Grid-connected lithium-ion battery energy storage system (BESS) plays a crucial role in providing grid inertia support. However, existing equivalent circuit models (ECM) cannot accurately represent the battery's impedance in the inertia support working condition (ISWC). Thus, this article proposes a novel negative resistor-based ECM for BESS in ISWC. First, the principle of grid inertia support of BESS is analyzed, and the typical sustaining current profiles in ISWC are obtained. Then, the distinguished battery impedance characteristic in ISWC is experimentally obtained. To capture the special impedance characteristic, a resistor-capacitor (RC) branch with negative resistor is added to the 1-RC model (Thevenin model). Moreover, based on the difference in impedance characteristics between the conventional RC branch and the negative resistor RC branch, this article proposes a parameter separation method based on impedance characteristics to obtain an improved ECM. A comparative analysis reveals that the improved ECM outperforms conventional models, achieving an average relative error below 0.06% compared to experimental results. The improved ECM offers a simple structure, reduced computational burden, and enables rapid analysis of battery state in ISWC. Furthermore, a new simplified negative resistor-based model without altering the conventional RC model's structure is proposed. It is simple in structure with only three parameters, significantly reducing computational costs while maintaining accuracy.