How much energy is saved when using luggage with wheels? How much energy is saved when using luggage with wheels as opposed to carrying the luggage? Thanks.
 A: There is no fundamental physics reason why carrying a bag should use more energy than wheeling it. Whether you be carrying a bag or wheeling it, you are essentially sliding it along a line of almost constant gravitational potential (aside from a little jiggling up and down with your stride), so the bag's total energy isn't changing and in theory does not need to be topped up.
The reason why animals like us expend more energy carrying than wheeling their bags is wholly to do with how animal muscles work. If you could clamp your hand shut around the handle, you would expend no effort carrying the bag. A muscle, even though it does no work on the bag, must be constantly supplied with energy in various chemical forms by the body to stay in a contracted state. This work is dissipated as heat in your forearm muscles as they holds the hand clamped shut. The process is explained in more detail in Malabarba's Answer to the Physics SE Question, "Why does holding something up cost energy while no work is being done?". 
A further two factors are that:


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*In the contracted state, muscles are metabolising ATP and other energy stores and outputting waste, which swiftly "poisons" the tissue if it builds up faster than the circulatory system can get rid of it, leading to a lowered ability of the muscle to work and "tiring";

*The contracted state of a muscle, and the accompanying stretched state of connective tissue actually hinders blood circulation to both. 
Your muscles and connective tissue can brook neither of these things for long without needing a rest: this is why your arm feels like it's going to drop off if you need to walk without a wheeled truck for you bags for long.
AS for the exact amount of energy used by a muscle to stay in the contracted state, i.e. a quantitative answer to your question, you may need to put this to the biology stack exchange.
