Optimal unit commitment by branch-and-bound exploiting dual optimization conditions (Q2740061)

The authors consider the unit commitment problem in combination with the economic dispatch problem as the following nonlinear mixed integer mathematical programming problem NEWLINE\[NEWLINE \min\limits_{p,u}[\sum\limits_{i=1}^{I}\sum\limits_{k=1}^{K}(\alpha_ {k}^{2}(p_{i,k})^2+ \alpha_{k}^{1}p_{i,k}+ \alpha_{k}^{0})u_{i,k}+\sum\limits_{i=1}^{I}\sum\limits_{k=1}^{K}( 1-u_{i-1,k})u_{i,k}\gamma_{k}] NEWLINE\]NEWLINE NEWLINE\[NEWLINE s.t.\quad \sum\limits_{k=1}^{K}p_{i,k}=p_{i,D};\;\sum\limits_{k=1}^{K}\bar p_{k}u_{i,k}\geq p_{i,R};\;\underline{p}_{k}u_{i,k}\leq p_{i,k}\leq\bar p_{k}u_{i,k};\;u_{i,k}\in\{0,1\}, NEWLINE\]NEWLINE where \(p_{i,k}\) is the power production for unit \(k\) in time interval \(i\); \(u_{i,k}=1\) if unit \(k\) is producing in time interval \(i\) and \(u_{i,k}=0\) if not producing; \((\alpha_{k}^{2}(p_{i,k})^2+ \alpha_{k}^{1}p_{i,k}+ \alpha_{k}^{0})u_{i,k},\;\alpha_{k}^{2}>0\) is the cost for producing power in a production unit; \((1-u_{i-1,k})u_{i,k}\gamma_{k}\) is the start-up cost, \(p_{i,D}\) is the power demand and \(p_{i,R}\) is the reserve requirement. In this paper the branch-and-bound algorithm is presented. Applying the Lagrangian relaxation the convex but non-smooth dual problem is obtained. The lower bounds on the optimal function value are computed from the dual objective function. Some computational results are also given.