The Hydro Unit Commitment problem (HUC) specific to hydroelectric units is part of the electricity production planning problem, called Unit Commitment Problem (UCP). More specifically, the case studied is that of the HUC with a single unit, denoted 1-HUC. The unit is located between two reservoirs. The time is discretized in time periods. The unit operates at a finite number of points defined as pairs of the generated power and the corresponding water flow. Several constraints are considered. Each reservoir has an initial volume, as well as a maximum and minimum volume per time period. The unit has both ramping and min-up/min-down constraints limiting the flow rate variation between two consecutive time periods. At each time period, there is an additional positive, negative or zero intake of water in the reservoirs. The case of a pricetaker revenue maximization problem is considered. An efficient A* algorithm is first proposed to solve a simplified variant of the 1-HUC without ramping nor min-up/min-down constraints. The algorithm is very efficient to solve difficult instances and is shown to have a more stable behavior than CPLEX on realistic EDF instances. For the complete variant of the 1-HUC including both the ramping and min-up/min-down constraints the A* algorithm, as well as standard resourceconstrained shortest path approaches are shown to be ineffective. The latter change is due to the weakening of the dominance rules. To overcome this difficulty, we propose a two-phase algorithm inspired by algorithms used in bi-criteria optimization. The first phase solves extremely fast simple shortest path problems while the second phase is based on a K-best solutions algorithm to enumerate feasible solutions in a restricted search space. We compare the dedicated two-phase algorithm with a mixed-integer linear program solved by CPLEX and show that the proposed algorithm outperforms CPLEX on instances with 48 time periods, 20 operating points and a small price variability.