# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 3 Simulation Results
### Tests under Dynamic Conditions
#### 3.3.3 Step Tests  
In these tests, the applied input signal is sinusoidal and positive sequence with rated frequency, but it exhibits a step variation of either its magnitude or phase angle in a predetermined time instant \(t_{step}\). The target is assessing the behavior of the PMU algorithms in the presence of fast transients, such as those triggered by switching and faults. As prescribed by [7], step magnitudes are 10% in magnitude and 10\({}^{\circ}\) in phase; performance are evaluated thus assessing the times required by TVE, FE and RFE to return below the corresponding steady-state limits (response times), the delay time and the magnitude of overshoot/undershoot which may be present in the estimates. Figure 21 shows the magnitude of the positive sequence synchrophasor estimated by the different algorithms in the presence of a \(+10\%\) amplitude step, starting from 1 p.u. magnitude. All the algorithms have negligible delay time thanks to a proper compensation of the group delay: halfway of the step is reached within 1 sample (100 \(\upmu\)s) from the step instant \(t_{\text{step}}=1\) s. The magnitude estimate obtained with the SV-TF method exhibits undershoot in the pre-transition region and overshoot in the post-transition region (as evident from Figure 21) with a maximum of 8.08% of the step size. Such effect, which is due to the characteristics of the TF filters, can be mitigated by using Hann weighting as discussed above. All the other algorithms do not show significant under/overshoot: they are easily within the limit indicated by the standard (5% of the step magnitude). The TVE, FE and RFE response times for all the algorithms reported in Table 1. It is worth highlighting that SV-F and IEC-P have zero error both in frequency and ROCOF since in this case phase angle is decoupled from amplitude estimate. SV-TF has response times that strongly depend on the order of expansion and on the possible use of weights, which can easily lead to maximum \(|\)FE\(|\) and \(|\)RFE\(|\) lower than the corresponding steady-state limits: in that case, frequency and ROCOF response times becomes equal to zero. Figure 20: TVE as a function of time under frequency ramp conditions. Figure 22 reports the positive sequence synchrophasor phase angle estimated by the considered algorithms when a \(-10^{\circ}\) phase step is applied. Analogously to amplitude step test, overshoot and undershoot are noticeable only for SV-TF algorithm, which leads to a maximum value of 8.1% with respect to the step magnitude, thus above the limit of the standard. Response times are reported in Table 2, where SV-TF shows the longest TVE response time because of the secondary lobes present in the time evolution of TVE (see Figure 23) that go beyond the 1% steady-state limit prescribed by [7]. The Hann weighting allows improving SV-TF dynamic response, leading to a TVE response time of 15.0 ms and to a maximum under/overshoot of 4.4%. In addition, it also allows reducing the frequency and ROCOF response times, which, as reported in Table 2, are very close to the window length for all the algorithms (namely two and three nominal cycles for IEC-P and the other methods, respectively). In fact, FE and RFE promptly react to the abrupt phase change as soon as the step enters in the sample window, resulting in large deviations from actual frequency and ROCOF (up to about 1 Hz and 80 Hz/s for \(|\)FE\(|\) and \(|\)RFE\(|\), respectively). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{3}{c}{**Response Time [ms]**} \\ \cline{2-4}  & **TVE** & **Frequency** & **ROCOF** \\ \hline IEC-P & 21.7 & 0.0 & 0.0 \\ SV-F & 21.6 & 0.0 & 0.0 \\ SV-TF & 22.6 & 52.3 & 58.7 \\ SV-IpDFT & 28.4 & 52.5 & 58.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Response times for \(+10\%\) magnitude step test. Figure 21: Positive sequence synchrophasor magnitude estimation, 10 % amplitude step. Figure 23: TVE in the presence of a \(-10^{\circ}\) phase step.