In the electricity sector the carbon tax is a common environmental policy aiming to reduce CO2 emissions, but is often regarded as economically unfriendly, especially for areas relying on coal-fire and other carbon-intensive generators. A power grid utilizing an energy storage system can be a promising solution to alleviate the regional economy pressure in a grid where the carbon tax is enforced. With the increasing exploitation of clean energy, e.g., solar and wind power, in this work, we characterize the stochastic emission-aware economic dispatch with a storage system utilizing two frameworks, namely a chance-constrained framework and a robust optimization framework. We highlight their differences and connections by studying the trade-offs between robustness and overall cost. Specifically, we bridge the two frameworks with a novel distributed robust optimization framework that considers practical bounds to estimate the optimal system performance under the reliability requirement. Numerical studies on the six-bus model and the IEEE-118 bus model further justify our findings.