## 3 Generation of Scenarios by Utilizing a Seasonal ARIMA Model  
The probabilistic configuration of the stochastic process \(Y\) can be characterized through the identification of the combined distribution of its arbitrary variables. This demonstrates both the connections which occur between all of the variables (also known as statistical dependencies) and the probabilistic way of every random variable on its marginal distributions. In real circumstances, the identification of the combined distribution is usually a challenging and exhaustive effort. However, this can be accomplished if the following postulations are made [50].  
1. The considered combined distribution has to be a multivariate Gaussian distribution and has to be assessed by identifying the average vector and the random variables' variance-covariance matrix constituting the stochastic method.
2. The analyzed stochastic method is deemed stationary, signifying that the variance-covariance matrix and the mean vector do not depend on time \(t\).  
As a result, all autoregressive moving average (ARMA) models rely on the above-mentioned two assumptions. On the other hand, such principles are not followed in certain stochastic processes, thus, with the purpose of reaching the wanted proprieties, a few alterations are necessary to be made to the process. Furthermore, occasionally, several time series events, such as monthly, indicate a seasonality trend which signifies a relation among observations registered in the course of a comparable period of time in consecutive time series. The seasonal tie is not the only one, another connection is the relation among observations noticed in the course of a successive period. Such an observation can be noticed, for instance, for the load demand in certain smaller periods during the course of the month which indicates a closely matching profile each week and each day, thus creating an example of both weekly and daily seasonality. For the case in this study, the quotiian seasonality shows that a seasonal pattern of a 24 order is considered for an hourly load demand series, thus, signifying that the demand of day \(d-1\) at hour \(h\) is similar to the load of day \(d\) at hour \(h\). This similarity is easily verified in cases of a weekly seasonality, in which the seasonality order is consequently \(24\times 7=168\). For such types of conditions, a seasonal autoregressive integrated moving average method (SARIMA) becomes much better suited in detriment of ARMA. This is due to the fact that SARIMA, by being an improvement of the ARMA model, takes into account the potential seasonal unit roots and, specially, the seasonality [51]. Given a stochastic problem featuring a seasonality of order \(S\), the overall equation of the SARIMA method with the subsequent parameters (_p,d,q_) \(\times\) (_P,D,Q_)\({}_{\rm s}\) is represented by the following expression.  
\[(1-\sum_{j=1}^{p}\phi_{j}B^{j})(1-\sum_{j=1}^{p}\varphi_{j}B^{jS})(1-B)^{d}(1-B^{ S})^{D}y_{t}= (1-\sum_{j=1}^{q}\theta_{j}B^{j})(1-\sum_{j=1}^{Q}\theta_{j}B^{jS})\varepsilon_ {t} \tag{5}\]  
With a seasonal component of \(P\) the autoregressive parameters are \(\varphi_{1}\), \(\varphi_{2}\),..., \(\varphi_{P}\), with a \(Q\) moving average parameters \(\theta_{1}\), \(\theta_{2}\),..., \(\theta_{Q}\) and a with a differentiation order \(D\). For the case of this study, the load demand and the PV generation, by displaying time-varying features, are described by the SARIMA method.