Network virtualization has been widely considered to improve the resource efficiency of network infrastructure by allowing multiple virtual networks to coexist on a shared substrate network. With the exponential growth of Internet traffic, the network energy consumption incurs a considerable increase around the world. In this paper, we consider energy-aware virtual optical network embedding (EA-VONE) issue in flexible-grid elastic optical networks (EONs), while sliceable transponders (TPs) are assumed to be equipped in each node. First, an integer linear programming (ILP) model for the energy-minimized VONE is developed to optimally solve this problem. Due to the non-scalability of the ILP model, we then design two energy-saving policies for data centers (DCs) and TPs in the process of the node mapping and link mapping, respectively. Based on the two policies, two heuristic schemes are also developed: 1) DC-EA scheme, which only considers DC energy-saving in the node mapping procedure and 2) DC&TP-EA scheme, which simultaneously considers the energy-saving for both the DCs in the node mapping and TPs in the link mapping. Moreover, offline and online EA-VONE algorithms are developed. To investigate the benefits of the two energy-saving (i.e., DC-EA and DC&TP-EA) schemes, a benchmark scheme is also realized, which only maintains traffic-balancing (TB) policy without any energy-saving consideration. The simulation results indicate that our proposed DC&TP-EA scheme achieves the maximum power saving efficiency compared with the DC-EA and TB schemes. Also, in the DC-EA and DC&TP-EA schemes, the effect of DC energy saving is very remarkable when the traffic load is smaller. With the increase of traffic load, the DC energy saving may matter less, meanwhile, the TP energy-saving plays a more and more important role. Moreover, both the DC&TP-EA and DC-EA schemes maintain similar blocking performance as that of the TB scheme, which shows the superiority of our proposed schemes.