# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 2 Measurement Algorithms
### IEC/IEEE 60255-118-1 P-Class Reference Algorithm
Let us assume sinusoidal steady-state operation with frequency equal to \(f_{0}\): \(\hat{X}_{p}(t)\) in (7) is constant, while the term involving \(\hat{X}_{p}^{*}(t)\) rotates with angular speed \(-2\omega_{0}\). It can be easily proven that the employed filter has frequency response zeros located at multiples of \(f_{0}\). Hence, under these conditions it is able to completely cancel out the term rotating at \(-2\omega_{0}\). Since the DC gain of the triangular window FIR filter defined as in (9) is \(G=M_{C}\), the estimate provided by (8) is theoretically exact. In this algorithm, a three-phase definition of frequency is employed in order to extract a unique value from the three phases. In particular, the frequency deviation \(\Delta f\) corresponds to the rotational speed (in hertz) of the positive sequence synchrophasor, whose estimate \(\hat{X}_{+}^{\prime}\) is obtained by applying the unitary Fortescue transformation (6):  
\[\hat{X}_{+}^{\prime}(t_{r})=\hat{X}_{+}^{\prime}(t_{r})e^{i\hat{\varphi}_{+}( t_{r})}=\frac{\hat{X}_{a}^{\prime}(t_{r})+\hat{x}\hat{X}_{b}^{\prime}(t_{r})+ \hat{\alpha}^{2}\hat{X}_{c}^{\prime}(t_{r})}{\sqrt{3}} \tag{11}\]  
From a physical point of view, this definition of frequency corresponds to the electrical speed of the air gap magnetic field in an ideal, symmetric three-phase machine whose winding magnetomotive force is purely positive sequence and expressed by the synchrophasor \(\hat{X}_{+}\). Frequency deviation and ROCOF measurements are obtained as first and second order discrete-time derivatives of the estimated phase angle \(\hat{\varphi}_{+}\). For this purpose, let us firstly compute \(\hat{\varphi}_{+}\) in the time instants \(t_{r}-ST_{s}\) and \(t_{r}+ST_{s}\), where \(S\leq Q\) is a positive integer; after that, frequency and ROCOF estimates result:  
\[\hat{f}(t_{r})=\hat{\Delta f}(t_{r})+f_{0}=\frac{\hat{\varphi}_{+}(t_{r}+ST_{s })-\hat{\varphi}_{+}(t_{r}-ST_{s})}{4\pi ST_{s}}+f_{0} \tag{12}\]  
\[\widehat{\text{ROCOF}}(t_{r})=\frac{\hat{\varphi}_{+}(t_{r}+ST_{s})+\hat{ \varphi}_{+}(t_{r}-ST_{s})-2\hat{\varphi}_{+}(t_{r})}{2\pi(ST_{s})^{2}} \tag{13}\]  
Therefore, (12) and (13) correspond to the application of two linear-phase FIR filters \(D_{S}^{1}\) and \(D_{S}^{2}\) to \(\hat{\varphi}_{+}/(2\pi)\), respectively. Now, let us suppose that the input signals \(x_{p}(t)\) are pure sinewaves whose frequency \(f\) is different from \(f_{0}\). In this case, the phase \(p\) synchrophasor is:  
\[\hat{X}_{p}(t)=X_{p}e^{i((\omega-\omega_{0})t+\varphi_{ph,p})}=\hat{X}_{ph,p} e^{i(\omega-\omega_{0})t} \tag{14}\]  
where \(\hat{X}_{ph,p}\) represents the usual phasor at the frequency \(f\). Hence, the synchrophasor is a complex number having the same magnitude as the phasor, but its phase angle is time-varying, since it rotates in the complex plane with angular speed equal to the difference between \(\omega\) and \(\omega_{0}\). In particular, the real and imaginary parts of \(\hat{X}_{p}(t)\) are quadrature sinewaves whose frequencies are equal to the frequency deviation \(\Delta f\). From (7), the signal to be filtered is made of two contributions:  
Figure 1.0  
Introducing \(W_{tri}(j\omega)\) as the frequency response of the previously defined triangular window filter (real-valued and identical to the amplitude response), the evaluated synchrophasor results:  
Figure 1.1  
reminding that \(2\pi\Delta f=\omega-\omega_{0}\). The first thing to be noticed is that the term proportional to \(\hat{X}_{ph,p}^{*}\) is attenuated but not completely rejected by the filter, since its rotational speed does not correspond to a zero of the filter. Furthermore, even neglecting this effect, the magnitude of the synchrophasor estimate is biased: in fact \(W_{tri}(j2\pi\Delta f)\
eq G\) when \(\Delta f\
eq 0\). This effect is significant even for small values of the frequency deviation. However, having measured \(\widehat{\Delta f}\), it can be compensated since the filter response is known; a more accurate synchrophasor estimate \(\hat{X}_{p}\) can thus be obtained. In this respect, ref.