# Feasibility Analysis of Online Uncertainty Metrics for PMU-Based State Estimation
## 1 Introduction  
In recent years, the ever-increasing penetration of renewable energy sources and distributed generation is pushing for the development and deployment of more sophisticated and precise control techniques [1; 2; 3]. Due to the loss of rotational inertia in favor of inverter-connected facilities, modern power systems are subject to strongly non-stationary operating conditions, characterized by fast dynamics and wide-band spectrum [4]. In this context, the measurement infrastructure is expected to become the backbone of many monitoring and control applications [5], as state estimation [6], fault location [7], loss-of-mains detection [8], and under-frequency load shedding [9]. To this end, Phasor Measurement Units (PMUs) represent a promising solution due to their measurement accuracy and time synchronization [10; 11]. In compliance with the recent IEC Std 60255-118-1 (briefly, IEC Std) [12], PMUs measure the phasor, frequency, and rate of change of frequency (ROCOF) of the fundamental component with reporting rates in the order of tens of frames per second. The National Metrology Institutes (NMIs) are responsible for the rigorous characterization and precise calibration of PMUs [13; 14]. In this sense, the IEC Std introduces two performance classes, P and M, intended for protection and measurement applications. The first one has higher responsiveness, whereas the second one is required higher accuracy. The performance assessment relies on metrics specifically defined to quantify the estimation accuracy, reporting latency, and response time of the PMU. In particular, the estimation accuracy is defined in terms of Total Vector Error (TVE), Frequency Error (FE), and ROCOF Error (RFE). However, the entire IEC Std theoretical framework relies on a static and stationary signal model that cannot fully represent the dynamics of modern power systems. Hence, recent literature has explored extending or overcoming the phasor signal model in favor of more sophisticated approaches capable of properly capturing wide-bandwidth phenomena [15; 16]. Nevertheless, the transition from PMUs (and similar phasor-based devices) to a new generation of instruments is neither envisioned nor expected to be completed for a long time. Therefore, the role of NMIs is to evaluate the measurement uncertainty caused by the modeling discrepancy between the signal model and the signal under test and determine plausible confidence intervals for PMU-based control applications. A typical example in this sense is represented by the regulations regarding ROCOF withstand capability for connecting new power-generating modules or demand units [17]. An alternative approach is to identify new performance metrics that do not suffer from the same definitional uncertainty [18], e.g., by considering the signal in the time domain where the only limitation is represented by the amplitude and time resolution of the acquired time series. In this context, two promising metrics have been introduced, namely the Goodness-of-Fit (GoF) [19] and the Normalized Root Mean Squared Error (nRMSE) [20]. Similarly, both these metrics quantify how much of the original signal energy has been neglected by the corresponding phasor representation [21]. The larger the residual energy, the lower the reliability of the PMU estimates. A recent publication has discussed the variation range and the estimation uncertainty inherent in these metrics. It has proposed a possible standard amendment that would allow transmitting its value with the PMU estimates [22]. Indeed, the IEC Std format for the measurement data packet already contains a few bit-mapped flags defining the instrument's internal state and the measurement quality info. Without affecting the overall packet size, it would be possible to transmit the reliability metric value at the cost of an extra 4 bytes and thus make it available to the applications processing the PMU estimates. The knowledge of this reliability metric opens up new possibilities at different network levels.