Stability is an essential property for any meaningful operation of a dynamic system. While it is highly desirable to prove stability analytically, this is a challenging task. For linear systems, a comprehensive theory usually allows stability to be proven (e.g. through eigenvalue analysis, gain and phase margins or the Nyquist criterion), but no such theoretical framework exists for nonlinear systems. In the realm of nonlinear systems, only special methods are available for certain system classes (e.g. the Popov criterion). There is also the Lyapunov theory that can be applied to broad range of systems by investigating the system behavior in an abstract way with respect to its inherent energy. However, to do so requires a Lyapunov function with specific properties, and it is unclear how such a function can be identified.

This paper proposes an approach to construct a Lyapunov function using the Hamiltonian form of the system under investigation. The Hamiltonian form is a special structure of the system's differential equations that includes a Hamilton function. If the system can be transformed into this form, the Hamilton function can be used to derive a candidate Lyapunov function. Due to the special structure of the system in Hamilton form, it is straightforward to determine whether the candidate Lyapunov function is indeed a Lyapunov function and whether the stability of the system can be proven in this manner.

This methodology is applied to grid-forming inverters by considering two interconnected inverters operating with standard droop control strategies. The approach leads to a successful stability proof when transmission line dynamics and the resistive part of the line are neglected. Based on this, efforts were made to extend the method to include these effects in the stability investigation. Taking the transmission line dynamics into account results in a higher-order system since the current signals must be included in the state vector. This significantly increases the complexity of the investigation, and while it is still possible to transform the system into a Hamiltonian form, it is not possible to prove its stability due to the system now performing oscillations. These oscillations are caused by a lack of damping due to the resistive part of the transmission line still being neglected. While the proposed method allows damping to be included, this is only possible under very strict limitations, which hinders the meaningful application of the method.

In conclusion, this work presents results for analytical stability investigation for inverter systems under simplifications, but also demonstrates the challenges of extending the approach. Further research is needed, especially with regard to including the resistive part of the transmission line in the investigation.