# Feasibility Analysis of Online Uncertainty Metrics for PMU-Based State Estimation
## 3 Online Performance Assessment  
In the considered low-inertia scenario, a generic power signal can be represented by a non-linear dynamic model:  
\[x(t)=A\cdot(1+\varepsilon_{A}(t))\cdot\cos\bigl{(}2\pi ft+\varphi+\varepsilon_{ \varphi}(t)\bigr{)}+\eta(t)+z(t) \tag{1}\]  
where \(A\), \(f\), and \(\varphi\) are the fundamental component's amplitude, frequency, and initial phase, respectively. The time-varying terms \(\varepsilon_{A}\) and \(\varepsilon_{\varphi}\) account for amplitude and phase dynamics in terms of polynomial, exponential, or modulation trends. The additive terms \(\eta\) and \(z\) represent the spurious contribution of narrow- and wide-band disturbances, respectively. More in detail, the first refers to the combination of harmonic and inter-harmonic terms, whereas the second accounts for continuous-spectrum components as white or colored noise, decaying DC, or transients. In any PMU-based measurement system, the first step of the measurement chain consists of the acquisition process:  
\[x[n]\simeq x(t=nT_{\rm s}),\quad T_{\rm s}=F_{\rm s}^{-1},n=1,\ldots N_{\rm s} \tag{2}\]  
where \(F_{\rm s}\) is the sampling rate, and \(N_{\rm s}\) is the sample length. It is worth noticing that such equality represents just an approximation, as it does not account for the quantization errors, the distortion introduced by the instrument transformer and the analog front-end, or the phase noise caused by the non-perfectly uniform sampling rate [30]. On the other hand, the dynamics considered in this paper are likely to affect the considered performance metric more significantly, as further proven in the following Section, making these uncertainty contributions negligible for the considered application. Given the acquired sample series, the PMU is required to estimate the synchrophasor \(\hat{p}\), frequency \(\hat{f}\), and ROCOF \(\hat{R}_{f}\) associated with the fundamental component:  
\[\hat{p}[m]=\hat{A}[m]\cdot\exp^{-j\left(2\pi(f[m]-f_{0})mT_{\rm r}+\hat{\varphi }[m]\right)} \tag{3}\]  
where the superscript indicates the estimated parameters, while \(T_{r}\) and \(m\) are the reporting period and the reporting index, respectively. The subtraction by the system-rated frequency \(f_{0}\) allows for expressing the phase contribution due to off-nominal signal frequencies. Figure 1: Classifications of the uncertainty contributions inherent in a PMU-based measurement system are intrinsic, measurement, and overall uncertainty. It is worth observing that the phasor signal model assumes that the signal energy is stationary within the considered observation interval and that the signal energy is mostly concentrated in a narrow bandwidth around the fundamental frequency. When these assumptions are not met (e.g., during an instantaneous step change of amplitude or phase), the PMU estimates suffer from definitional uncertainty due to the model inconsistency between the spectral properties of the signal under test and its phasor representation [31]. Consequently, the recent literature has discussed the metrological significance of standard performance metrics in real-world operating conditions and approaches for assessing PMU reliability during transient conditions. In particular, two novel metrics have been introduced: the GoF and the nRMSE. Both are defined in the time domain and do not rely on the phasor signal model, and they do not introduce any constraint regarding the spectral bandwidth of the observed phenomenon.