Abstract：The frequency-adaptive dual second-order generalized integrator phase-locked loop (FA- DSOGI-PLL) is widely used for grid synchronization in three-phase unbalanced systems. However, its parameter design faces significant challenges. Existing approaches often neglect the coupling effect introduced by the frequency-adaptive (FA) loop, thereby approximating the system as a fixed-frequency DSOGI-PLL (FF- DSOGI-PLL) or a low-pass filter-based SRF-PLL (LSRF-PLL) for design. While effective at lower bandwidths, these approximations degrade parameter robustness and introduce performance imbalances as bandwidth requirements increase. This paper proposes a novel parameter design method for the FA-DSOGI-PLL. Firstly, to accurately capture the FA loop's influence, the conventional SOGI model was improved by accounting for a time-varying resonant frequency, thereby addressing inaccuracies arising when traditional linearization mistakenly characterizes the FA-DSOGI-PLL as a non-minimum-phase system. Then, a triple-input triple-output (TITO) small-signal model was established, and the conditions for simplifying it to a double-input double-output (DIDO) model were specified. Thus, transfer functions were derived that explicitly account for the coupling among voltage magnitude, frequency, and phase-angle disturbances. Secondly, steady-state and dynamic performance were analyzed. Based on the coupling of the FA loop, the approximate boundary between FA-DSOGI-PLL and FF-DSOGI-PLL was quantified under the single degree of freedom of the natural frequency fn of the SRF-PLL. The bandwidth limit of FA-DSOGI-PLL under the two degrees of freedom of the damping coefficient k of SOGI and the natural frequency fn of SRF-PLL was analyzed, along with coupling effects among disturbances. The advantages and limitations of introducing the damping ratio ζ of the SRF-PLL to extend bandwidth were discussed. Finally, a parameter design method that accounts for the approximate boundary was proposed. The following conclusions can be drawn. (1) An improved SOGI model accounting for time-varying resonant frequency was established to reflect the coupling effect of the FA loop. The amplitude and phase angle constraints for approximating FA-DSOGI-PLL to FF-DSOGI-PLL under the single fn degree of freedom are quantified, and a calculation formula for the approximation boundary is provided. (2) Using the improved model, the bandwidth is limited under different parameter degrees of freedom. Three direct causes of the difficulty in balancing dynamic-steady-state performance and parameter design are identified: the coupling of PM, GM, and bandwidth with the single fn degree of freedom limits the validity of approximations at lower bandwidth; the intersection of PM and GM under k and fn degrees of freedom constrains the bandwidth, with k having opposite effects on frequency/phase and voltage-magnitude disturbance responses; although the ζ degree of freedom extends bandwidth for frequency tracking, it suffers from slow phase error elimination during frequency deviations. (3) The proposed parameter optimization design method extends the bandwidth of FA-DSOGI-PLL beyond the approximate boundary while maintaining stability margin and harmonic rejection, thereby improving dynamic and steady-state performance under frequency-adaptive conditions. It balances filtering and voltage magnitude tracking, meets stability requirements, and enhances bandwidth. The presented analysis framework and design method can be extended to improved DSOGI-PLLs and other advanced PLLs.
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