# Lessons for Data-Driven Modelling from Harmonics in the Norwegian Grid
## 3 Results
### Temporal Distribution of THD Excursions  
Previous work [60] suggests that events in the power grid are not uniformly distributed in time, but rather occur in clusters. While not having access to event data, we can analyze the temporal distribution of THD excursions (values \(>99\) percentile) for the three sites on a signal downsampled a time-resolution of one minute. For each site, there are a total  
Figure 8: Statistical descriptors of the THD and selected harmonics (rows) aggregated per month for the three sites (columns). For each month, we indicate the median as well as 1, 10, 25, 75, 90, and 99 percentile (see legend). Grey shaded bands indicate the passage of one year. \(\sim\)\(2\times 10^{6}\) samples (minutes). For the 22, 66, and 300 kV sites, we find 34362, 39015, and 22063 min with excursions, respectively. Figure 9 show a temporal scatterplot of THD excursions as well as the fractal dimension \(\mathcal{D}\) as a function of time-scale \(s\), on the left and right panel respectively. We find that--irrespective of voltage level--the fractal dimension \(\mathcal{D}\) depends on the time-scale. In particular:  
1. At timescales \(300\) s \(<s\leq 10^{5}\) s (a few days), we find \(\mathcal{D}\sim 0.34<1\) (with slight variations across voltage levels, but a goodness of fit \(R\sim 0.99\) for each level). This suggest multi-scale substructure of THD excursions in time. Visually, this is manifested as clumping of THD excursions (see Figure 9, lower panel). Clusters also vary in size (duration) and can be decomposed into further (sub-)clusters;
2. At time-scale \(s\geq 10^{5}\) s, we find \(\mathcal{D}\sim 1\) (\(R\sim 0.99\)). This suggests no (or at least very little) temporal substructure in the distribution of THD excursions. Visually speaking, there are long sequences of THD excursions with similar timing (Figure 9, upper panel). There are only occasional large gaps in time. The fractal dimension is essentially a measure of signal roughness. At timescales \(\gtrsim 10^{5}\) s, our binary signal of THD excursions is fairly smooth (visually, excursion appear equally spaced in time). Statistical measures will weakly depend on the timescale over which they are calculated. For example, the mean return interval computed over a period of a weeks, a months, or years will be similar. At timescales \(\lesssim 10^{5}\) s, excursions clump together (the signal is rough) so that statistical measures will strongly depend on the timescale over which they are calculated. The mean return intervals computed over a few minutes or over a few hours will be different. Published works using fractal analysis apply a variety of measures and is therefore difficult to compare to. For example, refs. [76; 77] compute the fractal dimensions of one-dimensional time-series (of power and current, respectively) to detect the presence of (artificially induced) events and loads. Closest to our work is [78], which computes the Hurst exponent \(\mathcal{H}\) of a (presumably binarized) power fault time-series to determine the timescales over which faults in transmission (and distribution systems) are correlated. While \(\mathcal{D}\) measures local roughness, \(\mathcal{H}\) measures long-term correlations. (Values \(\mathcal{H}>0.5\) indicate long-term dependencies, while \(\mathcal{H}<0.5\) indicates rapid mean reversion.) For self-similar (self-affine) processes, \(\mathcal{D}=n+1-\mathcal{H}\) (\(n=0\) for a binarized time-series), so that locally defined roughness can be related to long-term correlations [79]. For \(\mathcal{D}=0.34\) and \(\mathcal{D}=1\), we find \(\mathcal{H}=0.66\) and \(\mathcal{H}=0\), respectively. In our case, this means that long-term correlations are limited to timescales \(\lesssim 10^{5}\) s. In other words, the autocorrelation drops off at lags \(\gtrsim 10^{5}\) s and phenomena longer than a few days ago do not influence THD excursions. By directly computing \(\mathcal{H}\), ref.