# A Systematic Control Design Method with Active Damping Control in Voltage Source Converters
## 3 Design of the Control Gain Matrix
### Robustness Analysis  
The previous design process can provide sufficient stability margins. However, these excellent stability margins are guaranteed for the nominal values of system parameters. However, some of these parameters may not be constant during the inverter operation. For example, their operating-point conditions continuously change during the day, and the grid SCR can vary over a wide range due to variations of the connected load and generators. In this respect, a robustness assessment based on the eigenvalue calculation was carried out to examine the system's robustness under these uncertainties. In such a way, the variations in the critical mode are studied for different values of grid SCR and operating point conditions. In addition, a comparison between the proposed and the conventional control methods is also given. The stability margins of the proposed control method are calculated based on the linearized model in (6) and feedback control gain in (10) \(\left(A_{cl}=A-BK_{f}\right)\). It is worth remarking that the control gain matrix in (10) is calculated for the nominal power (\(P_{n}\)), the average value of grid SCR (SCR = 5.5), and \(f_{PLL}\) = 10 Hz. Then the control gain matrix is kept constant during the robustness assessment. Moreover, the critical mode of a conventional control system, as shown in Figure 2, is determined based on the linearized model represented in Reference [25]. Therefore, it is not repeated, and the interested reader can refer to it for more details. Figure 7 investigates the impact of grid SCR uncertainties (1 < SCR < 10), different PLL bandwidths (8 Hz <\(f_{PLL}\) < 80 Hz), and inverter power levels (0.01 \(P_{n}\) < \(P\) < \(P_{n}\)) on the critical mode under both proposed and conventional control methods. The following conclusions can be drawn from this figure:  
Figure 7: Critical mode locations of a grid-connected VSC in the complex s-plane: (**a**) impact of grid SCR variations P = Pn, \(f\)PLL = 10 Hz (conventional controller), and 50 Hz (proposed controller); (**b**) impact of different PLL bandwidths (P = Pn, SCR = 1.5); and (**c**) impact of inverter power level SCR = 1.5, \(f\)PLL = 10 Hz (conventional controller), and 50 Hz (proposed controller). (1) In Figure 7a, the grid SCR varies from 10 (strong grid condition) to 1 (ultra-weak grid condition); thus, the assessment results can cover all possible situations in practice. As it can be seen, under a lower grid SCR, the critical mode moves to a hazardous area under the conventional control method. Moreover, it covers a larger area, meaning higher sensitivity to grid SCR variation. For example, as shown in Figure 7a, the system equipped with the conventional control system becomes unstable under a SCR lower than 1.37, even for a low PLL bandwidth (\(f_{PLL}=10\) Hz). In contrast, the proposed control method employs a higher PLL bandwidth (\(f_{PLL}=50\) Hz) and provides excellent stability margins under a wide range of SCR variations, from 10 to 1. The maximum damping factor is lower than \(-30\) rad/s under different grid conditions. It is worth mentioning that employing a high PLL bandwidth provided by the proposed control method is essential for good FRT capability and disturbance rejection. (2) Figure 7b investigates the impact of different PLL bandwidths under the SCR = 1.5. The system under the conventional control method becomes unstable for PLL bandwidths higher than 17 Hz, while the proposed control method remains stable for all considered ones. Moreover, the critical mode covers a large area under the conventional ones, meaning higher sensitivity to the PLL bandwidths and severe control loops interaction. In contrast, the impact of the synchronization block on the system's critical mode is considerably reduced under the proposed control method.