# A Systematic Control Design Method with Active Damping Control in Voltage Source Converters
## 2 System Description and Modeling
### Derivation of Proposed Control Method
\begin{array}{l}\dot{\gamma}_{q}=v_{fg}\\ \dot{\delta}=k_{pp}v_{fg}+k_{ip}\gamma_{q}\\ \dot{\gamma}_{dc}=v_{dcref}-v_{dc}\\ \dot{\gamma}_{g}=v_{ref}-v_{fd}\\ \dot{\gamma}_{id}=-\gamma_{dc}-i_{fd}+u_{1d}\\ \gamma_{iq}=-\gamma_{g}-i_{q}+u_{1q}\\ v_{ref,d}=\gamma_{id}+u_{2d}\\ v_{ref,q}=\gamma_{iq}+u_{2q}\\ \dot{x}_{dd}=(-2/T_{d})x_{dd}+v_{ref,d}\\ \dot{x}_{dq}=(-2/T_{d})x_{dq}+v_{ref,q}\\ v_{d}=(4/T_{d})x_{dd}-v_{ref,d}\\ v_{q}=(4/T_{d})x_{dq}-v_{ref,q}\\ i_{fd}={L_{1}}^{-1}\Big{(}L_{1}\omega i_{fq}+v_{d}-v_{fd}\Big{)}\\ i_{fq}={L_{1}}^{-1}\Big{(}-L_{1}\omega_{1}i_{fd}+v_{q}-v_{fd}\Big{)}\\ \dot{v}_{fd}={C_{f}}^{-1}\Big{(}{C_{f}}\omega_{1}v_{fq}+i_{fd}-cos(\delta)i^{ \prime}_{gd}-sin(\delta)i^{\prime}_{gq}\Big{)}\\ \dot{v}_{fg}={C_{f}}^{-1}\Big{(}-{C_{f}}\omega_{1}v_{fq}+i_{fg}+sin(\delta)i^{ \prime}_{gd}-cos(\delta)i^{\prime}_{gq}\Big{)}\\ i^{\prime}_{gd}={L_{g}}^{-1}\Big{(}{L_{g}}\omega_{1}i^{\prime}_{g}+cos(\delta )v_{fd}-sin(\delta)v_{fg}-v^{\prime}_{gd}\Big{)}\\ i^{\prime}_{gq}={L_{g}}^{-1}\Big{(}-{L_{g}}\omega_{1}i^{\prime}_{g}+sin(\delta )v_{fd}+cos(\delta)v_{fq}-v^{\prime}_{gq}\Big{)}\\ \dot{v}_{dc}=\frac{1}{{\dot{\Delta}}_{dc}}\frac{v_{fd}i_{fd}+v_{fg}i_{q}}{v_{ dc}}+\frac{1}{{\dot{\Delta}}_{dc}}i_{dc}\end{array}  \tag{2}\]  
where \(L_{g}=L_{g1}+L_{g2}\), and \(\gamma_{q},\ \gamma_{dc},\ \gamma_{g},\ \gamma_{id},\) and \(\gamma_{iq}\) are integral states for the PLL, DVC, AVC, and current controller (CC), respectively. It is worth noting that the converter is assumed to be lossless, and the delay is approximated by the following transfer function, using first-order Pade approximation [26]:  
\[ \begin{array}{l}\frac{v}{v_{ref}}=e^{(-T_{d}s)}\approx\frac{2-T_{d}s}{2+T_{ d}s}\rightarrow\left\{\begin{array}{l}\dot{x}_{d}=(-2/T_{d})x_{d}+v_{ ref}\\ v=(4/T_{d})x_{d}-v_{ref}\end{array}\right.\end{array}  \tag{3}\]  
Furthermore, two _dq_-frames are considered to include the impact of PLL in the system model and analysis. The global _dq_-frame is aligned with the infinite bus voltage, and the converter or local dq-frame is aligned with the PCC voltage. Signals in the global and the converter _dq_ frames are represented with and without the apostrophe superscript (), respectively [27]. The nonlinear model can be rewritten in the following compact form:  
\[ \begin{array}{l}\dot{x}=f(x,u,d)\end{array}  \tag{4}\]  
Here the system state (\(x\)) and the control and disturbance inputs (\(u\) and \(d\)) are as follows:  
\[\begin{array}{l}x=\left[ \begin{array}{cccccccccccccccc}\gamma_{id}&\gamma _{iq}&x_{dd}&x_{dq}&\gamma_{dc}&\gamma_{g}&\gamma_{q}&\delta&i_{fd}&i_{q}&v_{fd }&v_{fg}&i^{\prime}_{gd}&i^{\prime}_{gq}&\Delta\alpha_{dc}\end{array} \right] ^{T}\\ u=\left[\begin{array}{cccccccccccc}\gamma_{id}&\gamma_{iq}&x_{dd}&x_{dq}& \gamma_{dc}&\gamma  
A linearized state-space model around an equilibrium point (\(x_{e}\), \(u_{e}\), and \(d_{e}\)) is required, which facilitates designing a state feedback control for a grid-connected VSC. The following equations can be used to calculate the linearized model:  
\[\dot{x}=Ax+Bu+Dd \tag{6}\]  
\[A=\frac{\partial f}{\partial x}\Big{|}x=x_{e},u=u_{e},d=d_{e} \tag{7}\]  
\[B=\frac{\partial f}{\partial u}\Big{|}x=x_{e},u=u_{e},d=d_{e}\] \[D=\frac{\partial f}{\partial d}\Big{|}x=x_{e},u=u_{e},d=d_{e}\]