# A Novel Hybrid Harris Hawk-Arithmetic Optimization Algorithm for Industrial Wireless Mesh Networks
## 4 Results and Discussion
### Performance Evaluation on Pressure Process Control  
In the numerical analysis, the comparison was carried out for the process rise time (\(t_{r}\)), settling time before (\(t_{s1}\)) disturbance, overshoot (\(\%OS\)), and settling time after (\(t_{s2}\)) disturbance injection. In order to evaluate the ability of the FOPPI to decrease the stochastic disturbance and to track the set-point effectively, an external disturbance of 35% was injected at 100 s in the process feedback loop. Table 9 shows the quantitative analysis of the optimized FOPPI controller using different optimization techniques, and the comparative analysis is shown in Figure 10. Here, controller parameters \(K_{p}\), \(T_{i}\) and \(\lambda\) were obtained  
Figure 9: WMN network connectivity and its coverage area for various algorithms. (**a**) Initial Network; (**b**) AOA Optimized WMN; (**c**) MFO Optimized WMN; (**d**) SCA Optimized WMN; (**e**) GWO Optimized WMN; (**f**) WOA Optimized WMN; (**g**) HHO Optimized WMN; (**h**) HHAOA Optimized WMN. based on the ITAE value and integral time was obtained by \(T_{i}=\frac{K_{p}}{K_{i}^{*}}\), where \(K_{i}\) is the integral gain. The numerical results from the table show that the MFO has the fastest rise time of 0.7552 s, followed by the AOA, HHO, WOA, and SCA with values of 0.8404, 0.8411, and 0.9381 s, respectively. Here the GWO has the slowest rise time of 5.1078 s which is 576.351% slower performance than the fastest MFO. The proposed HHAOA optimized FOPPI had the second-last rise time of 1.4504 s which is almost two times slower than the MFO. During the settling time before disturbance (\(t_{s1}\)), even with the slowest rise time, the HHAOA-optimized FOPPI managed to settle faster at 37.2265 s which is 29.7129 s faster than the slowest settled SCA with the settling at 66.9394 s. The result shows that the proposed HHAOA has 79.816% faster performance, which is twice as fast as SCA. The HHO comes second fastest, settling at 49.2073 s, followed by GWO, WOA, MFO, and AOA with the respective values of 52.9052, 54.2561, 57.5289, and 59.6059 s. While observing the settling time \(t_{s2}\) after disturbance injection performance, once again, the  
Figure 10: Set-point tracking and disturbance rejection analysis for optimal FOPPI controller. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Optimization Type** & \(K_{p}\) & \(K_{i}\) & \(\Lambda\) & \(t_{r}\) & \(t_{s1}\) & \(t_{s1}\) & \%**OS** \\ \hline AOA & 1.207 & 0.836 & 0.97 & 0.8404 & 59.6059 & 142.2151 & 2.8723 \\ MFO & 2.713 & 1.183 & 0.98 & 0.7552 & 57.5289 & 140.1284 & 18.9260 \\ SCA & 1.881 & 1.062 & 0.96 & 1.0371 & 66.9394 & 151.0140 & 23.3440 \\ GWO & 1.672 & 0.714 & 0.96 & 51.078 & 52.9052 & 135.1132 & 2.5131 \\ WOA & 2.015 & 0.615 & 0.99 & 0.9381 & 54.2561 & 137.2006 & 2.6118 \\ HHO & 2.427 & 1.031 & 0.99 & 0.8411 & 49.2073 & 130.0259 & 4.8653 \\ HHAOA & 2.359 & 0.901 & 0.98 & 1.4504 & 37.2265 & 119.2158 & 4.7061 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Experimental quantitative analysis of various optimized FOPPI controller on the pressure process. HHAOA settled faster at 119.2158, which is 34.7982 s ahead of SCA (151.0140s). The same settling time trend is also observed in this, with the HHO settling at 130.0259 s followed by GWO settling at 135.1132 s. The second last rank is for AOA settling at 142.2151 s. In the peak overshoot performance, the HHAOA has 4.7601%, corresponding to a 396.037% reduction of the value, while comparing the SCA at 23.3340%. Amazingly, the GWO has the least peak overshoot value of 2.5131%, followed by WOA, AOA, HHO, and HHAOA, respectively. In this numerical performance analysis, in most of the cases, the Friedman ranking order is observed. Based on the comparison results, it is evident that the FOPPI controller optimized by HHAOA is more efficient in producing a stable control signal and rejecting disturbances. However, it is worth noting that the HHAOA may have a slower rise time when tracking the initial set-point, which is noticeable.