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In this paper, we investigate both static and dynamic traffic grooming problems in a wavelength routing network, so as to minimize the total energy consumption of the core network, with the additional consideration of the holding times of the lightpaths and connection requests. In the static case, all connection requests with their setup and tear-down times are known in advance. We formulate an integer linear programming to minimize the energy consumption. In the dynamic case, we adopt a layered graph model called grooming graph and propose a new traffic grooming heuristics called time-aware traffic grooming, which takes the holding time of a new arrival connection request and the remaining holding time of existing lightpaths into consideration. We compare different traffic grooming policies, in terms of the energy efficiency, under various traffic loads. The results provide implications to select the most energy-efficient traffic grooming policies under various scenarios.