## 8 Conclusions  
In this study, a different representation of the expressions for the unbalanced power vectors and apparent power vectors of a linear three-phase electrical system unbalanced in currents and voltages has been proposed. It was verified that the modulus of apparent power and unbalanced power is the same as that obtained by the Buchholz method. The proposed vectors allow the evaluation and determination of the unbalanced power anywhere in an electrical system, without requiring the use of equivalent circuits at the nodes of the system. Using the proposed vectors, it has been shown that it is sufficient to calculate the voltages at nodes and line currents at loads to determine the unbalanced \begin{table}
\begin{tabular}{c c c c c} \hline  & \(S_{\text{p+}}\) & \(S_{\text{p-}}\) & \(S_{\text{p0}}\) & \(S_{1}\) \\ \hline  & **Modulus** & **Modulus** & **Modulus** & **Modulus** \\ \hline Load 1 & 36,095.29 & 3834.01 & 1160.91 & 36,316.91 \\ Load 2 & 7609.68 & 974.85 & 57.79 & 7672.08 \\ Load 3 & 8210.98 & 1051.88 & 62.35 & 8278.31 \\ Line 1-2 & 13,194.66 & 1925.28 & 1708.68 & 13,443.41 \\ Line 2-3 & 1309.27 & 238.60 & 526.92 & 1431.35 \\ Bus 1 & 64,920.90 & 7096.78 & 3250.72 & 65,388.49 \\ Bus 2 & 51,790.99 & 5501.19 & 1665.72 & 52,108.97 \\ Bus 3 & 15,661.52 & 2006.34 & 118.93 & 15,789.96 \\ \hline \end{tabular}
\end{table} 
Table 10: Results of apparent powers. \begin{table}
\begin{tabular}{c c c c c c} \hline  & \multicolumn{2}{c}{**Phase A**} & \multicolumn{2}{c}{**Phase B**} & \multicolumn{2}{c}{**Phase C**} \\ \hline  & \(P_{a0}^{a}\) & \(Q_{a0}^{a}\) & \(P_{b0}^{b}\) & \(Q_{b0}^{b}\) & \(P_{c0}^{c}\) & \(Q_{c0}^{c}\) \\ \hline Load 1 & 159.55 & \(-\)393.83 & 216.45 & 158.84 & \(-\)376.01 & 234.98 \\ Load 2 & 12.52 & \(-\)0.79 & \(-\)7.95 & 23.12 & \(-\)12.77 & \(-\)13.96 \\ Load 3 & 17.21 & \(-\)3.33 & \(-\)10.96 & 21.96 & \(-\)6.25 & \(-\)18.63 \\ Line 1-2 & \(-\)129.46 & \(-\)556.77 & 501.75 & 44.43 & \(-\)323.36 & 536.80 \\ Line 2-3 & \(-\)16.78 & \(-\)131.93 & 198.80 & 85.98 & \(-\)145.32 & 82.65 \\ Bus 1 & 43.05 & \(-\)1086.64 & 898.09 & 334.34 & \(-\)863.70 & 821.85 \\ Bus 2 & 172.51 & \(-\)529.87 & 396.34 & 289.90 & \(-\)540.34 & 285.05 \\ Bus 3 & 29.74 & \(-\)4.12 & \(-\)18.91 & 45.08 & \(-\)19.02 & \(-\)32.59 \\ \hline \end{tabular}
\end{table} 
Table 9: Components of the apparent power vector and unbalanced power due to zero-sequence voltage. power flows. Each of these vectors contains six components, distributed in two components per phase that are expressed in terms of active and reactive powers. It has been proven that in the assumption of two loads connected in parallel, through the components of the vectors of each one, the resulting vectors of the group of loads are easily determined, regardless of the characteristics and type of connection of the loads. This allows us to quickly see if the unbalance power will increase or decrease by coupling the loads in parallel at the connection bus. This analysis cannot be performed directly from the currents and voltages, mainly when the voltages are unbalanced. Similarly, this advantage extends to the analysis of the unbalance power flows between various buses in the system to correctly assess the losses in the lines. To validate the applicability of the proposed vectors and improve comprehension, a practical case of a three-phase four-wire linear system with three buses was developed and studied. Conceptualization, P.A.B., R.M., and J.M.D. ; methodology, P.A.B., R.M.-M., J.M.D., and R.M. ; validation, P.A.B., R.M.-M., and R.M. ; formal analysis, P.A.B. and R.M.-M.; investigation, P.A.B., R.M.-M., J.M.D., and R.M. ; resources, P.A.B. and R.M.-M.; data curation, P.A.B. and R.M.-M.; writing--original draft preparation, J.M.D. and R.M. ; writing--review and editing, R.M.-M., J.M.D., and R.M. ; visualization, P.A.B. and R.M.-M.; supervision, J.M.D. and R.M. ; project administration, J.M.D. All authors have read and agreed to the published version of the manuscript.