# Statistical Characterization of Supraharmonics in Low-Voltage Distribution Networks
## 3 Results
### TSHD Characterization  
The TSHDavg and TSHDpeak evaluated at the eight locations are characterized through the Norm, LogN, Weib, Burr, and MixN distributions. As an example of the obtained results, the estimated cumulative distribution functions (CDFs) of the TSHDavg and TSHDpeak for two locations (Alameda and Wohlen) are illustrated in Figure 1. The empirical CDF shapes of the TSHD data at Alameda clearly demonstrate the presence of regimes, which can be adequately modeled only through the multimodal MixN distribution. Particularly, the TSHDavg is captured only through a MixN with \(C=3\) components. This behavior, however, is not generalizable, as, for example, the TSHDpeak at Wohlen has a cleaner behavior which can also be captured by the unimodal distributions. The quantitative assessment of the GOF is reported in Table 1 through the ADC. Bold values indicate the best fit. The MixN distribution with \(C=3\) components is always the best pick as it returns the greatest ADC in all the considered locations and scenarios. Figure 1: Estimated and empirical CDFs of the: (**a**) \(\mathrm{TSHD_{avg}}\) at Alameda (CA); (**b**) \(\mathrm{TSHD_{peak}}\) at Alameda (CA); (**c**) \(\mathrm{TSHD_{avg}}\) at Wohlen (Switzerland); (**d**) \(\mathrm{TSHD_{peak}}\) at Wohlen (Switzerland). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Location**} & \multirow{2}{*}{**Index**} & \multicolumn{6}{c}{**TSHD Distribution**} \\ \cline{3-8}  & & **Norm** & **LogN** & **Weib** & **Burr** & **MixN _C = 2_ & **MixN _C = 3_ \\ \hline Austin & ADC (avg) & 0.9913 & 0.9847 & 0.9787 & 0.9974 & **0.9999** & **0.9999** \\ \hline Delray Beach & ADC (avg) & 0.9326 & 0.8770 & 0.9556 & 0.9556 & 0.9971 & **0.9998** \\ \hline Murphys & ADC (avg) & 0.8469 & 0.8944 & 0.7542 & 0.9933 & **0.9971** & **0.9971** \\  & ADC (peak) & 0.8409 & 0.9174 & 0.7812 & 0.9957 & 0.9837 & **0.9987** \\ \hline Alameda & ADC (avg) & 0.9329 & 0.9357 & 0.9402 & 0.9411 & 0.9880 & **0.9965** \\  & ADC (peak) & 0.4299 & 0.7281 & 0.6613 & 0.8820 & **0.9140** & **0.9140** \\ \hline Wohlen & ADC (avg) & 0.9174 & 0.9418 & 0.9172 & 0.9792 & 0.9955 & **0.9999** \\  & ADC (peak) & 0.8794 & 0.9402 & 0.8694 & 0.9919 & 0.9854 & **0.9991** \\ \hline Rochford & ADC (avg) & 0.7626 & 0.8643 & 0.8495 & 0.9082 & 0.9640 & **0.9958** \\  & ADC (peak) & 0.8037 & 0.8900 & 0.8789 & 0.9088 & 0.9707 & **0.9963** \\ \hline Amsterdam & ADC (avg) & 0.9857 & 0.9640 & 0.9878 & 0.9878 & 0.9896 & **0.9988** \\  & ADC (peak) & 0.9873 & 0.9430 & 0.9532 & 0.9846 & 0.9929 & **0.9945** \\ \hline Skelleftea & ADC (avg) & 0.9894 & 0.9949 & 0.9331 & 0.9988 & **0.9999** & **0.9999** \\  & ADC (peak) & 0.5604 & 0.9127 & 0.4604 & 0.9944 & 0.9982 & **0.9995** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Adjusted determination coefficient of the TSHD distributions. Bold values indicate the best fit. In four case studies, the MixN with \(C=2\) components has equal performance, denoting that a bimodal distribution is, however, effective when two regimes can be individuated. Restricting the analysis to the unimodal distributions, only the Burr distribution has overall the best characterization skill; in two case studies, i.e., \(\text{TSHD}_{\text{peak}}\) at Murphys and Wohlen, the Burr performs also better than the MixN with \(C=2\) components. It is worth noting that the most common distributions, such as the Norm and the Weib, perform very poorly, especially in characterizing the \(\text{TSHD}_{\text{peak}}\). The Q-Q plot of the \(\text{TSHD}_{\text{avg}}\) and \(\text{TSHD}_{\text{peak}}\) for Alameda and Wohlen is illustrated in Figure 2; for the sake of clarity, only the Burr distribution and the MixN distribution with \(C=3\) components are considered in the Q-Q plots. The Q-Q plots of the MixN distribution with \(C=3\) components are closer to resemble the bisector line, compared to the Q-Q plots of the Burr distribution, which cannot even be assimilated to a straight line.