## 2 Methodology
### Continuous Power Consumption Household Appliances  
The starting data corresponds to each 10-min period of the day, from the one included in 00:00-00:10 to the one corresponding to 23:50-00:00. The distribution of consumption for each period is evaluated. First, every power consumption set of values will be evaluated using the Kolmogorov-Smirnov test [44] to check whether they follow a Normal distribution or not. First, the mean (\(\mu\)) and standard deviation (\(\sigma\)) of each sample are calculated, and a theoretical normal cumulative distribution function, expressed in Equation (1), is used. \[F_{O}(x)=\int_{\infty}^{x}\biggl{(}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{1 }{2}\left(\frac{x-\mu}{\sigma}\right)^{2}}\biggr{)}dx \tag{1}\]  
This theoretical cumulative function is compared with the observed cumulative frequency (\(\hat{F}_{n}(x)\)). At first, the maximum upper difference (\(D^{+}\)) is obtained, as in Equation (2). \[D^{+}=\underset{1\leq i\leq n}{max}\bigl{\{}\hat{F}_{n}(x_{i})-F_{O}(x_{i}) \bigr{\}} \tag{2}\]  
Subsequently, the maximum lower difference is calculated (\(D^{-}\)), as it is expressed in Equation (3). \[D^{-}=\underset{1\leq i\leq n}{max}\bigl{\{}F_{O}(x_{i})-\hat{F}_{n}(x_{i-1}) \bigr{\}} \tag{3}\]  
The maximum value between the upper difference and the lower difference is the maximum absolute difference (\(D\)) between the theoretical cumulative function and the observed cumulative frequency, as in Equation (4). \[D=max\bigl{\{}D^{+},D^{-}\bigr{\}} \tag{4}\]  
\(D\) is compared with \(D_{a}\), the maximum difference allowed according to the level of significance (\(\alpha\)) and the type of distribution. \(D_{a}\), Equation (5), is calculated by checking Tables 1 and 2 for the values of \(c_{a}\) and \(k(n)\). \[D_{a}=\frac{c_{a}}{k(n)} \tag{5}\]The significance selected for these distributions is 0.05 (95% of confidence). In Table 1, the coefficient of significance (\(c_{a}\)) is selected according to the model, its amount of data if the model is Weibull and the level of the significance. The significance selected for these distributions is 0.05 (95% of confidence). In Table 2, the expression to calculate \(k(n)\) is selected according to the distribution, and subsequently, this last parameter is calculated with the amount of data. In this way, \(D_{\alpha}\) is calculated. As it can be seen, in Table 2, the expression of \(k(n)\) is different in function of the distribution model, from Normal distribution to Weibull distribution. In the case of \(D<D_{\alpha}\), the null hypothesis H\({}_{0}\) of Normality is accepted so that the corresponding distribution would follow a Normal distribution. 10-min periods whose distribution rejects the null hypothesis of Normality are assessed in order to check if they follow an Exponential distribution with an analogous process to the previous one. The difference is the cumulative distribution function, expressed in Equation (6), where \(\lambda\) is the rate parameter, as well as the different values of \(c_{a}\) and the way \(k(n)\) is calculated according to Table 2.  
\[F_{O}(x)=1-e^{-\lambda x} \tag{6}\]  
Again, it is checked for the Exponential distribution condition, and the unfitted sets are assessed against a Weibull distribution using the same process with its respective cumulative distribution function, formulated in Equation (7), where \(\beta\) is the scale factor and \(\alpha\) is the shape factor. \[F_{O}(x)=1-e^{-(\beta x)^{\alpha}} \tag{7}\]  
The sets of values that are still not associated with any of the three distributions tested so far, will be subjected to an evaluation of the Lognormal, Logistic, Loglogistic, Gamma, Generalized Pareto and Rayleigh distributions until one of these is found to be correct. The method used for the last six mentioned distributions will be Pearson's X\({}^{2}\) test [45].