This research addresses the challenge of integrating forecasting and optimization in energy management systems, focusing on the impacts of switching costs, forecast accuracy, and stability. It proposes a novel framework for analyzing online optimization problems with switching costs and enabled by deterministic and probabilistic forecasts. Through empirical evaluation and theoretical analysis, the research reveals the balance between forecast accuracy, stability, and switching costs in shaping policy performance. Conducted in the context of battery scheduling within energy management applications, it introduces a metric for evaluating probabilistic forecast stability and examines the effects of forecast accuracy and stability on optimization outcomes using the real-world case of the Citylearn 2022 competition. Findings indicate that switching costs significantly influence the trade-off between forecast accuracy and stability, highlighting the importance of integrated systems that enable collaboration between forecasting and operational units for improved decision-making. The study shows that committing to a policy for longer periods can be advantageous over frequent updates. Results also show a correlation between forecast stability and policy performance, suggesting that stable forecasts can mitigate switching costs. The proposed framework provides valuable insights for energy sector decision-makers and forecast practitioners when designing the operation of an energy management system.