## 4 Distortion Power Bivector  
### Power Multivector  
Consider an arbitrary nonlinear single-phase system supplied by the voltage _u(t)_. The resulting current has an instantaneous value given by _i(t)_. The instantaneous voltage and current signals have been defined in the section above. In order to represent the concept of power multivector in GCGA, we introduce a new rule for multiplying geometric-phasors (complex-vectors). This rule is the new geometric product (10), where \(\Re\)[32] is an application over complex planes and \"\(\otimes\)\" is the standard geometric product. Apparent power can thus be expressed as a multivector \(\widetilde{S}\) in \(\{\mathcal{CG}_{n}\}\), generated by \"\(\odot\)\" of the voltage and conjugate current geometric phasors:  
\[\widetilde{S}=\left\{\frac{\widetilde{U}\cdot\widetilde{I}^{*}}{\widetilde{ \Omega}^{\bullet}}+\ \frac{\widetilde{U}\wedge\widetilde{I}^{*}}{\widetilde{\Omega}^{\wedge}}\ \right\} \tag{17}\]  
and which obeys the usual conservation law [33]. From Equation (17), one easily defines that power multivector is given by:  
\[ \begin{split}\widetilde{S}=\sum\limits_{\begin{subarray}{c}p\in N \cup L\\ q\in N\cup M\end{subarray}}\widetilde{U}_{p}\odot\ \widetilde{I}^{*}_{q}\left(\sum\limits_{p=q}U_{p}I_{p}\cos\phi_{p}+j\sum \limits_{p=q}U_{p}I_{p}\sin\phi_{p}\right)\sigma_{0}+\sum\limits_{ \begin{subarray}{c}p<q\\ p\
eq\mathcal{C}_{N}\end{subarray}}\left\{e^{j(\alpha_{p}-\alpha_{q})}\underbrace{ \left(U_{p}I_{q}e^{j\phi_{q}}-U_{q}I_{p}e^{j\phi_{p}}\right)\sigma_{pq} \right)}_{\widetilde{D}_{pq,Lin}}\\ +\sum\limits_{\begin{subarray}{c}p\in L\cup M,q\in M\\ p\in L,q\in N\end{subarray}}e^{i\alpha_{p}}\underbrace{U_{p}I_{q}e^{-j\beta_{ q}}\sigma_{pq}}_{\widetilde{D}_{pq,Nonlin}}=\widetilde{P}+j\widetilde{Q}+ \widetilde{\Delta}\end{split}  \tag{18}\]  
which consist of a complex-scalar part \(\widetilde{\Omega}^{\bullet}=\widetilde{P}+j\widetilde{Q}\) and a complex-bivector part \(\widetilde{\Omega}^{\wedge}=\widetilde{\Delta}\). In Equation (18), \"\(\odot\)\" is the new \"generalized complex geometric product\" [32]. The components \(\widetilde{P}\), \(\widetilde{Q}\), \(\widetilde{\Delta}\) in Equation (18), can be represented as:  
\[\widetilde{P}=Re\Big{[}\widetilde{U}\cdot\widetilde{I}^{*}\Big{]}=\sum\limits _{p\in N}U_{p}I_{p}cos\phi_{p}\sigma_{0} \tag{19}\]  
\[\widetilde{Q}=Im\Big{[}\widetilde{U}\cdot\widetilde{I}^{*}\Big{]}=\sum\limits _{p\in N}U_{p}I_{p}sin\phi_{p}\sigma_{0} \tag{20}\]  
\[\widetilde{\Delta}=\sum\limits_{ \begin{subarray}{c}p<q\\ p,q\in N\end{subarray} }\Re\Big{(}\widetilde{U}_{p}\wedge\widetilde{I}^{*}_{q} \Big{)}\sigma_{pq}+\sum\limits_{ \begin{subarray}{c}p\in L\cup N,q\in M\\ p\in L,q\in N\end{subarray} }e^{i\alpha_{p}}U_{p}I_{q}e^{-j\beta_{q}} \tag{21}\]  
The generic term \(\widetilde{D}_{pq}\) (18) is a bivector named distortion power term and it is related with the generic term \(\widetilde{\Delta}_{pq}\) named _rotated distortion_ power term:  
\[\widetilde{\Delta}_{pq}=e^{j(\alpha_{p}-\alpha_{q})}\widetilde{D}_{pq}\Rightarrow \|\widetilde{\Delta}_{pq}\|=\|\widetilde{D}_{pq}\| \tag{22}\]  
The \(\widetilde{\Delta}_{pq}\) terms in Equations (18) and (21), can be collected into four groups and written in a short notation related with the sets \(L\), \(N\) and \(M\) as:  
\[\widetilde{\Delta}^{N,N}=\sum\limits_{p,q\in N}\widetilde{\Delta}_{pq} \Rightarrow\widetilde{\Delta}_{pq}=e^{j(\alpha_{p}-\alpha_{q})}\widetilde{D}_ {pq,Lin} \tag{23}\]  
\[\widetilde{\Delta}^{L,N}=\sum\limits_{ \begin{subarray}{c}p\in L\\ q\in N\end{subarray} }\widetilde{\Delta}_{pq}\Rightarrow\widetilde{\Delta}_{ pq}=e^{j(\alpha_{p}-\alpha_{q})}\widetilde{D}_{pq,Nonlin} \tag{24}\]\[\widetilde{\Delta}^{L,M}=\sum_{ \begin{subarray}{c}p\in L\\ q\in M\end{subarray} }\widetilde{\Delta}_{pq}\Rightarrow\widetilde{\Delta}_{pq}=e^ {i\pi_{p}}\widetilde{D}_{pq,Nonlin} \tag{25}\]  
\[\widetilde{\Delta}^{N,M}=\sum_{ \begin{subarray}{c}p\in N\\ q\in M\end{subarray}