# An Optimal Power Control Strategy for Grid-Following Inverters in a Synchronous Frame
## 4 Simulation and Experimental Results
### Robustness and Stability Analysis  
The continuous LQR controller has guaranteed robustness properties such as infinity gain margin and a minimum phase margin of \(60^{\circ}\)[18]. However, these margins become reduced when the LQR controller is discretized. Studies performed in [33] and [34] show that minimum robustness margins for the discrete LQR are reduced depending on many factors such as eigenvalue location, controllability, and observability. To perform robustness and stability analysis on the LQR-ORT controller, the open-loop system \(\mathrm{L}(\mathrm{z})~{}=~{}\mathrm{K}(\mathrm{z})\mathrm{G}(\mathrm{z})\) shown in (25) was used.  
\[\mathrm{L}~{}=~{}\mathrm{K}_{\mathrm{d}}\Big{(}\mathrm{z}\mathrm{I}-\overline {\mathrm{A}}_{\mathrm{dq}}\Big{)}^{-1}\overline{\mathrm{B}}_{\mathrm{Idq}} \tag{25}\]  
Stability margins were calculated using the disk margin method [35]. The disk margin method is used for estimating structured robustness under multiplicative uncertainties for MIMO systems. The disk-based margins are calculated considering all loop interactions and frequencies. Thus, the open-loop system \(\mathrm{L}\) showed a gain margin of 12.03 dB and a phase margin of 52.23\({}^{\circ}\), which is appropriate for inverter control applications.  
From (19), it can be inferred that the stability of the power contribution of \(\mathrm{V}_{\mathrm{dq}}\) depends on the eigenvalue locations of the closed-loop state matrix \(\left(\overline{\mathrm{A}}_{\mathrm{T}}-\overline{\mathrm{B}}_{\mathrm{T}} \mathrm{K}_{\mathrm{d}}\right)\). Thus, the stability of the complete closed-loop system is assessed by analyzing the return difference \(\left(\mathrm{I}+\mathrm{L}\right)^{-1}\), which determines the closed-loop eigenvalues.  
\[\lambda\Big{(}\left(\mathrm{I}+\mathrm{L}\right)^{-1}\Big{)}=~{}\lambda\Big{[} \overline{\mathrm{A}}_{\mathrm{T}}-\overline{\mathrm{B}}_{\mathrm{T}} \mathrm{K}_{\mathrm{d}}\Big{\}} \tag{26}\]  
The operator \(\lambda\{\cdot\}\) refers to the eigenvalue computation.  
To assess robustness against parameter variations, the components in the _LCL_ filter were defined as stochastic elements with a uniform variation of \(\pm 65\%\) around parameters shown in Table 1. Thus, a Monte Carlo stability test was performed with 50 random instances of \(\mathrm{L}\) based on parameter variations using the same controller \(\mathrm{K}_{\mathrm{d}}\)[36]. This way, all interactions between parameter variations were evaluated instead of varying one parameter at the time. The continuous-time closed-loop eigenvalues of the nominal system and the system under component variations using the LQR-ORT controller are shown in Figure 4. The color bar indicates the maximum variation among each set of component values. Thus, the closed-loop eigenvalues tend to become unstable after a component deviation of \(-60\%\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Grid Voltage & \(V\) & 120 Vrms \\ DC bus Voltage & \(V_{dc}\) & 350 V \\ Grid Frequency & \(f\) & 60 Hz \\ Output Inductance & \(L_{c}\) & 1.8 mH \\ Input Inductance & \(L_{i}\) & 1.8 mH \\ Filter Capacitance & \(C\) & 8.8 \(\mathrm{\SIUnitSymbolFace}\) \\ Switching Frequency & \(f_{s}\) & 10 kHz \\ Sampling Period & \(T_{s}\) & 100 \(\mathrm{\SIUnitSymbolFace}\) \\ \hline Error Weighting Matrix & \(Q_{P}\) & \(5000\times I_{2\times 2}\) \\ Input Weighting Matrix & \(R_{P}\) & \(0.2\times I_{2\times 2}\) \\ Inner Integrator Gain & \(K_{i}\) & 1 \\ Outer Integrator Gain & \(K_{s}\) & 5 \\ SOCI gain & \(K_{SOGI}\) & 0.7 \\ PLL Proportional Gain & \(K_{pPLL}\) & 0.28307 \\ PLL Integral Gain & \(K_{ipPLL}\) & 7.5102 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter Specification for the Linear Quadratic Regulator with Optimal Reference Tracking (LQR-ORT) Controller.