## 2 Command Feedforward Control for a GCI Current Control System
\[\frac{\mathrm{i_{g}}}{\mathrm{i_{g}^{*}}} = \frac{R_{\mathrm{d}}\mathrm{\text{-}}\mathrm{C_{f}}K_{\mathrm{p} }\mathrm{s^{2}}\mathrm{+}(K_{\mathrm{p}}\mathrm{+}R_{\mathrm{d}}\mathrm{ \text{-}}\mathrm{C_{f}}\mathrm{\text{-}}K_{\mathrm{i}})\mathrm{s}\mathrm{+}K_{ \mathrm{i}}}{L_{1}\mathrm{\text{-}}L_{2}\mathrm{\text{-}}\mathrm{C_{f}} \mathrm{\text{-}}\mathrm{s^{4}}\mathrm{+}R_{\mathrm{d}}\mathrm{\text{-}} \mathrm{C_{f}}(L_{1}\mathrm{+}L_{2})\mathrm{s^{3}}\mathrm{+}(L_{1}\mathrm{+}L _{2}\mathrm{+}R_{\mathrm{d}}\mathrm{\text{-}}\mathrm{C_{f}}\mathrm{\text{-}}K _{\mathrm{p}})\mathrm{\text{-}}\mathrm{s^{2}}\mathrm{+}(K_{\mathrm{p}}\mathrm{ +}R_{\mathrm{d}}\mathrm{\text{-}}\mathrm{C_{f}}\mathrm{\text{-}}K_{\mathrm{i}} )\mathrm{s}\mathrm{+}K_{\mathrm{i}}} \tag{4}\] \[\frac{\mathrm{i_{g}}}{\mathrm{i_{g}}}= \frac{R_{\mathrm{d}}\mathrm{\text{-}}\mathrm{C_{f}}(L_{1}\mathrm{+ }L_{2})\mathrm{s^{3}}\mathrm{+}(L_{1}\mathrm{+}L_{2}\mathrm{+}R_{\mathrm{d}} \mathrm{\text{-}}\mathrm{C_{f}}K_{\mathrm{p}})\mathrm{\text{-}}\mathrm{s^{2}} \mathrm{+}(K_{\mathrm{p}}\mathrm{+}R_{\mathrm{d}}\mathrm{\text{-}}\mathrm{C_{f }}\mathrm{\text{-}}K_{\mathrm{i}})\mathrm{s}\mathrm{+}K_{\mathrm{i}}}{L_{1} \mathrm{\text{-}}L_{2}\mathrm{\text{-}}\mathrm{C_{f}}\mathrm{\text{-}} \mathrm{s^{4}}\mathrm{+}R_{\mathrm{d}}\mathrm{\text{-}}\mathrm{C_{f}}(L_{1} \mathrm{+}L_{2})\mathrm{s^{3}}\mathrm{+}(L_{1}\mathrm{+}L_{2}\mathrm{+}R_{ \mathrm{d}}\mathrm{\text{-}}\mathrm{C_{f}}\mathrm{\text{-}}K_{\mathrm{p}}) \mathrm{\text{-}}\mathrm{s^{2}}\mathrm{+}(K_{\mathrm{p}}\mathrm{+}R_{\mathrm{d }}\mathrm{\text{-}}\mathrm{C_{f}}\mathrm{\text{-}}K_{\mathrm{i}})\mathrm{s} \mathrm{+}K_{\mathrm{i}}} \tag{5}\]  
Applying Equations (4) and (5) indicates that steady state error and phase delay are expected as CFFC is applied. Using Equations (4) and (5), the stability and frequency response of the GCI current control system may be simulated and analyzed. For stability analysis of the GCI current control system with CFFC, root locus of the GCI current control system is implemented. The simulation results of the root locus implementation are shown in Figure 3.  
Figure 3: Root locus of the GCI current control system applying the CFF control. Figure 2: Block diagram of a GCI current control system with command feedforward control (CFFC). The root locus in Figure 3 shows the movement of the system's poles with respect to variation in the proportional controller gain \(\text{K}_{\text{p}}\). Applying the root locus, it is apparent that faster transient responses can be achieved, but that the GCI current control system becomes unstable as \(\text{K}_{\text{p}}\) increases. Accordingly, GCI current controller gains should be determined after accounting for the transient dynamic performance and stable operation of the GCI current control system using a CFFC with the root locus. Figure 4 shows the frequency response of the GCI current control system with and without the CFFC applied. Considering the magnitude of the frequency response, applying the CFFC extends the bandwidth of GCI current control system. Extending the bandwidth will likely result in faster transient dynamic performance and less steady state error. Considering the phase side of the frequency response, applying the CFFC is expected to result in less phase lagging. Based on the root locus and the frequency response, the benefits of the proposed GCI current control--improved stability and command tracking performance--are verified.