## 1 Introduction
Multi-frequency systems continue to develop as an emerging field because multiple challenges persist despite recent advancements. Research in this field primarily investigates conceptual feasibility through studies that use single-phase systems and simplified conditions [4; 7]. Three-phase implementations and complex harmonic combinations became the focus of research only in recent times. The integration of distributed generation systems and storage facilities into multi-frequency power grids has not received enough research attention. The integration of PV panels and batteries into multi-frequency buses has been proposed, but effective control systems for this application are hard to find. The analysis of PV and BESSs connected through a DC link requires studying AC-side multiple frequency power interactions, DC-link behavior, and device connectivity. The integration of these elements requires interdisciplinary control approaches that combine power quality management with power electronics and microgrid energy management principles [1; 2; 8]. A low-voltage DC link enables a hybrid AC/DC microgrid that combines the strengths of both systems. The main AC grid is connected to local AC networks via a back-to-backconverter, and the DC link integrates PV, batteries, other DC sources, or DC networks via DC/DC converters. This architecture improves efficiency, simplifies power flow coordination, and enables bidirectional energy exchange between AC and DC subsystems [3; 9]. The stability of the DC link becomes a problem when multiple converters, such as PV inverters and BESS converters, are exchanging power. Several control strategies have been proposed to coordinate these units. A unified control method has been proposed to enable smooth transitions between grid-connected and islanded modes, thus providing resiliency during grid outages and seamless resynchronization when the grid returns [10; 11]. Integrating a photovoltaic source and a battery energy storage system into a shared DC-link improves microgrid performance by enabling local renewable supply and buffering power fluctuations. The PV operates under maximum power point tracking (MPPT), while the BESS smooths DC-link variations by charging or discharging. To ensure stable operation in both grid-connected and autonomous modes, mode-adaptive control is used: the battery inverter operates in current-control mode during grid connection and switches to voltage-control mode during islanding [12]. Energy management strategies are critical for real-time control of BESS charging and discharging. Hierarchical control methods allocate power between PV and BESS based on forecasts and demand [13], while centralized EMS monitors battery state-of-charge and DC-link status to optimize energy flow and support peak shaving [14]. These strategies ensure that the PV and battery jointly maintain DC-link stability, store surplus energy during peak production, and supply power during high demand or transient events [15]. The stability of DC-link voltage represents a fundamental operational challenge for standalone and grid-forming inverter systems, especially when designers implement reduced-capacitance solutions to enhance reliability and lifespan. Research indicates that sliding mode control successfully controls voltage instabilities that occur from small film capacitors in standalone photovoltaic systems to improve system dynamics and maintain power quality [16; 17]. The improper adjustment of PI-based DC-link controllers in grid-forming inverters creates transient instability because unstable limit cycles appear even though small-signal stability remains intact. A bifurcation-based analysis was proposed to guide parametric selection and ensure synchronism under large disturbances [18]. An important aspect is the control of the BESS converter, typically a bidirectional DC/DC unit connected to the DC-link. Its role is to regulate battery charging and support DC-link voltage stability alongside the grid-tied inverter.