# Feasibility Analysis of Online Uncertainty Metrics for PMU-Based State Estimation
## 5 Simulation Results
### PMU-Based State Estimation
Consequently, both static and recursive SE methods allow for associating a weight to each measurement based on some prior information. More specifically, by adjusting the measurement noise covariance matrix \(W\), it is possible to associate different weights to each measurement, in order to reflect the specific PMU reliability. In this sense, it is common practice to use the IT class information to define the most suitable weight for each PMU measurement [43]. It is accordingly assumed that the measurement errors belong to a Gaussian distribution whose variance is derived from the maximum allowed systematic error, as given in Table 1. However, this information is fixed and does not depend on the current system conditions. Therefore in this paper, we investigate whether the application of online performance metrics could be beneficial in terms of state estimation accuracy and robustness. In Figure 7a, we present the voltage waveform as acquired at node 4 of the IEEE 14 bus network. The effect of the dynamic load model is evident in the amplitude level fluctuations. In Figure 7b, the corresponding nRMSE computed according to the static and dynamic phasor formulations are represented in blue and red, respectively. Once again, the static one is more sensitive to the parameter time variations, whereas the dynamic one exhibits a nearly constant value (mainly dependent on the narrow- and wide-band distortion levels). Based on the static nRMSE, the PMU-based measurements would be considered highly reliable only in the time interval between \(t=1\) and \(t=2.5\) s. Conversely, the dynamic nRMSE keeps a nearly constant value over the considered time span. It is thus reasonable to say that an augmented signal model (that accounts also for higher derivative order terms) is capable of fully representing the system dynamics. On the other hand, it is worth remembering that the quantities involved in the SE are the ones associated with the zero-th-order expansion. Therefore, the dynamic nRMSE may produce a misleading evaluation of the actual PMU reliability. In the absence of ground-truth reference values, a Load Flow (LF) analysis has been conducted to define the instantaneous value of voltage and current at each reporting time instant. By subtracting them from the simulated waveforms, it is possible to derive the actual measurement noise and verify the assumptions introduced in (7). In this regard, Figure 8 presents the histogram of the measurement noise of the voltage waveform. The red dashed line indicates the best fit with a normal distribution. It is interesting to observe how the distribution is quite consistent with the modeling assumptions. The mean value is equal to zero, and the probability density function is consistent with a normal distribution. Therefore, the combined effect of IT systematic errors, additive measurement noise, and harmonic distortion, can be well approximated by a Gaussian random variable. In the context of PMU-based state estimation, WLS and DKF represent the most widely used solutions. Indeed, PMU-based state estimation relies on strong assumptions such as the exact knowledge of the network topology and its full observability, but also the fact that measurement errors are uncorrelated with the true state. In such a condition, WLS and DKF represent ideal estimators as they allow to account for the network topology and map the expected measurement uncertainty as an equivalent random variable, normally distributed and with a variance consistent with the expected noise level. The literature has recently proposed several modifications of these two approaches, in order to make them more resilient to possible outliers or bad data. In this sense, a promising solution could be identified in the Least Absolute Value (LAV) estimator, which is less sensitive to leverage measurements. However, from the perspective of the present paper, we preferred to focus on WLS and DKF, where the mapping of the statistical distribution of the expected errors is more straightforward and intuitive.