# Large-Scale Grid Integration of Renewable Energy Resources with a Double Synchronous Controller
## 5 The Power Sharing Capability  
In this section, the mathematical representations of the NSC and the DSC closed-loop models, given in Figures 2 and 3, respectively, are considered to analyze the ability of these controllers at tracking the reference values of active and reactive power. By referring to Figure 2, the relation between active and reactive power and their reference values for the NSC can be:  
\[\frac{P}{P^{*}} = \frac{k_{pp}s+k_{ip}}{(L/R)s^{2}+\Big{(}1+k_{pp}\Big{)}s+k_{ip}} \tag{16}\] \[\frac{Q}{Q^{*}} = \frac{k_{pq}s+k_{iq}}{(L/R)s^{2}+\Big{(}1+k_{pq}\Big{)}s+k_{iq}} \tag{17}\]  
Also, by using Figure 3, the relationship between active and reactive power and their reference values for the DSC can be achieved as (18) and (19):  
\[\frac{P}{P^{*}} = \frac{\beta_{2}\Big{(}k_{pq}s+k_{iq}\Big{)}}{(L/R)s^{2}+\Big{(}1 +\beta_{2}k_{pq}\Big{)}s+\beta_{2}k_{iq}}\] \[\frac{Q}{Q^{*}} = \frac{\beta_{2}\Big{(}k_{pq}s+k_{iq}\Big{)}}{(L/R)s^{2}+\Big{(}1 +\beta_{2}k_{pq}\Big{)}s+\beta_{2}k_{iq}} \tag{18}\] \[+ \Bigg{[}\begin{array}{l}\beta_{2}k_{iq}k_{ip}\alpha_{1}+\Big{(} 1+\beta_{2}k_{pq}\Big{)}(L/R)+(L/R)\alpha_{2}+\Big{(}1+\beta_{2}k_{pq}\Big{)} k_{pp}\alpha_{1}+\\ \\ (L/R)\Big{(}k_{ip}\alpha_{1}+k_{pp}\alpha_{1}\alpha_{2}\Big{)}-(L/R)\alpha_{1} k_{pp}\end{array}\Bigg{]}s^{3}\] \[+ \Bigg{[}\begin{array}{l}\beta_{2}k_{iq}(1+(L/R)\alpha_{2})+ \Big{(}1+\beta_{2}k_{pq}\Big{)}\omega_{2}+\beta_{2}k_{iq}k_{pp}\alpha_{1}+\\ \\ \Big{(}1+\beta_{2}k_{pq}\Big{)}\Big{(}k_{ip}\alpha_{1}+k_{pp}\alpha_{1}\omega_{ 2}\Big{)}+(L/R)k_{ip}\alpha_{1}\omega_{2}+\\ \\ \beta_{2}\omega_{1}Q^{*}(L/R)\beta_{1}k_{pq}-\Big{(}1+\beta_{2}k_{pq}\Big{)} \omega_{1}k_{pp}-(L/R)\omega_{1}k_{ip}\end{array}\Bigg{]}s^{2}\] \[+ \Bigg{[}\begin{array}{l}\beta_{2}k_{iq}\omega_{2}+\beta_{2}k_{iq} \Big{(}k_{ip}\alpha_{1}+k_{pp}\alpha_{1}\omega_{2}\Big{)}+\\ \\ \Big{(}1+\beta_{2}k_{pq}\Big{)}k_{ip}\alpha_{1}\omega_{2}+\beta_{2}\omega_{1}Q^ {*}(L/R)\beta_{1}k_{iq}\\ \\ -\beta_{2}k_{iq}k_{pp}\omega_{1}-\Big{(}1+\beta_{2}k_{pq}\Big{)}\omega_{1}k_{ip }\end{array}\Bigg{]}s\] \[+ \beta_{2}k_{iq}k_{ip}\alpha_{1}\omega_{2}-\beta_{2}k_{iq}\omega_{ 1}k_{ip}\] \[\frac{Q}{Q^{*}} = \frac{\beta_{2}\Big{(}k_{pq}s+k_{iq}\Big{)}}{(L/R)s^{2}+\Big{(}1 +\beta_{2}k_{pq}\Big{)}s+\beta_{2}k_{iq}} \tag{19}\]The Bode diagrams of Figure 7 ascertain more the aforementioned points. Notably, the angles of the DSC transfer functions for both active and reactive power are nearly equal to zero. However, this angle is about eight degrees for both active and reactive power of the NSC that leads to a decrease the precise performance of the NSC.  
Figure 7: The Bode diagrams of: (**a**) P/P* and (**b**) Q/Q* of the NSC and the DSC.  
Choosing various values for virtual inertia can impact on the performance of the power-sharing of the DSC that is investigated in this section. By considering the conditions of J1 > J2 > J3 > J4, the Nyquist diagrams of P/P* and Q/Q* for the DSC are plotted, as shown in Figure 8.  
It can be seen that the ability of the DSC to track the reference value of active power is noticeably decreased as the virtual inertia decreases. The negative effect of decreasing virtual inertia is more for reactive power, as shown in Figure 8b.  
Figure 8: Various values of virtual inertia for the DSC: (**a**) P/P* and (**b**) Q/Q*.