# Instantaneous Reactive Power Theory in the Geometric Algebra Framework
## 5 Application Cases
### Case 2: An Industrial Three-Phase System  
In the following, a real engineering system is presented. This shows more clearly the effectiveness of the proposed method. An unbalanced and distorted three-phase system with neutral conductor is considered. An asymmetrical nonlinear load is formed by three bridge rectifiers connected in a star with an accessible neutral. For this purpose, each RL branch on the dc side of each rectifier has different parameter values according to Table 4. Figure 8 shows a schematic of the implemented power system. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **RMS Voltage Values (V)** & & **Rectifiers** \\ \hline
**Phase j** & **Harmonic h = 1** & **Harmonic h = 5** & **Rj (f)** & **Lj (H)** \\ \hline
1 & 230 & 230 \(\times\) 1 \(\times\) 0.05 & 20 & 0.01 \\
2 & 230 \(\times\) 1.05 & 230 \(\times\) 1.05 \(\times\) 0.05 & 25 & 0.01 \\
3 & 230 \(\times\) 0.95 & 230 \(\times\) 0.95 \(\times\) 0.05 & 15 & 0.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: RMS values of the supply voltage and RL branch parameters on the dc side of the rectifiers. Figure 7: Power variables before/after TAC (t = 0.02 \(s\)): (**a**) instantaneous power, (**b**) instantaneous reactive power, (**c**) norm of instantaneous power multivector. A three-phase unbalanced voltage set with distortion is applied to the load; Table 4. Figure 9 shows the supply voltage waveforms. The two types of compensation established in Section 4 are applied. As previously established, in TIC the compensation occurs instantaneously from the very moment the APLC operates. In TAC there will be a delay since it is necessary to obtain the averaged value of the load active power. It is common practice for the APLC control circuit to include a low-pass filter (LPF) to perform this task, [1]. Here, a second-order Butterworth filter is used; this implies a delay in the response to TAC which, in this case, is of the order of one and a half cycles. Results of the waveforms of interest are presented below. Figure 10a presents the waveforms corresponding to the supply current before/after TIC; the compensator is switched on at instant t = 0.06 s. From that moment on, the currents transport only the instantaneous power; they do not transfer instantaneous reactive power between them, reducing, therefore, the value of their instantaneous norm. Figure 10b shows the supply current waveforms before/after TAC. The compensated currents can be seen after the LPF settling time has elapsed. In this case, the currents transport only the average power absorbed by the load. Figure 11 shows the power balance before/after compensation from the norm of the instantaneous power multivector, s\({}_{\text{s}}\). Figure 11a presents s\({}_{\text{s}}\) before/after TIC; from 0.06 s, s\({}_{\text{s}}\) is identified with instantaneous power (does not include instantaneous reactive power component). Figure 11b shows s\({}_{\text{s}}\) before/after TAC; from the APLC connection and after the LPF settling time, s\({}_{\text{s}}\) is identified with the average power (does not include instantaneous reactive power and does not include the oscillatory part of the instantaneous power). Finally, Figure 12 shows the waveforms of the neutral current. Figure 12a presents the neutral current before/after TIC, and Figure 12b shows the neutral current before/after TAC. In both cases there is a significant mitigation of the current after compensation; the small value observed in the figure corresponds to the harmonics still present in the sum of the supply currents after compensation. Figure 8: Scheme of an unbalanced non-sinusoidal compensated three-phase system. Figure 9: Voltages of a distorted and unbalanced 3-phase system; phase 1 is highlighted. Figure 11: (**a**) Norm of instantaneous power multivector, s\({}_{\text{s}}\), before/after instantaneous compensation (TIC); s\({}_{\text{s}}\) only includes the instantaneous power after compensation.