**6. Conclusions**

The diagnostic condition of mechanical systems is that the training data are sufficient and the samples of different categories in the training data have a balanced distribution. In the whole life cycle of a high-voltage circuit breaker system, it is in normal state most of the time, and the number of normal samples in the actually monitored signal will be more than the number of fault samples, resulting in the imbalance of training data and unknown distribution. Thus, for the problem of large error in the residual lifetime prognosis of power systems, a novel residual lifetime prognosis model based on HOHSMM and polynomial fitting was proposed. Based on HSMM, an order reduction method and composite node mechanism of HOHSMM based on permutation were proposed. The order reduction method of the permutation and combination model is simple and intuitive and uses the definition of the high-order hidden Markov group model. The high-order model can be transformed into the corresponding first-order model by changing the observation angle, and the solution of the three problems of the low-order model can be used in the high-order complex model. The intelligent optimization algorithm group can be used to replace the EM algorithm to estimate the parameters and optimize the structure of the proposed model, and the simplification of the topology of the high-order model by the intelligent optimization algorithm can be realized. The complex dependency information in the high-order model is transferred to the deformed parameter group. It effectively simplifies the model and provides a new idea for the study of this kind of model. Finally, a case was studied to verify the proposed model. From the experimental results, the comparison between the proposed model and HSMM showed several advantages of the proposed model, indicating that the remaining life prediction based on polynomial fitting has better performance for the health prognosis problem of the power system.

**Author Contributions:** Conceptualization, Q.L., D.L. and W.L.; methodology, T.X.; investigation, Q.L. and D.L.; resources, J.L.; writing—original draft preparation, Q.L.; writing—review and editing, D.L.; visualization, W.L.; supervision, T.X.; funding acquisition, Q.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China(Nos. 71632008, 71840003, and 51875359), the Natural Science Foundation of Shanghai (No. 19ZR1435600 and 20ZR1428600), the Humanity and Social Science Planning Foundation of the Ministry of Education of China (No. 20YJAZH068), and the Science and Technology Development Project of the University of Shanghai for Technology and Science (No. 2020KJFZ038).

**Acknowledgments:** The authors are indebted to the reviewers and the editors for their constructive comments which greatly improved the contents and exposition of this paper.

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