# An Optimal Power Control Strategy for Grid-Following Inverters in a Synchronous Frame
## 4 Simulation and Experimental Results
### Experimental Results  
Experimental results were obtained using the setup shown in Figure 8. This setup uses four Danfoss 2.2 kVA inverters with _LCL_ filters and four sensor boxes with voltage and current LEM sensors. These four inverters may be dynamically connected to the AC bus. The AC bus may be also dynamically connected or disconnected from the main grid using solid-state relays handled with a PLC. To implement control algorithms, a dSPACE 1006 simulator was used to read signals from the sensor boxes and send the 10 kHz PWM pulses via optical fiber. Experimental results for the LQR-ORT and the PR-Droop controller are shown in Figure 9. First, the inverter started without injecting energy to the main grid. A step reference of 300 W and 200 Var was set at t\({}_{1}=~{}0.35\) s and t\({}_{2}=~{}1.05\) s, respectively. The LQR-ORT shows lower settling time, less noise, and a smoother transient response under a step reference compared to the PR-Droop controller. In addition, the LQR-ORT controller shows a negligible steady-state error compared to the PR-Droop controller. In addition, at t\({}_{1}\) and t\({}_{2}\), the LQR-ORT controller shows better decoupling compared to the PR-Droop controller. The decoupling of the LQR-ORT controller is produced by the _dq_ transformation and the intrinsic robustness properties of the LQR controllers. In addition, the PR-Droop controller has less decoupling because droop \((\omega-V)\) curves are not completely independent [5]. The normalized cost was calculated using (12) divided by the highest achieved cost during the simulation. The normalized cost of the PR-Droop controller is about 2.92 times higher compared to the LQR-ORT controller. This implies that the LQR-ORT controller spends less energy tracking reference signals. To evaluate the performance of the LQR-ORT controller under unbalanced and nonlinear loads, the scheme shown in Figure 10 was implemented. This scheme used a solid-state demultiplexer symbolized as \"switch 2\" that selected between linear balanced, linear unbalanced, and nonlinear loads. The linear load was selected to be an RL series circuit with R = 171.42 \(\Omega\) and L = 0.4571 H. The linear unbalanced load was selected as an RL series circuit with R\({}_{\text{u}}\) = {171.42, 200, 600} \(\Omega\) and L = 0.4571 H. Finally, for the unbalanced load, a three-phase noncontrolled rectifier with an RLC parallel circuit with R\({}_{\text{NL}}\) = 1200 \(\Omega\), C\({}_{\text{NL}}\) = 100 \(\mu\)F, and L\({}_{\text{NL}}\) = 2 mH was used. Figure 8: Photo of the experimental setup [37]. The results of this experiment are shown in Figure 11. First, the inverter injected 400 W and 400 Var to the main grid with the linear balanced load connected to the output of the inverter (Figure 11a,b). At this time, the output current of the inverter had a Total Harmonic Distortion (THD) of 10.8% and the balanced linear load had a THD of 6.71%. At \(t=0.7\) s, the unbalanced load was connected. Figure 10: Experimental scheme for the evaluation of the performance of the LQR-ORT controller under unbalanced and nonlinear loads. Figure 9: Experimental results comparing the LQR-ORT controller performance with the PR-Droop controller using the same reference signals. (**a**) LQR-ORT response. (**b**) PR-Droop response (**c**) LQ-Cost values. This generated a small disturbance in the active power, which was restored in less than 0.2 s. In addition, the THD of the output current and the load decreased to 10.7% and 6.55%, respectively (Figure 11c). The low disturbance in the injected power and the decrease in the THD indicate that the LQR-ORT controller is robust under unbalanced loads. Finally, at \(t=2.5\) s, the nonlinear load was connected. This generated a slightly larger disturbance in the active power, which was restored in less than 0.2 s as well. In addition, the THD of the output current increased to 10.98% and the load current THD increased to 126.02%. (Figure 11d).