# Feasibility Analysis of Online Uncertainty Metrics for PMU-Based State Estimation
## 4 State Estimation Scenario  
In the MATLAB Simulink programming environment, we reproduce the IEEE 14-bus test case. For completeness, the one-line diagram of the network is reported in Figure 2. It is worth noticing that the IEEE 14-bus test case represents a portion of the American electric power system (in the Midwestern US) as of February 1962. It consists of fourteen buses and five generators (without loss of generality, the nominal system rate has been set equal to 50 Hz. Nevertheless, similar results are expected in a 60-Hz scenario), and 11 loads [27]. In each bus, we deployed a PMU model that measures both voltage and current phasors. In more detail, the PMU implements a dynamic phasor estimator, namely the Compressive Sensing Taylor-Fourier Multifrequency (CS-TFM) model, that guarantees a remarkable performance in the presence of time-varying and highly distorted conditions. For this analysis, the PMU is set in P-class configuration, with a sampling rate of 5 kHz, an observation interval of 60 ms, and a reporting rate of 50 frames per second (fps). The CS-TFM is a dynamic estimator, i.e., it recovers not only the fundamental synchronasor but also its first- and second-order derivatives as computed around the re porting time instant. The time-domain trend of the fundamental component can be thus computed in two ways:  
* _static:_ considering only the zero-order parameters, i.e., amplitude, frequency, phase, and ROCOF, as in (4);
* _dynamic:_ considering all the derivative terms, i.e., accounting also for the parameter variations within the considered window. As further discussed in the following Section, the different formulations result in different nRMSEs and consequent considerations. In the _static_ case, we measure the estimation reliability of the IEC Std parameters, whereas in the _dynamic_ case, we evaluate how much the dynamic model is capable of tracking their time variations. A window of 60 ms is required to guarantee optimal performance. In a Real-Time implementation, the estimation time is circa 20 \(\upmu\)s.  
The measurement uncertainty inherent in the PMU acquisition process is modeled through two independent contributions: the distortion due to wide- and narrow-band disturbances and the systematic error introduced by the instrument transformer. The wide-band distortion is modeled as additive white Gaussian noise, such that the Signal-to-Noise ratio is equal to 45 and 35 dB for voltage and current waveforms, respectively. Such values are taken from an experimental data set of real-world waveforms directly acquired on the EPFL campus. In a similar way, the narrow-band distortion is modeled as a linear combination of the first 40 harmonic terms, whose amplitude is randomly selected in order to reproduce an overall Total Harmonic Distortion (THD) ratio of 0.75 and 14.50% for voltage and current waveforms, respectively, as inferred from the spectral analysis performed on the same data set [32]. As regards the instrument transformers, in this stage of the research, we consider only the systematic error contributions, as we assume that the non-linear effects are covered by noise and harmonics [33]. For this analysis, we consider two performance classes for both current and voltage transformers. Table 1 reports the maximum systematic error in terms of ratio \(\Delta\varepsilon\) and phase error \(\Delta\varphi\) for the different transformer classes and the interested nodes. From an implementation point of view, systematic errors are applied to each PMU estimate, assuming a uniform distribution with a variance equal to one-third of the squared maximum error [34]. Figure 2: Network diagram of the IEEE 14-bus test case. In order to reproduce dynamic operating conditions, the loads have been simulated according to the EPRI LOADSYN model [35].