Sharp formulations of nonconvex piecewise linear functions to solve the economic dispatch problem with valve-point effects

•New mathematical programming techniques to solve the nonconvex ED are proposed.•A new breakpoint assignment method for piecewise linear approximations is presented.•CC, DCC, MC, Inc, and Log techniques are used to model the ED with valve effects.•Sharp and locally ideal properties of the mixed-integer models are discussed.•Converging consistently to a unique high-quality solution can be guaranteed.•Two reformulation methods are used to handle non-differentiable absolute value terms. Wire drawing effects in a unit with several steam admission valves produce some ripples in the unit cost characteristic. Representing the ripples as sinusoidal terms gives rise to a nonconvex cost characteristic. This paper focuses on the solution of the economic dispatch (ED) problem with the nonconvex cost functions. To this end, this work presents a new framework in which, firstly, the complex problem is tightly transformed into a set of linear programming (LP) problems using an iterative piecewise linear function (PLF) approximation especially developed for the nonconvex terms. Subsequently, five strong models including convex combination, disaggregated convex combination, multiple choice, incremental, and logarithmic techniques are proposed to represent the set of built LP problems as a mixed-integer programming (MIP) problem which efficiently can be solved by a robust powerful algorithm. Then, the obtained solution from the MIP algorithm is used to warm-start an enhanced nonlinear programming (NLP) model and to further improve the solution. In extensive experiments, we compare the performance of the presented methods in terms of solution quality, computational efficiency, and consistency with other approaches in the literature of the nonconvex ED. The tight PLF approximation and the five sharp and generally locally ideal MIP formulation techniques in conjunction with the existing efficient MIP and NLP algorithms result in the strong solution framework which can find a global or a virtually global solution and surpass the state-of-the-art methods in the area of the nonconvex ED in all considered case studies.