## 4 Results and Discussion
### Comparison Between the Training and Synthetic Data  
#### 3.2.1 Case Study 1: Generation of Synthetic Power Consumption Data for a Domestic Consumer  
As already stated, the investigated load profile of the domestic consumer is characterized by uneven consumption during the days, which can be explained by the lifestyle of the people living there. To investigate the performance of the applied approach, the load profile was evaluated using a different number of states varying between 10 and 26. For each number of states:  
Figure 6: Sample datasets with the power consumption of (**a**) a domestic house; (**b**) a pig farm; (**c**) a printing house. * Matrices were generated according to Step 2 of the methodology. * 10,000 days of synthetic data were generated according to Step 3.1 from the methodology to make sure the long-term trend of the generated data has reached the final probability distribution vector. * The matrices of the synthetic and of the original data were compared using the Frobenius distance and \(R^{2}\), according to Step 3.2 of the methodology. * The average Frobenius distance and \(R^{2}\) were estimated for each hour of the day. * The relative difference between the variance and standard deviation of the original and the synthetic data was obtained. Initially, the Frobenius distances and the \(R^{2}\) values were estimated for each hour of the day and each number of states. For example, Figure 7 presents the situation with 12 states. Next, their average values were estimated for each number of states, with the results summarized in Figure 8. It can be seen that the average Frobenius distance of the hourly matrices varied between 0.22 and 1.03 and had a minimum at 12 states, as well as local minimums at 17 states and 23 states. Similarly, the average Frobenius distance for the hour-change varied between 0.08 and 0.99 and had a minimum at 12 states and local minimums at 18 states, 21 states, and 24 states. The average \(R^{2}\) of the hourly matrices varied between 0.76 and 0.96 and had maximums at 12, 13, 17, and 18 states. Similarly, the average \(R^{2}\) of the hour-change matrices varied between 0.74 and 0.98 and had a maximum at 12 states and local maximums at 18 and 24 states. Regarding the statistical analysis, it can be observed that the relative error of the variance varied between 2.54% and 19.59% and had a minimum at 10 states. The errors remained below 6% between 13 and 18 states, as well as at 24 states. Similarly, the relative error of the standard deviation varied between 1.28% and 10.33%. It had a minimum of 10 states and local minimums of 14, 18, 20, and 24 states. A good candidate in terms of the number of states is expected to have low Frobenius distances, low variances, and low standard deviations, as well as \(R^{2}\)s near 1. Since there is no perfect combination, the following situations were investigated more closely (Table 2):  
* The generated data with 12 states, which was characterized by the lowest Frobenius distances and the highest \(R^{2}\) values, but with higher relative error for the variance and standard deviation. * The generated data with 18 states, which was characterized by a local minimum of the Frobenius distances, high \(R^{2}\) values, and local minimums of the statistical measures. * The generated data with 24 states, which was characterized by local minimums of the Frobenius distances and the statistical measures, as well as high \(R^{2}\) values. Figure 7: Hourly Frobenius distances and \(R^{2}\) coefficients of the generated data with 12 states. It can be seen that the data of the domestic consumer generated with a higher number of states had better statistical metrics, but slightly worse probabilistic ones; i.e., the choice of an optimal solution is a tradeoff between them. To better understand these three situations, 7 days of data were generated (Step 4 from the methodology), respectively, with 12 (Figure 9b), 18 states (Figure 9c), and 24 states (Figure 9d) and visually compared with 7 days from the original dataset (Figure 9a).