# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 3 Simulation Results  
### Implementation of the Algorithms  
The target of this section is comparing the capabilities of the previously discussed PMU algorithms in estimating the positive sequence synchrophasor, frequency and ROCOF under different conditions. The tests waveforms follow the standard [7], with reference to P compliance class and maximum reporting rate, but additional cases are also included. Accuracy of the estimates is quantified in terms of total vector error (TVE), absolute frequency error (\(|\text{FE}\,|\)) and absolute ROCOF error (\(|\text{RFE}\,|\)) defined, with reference to the generic measurement instant \(t_{r}\), as:  
\[\begin{split}\text{TVE}(t_{r})=\frac{|\hat{X}_{+}(t_{r})-\hat{X}_ {+}(t_{r})|}{X_{+}(t_{r})}\\ |\text{FE}(t_{r})|=|\hat{f}(t_{r})-f(t_{r})|\\ |\text{RFE}(t_{r})|=|\widehat{\text{ROCOF}}(t_{r})-\text{ROCOF}(t _{r})|\end{split} \tag{48}\]  
The considered algorithms have been applied with the same sampling rate \(f_{s}=200\cdot f_{0}\); a reporting interval equal to the sampling interval (that is \(Q=1\), \(T_{RR}=T_{s}\)) is employed for a more accurate comparison. In order to have similar latencies, so that the comparison is more significant, all the methods except for IEC-P have been implemented to use three nominal cycles for each synchrophasor estimation. Filter design has key importance in the performance achieved by the SV-F algorithm. The first filter \(H\) is a 181 tap lowpass linear-phase equiripple having 2 Hz passband frequency with \(2\cdot 10^{-3}\) passband ripple, 50 Hz stopband frequency and 0.03 stopband ripple. Filters \(M\) and \(P\) are identical 421 tap lowpass equiripple linear-phase filters with 2 Hz passband frequency, 0.01 passband ripple, 50 Hz stopband frequency and 0.03 stopband ripple. All the coefficients of \(H\), \(M\) and \(P\) have been scaled in order to have unit DC gain; this ensures theoretically zero error when the input is sinusoidal and positive sequence, also under off-nominal frequency. Filter \(F\) is a 421 tap linear-phase, equiripple partial-band differentiator. Its coefficients have been scaled so that the output is one when the input is \(x[n]=nT_{s}\); it results in theoretically zero frequency estimation error when input is sinusoidal, positive sequence but frequency is different from its rated value. Finally, filter \(R\) is a 421 tap, linear-phase, equiripple 2 nd order partial-band differentiator. Its coefficients have been scaled in order to return unitary output when input is \(x[n]=(nT_{s})^{2}/2\); this corresponds to zero RFE in the presence of a purely positive sequence input exhibiting a frequency ramp. The SV-TF estimator has been designed in order to exploit the additional flexibility enabled by the native three-phase approach: \(K_{+}=3\) guarantees a good modeling of the time-dependency of the positive sequence synchrophasor, while \(K_{-}=1\) is generally sufficient to prevent the infiltration of the negative sequence term in the positive sequence synchrophasor estimate. When considering the IEC-P and SV-IpDFT methods, \(S=1\) is employed, namely the finite differences in the reporting instant \(t_{r}\) are computed from the estimates in \(t_{r}-T_{s}\) and \(t_{r}+T_{s}\). Finally, a periodic Hann window is used for the implementation of the SV-IpDFT algorithm.