## 1 Introduction  
With the widespread integration of distributed energy sources supply to the power grid, the penetration rate of power electronic equipment in the electronic power system is gradually increased. This phenomenon leads to the low-inertia, weak damping, and weak frequency support characteristics of the traditional power system [1, 2], significantly affecting its stability and security. To address the issue of grid instability and frequency oscillations arising from the high proportion of renewable energy [3, 4], some scholars, drawing from the operational characteristics of synchronous generators, introduced virtual inertia and damping links through the simulation of its power swing equation. Based on this, VSG control technology is proposed [5, 6]. This control strategy makes the grid-connected converter capable of primary frequency regulation (PFR) and one-time voltage regulation, improving the anti-interference ability of the system [7]. Introducing virtual inertia and damping mechanisms is an effective strategy for improving the system's PFR capability and enhancing the dynamic stability of the system's frequency. However, the traditional virtual inertia control strategy alters the system characteristics of the TVSG's APL from first-order to second-order oscillatory characteristics, resulting in transient oscillations within the VSG's grid-connected system during active power and grid frequency disturbances [8], weakening the dynamics of the APL system. The introduction of damping mechanisms can be viewed as an equivalent to increasing PFR parameters; however, larger damping parameters may result in greater steady-state deviations within the APL [9; 10]. Therefore, the TVSG based on a fixed damping strategy contradicts transient performance and steady-state performance, which cannot eliminate the problem of active steady-state deviation caused by damping characteristics while suppressing transient oscillation [11]. Researchers have embarked on an extensive array of studies to tackle the identified technical difficulties and refine the active power response performance within the TVSG. These investigations are categorically divided into two distinct approaches: the adaptive virtual inertia damping methodology [12] and the equivalent damping ratio strategy [13]. To enhance the response rate of TVSG steady-state frequency tracking, a VSG control strategy based on the intermittent variation in virtual inertia is proposed in [14]. However, this intermittent variation introduces redundant nonlinear characteristics that adversely affect the system's stability. The authors of [15] introduce a fuzzy controller into the VSG power swing equation to enhance the system's response performance during transient processes while also addressing the adverse effects of discontinuous virtual inertia settings on system stability. The authors of [16; 17] propose a VSG control strategy incorporating synergistic adaptive virtual inertia and damping parameters, which not only enhances the transient stability of the power outputs of the VSG system but also further mitigates frequency deviations resulting from active power disturbances, thereby improving the dynamics of the APL system. However, it is important to note that most aforementioned studies are predicated on stable operating conditions of the grid, concentrating on enhancing system stability under small disturbances, and thus failing to address the steady-state deviations arising from the coupling of damping and frequency modulation parameters. Furthermore, the improper selection of adaptive parameter ranges can adversely impact the steady-state performance of VSG grid-connected systems. In the context of equivalent damping ratio methodologies, the authors of [18] conducted a qualitative analysis of the fault limit clearing time and the fault limit clearing angle in the VSG and established the influence mechanisms between control parameters and the transient characteristics of the APL system in the TVSG.