## 3 Compensation in Three-Phase Three-Wire Systems with Balanced Voltages
### Negative-Sequence Current Compensator  
As Figure 2 shows, to compensate for the unbalanced apparent power \(S_{u+}^{-}\), we use the NSCC compensator from reactive elements (coils and/or capacitors). In this case, the compensator reactance values (\(X_{aF}\), \(X_{bF}\) and \(X_{cF}\)) are different in each phase, because the positive-sequence voltage and negative-sequence current have different sequences. In our case, the objective is to compensate for the unbalanced powers due to the negative-sequence currents of the load (\(I_{aL-}\), \(I_{bL-}\) and \(I_{cL-}\)). Therefore, the currents consumed by the SVC compensator are equal to \(I_{aL-}\), \(I_{bL-}\) and \(I_{cL-}\), but in opposite directions, according to Equation (15). \[I_{aF}=-I_{aL-}\ I_{bF}=-I_{bL-}\ I_{cF}=-I_{cL-} \tag{15}\]  
Considering reference point \"0\" for all voltages, and applying [PERSON]'s second law to the compensator, the Equations (16)-(18) are obtained. The positive-sequence voltages are used because they are balanced. On the other hand, the powers in each phase consumed by the NSCC compensator at the balanced voltages are due to voltages and currents of different sequences; therefore, the sum of the values of all phases is null, complying with Equation (19). \[\frac{V_{a+}-X_{aF}I_{aF}}{}=\frac{V_{NF}}{} \tag{16}\]  
\[\frac{V_{b+}-X_{bF}I_{cF}}{}=\frac{V_{NF}}{} \tag{17}\]  
\[\frac{V_{c+}-X_{cF}I_{cF}}{}=\frac{V_{NF}}{} \tag{18}\]  
\[\frac{X_{aF}+X_{bF}+X_{bF}}=0 \tag{19}\]  
Equations (16)-(19) form a system of four equations with complex numbers, where \(X_{aF}\), \(X_{bF}\), \(X_{bF}\) and \(V_{NF}\) are unknown. Solving the system of equations would directly provide the values of the reactors of the NSCC compensator. We use another approach to solve the system, to obtain simpler and quicker expressions. Considering that Equation (16) refers to Phase \(A\), and decomposing this expression into real and imaginary parts, Equations (20) and (21) are obtained. The angle of \(X_{aF}\) is \(\frac{\pi}{2}\) when it is a coil, and \(-\frac{\pi}{2}\) when it is a capacitor. We consider that the load is inductive, and therefore, the reactance of the compensator, \(X_{aF}\), is capacitive and its angle is \(-\frac{\pi}{2}\). \[V_{a+}cos\ \alpha_{a+}-X_{aF}I_{aF}cos\left(\beta_{aF}-\frac{\pi}{2}\right)=V_{ NF}cos\ \alpha_{NF} \tag{20}\]  
\[V_{a+}sin\ \alpha_{a+}-X_{aF}I_{aF}sin\left(\beta_{aF}-\frac{\pi}{2}\right)=V_{ NF-}sin\ \alpha_{NF-} \tag{21}\]  
Squaring Equations (20) and (21), adding both equations, and regrouping the terms, we obtain (22). To simplify, we consider that \(\alpha_{a+}=0\). \[I_{aF}{}^{2}X_{aF}{}^{2}-2V_{a+}I_{aF}X_{aF}cos\left(\beta_{aF}-\frac{\pi}{2} \right)+\left(V_{a+}{}^{2}-V_{NF}{}^{2}\right)=0 \tag{22}\]Developing the same procedure for Phase \(B\) from Equation (17) and for Phase \(C\) from Equations (18), (23) and (24) are obtained. \[{I_{\text{bf}}}^{2}{X_{\text{bf}}}^{2}-2{V_{\text{bf}}}_{+}{I_{\text{bf}}}{X_{ \text{bf}}}{X_{\text{bf}}}cos\left({\beta_{\text{bf}}}-\frac{\pi}{2}\right)+ \left({V_{\text{bf}}}_{+}{}^{2}-{V_{\text{bf}}}^{2}\right)=0 \tag{23}\]  
\[{I_{\text{cf}}}^{2}{X_{\text{cf}}}^{2}-2{V_{\text{cf}}}{X_{\text{cf}}}cos \left({\beta_{\text{cf}}}-\frac{\pi}{2}\right)+\left({V_{\text{cf}}}^{2}-{V_{ \text{NF}}}^{2}\right)=0 \tag{24}\]  
To include the condition defined in Equation (19), we graphically analyse different case studies, where we vary the characteristics of the loads as well as the line-to-line voltages. Figure 3 shows the relation between the difference in the modules \(V_{z+}-V_{\text{NF}}\) (for \(z=\{a,b,\ c\}\)), and the sum of the compensator reactances \(X_{\text{af}}\), \(X_{\text{bf}}\) and \(X_{\text{cf}}\). It is observed that regardless of the characteristics of the load and the voltages, the sum of the reactances is null when the difference in the modules \(V_{z+}-V_{\text{NF}}=0\). This implies that Equation (19) is true when the modules of \(V_{z+}\) and \(V_{\text{NF}}\) are equal.