In this work, we consider a scheduling problem faced by production companies with large electricity consumption. Due to the contract with the electric utility, the production companies are obligated to comply with the total energy consumption limits in the specified time intervals (usually 15-minutes long), otherwise, the companies pay substantial penalty fees. Although it is possible to design production schedules that consider these limits as hard constraints, uncertainties occurring during the execution of the schedules are usually not taken into account. This may lead to situations in which the unexpected delays of the operations cause the violations of the energy consumption limits. Our goal is to design robust production schedules pro-actively guaranteeing that the energy consumption limits are not violated for the given set of uncertainty scenarios. We consider scheduling on one machine with release times of the operations and total tardiness as the objective function. To tackle this problem, we first propose a pseudo-polynomial algorithm for finding the optimal robust schedule for the given permutation of the operations. This algorithm is then utilised in three different algorithms for finding the optimal permutation: two exact (Branch-and-Bound and logic-based Benders decomposition) and one heuristic algorithm (tabu search). All the algorithms were experimentally evaluated on random instances with different sizes of the uncertainty scenarios set. Using the tabu search algorithm, we are able to solve large instances within one minute.