# Wind Turbine Power Curve Modeling with a Hybrid Machine Learning Technique
## 3 Proposed Wind Power Curve Model
### Extreme Learning Machine and Its Variants
#### 3.2.1 Extreme Learning Machine  
ELM is a new single hidden layer feed-forward neural network that was first proposed by [PERSON] et al. [35] in 2004. Given the \(i\)th sample pair \(\{X_{i},T_{i}\}(i=1,\cdots,N)\), where \(X_{i}=\left[x_{i1},x_{i2},\cdots,x_{iD}\right]^{T}\in R^{D}\) is the output of ELM with \(M\) hidden nodes is given by  
\[f(X_{i})=\sum_{m=1}^{M}\beta_{m}\,H(\alpha_{m},X_{i},\alpha_{m}) \tag{6}\]  
where \(H(\cdot)\) is the activation function, \(\beta_{m}\) is the \(m\)th output weight, \(\alpha_{m}\in R^{D}\) is the input weight vector, and \(\alpha_{m}\) is the corresponding bias. Equation (6) can be re-written as a matrix form,  
\[\hat{T}=H\beta \tag{7}\]  
where \(\beta=\left[\beta_{1},\beta_{2},\cdots,\beta_{M}\right]^{T}\in R^{M}\) is the output weight vector, the output vector \(\hat{T}=\left[f(X_{i}),\cdots,f(X_{N})\right]^{T}\in R^{N},\;H\in R^{N\times M}\) is the hidden layer output matrix, which is represented as  
\[=\left[\begin{array}{cccc}H(\alpha,X_{1},\alpha_{1})&\cdots&H(\alpha,X_{1},\alpha_{M})\\ \vdots&\ddots&\vdots\\ H(\alpha,X_{N},\alpha_{1})&\cdots&H(\alpha,X_{N},\alpha_{M})\end{array}\right] \tag{8}\]  
The optimization of ELM is to solve the following minimization problem,  
\[\min_{\beta}\lVert H\beta-T\rVert_{2}^{2}=\min_{\beta}\lVert H\beta-T\rVert_ {2}^{2} \tag{9}\]  
where \(T=\left[T_{1},\cdots,T_{N}\right]^{T}\in R^{N}\) is the observed target matrix, \(\lVert\cdot\rVert_{2}^{2}\) denotes the squared \(l_{2}\)-norm of a vector. Then, the output weight vector \(\beta\) can be estimated by  
\[\beta=H^{\dagger}T \tag{10}\]  
where \(H^{\dagger}\) denotes the Moore-Penrose generalized inverse of \(H\).  
function can be modified by replacing \(\|E\|_{2}^{2}\) with \(\|E\|_{0}\). However, the modified objective function becomes a non-convex problem, which is difficult to be optimized. With this in mind, [PERSON] and [PERSON] [37] replaced \(\|\varepsilon\|_{0}\) by the \(l_{1}\)-norm-based loss function, \(\|E\|_{1}\). Thus, the objective function is given by  
\[\begin{array}{c}\underset{\beta}{\operatorname{argmin}}\ \|\varepsilon\|_{1}+\frac{1}{C}\| \beta\|_{2}^{2}\\ s.t.\ \ T=H\beta+\varepsilon\end{array} \tag{18}\]  
For Equation (18), it not only guarantees the sparsity, but also leads to the overall minimization convex [37]. Then, the Augmented Lagrange Multiplie (ALM) method is used to optimize Equation (18), and the corresponding Lagrangian function is expressed as  
\[L(\varepsilon,\beta,\alpha)=\|\varepsilon\|_{1}+\frac{1}{C}\|\beta\|_{2}^{2}+ \alpha^{T}(T-H\beta-\varepsilon)+\frac{\mu}{2}\|T-H\beta-\varepsilon\|_{2}^{2} \tag{19}\]  
where \(\mu\) is a penalty parameter and set as \(\mu=2N/\|T\|_{1}\) in [37]. The optimal solutions for \(\varepsilon\), \(\beta\), \(\alpha\) can be obtained by iteratively minimizing Equation (19). The corresponding iterative functions are  
\[\begin{array}{c}\beta_{k+1}=\Big{(}H^{T}H+\frac{2}{C\mu l}\Big{)}^{-1}H^{T} \Big{(}T-\varepsilon_{k}+\frac{\alpha_{k}}{\mu}\Big{)}\\ \varepsilon_{k+1}=max\Big{\{}\Big{|}T-H\beta_{k+1}+\frac{\alpha_{k}}{\mu}\Big{|} -\frac{1}{\mu},0\Big{\}}\circ sign\Big{(}T-H\beta_{k+1}+\frac{\alpha_{k}}{\mu }\Big{)}\\ \alpha_{k+1}=\alpha_{k}+\mu(T-H\beta_{k+1}-\varepsilon_{k+1})\end{array} \tag{20}\]  
where \(\circ\) means the element-wise multiplication, and \(sign\)\((\cdot)\) is a sign function. The optimal output weights \(\beta\) can be obtained when the number of iterations reaches a predefined maximum iteration [37].