# Feasibility Analysis of Online Uncertainty Metrics for PMU-Based State Estimation
## 5 Simulation Results
### PMU-Based State Estimation
Figure 7: In (**a**), voltage signal as acquired by the PMU at node 4. In (**b**), nRMSE is measured based on static (blue) and dynamic (red) phasor representation. In this context, we ran the WLS and DKF state estimation with three different configurations of the measurement covariance matrix. As shown in Figure 9a, the noise covariance is set based on the systematic error variances, or equal to the inverse of the nRMSE in its static or dynamic formulation. In this regard, it is interesting to observe how the IT class and the dynamic nRMSE provide comparable values, whereas the static one associates a lower weight to any measurement taken during a non-stationary condition. Figure 9b,c shows the corresponding SE results as compared against the LF reference value for WLS and DKF, respectively. The similarity between the two estimation results descends directly from the adopted process model in (9), which does not yield any significant advantage with respect to the WLS approach. Figure 8: Statistical distribution of measurement noise as derived from simulated voltage signal and LF-based ground-truth values. The red dashed line represents the best fit with a Gaussian distribution. Figure 9: In (**a**), measurement weights are based on VT class (green), static (blue), and dynamic (red) nRMSE. In (**b**,**c**), WLS and DKF estimates of voltage real part in node 4 against the load flow (black) reference values. In terms of estimation errors, it is worth noticing how the dynamic formulation does not produce any improvement with respect to the IT class information, whereas the static one allows for a slight reduction of the noisy oscillations. In a more quantitative way, Figure 10 shows the statistical distribution of the estimation errors as a function of the selected configuration. The box plot refers to the WLS case, but similar results can be obtained in the DKF one. The static nRMSE allows for higher noise rejection in the non-stationary portions of the signals. On the overall considered time interval, though, the RMS error for the three different configurations is quite similar, ranging from 700, 750, and 500 ppm for the class, dynamic, and static configurations, respectively. Based on the obtained state estimates, it is also possible to verify the assumptions on the measurement noise. In this regard, Figure 11 shows the statistical distribution as inferred from the DKF estimates. The red dashed line indicates the best fit with a normal distribution. In this case, it is worth noticing how the obtained probability density function does not comply with the assumed Gaussian distribution and is strongly affected by the non-compensated effect of the IT systematic errors. In a more quantitative way, the coherence between the measurement noise and its mathematical approximation in the covariance matrix is below 5%. It is thus reasonable to say that the Gaussian assumption is only a limited approximation, and the estimation process cannot be modeled as a linear and predictable process. Figure 11: Statistical distribution of measurement noise as inferred from DKF estimates. The red dashed line represents the best fit with a Gaussian distribution. Figure 10: Statistical distribution of the estimation errors in the WLS case as a function of the weight configuration. This is further proof of the fact that the nRMSE can quantify the uncertainty contributions relative to the PMU acquisition and processing, but cannot account for external contributions, due to other components of the measurement chain. Namely, its application in SE produces a limited performance improvement but represents a promising first effort to associate each measurement value reliability indices that depend not on fixed a priori information, but on the actual operating condition of the instrument.