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287701 A Global Optimization Approach for the Short-Term Scheduling of Hydro Power Generation

The paradigm of production and consumption of electricity has been changing due to increasing electricity demand, time variable electricity prices for industry, the growing penetration of renewable sources, and due to public policies regarding supply security and environmental impact. Hydro power is a widely used form of renewable energy with clear advantages in terms of no fuel consumption, and no direct waste or CO2 emissions. Furthermore, hydro plants have high flexibility to be re-scheduled and the capability to store energy using pumped hydro storage in coordination with wind power generation.

This paper addresses the optimization of the short-term scheduling of a cascade of hydro plants using detailed and accurate models, and a deterministic global optimization approach to deal with the non-convexities. Cascades are represented by integrated systems of hydro plants involving plants with reservoirs and plants without the capacity to store water, the so-called run-off-the-river plants. The need for this integrated and necessarily more complex approach derives from the strong interaction that arises between upstream and downstream plants caused by river connection and water releases. The scheduling of a hydro plant involves the determination of the start and duration of the operation of each turbine for the next day, with the time horizon divided into periods of one hour. The operation of each turbine is characterized by its output flow and the respective energy generated for each period of one hour that maximizes the operating profit of the cascade subject to the water management constraints of each reservoir and limits on the performance of the turbines.

Most of the scheduling models for hydro power generation proposed in the literature are based on Mixed Integer Linear Programming (MILP) models, where simplified expressions are used to calculate the power generated by the hydro plants and the forebay and tailrace levels, or where piecewise linear approximations are considered to replace nonlinear functions (Conejo et al. 2002, Borghetti et al. 2008). In addition, most models include a single turbine per plant. Recently, some authors have proposed more detailed models involving nonlinearities in the calculation of the power output, resulting in Mixed Integer NonLinear Programming (MINLP) models that are solved without guarantee of global optimality (Catalão et al. 2009, Diaz et al. 2011).

In this work hydro plants are considered, where the majority has a set of turbines in parallel linked to the same reservoir, but with different specifications in terms of maximum turbined flow and power output. A model is developed, which is based on a discrete time formulation with a time horizon of one day, divided in time periods of one hour, involving a) the respective time dependent constraints for mass balances of water, and power generated; b) the topological, reservoir and plant constraints; c) nonlinear functions to determine the power generated of each turbine; and d) polynomial functions to calculate the forebay and tailrace levels. Furthermore, the power output of each turbine is a function of three variables: 1) the flow rate of water discharged by the turbine; 2) the tailrace level of water downstream the plant; and 3) the forebay level of water in the reservoir. Therefore, the short-term scheduling results in a detailed MINLP problem, involving bi-linear terms and fourth degree polynomials.

In order to find the global optimum within a pre-specified tolerance, a Branch and Bound framework is proposed, which is based on the solution of an MILP overestimator model and of a MINLP model in each node of the tree. In the MILP model the bilinear terms are replaced by piecewise linear estimators defining their convex hull envelopes over partitions (McCormick, 1976, Sherali and Alameddine 1992, Bergamini et al. 2005), while the fourth degree polynomials are relaxed using valid over and under estimators. The formulation of the MILP model is improved by enforcing symmetry breaking constraints on the turbines with the same specifications that operate in parallel, and by a specific partition for the relaxations of the bilinear terms. This partition is constructed taking into consideration that one of the variables involved in the bilinear term is a semi-continuous variable, and therefore forces the minimum positive value of this variable to be a bound of the first region of the partition.

The MILP model provides a tight linear overestimation of the nonconvex region of the original MINLP problem and thus a valid upper bound of the objective function on each node of the Branch and Bound framework. The lower bound of the original MINLP problem is obtained by fixing the binary variables associated with the turbines at the value of the MILP solution if they are equal to one, leaving the variables equal to zero free, and solving a reduced MINLP problem. The larger difference between the non-linear term and its approximation in the MILP model is used to select the branching variable. The search stops once the pre-specified global optimality has been reached.

In order to assess the performance of the proposed framework, three case studies consisting on cascades with a different number of hydro plants and turbines with varied specifications are presented. The results show that the proposed MILP over estimator leads to a tighter upper bound, and then small gaps between the upper and lower bound can be achieved. In addition, the best solutions obtained represent an improved profit when compared with solutions obtained with local solvers for MINLP problems. The performance of the proposed framework is also generally superior to the one obtained with the available solvers for global optimization.

**References**

Bergamini, M. L., Aguirre, P., & Grossmann, I. E. (2005). Logic-based outer approximation for globally optimal synthesis of process networks. Computers and Chemical Engineering., 29, 1914–1933.

Borghetti, A. , D’Ambrosio C., Lodi A., Martello S., 2008, An MILP approach for short-term hydro scheduling and unit commitment with head-dependent reservoir, IEEE Trans. Power Syst., 23,3,1115-1124.

Catalão, J.P.S., Mariano, S.J.P.S., Mendes, V.M.F., and L.A.F.M. Ferreira, 2009, Scheduling of head-sensitive cascade hydro systems: A nonlimear approach, IEEE Trans. Power Syst., 24,1, 336-346.

Conejo, A.J. , Arroyo, J.M., Contreras J., Villamor, F.A., 2002, Self-Scheduling of a Hydro Producer in a Pool-Based Electricity Market. IEEE Trans. Power Syst., 17, 4, 1265-1272.

Díaz, F.J., Contreras, J., Munoz, J. I., Pozo D., 2011, Optimal scheduling of a price taker cascade reservoir system in a pool-based electricity market, IEEE Trans. Power Syst., 26,2,604-615.

McCormick, G. P. (1976). Computability of global solutions to factorable nonconvex programs. Part I. Convex underestimating problems. Mathematical Programming, 10, 146–175.

Sherali, H. D., & Alameddine, A. (1992). A new reformulation linearization technique for bilinear programming problems. Journal of Global Optimization, 2, 379–410.

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