In this paper, a novel Lyapunov–Krasovskii stable fuzzy proportional-integral-derivative (PID) (FPID) controller is introduced for load frequency control of a time-delayed micro-grid (MG) system that benefits from a fuel cell unit, wind turbine generator and plug-in electric vehicles.
Using the Lyapunov–Krasovskii theorem, the adaptation laws for the consequent parameters and output scaling factors of the FPID controller are developed in such a way that an upper limit (the maximum permissible value) for time delay is introduced for the stability of the closed-loop MG system. In this way, there is a stable FPID controller, the adaptive parameters of which are bounded. In the obtained adaptation laws and the way of stability analyses, there is no need to approximate the nonlinear model of the controlled system, which makes the implementation process of the proposed adaptive FPID controller much simpler.
It has been shown that for a different amount of time delay and intermittent resources/loads, the proposed adaptive FPID controller is able to enforce the frequency deviations to zero with better performance and a less amount of energy. In the proposed FPID controller, the increase in the amount of time delay leads to a small increase in the amount of overshoot/undershoot and settling time values, which indicate that the proposed controller is robust to the time delay changes.
Although the designed FPID controllers in the literature are very efficient in being applied to the uncertain and nonlinear systems, they suffer from stability problems. In this paper, the stability of the FPID controller has been examined in applying to the frequency control of a nonlinear input-delayed MG system. Based on the Lyapunov–Krasovskii theorem and using rigorous mathematical analyses, the stability conditions and the adaptation laws for the parameters of the FPID controller have been obtained in the presence of input delay and nonlinearities of the MG system.
