# IpDFT-Tuned Estimation Algorithms for PMUs: Overview and Performance Comparison
## 2 Theoretical Background
### Signal Model and Basic IpDFT Algorithm  
Since the current or voltage AC waveform parameters change with time, a general AC waveform model is:  
\[y(t)=s(t)+\eta(t)=a_{1}(t)\cos(2\pi f_{1}(t)t+\phi_{1}(t))+\eta(t) \tag{1}\]  
where \(s(t)=a_{1}(t)\cos(2\pi f_{1}t+\phi_{1}(t))\) is the fundamental sinewave (with amplitude, phase, and frequency at a given time \(t\) represented by \(a_{1}(t)\), \(\phi_{1}(t)\), and \(f_{1}(t)\), respectively) and \(\eta(t)\) includes all possible disturbances, such as harmonics, inter-harmonics, and wideband noise. The sine-wave dynamic phasor is defined as \(p(t)\stackrel{{\rm def}}{{=}}a_{1}(t)e^{\phi_{1}(t)}\). It is called a synchrophasor when it is referred to timestamped UTC instants. If waveform (1) is sampled by a PMU at a rate \(f_{s}\) and an odd number of samples \(M=2N_{h}+1\) is acquired, the reference time \(t_{r}\) for synchrophasor estimation (with \(r\) being integer) can be chosen exactly in the middle of the \(r\)-th digitized data record, which is given by:  
\[y_{r}(m)=a_{1}(rL+m)\cos\left(2\pi\frac{f_{1}(rL+m)}{f_{s}}(rL+m)+\phi_{1}(rL +m)\right)+\eta(rL+m) \tag{2}\]  
for \(m=-N_{h},\ldots,0,\ldots,N_{h}\), where \(L\) is the distance (in samples) between subsequent data records. Note that, in order to ensure a continuous data acquisition, \(L\) can range from 1 (if the observation intervals shift by 1 sample at a time) to \(M\) (when consecutive disjoint observation intervals are considered). Thus, the objective of the estimation algorithms for PMUs is to determine the values of parameters \(a_{1}(t_{r})\), \(\phi_{1}(t_{r})\) and \(f_{1}(t_{r})\) and \(\left.\frac{df_{1}(t)}{dt}\right|_{t_{r}}\) at the chosen UTC reference time within the data record (2). Such data have to be transferred to the PDC at a given Reporting Rate\(RR\). For this reason, in practice it is often reasonable to choose \(L=\frac{f_{s}}{RR}\). In the following, the waveform parameters of interest within the \(r\)-th data record will be shortly denoted as \(a_{1,r}\), \(\phi_{1,r}f_{1,r}\) and \(ROCOF_{1,r}\), respectively. The ratio between the fundamental AC frequency \(f_{1,r}\) and the sampling rate \(f_{s}\) can be expressed as:  
\[\frac{f_{1,r}}{f_{s}}=\frac{\
u_{1,r}}{M}=\frac{J_{1}+\delta_{1,r}}{M}, \tag{3}\]  
where \(\
u_{1,r}=J_{1}+\delta_{1,r}\) is the sinewave frequency expressed in bins, i.e., the number of acquired waveform cycles. Expression (3) results from the sum of an integer part \(J_{1}\) and a fractional part \(\delta_{1,r}\) (with \(-0.5\leq\delta_{1,r}<0.5\)). While \(J_{1}\) typically does not change from record to record, the value of \(\delta_{1,r}\) changes as a function of time due to frequency fluctuations, and becomes zero only if coherent sampling is performed. The sampling rate is selected so that the following condition is fulfilled:  
\[\frac{f_{n}}{f_{s}}=\frac{J_{1}}{M-1}. \tag{4}\]  
where \(f_{n}\) is the nominal sinewave frequency (i.e., 50 Hz or 60 Hz). Notice that the value of \(J_{1}\) coincides also with the number of cycles acquired at the nominal frequency. In practice, a sinewave is usually sampled non-coherently (i.e., \(\delta_{1,r}\
eq 0\)). Therefore, the discrete spectrum of the acquired signal is affected by the spectral leakage due to the finite duration of the considered observation interval. The Maximum Sidelobe Decay (MSD) windows are often employed to reduce the impact of leakage on the estimated parameters [34; 35]. The expression of a generic MSD window is:  
\[w(m)=\frac{1}{2^{2H-2}}\left[\mathrm{C}_{2H-2}^{H-1}\cos\left(\frac{2\pi}{M}m \right)+2\sum_{h=1}^{H-1}\mathrm{C}_{2H-2}^{H-h-1}\cos\left(\frac{2\pi h}{M}m \right)\right],\ m=-N_{h},\ldots,0,\ldots,N_{h} \tag{5}\]  
where \(H\geq 2\) is the number of window terms and \(\mathrm{C}_{n}^{H}=\frac{n\,!}{(n-q)\,!\,q\,!}\). In particular, the 2-term MSD (or Hann) window will be used in this paper.