International Journal of Electrical Power & Energy Systems
A computational study of a stochastic optimization model for long term hydrothermal scheduling
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.
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2019, Renewable EnergyCitation 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.