## 5 Optimisation of a Wind Farm--Kinetic Energy Storage System
### Optimisation Calculations  
A search for the structure of a wind farm--kinetic energy storage system which minimises the capacity of energy storage \(A_{FESS}\) (Equation (11)) for the assumed output parameters of the system, was carried out for wind farms with two power values: \(P_{\text{WFN(1)}}\) = 5 MW and \(P_{\text{WFN(2)}}\) = 10 MW (indices expressed as integers stand for a parameter option). Six time values were taken into account concerning power generation periods with power values below \(P_{\text{3 min}}\): \(T_{max(1)}\) = 5 min, \(T_{max(2)}\) = 10 min, \(T_{max(3)}\) = 15 min, \(T_{max(4)}\) = 20 min, \(T_{max(5)}\) = 25 min and \(T_{max(6)}\) = 30 min; two minimum powervalues \(P_{3 min(1)}\) = 10%\(P_{\text{WFn}}\), and \(P_{3 min(2)}\) = 20%\(P_{\text{WFn}}\); and four values of the acceptable total time: \(\text{T}_{\text{sumTmax(1)}}\) = 75 h, \(\text{T}_{\text{sumTmax(2)}}\) = 100 h, \(T_{sumTmax(3)}\) = 125 h, and \(T_{sumTmax(4)}\) = 150 h.  
The calculations were carried out using the developed optimisation algorithm and application described in Section 5.2 in order to conduct a preliminary study involving changes in the AG parameters to solve the test task. The obtained results helped to draw conclusions relating to the value of the algorithm parameters which contribute to the final solution before the maximum number of iterations, and reduce computing time to ca. 3 h. Finally, a genetic algorithm was applied using the remainder selection method, the Gaussian mutation and elite strategy with a transfer of three best individuals. The number of individuals \(N_{\text{P}}\) = 50 and the number of generations \(N_{\text{g}}\) = 50 were established experimentally. Figure 6 presents the changes in the value of the best individual's (solution) adaptation as a function of the generation number for five activations of the algorithm (Figure 6b) for a farm with rated power \(P_{\text{WFn(2)}}\) = 10 MW, power \(P_{3 min(2)}\) = 20%\(P_{\text{WFn}}\) = 2 MW, \(T_{max(5)}\) = 25 min, and \(T_{sumTmax(1)}\) = 75 h.  
Figure 7 presents the results of optimisation calculations as changes in the minimum capacity of a kinetic storagesystem which meets the required parameters of a system controlling the flow of energy between the farm, storage and power grid as a function of time \(T_{max}\) for two wind farms with power \(P_{\text{WFn(1)}}\) = 5 MW (Figure 7a) and power \(P_{\text{WFn(2)}}\) = 10 MW (Figure 7b). In each case, the calculations were made for two minimum power values \(P_{3 min(1)}\) = 10%\(P_{\text{WFn}}\) and \(P_{3 min(2)}\) = 20%\(P_{\text{WFn}}\) and \(T_{sumTmax(1)}\) = 100 h.  
An essential issue for the analysed class of systems is to identify the relationship between the minimum capacity of the storage \(A_{FESSmin}\) and the time of eliminated outages \(T_{max}\) depending on the maximum total time \(T_{sumTmax}\). Optimisation calculations representing the aforementioned task were performed for a farm with power \(P_{\text{WFn(2)}}\) = 10 MW, two power values \(P_{3 min(1)}\) = 10%\(P_{\text{WFn}}\) = 1 MW and \(P_{3 min(2)}\) = 20%\(P_{\text{WFn}}\) = 2 MW and four times \(T_{sumTmax}\):\(T_{sumTmax(1)}\) = 75 h, \(T_{sumTmax(2)}\) = 100 h, \(T_{sumTmax(3)}\) = 125 h and \(T_{sumTmax(4)}\) = 150 h. The results (value of the minimum capacity of energy storage \(A_{FESSmin}\) meeting the assumed algorithm of the system operation as a function of time \(T_{max}\)) are presented in Figure 8a (\(P_{3 min(1)}\) = 10%\(P_{\text{WFn}}\) = 1 MW) and 8b (\(P_{3 min(2)}\) = 20%\(P_{\text{WFn}}\) = 2 MW). Figure 6: Changes in the value of the function of the best individual’s adaptation as a function of the generation number for a farm power \(P_{\text{WFn(1)}}\) = 10 MW, power \(P_{3 min(2)}\) = 20%\(P_{\text{WFn}}\) = 2 MW, \(T_{max(3)}\) = 25 min and \(T_{sumTmax(1)}\) = 75 h: (**a**) for five activations of the genetic algorithm; (**b**) mean value from ten activations of the genetic algorithm.