Recently increasing penetration of renewable energy generation brings challenges for power system operators to perform efficient power generation daily scheduling, due to the intermittent nature of the renewable generation and discrete decisions of each generation unit. Among all aspects to be considered, unit commitment polytope is fundamental and embedded in the models at different stages of power system planning and operations. In this paper, we focus on deriving polynomial time algorithms for the unit commitment problems with general convex cost function and piecewise linear cost function respectively. We refine an $\mathcal{O}(T^3)$ time, where $T$ represents the number of time periods, algorithm for the deterministic unit commitment problem with general convex cost function and accordingly develop an extended formulation in a higher dimensional space that provides integral solutions in which the physical meanings of the decision variables are described. Furthermore, for the case in which the cost function is piecewise linear, by exploring the optimality conditions, we derive more efficient algorithms for both deterministic (i.e., $\mathcal{O}(T)$ time) and stochastic (i.e., $\mathcal{O}(N)$ time, where $N$ represents the number of nodes in the stochastic scenario tree) unit commitment problems. We also develop the corresponding extended formulations for both deterministic and stochastic unit commitment problems that provide integral solutions. Similarly, physical meanings of the decision variables are explored to show the insights of the new modeling approach.
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