Current network infrastructures are over‐provisioned and thus exhibit poor power efficiency at low traffic load. We consider networks consisting of bundled links, whereby each link has one or more physical cables that can be switched off independently. The problem at hand is to switch off redundant nodes and cables during off‐peak periods, while retaining the quality of services provided to existing traffic demands. Unfortunately, the problem to maximally shutdown redundant nodes and cables is Non‐deterministic Polynomial‐time (NP)‐complete. Henceforth, we design a fast heuristic, called Multiple Paths by Shortest Path First (MSPF), that aims to maximise the number of switched‐off nodes and cables subject to satisfying maximum link utilisation (MLU) and path length (PL) constraints. 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 when each link has one cable, and Fast Greedy Heuristic (FGH), which supports bundled links but only for networks without MLU and PL constraints. On the Sprint network, MSPF can save on average 5% more power as compared to GreenTE while incurring only 1% of GreenTE's running time. While yielding equivalent power savings, MSPF requires only 0.35% of FGH's running time. Finally, setting MLU to at most 50% and PL to no longer than the network diameter, MSPF reduces the power usage of the GÉANT topology by up to 91% for links consisting of 10 cables. For experiments on synthetic topologies with bundle size of 30, MSPF yields a power saving of 89.81%. Copyright © 2014 John Wiley & Sons, Ltd.