# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 2 Measurement Algorithms
### Space Vector Algorithm  
PMU algorithms based on the SV transformation of the phase quantities and filtering of the resulting signal when decomposed in rectangular and polar coordinates have been recently proposed by the authors in [23; 24]. The starting point is the usual measurement model (1), here rewritten by using the vector notation in order to consider the three phases simultaneously:  
\[\mathbf{x}_{abc}(t)=\mathbf{x}_{1,abc}(t)+\mathbf{d}_{abc}(t) \tag{19}\]where:  
\[\mathbf{x}_{1,abc}(t)=\left[ \begin{array}{c}x_{1,a}(t)\\ x_{1,b}(t)\\ x_{1,c}(t)\end{array} \right]\qquad\qquad\qquad\mathbf{d}_{abc}(t)=\left[ \begin{array}{c}d_{a}(t)\\ d_{b}(t)\\ d_{c}(t)\end{array} \right] \tag{20}\]  
From the three-phase waveforms, it is possible to compute the SV \(\bar{x}_{SV}(t)\) in a rotating reference frame whose instantaneous angular position is \(\beta(t)\); it results:  
\[\bar{x}_{SV}(t)=\sqrt{\frac{2}{3}}\left[ \begin{array}{ccc}1&\bar{\alpha}& \bar{\alpha}^{2}\end{array} \right]\mathbf{x}_{abc}(t)\,e^{-j\beta(t)}=\bar{x}_ {SV_{0}}(t)\,e^{-j\beta(t)} \tag{21}\]  
where \(\bar{x}_{SV_{0}}\) is the SV in a stationary reference frame characterized by \(\beta=0\). Substituting (19) into (21) while using (3) and reminding (6) leads to:  
\[ \begin{split}&\bar{x}_{SV}(t)=\bar{X}_{+}(t)e^{j(\omega_{0}t- \beta(t))}+\bar{X}_{-}^{*}(t)e^{-j(\omega_{0}t+\beta(t))}+\bar{d}_{SV}(t)\\ &\bar{d}_{SV}(t)=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}1&\bar{\alpha }&\bar{\alpha}^{2}\end{array}\right]\mathbf{d}_{abc}(t)\,e^{-j\beta(t)}=\bar{ d}_{SV_{0}}(t)\,e^{-j\beta(t)}\end{split}  \tag{22}\]  
Three terms appear: the first one is related to the positive sequence synchrophasor, the second depends on the negative sequence synchrophasor, the last is produced by disturbances. Of course, the term due to the positive sequence synchrophasor is expected to be considerably higher with respect to the others. Furthermore, it is worth noting that the zero sequence synchrophasor does not affect the SV signal. Now, let us choose \(\beta(t)=\omega_{0}t\), namely performing the SV transformation in a reference frame whose angular speed corresponds to the rated angular frequency. In this case, the positive sequence synchrophasor produces a very low frequency contribution; instead, the term related to the negative sequence synchrophasor rotates with an angular speed close to \(2\omega_{0}\) in clockwise direction. Reminding that the spectral content of \(d_{p}(t)\) is assumed to be located away from \(f_{0}\), the spectrum of \(\bar{d}_{SV}(t)\) is well separated from that of the positive sequence synchrophasor. This suggests that an estimate of \(\bar{X}_{+}(t)\) can be extracted from \(\bar{x}_{SV}(t)\) by means of proper low-pass filtering, without explicitly computing per phase synchrophasors. Adopting the same three-phase definition of frequency deviation already employed by the IEC-P algorithm, \(\Delta f\) and ROCOF represent (except for a scale factor \(2\pi\)) the first and second order derivatives of the positive sequence synchrophasor phase angle. Therefore, their estimates can be obtained by filtering \(\bar{x}_{SV}\), computing its phase angle and performing numerical differentiations. The architecture of this approach corresponds to the block diagram reported in Figure 2; its practical implementation requires first of all to properly sample the phase signals with rate \(f_{s}\). From the acquired data, applying (21) with \(\beta(t)=\omega_{0}nT_{s}\) the samples of the SV signal \(\bar{x}_{SV}\) are obtained. A first low-pass filtering stage \(H\) with unit DC gain is applied to the real and imaginary parts of \(\bar{x}_{SV}\) in order to remove most of \(\bar{d}_{SV}\) as well as the term produced by the negative sequence. Both FIR and infinite impulse response (IIR) filters can be employed, even if best performance and greater flexibility is obtained with FIR design.