## 1 Introduction  
Ensuring operational continuity of modular multilevel converter (MMC)-based high-voltage direct current (HVDC) systems under DC faults is a critical engineering challenge [1, 2]. To address the DC side fault, the MMC is conventionally blocked, following by the trip-out of the circuit breaker [3]. However, full IGBT deactivation results in irreversible loss of converter control capability, cessation of power transmission, and prolonged system outage periods in HVDC networks [4], highlighting the need for advanced fault ride-through (FRT) solutions [5, 6]. The hybrid MMC represents a predominant architectural configuration among MMC variants with FRT capability, which has been commercially used in HVDC applications [7]. The hybrid MMC integrates full-bridge (FB) and half-bridge (HB) submodules (SMs). Leveraging the negative voltage generation capability of FBSMs, the hybrid MMC actively modulates DC-bus voltage to contain fault current progression. The preponderance of research is dedicated to devising control strategies that secure FRT functionality for hybrid MMC topologies. The existing literature [8, 9, 10, 11] employs the immediate activation of current regulation strategies upon DC fault detection, implementing zero or predefined reference values for DC current commands. A methodology presented in [12] focuses on source-side fault mitigation through current suppression techniques targeting line or converter currents. From the voltage regulation perspective, [13] introduces dynamic voltage coordination by aligning DC bus voltage with fault residual voltage, thereby facilitating seamless FRT implementation. Through implementation of an added control variable, [14] achieves zero-setpoint regulation for DC voltage constituents in fault-affected pole arms. While these investigations collectively advance FRT capability enhancement in hybrid MMC-HVDC systems through current/voltage manipulation strategies, the mathematical correlation between control parameter configurations and transient fault current characteristics remains insufficiently quantified. The establishment of mathematical correlations between control variables and transient fault characteristics constitutes a critical foundation for optimizing the FRT strategy in hybrid MMC-HVDC systems. There have been investigations to reveal that fault current characteristics exhibit strong dependence on control loop configurations. In [15], parametric adjustments in voltage modulation indices induce current variations; a DC modulation index reduction from 0.2 to 0.1 decreases fault current peaks from 3.28 p.u. to 2.75 p.u., illustrating the nonlinear sensitivity of transient responses to control parameter selection. In addition, the parameters of the proportional-integral (PI) controllers also have an effect on the fault current variation. As demonstrated in the fault case study of [16], the peak fault current magnitude exhibits a direct correlation with inner-loop controller reference setpoints. Specifically, a 0.83 p.u. reference configuration yields a maximum fault current of 1.1 p.u., whereas increasing the reference to 1.2 p.u. elevates the current peak to 1.5 p.u. The FRT control of the hybrid MMC must guarantee system resilience by maintaining transient currents below predefined acceptable thresholds under faults. Consequently, the development of robust control strategies requires systematic investigation through computational modeling to develop a quantitative relationship between the FRT control parameters and the fault current prior to commissioning of the converter. Existing studies predominantly employ electromagnetic transient simulation techniques to investigate fault current characteristics [12; 13; 17; 18].