# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 2 Measurement Algorithms
### Space Vector Taylor-Fourier Algorithm
\right] \tag{29}\]  
\[\bar{\mathbf{p}}_{\pm}(t_{r})=\left[\bar{X}_{+}^{(0)}(t_{r}),\cdots,\bar{X}_{ +}^{(K_{+})}(t_{r}),\bar{X}_{-}^{(0)*}(t_{r}),\cdots,\bar{X}_{-}^{(K_{-})*}(t _{r})\right]^{\mathsf{T}} \tag{30}\]  
\[\mathbf{A}_{+}= \begin{bmatrix}1&\frac{N_{w}-1}{2}T_{s}&\frac{\left(\frac{N_{w} -1}{2}T_{s}\right)^{2}}{2}&\cdots&\frac{\left(\frac{N_{w}-1}{2}T_{s}\right)^{ K_{+}}}{k!}\\ \vdots&\vdots&\vdots&\vdots\\ 1&0&0&0\\ \vdots&\vdots&\vdots&\vdots\\ 1&-\frac{N_{w}-1}{2}T_{s}&\frac{\left(-\frac{N_{w}-1}{2}T_{s}\right)^{2}}{2} &\cdots&\frac{\left(-\frac{N_{w}-1}{2}T_{s}\right)^{K_{+}}}{k!}\end{bmatrix}  \tag{31}\]  
\[\bar{\mathbf{\Phi}}= \begin{bmatrix}e^{i\omega_{0}\frac{N_{w}-1}{2}T_{s}}\\ &\ddots\\ &&1\\ &&&\ddots\\ &&&e^{-j\omega_{0}\frac{N_{w}-1}{2}T_{s}}\end{bmatrix} e^{j\omega_{0}t_{r}} \tag{32}\]  
and \(\mathbf{A}_{-}\) is obtained from (31) by replacing \(K_{+}\) with \(K_{-}\); the superscripts \({}^{\mathrm{H}}\) and \({}^{\mathsf{T}}\) indicate the Hermitian transpose and the transpose operators, respectively. \(\bar{\mathbf{p}}_{\pm}(t_{r})\) is the vector of the model parameters in the reporting instant. It can be estimated by minimizing the Euclidean norm of the vector of the differences between the samples of the SV signal and those obtained from the model, hence:  
\[\bar{\mathbf{p}}_{\pm}(t_{r})=\underset{\bar{\mathbf{p}}_{\pm}(t_{r})}{\arg \min}\big{\|}\bar{\mathbf{B}}_{\pm}\cdot\bar{\mathbf{p}}_{\pm}(t_{r})-\bar{ \mathbf{x}}_{SV_{0}}(t_{r})\big{\|} \tag{33}\]  
which corresponds to an ordinary least squares (LS) problem whose solution is:  
\[\hat{\mathbf{p}}_{\pm}(t_{r})=\bar{\mathbf{H}}\cdot\bar{\mathbf{x}}_{SV_{0}} (t_{r}) \tag{34}\]  
where \(\bar{\mathbf{H}}\) is a complex filter bank, namely its \(h\)th row \(\bar{\mathbf{H}}_{h,*}\) contains the complex-valued coefficients of the FIR filter that permits obtaining the \(h\)th element of \(\hat{\mathbf{p}}_{\pm}\). In the same fashion, \(\bar{\mathbf{H}}(j\omega)\) denotes the vector-valued frequency response of the filter bank; its \(h\)thelement is the frequency response of the FIR filter defined by the coefficients \(\hat{\mathbf{H}}_{l,*}\). Therefore, the estimated positive sequence synchrophasor is:  
\[\hat{\mathbb{X}}_{+}(t_{r})=\hat{\mathbf{H}}_{1,*}\cdot\bar{\mathbf{x}}_{5V_{0} }(t_{r}) \tag{35}\]  
Estimates can also be obtained by using a weighted LS (WLS) method [28] where the weights are given, for instance, by the squared coefficients of a cosine window. Up to now, the WLS estimator has never been applied in conjunction with the SV-TF approach. As previously mentioned, the TF expansion applied to the complex-valued SV signal permits selecting different expansion orders for the positive and negative sequence synchrophasor, which is not possible when considering the conventional implementation in the real-valued phase signals: this additional degree of freedom in the design of TF filters enables better performance. In general, increasing the positive sequence synchrophasor expansion order \(K_{+}\) is beneficial, since the model is able to better represent its dynamics. On the contrary, when the focus is measuring the positive sequence synchrophasor, choosing \(K_{-}=K_{+}\) often leads to an overparametrized model. In this case, the best choice is \(K_{-}<K_{+}\), since it increases the robustness with respect to noise and disturbances (which are bettered filtered thanks to the \"stiffer\" underlying model), without significant drawbacks.