again, Song et. al. [34], utilized the particle flow simulation method in conjunction with Greedy Algorithm.
Chen et. al. [36], used the Jensen model [33] in conjunction with nested GA for the improvement of results in a wind farm. He used two different heights for wind turbines in his study.
Gaumond et. al. [37], investigated the three main wind models while applied to an actual wind farm and concluded that the wake models were under-predicting the power produced.
A good formulation of the optimization problem has been discussed in [9]. Moreover a wide number of applications of hybrid algorithms have been illustrated in literature these include [38][39][40][41][42][43][44][45]. Rao  Salcedo-Sanz et. al. [39] used the HMA approach for the solution of the task assignment problem in Heterogenous Computing applications.
Shahsavari-Pour and Ghasemishabankareh [40], solved the FJSP (Flexible Job-Shop Scheduling Problem) by the use of novel HMA-NHGASA. He utilized three objective functions that minimized the total time of the operations, minimized the load on the most used machine and minimized the load on all the machines. His approach produced better results than classical results reported in literature.
Lozano and García-Martínez [41], proposed the combination of two algorithms, the ILSA (Iterative Local Search Algorithm) and an Evolutive Algorithm. These algorithms were used to reach better values of the parameters of intensification and diversification.
Leung et. al. [42], utilized the HMA for the solution of the knapsack packing problem. He utilized the Simulated Annealing and a greedy strategy for the solution of this problem.
Poorzahedy and Rouhani [43], was able to use a HMA for solving the network congestion problem by the use of GA, Simulated Annealing and Tabu Search. The results were better than the Ant Colony Optimization Algorithm.
Yi et. al. [44], proposed three HMAs for Engineering design optimization. His results were better than the Hybrid Differential Evolution Algorithms.
Fattahi et. al. [45], compared several Pareto based inventory control models by the use of HMA in his work.
A recent publication Massan, et. al. [46] may also be referenced for the application of the Differential Evolution Algorithm in this domain.
There are two main papers that have discussed the use of bi-algorithmic approach for wind farm optimization these are [4][5]. In these papers Wan et. al. [4][5] has proposed a similar approach to our research. Wan has used PSO algorithm along with Gaussian mutations (GPSO) to solve the constrained optimization problem.
Wan et. al. [5] has proposed that the global optimization and local optimization is reached by the use of two Gaussian mutation operators. The first Gaussian operator finds the global best solution while the local operator searches in the vicinity of the global best solution. In such a scenario, the local optima is efficiently searched by two levels of operation. It is widely known that GPSO is efficient and robust for finding the global optima and the simulation results show that there is a wide improvement in using the two operators [4][5].

Wind Farm Model
The wake model used shall be the Jensen model [33] as it gives the two constraints of optimization i.e.
(1) The power generated in terms of the wind speed (2) The cost model The same assumptions are taken as in previous studies [1][2][3],
In such layouts, each grid point has its own wind speed denoted by u i when this wind speed interacts with a wind turbine wake appears. This wake expands in a linear fashion behind the wind turbine in the wind direction. The wake expansion is given by the factor á that is derived from the hub height z and the terrain roughness z 0 .
The computer program shall be run so that multiple wakes are accounted for as well as the exact terrain roughness of that point.
The listed variables were utilized in the Matlab program, r r = Turbine rotor radius which is the radius of the wake X = The distance at which the wake is calculated r 1 = The radius of the wake at distance X According to Betz's theory [47] the wind speed after the rotor is given by:

FIG. 1. THE WAKE MODEL SCHEMATIC
Thus a value of U is obtained. Moreover, from [48] the axial induction factor a is derived from the Thrust Coefficient C T as follows We may now write r 1 as related to r r as: Where as, from [33] the calculation of the entertainment constant á is done for every grid point as follows: Where hub height z=60m and the value of the terrain of z i at the grid point yields the á for that point [9] It may be remarked that the lesser the z 0 the narrow the wake [9].
The equation for multiple wakes is denoted by: Where the available power can be calculated from equation: With the addition of efficiency the equation becomes: The power produced is reduced to The efficiency is given by the following equation: ( )

Cost Model
The cost model is generic and is given in literature as follows: (11) Where N t is the number of turbines [5].

The Optimization Problem
It is known that, a population of solutions to the optimization problem can be generated with k population: The bi-objective (cost and power) fitness values for the optimization problem at any instant, can be written as: Where P(n) and C(n) are values of power and cost of the wind farm at a certain instant in the iterative process. Where,

Power Produced, P(n) is taken from Equation (8) and the
Cost, C(n) from Equation (11). The optimization problem Equation (12) is subject to the constraints that: Where, L is the length and W is the width of the wind farm and D is the minimum distance between two adjacent wind turbines.

The Differential Evolution Algorithm
The DEA is an Evolutionary Algorithm that is related to Stochastic Search Algorithms. This algorithm searches a known domain for a global maxima or a minima. It is similar to GA, PSO Algorithm and the Evolutionary Strategy Algorithm as well as Evolutionary Programming Algorithms [49][50].
The DEA is similar to the GAs as it utilizes the main processes of selection-evaluation-recombination. This algorithm is outstanding as it adds a new dimension by looking for unique solutions by the process of recombination [51]. The process of recombination is facilitated by the addition of new species that are a result of a weighted difference of two old members that are recombined with the third member [50].
However, it has been found from the algorithm implementation that it is computationally expensive and nimble fine tuning of the parameters is required for better implementation [51].

The Firefly Algorithm
This algorithm is good at evaluating a cost or objective function under many constraints and the problem can either be linear or non-linear [27][28]. It functions in the stochastic domain which means that several solutions to the same problem may exist. However, if we sufficiently increase the number of iterations the problem may succumb to a single global solution set (Global minima or global maxima).
The basic working of the algorithm is very simple to understand. It mimics the mating behavior of Fireflies with the given sets of assumptions [27][28].
(1) All fireflies are equally attracted to each other which means there is a single gender.
(2) Brighter fireflies are more attractive. In this case the permeability of the medium may be altered to configure the algorithm properly. In the case of a tie i.e. if two fireflies have the same brightness then they shall move independently. adjoining fireflies [27].
The FA is more suitable for due to its simplicity and simple coding. It is useful for optimization problems as it mimics the intensity of the brightness that results in maximization of the objective function [1,[46][47][48].

Firefly Algorithm (Mathematical Model)
The inverse square law gives the light intensity: Giventhat: I(r) is light intensity,ris Distance, I 0 is Original light intensity,andγis Light absorption coefficient.
The objective function can be interchangeably used to denote the intensity of light of a firefly and attractiveness of one firefly to another. Both these terms are interchangeable and are dependent on the objective function [27][28]. In view of the above, if we replace the attractiveness with its proportional counterpart; intensity then The equation may be modified, as in [53],to read as follows: Moreover, the Cartesian distance existing between two fireflies i and j at x i and at x j is governed by the following equation, as in [52,[55][56].
( ) The term, x ik defines the kth value of the x i of the i th firefly.
The attractiveness of the i th and the j th fireflies results in the movement that is governed by the following equation: In the above equation x i and xj are just positions the second term is the attraction between the two and the last term gives the randomness factor in the above guess.

Meta-Heuristic Algorithm Implementation Methodology
The MHA has the intelligence to explore the search space effectively and not miss any points of the local optima.
At the global scale it is ingenious at escaping the local minima and astute enough to intensify the density of the turbines at high wind areas [57].
The proposed bi-algorithmic approach uses the two mainstream algorithms namely (1) The Differential Evolution Algorithm The Firefly Algorithm The working of the solution is discussed briefly as follows: A map of the region is taken and divided into (100x100) grid points. However, the DEA is allowed to work in a solution space of only (99.5x99.5) grid points. This is because the FA is allowed to function in (0.5x0.5) grid points. The resultant map can be used to identify the boundary and non-boundary regions of the solution set.
Then the HMA is run on the solution space in which the DEA is executed first to find the global optimization on n 2 area. In our present calculations a base/model area of (99.5x99.5) grid spaces in a 9900.25 km 2 area. The remaining space of 0.5x0.5 grid spaces is covered by the FA.
The HMAs does the initial calculations on the basis of the DEA and then does the Local Optimization automatically by the application of FA for each turbine position to find the best solution.
Thus, with the application of the two algorithms the global best solution can be enhanced by any scalar amount to become the global as well as local best position. Therefore, both algorithms working together in tandem can yield a better solution set in a piecewise manner.
The parameters of concern in the DEA are population size which is taken as 500 and the maximum time for the iteration which is taken as 900 seconds. Whereas the parameters of concern in the FA are taken as number of fireflies were taken as 20, Number of iterations were 100, Alpha was 0.5, Beta min was 0.2 and Gamma was 1. Fig. 2 gives a comparison of the power produced by the HMA as compared to the GA [2].

RESULTS AND DISCUSSION
After the simulation was run it was noticed that the results of HMA were better for higher number of turbines as compared to the GA. Only for 10, 16, 17, 18 and 19 turbines the results of GA [2] were better.
It was seen that HMA reported the same results for Power till the 9 th turbine was installed. At the installation of the 10 th turbine the HMA reported a value of 5183.99 kW for power whereas the GA [2] reported a value of 5184.00 kW.
It was also noticed that the Power obtained by using the HMA is measurably higher per unit turbine installed after the installation of the twentieth turbine.
The power difference is evident when the 20 th turbine is installed where by HMA we got a resultant power of 10,361.76 kW but with GA a lower value of 10,351.68 kW is obtained. This trend continues till the installation of the 100 th turbine where the HMA reports a higher amount of power produced of 50,846.92 as compared to the GA at 48,452.26 kW. In Table 1, it is clearly evident that a higher amount of peak power is produced with a higher number of turbines at a considerably lower cost as compared to the GA [2].
In Table 2, It is evident that the highest amount of power is produced by HMA as compared to GA [2] and Mosetti et. al. [13]. Table 3, when we compare the results obtained by Grady et. al. [16], Rajper and Amin by using GA [2] and by HMA it is evident that the latter is the best algorithm for evaluation of the objective function.

CONCLUSIONS
The algorithm implementation shows the comparison with the use of a single algorithm i.e. the GA and it shows that the HMA performs better.
This implementation is a significant step in towards learning the behavior of algorithms and their added advantages by using them together.
The main contribution of this work is that, the FA and the DEA have been used for the first time in tandem. This adds to a significant contribution by implementation of the both algorithms and the addition of complexity to the WTO problem.
This study has successfully calculated the total energy produced by a wind farm by using a HMA and compared it with the past approaches of using single algorithms. It is shown that there is a significant advantage of using the HMA as compared to the use of single algorithms.