Mathematical aspects of static and dynamic stability problems in power systems

A power system is modelled by a system of parameterised differential-algebraic equations (DAEs) when it is operating normally; it is modelled by time-piecewise DAEs when a transient process occurs. The normal operation of power systems is related to the static stability problem; while the transient process is related to the dynamic stability problem. After categorising the stability problems in power systems into static and dynamic cases, mathematical formulations are proposed and mathematical problems related to them are studied using different mathematical tools. In the static case, the determination of feasibility region is related to solving algebraic equations; while in the dynamic case, identifying attraction domain is related to solving a DAE. For a power system, the determination of feasibility region is usually reduced to optimisation problems, while the determination of attraction domain and critical clearing time is related to the study of transient energy functions.