Real-Time Hybrid Modeling of Francis Hydroturbine Dynamics via a Neural Controlled Differential Equation Approach

In recent years, deep learning has been widely applied to learning nonlinear dynamic models for the development of a digital twin system. However, most traditional deep learning frameworks, such as recurrent neural networks, convolutional neural networks, and multilayer perceptrons, find it difficult to learn continuous-time and nonlinear system models. To address this challenge, in this paper, a novel deep learning method called neural controlled differential equation has been proposed to model the unknown nonlinear dynamics of controlled continuous-time systems seen in Francis hydroturbines of hydropower systems. Following the development of discretized-model structures for the system using the first principles, a detailed learning algorithm is formulated that is integrated with the physical model of the hydroturbine. As a result, a hybrid modeling with effective learning capability is obtained. To test the effectiveness of the proposed learning algorithm, a set of operational data has been collected and used to train the nonlinear dynamics of the Francis hydroturbine, where the learning results of the two nonlinear dynamics, namely the mechanical torque and water flow dynamics, using the real data have indicated that the proposed method can accurately learn these unknown nonlinear dynamics in an online, adaptive way. Moreover, to address the overfitting problem that appears during the online training phase, we propose to apply a meta-learning technique to pre-train a meta-initial value for each parameter of the proposed neural controlled differential equations. It has been shown that the use of the meta-learning technique can reduce the prediction mean square error significantly by more than 60%.