# Wind Power Forecasting with Deep Learning Networks: Time-Series Forecasting +
## 2 Literature Review
### Differential Evolution Algorithm
For a given parameter vector \(V_{i}^{G}\), three vectors are selected randomly: \(X_{1}^{G}\), \(X_{2}^{G}\), and \(X_{r3}^{G}\), such that the indices \(i\), \(r_{1}\), \(r_{2}\), and \(r_{3}\) are distinct. First, the weighted difference of two of the vectors is added to form\(V_{i}^{G}\)  
\[V_{i}^{G}=X_{r1}^{G}+F\cdot\left(X_{r2}^{G}-X_{r3}^{G}\right), \tag{3}\]  
where \(F\) is the scaling factor that controls the amplification of the differential evolution, i.e., mutation scale; its value is located in \([0,\)\(2]\). Small values of \(F\) will lead to smaller mutation step sizes. Consequently, it will take longer for the algorithm to converge. Conversely, large values of \(F\) enable exploration, but can lead to the algorithm overshooting good optima. Thus, the value has to be small enough to enhance local exploration but also large enough to maintain diversity [25]. A well-known DE mutation operation is described as follows [26; 27]:  
\[V_{i}^{G}=V_{best}^{G}+F\cdot\left(V_{r1}^{G}-V_{r2}^{G}\right) \tag{4}\]  
\[V_{i}^{G}=V_{r1}^{G}+F\cdot\left(V_{r2}^{G}-V_{r3}^{G}\right)+F\cdot\left(V_{ r4}^{G}-V_{r5}^{G}\right) \tag{5}\]  
\[V_{i}^{G}=V_{i}^{G}+F\cdot\left(V_{r1}^{G}-V_{r1}^{G}\right)+F\cdot\left(V_{r2 }^{G}-V_{r3}^{G}\right), \tag{6}\]  
where \(r_{1}\), \(r_{2}\), \(r_{3}\), \(r_{4}\), and \(r_{5}\) are the distinct integers randomly generated from the range of \([1,\)\(N]\) and are not equal to \(i\), i.e., (\(r_{1}\
eq r_{2}\
eq r_{3}\
eq r_{4}\
eq r_{5}\
eq i\)). \(V_{best}^{G}\) is the best individual with the highest fitness value (objective value) at generation \(G\). (iii) CrossoverAfter mutation, a trial vector \(X_{i}^{G}=(X_{i,1}^{G},X_{i,2}^{G},\ldots,X_{i,D}^{G})\) is generated for each individual according to a binomial crossover operator on \(X_{i}^{G}\) and \(V_{i}^{G}\), as follows:  
\[U_{i,j}^{G}= \begin{cases}V_{i,j}^{G}&if\ \big{(}rand_{i,j}(0,1)\leq CR\ or\ j==j _{rand}\big{)}\\ X_{i,j}^{G}&otherwise\end{cases}  \tag{7}\]  
In this equation, _rand_ is a uniformly distributed random integer in the range of [1, \(D\)], which is generated for each individual. \(CR\) is the crossover rate, which is restricted in the range of [0, 1]. \(CR\) controls the number of elements that will adjust. Larger values of \(CR\) will lead to deriving more variation in the new population, therefore increasing it also increases exploration [25]. If the \(j\)-th variable \(U_{i,j}^{G}\) of the trial vector \(U_{i}^{G}\) violates the boundary constraints, it is reset as follows:  
\[U_{i,j}^{G}=X_{j}^{min}+rand(0,1)\cdot\Big{(}X_{j}^{max}-X_{j}^{min}\Big{)}. \tag{8}\]  
(iv) Selection  
The selection operator determines whether the target or trial vector survives and enters the next generation based on their fitness values. For a minimization problem, the decision vector with the lower fitness value (objective value) can enter the next generation, which can be expressed as follows:  
\[X_{i}^{G+1}= \begin{cases}V_{i}^{G}&if\ \big{(}fit\big{(}U_{i}^{G}\big{)}\leq fit \big{(}X_{i}^{G}\big{)}\\ X_{i}^{G}&otherwise\end{cases}  \tag{9}\]  
The process is repeated with the expectation, though it is not guaranteed, that a satisfactory solution will eventually be discovered.