*4.2. Physical Operative Constraint*

The energy storage capability of a SCES system is limited by the energy capabilities of the supercapacitor as well as for the admissible voltages in its terminals.

Figure 1 presents the dynamic behavior of the total energy stored in the SCES. In this figure, there are three critical points called O, P, and Q. Point O presents the minimum voltage value permissible in the supercapacitor terminals (*v*min *sc* ) that produce the value of the admissible minimum energy stored

(*E*min); this voltage value is the lower bound in the SCES operation to guarantee the controllability of the closed-loop system. Additionally, point Q is the upper bound of the energy storage variable, which reaches the maximum permissible energy stored in the supercapacitor device (*v*max *sc* ↔ *E*max), whereas point P represents some operating points between extreme points of O and Q.

 **Figure 1.** Behavior of the energy stored in the Supercapacitor energy storage (SCES).

Observe that point P allows positive or negative active power references. In the case in which *p* is positive, the energy stored decreases from point P to point O. In the case in which *p* is negative, the energy stored increases from point P to point Q. When point P is at the extreme points, we can conclude that if it is at point O, the reference for *p* can only be negative or zero, and if it is at point Q, it must be positive or zero. These are necessary conditions to preserve the useful life in the SCES.

### **5. Test System and Simulation Scenarios**
