# Instantaneous Reactive Power Theory in the Geometric Algebra Framework
## 3 The New Approach to Multi-Phase Systems
### Instantaneous Power Multivector in Multi-Phase Systems
Nevertheless, in the context of GA, the multivector \(\mathbf{s}\) consists of a 0-vector part identifying the instantaneous real power \(p\), and a 2-vector part \(\mathbf{q}\), which corresponds to a bivector of instantaneous reactive power. The 2-vector part consists of \(\mathrm{n}(\mathrm{n}-1)/2\) instantaneous reactive power components (59):  
\[\mathbf{q} =(u_{1}i_{2}-u_{2}i_{1})e_{12}+\ldots+(u_{1}i_{n}-u_{n}i_{1})e_{1n }+\ldots \tag{59}\] \[+(u_{2}i_{n}-u_{n}i_{2})e_{2n}+\ldots\] \[=q_{12}e_{12}+\ldots+q_{1n}e_{1n}+\ldots+q_{2n}e_{2n}+\ldots\]  
To obtain (59), (4) was taken into account. The \(\mathrm{n}(\mathrm{n}-1)/2\) terms constituting the instantaneous reactive power bivector \(\mathbf{q}\) are given by:  
\[q_{ij}= \begin{pmatrix}u_{i}i_{j}-i_{i}u_{j}\end{pmatrix} \quad\forall i,\;j=1, \;\ldots,\;n;\;i\prec j \tag{60}\]  
The terms of instantaneous reactive power (60) verify relation (61):  
\[\sum_{ \begin{subarray}{c}i,j\,k\,=\,1\\ i\,\
eq\,j\,\
eq\,k\end{subarray} }^{n}u_{k}q_{ij}=0 \tag{61}\]  
That is, only \((\mathrm{n}(\mathrm{n}-1)/2)-1\) instantaneous reactive power variables are independent. By definition of the bivector \(\mathbf{q}\), it is perpendicular to the vector \(\mathbf{u}\), and therefore the inner product of both (obtained as the semi-sum of the geometric products '\(\mathbf{qu}\)' and '\(\mathbf{qu}\)') is zero. Thus, from the latter relationship it follows that a multi-phase energy system is characterized by \(\mathrm{n}(\mathrm{n}-1)/2\) power variables; the instantaneous power \(p\) and \((\mathrm{n}(\mathrm{n}-1)/2)-1\) variables of instantaneous reactive power \(q_{ij}\). Theorem 4: The norm of the instantaneous power multivector represents the instantaneous apparent power \(s\), and verifies the relationship between instantaneous real power and instantaneous reactive power given by (62):  
\[s^{2}=p^{2}+q^{2} \tag{62}\]  
The proof is identical to that presented for Theorem 1. Thus, (63) is verified:  
\[\left\|\mathbf{s}\right\|^{2} =s^{2}=\left\langle\mathbf{s}\right\rangle_{0}\left\langle \mathbf{s}^{\dagger}\right\rangle_{0}+\left\langle\mathbf{s}\right\rangle_{2} \left\langle\mathbf{s}^{\dagger}\right\rangle_{2}\] (63) \[=p^{2}+\sum_{\begin{subarray}{c}i,j\,=\,1\\ i\,<\,j\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Relation (63) determines the partition of instantaneous apparent power. The first term represents the instantaneous power and the second, obtained from the norm of instantaneous reactive power bivector, (15), represents the instantaneous reactive power \(q\)[9]. The model established by (62) describes multi-phase systems according to the same approach followed in single phase systems. In the same way that was fixed for three-phase systems, the squared norm of the instantaneous power multivector has an orthogonal decomposition in terms of the instantaneous real power and instantaneous reactive power, of the same type as the decomposition of the apparent power depending on active power and reactive power, characteristic of the power analysis at a sinusoidal steady state.