# An Observer-Based Current Sensor-Less Control Scheme for Grid-Following Converters
## 5 Stability Analysis of the GF-VSC with the Proposed Observer
\begin{array}{c}\frac{dX(t)}{dt}=A_{2}X(t)+B_{u2}B_{new-X}X(t)+B_{u2}B_{new -\widehat{X}}\widehat{X}(t)+B_{u2}B_{new-Z}Z(t)\\ \frac{dX(t)}{dt}=A_{2}\widehat{X(t)}+B_{u2}B_{new-X}X(t)+B_{u2}B_{new-\widehat {X}}\widehat{X}(t)+B_{u2}B_{new-Z}Z(t)\\ +L\ C\Big{(}X(t)-\widehat{X}(t)\Big{)}\\ \frac{dZ(t)}{dt}=\bigg{(}K_{X}=\begin{bmatrix}0&0&-k_{I-Wc}\\ 0&0&0\end{bmatrix}\bigg{)}X(t)+\bigg{(}K_{\widehat{X}}=\begin{bmatrix}0&0&0\\ 0&-1.5v_{gd}k_{I-Q}&0\end{bmatrix}\bigg{)}\widehat{X}(t)\end{array}  \tag{33}\]  
The closed form of (31) is written in (34):  
\[\frac{d}{dt} \begin{bmatrix}X(t)\\ Z(t)\\ \widehat{X(t)}\end{bmatrix} = \begin{bmatrix}A_{2}+B_{u2}B_{new-X}&B_{u2}B_{new- z}&B_{u2}B_{new-\widehat{X}}\\ K_{X}&0&K_{\widehat{X}}\\ LC+B_{u2}B_{new-X}&B_{u2}B_{new-z}&A_{2}+B_{u2}B_{new-\widehat{X}}-LC\end{bmatrix} \begin{bmatrix}X(t)\\ Z(t)\\ \widehat{X(t)}\end{bmatrix}  \tag{34}\]  
The stability of the overall control system is dependent on the eigenvalues of the state matrix of (34). Given that \(A_{2}\) and \(B_{u2}\) are dependent on the operating points of GFL inverter (\(P_{dc}\), \(Q^{ref}\)), the eigenvalues of the state matrix of (34) should be analyzed in a full range of operating points. In addition, the eigenvalues of the conventional control system without an observer are presented for comparison purposes. In the case study and numerical analysis, a sample GFL-inverter, with the characteristic parameters given in Table 1, is used in the following. Figure 4 shows the poles of the GFL inverter with a conventional control system without using an observer in different operating points. From the figure, it can be seen that the system is a fifth-order system, and increasing the \(P_{dc}\) would shift the poles to the more negative values. In addition, \(Q^{ref}\) does not affect the poles, and hence, the poles are shown only for \(Q^{ref}=0\). Furthermore, all of the poles have a damping coefficient of \(\zeta=1\) regardless of the operating points. In the design of matrix \(L\), it is possible to place the observer poles of \(OBS_{poles}=Eigen(A_{2}|_{P_{min}}-LC)=[P_{1}\ P_{2}\ P_{3}]\) at any of the desired values. The negative poles with higher values provide a higher bandwidth and increased observer speed, which consequently decreases the estimation error in the presence of uncertainty in the model of the system. However, it may deteriorate the stability of the closed loop system in the presence of uncertainty. To further evaluate the effect of the observer poles on the stability of the overall system in various operating points, \(\left(P_{dc},Q^{ref}\right)\) are illustrated in Figure 5. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Nominal Power of VSC & 10 kW \\ Grid Frequency & 50 Hz \\ \(L_{f}\) & 8.6 mH \\ DC-link Capacitor & 200 uF \\ Grid impedance (\(Z_{s}\)) & \(0.1+j1\) \\ Grid Voltage & 380 V \\ \(K_{P-Wc}\) & 232/\(v_{gd}\) \\ \(K_{I-Wc}\) & 67/\(v_{gd}\) \\ \(K_{P-Q}\) & 0 \\ \(K_{I-Q}\) & 33/\(v_{gd}\) \\ \(K_{c}\) & 2000 \\ Observer Matrix L & \(1\times 10^{4}\)\([-3.08\ 6.76\ 0.065]^{\mathrm{T}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter Values of the studied system. Figure 4: The poles of the overall control system with conventional controller in different operating points without observer. Four cases for the poles of the observer are examined. Suppose that the four-observer matrix is designed in such a way that the poles of the observer are as follows. \[ \begin{array}{l}\text{Observer 1}:\;OBS_{Poles}=-2.5[1.1K_{c}\;1K_{c}\;0.9K_{c}]\\ \text{Observer 2}:\;OBS_{Poles}=-1.7[1.1K_{c}\;1K_{c}\;0.9K_{c}]\\ \text{Observer 3}:\;OBS_{Poles}=-1[1.1K_{c}\;1K_{c}\;0.9K_{c}]\\ \text{Observer 4}:\;OBS_{Poles}=-0.5[1.1K_{c}\;1K_{c}\;0.9K_{c}]\\ \end{array}  \tag{35}\]  
In this study, the position of the poles ([\(P_{1}\), \(P_{2}\), \(P_{3}\)]) are chosen depending on the current control-loop bandwidth (\(K_{c}\)), and \(P_{1}\), \(P_{2}\), and \(P_{3}\) are considered similar to each other.