*4.3. Simulation with the Help of Exponential Enclosures*

Figure 6 shows that the pseudo-state enclosures are significantly tightened by using the intersection of the novel exponential enclosures of Mittag-Leffler functions with the box-type ones that were solely used in Figure 5. This advantage cannot only be observed for the case of the wide initialization **x**(0) ∈ [**x**]1(0) but also for the tighter one **x**(0) ∈ [**x**]2(0).

The second advantage of this new enclosure approach is illustrated in Figure 7, where the required maximum numbers of evaluations of both (31) and (34) are illustrated for the ten different choices of the simulation horizon *T*. There, it can be seen clearly that the novel exponential enclosure approach allows to reduce the number of required iterations significantly for longer simulation horizons *T*. To make this comparison fair, the iterations have been stopped in all cases if the diameters of all [*λi*] *κ*+1 and [*λi*] *κ* deviate from each other by less than the value 10−4.

**Figure 6.** Use of exponential enclosures, intersected with the box-type ones, for the evaluation of the iteration Formula (34) for the computation of guaranteed pseudo-state enclosures: (**a**) State of charge *σ*(*t*) for **x**(0) ∈ [**x**]1(0); (**b**) State of charge *σ*(*t*) for **x**(0) ∈ [**x**]2(0); (**c**) Voltage *v*1(*t*) for **x**(0) ∈ [**x**]1(0); (**d**) Voltage *v*1(*t*) for **x**(0) ∈ [**x**]2(0); (**e**) Interval diameter for the enclosure of *σ*(*t*); (**f**) Interval diameter for the enclosure of *v*1(*t*).

As a summary, the new enclosure approach does not only allow to significantly reduce the overestimation in the computation of guaranteed bounds for the pseudo states but—at

least for the application scenario at hand—also reduces the computational effort by up to 80% due to significantly less required iterations.

**Figure 7.** Comparison of the required number of iterations for both box-type and exponential enclosures: (**a**) Required iterations for the initialization **x**(0) ∈ [**x**] <sup>1</sup>(0) (box-type enclosure); (**b**) Required iterations for the initialization **x**(0) ∈ [**x**] <sup>2</sup>(0) (box-type enclosure); (**c**) Required iterations for the initialization **x**(0) ∈ [**x**] <sup>1</sup>(0) (exponential enclosure); (**d**) Required iterations for the initialization **x**(0) ∈ [**x**] <sup>2</sup>(0) (exponential enclosure).
