# Instantaneous Reactive Power Theory in the Geometric Algebra Framework
## 1 Introduction
There, the proposed model, called Simple Kirchhoff Rotation (SKR), corresponds to an orthogonal transformation based on elementary rotations and is valid for any n-dimensional space. The use of GA makes it possible to visualize this new model from a geometric perspective. This methodology enables the generalization of the compensation process traditionally proposed by formulations based on Clarke, Park or space-vector transformations to systems with n phases. On the other hand, in [24] the analysis of power in n-phase systems in the GA environment is addressed with the aid of the Hilbert transform (HT). In fact, the introduction of a geometric vector defined as a combination of the signal and its HT for each phase current and voltage requires a Euclidean space of 2n dimensions. This makes it possible to identify a parallel geometric power and a quadrature geometric power for linear loads. However, the general analysis of nonlinear loads also requires extending the Euclidean space to 3n dimensions in order to include current harmonics not present in the voltage. This does not achieve an identification with the representative power and current terms of the IRPT; that is, it does not identify the instantaneous reactive power and the instantaneous reactive current of the IRPT. In this work, the GA framework is used to formulate the IRPT in a unified way, without the need to resort to any type of coordinate transformation or mathematical transform, applicable to power systems with any number of phases. The latter can be considered of great importance, since drives with more than three phases have emerged in recent years. In effect, the control of 5-phase, 12-phase, or 16-phase motors in wind turbine or electric vehicle applications is an ongoing field of research [25; 26]. The original contributions of this research are the following:  
* GA is used to carry out a unified analysis of the IRPT where it is only necessary to introduce a multidimensional power variable, the instantaneous power multivector. As in single-phase systems, the energy transfer between supply and load in an n-phase circuit can be analyzed from a single power variable. * IRPT is naturally extended within the GA framework to systems with any number of phases. Thus, three-phase systems emerge as a particular case of multi-phase analysis. * The compensation process is systematized from the new multivector power variable. Thus, a compensator power multivector makes it possible to establish instantaneous compensation and/or time-averaged compensation. Further elaboration of the previous paragraph is followed by clarification of the key idea of the paper. Instantaneous power is the only power variable necessary to describe the energy transfer between source and load in a single-phase system. The IRPT showed that, to explain the energy transfer between source and load in a system with more than one phase, the instantaneous power is not enough. This is true if an n-phase circuit is to be treated as an entire system and not as a combination of n independent circuits. In this paper it is shown that it is possible to analyze the power transfer between source and load in n-phase systems from a single power variable defined in the GA framework. The consistency with the power and current terms of the IRPT is shown. The paper is organized as follows. In Section 2, the basic definitions and theorems for the analysis of the IRPT in three-phase systems are presented, first establishing the most convenient instantaneous power terms, and then the current terms obtained from the power variables. In Section 3, the unified process used for three-phase systems is extended to systems with more than three phases. In Section 4, the basics of instantaneous and average value compensation under the GA are established. Section 5 includes two selected case studies corresponding to a five-phase system and a real engineering three-phase system. Finally, the main conclusions are given in Section 6.