# Gradient Formulation for the Stability of DC-Microgrids

## TL;DR

This paper introduces a novel non-linear stability analysis method for dc-microgrids using gradient systems, enabling efficient equilibrium calculation and stability conditions through convex optimization, with practical validation via simulations.

## Contribution

It presents a comprehensive stability analysis framework for dc-microgrids based on gradient system theory, addressing multiple aspects simultaneously.

## Key findings

- Demonstrates existence and uniqueness of equilibrium
- Provides numerical methods for equilibrium calculation
- Establishes conditions for global stability

## Abstract

This paper presents a non-linear stability analysis for dc-microgrids in both, interconnected mode and island operation with primary control. The proposed analysis is based on the fact that the dynamical model of the grid is a gradient system generated by a strongly convex function. The stability analysis is thus reduced to a series of convex optimization problems. The proposed method allows to: i) demonstrate the existence and uniqueness of the equilibrium ii) calculate this equilibrium numerically iii) give conditions for global stability using a Lyapunov function iv) estimate the attraction region. Previous works only address one of these aspects. Numeric calculations performed in cvx and simulations results in Matlab complement the analysis and demonstrate how to use this theoretical results in practical problems.