## 2 Methodologies
### Holt-Winters Forecasting Model  
Holt-Winters models are well-established models that have two seasonal variations: additive and multiplicative types. The additive method is appropriate for fitting a time series data with a constant seasonal variation. Whereas multiplicative method is appropriate for a time series with an increasing seasonal pattern relative to the data level. The generalized Holt-Winters model was reported in literature [19; 41; 43]. In this research, the generalized Holt-Winters model with multiple seasonality and initialization methods is used for forecasting the electrical generation of Kuwait until 2030.  
The Holt-Winters model consists of forecast equation and smoothing equations of the level, trend, and seasonality of the time series. Assuming that \(S_{t}\) represents the seasonality, \(T_{t}\) to represent the trend and that \(I_{t}\) corresponds seasonality, the additive trend multiplicative seasonality HW model can be defined as:  
\[S_{t}=a\Biggl{(}\frac{X_{t}}{\prod I_{t-s_{i}}^{i}}\Biggr{)}+(1-\alpha)(S_{t-1 }+T_{t-1}) \tag{7}\]  
\[T_{t}=\gamma(S_{t}-S_{t-1})+(1-\gamma)T_{t-1} \tag{8}\]  
\[I_{t}^{i}=\delta^{i}\Biggl{(}\frac{X_{t}}{S_{t}\prod_{j\
eq 1}I_{t-s_{j}}^{i}} \Biggr{)}+\Bigl{(}1-\delta^{i}\Bigr{)}I_{t-s_{i}}^{i} \tag{9}\]  
\[\hat{X}_{t}(k)=(S_{t}+kT_{t})\prod_{i}I_{t-s_{i+k}}^{i}+\varphi_{AR}^{k}(X_{t} -(S_{t-1}+kT_{t-1})\prod_{i}I_{t-s_{i}}^{i}) \tag{10}\]  
where \(S_{t}\) represents the level, \(T_{t}\) as the trend, \(I_{t}^{i}\) to correspond to multiple seasonality and \(\hat{X}_{t}(k)\) is the k-step ahead forecast. The smoothing parameters are defined as \(\alpha\) for the level smoothing, \(\gamma\) as the trend smoothing parameter, and \(\delta^{i}\) as the smoothing parameters of each seasonal pattern with cycle length of \(S_{i}\). The term \(\varphi_{AR}^{k}\) is an adjustment for the first autocorrelation error.  
There exist various initialization methods for the level, trend, and seasonality in order to enhance the forecast accuracy [19]. The trend is initialized as \(T_{0}=0\) for an additive trend. Whereas the level equation is initialized by the moving average value of the dataset and is obtained by:  
\[S_{0}=\frac{1}{S_{m}}[\frac{X_{1+s_{m}}-X_{1}}{s_{m}}+\frac{X_{2+s_{m}}-X_{2} }{s_{m}}+\ldots+\frac{X_{2s_{m}}-X_{s_{m}}}{s_{m}}] \tag{11}\]The seasonality is initialized by the method presented by [PERSON] and [PERSON] [39] and adopted by the National Institute of Standards (NIST) which depends on calculating the weights of the data series against the multiple seasonality pattern values. The proposed method is presented in the following steps:  
Step 1: Compute the yearly average as \(A_{m}^{i}\) for each seasonality of length \(s_{m}\) and has a pattern of \(n_{q}\) times in the dataset  
\[A_{m}^{i}=\frac{\sum_{j=1}^{s_{i}}X_{(m-1)s_{i}+j}}{s_{i}}\text{ for}m=1,\;2,\; \ldots,n_{q}\]  
Step 2: Divide the observations by the yearly averages as:  
Step 3: Write each seasonality as:  
\[I_{1-s_{i}}^{(i)}=\frac{\frac{X_{1}}{A_{1}^{i}}+\frac{X_{s_{i}+1}}{A_{2}^{i}}+ \cdots+\frac{X_{(m_{i}-1)s_{i}+1}}{A_{m_{i}}^{i}}}{m_{i}}\]  
\[I_{1-s_{i}}^{(i)}=\frac{\frac{X_{1}}{A_{1}^{i}}+\frac{X_{s_{i}+1}}{A_{2}^{i}}+ \cdots+\frac{X_{m_{i}s_{i}+1}}{A_{m_{i}}^{i}}}{m_{i}}\]  
Step 4: Write the seasonal indices as:  
\[I_{1-s_{i}}^{(i)}=\left\{\begin{array}{c}I_{t-s_{1}}^{\star(i)}i=1,\;t=1, \ldots,s_{i}\\ \frac{I_{t-s_{1}}^{(i)}}{\prod_{j=1}^{i-1}I_{t-s_{j}}^{(i)}}i>1,\;t=1,\;\ldots,s_{i}\end{array}\right.\]  
The simulation parameters are presented in Table 2 with the values of level, trend and seasonality smoothing parameters as \(\alpha,\;\gamma,\;\delta^{i}\) respectively. The seasonality smoothing parameters are presented as \(\delta^{y},\;\delta^{i}\) and \(\delta^{w}\) that corresponds to the yearly, quarterly, and weekly seasonality smoothing parameter values.