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Soft magnetic powder cores with their high saturation flux density and low core loss are excellent alternatives for filter inductors in inverter-based applications. However, their nonlinear current-dependent inductance characteristics pose a challenge for control of grid-connected inverter with LCL filter. In this paper, the variable inductance conundrum is discussed and a modified direct digital control method based on a variable-structure inductance estimation model that takes into consideration wide nonlinear variation in both inverter- and grid-side inductances is proposed. The proposed method is shown to have better grid-voltage harmonic rejection and improved stability margins. However, investigation of stability which is conventionally based on nominal values of filter inductors cannot predict instabilities over the entire range of inductance variation. Hence, a parametric approach to conventional stability methods with a parameter space defined by variation in actual and estimated inductance is explored in this paper. The effect of line impedance on stability is also investigated with impedance-based stability criterion by considering line inductance as an additional dimension in the parameter space. A pattern in stability margins is observed due to inductance variation, with inductance at its minimum being most vulnerable to instability. Experimental results measured from a 5 kW single-phase grid-connected inverter with various LCL filters have verified the feasibility of the proposed control method. The experimental results also match the analytical results with reasonable accuracy. [ABSTRACT FROM AUTHOR]