**1. Introduction**

Supercapacitors are promising energy storage technologies with high energy density and high charging/discharging capabilities [1]. They allow for the storage of electrical energy in electric fields, increasing their efficiency in comparison with mechanical or chemical devices [2]. Supercapacitor energy storage (SCES) and superconducting magnetic energy storage (SMES) are the only two devices that store energy in the form of electromagnetic fields [3]. Nevertheless, SCES systems are preferred because they work with voltage source converters [1], which are common and represent the cheapest option when compared with the current source converters in SMES applications [4]. Additionally, another advantage of using SCES over SMES systems is that the former does not require special thermal covers, such as those needed for cooling systems based on liquid hydrogen, helium, or nitrogen [3]. This increases the acquisition, installation, and maintenance of SMES systems.

The integration of SCES systems via voltage source converters allows controlling the active and reactive power independently by formulating a dynamic model in any reference frame (i.e., time domain or *abc*, Clark's or *αβ*, and Park's or *dq* reference frames, respectively) [5]. In addition, the dynamical model of the SCES system exhibits a nonlinear structure that makes it necessary to propose nonlinear controllers to deal with its operational goals. Multiple controllers for SCES

systems integration in electrical networks have been proposed such as feedback linearization methods [6], interconnection and damping PBC approaches [3,5], linear matrix inequalities [7], classical proportional-integral controls [8], adaptive predictive control [9], or proportional-integral passivity-based control (PI-PBC) methods [2], that typically control active and reactive power in an indirect form by controlling the currents on the AC side of the converter. Two interesting approaches based on IDA-PBC and PI-PBC approaches have been reported by [10,11] to control SMES and SCES systems in autonomous applications of single-phase microgrids; these PBC approaches take the advantages of the port-Hamiltonian modeling to propose asymptotically stable controllers of Lyapunov. Authors confirm that single-phase converters allow the control of active and reactive power independently; nevertheless, the main disadvantage is the dependence on the parameters of the control laws, which complicates their application over systems with parametric uncertainties.

Note that PBC approaches are preferred to operate SCES systems because their dynamic models exhibit a port-Hamiltonian (pH) structure in open-loop, which is a suitable structure that is employed in PBC designs. Because it allows proposing closed-loop control structures that can guarantee stable operation in the sense of Lyapunov [12]. Even though the PI-PBC method has been previously presented for SCES systems by [2], who presented a direct power control structure, we proposed a robust parametric approach that avoids prior knowledge on the system parameters (inductance, resistance, and supercapacitor values) and does not require the solution of additional differential equations. This is a gap that is ye<sup>t</sup> to be solved in specialized literature for SCES applications in AC distribution networks. In addition, the main advantage of using direct power control is that the state variables to design the controller are directly active and reactive power. Making more suitable the assignation of the references to control these variables in generation or load compensation applications, while classical approaches work with currents as state variables requiring additional steps regarding active and reactive power control.

The remainder of this paper is organized as follows: Section 2 presents a complete dynamical formulation of the SCES system interconnected to AC grids with a series resistive-inductive filter. In addition, we present its pH intrinsic formulation and its transformation from the current structure to the power ones. Section 3 presents the structure of the proposed PI-PBC approach, highlighting its independence regarding the filter parameters and provides a general proof to guarantee asymptotic convergence. Section 4 reports the general control structure as a function of active and reactive power measures as well as the physical constraints related to the integration of SCES in distribution networks. Section 5 presents the test system, simulating conditions, and numerical results with their corresponding analysis and discussion. Section 6 details the main conclusions derived from this research.
