Optimal placement of wind turbines within a wind farm considering multi-directional wind speed using two-stage genetic algorithm

Abstract

Most wind turbines within wind farms are set up to face a pre-determined wind direction. However, wind directions are intermittent in nature, leading to less electricity production capacity. This paper proposes an algorithm to solve the wind farm layout optimization problem considering multi-angular (MA) wind direction with the aim of maximizing the total power generated on wind farms and minimizing the cost of installation. A two-stage genetic algorithm (GA) equipped with complementary sampling and uniform crossover is used to evolve a MA layout that will yield optimal output regardless of the wind direction. In the first stage, the optimal wind turbine layouts for 8 different major wind directions were determined while the second stage allows each of the previously determined layouts to compete and inter-breed so as to evolve an optimal MA wind farm layout. The proposed MA wind farm layout is thereafter compared to other layouts whose turbines have focused site specific wind turbine orientation. The results reveal that the proposed wind farm layout improves wind power production capacity with minimum cost of installation compared to the layouts with site specific wind turbine layouts. This paper will find application at the planning stage of wind farm.

Keywords

optimal placement / wind turbines / wind direction / genetic algorithm / wake effect

Introduction

In recent times, the increase in global population, higher demand for better quality of life and reduction in the price of fuel among other factors have led to an increase in the exploration of carbon based energy resources [¹]. This exploration has led to a high global warming impact and far reaching climate challenges. In reaction to this, the global energy focus has been shifted towards renewable energy resources such as wind, solar, geothermal, hydropower and so on. Of these various renewable energy sources, wind energy is considered as one of the cheapest and most adopted renewable energy resources [²]. It is believed to have a long-term technical potential of supplying five times the current cumulative global energy demand [³].

In order to appreciably exploit wind energy, onshore and offshore wind farms have been setup in diverse locations. Most often, the wind turbines within wind farms are set up to face a pre-determined wind direction. In doing this, the collection of site specific data for wind speed and direction is needed so as to inform unidirectional wind turbine layout that will encourage yaw controlled maximum angle of attack which will minimize the effect of wake. Wake effect is one of the disadvantages of clustering wind turbines. The reason for this is that wind acquires turbulence and losses speed as it passes through each turbine. As a result, turbines downstream will only be exposed to turbulent winds with reduced speeds. The use of site specific unidirectional focus layouts is not optimal because wind directions are intermittent in nature. This intermittency could lead to less production capacity of the wind farms. If the wind direction veers off the expected path, the power loss and wake loss within the wind farm will increase, leading to inefficiencies in the operation of the wind farm. Hence, there is a need for a flexible multi-angular (MA) layout that takes into consideration various wind directions. One of the advantages of the MA layout is the optimal placement of wind turbines taking into consideration the wake and wind variation, in order to ensure the minimum cost per unit energy. The MA layout also introduces reliability into wind farms as it generates appreciable power at reduced cost of installation in times of unexpected wind direction.

Researchers have investigated optimal placement of wind turbines using different approaches and reported diverse results. The approaches can be classified into intuitive and concrete. The intuitive approach involves setting up layouts based on instinct and over time observations while the concrete approach is based on the use of systematic techniques in placing wind turbines on wind farms. Bansal et al. [⁴] have considered several test scenarios with the view to defining a yardstick for land requirement on wind farms. They suggest that 10 ha/MW can be taken as the land requirement of wind farms. But this is just for seat-of-the pant estimates. In another work, Patel has concluded that the optimal spacing of turbines is found in rows of 8–12 rotor diameters apart in the windward direction, and 1.5–3 rotor diameters in the crosswind direction [⁵]. However, Ammara et al. [⁶] have pointed out that the above intuitive micro-siting schemes could result in sparse wind farms that would inefficiently generate wind energy potential. A dense and staggered siting scheme that would use less land but yield production similar to the sparse schemes has then been proposed. This dense layout, however, has a reduction in overall efficiency. A wind farm layout optimization problem using a viral based optimization algorithm has been proposed by Ituarte-villareal and Espiritu [⁷]. They have used both fixed and variable number of turbines as input variables, and concluded that viral based optimization algorithm works best when there is no restriction on the number of turbines. As a result, their layout is limited. A genetic optimal micro-siting of wind farms using equilateral-triangle mesh has been carried out by Wang et al. It is concluded that setting up the possible location of turbines in a wind farm as a “honeycomb” mesh rather than the usual square cells is more optimal [⁸]. Similarly, Marmidis et al. [⁹] have worked on the same project description but applied a different methodology. In their work, optimal placement of wind turbines in a wind park using Monte Carlo simulation has been investigated and a lower fitness value at more generated power is achieved, but the case of unidirectional wind at constant wind speed is only considered.

In recent times, genetic algorithm (GA) is applied in solving wind farm optimization problems because of the following advantages: parallel search approach which reduces the possibility of getting stuck at a local optimum, probabilistic selection rules, and high flexibility [¹⁰]. The use of GA to determine the optimal placement of wind turbines within wind farms has been studied by Mosetti et al. [¹¹]. In their work, Jensen’s wake model has been adopted to account for downstream wind speed dissipation. A 2 km by 2 km coarse grid is divided into 200 m by 200 m, and 100 square cells are used for the study. The study, however, appears not to have run through sufficient generations to achieve convergence. About a decade later, Grady et al. have improved on the result obtained by Mosetti et al. Their work utilizes the GA approach to obtain the optimal placement of wind turbines for maximum production capacity [¹²] and uses the same coarse square grid (as Mosetti et al.) with 100 possible turbine locations. The same cost model, turbine characteristics and wake model are used and a better result is achieved with a lower fitness value and a higher output power. However, more wind turbines are required to achieve the result.

Mittal et al. [¹³] have proposed a novel hybrid optimization methodology to simultaneously determine the optimum total number of turbines to be placed in a wind farm along with their optimal locations. The proposed hybrid method combines the probabilistic genetic algorithms and deterministic gradient based optimization methods. The study reveals that the application of the proposed technique to representative case studies yields higher annual energy production compared to the results found by using two of the existing methods. There are commercial programs that have been developed to perform micro siting of wind turbines [¹⁴]. Some of these are WAsP [¹⁵], WindSim and Meteodyn. These software can perform wind resource assessment over the terrain under study by taking into account the wind climate observation supplied by the user. However, the results obtained by these methods have been proved not to be accurate enough when assessing complex terrain [¹⁴]. Various works on wind turbine micro-siting can be found in Refs. [¹⁶,¹⁷].

All the aforementioned studies are concerned with setting up of wind turbines to face a pre-determined wind direction within a wind farm. However, wind directions are intermittent in nature and vary from time to time. This unidirectional wind farm layout could lead to an inefficient production capability of the wind farms. In this paper, a two-stage genetic algorithm is developed to solve the wind farm optimization problem by evolving a MA layout for multi-directional wind at uniform speed. The results from the first stage, which are 8 unidirectional layouts together with their objective functions, are fed into the second stage. Consequently, the second stage evolves an averaged optimal layout in terms of cost per unit power that will perform well under multi-directional wind at uniform speed.

Methodology

In this section, the assumptions for wind farm layout optimization problem are presented, the wind regime layout is described, the wake model is established, and the cost model is analyzed.

Assumptions

To solve the wind farm layout optimization problem, a possible scenario with a number of assumptions is setup. The assumptions are as follows:

1) The wake is assumed to follow a linear pattern of expansion and the momentum inside it is conserved.

2) The wind farm is assumed to be flat with a surface roughness of 0.3 and it is to be subdivided into a 200 m by 200 m coarse grid, with 100 square cells.

3) A constant wind speed and similar wind turbine type is assumed.

4) The wind farm capital and power purchase agreements are neglected and therefore, in effect there is no limit on total power production and total cost of installation.

Wind regime

The total number of wind turbines and the optimal wind farm layout are to be determined so as to ensure the minimization of the cost per unit power for the entire wind farm. In this paper, the following multi-directional winds at uniform speed are considered. The wind directions are North (0°), North-East (NE) (45°), East (90°), South-East (SE) (135°), South (180°), South-West (SW) (225°), West (270°), and North-West (NW) (315°) at a wind speed of 12 m/s. The wind farm layout as well as the wind direction to be assumed is shown in Fig. 1.

Wake model

In this paper, the Jensen model was extended to MA wind directions and used to define the wake propagation. The wake model is similar to the one reported in Ref. [¹⁸]; therefore, there is a common ground for comparison with other layouts. The model hinges on conservation of momentum in the wake, contribution of tip vortices and neglected linear wake expansion [¹⁹]. Figure 2 demonstrates the schematic diagram for the wake generated by a single turbine. The model is only valid in the far wake: regions about 5 rotor diameters from the preceding turbines [¹⁷,²⁰].

In this model, the wind speed, u, downstream the turbine is given as

(1)u=u_{\mathrm{0}}\left\{\mathrm{1}-{\displaystyle \frac{\mathrm{2}a}{{\left[\mathrm{1}+\alpha \left(x/r_{\mathrm{1}}\right)\right]}^{\mathrm{2}}}}\right\},

where r₁ is the downstream radius and a is the entrainment constant. The axial induction factor, a, is related to the thrust coefficient, C_T as follows:

(2)a=\mathrm{0.5}\left(\mathrm{1}-\sqrt{\mathrm{1}-C_{\mathrm{T}}}\right).

The downstream radius, r₁, is given by

(3)r_{\mathrm{1}}=r_r\sqrt{{\displaystyle \frac{\mathrm{1}-a}{\mathrm{1}-\mathrm{2}a}}}.

The entrainment constant, a, is given by

(4)\alpha ={\displaystyle \frac{\mathrm{0.5}}{\ln \left(z/z_{\mathrm{0}}\right)}}.

The wake radius (r_w) at a distance, x, downstream is given by

(5)r_{\mathrm{w}}=R+\alpha x,

where R is the radius of the wind turbine.

When two or more wakes meet, they cause wake interaction whose influence on wind speeds is very crucial. It is assumed that the kinetic energy of the mixed wake is equal to the sum of the individual kinetic energy deficits. It is expressed as

(6){\left(\mathrm{1}-\frac{\left(\overline{\mathit{u}}\right)}{u_{\mathrm{0}}}\right)}^{\mathrm{2}}={\displaystyle \sum }_{i=\mathrm{1}}^N{\left(\mathrm{1}-{\displaystyle \frac{u_i}{u_{\mathrm{0}}}}\right)}^{\mathrm{2}}.

Cost model

The cost model is used to calculate the total cost of wind turbines in a wind farm. The cost model used is defined by Eq.(7) [⁹,¹¹,¹²,²¹]. For simplicity, the cost and maximum discount on a turbine are assumed to be 1 and 33.3% respectively. This is done such that only the number of turbines will be considered. Therefore, as the number of turbines purchased increases, the overall cost is reduced. The cost function is then modeled as

(7)\mathrm{Cost}=N\left({\displaystyle \frac{\mathrm{2}}{\mathrm{3}}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}{\mathrm{e}}^{{-\mathrm{0.00174}N}^{\mathrm{2}}}\right).

To show the maximum number of turbines attainable before any further increase becomes unadvisable especially for unidirectional layouts, the derivative of the cost function is plotted against the number of turbines, N in Fig. 3. From Fig. 3, it can be observed that the cost starts to increase monotonically at N>30. In effect, the cost of the (N+1)th turbine is more expensive compared to the Nth turbine. Additionally, the significance of this model will be seen in subsequent subsections as it also shows that as the cost of the turbine decreases, the cost per unit power (fitness value) decreases and vice-versa.

Power calculation

The power calculation derivable from wind farm is based on the cut-in and cut-out wind speeds of the individual wind turbine that makes up the wind farm. In this paper, the same wind turbine, Vestas V63, used in Refs. [²¹,²²] was used for analysis, which has the following characteristics: Hub height (z) of 60 m, rotor diameter of 40 m and thrust coefficient of 0.88. Since wind power is directly proportional to the cube of wind speed, it is generally given by

(8)P={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho A{u}^{\mathrm{3}}{C}_{\mathrm{p}},

where the coefficient of power, C_p, is assumed to be 40% [¹²,²¹]; r is the air density measured in kg/m³ which is 1.225 kg/m³; A is the rotor swept area measured in m²; and u is the wind speed measured in m/s. The power characteristic of the turbine is given by

(9)

The power curve showing the cut-in wind speed, cut-out wind speed, and power yield of Vestas V63 is as illustrated in Fig. 4.

Objective function

This is the function to be optimized. It is the mathematical expressions that defines the nature of the problem and assigns a fitness value to a set of variables [²³]. In this paper, there are two major objective functions, one for each stage. The objective function of stage 1 is given by

(10)\mathrm{Minimize}:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\mathrm{C}\mathrm{os}t\left(N\right)}{{\displaystyle \sum _{n=\mathrm{1}}^{N}P\left(u\right)}}},

\begin{array}{l}\mathrm{Subject\; to}:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{C}\mathrm{os}t\le C_{\max },1\le N\le 1\mathrm{00}\mathrm{;}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{P}_{\min }\le P\le P_{\max },u=\mathrm{12},\hfill \end{array}

where N is the number of wind turbines, C_max is the maximum cost that can be incurred when 100 turbines are bought to fill up the wind farm, P_max is the maximum power that can be generated by a turbine even when it is in the free stream, and P_min is the minimum power which only occurs when a turbine is absent from a position. The freestream wind speed, u, is at 12 m/s.

For stage 2, since each direction has its own direction dependent wake model, the objective function here finds the average in a bid to determine the lowest fitness value to satisfy all directions considered. It is worth mentioning that stage 1 is a special case of stage 2. The output of stage 2 must also satisfy the primary cost and power constraints of stage 1. The objective function is given by

(11)\mathrm{Minimize}:\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\displaystyle \frac{\mathrm{1}}{m}}{\displaystyle \sum _{i=\mathrm{1}}^m\frac{\mathrm{C}\mathrm{os}t{\left(N\right)}_i}{{\displaystyle \sum _{n=\mathrm{1}}^{N}P{\left(u\right)}_i}}}

\begin{array}{l}\mathrm{Subject\; to}:\mathrm{C}\mathrm{os}t\le C_{\max },1\le N\le \mathrm{1}\mathrm{00}\mathrm{;}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{P}_{\min }\le P\le P_{\max },u=\mathrm{12},\hfill \end{array}

where m is the number of wind directions under consideration. In this paper, 8 major directions are considered.

Optimization process

The optimization is conducted in two stages. Stage 1 is aimed at evolving the optimal layout for 8 different major wind directions (N, NE, E, SE, S, SW, W, NW) while stage 2 is aimed at evolving the MA layout based on the result obtained from stage 1.

Stage 1

A modified GA algorithm that could compute the wind speed, total power and the cost for wind farm layouts was developed for the optimization process and implemented in Matlab. The objective function had 100 binary variables, with 1 representing the presence of a turbine and 0 representing the absence. The initial random population was generated using complementary sampling and a uniform crossover approach to increase diversity as depicted in Fig. 5.

The first step as indicated in Fig. 5 is to define the GA parameters which include the number of variables (100), population size (300), mutation rate (0.1), number of generations (1500), stall generation limit (1000), and fitness limit (-∞). The reason for the selection of the various GA parameters are as follows:

1) Number of variables= 100

Since the wind farm is a 10×10 square grid, one variable was used to represent each square grid, as such the total number of variables is 100.

2) Population size= 300

The population must be able to sufficiently span the whole computation space; as such, it has a minimum theoretical value of √n where n is the number of variables [¹²]. In this case, n is 100; therefore, the minimum population size is 10, but the more the population size, the lower the chances of getting stuck in a local optimum, hence a population size of 300 proved to be large enough.

3) Mutation rate= 0.1

Mutation rate is a percentage, whose main purpose in the algorithm is to introduce diversity into the original population but if it is too high i.e., beyond 50%, it becomes unrealistic as it distracts the algorithm from converging to a global optimum. A 10% mutation rate seemed moderate. Additionally, there is no fixed value for mutation rate.

4) Number of generations= 1500

The minimum number of generations is theoretically defined as 200\sqrt{n} [¹²]. In this case, the minimum will be 2000 but after a few trials, it was discovered that 1500 was just sufficient.

5) Stall generation limit= 1000

This is a stopping criterion which was selected to minimize error. Basically, the algorithm stops when the average change in the fitness value over 1500 generations is less than the function tolerance which was defined as 1e⁻¹⁰.

6) Fitness Limit= –∞

This is an additional stopping criterion. It is a boundary for the lowest attainable fitness value. This gives the algorithm enough room to find the lowest possible fitness value.

The second step is to generate an initial random population using uniform random sampling. This method sometimes leads to oversampling of some regions and sparse sampling in others. To resolve this problem, an alternative approach which randomly generates 50% of the chromosomes and then toggle it to get the second half is applied [²³]. The next step is to decode the chromosomes using the quantization procedure. Each decoded chromosome is slotted into the objective function to determine its cost and to rank the chromosomes from best to worst.

If the optimization criteria (defined by the fitness limit, the number of generations and the stall generation limit) is satisfied, the result (chromosome with lowest cost) will be displayed. Otherwise, a 50% selection rate will be applied to determine the possible parents. Subsequently, rank weighted pairing will be applied to pick mates. This particular approach was chosen because it does not depend on the problem as it finds the selection probability directly from the rank, x, of the chromosome. The selection probability is defined as in Ref. [²³]:

(12)P_x={\displaystyle \frac{X_{\mathrm{keep}}-x+\mathrm{1}}{{\displaystyle \sum _{x=\mathrm{1}}^{X_{\mathrm{keep}}}x}}}={\displaystyle \frac{\mathrm{151}-x}{\mathrm{11325}}},

where X_keep refers to the chromosomes that will be retained for mating and is given by

X_{\mathrm{keep}}=X_{\mathrm{rate}}{N}_{\mathrm{pop}},

where X_rate is the selection rate and N_pop is the population size.

Table 1 lists a hypothetical case of 50% selection rate: the cumulative probability of a possible set of 20 selected chromosomes from a generation of 40 individuals. In order to select the possible parent, cumulative probability is computed for each of the member of the mating pool. A random number (between 0 and 1) is then generated and compared with the possible parents. Starting from the top of the list, the first chromosome with a cumulative probability greater than the previously generated random number is chosen. In order to select the second chromosome, another random number is generated and the mating pool is once again checked. The first chromosome with a cumulative probability greater than the random number is selected from the mating pool. This process together with subsequent uniform crossover is repeated until a new population with the same size as the initial population is arrived at. The next step is to perform uniform crossover. The uniform crossover approach increases diversity and scatters the potential offspring over a wider area of the cost surface. A mask is used in performing the uniform crossover. The mask is a binary row vector of the same length as the parents. The crossover is performed such that when the mask is 0, the corresponding bit in Parent₁ is passed to Offspring₁ and the corresponding bit in Parent₂ is passed to Offspring₂. When the bit in the mask is 1, the corresponding bit in Parent₁ is passed to Offspring₂ and the corresponding bit in Parent₂ is passed to Offspring₁. An example of uniform crossover is presented in Table 2. The penultimate step is to perform mutation on all but the best individual of the new generation. Applying a 10% mutation rate, the total number of mutations is given by

(13)\mathrm{Mutations}=\mu \left(N_{\mathrm{pop}}–1\right)N_{\mathrm{bits}}=\mathrm{299}\mathrm{0},

where µ is the mutation rate and N_bits is the number of bits per chromosome.

This simply means that 2990 bits selected randomly from the 29900 bits that can be mutated per generation will be toggled. The 0’s of such selected bits will be changed to 1 and vice-versa. Finally, the individuals will be decoded and ranked to choose the best solution and if the optimization criteria are not achieved, the process is repeated until they are achieved.

Stage 2

The objective functions for the 8 unidirectional optimal layouts obtained from stage 1 are fed into stage 2. Stage 2 is designed to satisfy an accumulation of the previous 8 wake models used in stage 1. A population of 292 individuals is generated through complementary sampling (in the same way as in stage 1) in addition to the 8 individuals from stage 1. They serve as the first population for stage 2. Each individual is decoded and the cost is evaluated for the 8 different directions. The costs, obtained by substituting each individual into each wake model, are then averaged. Based on this, the individuals are then ranked from best to worst, in an ascending order. Selection, reproduction, mutation and ranking are then repeated (following the same procedure as stage 1) until the stopping criteria are met. The flowchart in Fig. 6 shows the sequence of steps employed in the optimization process.

Results and discussion

The results for both the unidirectional and the MA layout of wind farm are presented in this section. Besides, the two layouts are compared to show the advantages of MA over the unidirectional layout. In addition, the results of sensitivity analysis are also presented.

Unidirectional layout

In determining the unidirectional layout, the wind was assumed from a single predetermined direction, as an example, the wind was assumed to come from the East with a wind speed of 12 m/s. The optimal unidirectional layout displayed in Fig. 7 was determined following the optimization process in stage 1. The result is depicted in Fig. 7. The corresponding power produced per turbine considering the eastward wind direction is depicted in Fig. 8.

From Fig. 7 and Fig. 8, the optimal number of wind turbines in the wind farm is 31. The following can be observed:

1) T1: The first turbine produces the least power (410.21 kW) compared to other turbines in the first column because it has 3 turbines directly ahead of it, while others have just two. It receives the weakest wind speed (11.08 m/s).

2) T2–T10: This set of turbines produces the same amount of power (447.83 kW) as they experience the same amount of wake dissipation.

3) T31: This turbine is next to turbine T1 in performance as it also produces a relatively small amount of power (411.68 kW). The reason for this is that it has two small spaced turbines ahead of it in the wind direction; hence, it experiences a strong wake.

4) T42–T50: This set of turbines yields the same amount of output power (469.79 kW) as they encounter the same wake.

5) T61: The small distance between T61 and T91 is responsible for creating a strong wake effect on T61, which leads to its lower power production (417.64 kW).

6) T91–T100: This last set of turbines produces the same amount of power (521.15 kW) as they are in the freestream.

It is further observed from Fig. 8 that turbines T91–T100 produce the highest power. This is expected as these wind turbines directly face the eastward wind direction without wake effect. Turbines T42–T50 produce the next highest wind power next to turbine T91–T100. The lower wind power experienced is the result of wake effect experienced by these turbines. The least power is produced by turbine T1.The reason for this is that there are three turbines ahead of the turbine in the eastward direction, thereby reducing the wind speed getting to turbine T1 due to the increased wake effect.

The overall fitness value, the total power generated from the wind farm, the optimal number of turbines, the efficiency of the optimal layout, and the convergence generation are tabulated in Table 3. It can be observed that the wind farm layout has a fitness value of 0.00153693 and a total power of 14.71 MW/a with 31 numbers of wind turbines. The wind farm has an efficiency of 91.051%. These are better than the values achieved by Grady et. al. [¹²]. The total power output of the wind farm is also higher than that of Grady et al. and Mosetti et al. [¹¹].

Proposed MA layout

Most proposed wind farm layouts have a predetermined wind direction. However, wind direction is not constant; it varies from time to time. Therefore any change in the direction of wind speed from the predetermined one will result in a high wake effect which will affect the efficiency of the wind farm in terms of wind power output and cost per unit power. Hence, there is the need for the development of the MA layout that has the capability to take into consideration many wind directions, i.e., cater for variation in of wind direction. The optimal MA layout is obtained from the output of the second stage of the optimization process. In this paper, 8 different major wind directions are considered in the optimization process. The result of the optimal layout is exhibited in Fig. 9.

The results reveal that the optimal number of wind turbine for the MA layout is 38 and are mainly located at the edges and at the center of the layout. This layout increases the reliability of the wind farm as it can respond favorably to the wind speed from any of the major wind directions. It is expected to have a better performance compared to other site specific unidirectional focus layouts in times of unexpected wind directions. The MA layout power curve which reflects the power produced by each turbine is depicted in Fig. 10. It can be seen that turbines T1, T10, T91, and T100 will produce the highest wind power on the overall considering the power aggregation from all the major wind direction. The reason for this is that the turbines are located at the edges of the wind farm which allows them to respond favorably to wind speed with little wake effect. Besides, turbines T45, T46, T55, and T56 will produce the least wind power. This is also expected as they will experience wake effect regardless of the direction of wind.

In order to show the significance and the merits of the MA layouts compared to predetermined unidirectional layout, 5 comparative analyses were conducted.

MA layout compared with SW unidirectional layout

Assuming the wind is coming from the SW without our prior knowledge. The result using MA layout is obtained and compared with

1) a layout in which the wind direction is known to be South-West before the evolvement of the wind farm layout (pre-determined wind direction) and the wind follows the predetermined direction.

2) a layout in which the wind direction is predetermined to follow SW and due to intermittency in wind direction, the direction of the wind changes to South rather than the predetermined South-West.

The result of the comparison is given in Table 4. It can be seen from Table 4 that the use of MA gives a better fitness value of 24.7%, an improved total power output of 27.3%, and an enhanced efficiency of 34% compared to a layout in which the wind veers off the expected direction due to the intermittent nature of wind. The result of a layout in which the wind follows the predetermined direction is only slightly better compared to the MA layout. However, wind direction is intermittent and will always veer off its path from time to time. The difference in result between the MA layout and the predetermined directional layout is insignificant. Hence, the MA layout is desirable since wind direction is not constant.

Comparison of overall cost per unit power and total power output of MA layout to other predetermined unidirectional layout

To further elucidate on the advantage of MA layout over the predetermined unidirectional layout, the wind direction is assumed to change through 8 major directions within a span of 24 h. The wind direction changes through North, South, East, West, NE, SE, SW and NW every 3 h over a period of 24 h. The cost per unit power (fitness value) and the total output power of the wind farm are thereafter determined. As an illustration, 4 cases are considered.

Case 1: MA layout (optimal) compared to the Southern unidirectional layout

A layout was evolved with a predetermined unidirectional wind from Southern direction. The layout was used over a period of 24 h with an assumption that wind direction changes every 3 h through the major wind directions. The aggregated wind farm output power and the cost per unit power of wind turbine were calculated using both the Southern layout and the MA (optimal) layout. The result is depicted in Fig. 11. It should be noted that the lower the cost per unit power of wind turbine is, the better the layout is. In addition, the better the total output of wind farm is, the better the wind farm layout is. From Fig. 11, it can be observed that the proposed MA layout performs better compared to the Southern unidirectional layout.

Case 2: Eastern unidirectional layout compared to MA layout

The performance of the optimal layout was also tested by comparing it with the unidirectional Eastern layout. The two layouts were exposed to the same 24 h wind condition with the wind direction changing every 3 h through the 8 major directions. The aggregated output power and cost per unit power when both the MA and the unidirectional layout are exposed to changing wind direction are depicted in Fig. 12.

It is seen from Fig. 12 that the optimal MA layout produces a better response as it has a higher generated power and a lower cost per unit power compared to the unidirectional Eastern layout.

Case 3: Western unidirectional layout compared with MA layout

Similarly, Western optimal layout is exposed to a 24 h wind pattern in which the wind direction changes every 3 h and then compared to the optimal MA layout. The cumulative power and cost per unit power over the duration of 24 hare plotted against the outputs of the MA optimal layout as depicted in Fig. 13.

Figure 13 indicates that MA layout produces more power at a lower cost per unit turbine compared to the Western layout.

Case 4: Northern unidirectional layout compared with MA layout

Unidirectional Northern layout is again compared to the optimal MA layout. The aggregate power and cost per unit power for the Northern optimal layout are plotted against the power yield and cost per unit power of the optimal MA layout. The MA optimal layout proves better as it yields more power at a cheaper cost as revealed in Fig. 14. The simulation result depicted in Fig. 14(a) shows that when the wind turbines are arranged in a unidirectional Norther layout, the daily total power is about 100 MW. However, when they are arranged in MA layout, the daily total power output is 130 MW. This indicates that there is an improvement in daily power production of about 30 MW when MA layout is utilized. Similarly, the daily cost per unit power of the wind turbines for Northern unidirectional layout is 0.016 while that of MA layout is 0.013, indicating a decrease of 0.003 in the daily cost per unit power.

Sensitivity analysis

Sensitivity analysis was performed on the resulting optimal MA layout to show the effect of wake on the position of wind turbine in the layout. Six representative wind turbines (T1, T2, T5, T55, T90, and T91) occupying different position in the layout were used for the analysis. The power yield of each of the representative turbine was extracted at varying wind speed (6–15 m/s) and assuming an Eastward wind direction. The result is depicted in Fig. 15.

From Fig. 15, it can be observed that for every wind speed coming through the Eastward direction, turbines T2 and T91 produce the highest wind power. The reason for this is that wind turbines in these positions are not affected by wake as they face the freestream of Eastward wind without disturbance. However, turbine T55 produces a lower wind power than turbines T2 and T91. This is the result of wake effect as T55 is located behind wind turbine T95. The only available wind for power generation is the distorted downstream wind coming behind wind turbine T95. Of the 6 representative turbines under consideration, it can be observed that T1 produces the least wind power. This is expected as there are 8 wind turbines (T11–T81) eastward of turbine T1 with limited spacing between turbines. This results in a high wake effect which tends to limit the power production capability of the turbine. Generally, as the number of wind turbine ahead of turbine Tx increases, the wake effect increases and hence less wind power is generated. Besides, as the space between wind turbines is closer, the wake effect increases.

In addition, sensitivity analysis was conducted for the six selected wind turbines to determine their cost per unit power at different wind speeds and the result is depicted in Fig. 16.

It is generally expected that the higher the power production capability of a wind turbine, the lower the cost per unit power of the turbine and vice versa. From Fig. 16, it is seen that turbine T1 is relatively the most expensive of the wind turbines. The reason for this is that turbine T1 experiences the highest wake effect which limits its wind power production capability. Moreover, the discount on turbine T1 is very small as it is the first turbine purchased in the layout. Turbine T91 produces the highest wind power and hence the cheapest wind turbine within the farm layout. The reason for this is that turbine T91 faces the freestream Eastward wind direction hence has zero wake disturbances. Moreover, it enjoys the best discount rate since it is one of the last 10 turbines purchased among the 38 installed in the wind farm layout. Turbine T2 also has an advantage in position as it is not affected by the wake effect; however, the unit cost per wind power is higher than that of T91 and T55, because of the difference in the discount rate, as it is the among the first sets of wind turbine purchased.

Conclusions

The MA layout which took into consideration variability of wind direction was developed and presented in this paper. The layout was evolved using the two-stage genetic algorithm. The power output and the cost per unit power of the layout was computed and compared with site specific unidirectional layout over a period of 24 h. From the result, it could be concluded that the proposed MA layout yielded a higher aggregated power and a lower cost per unit power when compared to the existing predetermined unidirectional layout. Since wind direction is intermittent and cannot be exactly predetermined, the MA layout is better on the aggregate compared to other predetermined unidirectional layout as the wake effect tends to reduce in this proposed layout. The sensitivity analysis reveals that turbines with low wake disturbance have a low cost per unit power because they produce high wind power, except if they have low discount during purchase.

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