# Feasibility Analysis of Online Uncertainty Metrics for PMU-Based State Estimation
## 5 Simulation Results  
### Sensitivity Analysis  
In the first test, we perform a sensitivity analysis of the nRMSE metric as a function of the test conditions and of the reconstruction model. For this analysis, we consider two test signals. The first one is in quasi-steady-state (SS) conditions. The amplitude and initial phase are stationary and equal to their nominal value (for simplicity, 1 pu and 0 rad, respectively), whereas the frequency is initiated at 50 Hz, but its time dependence is \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Class** & **\(\mathbf{\Delta\varepsilon}\) (\%)** & **\(\mathbf{\Delta\varepsilon}\) (mrad)** & **Nodes** \\ \hline \multirow{2}{*}{VT} & 0.1 & 0.1 & 1.45 & 1, 2, 5, 7, 8 \\  & 0.5 & 0.5 & 5.82 & 3, 4, 6, 9, 10–14 \\ \hline \multirow{2}{*}{CT} & 0.2 & 0.2 & 2.91 & 1, 2, 5, 7, 8 \\  & 0.5 & 0.5 & 8.73 & 3, 4, 6, 9, 10–14 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Instrument Transformer Classes. Figure 3: Diagram of the simulated dynamic EPRI LOADSYN model. modeled as a random walk process with a variance of 1 \(\upmu\)Hz. This reproduces the typical variability due to the continuous adjustment of the power generation and demand profiles. The second signal adds a further level of complexity by introducing a phase modulation (PM) with a modulation depth and frequency equal to \(\pi/10\) rad and 2 Hz, respectively. As a consequence, the erratic trend of the instantaneous frequency is super-posed to a sinusoidal oscillation of \(\pm 0.628\) Hz every 0.5 s. In this sense, the second signal accounts for the dynamics introduced by larger-scale imbalance events, e.g., during inter-area oscillations or load shedding and restoration operations. In this context, Figures 4 and 5 show the nRMSE (a) and TVE (b) distributions for the SS and PM signal, respectively. Both the plots are divided into two portions based on the VT class: 0.1 on the left and 0.5 on the right. Such distributions have been computed based on ten independent tests, each one of a duration of 5 s, resulting in a statistically relevant sample of 2500 estimates per metric. For the sake of brevity, the figures consider only the voltage signals, but similar considerations hold also for the current ones. In the SS case, the TVE is proportional to the systematic error contributions, whereas the nRMSE is dependent on the reconstruction model. Using the traditional static representation, we notice a clear differentiation between the two sensor classes. The dynamic representation, instead, allows for a more precise recovery of the original time domain signal, independently from the extent of systematic errors. In the PM case, the TVE distribution is wider for class 0.5, but the mean value is more or less coincident in both classes. This proves the scarce accuracy of the PMU-based estimates in such dynamic conditions. The nRMSE distributions present two totally inconsistent trends. The static representation mimics the TVE behavior, whereas the dynamic one presents results that are nearly coincident with the SS case. Figure 4: Steady-state conditions: comparison of nRMSE as computed on static and dynamic phasor representation (**a**) and corresponding TVE (**b**) as a function of the selected voltage transformer class. In Table 2, we evaluate in a more quantitative and rigorous way the correlation between TVE and nRMSE in the considered configurations. Based on the obtained distributions, we consider the Pearson correlation coefficient, as well as the corresponding \(p\)-value (null hypothesis: statistical independence of TVE and nRMSE) and the 95% confidence interval (defined as the variation range between upper and lower limit). As expected, the correlation is significantly higher for the static model, particularly when the systematic errors are lower. Nevertheless, it is also worth noticing that the addition of a single and not-so-fast dynamic, as in the PM case, produces a drastic drop in the correlation between nRMSE and TVE, even in the static representation.