A computational study of a stochastic optimization model for long term hydrothermal scheduling

https://doi.org/10.1016/j.ijepes.2012.06.021Get rights and content

Abstract

The long-term hydrothermal scheduling is one of the most important problems that must be solved in power systems area. This problem aims to obtain an optimal policy, under water resources uncertainty, for the hydro and thermal plants over a multi-annual planning horizon. The purpose of this paper is twofold. Firstly, we present a computational model which is in development for solving the long-term hydrothermal scheduling of the Brazilian hydrothermal power system. Secondly, it is described some modeling issues which may cause problems to achieve an optimal policy and are currently included in the official long-term optimization model of the Brazilian regulatory framework. This paper presents some solutions, for those problems, that were implemented on our model. We evaluate the solutions related to the original modeling and the modified one, in order to show evidences of the problems and that the solutions proposed are a good start point. To accomplish this evaluation, we consider in the model the whole Brazilian power system, with a reduced planning horizon.

Highlights

► We present the modeling used in the Brazilian LTHS problem. ► We show that some of the modeling aspects make the problem non-linear. ► We suggest some modeling strategies to deal with the non-linearity. ► We consider the whole Brazilian Power System in our computational results. ► We show the non-linearities consequences and the gain of the suggested approach.

Introduction

In a hydrothermal system with hydro resource predominance, the hydrothermal scheduling problem is a very complex task, given that all the particularities related to this problem cannot be accommodated in a single optimization model. In order to deal with this task, a chain of models is normally used, dividing the main problem into a hierarchy of smaller subproblems with different planning horizons and degrees of detail. Then, some information from the long-term scheduling problem [1], [2] is used as input to the more detailed mid-term problem [3], [4], [5] which, in turn, feed their results into the short-term scheduling problem [6], [7], [8], [9], [10], [11]. The information transferred from one problem to other can be some target of generation [12] or a function that describes the future operation cost with respect the storage level of water in the reservoirs [13].

In the long-term hydrothermal scheduling problem (LTHS), which is the focus of discussion in this paper, the main objective is to obtain a optimal policy that minimize the operation cost over a planning horizon, that considers 5–10 years, taking into account constraints related to the system and power plants. In this horizon, the model needs to take into consideration the water inflow seasonality and uncertainty, and the possibility to transfer water over the time by managing the reservoirs. Thus, depending on the electrical industry regulatory framework, the solution of this problem is useful for the Independent System Operator (ISO) or by the generation companies that own a mix of hydro and thermal plants, and need to submit bids (price and quantity) to the ISO.

Particularly, in the Brazilian hydrothermal power system, the ISO is responsible for the hydrothermal scheduling of the system and, as a consequence, it uses a chain of optimization models. In this context, LTHS problem is solved by a model called NEWAVE [14]. The main result of this model is the future cost function (FCF) which represents the system operation costs in the future as a function of the stored water and the water inflow.1 This FCF is used as boundary condition in the subsequent optimization model in the chain.

Several models have been proposed for solving the LTHS problem [15], [16], [17], [18], [19], [20], [21], [22]. However, most of them [15], [16], [17], [18], [19], [20] do not consider the specific characteristics of the Brazilian hydrothermal power system. Furthermore, NEWAVE has been continuously improved in order to comply with the growing requirements of the industry. Thus, based on our experience in other stochastic models [3], [23], [24], [25], [26], we are developing a model, called SMERA (Stochastic Model for Energy Resource Allocation), which allow us to suggest some improvements with respect the current model used in Brazil.

As it is detailed ahead, the LTHS is a large scale linear stochastic optimization problem, with temporal and spatial coupling. The stochastic characteristic arises from the fact that it is impossible to forecast future inflows accurately, and so these inflows are modeled as random variables with a known probability distribution. The temporal coupling is explained by the fact that the water can be managed over the time, observing the storage capacity of each reservoir. The spatial coupling is due to the fact that the dispatch of a particular hydro plant alters the inflow to the downstream plants, for example. The problem is represented by a linear model, because in this planning horizon (5–10 years) it is more important to represent the inflows uncertainties than the nonlinearities associated with hydro production and thermal costs [6]. Finally, the LTHS problem is considered to be a large scale problem due to the large number of hydro plants and time stages.

To include all the characteristics aforementioned, SMERA has being developed based on three main structures: (i) the Energy Equivalent Reservoir (EER) that aggregates a set of hydro plants in a smaller number of equivalent reservoirs in order to reduce the size of the optimization problem. In this case, the decisions variables are presented in terms of energy instead of water; (ii) the Periodic Autoregressive (PAR) model to generate the inflows scenarios; and (iii) the Stochastic Dual Dynamic Programming (SDDP) algorithm, responsible for calculating the optimal policy. These structures are detailed in the Section 3.

The first main contribution of this paper is offered by means of a discussion of the modeling used in the Brazilian LTHS problem. The EER and PAR models in use add nonlinearities to the problem, and as it can be seen in the computational results, these nonlinearities make the optimization problem nonconvex. The main consequence is that the SDDP cannot find an optimal policy, since this algorithm needs to solve a convex problem to find the optimal solution. The second main contribution is to present one strategy to deal with the EER’s nonlinearities and two alternative approaches for the PAR model.

The paper is organized as follows. In Section 2 we present a brief stochastic programming discussion just to introduce the reader some important concepts with respect to the scenarios tree. Then, Section 3 presents the modeling for the LTHS, more precisely, it is discussed the EER, the PAR model, and the SDDP algorithm. Section 4 highlights the nonlinear characteristics and how we are dealing with them in SMERA. Section 5 presents the results and analysis of the cases in study. Section 6 presents the final remarks.

Section snippets

Stochastic programming

In stochastic programming, uncertainty is modeled by random variables with known probability distributions. In a precise way, the random variables should be modeled by a continuous distribution, making it usually impossible to solve the problem [27]. In order to overcome this difficulty, the distribution is either discretized or sampled, for instance, by means of a Monte Carlo technique. Even in this case, the resulting problem may be very difficult to solve, given that it is necessary to find

SMERA model

The first important structure in the model is the representation of the reservoirs, in which a particular set of hydro plants are modeled as a single one by using the concept of Energy Equivalent Reservoir (EER). The second one is the Periodic Autoregressive (PAR) model, which is responsible for generating the scenario tree. The last one is strategy to solve the LTHS problem that is the Stochastic Dynamic Dual Programming (SDDP). All these structures are discussed in the following. The purpose

Nonlinear characteristics

As discussed previously, the SMERA is a linear model and its solution algorithm is based on this characteristic. More precisely, to use the SDDP algorithm the problem must be convex. However, as it is pointed out in this section, there are some nonlinearities that yields a nonconvex problem. These nonlinear characteristic are within the EER and PAR model.

Computational tests

In order to evaluate the impact of the nonlinearities in the LTHS problem and the proposed solutions, it is used the whole Brazilian interconnected system (139 hydro and 146 thermal plants). It was considered a 6 months horizon and with two energy inflows per stage, which gives us a scenario tree with 32 scenarios. Obviously, this scenario tree is insufficient to define a good operation policy, but it enables us to enumerate all scenarios

Conclusions

In this paper we presented the SMERA model, which has being developed by the authors for solving the LTHS problem. The theoretical references used in SMERA are the same of the NEWAVE. By means of this research project we found some shortcomings caused by nonlinearities introduced in the problem formulation, which may not guarantee that the SDDP algorithm produces an optimal policy. Therefore, we discussed those problems and presented possible solutions that were implemented so far. It is

Acknowledgement

Research supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Agência Nacional de Energia Elétrica (ANEEL) through projects of Research & Development.

References (36)

  • A.B. Philpott et al.

    On the convergence of stochastic dual dynamic programming and related methods

    Oper Res Lett

    (2008)
  • T. Homem-de-Mello et al.

    Sampling strategies and stopping criteria for stochastic dual dynamic programming: a case study in long-term hydrothermal scheduling

    Energy Syst

    (2011)
  • de Matos VL, Philpott AB, Finardi EC, Guan Z. Solving long-term hydrothermal scheduling problem. 17th power systems...
  • R.E.C. Gonçalves et al.

    Comparing stochastic optimization methods to solve the medium-term operation planning problem

    Comput Appl Math

    (2011)
  • E.C. Finardi et al.

    Solving the hydro unit commitment problem via dual decomposition and sequential quadratic programming

    IEEE Trans Power Syst

    (2006)
  • A.J. Wood et al.

    Power generation, operation, and control

    (1984)
  • M.V.F. Pereira et al.

    Multi-stage stochastic optimization applied to energy planning

    Math Program

    (1991)
  • Maceira MEP, Mercio CMVDB, Gorenstin GB, Cunha SHF, Suanno C, Sacramento MC, et al. Energy evaluation of the...
  • Cited by (54)

    • Assessing solution quality and computational performance in the long-term generation scheduling problem considering different hydro production function approaches

      2019, Renewable Energy
      Citation Excerpt :

      In order to obtain a simplified HPF model, two requirements should be considered: (i) the HPF model must be concave, to ensure convergence of the optimization algorithm; (ii) it must have as few constraints and variables as possible since usually many inflow scenarios are used in the SDDP. In this context, the simplest HPF modeling in LTGS is the one based on the Equivalent Energy Reservoir (EER) model [4] [7], [8]. Although the previous requirements (i) and (ii) are met, the quality of the policy obtained by the EER can be very poor since the aggregation of several reservoirs neglects the individual constraints of the hydro plants.

    View all citing articles on Scopus
    View full text