# IpDFT-Tuned Estimation Algorithms for PMUs: Overview and Performance Comparison
## 3 Estimation Results and Performance Analysis  
In this Section, the performances of the IpDFT-tuned algorithms presented in Section 2 are compared under multiple testing conditions. In particular, the maximum values of the following parameters are computed, i.e.,  
* the _Total Vector Errors_ (TVEs), namely the magnitude of the error vectors of the estimated synchrophasor relative to the actual synchrophasor values at the same UTC reference times; \begin{table}
\begin{tabular}{l l} \hline \hline
**Step** & Apply the IpDFT algorithm based on the Hann window to the \(r\)-th data record \(y_{r}(\cdot)\) and \\
**1:** & determine the estimator \(\hat{v}_{10,r}\) from (7). \\
**Step** & Mix the samples of the data record \(y_{r}(\cdot)\) with two quadrature sinewaves of unit amplitude \\
**2:** & and frequency \(\hat{f}_{1,r}=\hat{v}_{10,r},f_{s}/M\) to obtain the I/Q components of the collected signal. \\
**Step** & Apply a suitable low-pass filter to the I/Q components to remove disturbances and the \\
**3:** & double-frequency terms. \\
**Step** & Compute the estimators \(\hat{a}_{1,r}\) and \(\hat{\phi}_{1,r}\) by (29), and the estimators \(\hat{f}_{1,r}\) and \(RO\hat{C}OF_{1,r}\) with (30) \\
**4:** & and (31), respectively. \\
**Step** & Return the estimated parameters. \\
**5:** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Steps of the tuned DCF estimation algorithm for PMUs. the _Frequency Errors_ (FE), i.e., the differences between the estimated fundamental frequencies and the actual ones and
* the _Rate of change of Frequency Errors_ (RFEs) namely the differences between the estimated ROCOF values and the actual ones. Such parameters are estimated assuming \(L=120\) and a reporting rate \(RR=50\) readings/s. This is the largest mandatory rate reported in the Standard [10]. To this purpose, the results of the IpDFT-tuned estimation algorithms under study are compared over 1000 data records in the \(P\)_Class_ and \(M\)_Class_ steady-state and dynamic testing conditions specified in [10]. The initial phase values of all possible disturbances (e.g., harmonics, inter-harmonics or modulating tones) are varied randomly with a uniform probability density function in the range \([0,2\pi]\). Then, the TVE \(|\) FE \(|\) and \(|\) RFE \(|\) response times as well as the corresponding delay times and the maximum overshoots due to amplitude or phase steps in the worst-case \(P\)_Class_ and \(M\)_Class_ simulated transient conditions reported in [10] are summarized. All the simulated \(P\)_Class_ and \(M\)_Class_ tests are performed assuming the following common settings, i.e., nominal fundamental frequency \(f_{n}=50\) Hz, sampling frequency \(f_{s}=6\) kHz and observation interval length ranging from \(I_{1}=2\) or \(4\) nominal power line cycles for \(P\)_Class_ tests to \(J_{1}=6\) or \(8\) nominal power line cycles for \(M\)_Class_ tests. Such observation interval values were chosen as they provide a good trade-off between estimation accuracy and responsiveness, although with actual performances that may differ considerably from case to case. The preliminary IpDFT stage (common to all algorithms) is based on the 2-term MSD (or Hann) window defined in [35] with two DFT interpolation points [17]. Quite importantly, the TVE \(|\) FE \(|\), and \(|\) RFE \(|\) values achievable with the IpDFTc algorithm in steady-state tests with off-nominal frequency and harmonics are much lower than those obtained with the classic IpDFT estimator, as confirmed by Figure 1. In the other testing conditions instead, the results of the IpDFTc and the IpDFT almost coincide. Therefore, in the rest of this paper the latter ones are omitted for the sake of brevity. In Figure 1 the curves with different line styles and colors refer to the maximum TVE, \(|\) FE \(|\) and \(|\) RFE \(|\) values returned by the classic IpDFT and the IpDFTc estimators as a function of the off-nominal frequency deviation.