## 1 Introduction
While demonstrating satisfactory accuracy in current waveform reproduction, such numerical approaches impose computational burdens and present outcomes without elucidating the fundamental mechanisms governing the emergence of specific fault current characteristics [19], which motivates the exploration of analytical modeling methodologies. Notably, analytical approaches establish explicit correlations between parameters and fault current, thereby enabling proactive fault current management during system design optimization. Previous modeling attempts primarily represented MMC topologies as simplified RLC networks [15; 20; 21; 22]. However, those models fail to accurately characterize the hybrid MMC with fault control capability, as they omit the controlled capacitor discharge dynamics governed by FRT strategies. Considering the maximum fault current limitation capability, the hybrid MMC is equivalent to the constant-voltage source-series-RL circuit in [8; 23]. This method is still not able to capture the nonlinear dynamics inherent in hybrid MMC operation during fault transients when accounting for FRT control and controller saturation phenomena [24]. To elucidate fault phenomena arising from converter saturation induced by per-arm FBSMs negative voltage constraints [25], the fault response of a hybrid MMC using direct arm current control is divided into several fault stages, and a short-time numerical calculation with simplified AC control loops is adopted to account for saturation effects in [19]. Based on numerical simulations and Hilbert-Huang transformation analysis, [26] reveals that the hybrid MMC exhibit weakly nonlinear transient behavior during pole-to-ground FRT operation. Leveraging this characteristic, a linearized transient MMC model is established, yielding an approximate expression for DC fault current. This study prioritizes the characterization of fault current attenuation dynamics over precise analytical quantification. In addition, incorporating control strategies via converter transfer functions can reflect the control effect in analytical calculations [27, 28, 29]. The authors of [30] propose a complex frequency analytical technique for quantifying HVDC fault currents under FRT control, which can obtain the numerical solution of the branch currents. However, further investigation is warranted to elucidate the correlation between regulation parameters and fault current characteristics. To address the aforementioned challenges, this study introduces a quantitative fault current analysis methodology for hybrid MMCs that incorporates FRT control response. An equivalent circuit model of the hybrid MMC is developed based on its operational principles and control mechanism. Subsequently, quantitative relationships between fault current characteristics and DC controller parameters are established for distinct control modes. Control saturation effects are examined throughout all fault phases. The principal innovations of this work are threefold, as follows: (1) Circuit representation: Departing from conventional RLC models, the hybrid MMC is represented as an RL circuit in series with a time-variant controlled voltage source. This formulation effectively captures the converter's control response. (2) Parametric quantification: Accounting for differential transient behaviors in fault circuits, explicit analytical relationships between fault current response and control parameters are derived. (3) Saturation Effect Integration: The impact of control saturation on fault current evolution is characterized, enabling the precise determination of DC fault currents across sequential fault stages. The paper is organized as follows: Section 2 reviews conventional equivalent circuit representations for hybrid MMCs and discusses their inherent limitations. Section 3 develops a refined equivalent model of the hybrid MMC. Section 4 derives quantitative relationships between fault current characteristics and control parameters.