When considering the design of electricity-intensive industrial processes, a challenge is that future electricity prices are highly uncertain. Design decisions made before construction can affect operations decades into the future. We thus explore whether including electricity price uncertainty into the design process affects design decisions. We apply stochastic optimization to the design and operations of a chlor-alkali plant, an electrochemical process that produces chlorine, caustic soda, and hydrogen. Chlor-alkali production is electricity intensive and can be operated flexibly based on fluctuating electricity prices. We consider participation in the 5-minute real time market and consider each day as a scenario in the stochastic program. We find that flexible plant designs that oversize certain plant components can enhance participation in electricity markets and increase profits. When electricity-price uncertainty is considered by using stochastic optimization, the optimal system design includes fuel cell and hydrogen storage capacity, which allow the plant to hedge against price uncertainty. We furthermore find that scenario reduction techniques in our example approximate the expected objective function value well, but lead to error in terms of optimal design decision variables. This error ranges from not building some components (fuel cell and hydrogen storage capacity) to overestimating their capacities by 50%.