**5. Conclusions and Outlook on Future Work**

In this paper, a novel enclosure approach for the pseudo states of fractional-order differential equations has been presented. It is based on enclosing Mittag-Leffler functions by exponential enclosures instead of box-type enclosures employed so far in previous work. By using a close-to-life model for the charging/discharging dynamics of a Lithium-ion battery, it has been shown that this new enclosure technique leads not only to significantly tighter enclosures, yet preserving the guaranteed enclosure property, but also leads to a noticeable reduction in the computational effort by significantly less iterations required to obtain an identical enclosure quality.

Future work will aim at extending the presented approach to system models with non-commensurate orders. In addition, the approach will be included into the observerbased technique presented in [7] for the quantification of truncation errors which allows for resetting fractional integrators in a guaranteed way. In such a way, it is planned to make the proposed simulation approach applicable to tasks such as the parameter identification of fractional-order differential equations and to the identification of their initialization functions for *t* < 0. Moreover, the application to more complex dynamic models from the domains of electrochemical energy storage and energy conversion will be investigated.

**Funding:** This research received no external funding.

**Data Availability Statement:** Data are contained within the article.

**Conflicts of Interest:** The author declares no conflict of interest.
