# An Observer-Based Current Sensor-Less Control Scheme for Grid-Following Converters
## 5 Stability Analysis of the GF-VSC with the Proposed Observer  
The stable operation with a high-enough stability margin is the preliminary requirement of a given control system. In this section, the stability of the GFL-inverter with the proposed observer and control structure is analyzed, using the eigenvalue theory. To cover all of the dynamics of the system, the dynamic model of the grid-side current and the DC-link voltage of (14), the observer dynamic of (25), and the controllers of (21) and (23) are integrated and used to obtain the overall system model. It should be mentioned that, for stability analysis by eigen values, a linearized model of the overall system is needed. A linearized state-space model of the primary system was presented in (15), also, the linearized model of the observer in (25) is given by (27):  
\[\frac{\widehat{dX(t)}}{dt} =A_{2}\widehat{X(t)}+B_{u2} \begin{bmatrix}v_{td}\\ v_{tq}\end{bmatrix} +B_{d} \begin{bmatrix}v_{gd}\\ v_{gq}\\ P_{dc}\end{bmatrix} +L\Big{(}CX(t)-C\widehat{X(t)}\Big{)} \tag{27}\]  
Figure 3: Complete control block diagram of a GFL-inverter including (i) the proposed observer control scheme; (ii) DCLV controller; (iii) reactive power controller; (iv) current control loop; (v) measured DC-link voltage and grid-side voltages. Therefore, it is only needed to extract the state-space model of controllers, to complete the state-space model of the overall system shown in Figure 3. According to the block diagram in Figure 3, the two state variables of \(z_{1}\), \(z_{2}\) are defined to represent the controller in the state-space equations as follows:  
\[ \begin{array}{c}\frac{dz_{1}}{dt}=k_{I-Wc}\Big{(}W_{c}^{ref}-W_{c}\Big{)}\\ \frac{dz_{2}}{dt}=k_{I-Q}\Big{(}Q^{ref}-Q\Big{)}=k_{I-Q}\Big{(}Q^{ref}-1.5v_{gd }\hat{d}_{q}\Big{)}\end{array}  \tag{28}\]  
Furthermore, based on the block diagram of the system in Figure 3, the control variables of \(v_{td}\) and \(v_{tq}\) are as given in (29):  
\[ \begin{array}{c}v_{td}=v_{gd}+L_{f}v\hat{u_{q}}+R_{f}\hat{d_{q}}+L_{f}k_{C} \Big{(}\hat{u_{q}^{ref}}-\hat{i_{q}}\Big{)}\\ v_{tq}=v_{gq}-L_{f}v\hat{u_{d}}+R_{f}\hat{i_{q}}+L_{f}k_{C}\Big{(}\hat{u_{q}^{ ref}}-\hat{i_{q}}\Big{)}\end{array}  \tag{29}\]  
In which, the current references of \(\hat{u_{d}^{ref}}\) and \(\hat{u_{q}^{ref}}\) are given in (30):  
\[ \begin{array}{c}\hat{u_{d}^{ref}}=\frac{P_{L_{f}}}{1.5v_{gd}}+z_{1}+k_{P-Wc }\Big{(}W_{c}^{ref}-W_{c}\Big{)}\\ \hat{u_{q}^{ref}}=\frac{Q^{ref}}{1.5v_{gd}}+z_{2}+k_{P-Q}\Big{(}Q^{ref}-1.5v_{gd }\hat{i_{q}}\Big{)}\end{array}  \tag{30}\]  
Substituting (30) into (29), and with some simplification, the result is as given in (31):  
\[ \begin{array}{c}\begin{bmatrix}v_{td}\\ v_{tq}\end{bmatrix}=\begin{bmatrix}v_{gd}\\ v_{gd}\end{bmatrix}+B_{new-x}X+B_{new-\widehat{x}}\widehat{X}+B_{new-z}Z\end{array}  \tag{31}\]  
where,  
\[ \begin{array}{c}B_{new-x}=\begin{bmatrix}0&0&-L_{f}K_{C}K_{P-Wc}\\ 0&0&0\end{bmatrix}\\ B_{new-\widehat{x}}=\begin{bmatrix}-L_{f}K_{C}&L_{f}w&0\\ -L_{f}w&L_{f}K_{C}\Big{(}-1-1.5v_{gd}K_{P-Q}\Big{)}&0\end{bmatrix}\\ B_{new-z}=\begin{bmatrix}L_{f}K_{C}&0\\ 0&L_{f}K_{C}\end{bmatrix}\end{array}  \tag{32}\]  
It is worth noting that the inputs of the system \(W_{c}^{ref}\), \(Q^{ref}\) and \(P_{dc}\) do not affect the stability of the system, and they are eliminated in (31) for the stability analysis. Applying the control variables of (31) into the linearized model of the system and observer in (15) and (27), then by merging with the controller-state equations of (28), the resultant linearized model of the overall system is given in (33):  
\[