# Instantaneous Reactive Power Theory in the Geometric Algebra Framework
## 1 Introduction  
The development of the instantaneous reactive power theory (IRPT) has been linked from the beginning to the control of active power line conditioners (APLCs). In fact, the use of APLCs for the compensation of non-linear loads led to the proposal of a new power theory in the time domain. This has allowed for new control strategies for these devices, APLCs, which include in their constitution a dc/ac converter as a fundamental element of their power circuit. In addition, the IRPT also established new paradigms regarding the transfer of energy between source and load. This showed that instantaneous power was not the only power variable needed to describe such a process; a new power variable (in its original version) was postulated, the instantaneous imaginary power. Subsequently, different formulations have appeared over time, in which new variables of instantaneous reactive power were added, as collected in [1]. Originally, the so-named pq formulation was published, aimed at the analysis of three-wire three-phase systems. There, the instantaneous active current and the instantaneous reactive current were defined, obtained from the instantaneous power and the instantaneous imaginary power; this was revealed in [2]. Zero-sequence components were treated separately. Later, what has come to be known as the modified pq formulation was published, where the zero-sequence components were included in the global development. For this, instantaneous power and three components of instantaneous reactive power were considered as power variables; a complete exposition of both approaches can be found in [3]. These formulations have been established both in \(\alpha\beta 0\) coordinates and in phase coordinates [4,5]. Other developments, with less acceptance, have also been proposed,although all of them are based on the management of an instantaneous power variable, and two/three instantaneous reactive power variables. In that setting, the current components are obtained from the power variables in reference [6]. The implementation of different compensation strategies for non-linear and unbalanced loads, through the use of active compensation equipment (APF or APLCs), has been the largest field of the IRPT application, which continues today; see [7; 8; 9]. Moreover, although references [10; 11] perform an analysis for multi-phase systems, it cannot be considered a unified theory that is generated from a single generalized power variable. On the other hand, geometric algebra (GA) has been configured as a very useful tool for the analysis of multidimensional problems. It allows the introduction of an operation, the geometric product, which includes in its definition the inner product and the cross product. This means having multidimensional objects called multivectors that are independent of any coordinate system. The GA has its origin in the 19 th century, but it was not until [PERSON] [12] that it was systematically applied in different fields of science and technology [13; 14; 15]. One of the most widely applied disciplines today is cybernetics [16]. In this context, the GA provides a suitable mathematical framework for the development of a decomposition model for electrical power. One of the first works on the subject is [17], where an electrical power multivector for distorted single-phase systems is proposed. In [18], the same topic is advanced by introducing a complex GA. In both cases, an identification of the power terms collected in the frequency theories adopted up to that moment is achieved. In the same field, the works [19; 20; 21] can be considered, which reformulate circuit laws and electrical power within the framework of GA in an original way. However, none of these latter proposals support an extension to systems with more than one phase, which requires a new review of the matter. In this regard, new papers published recently have raised the issue from a new perspective [22; 23; 24]. In [23], a generalization of the transformations commonly used in electrical engineering is introduced.