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Seasonal thermal energy storage systems (STESSs) can shift the delivery of renewable energy sources and mitigate their uncertainty problems. However, to maximize the operational profit of STESSs and ensure their long-term profitability, control strategies that allow them to trade on wholesale electricity markets are required. While control strategies for STESSs have been proposed before, none of them addressed electricity market interaction and trading. In particular, due to the seasonal nature of STESSs, accounting for the long-term uncertainty in electricity prices has been very challenging. In this article, we develop the first control algorithms to control STESSs when interacting with different wholesale electricity markets. As different control solutions have different merits, we propose solutions based on model predictive control and solutions based on reinforcement learning. We show that this is critical since different markets require different control strategies: MPC strategies are better for day-ahead markets due to the flexibility of MPC, whereas reinforcement learning (RL) strategies are better for real-time markets because of fast computation times and better risk modeling. To study the proposed algorithms in a real-life setup, we consider a real STESS interacting with the day-ahead and imbalance markets in The Netherlands and Belgium. Based on the obtained results, we show that: 1) the developed controllers successfully maximize the profits of STESSs due to market trading and 2) the developed control strategies make STESSs important players in the energy transition: by optimally controlling STESSs and reacting to imbalances, STESSs help to reduce grid imbalances.