Current network infrastructures are over-provisioned and thus exhibit poor power efficiency at low traffic load. In this paper, we consider networks comprising of bundled links, whereby each link has one or more physical cables that can be switched off independently. The problem at hand is then to switch off redundant cables during off peak periods, while retaining the QoS provided to existing traffic demands. Unfortunately, the problem to maximally shutdown redundant cables is an NP-complete problem. Henceforth, we design a fast heuristic, called Multiple Paths by Shortest Path First (MSPF), that aims to maximize the number of switched-off cables subject to satisfying maximum link utilization (MLU) and end-to-end delay requirements. We have extensively evaluated the performance of MSPF on both real and synthetic topologies and traffic demands. Further, we have compared its performance against two state-of-the-art techniques: GreenTE usable only when each link has one cable, and FGH that supports bundled links but usable only for networks without MLU and delay constraints. MSPF improves the energy saving on average by 5% as compared to GreenTE incurring only 1% the CPU time. While yielding equivalent energy savings, MSPF requires only 0.35% of the running time of FGH. Finally, for MLU at most 50% and end-to-end delay no longer than the network diameter, MSPF reduces the power usage of the GÉANT topology up to 91% and bundled links consisting of ten cables.