## 4 Distortion Power Bivector
### Example 2  
Let a periodic \(n\)-sinusoidal voltage (Figure 7) with instantaneous value given by:  
\[u_{2}(t)=\sqrt{2}[200 sin(\omega t+53.1^{\circ})+100 sin(2\omega t+69.4^{\circ})] \tag{56}\]  
be applied to a hypothetical commercial building. The resulting current (Figure 7) has an instantaneous value:  
\[i_{2}(t)=\sqrt{2}[40 sin(\omega t)+11.7 sin(2\omega t)+10 sin(3\omega t+30^{\circ})] \tag{57}\]  
In this case, \(\alpha_{1}=53.1^{\circ}\), \(\alpha_{2}=69.4^{\circ}\), and the corresponding geometric phasors are respectively:  
\[\widetilde{\mathrm{U}}_{2}=200\mathrm{e}^{\mathrm{i}53.1}\sigma_{1}+100 \mathrm{e}^{\mathrm{i}69.4}\sigma_{2},\|\widetilde{\mathrm{U}}\|=223.60 \tag{58}\]  
\[u_{2}\widetilde{I}_{2}=40\mathrm{e}^{\mathrm{i}0}\sigma_{1}+11.71\mathrm{e}^{ \mathrm{i}0}\sigma_{2}+10\mathrm{e}^{\mathrm{i}30}\sigma_{3},\ \|\widetilde{I}\|=42.86 \tag{59}\]  
From Equation (18), the apparent power multivector may be written as:  
\[\widetilde{S}_{2}=(5215.4+j7494)\sigma_{0}+(-1797-j523.2)\sigma_{12}+(1840+j78 4.6)\sigma_{13}+(772.7+j634.7)\sigma_{23} \tag{60}\]  
where \(P_{2}=5215.4\), \(Q_{2}=7494\), \(\|\widetilde{\Delta}_{2}\|=2915.82\), \(\|\widetilde{S}_{2}\|=9584.3\) and power factor is \(\mathrm{PF}_{2}\)= 0.54. Table 2 and Figure 8, illustrate the simulation and geometric representation results.  
These two examples cannot be distinguished respect of the distortion power magnitudes of well-established distortion power approaches. One can show that in each example all the corresponding rotated distortion and distortion power magnitudes are the same:  
\[\|\widetilde{\Delta}_{12}\|=\|\widetilde{D}_{12}\|=1871, \tag{61a}\] \[\|\widetilde{\Delta}_{13}\|=\|\widetilde{D}_{13}\|=2000, \tag{61b}\]  
\begin{table}
\begin{tabular}{c c c} \hline _p = 1, q = 2_ & _p = 1, q = 3_ & _p = 2, q = 3_ \\ \hline \(\widetilde{\Delta}_{12}=-1797-j523.2\) & \(\widetilde{\Delta}_{13}=1840+j784.6\) & \(\widetilde{\Delta}_{23}=772.7+j634.7\) \\ \(\widetilde{D}_{12}=-1578-j1006\) & \(\widetilde{D}_{13}=1732-j1000\) & \(\widetilde{D}_{23}=866-j500\) \\ \(\|\widetilde{\Delta}_{12}\|=1871\) & \(\|\widetilde{\Delta}_{13}\|=2000\) & \(\|\widetilde{\Delta}_{23}\|=1000\) \\ \(\|\widetilde{D}_{12}\|=1871\) & \(\|\widetilde{D}_{13}\|=2000\) & \(\|\widetilde{D}_{23}\|=1000\) \\ \hline \end{tabular}
\end{table}
Table 2: Example 2. Simulation results.  
Figure 7: Distorted voltage and load current.  
\[\|\widetilde{\Delta}_{23}\|=\|\widetilde{D}_{23}\|=1000,\] (61c) and: \[\|\widetilde{\Delta}_{1}\|=\|\widetilde{D}_{1}\|=\|\widetilde{\Delta}_{2}\|=\| \widetilde{D}_{2}\|=2915.82. \tag{62}\]  
The corresponding rotated distortion power bivectors for the proposed examples can be calculated as:  
\[\widetilde{\Delta}_{1}=(-1578-j1006)\sigma_{12}+(1732-j1000)\sigma_{13}+~{}(866 -j500)\sigma_{23} \tag{63}\]  
and:  
\[\widetilde{\Delta}_{2}=(-1797-j523.2)\sigma_{12}+(1840+784.6)\sigma_{13}+(772.7+j634.7)\sigma_{23} \tag{64}\]  
Accordingly, it is coherent to assume that any differentiation between both buildings must be based on the information of distortion power bivector. Only this concept can distinguish one from the other, to energy building control because it considers, voltage phase angles, magnitude, direction, and sense as can be seen in geometric representation Figures 6 and 8.