## 1 Introduction  
The growing integration of inverter-based resources (IBRs) into power grids is substantially influencing system operation. The overall inertia of the system is diminishing due to the decommissioning of synchronous generators (SGs) and their subsequent replacement with IBRs [1]. This transition adversely affects system stability, impacting frequency stability [2, 3] and small-signal rotor angle stability [4]. Regarding small-signal rotor angle stability, the reduction in system inertia leads to the decreased damping of the system's electromechanical modes, typically within the frequency range of 0.2 Hz to 2 Hz [5, 6]. As power oscillates across transmission lines without proper damping, this effect also affects the dynamics of the power flow and reduces the transient stability margins [7]. In the coming years, the share of IBRs is expected to increase, and low-frequency oscillations (LFOs) may occur more frequently in power grids. Flexible AC transmission systems (FACTSs) based on power electronics have been the primary solution, which have been widely used to improve power system stability [6; 8]. For shunt-connected FACTSs, such as Static Var Compensators (SVCs) and STAT-COMs, power oscillation damping (POD) can be achieved by modifying the reactive power injection. This approach, which uses reactive power to damp power oscillations, is referred to as POD-Q [9]. Alternatively, when the control algorithm directly influences the voltage at the PCC without regulating reactive power exchange, the method is known as POD-V [10]. In predominantly reactive transmission lines, active power injection influences both the voltage angle and frequency at the point of common coupling (PCC). Thus, when active power is the controlled magnitude for power oscillation damping, it is denoted by the acronym POD-P. Besides FACTSs, POD-P algorithms have also been implemented on Voltage Source Converters, such as offshore wind farms [11], HVDC links [5], energy storage systems (ESSs), and renewable power plants [12; 13; 14]. New grid-forming (GFM) controls are emerging as an alternative to conventional grid-following (GFL) IBRs. As the GFM control provides inertia, the controller directly affects the small-signal rotor angle stability of power grids [15; 16; 17]. Thus, adopting the GFM approach could potentially solve or at least mitigate the LFO problem. However, the best way to address this issue is unclear. Some works focus on the power synchronization loop (PSL) of GFM controllers to improve the damping provision. For example, ref. [18] compares two different control architectures regarding their damping provision capability. When a GFM inverter is connected to a known power grid with poorly damped low-frequency modes, the PSL or the damping control algorithm can be adjusted to mitigate undesired small-signal interactions with the rest of the network [19; 20]. Following a similar philosophy, but without considering the power grid specific characteristics, ref. [21] proposes a design method to optimize the damping by tuning control parameters. However, changes in the grid over time or limitations inherent in this control method can make the inertia provision insufficient, and the implementation of additional POD algorithms in GFM devices may be required. Damping control methods play a very important role in suppressing the dynamic oscillations of GFM inverters' output active power and frequency. These strategies can generally be grouped into three main categories [22]: adaptive parameter tuning, feedback-based compensation, and feedforward compensation methods. The adaptive parameter tuning approach enhances the GFM inverters' ability to damp oscillations by dynamically adjusting parameters such as virtual inertia, virtual impedance, or those related to primary frequency regulation. For example, in [23; 24], the inertia is adapted in order to counteract oscillations. The first proposes to adapt the inertia-modifying droop constant, while the second proposes to modify it by changing the virtual impedance.