# Feasibility Analysis of Online Uncertainty Metrics for PMU-Based State Estimation
## 5 Simulation Results
### PMU-Based State Estimation  
In modern power systems, PMUs represent the backbone of many monitoring and control applications. Based on the prior knowledge of the network admittance matrix and topology, the PMU-based measurements can be used to retrieve the system states in all the network nodes. The recent literature has investigated several state estimation (SE) methods [39; 40; 41; 42] that can be classified into two main categories [25], namely the static methods as the WLS, and the recursive methods as the DKF. The SE application of PMU-based measurements yields several advantages. Indeed, PMU-based measurements are linear functions of the state variables, thus leading to a linear state estimation problem, solvable by non-iterative algorithms with reduced computational requirements. Moreover, PMU-based measurements are time-stamped and can be aggregated and time-aligned at a central data collection point. This ensures that every set of measurements given to the SE is composed of measurements taken at the same instant. Finally, the high PMU reporting rate allows for an SE characterized by high refresh rate and low latency, in line with the real-time monitoring requirements of modern power systems. In the present case, we consider a system whose states are the nodal voltage phasors in all the buses, and the SE is required to infer them from an incomplete set of measurements,  
Figure 6: In (**a**), voltage profile of a signal affected by contingencies in reduced inertia conditions. In (**b**,**c**), the static (blue) and dynamic nRMSE (red) as computed with a PMU reporting rate of 50 and 100 fps, respectively. namely voltage and current phasors in a subset of buses. Let us assume that the nodal admittance matrix is known exactly a priori. The measurement model is as follows:  
\[m=Hs+\varepsilon_{m},\qquad\varepsilon_{M}\in\mathcal{N}(0,W) \tag{7}\]  
where \(m\) and \(s\) are the vectors of measurements and states, respectively. Their relationship is defined by the measurement matrix \(H\), while \(\varepsilon_{M}\) accounts for the measurement uncertainty and is modeled as a purely additive random variable, normally distributed with zero mean and variance defined by the noise covariance matrix \(W\). As first approximation, it is reasonable to assume that the noise affecting each measurement is statistically independent, therefore, \(W\) consists of a diagonal matrix. In the context of DKF, the process model is defined as:  
\[m[i]=Am[i-1]+\varepsilon_{P}[i-1] \tag{8}\] \[\varepsilon_{P}\in\mathcal{N}(0,Q[i-1]),\quad\mathrm{E}( \varepsilon_{P}\varepsilon_{M}^{T})=0\]  
where \(i\) denotes the reporting index, the process matrix \(A\) defines the relationship between current and previous measurements, and \(\varepsilon_{P}\) accounts for the uncertainty of the process model. Due to the PMU high-reporting rate, it is reasonable to assume that the process matrix corresponds to the identity matrix, i.e., the system evolution is much slower than the measurement update rate. The process noise is modeled as a purely additive random variable, normally distributed with zero mean and a standard deviation that depends on the process noise covariance matrix \(Q\). Similarly, as per \(W\), also \(Q\) is defined as a diagonal matrix, i.e., the process noise insisting on each measurement is modeled as a statistically independent random variable. In this regard, it should be noticed that measurement and process noise are assumed to be perfectly uncorrelated. Based on (9), the PMU measurement process is approximated as an ARIMA model of order (0, 1, 0), also known as the random walk model. The variance of the process noise is initialized to a nominal value, namely \(10^{-8}\), and then iteratively updated as the sample variance of a circular buffer with the last ten estimates of the corresponding state. As mentioned in Section 2, the accuracy of PMU-based measurements is strongly dependent on the employed measurement chain components.