Optimisation of the Kaplan hydropower system via PID 2 and digital twin

This paper proposes a proportional–integral-double–derivative (PID2) optimisation method for the Kaplan hydropower system by building a digital twin. The study first uses one multilayer perceptron (MLP) to model the hydroturbine dynamic and then adopts three connected MLPs to model the generator dynamic, both in an open-loop fashion. Inspired by stochastic distribution control (SDC) theory, we regard the training of the turbine's neural network model as a process control problem, and we propose minimising entropy loss to update the network parameters. The next step is to build the digital twin by connecting the neural network models with a PID2 controller and a lead-lag exciter and run the whole model in a closed-loop fashion. After that, a binary search approach is applied to optimise the PID2 parameters based on the obtained digital twin model. The simulation results show that the proposed method can reduce the mean square tracking error by more than 90%. Furthermore, the method is extended to jointly optimise the PID2 controller and excitation system gains through multiobjective optimisation, leveraging Pareto frontier analysis to balance active power and voltage tracking performance. Simulation results confirm the effectiveness of the proposed method, achieving a 83.46% reduction in relative mean square error of active power, a 47.13% reduction in terminal voltage tracking error, and an 82.78% improvement in the overall scalarized objective.