Application of Differential Evolution for Wind Turbine Micrositing

WTM (Wind Turbine Micrositing) has been an important topic of discussion in recent times. A number of Evolutionary Algorithms have been applied to the WTM problem. The DEA (Differential Evolution Algorithm) is used for a bi-constrained optimization for getting maximum power production at the least cost from a 2x2 km space. It is shown that the DEA performs comparably to the GA (Genetic Algorithms) for wind farm optimization. The optimal configuration obtained enlists the number of turbines, the cost of power generated as well as the power produced. Moreover, this study is augmented by comparison with past approaches by using the GA for the same purpose.


INTRODUCTION
that a number of times that the turbine position if modifies optimizes the power produced from the wind farms.
WET (Wind Energy Technology) of today has come a long distance from its early days a decade ago. It is cost effective and more efficient due to the wide spread use of composite materials and cutting edge technology. The present study is an endeavour to improve the efficiency by systematically or recursively reducing the cost and increasing the power production [2]. Our work builds upon the work of a decade of research in this area as referenced in [3][4][5][6][7][8][9][10][11][12]. It is true that previous research on optimization of WTM was based on GAs but researchers have also tried other algorithms for the same problem and compared the results with GAs; for example: [13][14]1]. Literature shows that the results from other algorithms are in good comparison to those by GAs and in some instances are better [1].
Hence, in this work we have used DEA -which has encouraging stability properties as compared to GA for WTM.
The WTO problem was first tackled by Mosetti [15]. He based his calculation on the Jensen model due to the ease of calculation of multiple wake interaction. He evaluated the wake interaction on a wind farm for three scenarios, as outlined below: (1) Constant speed Wind from one direction (2) Constant speed Wind from multiple directions (3) Variable speed Wind from multiple directions All the studies in this domain followed the 2x2 km area plan of Mosetti [15]. It was clearly shown that the This work was followed up by Grady [3] by the use of GAs for the solution of the WTO problem [3]. He reported better results as compared to Mosetti. Improvements from Marmidis [16] followed and he used the Probabilistic Monte Carlo Simulation. Some significant results were also obtained by using modified objective function by Emami [17].
Then Mittal [16] used GA for the solution of the WTO problem and he evaluated the highest number of turbines that can be installed in a 2x2 km area with the highest power at the least cost of installing such turbines [4].
The remainder of this paper has been organized as follows. The wind farm optimization problem is reviewed next in section 2. This is followed by the DEA that is elucidated in section 3. The DEA is followed by a discussion of the parameters used in section 4. The data evaluation is carried out in section 5 and the conclusions are drawn in section 6.

THE WIND FARM OPTIMIZATION PROBLEM
The Jensen model is used for evaluation of the multiple wake interaction behind turbines. The basic principle of conservation of momentum is utilized within multiple wakes. For the sake of simplification, it neglects the effects of turbulence in the near and far wakes [19].
In this model we consider the widely used basic assumptions in which Rotor radius is equal to 40 m, the Hub height is equal to 60 m, and the Thrust coefficient is equal to 0.88 [19]. The wake expands onwards by beginning at the rotor radius r r , in a cone shaped trajectory in the wind direction and is denoted by r 1 , at a distance of X from the rotor [19].
The wind speed is given by the equation (1); behind the rotor [19]: Where U is the wind speed behind the rotor (located at some finite distance, X, u 0 is the initial wind speed, a is the the axial induction factor, a is the constant of entertainment, X is the distance and r 1 is the wake radius [19]. Fig. 1 Illustrates the wake effect as a cone shaped phenomenon starting at the rotor with radius r r and expanding to r 1 , at a distance of X [19]. Where, U 0 is the free stream wind speed. The following equation (2) gives the value of the axial induction factors as follows [19]: Where, the wake radius is given by Equation (3), where X is the downstream distance from the turbine with radius r 1 [19]: Hence, a, the entertainment constant, where z is the hub height (meters) and z 0 is the factor of terrain roughness (meters) which is about 0.3 mfor flat lands [19], is given by equation (4): The following equation gives the estimation of the wind speed and can be used to simulate multiple wakes located in tandem behind each another [19]: In Equation (5), N t gives the total number of wind turbines in the particular wake vortex, u 0 is the initial wind speed,

FIG. 1. WAKE EFFECT AS A CONE SHAPED PHENOMENON STARTING AT THE ROTOR
u i is the i th turbine wind speed, and finally the factor u gives the resultant wind speed after multiple wake interactions [19].
The Jensen model is useful for depicting the wake orders due to the placement of multiple turbines [19].

Power Calculation Equations
The Available Power is defined asequation (6) with reference to [19]: By incorporating the efficiency parameter, ç [19],equation (6) reduces to equation (7): Hence, we may solve to get equation (8): The optimization objective function for getting the maximum power is given in equation (9):

Efficiency Calculations
The Betz's limit gives the aerodynamic efficiency of the wind turbines as equation (10) [19]: Which may be written as equation (11)

Cost Model
A non-dimensional classical cost model equation (12), is used for the purposes of this study. It decrements the cost by one third with the addition of every new turbine [19]: (12) Which is subject to: The cost model used in this study assumes that operating costs are negligible [19].  [20].

THE DIFFERENTIAL EVOLUTION ALGORITHM (DEA)
The general behaviour of the DEA can be deduced from [21], as the DEA has significant advantages over GA when compared in terms of: (1) The speed of convergence of the algorithm The complexity of the code structure etc. (3) The overall accuracy as compared from both algorithms by nearness to the optimal solution (4) The stability of the solution set In various studies, it has been demonstrated that the greatest advantage of using the DEA is that the sample space is continuously improved with each run of the algorithm so that the average solution at any instant is the GA is known to falter for a less than optimal solution or a local optima due to premature convergence [21].
The DEA is much improved as compared to GA and other algorithms of the same class and is quoted in [22][23][24]. The DEA is robust and stable in problems having multiple dimensions, that are often multi-modal and have inherent noise that are otherwise tedious to resolve using other methods. The main reason for the use of the DEA is the Crossover Another impressive trait of differential evolution is that its parameters NP, CR and F are able to self-tune themselves in according to the requirements of the problem; a trait which is not available in other Metaheuristic Algorithms [25].
The DEA formulates its solutions on the basis of careful selection and then evaluation and finally by the process of recombination of the results. It is a self-adapting algorithm which is a crucial property that enables it to escape local minima or maxima very easily [26]. It takes three members of the species, takes a weighted difference of two of them and adds them to the third member to obtain a unique new member. The evaluation of the fitness of the new member introduced is done recursively with respect to the given objective function. The best member is selected on the basis of survival of the fittest and is hence a basic property of all evolutionary algorithms [20].
The DEA requires that the user properly configures the algorithm by appropriate representation, refining the selection process used and then setting the parameters of use [26]. Of these three processes the most important is the parameter setting which is of foremost concern before deploying the DEA [26].
It is also known as a directed parallel method of search.
Therefore, the total population is denoted by NP each having D-Dimensions in each generation, [13]: The DEA assumes that the NP remains the same during the iterative optimization process. It utilizes the Uniform Distribution to make the random guess about the next member of the population [13].
Hence, Uniform Distribution selects three members of the population and these are subjected to selective mutation and crossover to produce offspring. The process of selection yields the vector described by equation (13): x i,G, i= 1,2,3,.. NP (13) which is mutated by the process stated in equation (14): The following indexes r 1 , r 2 , r 3 ∈{1,2,3,.. NP} of the mutation vector are two dimensional integers in our case.
And it is given for a finite weighted value of F > 0 [13].
The DEA is further augmented by the use of crossover by having a trial vector: Therefore, j=1,2,..,D Hence, the DEA uses the function of r and b(j) to evaluate the value of the function in its jth value for a range of values between 1 and 0. In addition the crossover constant, CR is evaluated between the value of 1 and 0.
Thus generating an index rnbr(i) that decides whether crossover takes place for a minimum of one species during the algorithm implementation [13].
The cost function is compared with the resultant vector u i,G+1 and with the initial vector x i,G . If there is an improvement in the value then the value of u i,G+1 is carried over to the value of x i,G+1 otherwise the historical value of This algorithm is often written as (

PARAMETERS USED
In our present study the results of the DEA have been compared with the results of GAs and have been categorized on the basis of [14]: (1) Total Power dissipated (2) Cost per unit power and the Cost per turbine At the end of the simulation the results were obtained and matched with those of Rajper's work [14].
The values of the parameters for which the DEA code is run is as follows [14]: We have also followed the conventional cost analysis as also in Mittal [4] and Rajper [8] which is primarily done for the case of U 0 = 12 m/s. The configuration of the DEA is done as follows: Population size(nP) = 100 Feed slave process (feedSlaveProc) = 5 Maximum iterations (maxiter) = 900 Maximum time (maxtime) = 900

DATA EVALUATION
When the first turbine experiences a steady wind of The maximum power that can be harnessed by using GA's [4] is till the time 54 turbines are introduced to the simulation but a maximum of 81 turbines can be inserted by using DEA. After the installation of these maximum turbines the power produced reduces from the maximum value. Table 1 illustrates that it is possible to install more turbines by using the DEA in comparison to using the GA [14]. It is evident that more power can be reaped from the same area used by introducing 81 turbines.   by Marmidis [16] at 0.0014107 and by the use of GA's [14] at 0.001423. Last but not the least, the value of the efficiency of the DEA approach is the best at 99.6991%

Mehran University Research
as compared to Mittal [4] which was not reported and GA's at 98.721%.
Again in Table 5, it is evident that the DEA outperforms the comparative approaches at the introduction of the          T  f  o  .  o  N  E  D  y  b  r  e  w  o  P  A  G  y  b  r  e  w  o  P  E  D  y  b  t  s  o  C  A  G  y  b  t  s  o  C  E  D  y  b  y  c  n  e  i  c  i  f  f  E   .  1  5  6  9  .  6  1  0  ,  6  2  0  7  .  0  1  7  ,  5  2  9  3  1  3  1  0  0  .  0  6  9  2  3  1  0  0  .  0  0  6  0  4  8  9  .  0   .  2  5  7  4  .  5  2  6  ,  6  2  7  9  .  6  9  1  ,  6  2  9  7  0  3  1  0  0  .  0  3  9  2  3  1  The DEA approach is a stochastic one and has a faster convergence than the deterministic approach which describes the jumps in the algorithm results. The use of the DEA, rules out the manual approaches by application of the Finite Difference Method that would be tedious and almost impossible to compute given the vast number of calculations required.
It is concluded that the DEA approach used in this work and other recent approaches by firefly algorithm [1] and adjoint method [27] are better than the GAsand the finite difference methods [3]- [8] for Wind Farm Micrositing.

CONCLUSIONS
The performance of the DEA was evaluated as the number of turbines was increased. The DEA has proven itself to