# IpDFT-Tuned Estimation Algorithms for PMUs: Overview and Performance Comparison
## 3 Estimation Results and Performance Analysis
### \(P\)_Class_ and \(M\)_Class_ Steady-State and Dynamic Testing Conditions  
The main \(P\)_Class_ and \(M\)_Class_ steady-state and dynamic testing conditions specified in [10] are briefly recalled below. * Fundamental frequency off-nominal static deviations within \(\pm 2\) Hz (\(P\)_Class_) or \(\pm 5\) Hz (\(M\)_Class_);
* Harmonics from the 2 nd to the 50 th (taken one at a time) with amplitude set to 1% (\(P\)_Class_) or 10% (\(M\)_Class_) of the fundamental with the same off-nominal frequency deviations mentioned above;
* Amplitude modulation (AM) or phase modulation (PM) with modulation indexes equal to \(k_{a}=0.1\) or \(k_{p}=0.1\) rad, respectively, and frequency of the modulating tones equal to \(f_{a}=f_{p}=2\) Hz (\(P\)_Class_) or 5 Hz (\(M\)_Class_);
* Linear increment/decrement of the fundamental frequency at rates of \(\pm 1\) Hz/s within \(\pm 2\) Hz (\(P\)_Class_) or \(\pm 5\) Hz (\(M\)_Class_);
* White zero-mean Gaussian noise with variance such that the Signal-to-Noise Ratio (SNR) is 60 dB (this testing condition is not mentioned in the IEEE/IEC Standard,but it has a high practical relevance due to the strong impact of wideband noise on frequency and above all ROCOF estimation [39]);
* Finally, in the _M Class_ case only, a single inter-harmonic tone of amplitude equal to 10% of the fundamental and frequency changing within the bands [10 Hz, 25 Hz] or [75 Hz, 100 Hz] is generated for testing, assuming that the off-nominal fundamental frequency deviations are comprised within [\(-2.5\) Hz, \(2.5\) Hz] and that _RR_ = 50 readings/s. Table 4a,b shows the maximum values of TVE, \(\mid\)FE\(\mid\) and \(\mid\)RFE\(\mid\) (over 1000 records) defined as reported in the Standard and obtained over 2-cycle (a) or 4-cycle (b) observation intervals in the _P Class_ testing conditions summarized above. The _P Class_ limits mentioned in the Standard (when defined) are also reported. The values exceeding such limits are highlighted in bold, whereas the cells containing the best results obtained in each test are shaded. In the case study considered, both the IpDFTc and the tuned TWLS algorithm rely on the Hann window. In the DCF case instead, two different sets of results obtained using two different filters are shown. The first set of DCF results is obtained with 2-cycle-long and 4-cycle-long linear-phase Finite Impulse Response (FIR) filters designed by minimizing the least-squares errors with respect to the frequency response of an ideal low-pass filter. These filters will be shortly labeled as Least-Squares Filters (LSF) in the following. The second set of DCF results is obtained instead with the simple filter suggested in Annex D of the Standard [10], i.e., a filter with a triangular impulse response denoted simply as standard (STD) _P Class_ filter in the rest of this manuscript. Quite importantly, also the classic equiripple FIR filters resulting from Parks and McClellan minimax optimization were considered in the analysis. However, they led to worse performances in almost all testing conditions. Therefore, the results obtained with such filters are omitted for the sake of brevity. This is probably due to the fact that, unlike the LSF and STD filters, the constant ripple magnitude in the stopband cannot be further reduced unless the order of the filter is increased. As a consequence, possible crucial disturbances (e.g., large high-order harmonics) with frequencies close to the position of the peaks of the stopband ripples are not attenuated enough. Figure 1: Maximum TVE, \(\mid\)FE\(\mid\), and \(\mid\)RFE\(\mid\) values over 1000 records of 6-cycle and 8-cycle observation intervals obtained with the classic IpDFT and the IpDFTc estimators as a function of the off-nominal frequency deviations assuming that the first 50 harmonics of amplitude equal to 10% of the fundamental component affect the collected waveform.