Value Set-Based Method for Solving Stability Boundary of Systems With Uncertain Time Delays

Existing methods for computing stability boundary of systems with multiple uncertain time delays are either conservative or suffer from low computational efficiency. Addressing this challenge, a value set based method is proposed to solve stability boundary for time delays with high efficiency. Firstly, the characteristic quasi-polynomial of the system is constructed in the periodic multiplication between frequency and time delays (FDM) space. Subsequently, the characteristic quasi-polynomial is transformed by the value set approximations, which is then refined through parameter space subdivision to achieve the required accuracy. Finally, the stability boundary is solved by traversing the parameter space with a bisecting step size. This method is characterized by its efficiency without introducing conservativeness, which is achieved through the decreasing step size space traversal. Numerical examples and a case study based on a microgrid with three distributed generators (DGs) that introduce communication delays are served to validate the accuracy and efficiency.Note to Practitioners—This paper proposes a value set based method to solve stability boundary of the system with multiple uncertain time delays. The method is developed based on the necessary and sufficient criteria. The method leverages the periodicity of FDM and employs a variable step traversal algorithm to decrease the computational burden and achieve high efficiency. Notably, this method can be used for systems with multiple time delays, regardless of the range or quantity of the delays. This method is effective for analyzing systems with multiple time delays, offering support for controller design and enhancing system stability.