# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 2 Measurement Algorithms
### Space Vector IpDFT Algorithm
Similarly to the TF approach, the IpDFT algorithm can be favorably applied to the SV signal \(\hat{x}_{SV_{0}}\) in a stationary reference frame, as in the block diagram reported in Figure 4; this enables a direct estimation of the positive sequence synchrophasor. As in Section 2.4, \(\hat{x}_{SV_{0}}\) is computed from the phase waveforms by means of the SV transformation (21) with \(\beta=0\). Its model, which also in this case is assumed to be valid over the \(C\) nominal cycle interval centered around the reporting instant, results by applying to (38) the SV transformation on the same reference frame, hence:  
\[\hat{x}_{SV_{0}}(t) =\bar{X}_{ph,+}(t_{r})e^{i\omega(t_{r})t_{r}}e^{i(\omega(t_{r})- \omega_{0})t}e^{i\omega_{0}t}+\bar{X}_{ph,-}^{*}(t_{r})e^{-i\omega(t_{r})t_{r }}e^{-i(\omega(t_{r})-\omega_{0})t}e^{-i\omega_{0}t}+\bar{d}_{SV_{0}}(t)\] \[=\bar{X}_{+}(t_{r})e^{i\omega_{0}t}+\bar{X}_{-}^{*}(t_{r})e^{-i \omega_{0}t}+\bar{d}_{SV_{0}}(t) \tag{39}\]  
Figure 4: Block diagram of the SV-IpDFT algorithm. The next step is considering the \(N_{w}=CM_{C}\) samples of the SV signal collected around \(t_{r}\) and, having applied a smoothing window defined by the sequence \(w[n]\) (\(n=0,\ldots,N_{w}-1\)), computing the DFT bins \(\bar{X}_{SV_{0}}(k,t_{r})\) whose corresponding frequencies are close to \(f_{0}\), namely whose indexes \(k\) are near to \(C\). Let us assume that, as typically happens in practical implementations, \(M_{C}\) and thus also \(N_{w}\) are even numbers. \[\bar{X}_{SV_{0}}(k,t_{r})=\sum_{n=0}^{N_{w}-1}w[n]\bar{x}_{SV_{0}}(t_{r}+(n-N_ {w}/2)T_{s})e^{-j2\pi\frac{j\pi}{N_{w}}} \tag{40}\]  
The key underlying assumption of IpDFT algorithms is that long-range leakage is negligible for the considered indexes \(k\). When comparing (39) with (38), it is evident that this assumption is more easily met as long as the IpDFT algorithm is applied to the SV signal instead of a phase waveform. In fact, thanks to low unbalance level of power systems, in the first case the disturbance produced by the counter-rotating component is considerably lower in relative value. Therefore, neglecting spectral interference, it results (\(\omega=\omega(t_{r})\) in the following):  
\[\bar{X}_{SV_{0}}(k,t_{r}) \approx\bar{X}_{ph_{r}+}(t_{r})\bar{W}(k-\lambda)e^{-j\omega N_{w }T_{s}/2}\] \[=\bar{X}_{ph_{r}+}(t_{r})\bar{W}(k-\lambda)e^{-j\pi\lambda} \tag{41}\]  
where \(\bar{W}(\gamma)\) is the discrete-time Fourier transform of \(w[n]\) computed in the generalized bin \(\gamma\), while \(\lambda=fN_{w}T_{s}\). Now, let us suppose to have employed a periodic window having \(w[0]=0\): it is worth noting that this property holds true for many widely employed windows, even including those belonging to the Rife-Vincent class I family. In this case, it is possible to write:  
\[\bar{X}_{SV_{0}}(k,t_{r})\approx(-1)^{k}\bar{X}_{ph_{r}+}(t_{r})W(k-\lambda) \tag{42}\]  
where \(W(\gamma)\) is the amplitude response of the window. Let us assume that \(k_{0}\) is the closest integer to \(\lambda\), namely the index of the highest DFT component. Writing \(\lambda=k_{0}+\delta(t_{r})\) (\(|\delta(t_{r})|<1/2\)), \(\delta\) can be estimated from the ratio between the magnitudes of the two largest bins with the following equations:  
\[\xi(t_{r}) =\frac{|\bar{X}_{SV_{0}}(k_{0}+l(t_{r}),t_{r})|}{|\bar{X}_{SV_{0} }(k_{0},t_{r})|}\approx\frac{|W(l(t_{r})-\delta(t_{r}))|}{|W(-\delta(t_{r}))| }=g(\delta(t_{r})) \tag{43}\] \[=\frac{\sqrt{2}}{2}\Big{[}\hat{\mathbb{X}}_{p}(t_{r})e^{j\omega_ {0}t}+\hat{\mathbb{X}}_{p}^{*}(t_{r})e^{-j\omega_{0}t}\Big{]}+d_{p}(t) \tag{38}\]0  
where \(l(t_{r})=\text{sign}(|\bar{X}_{SV_{0}}(k_{0}+1,t_{r})|-|\bar{X}_{SV_{0}}(k_{0} -1,t_{r})|)\).