# Instantaneous Reactive Power Theory in the Geometric Algebra Framework
## 2 Instantaneous Reactive Power Theory in Three-Phase Systems: A New Approach  
### Instantaneous Power Multivector  
Let the three-phase system exist as shown in Figure 1; the voltage vector \(\mathbf{u}\) and current vector \(\mathbf{i}\) are defined in the form:  
\[ \begin{array}{l}\mathbf{u}=\left[\begin{array}{ccc}u_{1}&u_{2}&u_{3}\\ \end{array}\right]^{T}\\ \mathbf{i}=\left[\begin{array}{ccc}i_{1}&i_{2}&i_{3}\\ \end{array}\right]^{T}\end{array}  \tag{1}\]  
Both vectors include line-to-neutral voltage waveforms and line current waveforms. Each of the voltage and current waveforms are time dependent, although this is not explicitly written in most of the expressions in the text that follows. The vectors \(\mathbf{u}\) and \(\mathbf{i}\) can be considered elements of a three-dimensional Euclidean space generated by the orthonormal basis \(e_{j}\) with \(j=1\), \(2\), \(3\). Thus, the voltage and current vectors can be expressed as (2):  
\[ \begin{array}{l}\mathbf{u}=u_{1}e_{1}+u_{2}e_{2}+u_{3}e_{3}\\ \mathbf{i}=i_{1}e_{1}+i_{2}e_{2}+i_{3}e_{3}\end{array}  \tag{2}\]  
For the system of Figure 1, from relations (1)-(2), the following definition of power is established:  
**Definition 1**. : _The instantaneous power multivector \(\mathbf{s}\) is defined as the geometric product of the voltage and current vectors (3),_  
Figure 1: Phase-neutral voltages and line currents in a three-phase power system. \[\mathbf{s}=\mathbf{u}\mathbf{i}=\mathbf{u}\cdot\mathbf{i}+\mathbf{u}\wedge\mathbf{i} \tag{3}\]  
From (3) it follows that the geometric product of any two vectors is obtained from the sum of their inner product (\(\mathbf{u}\cdot\mathbf{i}\)) and their cross product (\(\mathbf{u}\wedge\mathbf{i}\)). Although a relation such as the one established in (3) would not make sense in ordinary algebra, it is of special relevance in the framework of geometric algebra (GA). In the geometric algebra environment, each of its entities is called a multivector and is obtained from the geometric products of the orthonormal basis vectors \(e_{j}\) of the support linear space. Thus, an operative expression for \(\mathbf{s(t)}\) is possible by performing the term-by-term geometric product of the vectors given in (2), taking into account the following properties of the basis vectors \(e_{j}\), [16]:  
\[ \begin{split}& e_{i}e_{j}=\begin{cases}1&i=j\\ e_{ij}&i\
eq j\end{cases}\\ & e_{ij}=e_{i}\cdot e_{j}+e_{i}\wedge e_{j}=e_{i}\wedge e_{j}=-e_{j}\wedge e _{i}=-e_{ji}\end{split}  \tag{4}\]  
That is, the geometric product of a basis vector with itself is a scalar or 0-vector, and the geometric product of two different basis vectors is a bivector or 2-vector, since its inner product is zero. From (3) and (4) we obtain the expression for the multivector \(\mathbf{s}\):  
\[\mathbf{s}=\mathbf{u}\mathbf{i}=p+q_{12}e_{12}+q_{31}e_{31}+q_{23}e_{23}=p+ \mathbf{q} \tag{5}\]  
In (5), the 0-vector part is the instantaneous power \(p\):  
\[p=u_{1}i_{1}+u_{2}i_{2}+u_{3}i_{3} \tag{6}\]  
Furthermore, the 2-vector part \(\mathbf{q}\), which will be named the instantaneous reactive power bivector, is formed by three instantaneous reactive power components \(q_{12}\), \(q_{31}\), \(q_{23}\), [5]:  
\[ \begin{split}\mathbf{q}&=(u_{1}i_{2}-u_{2}i_{1})e_{12}+(u_{ 3}i_{1}-u_{1}i_{3})e_{31}+(u_{2}i_{3}-u_{3}i_{2})e_{23}\\ &=q_{12}e_{12}+q_{31}e_{31}+q_{23}e_{23}\end{split}  \tag{7}\]  
In (5), it is observed that the multivector \(\mathbf{s}\) is constituted by the combination of two parts or blades, a 0-vector \(p\), that is, a blade of degree zero, and a 2-vector \(\mathbf{q}\), that is, a blade of degree two. In general, a multivector will be constituted by the combination of k-vector parts, with k from 0 to \(2^{\mathrm{n}}\), where \(\mathrm{n}\) is the dimension of the linear support space of the voltage and current vectors. In GA it is also common to use the symbol \(<>_{\mathrm{k}}\) to designate each one of the blades of degree k of a multivector.