## 1 Introduction  
The majority of low-voltage distribution systems are unbalanced, mainly owing to the use of single-phase loads. In transmission systems or high-voltage systems, these effects are also present but they are very small. The phenomenon of imbalance causes unbalanced powers. The unbalanced powers constitute inefficiencies in the system [1; 2; 3; 4] that cause increased losses in the lines and the malfunction of motors, generators, transformers, and protective equipment. In addition, the apparent power of the system is significantly increased [5; 6; 7; 8]. Therefore, the powers due to imbalances must be calculated properly to facilitate the reliable design of active, passive, or hybrid devices, which can be used to compensate for them. At present, there is no theory agreed on by all that allows an adequate evaluation of this type of phenomenon [9]. Most of the studies that exist focus on using the expressions proposed in the standard IEEE Std. 1459-2010 [10] and Buchholz [11]. These expressions use the Root Mean Square (RMS) values of the voltage and current that are expressed in symmetric or phase components. As a result, the moduli or RMS values of the apparent power and unbalanced power are obtained at a bus in the system [12]. These values are perfectly valid, but they are insufficient for accurate evaluation and analysis of the causes of the imbalance, as well as the place where they occur. In certain cases, the imbalances at a node of the system can help to compensate for some of the imbalances at another node [13; 14]. This type of case cannot be analysed correctly if we only know the moduli of the unbalanced powers, for example, when one knows the individual unbalance powers of two loads connected in parallel and one wants to calculate the unbalanced power that results from the coupling since the resulting unbalanced apparent power is not the arithmetic sum of both. There are other studies that have proposed phasor or vector expressions to evaluate these inefficiencies. In [15], vector apparent power is defined as the vector sum of total active, total reactive, and total distortion powers, which are the sum of respective values for each phase. It treats each phase as an independent single-phase system like arithmetic apparent power. In this study, the unbalance power is not proposed by means of a vector expression. In [16], the unbalanced power vectors due to voltage and current are defined. To consider the unbalanced voltages, they use the unbalance factors \(\delta_{-}=V_{-}/V_{+}\) and \(\delta_{0}=V_{0}/V_{+}\). In [17], the authors propose a new tool to phasorically represent the powers in the Clifford geometric space that is only applied to single-phase systems. In [18], the phasor of the total unbalance power is defined. It presents the same problem as [16] since they use the unbalance factors \(\delta_{-}\) and \(\delta_{0}\) to consider the unbalance power caused by the unbalanced voltages. Other authors extend their studies to the analysis of non-linear three-phase systems [19; 20]. The use of these expressions significantly improves the capacity in the analysis of unbalanced phenomena and adapts correctly to systems fed with balanced voltages. Its application in systems with unbalanced voltages is based on including scalar expressions, such as factors \(\delta_{-}\) and \(\delta_{0}\). These scalar values diminish the capacity of analysis of vector expressions, making necessary their determination at any node of the system as we cannot use the vector properties that relate one node to another. Therefore, the unbalanced powers caused by the asymmetry of the voltages are not adequately considered. For this reason, when the voltages are unbalanced, it is also not possible to determine whether the compensation of the unbalanced powers at one node of the system improves or worsens the rest of the system.