# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 3 Simulation Results
### Tests under Steady-State Conditions
#### 3.2.3 Unbalance  
Asymmetries between the three phases may be present in real-world applications: thus, assuming purely sinusoidal steady-state conditions, the three-phase input to the PMU is not purely positive sequence, but it may contain also negative and zero sequence contributions. From the considerations of Section 3.2.2, it can be noticed that, for all the algorithms, the zero sequence contribution is inherently rejected, while the negative sequence term results as a complex exponential disturbance, whose rotational speed  
Figure 8: Maximum | RFE| in the presence of 1% harmonic disturbances, \(f=49\) Hz. is \(-(\omega+\omega_{0})\), affecting the positive sequence synchrophasor, frequency and ROCOF estimates. Therefore, the target of these tests is assessing the capability of the different algorithms to provide accurate estimates also in the presence of three-phase asymmetry; for the sake of completeness, frequency \(f\) is swept in the range \([48,52]\) Hz, as in the previous tests. Accuracy is quantified by the maximum TVE, \(\mid\)FE\(\mid\) and \(\mid\)RFE\(\mid\) evaluated on a 2-s time interval for each test condition and PMU algorithm. It is worth noting that the IEC-P and SV-IPDFT methods are intrinsically immune to the infiltration of the negative sequence component when \(f=f_{0}\), thanks to the zeros of the employed triangular or Hann window. Also the SV-TF method results in virtually zero error in this case: in fact, the underlying model produces zeros located at \(-f_{0}\) in the frequency responses in the resulting complex-valued filters which are applied to the SV signal in order to obtain the positive sequence synchrophasor and its derivatives. First of all, amplitude unbalance has been considered: phase \(a\) magnitude is set 10% larger with respect to the others. TVE is still very low for all the algorithms and frequency values (below \(1.2\cdot 10^{-2}\)%) and hence not shown, while results in terms of FE and RFE are much more significant; they are reported in Figures 9 and 10 as functions of \(f\). As mentioned above, the negative sequence acts as a disturbance in the SV signal located at frequency \(-(f+f_{0})\) whose relative magnitude is 3.2% with respect to the positive sequence component; it has to be rejected by the filtering stages of the different algorithms. As expected, when \(f=f_{0}\) all the algorithms except for SV-F result in zero error and exhibit a monotonic increase of errors as long as \(f\) deviates from \(f_{0}\). For significant values of \(\Delta f\), SV-F becomes the most accurate algorithm, both for frequency and ROCOF measurement. The behavior of SV-IPDFT is fairly good as far as the frequency estimate (\(\mid\)FE\(\mid\) below 1 mHz), while \(\mid\)RFE\(\mid\) values are significant, exceeding 0.5 Hz/s; the reason is the spectral interference produced by the negative sequence term (it becomes larger as the difference between \(f\) and \(f_{0}\) increases) that the algorithm assumes to be negligible, whose impact is magnified by the numerical differentiation. The triangular window adopted by the IEC-P is not very effective in reducing the oscillation produced by the negative sequence component in the positive sequence synchrophasor phase angle estimate under off-nominal frequency conditions. When numerical first and, even more, second order discrete differentiation is applied to obtain frequency and ROCOF estimates, the magnitude of this oscillation increases noticeably: it is not surprising that the IEC-P method achieves the worst \(\mid\)RFE\(\mid\) values. The SV-TF method achieves the highest frequency error because of the response of the filter used for obtaining the positive sequence synchrophasor first-order derivative, which in turns is used for frequency estimation. Better results would have been obtained by designing a filter with a wider stopband around \(-f_{0}\).