## 1 Introduction  
Low voltage electrical systems usually operate unbalanced, especially in four-wire supply networks where three-phase and single-phase loads coexist. These imbalances are attributable to both the loads and the voltages; this imbalance manifests as an increase of the total apparent power with respect to the ideal power of a balanced system, which is characterised by the positive sequence active power. The difference with three-phase three-wire systems is that there exists a zero sequence of the line current or current flowing through the neutral wire. These unbalanced powers that appear can be studied from Buchholz's apparent power [1] expressed as a function of its symmetric components [2]. The first reactive power compensator was developed by [PERSON] in 1917 [3]. Subsequently research continued, until relatively recently, in the investigation of reactive compensation giving rise to several research papers (those worth mentioning include [4; 5; 6]). However, imbalances are responsible for increases in the total apparent power that the generator must deliver, increased line losses [7],heating [8], machine malfunctions [9], protection malfunctions [10], etc. These unbalanced apparent powers have already been established in the IEEE-Std 1459-2010 [11]. A three-phase four-wire system powered by an infinite short-circuit power generator would have negative- and zero-sequence voltages that generate unbalanced powers, which cannot be efficiently compensated for. However, there are regulatory standards on the quality of the electricity supply that limit the unbalanced voltages [12; 13]. The unbalanced currents are directly attributable to the characteristics of the load, although in systems with unbalanced voltages and balanced loads, unbalanced currents are also generated. From the above, the need to study the mechanisms to eliminate these inefficiencies is clear. There have been more papers aimed at eliminating inefficient apparent powers in three-phase three-wire systems than in four-wire systems. In this work, the three-phase four-wire systems are analyzed and a new methodology for the elimination of said powers is proposed, since in a previous work [14] the authors already resolved this problem for three-phase three-wire systems through passive compensators. It is clear that active filters are more effective than passive compensators. However, active filters are much more expensive and less robust than passive compensators. Nevertheless, the reactive power compensators remain valid solutions for applications in consumer and electricity distribution in those situations when the criterion regarding the costs of installing and operating the equipment is more important than the ones related to the reaction speed or the control accuracy. This is also the case of the equipment for power factor improvement and load balancing in a three-phase distribution network. Of the works aimed at three-phase four-wire systems, it is worth mentioning the ones discussed below. [PERSON] and [PERSON] [15] developed three-phase three-wire systems based on an earlier article by [PERSON] [16]. At first, they assumed that the voltage system was unbalanced, but it really used balanced voltage, much like [PERSON]. [PERSON] et al. [17], also working off [PERSON], extended the method to four-wire systems. They added the desired power factor to the equations of the compensators. However, their method was only valid for balanced voltages and presented infinite solutions. [PERSON] [18] presented a methodology to obtain a compensator for negative- and zero-sequence currents, and positive-sequence reactive currents. The compensator is only formed by capacitors. For this, they assumed the load to be very inductive. They based this on the theory of instantaneous reactive power (Akagi) and used [PERSON]'s transform matrix. However, it is only applicable to three-phase systems with balanced voltages. It is also necessary to know the value and characteristics of the load.