# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 3 Simulation Results
### Tests under Steady-State Conditions
#### 3.2.2 Harmonic Disturbances  
Another important set of tests is focused on assessing the impact of harmonic disturbances on positive sequence synchrophaser, frequency and ROCOF measurements obtained with the different algorithms. SV-based PMU algorithms (SV-F, SV-TF and SV-IpDFT) estimate the positive sequence synchrophaser, frequency and ROCOF by processing the SV signal. Therefore, in order to understand their behavior, it is significant to write the expression of \(\bar{x}_{SV}\) under steady-state conditions in the presence of harmonic disturbances. It results from (22) where:  
\[\bar{X}_{+}(t)=\bar{X}_{ph_{+}}e^{j(\omega-\omega_{0})t} \bar{X}_{-}(t)=\bar{X}_{ph_{-}}e^{j(\omega-\omega_{0})t} \tag{49}\]  
Figure 5: Maximum TVE results under off-nominal frequency conditions. \[\bar{d}_{SV_{0}}(t)=\sum_{h=2}^{\infty}\left[\bar{X}_{ph,h,+}e^{ih\omega t}+\bar{X} _{ph,h,-}^{*}e^{-ih\omega t}\right] \tag{50}\]  
\(\bar{X}_{ph,h,-}\) is the negative sequence phasor of the fundamental; \(\bar{X}_{ph,h,+}\) and \(\bar{X}_{ph,h,-}\) are, respectively, the positive and negative sequence phasors of the \(h\)th order harmonic. Therefore, positive sequence terms produce complex exponential contributions to \(\bar{x}_{SV}\) which rotate with angular speed \(h\omega\), hence in positive direction; conversely, negative sequence components are responsible for complex exponential terms which rotate with angular speed \(-h\omega\), thus in negative direction. Zero sequence terms do not appear in \(\bar{x}_{SV}\). As mentioned before, SV based PMU algorithms apply some kind of bandpass filtering to \(\bar{x}_{SV_{0}}\) in order to extract only the components whose rotational speeds are close to \(\omega_{0}\), thus representing the centre of the passband. It is worth noting that, for a given harmonic order \(h\), negative sequence components are easier to be filtered with respect to positive sequence terms, since they are \(2\omega_{0}\) farther with respect to the middle of the passband. Similar consideration applies also to the IEC-P method. In fact, with some manipulations, it is possible to show that the provided positive sequence synchrophasor estimate corresponds to the application of the triangular FIR filter to the SV signal whose instantaneous angular position of the reference frame is \(\beta(t)=\omega_{0}t\). In practical applications, the three-phase set is very close to be symmetric: this means that phase \(b\) and \(c\) waveforms are obtained by time-shifting that of phase \(a\) by \(-1/(3f)\) and \(1/(3f)\), respectively. Under this assumption, it can be shown that:  
* harmonics whose orders are multiple of three are purely zero sequence contributions;
* harmonics whose orders are given by \(h=3k+1\) are purely positive sequence contributions;
* harmonics whose orders are given by \(h=3k-1\) are purely negative sequence contributions. In order to test the performance of the different algorithms in the presence of harmonic disturbances, three-phase test signals characterized by a positive sequence contribution at frequency \(f\) with a superimposed symmetric \(h\)th order harmonic (\(h\) ranging from 2 to 50) having 1% relative amplitude have been applied. For each test signal and algorithm, maximum TVE, \(\mid\)FE\(\mid\) and \(\mid\)RFE\(\mid\) values over a 2-s time interval have been evaluated and compared. It is worth reminding that because of the aforementioned considerations, injected harmonics whose orders are multiple of 3 do not affect the estimates. A particular situation for methods IEC-P and SV-IPDFT occurs when frequency is equal to its rated value. In fact, they apply a frequency shift of \(-f_{0}\) to the SV signal and then either a two-cycle triangular or a three-cycle Hann window having frequency response zeros located at multiples of \(f_{0}\), thus in correspondence of the harmonics, which are fully removed. As a result, positive sequence synchrophasor, frequency and ROCOF estimates obtained with these techniques are intrinsically immune to harmonic disturbances when \(f=f_{0}\).