The WFLD problem is concerned with placing the wind turbines in an optimal configuration within a wind farm. This optimal placement is vital for minimizing power loss due to different phenomena such as turbulence, wake decay, and transmission line loss. In addition, an optimal placement of wind turbines significantly affects the costs associated with the installation, operations, and maintenance of these turbines. In broader terms, the wind farm layout design problem can be defined as follows [2]:

“Given the location of a wind farm and the number of wind turbines, the aim is to place the turbines in the wind farm such that the design objective(s) are optimized while satisfying the design constraint(s).”

A wind farm is categorized as a geographical area that has a strong wind corridor and is therefore capable of producing wind energy through the use of wind turbines. Studies have proposed several wind farm layout structures based on the nature and topography of the terrain. The most prevailing model for a wind farm used in several studies assumes a discrete search space, where the farm is constituted of a planar area in a square form. This square is divided into 100 equally sized cells in a 10 × 10 grid configuration as shown in Fig. 1. These 100 cells lead to 2100 possible solutions [2], thus classifying the WFLD problem as an NP-hard problem. Turbines are placed in the middle of a cell while ensuring that a distance of 3D to 5D (where D is the turbine diameter) is maintained between turbines in all directions.

Studies have proposed various wake-effect models, such as the Jensen model [62], the Katic model [63], the Frandsen model [64], two-dimensional wake model [65], three-dimensional wake model [66], Gaussian wake model [67], and PARK wake model [68]. In this study, we adopt the wake-effect model used in a recent study by Ju and Liu [47]. This wake effect model is derived from the two most established models which are the Jensen model [62] and the Katic model [63]. The study of Mosetti et al. [3] was the first to use the Jensen model and characterized the phenomenon of wake effect quantitatively. Furthermore, the model in Katic et al. [63] can strike a better balance between positive and negative errors and this wake model is widely used in literature [47].

Assume N turbines are to be placed in the wind farm. The prevailing wind has a speed of v0 as shown in Fig. 1. The turbines directly facing the wind are under no-wake effect. Therefore, wind speed remains unaffected at these turbines. Turbines that are affected by the wake encounter a wind speed of vi (i=1,2,…N) and vi < v0. The value of vi depends on whether a turbine is affected by a single wake or multiple wakes. If a turbine is affected by the wake of a single turbine, the mathematical representation to calculate this wake is given by the following equation [47]: (1)vi,j=v0[1−23(Rjrj)2]where vi,j represents wind speed at turbine i under the wake effect of turbine j, and Rj is the rotor radius of the wind turbine j. Furthermore, rj is the wake radius and is represented as follows [47]: (2)rj=Rj+αdi,j

In Eq. (2), α denotes the entrainment factor while di,j is the downward distance.

If a turbine i is affected by multiple wakes, the following equation is used to calculate this wake [47]. (3)vi=v0[1−∑j∈Φj(1−vi,jv0)2]where Φi is a set that includes all indices of turbines that are upwind of turbine i. A detailed discussion on the single and multiple wake models can be found in the study by Ju and Liu [47].

As stated earlier, the purpose of the present study is not to evaluate the optimization model or its effectiveness. Therefore, only necessary details are provided herein. For details of the optimization model, the interested reader is referred to the study by Ju and Liu [47].

The number of wind turbines in the wind farm is assumed to be fixed and the direction of the wind is known. With these conditions, the aim is to harness wind energy with maximum efficiency. Therefore, the objective function is defined to maximize the conversion efficiency, represented as follows [47]: (4)maximize Efficiency=PcurrentPtotalwhere Pcurrent is the total power generated by turbines under the wake effect in the current layout and Ptotal is the ideal total power generated by all turbines without the impact of any wake.

The genetic algorithm is an iterative algorithm based on the theory of evolution. The algorithm was originally proposed by Fraser [69] in 1957 but received popularity through the work by Holland [70]. GAs operate on a set of solutions in parallel. The set of solutions is referred to as a population. Each solution, commonly referred to as a chromosome, is represented by a string of symbols. A chromosome consists of individual elements known as genes. During each iteration, a new set of chromosomes, called offspring, is generated.

The effectiveness of search in GA depends on two processes: exploration and exploitation [71]. Exploration is concerned with traversing the search space with the aim of discovering new regions therein. Exploitation is connected with fine-turning the search, focusing on a directed search in a smaller region of the search space. To make GA carry out the search efficiently, a balance between both processes is crucial during the search. A high level of exploration can lead to inefficient search since the algorithm is prone to missing good solutions, resulting in a higher execution time to reach convergence. In contrast, a high level of exploitation can cause a loss of population diversity and may lead to premature convergence, thus resulting in low-quality solutions [72,73].

The exploration and exploitation in GA are controlled by two distinct mechanisms known as crossover and mutation. Crossover is responsible for exploitation and is controlled by a parameter called the ‘crossover rate’. Similarly, exploration is controlled by another parameter known as the ‘mutation rate’. The values of these two parameters are user-defined. For GA to reach convergence to produce an optimal solution, an ample amount of time (and equivalently, the number of iterations) is required so that both exploration and exploitation are fully utilized. However, as stated in Section 1, both over-execution and under-execution of an NIA, vis-a`-vis a GA, are not desired. This leads to the incorporation of different termination criteria in GA as discussed below. The objective is to determine which termination criterion best achieves just the right number of iterations. Fig. 2 shows the flowchart of GA employed for the WFLD problem.

As mentioned in Section 1, several termination criteria have been proposed in the literature, but not utilized in the context of the WFLD problem. Some well-known examples are Standard Deviation, Best-Worst, Running Mean, Phi, Pop-Var, and Hitting bound criteria. These are explained below.

The Standard deviation (St dev) based termination criterion [61] measures the standard deviation of the fitness values of all solutions in the current iteration. The execution stops if this standard deviation is equal to or less than a predefined threshold value λ ≥ 0 set by the user.

The best-worst termination criterion [61] works on the difference between the best and the worst fitness values in the current iteration. Once this difference is equal to or less than a predefined threshold value λ≥ 0, the execution of GA stops. This threshold is predefined by the user.

The running mean termination criterion [61] is fulfilled if the difference between the best fitness value of the current iteration and the average of the best fitness values in the last tlast iterations is equal to or less than a given threshold λ ≥ 0.

The Phi termination criterion [61] is satisfied if the quotient of the best fitness value and the mean of all fitness values of the current iteration is equal to or greater than a given threshold 1 − λ, where 1 ≫ λ ≥ 0. Once this condition is satisfied, the algorithm stops.

The Pop-var termination criterion [26] deals with the variance in the current population. The algorithm terminates once the variance of fitness values of all individuals in the current population becomes equal to or less than a given threshold 1 ≫ λ ≥ 0.

In the hitting bound termination criterion [61], the best value of the fitness function is defined by the user. Once this best fitness value by any solution is obtained, the algorithm terminates. In this case, it is expected that the found solution is close enough or equal to the known global optimum.

Table 2 summarizes the traits used in different termination criteria. Based on the information in the table, the termination criteria can be divided into two groups. The first group consists of criteria that have a single trait. Examples of these criteria are St dev, Pop-var, and Hitting bound. The second group consists of the criteria having two traits. Best-worst, Running mean, and Phi are examples of these termination criteria.

One obvious and common limitation of most of the aforementioned termination criteria is that the threshold value of λ that triggers the termination of GA has to be defined by the user. In most cases, determination of the best threshold value of λ is cumbersome. In addition, certain other factors, such as the standard deviation of the population, the difference between the best and worst fitness values, the mean fitness of the solutions in the current iteration, and other simulation-dependent parameters, are not known to the user a priori. In the context of the WFLD problem, getting specific information such as the optimal fitness value, the iteration best/worst values, defining the bound, identifying a certain standard deviation or population variance, and other such variables are not possible until the simulation is carried out. This again raises the question of how to terminate the algorithm if problem-specific information is not available or not known. This issue was somewhat addressed by Mohandes et el. [60] by adopting an approach in which the uncertainties about the termination process are at least minimized, if not completely alleviated. They achieved this by utilizing the K-iterations (KIt) termination as identified in Section 1. The KIt criterion ensures termination at a definite instance, provided that the algorithm reaches a stable state where no further improvements are observed in the solution. However, KIt itself cannot be used alone since it is not known in advance as to when the algorithm would achieve convergence. This led to the development of hybrid termination criteria. A brief discussion of these hybrid termination criteria is provided below for the sake of completeness. Further details can be found in Mohandes et al. [60].

The modified criteria evolve from the hybridization between the basic criterion and KIt. With regard to the hybrid standard deviation criterion, the standard deviation of the fitness values of all solutions in the current iteration is calculated. These fitness values are recorded for each iteration. If the standard deviation does not change (remains constant) for K consecutive iterations, then it signifies that the algorithm has reached convergence and no further diversity is possible in the population. Eventually, the execution of the algorithm stops. In the same sense, the hybrid best-worst approach monitors the difference between the best and worst fitness values in the current iteration. The algorithm terminates if the difference between these two extremes does not change for K consecutive iterations. The hybrid running-mean criterion is satisfied if the difference between the best fitness value of the current iteration and the average of the best fitness values in the last tlast iterations does not change for k consecutive iterations. In the hybrid Phi criterion, termination is reached if the quotient of the best fitness value and the mean of all fitness values of the current iteration remains the same for K consecutive iterations. Likewise, in the hybrid Pop-var approach, the algorithm terminates when the variance of the fitness of solutions in the current population does not change for K consecutive iterations. Finally, in the hybrid hitting bound criterion, the user defines the threshold of the best fitness value. Once this threshold is achieved by any solution and remains constant for K consecutive iterations, the algorithm terminates.

The membership function for Efficiency, μE(x), can be defined as follows. The two extreme values of Efficiency ‘Emax’ and ‘Emin’ define the maximum (ideal) and minimum values, respectively. Emax is taken as 1 while Emin is set at 0. The membership function for Efficiency is mathematically represented as follows. (6)μE(x)={1if E(x)≥EmaxE(x)−EminEmax−Eminif Emin≤E(x)<Emax0if E(x)<Eminwhere the term E(x) represents the Efficiency of the current solution.

The membership function for Execution time, μT(x), can be defined as follows. The values ‘Tmax’ and ‘Tmin’ define the maximum and minimum (ideal) execution time, respectively. The minimum value, Tmin is set at 0, indicating that the execution takes no time at all, which is an ideal (but hypothetical) scenario. The maximum time, Tmax is taken to be the highest execution time taken by any of the termination criteria. Then, the membership value for Execution time is determined as follows.

(7)μT(x)={1if T(x)≤TminTmax−T(x)Tmax−Tminif Tmin<T(x)≤Tmax0if T(x)>Tmaxwhere the term T(x) represents the execution time of the current solution.

Tables 4–6 show the results obtained for Efficiency for the six termination criteria for Case 1, Case 2, and Case 3, respectively. With regard to Case 1, the results presented in Table 4 are adopted from Mohandes et al. [60] and are presented here for a comprehensive analysis. It is observed from Table 4 that the running mean termination criterion produced the best results. More specifically, the maximum values (0.821), minimum values (0.792) and average values (0.803) are all on the high sides compared to the results of the other termination criteria. However, the standard deviation (with a value of 0.0083) is relatively higher than many other termination criteria. On the other hand, if we look at the results produced by the St dev criterion, the results indicate a stable behavior since the standard deviation of the 30 runs is the lowest among all criteria, with a value of 0.0055. However, the maximum, minimum, and average values obtained by the St dev criterion are not impressive, as they fall on the lower side compared to the results for several other termination criteria. In short, the St dev criterion achieved improvements in the range of 0.75% to 5.60%.

For Case 2, the results in Table 5 indicate that the Phi termination criterion was the best, with the highest average efficiency of 0.730. Furthermore, the standard deviation associated with the results of the criterion Phi is also the lowest among all other criteria, with a value of 0.0040. This performance translates to improvements in the range of 1.23% to 2.05% achieved by Phi.

Finally, the termination criterion Phi is also the best for Case 3, as revealed by the results in Table 6. The average conversion efficiency for Phi is 0.809, with a standard deviation of 0.0052. Although this standard deviation is the highest among all criteria, it should be noted that even the minimum of Phi is greater than the maximum values obtained by all other criteria. Therefore, Phi is undoubtedly the best for Case 3, showing an improvement in the range of 3.34% to 4.70%.

Fig. 3 provides more details about the different termination criteria. The figure illustrates the distribution of the best solution for 30 runs with regard to each criterion for the three test cases. These solutions are distributed in five ranges of efficiency, which are 0.7 to 0.729, 0.73 to 0.759, 0.76 to 0.789, 0.79 to 0.819, and 0.82 to 0.849. These five efficiency ranges were carefully chosen to represent the sensitivity of the efficiency change for each termination criterion. As observed in Fig. 3a, Running mean has solutions in the topmost efficiency range (i.e., 0.82 to 0.849). That is, out of the 30 solutions generated during the 30 runs, Running mean had 2 solutions in the top efficiency range, while 28 solutions were in the second topmost range (0.79 to 0.819). Furthermore, the closest contenders to Running mean are the St dev and Hitting bound criteria which produced 29 solutions each in the second topmost range, with one solution each in the middle-efficiency range (0.76 to 0.789). In contrast, Pop-var showed the worst performance since the criterion resulted in no solution in the topmost efficiency range, with solutions distributed in the second topmost and the middle-efficiency range. However, in Case 2, all criteria (with the exception of Phi) produced huge number of solutions in the lowest range of 0.7 to 0.729, as can be seen in Fig. 3b. In contrast, Phi is the only criterion that had the majority of results in a higher efficiency range of 0.73 to 0.759. Finally, for Case 3, Fig. 3c clearly illustrates that only the Phi was able to generate all results in the second topmost efficiency range (0.79 to 0.819), while all other criteria produced results in the lower range of 0.76 to 0.789.

Fig. 4 shows the progression of efficiency vs. iteration for the three cases. The plots show the progression of the best run (out of the 30 runs) for each termination criterion. For Case 1 (Fig. 4a), it is observed that Running mean was able to achieve the best efficiency as the criterion allowed the algorithm to execute for a sufficient amount of time. In contrast, Pop-var caused GA to terminate in the very early stage of execution which did not allow the algorithm to properly search the solution space. With regard to Case 2, Best-worst achieved convergence in the very early stage, whereas Phi was also able to reach almost the same efficiency level, but at a later stage, as shown in Fig. 4b. In contrast, Pop-var made the algorithm execute for a very long duration, yet did not achieve the high efficiency level as that of other termination criteria. For Case 3, the best performance of Phi is visible in Fig. 4c where the criterion achieved the highest efficiency. This high level of efficiency was achievable since GA had ample time to converge.

Based on the above observations, it can be fairly claimed that for Case 1, the Running mean termination criterion was the best among all since it has generated most of the results with the highest efficiency while maintaining a stable behavior as suggested by its small standard deviation associated with the results. Although St dev criterion had the lowest standard deviation, it was not able to produce enough results in the high-efficiency ranges. In contrast, the worst performer was Pop-var; despite a relatively low standard deviation, the solutions produced by the criterion in terms of efficiency were of the worst quality. For Cases 2 and 3, the Phi criterion was the best. For Case 2, the Best-worst approach was the worst, although comparable to Hitting bound, and for Case 3, Hitting bound was the worst.

In order to further strengthen the claims about the best performers, statistical validation was also carried out. The p-values are shown for all three cases in Table 7. For Case 1, Running mean reflected the best output as reported in Table 4, and therefore this termination criterion was used as the benchmark. The p-values in Table 7 indicate that the fitness values (i.e., efficiency) of Running mean were statistically significant as compared to the other termination criteria since all p-values were less than 0.05. Similarly, for Cases 2 and 3, the Phi termination criterion was the best (statistically significant) among all other termination criteria. These outcomes prove the hypothesis that the termination condition has an impact on the quality of solutions produced in terms of efficiency.

With regard to the impact of the termination criteria on the execution time, results of the 30 runs are summarized in Tables 8–10 for Case 1, Case 2, and Case 3, respectively. For Case 1 (adopted from Mohandes et al. [60]), it is visible from Table 8 that termination criterion Phi resulted in the lowest execution time with an average of 164.2 s with a standard deviation of just 5.1, which is also the lowest among all criteria. Although Pop-var had a minimum execution time of 90.9 s, the criterion cannot be declared as the best since the average execution time (187.7 s) and standard deviation of 56.4 s are much higher than that of Phi. Furthermore, the worst performer was the Running mean as the average execution time was the highest (814.5 s) with the highest standard deviation of 210 s. In terms of percentage improvements, Phi achieved values in the range of 14.31% to 396.04%.

For Case 2, the Best-worst criterion resulted in the minimum execution time with an average of 228.5 s. The criterion also showed a standard deviation of 25.5 s, which was the lowest. The worst performer for Case 2 was Pop-var with an average execution time of 803.9 s and a standard deviation of 631.7 s. The percentage improvement achieved by Best-worst falls in the range of 13.83% to 251.82%.

Finally, for Case 3, the criterion Hitting bound displayed the lowest average execution time of 669.0 s, while Phi was the worst with the highest average execution time of 5175.4 s. The standard deviation of Hitting bound was the lowest with 232.9 s, while that of Phi was 277.5 s. One interesting observation is for the Running mean criterion which has the highest run time of 11894.7 s, which is the highest among all criteria, as well as the highest standard deviation of 2103 s, indicating an unstable behavior of the criterion. This translates into improvements between 37. 01% and 673. 60% achieved by Hitting bound in comparison with other criteria.

A further analysis of the results in Fig. 5 gives more insight into the behavior of the criteria. For Case 1, Fig. 5a indicates that all solutions for Phi were spread in the low execution time spectrum (0 to 249 s), while the worst performer was Running mean with most of the solutions in the high execution time range (with 6 solutions in the range of 1000 to 1249 s and 13 solutions in the range of 750 to 999 s). For Case 2, as shown in Fig. 5b, only Best-worst had all its solutions in the range of 0–299 s, while Pop-var had some solutions above 1200 s. However, most of the solutions of Pop-var were spread among different ranges. For Case 3, Hitting-bound was the best since it had almost all its solutions in the range of 0 to 999 s, while all other criteria had solutions in higher ranges, particularly in the “Above 4000” s, as shown in Fig. 5c.

To obtain a more accurate picture, the statistical analysis in Table 11 reveals that for Case 1, the Phi criterion was the best among other criteria since the p-values of the criterion were always less than 0.05. This indicated that the difference in the runtime by Phi and other criteria was statistically significant. In Case 2, the difference in execution time between the Best-worst criterion and other criteria was statistically significant, indicating that the criterion was the best. An exception here was the Hitting-bound criterion where the p-value was 0.158865, indicating that both Best-worst and Hitting-bound were equally good, in statistical terms. Finally, for Case 3, the Hitting-bound criterion performed the best in terms of statistical significance, since all p-values were less than 0.05. Statistical testing indicates that termination criteria have an impact on the execution time of the algorithm.

The results presented in Sections 5.1 and 5.2 suggest that in terms of conversion efficiency, Running mean was the best in Case 1, while Phi was the best in Cases 2 and 3. On the other hand, in terms of execution time, Phi was the best for Case 1, Best-worst for Case 2, and Hitting-bound for Case 3. This points towards a conflicting situation, whereby the decision about the best termination criterion with regard to both conversion efficiency and execution time couldn’t be clearly identified. As such, fuzzy logic-based approach proposed in Section 4 was employed to reach a concrete decision.

Tables 12–14 present the results of aggregation of Efficiency and Execution time using the fuzzy arithmetic mean operator, represented by μC(x) in Eq. (5), for Cases 1, 2, and 3, respectively. The results in Table 12 indicate that for Case 1, Phi turned out to be the overall best termination criterion since it had the fuzzy value (average) of 0.82 which was the highest among the fuzzy values of other criteria. This translates to an improvement in the range of 2.44% to 32.93% by Phi compared to other termination criteria. For Case 2, Table 13 signifies that Best-worst was the best option with a fuzzy value of 0.83. Although Hitting-bound is also comparable with a fuzzy value of 0.82, the criterion had a high standard deviation of 0.015 compared to 0.005 of Best-worst. The improvement achieved by Best-worst was in the range of 1.2% to 9.64%. As for Case 3, the best performer was Hitting-bound with a fuzzy value of 0.86 and a standard deviation of 0.010 as shown in Table 14. For Case 3, Best-worst was also the closest contender with a fuzzy value of 0.85 and a standard deviation of 0.012. Overall, Hitting-bound was the best performer for Case 3, reflecting an improvement between 1.16% to 20.93% compared to the other termination criteria.

In terms of statistical validation, Table 15 indicates that Phi produced statistically significant results for Case 1 compared to all other criteria. For Case 2, Best-worst was the best, with the exception of Hitting-bound since the p-value was 0.298952, indicating that both Best-worst and Hitting-bound were equally good in terms of statistical significance. For Case 3, Hitting-bound was the best among all options.

Based on the statistical analysis presented in Tables 7, 11, and 15, a summary of the best termination criteria is presented in Table 16. It is observed from Table 16 that for the given test scenarios, Running mean, Phi, Best-worst, and Hitting-bound produced the best results. One possible explanation for the superior performance of Phi, Best-worst, and Running mean can be attributed to the very structure of the said termination criteria. Note that the three termination criteria are governed by two traits as highlighted in Table 2. Furthermore, the common trait among the three criteria is the best fitness value. An exception was the Hitting-bound criterion, which is based on a single attribute. However, in general, termination criteria with a single trait showed poorer performance compared to termination criteria with multiple traits.

Another observation in Table 16 is that the overall decision (i.e., fuzzy decision criterion) is more influenced by the objective time objective rather than the fitness objective. That is, the best termination criterion for each case is the same for the time objective, as well as the fuzzy decision. This is interesting since both efficiency and time criteria were given equal preference in the decision-making using the arithmetic mean operator (Eq. (5)). To an extent, this also highlights the importance of the execution time in the optimization process for the underlying problem.