## 4 Compensation with Unbalanced Voltages  
Let us consider that the line-to-line voltages of the system shown in Figure 2 are unbalanced. Under these conditions, the apparent power \(S_{T}\) includes all terms of powers defined in Equation (12). As an infinite power system is considered, the values of the line-to-line voltages on the bus do not change, therefore, we obtain a negative-sequence line-to-line voltage with constant value. As a result, the apparent power \(S_{u-}^{+}\) cannot be compensated, as the bus imposes the negative-sequence line-to-line voltage. The value of \(S_{u-}^{+}\) once connected with the SVC and NSCC compensators is modified based on the new value of the positive-sequence current on the bus, which is delivered by the network, and is lower than its initial value. In contrast to systems with balanced voltages, the behaviour of the SVC and NSCC compensators are not independent. There is an interaction between both compensators, so that the design of one of them affects the other, and vice versa. The reason for this is evident, when the voltages are unbalanced, the SVC compensator consumes a positive-sequence and negative-sequence current. This negative-sequence current must be considered in the design of the NSCC compensator. The NSCC compensator reaction is similar at unbalanced voltages, it consumes a positive-sequence and negative-sequence current. This positive-sequence current must be considered in the SVC compensator design. This dependence of one compensator on another makes the iteration necessary, to determine the reactance values of both compensators. The number of iterations required to converge on the optimal solution depends on the degree of unbalance of the voltages. The proposed calculation method allows us to obtain the optimal solution without having to iterate. For this purpose, we analyse the system and behaviour of the SVC and NSCC compensators at unbalanced voltages, from which the following considerations are deduced:  
* In a three-wire system with unbalanced voltages, the total active power \(P\) consumed by the system is determined by the sum of the active positive- and negative-sequence powers, that is: \(P=P_{+}+P_{-}\). This power is unidirectional, and the SVC and NSCC compensators can only compensate powers with a null average value because they are configured from reactive elements (coils and/or capacitors). For this reason, the value of \(P\) must be constant and continue to be supplied by the network or generator. The NSCC compensator is responsible for maintaining a constant \(P\). This compensator consumes a positive-sequence active power in each phase \(P_{zF+}\) of the same value as that of the negative-sequence active power consumed by load and a negative-sequence active power in each phase \(P_{zF-}\) in the opposite direction, that is, \(P_{zF+}=P_{zL-}=-P_{zF-}\). Therefore, \(P_{zF-}\) will compensate \(P_{zL-}\) and \(P=P_{zL+}+P_{zF+}\). The SVC compensator consumes neither positive-sequence nor negative-sequence active power, because the voltages and currents of the same sequence are out of phase, \(\pm\frac{\pi}{2}\), because the SVC compensator reactances have the same value. The sign depends on the inductive or capacitive nature of the system to be compensated and will be the same for the values of each sequence. * The total reactive power \(Q\) consumed by the system is determined by the sum of the positive-sequence reactive power \(Q_{+}\) and the negative-sequence reactive power \(Q_{-}\); therefore, \(Q=Q_{+}+Q_{-}\). At unbalanced voltages, both compensators consume positive- and negative-sequence reactive powers. The optimal solution is one whose result cancels the reactive powers \(Q_{+}\) and \(Q_{-}\); therefore, \(Q=0\). It is observed, from analysing both compensators to obtain the optimal solution, that in the SVC compensator, the relation (33) is fulfilled, and in the NSCC compensator, the Equation (34) is fulfilled, where \(\delta_{-}=\frac{V_{-}}{V_{+}}\).