# Low-Latency, Three-Phase PMU Algorithms: Review and Performance Comparison
## 2 Measurement Algorithms
### Common Background on Synchrophasor Measurements  
Let us consider a three-phase power system characterized by the rated frequency \(f_{0}\), corresponding to the angular frequency \(\omega_{0}\). The synchrophasor approach is based on the following quasi-steady-state model of the generic phase \(p\) signal \(x_{p}(t)\) (with \(p\in\{a,b,c\}\)), which may represent the voltage or current waveform measured in a node as a function of a shared time coordinate \(t\):  
\[ \begin{split} x_{p}(t)=x_{1,p}(t)+d_{p}(t)\\ x_{1,p}(t)=\sqrt{2}X_{p}(t)\cos(\theta_{p}(t))=\sqrt{2}X_{p}(t) \cos(\omega_{0}t+\varphi_{p}(t))\end{split}  \tag{1}\]  
Basically, the generic phase \(p\) waveform consists of a main term \(x_{1,p}(t)\), which is assumed to be a modulated sinewave whose peak amplitude \(\sqrt{2}X_{p}(t)\) and phase angle \(\varphi_{p}(t)\) are slowly-varying with respect to the rated frequency. \(d_{p}(t)\) represents a second contribution containing the other components that may be present in the signal, which are considered as disturbances (e.g., harmonics and interharmonics); its magnitude is supposed to be considerably lower than that of \(x_{1,p}(t)\), while its spectral content near \(f_{0}\) is assumed to be negligible. Starting from these considerations, the corresponding phase \(p\) (dynamic) synchrophasor \(\bar{X}_{p}(t)\) is defined as:  
\[\bar{X}_{p}(t)\triangleq X_{p}(t)e^{j\varphi_{p}(t)} \tag{2}\]  
In the following, overbar is used to highlight complex-valued quantities; for a lighter notation, a previously defined complex variable written without overbar denotes its magnitude. From an operative point of view, the synchrophasor is a possibly time-varying complex-valued number associated to an electric quantity of an AC power system that, under the previous assumptions, is able to carry its most relevant informative content. Using the [PERSON]'s formula, an expression of \(x_{1,p}(t)\) where the synchrophasor explicitly appears can be easily obtained:  
\[x_{1,p}(t)=\frac{\sqrt{2}}{2}\Big{[}\bar{X}_{p}(t)e^{i\omega_{0}t}+\bar{X}_{p}^{ *}(t)e^{-i\omega_{0}t}\Big{]} \tag{3}\]  
where \({}^{*}\) denotes the complex conjugate operator. From another point of view, \(x_{1,p}(t)\) has been decomposed into the sum of two counter-rotating vectors in the complex plane: one with positive, the other with negative angular speed, which is often known as image. It is worth noting that when assuming sinusoidal steady-state conditions at the rated frequency, the synchrophasor corresponds to the usual phasor. Considering once again the waveform model (1), the frequency \(f(t)\) (corresponding to the angular frequency \(\omega(t)\)) and its rate of change ROCOF\((t)\) are defined as:  
\[f(t)=\frac{1}{2\pi}\frac{d\theta_{p}(t)}{dt}=\frac{1}{2\pi}\frac{d\varphi_{p}( t)}{dt}+f_{0}=\Delta f(t)+f_{0} \tag{4}\]  
\[\text{ROCOF}(t)=\frac{df(t)}{dt}=\frac{1}{2\pi}\frac{d^{2}\theta_{p}(t)}{dt^{2 }}=\frac{1}{2\pi}\frac{d^{2}\varphi_{p}(t)}{dt^{2}} \tag{5}\]  
\(\Delta f(t)\) is also introduced: it represents the frequency deviation with respect to rated conditions, which is the rotational speed of the synchrophasor in the complex plane divided by \(2\pi\). The target of a PMU is sampling the three-phase waveform of an electrical quantity and, by means of a proper algorithm, extracting the corresponding synchrophasors, frequency and ROCOF at discrete and fixed time instants \(t_{r}=iT_{RR}\) (with \(i\) integer, see [7]), hence multiples of the reporting interval \(T_{RR}\). Since \(T_{RR}\) is generally an integer multiple of \(T_{0}=1/f_{0}\), \(\theta_{p}(t_{r})=\varphi_{p}(t_{r})\) when phase angle wrapping is considered. The algorithm should be able to track dynamic variations as well as rejecting disturbances. The study of three-phase systems is more efficient when it is performed by using the symmetrical components.