# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 2 Measurement Algorithms
### IEC/IEEE 60255-118-1 P-Class Reference Algorithm  
The IEC Std suggests a measurement model for synchrophasor estimation and two PMU algorithms, one for class P, intended for applications requiring fast response, and another one for class M, designed for measurement applications. Both of them are based on the same concept of quadrature demodulation, which is briefly summarized in the following. Let us substitute (3) in the decomposition (1) of the phase \(p\) waveform \(x_{p}(t)\) and let us multiply by \(e^{-j\omega_{0}t}\), namely a unitary magnitude vector whose rotational speed in the complex plane corresponds to the rated angular frequency of the power system. The signal \(\bar{x}_{p,d}(t)\) is thus obtained, which results:  
\[\bar{x}_{p,d}(t)=\frac{\sqrt{2}}{2}\bar{X}_{p}(t)+\frac{\sqrt{2}}{2}\bar{X}_{p} ^{*}(t)e^{-j\omega_{0}t}+d_{p}(t)e^{-j\omega_{0}t} \tag{7}\]  
From the previous expression, \(\bar{x}_{p,d}(t)\) is a complex-valued signal consisting of three contributions. Except for a constant scale factor equal to \(\sqrt{2}/2\), the first term coincides with the quantity of interest, i.e., the synchrophasor, that--based on the assumptions introduced in Section 2.1--is expected to have a spectral content restricted at very low frequencies. The second term is generated by the counter-rotating image component in (3), and is represented by the same scale factor multiplied by the synchrophasor conjugate and an unitary vector, that is rotating clockwise in the complex plane with angular speed equal to \(2\omega_{0}\); therefore, its frequency content is confined around \(-2f_{0}\). The last term is the disturbance multiplied by an unitary vector rotating in the complex plane with angular speed equal to \(\omega_{0}\); reminding the hypothesis about \(d_{p}(t)\), the spectral content of this third contribution is virtually entirely located away from zero. Thanks to the spectral separation of these terms, the synchrophasor can be extracted from \(\bar{x}_{p,d}(t)\) through proper low-pass filtering. In this respect, it should be stressed that it is particularly important to somehow attenuate the second contribution (namely that involving the conjugate of the synchrophasor): it represents a very large unwanted component, since it has the same magnitude as the low-frequency term to be extracted. After having explained the basic concepts, we focus on the details of the P-class algorithm reported in [7] (from here on IEC-P algorithm for the sake of brevity), which will be considered as a benchmark for the comparison among the different low-latency techniques; its block diagram is reported in Figure 1.  
Let us suppose that the phase quantities have been sampled with rate \(f_{s}=M_{\text{C}}f_{0}\) (the positive integer \(M_{\text{C}}\) is the number of samples per nominal cycle) corresponding to the sampling interval \(T_{s}=1/f_{s}\); it is also assumed that \(T_{RR}=QT_{s}\), with \(Q\) positive integer. For each phase \(p\), a first estimate \(\hat{\bar{X}}_{p}^{\prime}\) of the synchrophasor in the generic reporting instant \(t_{r}=iT_{RR}\) is obtained by means of the following expression, which corresponds to multiplication by the rotating exponential and finite impulse response (FIR) filtering:  
Figure 1: Block diagram of the IEC-P algorithm. \[\hat{X}_{p}^{\prime}(t_{r})=\hat{X}_{p}^{\prime}(t_{r})e^{i\phi_{p}(t_{r})}=\frac{ \sqrt{2}}{G}\Big{(}\hat{x}_{p,d}*w_{tri}\Big{)}(t_{r})=\frac{\sqrt{2}}{G}\sum_{n =-M_{C}+1}^{M_{C}-1}x_{p}(t_{r}-nT_{s})w_{tri}[n]e^{-i\omega_{0}(t_{r}-nT_{s})} \tag{8}\]  
where \(*\) indicates the convolution sum; from here on, hat is used to denote estimated values. Filter coefficients \(w_{tri}[n]\) correspond to the \(2M_{C}-1\) sample triangular window:  
\[w_{tri}[n]=1-\frac{|n|}{M_{C}}\text{, for }n=-M_{C}+1,\ldots,M_{C}-1 \tag{9}\]  
and the constant \(G\) is its DC gain:  
\[G=\sum_{n=-M_{C}+1}^{M_{C}-1}w_{tri}[n]=M_{C} \tag{10}\]  
It is worth noting that the low-pass filter defined by (9) is linear-phase, non causal, with zero group delay.