Solving the unit commitment problem of hydropower plants via Lagrangian Relaxation and Sequential Quadratic Programming

Abstract

Abstract.We consider the optimal scheduling of hydropower plants in a hydrothermal interconnected system. This problem, of outmost importance for large-scale power systems with a high proportion of hydraulic generation, requires a detailed description of the so-called hydro unit production function. In our model, we relate the amount of generated hydropower to nonlinear tailrace levels; we also take into account hydraulic losses, turbine-generator efficiencies, as well as multiple 0-1 states associated with forbidden operation zones. Forbidden zones are crucial to avoid nasty phenomena such as mechanical vibrations in the turbine, cavitation, and low efficiency levels. The minimization of operating costs subject to such detailed constraints results in a large-scale mixed-integer nonlinear programming problem. By means of Lagrangian Relaxation, the original problem is split into a sequence of smaller and easy-to-solve subproblems, coordinated by a dual master program. In order to deal better with the combinatorial aspect introduced by the forbidden zones, we derive three different decomposition strategies, applicable to various configurations of hydro plants (with few or many units, which can be identical or different). We use a Sequential Quadratic Programming algorithm to solve nonlinear subproblems. We assess our approach on a real-life hydroelectric configuration extracted from the south sub region of the Brazilian hydrothermal power system.

An MILP Approach for Short-Term Hydro Scheduling and Unit Commitment With Head-Dependent Reservoir

“…This task is far from being trivial for various reasons. It is clear that the model would require the continuity equations of the hydro reservoirs, taking into account the relevant constraints such as branch flow limits and water travel time (see, e.g., [7]- [9], [11], [13]). The main drawback, however, could be that a cascade hydro system model for, say, power plants would multiply by the number of variables and constraints of BDLM+, and the computational experiments of Section IV have shown that the performance of the model is heavily affected by its size.…”

“…In [6] a multistage looping optimization algorithm was proposed for the development of the optimal bidding strategies of an individual pumped-storage unit owner in a competitive electricity market. In [7] the large-scale mixed-integer NLP problem of determining the optimal scheduling of hydropower plants in a hydrothermal interconnected system is considered: the authors use Lagrangian relaxation decomposition strategies, and a sequential quadratic programming algorithm to solve nonlinear subproblems. Various mixed-integer linear programming (MILP) approaches have been presented in the literature: for example, [8] and [9] used the interior point method within a branch-and-bound algorithm, while [10]- [13] used the Ilog-Cplex [14] MILP solver under GAMS.…”

Application-oriented mixed integer non-linear programming

“…This task is far from being trivial for various reasons. It is clear that the model would require the continuity equations of the hydro reservoirs, taking into account the relevant constraints such as branch flow limits and water travel time (see, e.g., [37,38,59,39,49]). The main drawback, however, could be that a cascade hydro system model for, say, k power plants would multiply by k the number of variables and constraints of BDLM+, and the computational experiments of Section 6.4 have shown that the performance of the model is heavily affected by its size.…”

“…In [88] a multistage looping optimization algorithm was proposed for the development of the optimal bidding strategies of an individual pumped-storage unit owner in a competitive electricity market. In [49] the large-scale mixed-integer NLP problem of determining the optimal scheduling of hydropower plants in a hydrothermal interconnected system is considered: the authors use Lagrangian relaxation decomposition strategies, and a sequential quadratic programming algorithm to solve non-linear subproblems. Various Mixed Integer Linear Programming (MILP) approaches have been presented in the literature: for example, [37] and [38] used the Interior Point method within a Branch-and-Bound algorithm, while [11], [59], [60] and [39] used the Ilog-Cplex [71] MILP solver under GAMS.…”

Peak Operation of Cascaded Hydropower Plants Serving Multiple Provinces

“…Mathematically, the above problem is a highly nonlinear and nonconvex optimization problem with the nonlinear objective function of peak shaving and large amounts of spatial-temporal coupling system and plant operation constraints [6,7]. The simplification of such complex objective functions and constraints make it hard to obtain directly an analytic solution or a discrete optimum using linear programming [8,9], nonlinear programming [10][11][12], or dynamic programming [13][14][15]. Moreover, these analytic methods are closely dependent on the computable requirements of available commercial software.…”