Parallel computing applied to the stochastic dynamic programming for long term operation planning of hydrothermal power systems

Abstract

In this paper, parallel processing techniques are employed to improve the performance of the stochastic dynamic programming applied to the long term operation planning of electrical power system. The hydroelectric plants are grouped into energy equivalent reservoirs and the expected cost functions are modeled by a piecewise linear approximation, by means of the Convex Hull algorithm. In order to validate the proposed methodology, data from the Brazilian electrical power system is utilized.

Introduction

The electric power system operation planning aims at determining the hydro and thermoelectric generation targets, which minimizes the expected operation cost of the whole system in a given horizon (i.e. 5 years), while minimum quality and reliability standards are met in Pereira (1989).

In hydrothermal systems operation planning, such a goal is attained by dispatching hydroelectric power plants as much as possible, whereas other energy resources, such as thermal, are used to match the demand and, consequently, improving the overall system reliability.

The Brazilian Power System (BPS) is an example of a large scale hydrothermal system, with almost 75% of its installed capacity based on hydroelectric generation. In the BPS the operation planning is categorized according to the study horizon, which range from the dispatch (a week ahead) to the long term operation planning (5 years ahead). In the hydrothermal dispatch category, the constraints are represented in detail, as the reservoir operational restrictions are known, while the stochasticity of the water inflow has little impact. The uncertainty of the water inflows becomes more significant as the considered horizon increases, for instance, in the long term operation planning (see Maceira et al., 2002).

Other aspects also influences the long term operation planning (LTOP), such as spatial and temporal coupling. A common practice in large watersheds is erecting several hydro dams in the same river basin in order to increase the electricity production (Grygier and Stedinger, 1985). This implies that the operation of an upstream reservoir interferes in the operation of a downstream one (see Xi et al., 1999, Terry et al., 1986). Hence, spatial coupling among reservoirs need to be taken into account. Moreover, the decision to use hydro resources in the present may incur a higher operational cost in the following months, due to the need of dispatching expensive thermal generators or even shedding load (Silva, 2001).

In a general manner, the dynamic programming is a technique used to solve sequential optimization problems by enumerating states that are likely to happen in each stage (see Bellman, 1957, Ferrero et al., 1998). In the dynamic programming sense, the optimum operation policy consists of a series of decisions made in each stage, so an objective can be reached. In this way, present decisions will have direct impact on future ones. In the hydrothermal system operation planning problem, possible water storage levels in each reservoir represent states and a month a stage, while the shares of hydro and thermal generation are decisions. The objective is to minimize the expected operation cost during the planning horizon. However, the inclusion of water inflows turns the original problem into a stochastic one, which can be tackled by the Stochastic Dynamic Programming (SDP) (Archibald et al., 2006, Zambelli et al., 2006).

A drawback that can be readily seen from the SDP approach is the curse of dimensionality caused by the exponential increase of the computational effort experienced whenever the number of states (e.g., water reservoirs) increases. A common practice in mitigating such a hurdle consists in aggregating a large number of reservoirs into equivalent ones as presented by Arvanitidis and Rosing, 1970a, Arvanitidits and Rosing, 1970b.

In addition to the aforementioned equivalent reservoir method, recent advances in parallel computing architectures also come to play a fundamental role in making the LTOP solvable within practical time limits (Silva and Finardi, 2003).

A significant example to be considered is the Brazilian hydrothermal system. In the mid 1970s, with the increased amount of generation and transmission interconnections of the BPS, the SDP approach used in the optimal LTOP problem was replaced by a Stochastic Dual Dynamic Programming (SDDP) algorithm together with a Bender’s Decomposition approach (Pereira and Pinto, 1985), in order to overcome the SDP computational burden. Since then the SDP was considered to be impractical to be used to solve the operation planning problem and the SDDP algorithm is still in use today with some attempts to improve the SDDP is presented like the ones seen in Shapiro (2011) and Shapiro et al. (2013).

Many attempts to reduce the computational time of Dynamic Programming applications can be seen on Rodriguez et al. (1996) and Stivala et al. (2010). A broad set of solutions concerning power system applications have been recently proposed using parallel processing. Among them, there can be highlighted some applications in the hydrothermal system operation planning. Tang and Luh, 1995, Pinto, 2011 and Pinto et al. (2009), present a parallel processing approach to solve the short-term operation planning. Considering the long-term horizon Añó et al. (1999) present an attempt to optimal operation of large scale systems considering individual power plants representation. Additionally, another parallel approach is proposed by Silva and Finardi (2003) for the SDDP method to solve the BPS operation planning problem.

In this paper, two parallel programming approaches are adopted to reduce the computational time of the SDP applied to the LTOP problem. The proposed methodology makes use of the Convex Hull algorithm for modeling the expected cost-to-go function (ECF), as reported by Dias et al. (2010) to solve the LTOP problem. A case study based on the Brazilian hydrothermal system is presented.

The paper is outlined as follows. Section 2 presents the hydrothermal systems stochastic dynamic programming Problem formulation. In Section 3, the Convex Hull algorithm is presented applied to the SDP problem solution. Section 4 details the parallelization strategies for the LTOP problem and in Section 5 the proposed methodology is tested with two real systems study case, based on the Brazilian hydrothermal system. Finally conclusions are drawn in Section 6.