Hydropower plants play a crucial role in the power system facing ambitious renewable energy targets. Due to their inherent controllability, they are well suited to provide flexibility to the grid. However, an increased flexibility provision leads to a prolonged operation in off-design conditions and more frequent hydraulic transient, notoriously detrimental in terms of dynamic loads induced on the hydraulic machines. The aim of this paper is to provide a methodology to estimate the damage on Pelton turbines due to the accumulated mechanical stress during their start-up sequence. Transient sequences, as the back-to-back start-up of centrifugal pumps in a pumped-storage power plant, requires the hydraulic machine to operate in conditions featuring high mechanical torque values, leading to an increased structural damage with respect to the nominal operating point. In view of the above, the paper proposes a polynomial optimization algorithm to decrease the fatigue-related operational cost induced on the runner during a start-up sequence. The performance of the algorithm is assessed on a real test case. The presented algorithm identifies the admissible injector opening rate as the crucial parameter in the start-up computation, and shows the dependence of the optimal opening law on the admissible rate with the bucket resonance characteristics.