Using Semidefinite Relaxation to Solve the Day-Ahead Hydro Unit Commitment Problem

Abstract

In this paper, we present a mixed-integer formulation with nonlinear production functions of the day-ahead hydro unit commitment problem for hydro-dominant power systems. The objective is to minimize unit startup and shutdown costs and maximize water use efficiency in the daily operation of multiple hydro plants. Determination of hourly generation dispatches and unit configurations is subject to reservoir dynamics, power balance, as well as target constraints coupling short-term generation with long-term reservoir operation goals. In addition, ac power flow constraints are considered. An equivalent quadratically-constrained quadratic formulation is approximated by successive convex semidefinite programming relaxations in a branch-and-bound algorithm for optimal solutions. Numerical case studies for several system configurations are presented to illustrate the effectiveness of the relaxation algorithm.

Financial storage rights for hydroelectricity

“…Depending on reservoir geometry, its relative capacity with respect to water discharge, and turbine type, head and efficiency factors might be more prominently influenced by tailrace height, as forebay height changes slowly in timescales of hours or days, especially and more significantly in very large upper reservoirs. In this case, holding forebay height constant in (1) is a common approach [21], [22], [23], although the modeling of head variability by forebay height is not uncommon [24], [25].…”

A Low-Rank Rounding Heuristic for Semidefinite Relaxation of Hydro Unit Commitment Problems

“…It is based on the quadratic relaxation of discrete variables of the day-ahead HUC problem with hourly discretization [11], such that:…”

“…In this paper a quadratic model [11] of the hydroelectric production function is used, such that water discharge can be formulated as a function of active power generation, i.e:…”

“…where N U t,h is the number of unit configurations available for commitment, and y t,h ≥ 0 is the number of units scheduled for startup. In order to solve this mixed-integer, quadratically-constrained quadratic program (MIQCQP) in a computationally efficient manner, we initially solve a semidefinite programming-based (SDP) convex relaxation [12] of the problem with AC power flow formulation, as originally proposed in [11]. This convex SDP relaxation has been previously shown to guarantee valid lower bounds with relatively small gaps.…”

Sequential Relaxation of Unit Commitment with AC Transmission Constraints

“…Conic inequalities are proposed in [14], [35]- [37] to strengthen convex relaxations in the presence of nonlinear cost functions. In [38], a combination of semidefinite programming relaxation and branch-and-bound is used to solve the day-ahead hydro unit commitment problem. In [39], [40], reformulation-linearization cuts are proposed to strengthen semidefinite programming relaxations of unit commitment.…”