# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 3 Simulation Results
### Tests under Steady-State Conditions
#### 3.2.4 Wideband Noise  
Tests with additive uniform white noise permit to verify the immunity of the algorithms to disturbances that are not narrowband (such as unbalance and harmonics discussed in Sections 3.2.2 and 3.2.3) but correspond instead to wideband noise which may be due to different sources, such as thermal noise of analog signal conditioning stages and  
Figure 11: Maximum \(|\)FE\(|\) results under off-nominal frequency conditions in the presence of phase-angle unbalance.  
Figure 12: Maximum \(|\)RFE\(|\) results under off-nominal frequency conditions in the presence of phase-angle unbalance.  
quantization noise of digitization stages. Purely sinusoidal, positive sequence input at nominal frequency has been considered and independent zero-mean uniform noise has been superimposed to the samples. This permits to highlight the impact of the noise alone having considered different yet realistic signal-to-noise ratios (SNRs). TVE, FE and RFE have been evaluated over a 20-s test duration, and their Root Mean Square (RMS) values have been computed. Maximum errors are not considered in this case, since they have intrinsically a considerably worse statistical behavior. Figure 13 reports the impact of noise on the RMS value of TVE. It is clear that the impact depends on the equivalent noise bandwidth of the filters or cascades of filters used for synchrophasor estimation; as expected, the error increases linearly with the noise level. In this respect, SV-IpDFT (characterized by the narrowest equivalent bandwidth) achieves the lowest error values. On the opposite side, the SV-TF technique suffers from the largest TVEs: it is somewhat expected, since TF filters have been designed with dynamic performance in mind. In general, increasing the expansion order widens the equivalent bandwidth and thus also the sensitivity to noise. Furthermore, it is interesting to notice that the aforementioned Hann weighting in SV-TF leads, in the presence of wideband noise, to an error which is 25% higher, meaning that the improved characteristics of dynamic synchrophasor tracking have, as a side effect, lower noise immunity.  
Figures 14 and 15 report the RMS errors for frequency and ROCOF, respectively. Also in this case, performance depends on the equivalent noise bandwidths (this applies to linear but also nonlinear algorithms trough small-signal linearization) that characterize the different methods for estimating frequency and ROCOF, and as expected the values of the errors are proportional to the noise level. It can be noticed that the differences among the algorithms under test are larger when RMS FE and RFE are considered instead of the RMS TVE.  
Figure 13: RMS TVE in the presence of additive wideband noise at different SNRs.  
When looking at the RMS FE, the SV-F algorithm results in the lowest error values, while the other techniques achieve similar performance (SV-TF is marginally the worst). Focusing on the RMS RFE, it is clear that IEC-P and SV-IpDFT suffer from the largest errors: the reason is that ROCOF is obtained with first (SV-IpDFT) or even second-order (IEC-P) discrete-time derivatives, whose outputs are notably highly sensitive to the presence of input noise. The SV-TF with the selected expansion orders provides the best ROCOF estimate in the presence of noise, with SV-F resulting in slightly higher RMS RFE. Also  
Figure 14: RMS FE in the presence of additive wideband noise at different SNRs.  
Figure 15: RMS RFE in the presence of additive wideband noise at different SNRs.  
in this case, Hann weighting for SV-TF leads to significantly worse errors, with a growth up to 78% and 176% for \(|\)FE\(|\) and \(|\)RFE\(|\), respectively. This result highlights how the values of the adopted weighting coefficients must be carefully considered: they can be very helpful for synchrophasor estimation algorithm design (as proven in previous sections), but only if high SNR is expected.