# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 2 Measurement Algorithms
### Space Vector IpDFT Algorithm  
IpDFT algorithms based on proper weighting windows [29, 30] have been widely used to measure spectral components and frequencies of electrical signals [31]. More recently, several PMU techniques exploiting this approach have been proposed in the literature [16, 32, 33]. In order to obtain a synchrophasor estimate, they require observing the phase signal \(x_{p}(t)\) over a time interval centered around the reporting instant \(t_{r}\) and corresponding to an integer number \(C\geq 2\) of rated cycles; the sampling frequency \(f_{s}\) is multiple of the rated frequency \(f_{0}\), so that \(M_{C}\) samples per nominal cycle are collected. The underlying signal model is very similar to (1), but in this case the amplitude and frequency (defined as in (4)) are assumed to be constant within the analyzed time interval. Therefore, it results:  
\[x_{p}(t) =\frac{\sqrt{2}}{2}\Big{[}\hat{\mathbb{X}}_{ph,p}(t_{r})e^{-j \omega(t_{r})t_{r}}e^{j(\omega(t_{r})-\omega_{0})t}e^{j\omega_{0}t}+\hat{ \mathbb{X}}_{ph,p}^{*}(t_{r})e^{j\omega(t_{r})t_{r}}e^{-j(\omega(t_{r})-\omega _{0})t}e^{-j\omega_{0}t}\Big{]}+d_{p}(t)\] \[=\frac{\sqrt{2}}{2}\Big{[}\hat{\mathbb{X}}_{p}(t_{r})e^{j\omega_ {0}t}+\hat{\mathbb{X}}_{p}^{*}(t_{r})e^{-j\omega_{0}t}\Big{]}+d_{p}(t) \tag{38}\]  
The first step of the IpDFT algorithm is applying a proper tapering window to the collected samples. Assuming that spectral interference is negligible, there should be at least two DFT bins produced by the signal component rotating with angular frequency \(\omega\) (namely the one related to the synchrophasor to be estimated) which falls under the main lobe of the window. Since its shape is known, the ratio between the magnitudes of these components allows estimating the frequency deviation \(\Delta f\) and, in turns, also the synchrophasor. In particular, the knowledge of \(\Delta f\) allows compensating the effect of scalloping loss which may undermine the evaluated magnitude under off-nominal frequency conditions. As mentioned above, the method properly works if the two considered DFT bins are not affected by spectral interference: this may be produced by the disturbance \(d_{p}(t)\), but also by the image component (namely the counter-rotating term appearing in (38)): specific care has to be taken, since it has the same magnitude as the term to be evaluated. This effect can be reduced by selecting a proper window (e.g., Rife-Vincent type I, characterized by maximum asymptotic decay of the sidelobes) and increasing the observation interval, even if the latter expedient results in several drawbacks. First of all, algorithm latency and computational burden are increased; secondly, the assumption of having constant signal parameters over a longer time window becomes harder to be met, thus it may seriously jeopardize the achieved dynamic performance. Having estimated the three synchrophasors of the phase quantities, the positive sequence synchrophasor \(\widehat{X}_{+}\) is evaluated through the Fortescue transformation (6). From the previous considerations, the estimates \(\hat{\mathcal{X}}_{p}\) of the phase synchrophasors may contain disturbances due to the infiltration of the image components. Similarly to what explained in Section 2.2, this effect is significantly attenuated in the positive sequence synchrophasor measurement \(\hat{\mathcal{X}}_{+}\) thanks to the weak unbalance level of real-world three-phase quantities. However, explaining how this cancellation occurs is troublesome, since the IpDFT algorithm is inherently nonlinear and closed-form expressions cannot be easily derived. Finally, it is also worth noting that applying the IpDFT algorithm to the phase signals produces three (different) frequency estimates, which have to be processed (averaged) in order to obtain a unique value.