# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 3 Simulation Results
### Tests under Dynamic Conditions
#### 3.3.1 Amplitude and Phase-Angle Modulations  
Electrical signals in the presence of power system oscillations can be modeled as in (1). Therefore, the tracking capability of PMUs under these conditions can be assessed by applying three-phase balanced sinewaves with sinusoidal amplitude or phase angle modulation. \[\mathbf{x}_{abc}(t)=\sqrt{2}X(1+k_{x}\cos(2\pi f_{m}t))\left[ \begin{array}{c} \cos(\omega_{0}t+k_{x}\cos(2\pi f_{m}t-\pi))\\ \cos(\omega_{0}t-\frac{2}{3}\pi+k_{x}\cos(2\pi f_{m}t-\pi))\\ \cos(\omega_{0}t+\frac{2}{3}\pi+k_{x}\cos(2\pi f_{m}t-\pi))\end{array} \right] \tag{51}\]  
where \(k_{x}\) and \(k_{x}\) are the amplitude and phase-angle modulation indexes, respectively, \(X\) is the unmodulated RMS amplitude and \(f_{m}\) is the modulation frequency. Applying the SV transformation to (51), it results:  
\[\bar{x}_{SV}(t)=\sqrt{3}X(1+k_{x}\cos(2\pi f_{m}t))e^{j(k_{x}\cos(2\pi f_{m}t- \pi)+\omega_{0}t-\beta(t))}=\bar{X}_{+}(t)e^{j(\omega_{0}t-\beta(t))} \tag{52}\]  
Tests are performed with different values of \(f_{m}\), which is swept from 0 (unmodulated) to 2 Hz with 0.1 Hz step in order to evaluate the measurement bandwidths of the different techniques. For each algorithm and condition, accuracy is expressed by the maximum values of \(\text{TVE},\ |\ \text{FE}\ |\ \text{and}\ |\ \text{RFE}\ |\ \text{over a}\ 10\text{-s}\) test duration. Let us firstly consider amplitude modulated signals, with \(k_{x}=0.1\) and \(k_{a}=0\); in this case, the algorithms are required to extract a bandpass signal from \(\bar{x}_{SV}\), whose spectral content is located between \(f_{0}\pm f_{m}\). This is evident for SV based algorithms, but it is true also for the IEC-P method as long as the positive sequence synchrophasor, frequency and ROCOF measurements are considered. The SV-TF immediately appears as perfectly suited to cope with this test, since TF filters are designed with dynamic performance in mind; in particular, the filter that allows positive sequence synchrophasor measurement exhibits a flat frequency response around \(f_{0}\) (due to multiple null derivatives according to the selected expansion order). Figure 16 reports the TVE values for all the algorithms; as expected, errors monotonically increase with the modulation frequency and, furthermore, it is clearly noticeable that SV-TF provides the most accurate synchrophasor measurement. IEC-P and SV-F achieve very close error values, but it should be highlighted that the results obtained with SV-F strongly depend on the filter design, which is obtained as a tradeoff between steady-state and dynamic performance according to the specific needs. It is not surprising that the highest error values are reached by the SV-IpDFT method, which relies on a steady-state model of the SV signal. Frequency errors are either small or negligible (<2 \(\cdot\) 10\({}^{-3}\) mHz) for all algorithms but SV-IpDFT, which shows higher \(\mid\)FE\(\mid\) up to 13.25 mHz due to its underlying steady-state assumption that is not able to effectively cope with amplitude modulated input signals. As a consequence, \(\mid\)RFE\(\mid\) achieved by SV-IpDFT is also huge (it can easily go beyond 10 Hz/s, when \(f_{m}\) gets larger than a few tenths of hertz). Conversely, the other algorithms result in negligible RFE values (order of magnitude 10\({}^{-5}\) Hz/s, or less). More varied is the performance of the methods under phase angle modulations, as shown in Figures 17-19. Again, the dynamic model which the SV-TF is based on results in the most accurate synchrophasor estimations: the TF filters preserve the bandpass characteristics of the SV signal also in this case. As for the other algorithms, TVE values are low, but they show how the synchrophasor tracking properties can be significantly different. In particular, SV-IpDFT shows the highest errors because of the steady-state signal representation, which shows its major weakness under dynamic conditions.