## 3 Materials and Methods
### Proposed Hybrid Model  
The proposed model considers the structure of a power grid, as well as historical demand data. Based on this information, the parameters of the alpha stable distribution are estimated. Subsequently, using the Chambers-Mallows-Stuck method [42], stable random values simulate as much data as necessary. Using the simulated information as a training base, a GAM model is fitted to estimate the demand for electricity in the desired time horizon. The conceptual proposal is presented in Figure 1.  
The structure of the proposed model is based on the sequential estimation of a set of parameters for simulating the power system test cases. The first step is to fit the historical data from the reference power system to the stable alpha distribution using the regression method.  
Let \((X_{t})\) be a random variable that represents the demand for electrical energy for a period of time \((t)\). With these historical data, the parameters of the stable alpha distribution are estimated by the regression method using the following equations [35, 36, 46, 47].  
Estimation of parameters \(\alpha\)_and_\(\gamma\), is presented in Equation (22).  
\[y_{k}=m+\alpha_{k}+\varepsilon_{k},\;k=1,\;2,\ldots,\;K \tag{22}\]  
where, \(y_{k}=\log(-\log\left(\varphi(t_{k})\right)^{2}\), \(m=\log\left(2\gamma^{\alpha}\right)\), \(w_{k}=\log\left(t_{k}\right)\), \(\varepsilon_{k}\) is the random error, and \(K\) of the historical data seats \(t_{k}=\frac{\pi k}{25}\;(k=1,\;2,\;\ldots,\;K)\). While estimations of \(\alpha\)_and_\(\gamma\) is a possible estimate of the symmetric parameter \((\beta)\) and location parameter based on Equation (23).  
\[Z_{l}=\mu_{\mu l}-\beta\gamma^{\alpha}\tan\left(\frac{\pi\alpha}{2}\right)sgn (\mu_{l})\big{|}\mu_{l}\big{|}^{\alpha}+\eta_{l},\;l=1,\;2,\;\ldots,\;L \tag{23}\]  
where, \(Z_{l}=g_{n}(\mu_{l})+\pi k_{n}(\mu_{l}),g_{n}(\mu)=\arctan\frac{(Im(\varphi( \mu))}{Re(\varphi(\mu))}\), \(\eta_{k}\) is the random error, and \(L\) of the real data sets \(\mu_{l}=\frac{\pi l}{50}\;(l=1,\;2,\;\ldots,\;L)\). The results of this stage are the parameters of the stable alpha distribution for the historical data series \(X_{t}\sim S(\hat{x},\;\hat{\beta},\;\hat{\gamma},\;\hat{\delta})\). Using Equations (18) and (19) and the estimated stable alpha distribution parameters, stable alpha random numbers can be generated \(\hat{X}_{n}\sim S(\hat{x},\;\hat{\beta},\;\hat{\gamma},\;\hat{\delta})\). In order to estimate the GAM model, information on the operation of the power system is considered as the price of energy and the price of fuels for the time horizon to be modeled. In this case, fuel prices and the local marginal price will be used so have \(P\hat{L}M_{i}\) and \(PC\hat{O}MB_{i}\). The specification of the GAM model is presented in Equation (24) [45].  
\[\hat{X}_{i}=\beta_{0}+f_{1}(P\hat{L}M_{2})+f_{2}(PC\hat{O}MB_{i3})+\varepsilon_ {i} \tag{24}\]  
where \(f_{1}\) and \(f_{2}\) are smooth non-linear functions (could be natural splines, step functions, polynomial, smoothing splines, or basis functions).  
The objective of the proposed model is to be able to have as many instances as possible, as close as possible to reality, in order to validate optimization algorithms for the solution of the economic dispatch problem. One of the advantages of the hybrid model is that long-term horizons can be simulated, such as a full year of operation of the power system.