## 4 Distortion Power Bivector
### Geometric Interpretation of Distortion Power Bivectors  
In the \((\mathcal{CG}_{n},\odot)\) algebraic structure, the bivectorial part \(\widetilde{\Omega}^{\wedge}\) of Equations (18) and (21), can be represented as a linear combination of basis bivectors, each encoding a distinct independent \(pq\)-plane involving cross-frequency products (\(p\
eq q\)) in \(n\)-dimensional space.  
Thus, Figure 2 shows the geometric interpretation into generic \(pq\)-planes of \(\widetilde{\Delta}_{pq}\)_rotated distortion power_ and the corresponding \(\widetilde{D}_{pq}\)_distortion power_ term when \(p,q\in N,q\
eq p\) and/or \(p\in L,\ q\in N\). Therefore, it is seen that Figure 3 shows, into generic \(pq\)-planes, the geometric interpretation of \(\widetilde{\Delta}_{pq}\)_rotated distortion power_ and the corresponding \(\widetilde{D}_{pq}\)_distortion power_ term when \(p\in L\cup N,q\in M\). It can be seen from Figures 2 and 3, that the geometric interpretation of Equation (21) is based on the association of complex planes \((\mathcal{C})\) to each multivectorial element of Clifford basis \(\mathcal{G}_{n}\).  
Note in Figure 2 that:  
\[Re\Big{\{}\widetilde{\Delta}_{pq}\Big{\}}=Re\Big{\{}e^{i(\alpha_{p}-\alpha_{q})} \widetilde{D}_{pq}\Big{\}}\
eq\widetilde{\Delta}_{pq,Act} \tag{44}\]  
\[Im\Big{\{}\widetilde{\Delta}_{pq}\Big{\}}=Im\Big{\{}e^{i(\alpha_{p}-\alpha_{q}) }\widetilde{D}_{pq}\Big{\}}\
eq\widetilde{\Delta}_{pq,Re\,act} \tag{45}\]  
Similarly, in Figure 3:  
\[Re\Big{\{}\widetilde{\Delta}_{pq}\Big{\}}=Re\Big{\{}e^{i\alpha_{p}}\widetilde {D}_{pq}\Big{\}}\
eq\widetilde{\Delta}_{pq,Act} \tag{46}\]  
\[Im\Big{\{}\widetilde{\Delta}_{pq}\Big{\}}=Im\Big{\{}e^{i\alpha_{p}}\widetilde {D}_{pq}\Big{\}}\
eq\widetilde{\Delta}_{pq,Re\,act} \tag{47}\]  
The corresponding geometric phasors are respectively:  
\[\widetilde{U}_{1}=200e^{i0}\sigma_{1}+100e^{i0}\sigma_{2},\ \left|\widetilde{U} \right|=223.6 \tag{53}\]  
\[\widetilde{I}_{1}=40e^{-\beta 53.1}\sigma_{1}+11.71e^{-j69.4}\sigma_{2}+10e^{j30} \sigma_{3}\,\ \left|\widetilde{I}\right|=42.86 \tag{54}\]  
From Equation (18), the apparent power multivector may be written as:  
\[\widetilde{S}_{1}=(5215.4+j7494)\sigma_{0}+(-1578-j1006)\sigma_{12}++(1732-j100 0)\sigma_{13}+(866-j500)\sigma_{23} \tag{55}\]  
where \(P_{1}=5215.4\), \(Q_{1}=7494\), \(\left\|\widetilde{A}_{1}\right\|=2915.82\), \(\left\|\widetilde{S}_{1}\right\|=9584.3\) and power factor is PF\({}_{1}\)= 0.54  
Table 1 and Figure 6, illustrate the simulation and geometric representation results.  
\begin{table}
\begin{tabular}{c c c} \hline \(\boldsymbol{p=1,q=2}\) & \(\boldsymbol{p=1,q=3}\) & \(\boldsymbol{p=2,q=3}\) \\ \hline \(\widetilde{\Delta}_{12}=-1578-j1006\) & \(\widetilde{\Delta}_{13}=1732-j1000\) & \(\widetilde{\Delta}_{23}=866-j500\) \\ \(\widetilde{D}_{12}=-1578-j1006\) & \(\widetilde{D}_{13}=1732-j1000\) & \(\widetilde{D}_{23}=866-j500\) \\ \(\left\|\widetilde{\Delta}_{12}\right\|=1871\) & \(\left\|\widetilde{\Delta}_{13}\right\|=2000\) & \(\left\|\widetilde{\Delta}_{23}\right\|=1000\) \\ \(\left|\widetilde{D}_{12}\right|=1871\) & \(\left|\widetilde{D}_{13}\right|=2000\) & \(\left|\widetilde{D}_{23}\right|=1000\) \\ \hline \end{tabular}
\end{table}
Table 1: Example 1. Simulation results.  
Figure 5: Distorted voltage and load current.