<p>With the increasing global emphasis on energy independence and the integration of renewable energy sources, ensuring the stability of grid-forming energy storage systems in standalone networks has become a critical challenge. This study investigates self-excited oscillations observed in standalone grid-forming energy storage systems, triggered by the saturation characteristics of transformers during operation. A single-machine equivalent model of the standalone system is developed, and it is shown that traditional linearized methods, such as small-signal analysis and impedance-based analysis, fail to accurately capture the oscillation mechanisms under these nonlinear conditions. To address this, the describing function method is applied to analyze the underlying oscillation mechanisms. The root cause of the oscillations is further validated through hardware-in-the-loop experiments, demonstrating the accuracy of the proposed analysis approach. Moreover, effective suppression strategies are developed and validated, providing a practical solution to enhance system stability.</p>