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## Hydro generation scheduling: non-linear programming and optimality conditions

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A mathematical analysis based on hydroelectric plant characteristics, i.e. reservoir and tailrace geometry, is the central chapter's subject. Deterministic mid-short term scheduling planning could be solved to optimality if generation function is at least biconcave, i.e. a discontinuous reservoir and a uniform tailrace. Although the formulation is not proved to be unimodal, a B&B method based on concave envelope relaxation converges to global optimum. In a broader sense, increasing property is established for hydro-generation function. As the formulation is linearly constrained, i.e. only linear inequalities, special algorithms could exploit boundary of the feasible set (a polytope) to find a global optimum. Unimodularity property could also be studied in general case, i.e. arbitrary reservoir and tailrace height.

Chapter Contents:

• 8.1 Introduction
• 8.2 Hydropower generation function
• 8.2.1 Physical properties of geometric functions
• 8.2.1.1 Increasing property
• 8.2.1.2 Convexity and concavity
• 8.2.2 Special cases of geometric functions
• 8.2.2.1 Examples
• 8.2.3 Mathematical properties
• 8.2.3.1 Run-off-the-river plant with variable tailrace height
• 8.2.3.2 Storage plant with constant tailrace height
• 8.2.3.3 Stair-like reservoir with a uniform tailrace
• 8.2.3.4 Storage plant with variable tailrace height
• 8.3 Water conservation and discharge limits
• 8.3.1 Head sensitive discharge limits
• 8.5 Global optimization approach
• 8.5.1 Computational results
• 8.6 Conclusions
• References

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