## 6 Results  
We used a set of 60 synthesized TSs as model data, each containing numerical values for seven periods. Such a small TS length was chosen because the result was the TS length corresponding to the real analyzed energy consumption in residential buildings in cold weather. The first example's time series (TS) was part of six clusters formed by transforming six base patterns (TSs) using periodic functions like sine and cosine. We warped the original model set of TSs to assign it to a new cluster. A visualization of the initial set of TSs of six clusters is presented in Figure 4(a), where time is a countdown of time and value is the value of the TS element. The graphs of different clusters are indicated in various colors, while the graphs of TSs forming the initial clusters are grouped and create a \"thick\" line. The distorted graph was obtained based on the values of the elements of one of the TS that belonged to the cluster whose TSs were located at the bottom of the group by changing their values starting from the third time point. The new TSs should have formed a new cluster. The algorithm confirmed this fact as seen in further detail in the following. Figure 4(b) shows the normalized initial set of TSs and the normalized distorted TSs. Using the algorithm on a normalized initial set of time frames with t = (3.7) allowed us to find six groups, with time frame lengths of three, five, six, and seven. The best number of six clusters was determined by minimizing the XB index. The XB index made it possible to determine that the optimal number of clusters was five, with the number of time samples equal to four. Poor separability with a sample count equal to four can explain these clustering results. It is important to note that for TSs with lengths three and four, both clustering quality indicators (XB index) agreed on the best number of clusters. Using the algorithm on a normalized skewed set of time frames helped us find six clusters when there were three time samples and five clusters when there were four time samples. Applying the algorithm on a normalized skewed set of time frames made it possible to identify six clusters with the number of time samples equal to three and five clusters with several time counts equal to four. For TSs of length three and four, both clustering quality indicators (XB index) gave a consistent solution for the optimal number of clusters. Figure 5: Distorted TS model set visualization. With several time samples equal to five, six, and seven, the XB index determined that seven was the optimal number of clusters. The distorted graph was obtained based on the values of the elements of one of the TSs that belonged to the cluster whose TSs were located at the bottom of the group by changing their values, starting from the third time point. The new TSs should have formed a new cluster. We further confirmed this fact using the algorithm. Figure 5b shows the normalized initial set of TS and the normalized distorted TS. Implementing the algorithm on the normalized initial set of time frames with the number of time samples t = (3.7) made it possible to identify six clusters with time frame lengths equal to five, six, and seven. The use of the Hurst index (XB) at four time points allowed us to determine that the optimal number of clusters was five (Table 1 and Figure 6). Testing of the proposed algorithms with a two-stage method of TS variability was carried out based on a real group of indicators when studying the location and operation of small photovoltaic installations for household consumers in sparsely populated areas in southeastern Bulgaria for the period from 2021 to 2024.