## 2 Genetic Algorithm Dimensioning
### Mutation Rate  
[PERSON] suggested multiple formulations to optimally determine the mutation rate of a GA. Mutation is fundamental in a GA as it increases the coverage of the design space. [PERSON] suggests multiple equations to be used to calculate mutation rates, shown below:  
\[P_{mut}{}^{Max}=1-2^{-\frac{1}{N_{b}}}, \tag{4}\]  
\[P_{mut}{}^{Min}=1-\left(\frac{N_{i}-1}{N_{i}}\right)^{\frac{1}{N_{b}}}, \tag{5}\]  
\[P_{mut}{}^{nat}\approx 1-0.82^{\frac{1}{N_{b}}}, \tag{6}\]  
where \(P_{mut}{}^{Max}\), \(P_{mut}{}^{Min}\), and \(P_{mut}{}^{nat}\) being the maximum, minimum, and suggested mutation rate for every bit for the algorithm at each iteration. The minimum mutation rate is calculated assuming an  
Figure 1: Growth of population size requirement with increasing chromosome bit size.  
Figure 2: Integer population size against total bits for total bits selection.  
average of one chromosome mutating at each generation. The maximum mutation rate is set as the minimum required to have half the bits mutating at each generation. The suggested value is somewhat in-between the extremes and is defined as the maximum mutation probability that features negligible chances of two mutated chromosomes to perform a crossover with each other.  
While this approach is the purely theoretical optimum, it was not practically viable. It served more as a valid guideline. More recent literature [19] showed that algorithm performance is not sensitive to mutation rate in terms of quality of results, however lower mutation rates achieve faster convergences. Literature also suggests a chromosome-based mutation rate, with complete regeneration in case of mutation, with one full reset per generation. The mutation rate in this case that would correspond to this value is 4.76%. Later analysis on GAs featuring Elitism (see Section 2.5) identified a low genetic variability, requiring a higher mutation rate. Figure 3 provides a visualisation of increased genetic variance in the population with increased mutation rate. Red data points are relative to the first 15 generations of the GA, run with 4.76% entire-bit mutation rate (\(p^{\prime}_{mut}\)). Black data points are representative of the final results, run with a value for mutation probability of 15.10%. Generation 13 onwards show a complete convergence for low mutation rates, and the algorithm is essentially relying on a mutation to improve on fitness values, thus operating as an incredibly inefficient Monte-Carlo-style search. Mutation was also chosen to not affect the elite member, as to not lose any GA progress to chance. This approach prevented the process from being chaotic, thus higher variance was effectively positive. The metric for genetic variability was chosen to be power generated, as it was a secondary objective of the optimization. Results would still be influenced by this parameter, while not as heavily as efficiency. This gave the algorithm the ability to pursue the lowest energy path to the optimal power generated value, which was reached after 13 generations. In other words, a low mutation rate meant that essentially all turbines were generating the same amount of power after an extremely low amount of design space exploration, and that was determined by the elite member's power generation value, which in turn was determined by the random initialization. By observing Figure 3, one can see the gradual reduction of variance around the elite members for every generation (at around 1.81 kW).