Dynamic Behavior Modeling of Price-Responsive Flexible Loads: A Deep Learning-Parameterized Inverse Optimization Approach

Accurately modeling the behavior of price-responsive flexible loads (PRFLs) is increasingly critical for enhancing power system flexibility. While existing inverse optimization methods can achieve flexible load modeling using only external price-response data, the surrogate model with time-invariant parameters severely limits its ability to capture dynamic behaviors induced by user demands, weather conditions, etc. In contrast to mainstream static modeling formulations, this paper proposes a Deep Learning-parameterized Inverse Optimization (DL-IO) method that enables dynamic surrogate modeling of PRFLs. Specifically, a dynamic virtual energy storage (DVES) model is formulated as the surrogate model, with its parameters represented by multi-task deep learning to exhibit time-varying properties. Then, the DVES model is integrated into the DL-IO framework, enabling the approximation of response behaviors for thermostatically controlled loads (TCLs) and electric vehicles (EVs) based solely on price-response data and public information. To solve the resulting bi-level DL-IO problem, it is reformulated as an empirical risk minimization task, which consists of a deep learning-based parameter estimation layer and an optimization-based response estimation layer. An analytical end-to-end gradient is then derived and backpropagated through both layers to achieve behavior learning. Numerical tests indicate that the proposed modeling method outperforms existing inverse optimization and data-driven regression approaches.