*4.2. Case Study 2*

Another criterion applies to vindicate the achievement of the derived controller that is tracking the voltage of the DC link under constant and changing irradiation levels of the PV source. After analysis, it was observed that the HHO-AFOPID again satisfies the robustness. In Figure 6a, it shows that the proposed controller is the best one to meet the aim of the constant irradiation condition, and in Figure 6b, satisfactory performance is observed under varying irradiation conditions.

**Figure 6.** Features of the voltage of the dc link (**a**) under constant irradiation level and (**b**) under changing irradiation levels.

#### *4.3. Case Study 3*

Additionally, the system is evaluated within partial shading circumstances. When compared to other controllers, the suggested controller performs better in this instance and extracts 100 kW from the system. In this instance, a 4 × 1 array configuration with a right-skewed half-plane MPP position is chosen. This is carried out to confirm that the proposed HHO-AFOPID MPPT control topology is repeatable and can successfully manage the partial shading condition. In addition to this, another goal is to show how the methodology with skewed global MPP varies in performance on the characteristics graph. The power to validate competing methodologies is displayed in the comparison in Figure 7.

**Figure 7.** Experiment under the partial shading condition.

The methodical details are revealed in Table 4. Due to the inclusion of the method of the fractional order, it is clear from the analytical observation that HHO-AFOPID exhibits the lowest fitness function and the shortest time of convergence, which highlights its superior performance index and robustness compared to other control methodologies


**Table 4.** Comparative performance analysis of different controllers.

#### *4.4. Case Study 4*

Power quality was also assessed in terms of the voltage variation, THD, and frequency for verifying the controller's effectiveness. HHO-AFOPID exhibits the best outcome and upholds its reputation in this instance as well.

#### 4.4.1. Voltage Deviation

Power quality is one of the main issues of grid-connected systems. The root mean square (RMS) value of the voltage can be expressed as an equation based on the peak value and sample/cycle (61) [20]:

$$v\_i^{rms} = \sqrt{\frac{1}{M} \sum\_{k=1}^{i+M-1} v\_k^2} \tag{61}$$

*M* = sample/cycle of the initial;

*vk* = *k*-th specimen of the registered potential waveform;

*vrms <sup>i</sup>* = *i*-th specimen of the measured r.m.s voltage.

The value of the root mean square voltage lags behind the phase voltage by (*M* − 1) cycles since there are *M* cycles per second.
