*6.2. Scenario 2*

This scenario investigates the ability of the SCES system using the DPC model to compensate for the power oscillations in the test system. Therefore, two SCES systems are considered, located at buses 1 and 5 (see Figure 2). The first SCES system relieves the power oscillation introduced by the wind power generator at bus 2. Here, we assume that the SCES system must keep active power at 28 kW and supply all the requirements of the reactive power of the generator. In contrast, the second SCES system compensates for the power oscillations provided by DL2 demands at bus 4, maintaining its active and reactive power at 30 kW and zero, respectively.

Figures 6 and 7 illustrate the dynamic behaviors at the DC side of the first and second SCES systems, respectively. In addition, their respective active and reactive powers are also plotted.

**Figure 6.** Dynamic behavior of the first SCES system for S2: (**a**) supercapacitor voltage; (**b**) active power provided; and (**c**) reactive power absorbed.

Observe in Figure 6 that both controllers maintain the control objectives (*p* = 30 kW and *q* = 0 kvar). However, the PI-PBC method continues presenting a better performance with lower ripples for the active and reactive power. Moreover, IDA-PBC has a steady-state error for active power of around 25 kW.

Note in Figure 7 that the proposed controller continues to show an enhanced response of active power, without steady-state error, as presented with the IDA-PBC. This behavior occurs because the IDA-PBC works as a proportional control. In contrast, the proposed controller includes an integral action that removes this error.

**Figure 7.** Dynamic behavior of the second SCES system for S2: (**a**) supercapacitor voltage; (**b**) active power provided; and (**c**) reactive power absorbed.

Complementary Analysis

Mean absolute error (MAE) and integral of time multiply absolute error (ITAE) (for the active and reactive power), and total harmonic distortion (THD) (for the AC currents) are used to quantify the performance of the controllers. Table 1 depicts these indexes for each scenario analyzed.

In Table 1, both controllers have a notable low MAE and ITAE. Nevertheless, the PI-PBC approach performs better for tracking power references than the IDA-PBC approach. This finding is supported by the reduction of MAE*p* and MAE*q* by 12.1% and 21.8% in the worst-case scenario (see first and the second column in Table 1), respectively. While ITAE*p* and ITAE*q* were reduced by 16.9% and 21.25% in the worst-case scenario, respectively.


**Table 1.** Performance Indexes.

In Table 1, it can be observed that both controllers meet the THD limits for power electronic converters established in Standard IEEE-1547 [21] even though the proposed controller has lower THD than the IDA-PBC approach. This entails that the PI-PBC approach presents better wave quality and lower losses.

The robustness of the proposed controller is investigated by applying a sensitive analysis for filter parameters. We assume that there are RL-filter mismatches with a variation of ±50% and ±40% for R and L parameters, respectively, when active power must be kept in 10 kW. For this test, we generate 100 random mismatches with uniform distribution. Figure 8 depicts the error mean value of the active power Δ *Pmean* against plant-model mismatches. Observe in this figure, that the IDA-PBC approach has a greater variation for Δ *Pmean* than the PI-PBC approach. This demonstrates that the proposed controller presents a better performance when there is a plant-model mismatch. In Figure 8, it can also be noted that the resistance variations do not influence over the controllers performance, while inductance variations do affect the performance of the controllers. This effect tends to be linear and is greater when the IDA-PBC approach is implemented.

**Remark 6.** *The proposed controller does not show a remarkable difference in performance according to the IDA-PBC approach, which is easy to implement. This is because it only needs proportional-integral action and does not depend on the system parameters, while the IDA-PBC approach requires the system parameters and derivative calculation from the references to be applied.*
