A robust optimal design method based on the minimax regret criterion is proposed to determine equipment capacities of energy supply systems so that they are robust economically against the uncertainty in energy demands. Equipment capacities and utility contract demands as well as energy flow rates are determined to minimize the maximum regret in the annual total cost while satisfying all the possible energy demands. This optimization problem is formulated as a kind of multilevel linear programming problem, and its solution is obtained by repeatedly evaluating lower and upper bounds for the maximum regret in the annual total cost. Through a numerical case study on a cogeneration system, it is shown that determining equipment capacities appropriately is more important under uncertain energy demands rather than certain ones. It is also shown that the ratio of power generating capacity to total power supply capacity decreases with an increase of the uncertainty in energy demands.