# Instantaneous Reactive Power Theory in the Geometric Algebra Framework
## 5 Application Cases
### Case 1: A Five-Phase System Example  
In this section, a five-phase system with unbalanced and distorted voltages and currents is used. The voltages include the 50 Hz fundamental harmonic and a fifth-order harmonic. In order to provide details of the simulation example, the following nomenclature has been used. Thus, (88) represents a generic expression for the voltage waveform of one of the phases \(j=1,\ldots,5\):  
\[v_{sj}(t)=\sqrt{2}\sum_{h=1,5}\sum_{j=1}^{5}V_{hj}\cos\left(h\left(\omega t+ \theta_{j}\right)\right) \tag{88}\]  
To express unbalance and distortion, \(VU_{j}\) and \(VHF_{h}\) factors, respectively, are introduced as indicated by (89):  
\[V_{hj}=VU_{j}\times VHF_{h} \tag{89}\]  
Tables 1 and 2 present the values of each of these factors. \begin{table}
\begin{tabular}{c c c} \hline \hline
**j** & \(\mathbf{VU_{j}}\) & \(\mathbf{CU_{j}}\) \\ \hline
1 & 1 & 1 \\
2 & 1.05 & 1.1 \\
3 & 0.95 & 0.9 \\
4 & 1.1 & 1.05 \\
5 & 0.9 & 0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Factors used to express voltage and current unbalance. \begin{table}
\begin{tabular}{c c c} \hline \hline
**h** & \(\mathbf{VHF_{h}}\) & \(\mathbf{CHF_{h}}\) \\ \hline
1 & 1 & 1 \\
5 & 0.1 & 0.5 \\
7 & - & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Factors used to express harmonics of voltage and current. Similarly, for the supply current, the 50 Hz fundamental harmonic, and fifth- and seventh-order harmonics, are considered. The subscript \(h\) is reserved to indicate the order of the harmonic. A generic waveform for one of the phases is:  
\[i_{\text{sj}}(t)=\sqrt{2}\sum_{h=1,5,7}\sum_{j=1}^{5}I_{hj}\cos\left(h\left( \omega t+\theta_{j}\right)+\varphi_{j}\right) \tag{90}\]  
with:  
\[I_{hj}=CU_{j}\times CHF_{h} \tag{91}\]  
where the \(CU_{j}\) and \(CHF_{h}\) factors are introduced to account for unbalanced and distorted currents. Likewise, Tables 1 and 2 record the values that are adopted for these factors in this application example. The phase angles for the voltage and current waveforms are listed in Table 3.  
Figure 4 shows the five-phase voltage system (unbalanced and distorted) in which the voltage of phase 1 is highlighted. The two types of compensation indicated in Section 4 are considered. The time instant t = 0.02 s is established as the one where the compensation is applied. Figure 5a shows the current waveforms before and after TIC. In the same way, Figure 5b presents the current waveforms before and after TAC. Figure 5a shows how the source current after TIC is the instantaneous power current, and therefore does not transfer instantaneous reactive power. In the same way, Figure 5b shows how, after TAC, the source current does not transfer instantaneous reactive power nor does it transfer the oscillatory part of the instantaneous power. The current transfers only the average power and reproduces the unbalance and distortion of the voltage. The balance of the power variables before and after TIC is presented in Figure 6. Figure 6a shows how the instantaneous power is the same before/after TIC. In contrast, Figure 6b shows how the instantaneous reactive power after TIC cancels out. Thus, the instantaneous power multivector norm coincides with the instantaneous power after TIC; Figure 6c. Figure 4: Voltages of a distorted and unbalanced five-phase system; phase 1 is highlighted. \begin{table}
\begin{tabular}{c c c} \hline \hline
**j** & \(\theta_{\text{j}}\) & \(\varphi_{\text{j}}\) \\ \hline
1 & 0 & \(-\pi/6\) \\
2 & \(-2\pi/5\) & \(-2\pi/6\) \\
3 & \(-4\pi/5\) & \(-1.5\pi/6\) \\
4 & \(-6\pi/5\) & \(-1.8\pi/6\) \\
5 & \(-8\pi/5\) & \(-1.2\pi/6\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Phase angles of line voltages and currents. The balance of the power variables before and after TAC is presented in Figure 7. Figure 7a shows how the instantaneous power after TAC is constant: average power (active power). Figure 7b presents how the instantaneous reactive power is null after TAC, the  
Figure 5: (**a**) Supply currents before/after instantaneous compensation (TIC).