# A Systematic Control Design Method with Active Damping Control in Voltage Source Converters
## 2 System Description and Modeling
### Derivation of Proposed Control Method  
The main aims of the grid-side VSC are to inject the produced power by renewable energy systems into the power grid with the sinusoidal current. However, the voltage compensation of the PCC is also necessary when connected to a weak grid. The conventional control systems of grid-side VSC are shown in Figure 2. The conventional one consists of five main parts, including the dc-link voltage controller (DVC), AC voltage magnitude controller (AVC), current controller (CC), voltage feedforward controller (VFF), and a synchronous reference frame phase-locked loop (SRF-PLL). It is worth remarking that they all (except for VFF) employ a proportional-integral (PI) controller to ensure good dynamic response and zero steady-state tracking error in the synchronous reference frame. Moreover, grid-side current feedback and voltage feedforward control through a high-pass filter are provided to damp the LCL filter resonances and better disturbance  
Figure 1: Single line diagram of an LCL-filtered grid-connected three-phase voltage source converter (VSC). rejection. In this work, a first-order high-pass filter is considered for VFF, which is expressed as follows [25]:  
\[VFF=\frac{k_{a}s}{s+\omega_{a}} \tag{1}\]  
As it can be seen, the conventional structure employs many control loops with at least eleven control gains which should be designed carefully to provide a proper response and stability margins and also avoid control loops' interactions. Moreover, there are many states in the system that can be appropriately utilized. It seems proper that employing them may remarkably increase the system's robustness and performance, as shown in Figure 3. Here, \(K_{f}\) and \(x\) are the active damping gain matrix and system state's vector, respectively. However, it may add extra feedback gains and increase complexity to calculate the optimal control gains. The control structure in Figure 3 can be simplified as follows. Since \(k_{pd}\) and \(K_{fl}\)(1,15) are multiplied by \(\Delta v_{dc}\) and provide the same damping, one can be ignored to reduce the number of tuning parameters. The same can also be concluded for CC, AVC, and integral gains. The resultant simplified control structure is depicted in Figure 4.  
By comparing Figures 3 and 4, it can be seen that duplicate coefficients are ignored, and all control gains of different control loops and state feedback-based active damping are lumped in the control gain matrix (\(K_{f}\)). Therefore, the proposed control method can utilize the maximum capability of the circuit and control systems and employs all available states (\(x\)) to improve the system's response. Moreover, a high-pass filter-based VFF, which is necessary to damp LCL filter responses under weak grid conditions, is also replaced with the simpler constant feedback controller. It is shown here that, although a dynamic controller is not considered for the VFF, the proposed control structure can still provide better robustness under weak grid conditions, as another noteworthy feature. The remaining question is how the control gain matrix can be obtained and based on which criteria. In the following, at first, a proper representation of the system equations is provided. Then, based on the proposed system model, a design procedure is presented to obtain the control gain matrix optimally and simply. Figure 2: The conventional control method. VFF, voltage feedforward controller; DVC, dc-link voltage controller; AVC, ac voltage-magnitude controller; CC, current controller; PLL, phase-locked loop. Figure 4: The proposed control method. Figure 3: The proposed solution is to include state-feedback-based active damping in the inner and outer control loops. Proposed Nonlinear and Linearized State-Space Model of a Grid-Connected VSC for Optimal Control System Design  
Considering the active power-balance equation, Park transformation, and control delay in the digital implementation in Figures 1 and 4, the proposed state-space nonlinear model of the overall system is determined as follows:  
\[