Abstract
The mathematical theory for optimal switching is by now relatively well developed, but the number of concrete applications of this theoretical framework remains few. In this paper, we bridge parts of this gap by applying optimal switching theory to a conceptual production planning problem related to hydropower. In particular, we study two examples of small run-of-river hydropower plants and provide an outline of how optimal switching can be used to create fully automatic production schemes for these. Along the way, we create a new model for random flow of water based on stochastic differential equations and fit this model to historical data. We benchmark the performance of our model using actual flow data from a small river in Sweden and find that our production scheme lies close to the optimal, within 2 and 5 %, respectively, in a long term investigation of the two plants considered.
Highlights
Small-scale hydropower plants are in many cases of “run-of-river” type (ROR) meaning that any dam or barrage is small, usually just a weir, and generally little or no water can be stored
We benchmark the performance of our model using actual flow data from a small river in Sweden and find that our production scheme lies close to the optimal, within 2 and 5 %, respectively, in a long term investigation of the two plants considered
Leap days are excluded in favour of a coherent presentation of the results
Summary
Small-scale hydropower plants are in many cases of “run-of-river” type (ROR) meaning that any dam or barrage is small, usually just a weir, and generally little or no water can be stored. ROR hydropower plants are common in smaller rivers and exist in larger sizes such as the Niagara Falls hydroelectric plants (Canada/USA), the Chief Joseph dam on the Columbia River (Washington, USA) or the Saint Marys Falls hydropower plant in Sault Sainte Marie (Michigan, USA). The optimal management strategy can be found trivially by starting the machine when flow is sufficient and stopping it when flow is insufficient. In rivers with large and rapid flow fluctuations, which is typically the case in smaller unregulated rivers, such strategies can lead to a large number of changes in the production, the cost of which can not be neglected; for example, starting and stopping the turbines induces wear and tear on the machines and may require intervention from personnel. The major breakdown in the Akkats hydropower plant (Lule river, Sweden) 2002 was caused by a turbine being stopped too quickly, resulting in rushing water destroying the foundation of both the turbine and the generator (Yang 2010; Yang et al 2018)
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