A comparison of the homotopy method with linearisation approaches for a non-linear optimization problem of operations in a reservoir cascade

Reservoir systems are often operated for multiple purposes. This can result conflicting operational goals. To efficiently control these systems and to satisfy the different interests as good as possible, mathematical optimization models can be used to support operational decisions. Common approaches for reservoir optimization apply linear optimization techniques. However, real-world systems often require non-linear functions to describe the relation between water level and volume in a reservoir or to account for the hydropower equation. When the non-linear equations form a non-convex optimization problem, the problem is not necessarily solved to a global optimum. Piecewise-linear or linear formulations of the non-linear equations are a common way to address non-linear non-convex optimization problems. In this paper, the novel homotopy method is compared with two established approaches—the piecewise-linear and the linear approximation—to account for non-linear components in the optimization problem. The analysis is carried out for a cascade of three reservoirs under two scenarios—a flood scenario and a load balance scenario. The optimization software is the open source package RTC-Tools 2.4. Compared to the piecewise-linear and the linear approach the homotopy method shows a better accuracy for the analysed cases, because the method solves the flow equations within the optimization in a non-simplified form. Different to the piecewise-linear and the linear approach, however, the homotopy method does not guarantee a global optimum. The solution is still path-stable, which is a basic pre-requisite for its application in an operational context of hydropwer scheduling. Compared to the piecewise-linear approach, the homotopy method is easier to implement under the condition that the software supports the method.