# 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
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**(a)**} & \multicolumn{4}{c}{**(a)**} \\ \cline{2-13}  & & \multicolumn{4}{c}{**TVE\({}_{\text{max}}\) (\%)**} & \multicolumn{4}{c}{**1 FE1\({}_{\text{max}}\) (minHz)**} & \multicolumn{4}{c}{**1 RF1\({}_{\text{max}}\) (Hz/s)**} \\ \cline{2-13}
**Test Type** & **Lim. ** & **IpDFF** & \multicolumn{4}{c}{**Tuned**} & \multicolumn{4}{c}{**DCF**} & \multicolumn{4}{c}{**Tuned**} & \multicolumn{4}{c}{**DCF**} \\  & & & **TUNS** & **LSF** & **STD** & **TUNS** & **LSF** & **STD** & **TUNS** & **LSF** & **STD** & **TUNS** & **LSF** & **STD** \\ \hline Freq. dev. only (\(\geq\)2 Hz) & 1 & 0.01 & 0.00 & 0.38 & 0.17 & 5 & 8.9 & 17.7 & 13.5 & 72.0 & 0.4 & 0.2 & 0.0 & 8.1 & **4.2** \\ \hline Freq. dev. \(\sim\) 15-harmonics & 1 & 0.01 & 0.74 & 0.38 & 0.17 & 5 & **10.6** & **141** & **141** & **7.5** & 0.4 & 0.3 & **54.7** & **8.4** & **4.8** \\ \hline AM (\(f_{e}\) = 2 Hz, \(k_{e}\), \(\sim\) 10\%) & 3 & 0.04 & 0.00 & 0.36 & 0.05 & 60 & 8.0 & 0.0 & 0.0 & 0.0 & 2.3 & 0.1 & 0.0 & 0.0 & 0.0 \\ \hline PM (\(f_{e}\) = 2 Hz, \(k_{e}\), \(\sim\) 0.14) & 3 & 0.04 & 0.00 & 0.37 & 0.03 & 60 & 1.8 & 0.5 & 1.6 & 0.7 & 2.3 & 0.0 & 0.0 & 0.0 & 0.0 \\ \hline Freq. ramp (\(\geq\) 42 Hz, \(\pm\) 0.11 Hz/s) & 1 & 0.02 & 0.00 & 0.40 & 0.19 & 10 & 54 & **10.8** & **13.9** & 7.2 & 0.4 & 0.1 & 0.0 & **8.0** & **4.2** \\ \hline AWGN (SNR = 60 dB) & - & 0.00 & 0.04 & 0.36 & 0.03 & - & 14.1 & 24.3 & 2.7 & 2.5 & - & 0.6 & 1.4 & 0.3 & 0.3 \\ \hline \multicolumn{13}{c}{(b)} & \multicolumn{4}{c}{**TVE\({}_{\text{max}}\) (\%)**} & \multicolumn{4}{c}{**1 FE1\({}_{\text{max}}\) (minHz)**} & \multicolumn{4}{c}{**1 RF1\({}_{\text{max}}\) (Hz/s)**} \\ \cline{2-13}
**Test Type** & **Lim. ** & **IpDFF** & \multicolumn{4}{c}{**DCF**} & \multicolumn{4}{c}{**Tuned**} & \multicolumn{4}{c}{**DCF**} & \multicolumn{4}{c}{**1 RF1\({}_{\text{max}}\) (Hz/s)**} \\  & & **TUNS** & **LSF** & **STD** & **TUNS** & **LSF** & **STD** & **TUNS** & **LSF** & **STD** & **TUNS** & **LSF** & **STD** \\ \hline Freq. dev. \(\sim\) 15-harmonics & 1 & 0.00 & 0.03 & 0.06 & 0.16 & 5 & 0.1 & 0.1 & 0.4 & 6.3 & 0.4 & 0.0 & 0.0 & 0.3 & **3.8** \\ \hline Fast (\% = 2 Hz, \(k_{e}\) = 10\%) & 3 & 0.15 & 0.00 & 0.09 & 0.15 & 60 & 14.3 & 0.0 & 0.0 & 0.1 & 2.3 & 0.2 & 0.0 & 0.0 & 0.3 \\ \hline PM (\(f_{e}\) = 2 Hz, \(k_{e}\), \(\sim\) 0.14) & 3 & 0.16 & 0.00 & 0.06 & 0.12 & 60 & 2.1 & 1.8 & 0.4 & 2.0 & 2.3 & 0.0 & 0.0 & 0.0 & 0.0 \\ \hline Freq. ramp (\(\geq\) 42 Hz, \(\The results in Table 4a,b can be summarized as follows. * As far as synrophasor estimation is concerned, all algorithms return maximum TVE values compliant with the Standard even over observation intervals so short as two nominal power line cycles. However, the IpDFTc and the tuned-TWLS algorithms provide the lowest TVE values. In particular, the IpDFTc algorithm is generally preferable in steady-state testing conditions and in the presence of wideband noise, whereas the tuned TWLS estimator is superior when dynamic disturbances (i.e., modulations and frequency ramps) are considered. * The estimation of the fundamental frequency is quite problematic. If just two cycles are observed, no estimator is able to meet the _P Class_\(|\)FE\(|\) limits reported in the Standard in all the testing conditions. In fact, further simulations showed that at least 3-cycle observation intervals are needed. In this scenario, the DCF approach with the STD filter seems to be globally preferable. If 4-cycle observation intervals are used, all estimators return results compliant with the Standard. The \(|\)FE\(|\) values are indeed comparable in most cases. However, the tuned TWLS approach appears slightly better than the others (and particularly much better than the IpDFTc under the effect of modulations). * Finally, as far as ROCOF estimation is concerned, the IpDFTc estimator is the only one that is able to meet the _P Class_\(|\)RFE\(|\) limits reported in the Standard by using both 2-cycle and 4-cycle observation intervals.