# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 2 Measurement Algorithms
### Space Vector Algorithm
The samples of \(\bar{x}_{SV,f}\) are thus obtained and they have to be further processed in order to compensate for the group and phase delay introduced by \(H\); it is extremely simple when a linear-phase FIR filter is adopted. After that, \(\bar{x}_{SV,f}\) is decomposed into its magnitude \(x_{SV,f}\) and argument \(\varphi_{SV,f}\). The estimate \(\hat{\varphi}_{+}(t_{r})\) of the positive sequence synchrophasor phase angle in the generic reporting instant \(t_{r}\) is obtained by applying to \(\varphi_{SV,f}\) a linear-phase FIR low-pass filter \(P\) (characterized by unit DC gain, designed to attenuate the residual impact of \(\bar{d}_{SV}\) and of the oscillating term produced by \(\bar{X}_{-}\)) and compensating its delay. Its bandwidth should be sufficient to properly follow the phase angle dynamics, which are typically rather slow. Similarly, frequency deviation and ROCOF measurements in the reporting instants (\(\widehat{A\eta}(t_{r})\) and \(\widehat{\text{ROCOF}}(t_{r})\), \(t_{r}=QT_{s}\)) are computed by filtering \(\sigma_{SV,f}\) with linear-phase, partial band first and second-order FIR differentiators (\(F\) and \(R\) respectively) and compensating the introduced delays. Their frequency responses should match those of ideal first and second order differentiators only near zero (just slow frequency variations have to be accurately tracked), while they should have considerably lower magnitudes at higher frequencies in order to achieve good disturbance rejection and noise immunity. An estimate \(\hat{X}^{\prime}_{+}\) of the positive sequence synchrophasor magnitude can be obtained by applying a low-pass filter \(M\) to \(x_{SV,f}\) while taking into account the introduced delay. Similarly to \(P\), \(M\) should have unitary DC gain and should be designed to reject the infiltration of \(\hat{d}_{SV}\) as well as the term proportional to \(\widehat{X}_{-}\). Its bandwidth should be large enough to properly follow amplitude modulations, which are fairly slow in practical power systems. However, it should be noticed that \(\hat{X}^{\prime}_{+}\) may be significantly biased under off-nominal frequency conditions, similarly to what happens with the IEC-P algorithm discussed in the previous subsection. Assuming perfectly sinusoidal, positive sequence input with angular frequency \(\omega\
eq\omega_{0}\), \(\bar{x}_{SV,f}\) results:  
\[\bar{x}_{SV,f}(nT_{s})=H(j2\pi\Delta f)\bar{X}_{+}(nT_{s})=H(j2\pi\Delta f) \bar{X}_{ph,+}e^{j(\omega-\omega_{0})nT_{s}} \tag{23}\]  
where \(\bar{X}_{ph,+}\) is the positive sequence phasor while filter \(H\) is assumed to be a linear-phase FIR filter having amplitude response \(H(j\omega)\) For a linear-phase filter \(H\), the relationship between its frequency response \(\hat{H}(j\omega)\) and amplitude response \(H(j\omega)\) is: \(\hat{H}(j\omega)=H(j\omega)e^{-j\omega nT_{s}}\), where \(\tau_{H}\) is the group delay. According to the procedure described in the previous lines, it is easy to derive the magnitude estimate \(\hat{X}_{+}=H(j2\pi\Delta f)X_{+}\), reminding that the DC gain of \(M\) is unitary by assumption. Similarly to what explained in Section 2.2, the effect of filter \(H\) can be removed since its response is known and a measurement of the frequency deviation is available. Hence a better estimate \(\hat{X}_{+}\) of the positive sequence synchrophasor magnitude is:  
\[\hat{X}_{+}(t_{r})=\frac{\hat{X}^{\prime}_{+}(t_{r})}{H(j2\pi\widehat{\Delta f }(t_{r}))} \tag{24}\]  
One of the main advantages of this SV-based approach is its flexibility: achieved performance strongly depends on the filters \(H\), \(M\), \(P\), \(F\) and \(R\), which can be tuned in order to reach predetermined goals in terms of accuracy, latency and responsiveness. It is worth highlighting that closed-form expressions provided in [24] allow predicting the results of  
Figure 2: Block diagram of the SV-F algorithm. the P- and M-class compliance tests prescribed by [7], thus substantially helping the design of the aforementioned filters.