Load serving entities with storage units reach sizes and performances that\ncan significantly impact clearing prices in electricity markets. Nevertheless,\nprice endogeneity is rarely considered in storage bidding strategies and\nmodeling the electricity market is a challenging task. Meanwhile, model-free\nreinforcement learning such as the Actor-Critic are becoming increasingly\npopular for designing energy system controllers. Yet implementation frequently\nrequires lengthy, data-intense, and unsafe trial-and-error training. To fill\nthese gaps, we implement an online Supervised Actor-Critic (SAC) algorithm,\nsupervised with a model-based controller -- Model Predictive Control (MPC). The\nenergy storage agent is trained with this algorithm to optimally bid while\nlearning and adjusting to its impact on the market clearing prices. We compare\nthe supervised Actor-Critic algorithm with the MPC algorithm as a supervisor,\nfinding that the former reaps higher profits via learning. Our contribution,\nthus, is an online and safe SAC algorithm that outperforms the current\nmodel-based state-of-the-art.\n