**6. Conclusions**

Here, we analyzed the shortcomings of the basic HHO algorithm and applied the chaotic mapping population initialization, adaptive weighting, variable spiral position update and optimal neighborhood disturbance mechanisms to the classical HHO algorithm, in which the Gauss chaotic mapping population initialization increased the coverage of the solution space by the initial solution of the algorithm, the adaptive weighting mechanism sped up the movement of Harris hawk populations to the optimal solution, and the variable spiral position update increased the ability of Harris hawk populations. The optimal neighborhood disturbance mechanism helped the improved algorithm to increase the algorithm's global search capability and avoided premature maturity. To verify the optimal performance of the four strategies, the experiments were separated into two groups.

First, seven commonly used chaotic mappings were selected for the population initialization of the HHO algorithm, including Sinusoidal, Tent, Kent, Cubic, Logistic, Gauss, and Circle mappings. The HHO algorithm's performance after population initialization of each of these seven mappings was evaluated. The HHO algorithm's performance after population initialization of Gauss mapping was significantly better than that of the HHO algorithm after population initialization of other mappings in terms of solution accuracy. Second, based on the results of the first set of experiments, the Gauss mapping was used for population initialization, and adaptive weights, variable spiral position update, and optimal neighborhood disturbance mechanisms were introduced into the algorithm after population initialization. Next, CSHHO was compared with other classical algorithms including WOA, SCA, CSO and advanced algorithms including GCHHO and DEPSOASS and Improved GSA and DGOBLFOA based on 23 classical test functions and the means and standard deviations of all algorithms were analyzed. Subsequently, each algorithm's

performance was evaluated comprehensively using Friedman's test and the Bonferroni– Holm corrected Wilcoxon signed-rank test with 5% confidence level, where numerical analysis concluded that CSHHO outperformed the other algorithms. In detail, analyzing the experimental results of this work, in the population initialization phase, Gauss chaos mapping had the best results in F2, F6, F12, F17, and F23 test functions, and comparing the results of the remaining six chaotic mappings, Gauss chaos mapping obtained the most optimal solutions; CSHHO algorithm outperformed HHO in 17 benchmark functions out of 23 classical test functions, outperformed WOA in 21 results, SCA in 23 results, and CSO in 22 experiments. It outperformed GCHHO in 9 results. Meanwhile, in the statistical experiments of advanced meta-heuristic and classical meta-heuristic, the ARVs obtained by CSHHO were 1.93 and 2.57, respectively, which were lower than the values obtained by other pairwise meta-heuristics in the same group of experiments. Additionally, dimensional scalability tests were conducted for CSHHO on the IEEECEC2017 dataset, including 50 and 100 dimensions, and the results showed that the improved optimizer effectively handled high-dimensional data with good stability. Meanwhile, in the statistical experiments of advanced meta-heuristic and classical meta-heuristic, CSHHO obtained ARVs of 1.70 and 2.57, respectively, which were lower than the values obtained by other meta-heuristic algorithms in the same set of experiments. Furthermore, dimensional scalability tests were conducted for CSHHO on the IEEE CEC 2017 dataset, including 50 and 100 dimensions, and the results showed that the improved optimizer effectively handled high-dimensional data with excellent stability.

Here, the CSHHO algorithm was also applied to the engineering problem of reactive power output modeling of the synchronous condenser. In view of the defects of the many calculations and low accuracy of the traditional reactive power output modeling method of the synchronous condenser, CSHHO-LSSVM was used to model the reactive power output of the synchronous condenser based on the advantages of LSSVM, which was not easy to fall into local minimum and had strong generalization ability, and CSHHO had high search accuracy and strong global search ability. The excitation current and excitation voltage of the synchronous condenser were used as the input of the LSSVM model, and the reactive power and system voltage were used as the LSSVM model's output. CSHHO was used to find the optimal values of the penalty parameter, kernel function parameter, and loss function parameter of LSSVM. The experiment showed that the CSHHHO-LSSVM model had better accuracy and better regression fitting performance compared with LSSVM.

In future work, we will try to improve the convergence speed and search accuracy of the algorithm and balance the exploration and exploitation phases of the algorithm to obtain better search performance. Additionally, the next step will be to investigate how CSHHO can be used to solve multi-objective optimization problems. In addition, CSHHO can also be used for evolutionary ML, such as extreme learning machines and parameter tuning of convolution neural networks. Other problems include grid scheduling and 3D multi-objective tracking.

**Author Contributions:** Methodology, C.W.; resources, S.J.; funding acquisition, S.J., R.G., Y.L. and Q.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is supported by the National Natural Science Foundation of China (No. 61871318), the Shaanxi Provincial Key Research and Development Project (No. 2019GY-099) and Open project of Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing (No. 2020CP10).

**Institutional Review Board Statement:** The study did not involve human and animals.

**Informed Consent Statement:** The study did not involve human and animals.

**Data Availability Statement:** In the paper, all the data generation information has been given in detail in the related chapter.

**Acknowledgments:** The author thanks the referees for detailed and constructive criticism of the original manuscript.

#### **Conflicts of Interest:** The authors declare no conflict of interest.
