Innovative Applications of O.R.
A stochastic program with time series and affine decision rules for the reservoir management problem

https://doi.org/10.1016/j.ejor.2017.12.007Get rights and content

Highlights

  • We study the reservoir management problem with stochastic inflows.

  • We combine popular time series models and affine decision rules.

  • We extend the approach to consider heteroscedasticity.

  • We conduct realistic numerical experiments with encouraging results.

Abstract

This paper proposes a multi-stage stochastic programming formulation for the reservoir management problem. Our problem specifically consists in minimizing the risk of floods over a fixed time horizon for a multi-reservoir hydro-electrical complex. We consider well-studied linear time series models and enhance the approach to consider heteroscedasticity. Using these stochastic processes under very general distributional assumptions, we efficiently model the support of the joint conditional distribution of the random inflows and update these sets as new data are assimilated. Using robust optimization techniques and affine decision rules, we embed these time series in a tractable convex program. This allows us to obtain good quality solutions rapidly and test our model in a realistic simulation framework using a rolling horizon approach. Finally, we study a river system in western Québec and perform various numerical experiments based on different inflow generators.

Introduction

This paper considers the problem of minimizing the risk of floods for a multi-reservoir system over a fixed time horizon subject to uncertainty on inflows while respecting tight operational constraints on total storage, spills, releases, water balance and additional physical constraints. This reservoir management problem is of vital importance to various real sites that are close to human habitations and that are prone to flooding (Castelletti, Galetti, Restelli, Soncini-Sessa, 2010, Gauvin, Delage, Gendreau, 2017, Pianosi, Soncini-Sessa, 2009, Quentin, Côté, Robert, 2014).

The deterministic version of the problem already poses serious challenges since operators must consider complex non-linear phenomena related to the physical nature of the system (Labadie, 2004). The interconnection of the catchment also complicates decisions as upstream releases affect downstream volumes and flows. This issue is particularly important for large catchments where there can be long water delays (Gauvin et al., 2017). Considering uncertainty significantly increases theses difficulties since the sequential decision-making under uncertainty represents a huge theoretical obstacle in itself (Dyer & Stougie, 2006).

In order to solve this problem, we propose a multi-stage stochastic program based on affine decision rules and well-known time series models. Our approach leverages techniques from stochastic programming, stochastic processes and robust optimization.

Starting with the pioneering work of Ben-Tal, Goryashko, Guslitzer, and Nemirovski (2004), adjustable robust optimization based on affine decision rules has emerged as a viable approach for dynamic problems where uncertainty is progressively revealed. The approach has been shown capable of finding good quality solutions to large multi-stage stochastic problems that would otherwise be unmanageable to traditional methods such as stochastic dynamic programming.

These techniques have been applied to the reservoir management problems with a varying degree of success. The authors of Apparigliato (2008) and Pan, Housh, Liu, Cai, and Chen (2015) use this framework to maximize the expected electric production for a multi-period and multi-reservoir hydro-electric complex while Gauvin et al. (2017) minimize the risk of floods by adopting a risk averse approach that explicitly considers the multidimensional nature of the problem subject to more realistic operating constraints.

Although some of these studies use elaborate decision rules based on works such as Chen, Sim, Sun, and Zhang (2008); Georghiou, Wiesemann, and Kuhn (2014); Goh and Sim (2010), they only consider very simplified representations of the underlying stochastic process and generally omit serial correlation. However, the importance of the persistence of inflows can play a crucial factor in hydrological modelling for stochastic optimization problems, particularly when daily inflows need to be considered. Authors like Pianosi and Soncini-Sessa (2009); Turgeon (2005) argue that serial correlation of high order is important to consider inflows that are high or low on many consecutive days and risk producing a flood or low baseflow.

This paper addresses the issue by developing a dynamic robust uncertainty set that takes into consideration the dynamic structure and serial correlation of the inflow process. We show that under certain conditions, these sets correspond to the support of the joint conditional distribution of uncorrelated random variables that determine the inflows over a given horizon. Our work shares similarities with the paper of Lorca and Sun (2015) who propose dynamic uncertainty sets based on time series models for a 2-stage economic dispatch problem in the presence of high wind penetration. Like these authors, we take advantage of the dynamic adaptability of the uncertainty sets by incorporating our model in a realistic simulation framework with rolling horizon.

Nonetheless, we give significantly more details on the construction of these uncertainty sets for general univariate ARMA models and provide key insights which are of value to practitioners and academics alike. We also consider the case of heteroscedasticity which is empirically observed in various inflow time series (Romanowicz, Young, & Beven, 2006). Although we minimize the risk of floods, our work can be directly extended to electricity generation, under the hypothesis that head is constant and that the production function can be modeled as a piecewise linear function.

Our model considers ARMA and GARCH models of any order without increasing the complexity of the problem. This is a huge improvement over stochastic dynamic programming (SDP) methods, which have historically been the most popular techniques used for reservoir management both in academia and in practice (see Castelletti, Pianosi, and Soncini-Sessa (2008); Labadie (2004); Turgeon (2005); Yeh (1985) and references therein). Although these methods can deal with more realistic non-convex optimization problems and provide excellent closed-loop policies, they can usually only consider serial correlation through autoregressive models of small order since higher order models require increasing the state-space, which quickly leads to numerical intractability; particularly for multi-reservoir operations (Cervellera, Chen, Wen, 2006, Tilmant, Kelman, 2007, Turgeon, 2007).

Considering heteroscedasticity further increases the state dimension and the resulting computational burden for SDP. As such, we are only aware of the work of Pianosi and Soncini-Sessa (2009) that is capable of considering this phenomenon with this methodology. Furthermore, these authors only manage to incorporate heteroscedasticity in a reduced model used for on-line computations.

Numerous refinements of classical SDP have emerged to circumvent some of these difficulties. Neuro-dynamic programming (Castelletti et al., 2010), sampling SDP (Stedinger & Faber, 2001) and more elaborate discretization schemes (Cervellera, Chen, Wen, 2006, Zéphyr, Lang, Lamond, Côté, 2016) have namely been applied successfully to large reservoir systems while explicitly or implicitely considering high order serial correlation. However, most of these methods still require simplifications of the river dynamics and inflow representation as well as discretization of decisions, which is not the case of our approach.

Other works based on SDP, such as Côté, Haguma, Leconte, and Krau (2011); Stedinger and Faber (2001); Tejada-Guibert, Johnson, and Stedinger (1995); Turgeon (2005), have focused on various low-dimensional hydrological variables such as seasonal forecasts, additional exogenous information like soil moisture and linear combinations of past inflows. Although these aggregate hydrological variables improve the solution quality without excessive computational requirements, they often rely on distributional assumptions such as normality that are not verified in practice or exogenous data that may be difficult to obtain. Our model does not suffer from such limitations.

In recent years, the stochastic dual dynamic programming (SDDP) method has emerged as an effective algorithm capable of successfully tackling multi-dimensional stochastic reservoirs problems (Gjelsvik, Mo, Haugstad, 2010, Pereira, Pinto, 1991, Phillpot, Guan, 2008, Rougé, Tilmant, 2016, Shapiro, Tekaya, da Costa, Soares, 2013, Tilmant, Kelman, 2007). Moreover, Maceira and Damázio (2004) illustrate that this method can consider multiple lag autoregressive processes without excessively increasing the size of the problem for the aggregated 4-state Brazilian hydro-thermal system. However, the algorithm can exhibit relatively slow convergence (Shapiro et al., 2013). To avoid this issue, various studies only consider a limited number of discrete scenarios ( < 100) (Rougé, Tilmant, 2016, Tilmant, Kelman, 2007), but this only leads to approximate solutions of questionable quality (Shapiro, 2011).

We also mention stochastic programming methods based on decision trees, which are also theoretically capable of explicitly handling highly persistent inflows (Carpentier, Gendreau, Bastin, 2013, Fleten, K.J, Näsäkkälä, 2011, Gonçalves, Gendreau, Finardi, 2013). These models are simple to implement when an existing deterministic model already exists and are intuitive to use and interpret. Unfortunately, they display exponential growth in complexity as a function of the time horizon. Therefore, they are usually limited to small decision trees. Gauvin et al. (2017) shows that these methods can be considerably more computationally intensive than corresponding stochastic programs based on decision rules.

The main advantage of our method with respect to SDDP and scenario-tree based stochastic program is its limited distributional assumptions. Whereas these competing methods requires using specific distributions from which it is possible to sample, our approach only demands hypothesis on the first 2 moments of the random variables as well as the correlation between them. As observed by Pan et al. (2015), this is likely to lead to solutions that are more robust when tested on out-of-sample scenarios, which is of prime concern when data availability is limited. Another shortcoming of SDDP and tree-based stochastic programming compared with decision rules and SDP are their inability to provide explicit policies that can be directly used by decision makers. Nonetheless, this is usually not a critical point, particularly when rolling horizon simulations are considered.

The paper is structured as follows. The model for the deterministic reservoir management problem and the stochastic version based on affine decision rules are presented in Section (2). Section (3) discusses inflow representation and general univariate ARMA models and the resulting conditional supports. It then studies heteroscedastic time series and their impact on the model formulation. Section (4) explains the solution procedure and simulation framework while Section (5) studies a river in western Québec. Conclusions are drawn in Section (6).

The sets Z+={0,1,2,} and Z={0,1,2,} represent the non-negative and non-positive integers while S+n is the cone of square n × n semi-definite matrices. The set T={1,,T} represents the entire horizon of T periods and L={0,,L1} denotes a limited look-ahead horizon for some L < T.

Let (Ω,G,{Gt},P) be a filtered probability space where {Gt} is a collection of σ-algebras representing some information available at time tT where G0={Ω,} and GT=G. We let E[ · ] denote mathematical expectation while E[·|A] represents conditional expectation given any σ-algebra AG. Both expectations are taken with respect to P, the base probability measure on G.

For the real random variables X, σ(X) represents the σ-algebra generated by X (Billingsley, 1995). We abuse language and refer to the support of X as the smallest closed set on which X takes values with probability one. For any discrete time real valued stochastic process {Xt}tZ, we denote the RL valued random vector (Xt,,Xt+L1)X[t,t+L1] for any tZ and LN with the special notation X[t+L1] if t=1. For simplicity, we abuse notation and do not distinguish a random variable from a given realization.

Section snippets

Deterministic look-ahead model for flood minimization

Before describing our stochastic reservoir management problem with affine decision rules, we describe the deterministic version. A similar model is described in Gauvin et al. (2017), but we present it here to make the paper self-contained. We write the problem in a “look-ahead” form to facilitate its integration in the rolling horizon framework presented in Section (4).

At the beginning of time tT, we seek a vector of decisions for each future time τ{t,,t+L1} that will minimize a coarse

General inflow representation

The quality of the solutions returned by solving the lookahead problem SPt crucially depends on the representation of the underlying stochastic process {ξt}. Assuming simple independent time series will likely lead to poor quality solutions in the presence of significant serial correlation. However, we also want to maintain the tractability of the overall linear program considering affine decision rules.

In order to achieve these conflicting objectives, we assume that at the beginning of each

Monte Carlo simulation and rolling horizon framework

Solving the stochastic version of problem (1a)–(1j) with affine decision rules at the beginning of time 1 for L=T provides an upper bound on the value of the “true” problem over the horizon T when various hypotheses on ϱt and ξt are verified. However, simulating the behaviour of the system with a given distribution can give a better assessment of the real performance of these decisions. Using random variables that violate the support assumptions also provides interesting robustness tests.

The river system

We apply our methodology to the Gatineau river in Québec. This hydro electrical complex is part of the larger Outaouais river basin and is managed by Hydro-Qu ebec, the largest hydroelectricity producer in Canada (Hydro-Québec, 2012). It is composed of 3 run-of-the-river plants with relatively small productive capacity and 5 reservoirs, of which only Baskatong and Cabonga have significant capacity (see Fig. 2).

The Gatineau represents an excellent case study as it runs near the small town of

Conclusion

In conclusion, we illustrate the importance and value of considering the persistence of inflows for the reservoir management problem. Repeatedly solving a simple data-driven stochastic lookahead model using affine decision rules and ARMA and GARCH time series model provides good quality solutions for a problem that would otherwise be intractable for classical SDP and that would require considerably more distributional assumptions for SDDP or tree-based stochastic programming.

We give detailed

Acknowledgments

The authors would like to thank Charles Audet, Fabian Bastin, Angelos Georghiou, Stein-Erick Fleten and Amaury Tilmant for valuable discussion as well as everyone at Hydro-Quẽbec and IREQ for their ongoing support, particularly Grégory Émiel, Louis Delorme, Laura Fagherazzi and Pierre-Marc Rondeau. The comments of two anonymous referees also greatly improved the quality of this paper. This research was supported by the Natural Sciences and Engineering Reasearch Council of Canada (NSERC) and

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