## 5 Simulation Results and Discussion  
Using the robust control law presented in the previous sections for synthesizing gains of the distributed energy system, this section groups results from simulations tests. The numerical values of electrical components, LCL filter and resistances and capacitances parameters associated with each load situation for design purposes of the DER controller are listed in Table 1. The reference signal to be followed by the output voltage is sinusoidal with an RMS value of 127 V and frequency of 60 Hz. The inverter switching frequency is 21.6 kHz, 360 times greater than the frequency of the reference signal. The simulation environment is given by Matlab/Simulink for the DER system with linear and non-linear test loads, including the generation of PWM signal for inverter switching in a half bridge and DER control loop, with a controller properly discretized by zero-order hold with a sampling frequency of 21.6 kHz as shown in Figure 4. The LMI parameters chosen in this paper to improve the dynamic response and stability in steady-state are shown in Table 2.  
Figure 4: General scheme of the distributed energy resource (DER) system with the proposed control. Figures 5 and 6 show the voltage and current curve under load steps in the period of two seconds considering additional load steps of 20% at \(t=0.2\) s and of 80% at \(t=0.6\) s. Following by a subtractive step of \(t=80\%\) at \(t=1.4\) s and, finally, a subtractive step of 20% at \(t=1.8\) s. In Figure 5a, a cycle (16.67 ms) of the voltage (127 V RMS) and output current (Figure 6a) of the DER system with 100% linear load. In this figure it is evident how the voltage curve fits almost perfectly over the reference curve. Similarly for the current curve, which has an almost sinusoidal appearance with and without controller with fundamental frequency and a harmonics compensator (HC). Figures 5b and 6b show voltage and output current curves from the DER system with 100% non-linear load with HC, differently as for linear load, in non-linear load, the deformation of the current curve is more pronounced, which is caused by this type of load. Figures 5c and 6c show the simulation results for voltage and current without HC in the controller; from these figures we can seen that the non-linear load creates a voltage deviation voltage that increases proportionally to the connected step load. In the linear and non-linear load with HC, the shape of the current is slightly deformed compared with 100% non-linear load without HC. However, both non-linear loads present small transients and overshoots for additive and subtractive load steps contrary to the linear load, which has voltage and current responses that are perfectly sinusoidal. Figure 7 shows the values of the empty RMS voltage, with 20% and 100% linear load, and with 20% and 100% non-linear load, respectively, of the DER system. In this study, we take account of the steady state voltage without the transients of connected and disconnected loads. For the linear load, we observe that the RMS value of the voltage remains satisfying the most stringent standard, which is also the case for the non-linear load with the controller use HC. However, for the non-linear load without HC, when the addition of 100% non-linear load occurs, the RMS value of the voltage slightly exceeds (2%) the limit of the IEEE 944 standard. Figure 8 shows the THD of the DER system output voltage for all additive and subtractive steps with different periods for linear and non-linear loads. In the cases where the PR controllers are used with harmonic compensators, the THD with 25% and 100% linear/non-linear loads varying meet the IEEE and IEC standards for the proposed LMI approaches, as shown in Figure 8a,b. However, when the PR controllers are used without harmonic