Parameter uncertainty analysis of reservoir operating rules based on implicit stochastic optimization
Introduction
Reservoirs are typically operated using operating rules, which specify operational decisions (e.g., releases) as a function of available information (Li et al., 2010b, Oliveira and Loucks, 1997), such as the current reservoir water level and the hydro-meteorological conditions (Guo et al., 2004). Implicit stochastic optimization (ISO) (Young, 1967) is an efficient alternative to explicit stochastic optimization (Stedinger et al., 1984, Tilmant et al., 2008) for considering hydrologic stochasticity, and has often been used in studies to derive optimal reservoir operating rules (e.g., Bhaskar and Whitlatch Jr., 1980, Celeste and Billib, 2009, Labadie, 2004, Lund and Ferreira, 1996, Koutsoyiannis and Economou, 2003, Rani and Moreira, 2010). ISO enables most characteristics of stochastic inflows, including spatial and temporal correlations among unregulated runoff, to be implicitly incorporated into reservoir operations modeling by inputting observed or synthetic samples (Labadie, 2004). As a result, ISO has become one of the most reliable methods of reservoir modeling (Celeste and Billib, 2009, Rani and Moreira, 2010, Simonovic, 1987, Wurbs, 1993).
Two approaches can be used to derive optimal operating rules within an ISO framework when the functional form of the operating rules has been pre-determined: fitting methods and simulation-based optimization (SBO) methods. With fitting methods, the reservoir operating rules are derived from deterministic optimization models using either linear regression (LR) (Bhaskar and Whitlatch Jr., 1980, Young, 1967) or nonlinear fitting methods, such as nonlinear regression (Young, 1967), artificial neural networks (Liu et al., 2006), fuzzy inference (Chang and Chang, 2001, Han et al., 2012) and decision trees (Wei and Hsu, 2008). However, the maximum goodness-of-fit criterion for establishing the operating rules may not always be appropriate to produce the best rules (Bhaskar and Whitlatch Jr., 1980).
Partly because of the deficiencies in fitting methods, SBO (Koutsoyiannis and Economou, 2003, Nalbantis and Koutsoyiannis, 1997) has become more widely used (Celeste and Billib, 2009, Rani and Moreira, 2010). SBO methods directly optimize performance measures, such as maximizing profits (e.g., hydropower generation) or minimizing loss (e.g., flood risk), by adjusting operating rule parameters as decision variables in an iterative simulation-based search algorithm (Chen et al., 2007, Koutsoyiannis and Economou, 2003, Ngo et al., 2007). Determination of the operating rule parameters is similar to that of parameter calibration in a hydrologic model (Table 1), in which the objective functions are the maximization of net benefits in reservoir operations and matching the observed data in a hydrologic model, respectively. Indeed, both problems are inverse problems, which are a general framework to convert observed measurements into information describing a system.
ISO involves uncertainty in the parameters characterizing the optimality of derived operating rules because of the uncertainty inherent in reservoir operations models. There is intrinsic uncertainty in the dynamics of inflow processes and uncertainty in knowledge of physical parameters such as hydro-meteorological parameters and reservoir characteristics (e.g., reservoir elevation-storage relationship, and uncertainty in the economic processes and parameters that describe operational performance). Specifically, it is difficult to identify the actual optimal operating rules using ISO because the rules depend on the historical or simulated inflow. As a result, “errors” always exist between the true optimal and estimated decision. When these methods are used, it is desirable to determine a method for analysis and evaluation of the resulting uncertainty in the derived reservoir operating rule parameters. This study deals with analysis of uncertainty associated with reservoir operating rule parameters, which has seldom been addressed in the literature.
Analysis of operating rule parameter uncertainty is required for generating robust reservoir operation rules, especially from the following two aspects (Liu et al., 2011b). (1) Alternative rules: optimization should not be used to find the best solution, but rather to identify a relatively small number of good alternatives that can later be tested, evaluated and improved (Loucks and van Beek, 2005). In contrast to identifying an optimal parameter for reservoir operating rules, uncertainty analysis assumes these parameters are random variables and therefore provides a set of decisions and their confidence intervals. These confidence intervals provide more information, including the robust decision (say median) and the probability coverage for the best decision, than a single decision. (2) Sensitivity analysis: uncertainty analysis can be used to determine how sensitive an objective is to variations in the reservoir decisions, and hence the critical and important periods for reservoir operation can be identified by analyzing the confidence interval.
Uncertainty analysis has received increasing attention in water resources research over the last two decades (Montanari, 2007, Pappenberger and Beven, 2006). Most of these studies have focused on hydrologic models (e.g., Beven and Freer, 2001, McMillan and Clark, 2009, Tolson and Shoemaker, 2008, Vrugt et al., 2003, Yang et al., 2008, Zhang et al., 2013), including groundwater models (Mugunthan and Shoemaker, 2006) and, frequency analysis (El Adlouni and Ouarda, 2009, Reis Jr. and Stedinger, 2005) and water quality (Deviney Jr. et al., 2012, Xu and Qin, 2013). In general, uncertainty analysis in model prediction (simulation) involves the quantification of uncertainty in the model inputs, parameters, structure, and observations (Liu and Gupta, 2007). In this study, we only discuss parameter uncertainty.
Because of the parallels between parameter estimation for both hydrologic models and operating rule parameter derivation models (Table 1), uncertainty analysis approaches applied into the former also can be used to the latter. Two widely used mutually independent uncertainty analysis approaches (Montanari et al., 2009), nonprobabilistic generalized likelihood uncertainty estimation (GLUE) (Beven and Binley, 1992) and probabilistic Bayesian methods, were implemented for uncertainty analysis in this study. The popular Markov Chain Monte Carlo (MCMC) (Chib and Greenberg, 1995) algorithm has been used for the probabilistic Bayesian inference. GLUE was proposed for the investigation of the hydrologic modeling uncertainty by producing the prediction limits for the modeled streamflow series and a set of behavioral parameter sets (Beven and Binley, 1992), while MCMC was primarily used to simulate observations from unwieldy distributions by constructing a Markov Chain as samplers (Chib and Greenberg, 1995).
The purpose of this paper is to demonstrate how parameter uncertainty analysis can be applied to reservoir operating rules and highlight the importance of such analysis for generating more robust reservoir operation rules. The remainder of this paper is organized as follows. In Section 2, we describe optimal operating rules and present two alternative methods of estimating rule parameter uncertainty, LR and Bayesian simulation (BS). The BS method is implemented with GLUE and the MCMC algorithm. Section 3 describes a case study application to China’s Three Gorges Reservoir (TGR). Section 4 provides further discussion of the analysis methods and the implication of the analysis results for more effective solutions to reservoir operations. Finally, conclusions are given in Section 4.
Section snippets
Methodology
Two approaches for analyzing the uncertainty associated with derived reservoir operating rules are presented and assessed: LR and BS. Assessing these methods depends on knowledge of the theoretical optimal releases. Therefore, a reservoir operation optimization model is first introduced. This is followed by a description of operating rules, which are often used in practice. The uncertainty analysis methods are then described.
The Three Gorges Reservoir
The Three Gorges Reservoir (TGR) is the largest and most important reservoir along China’s longest river, the Yangtze River (Fig. 2) and is currently the largest multipurpose hydro-development project ever built. The TGR receives inflow from a drainage area of approximately 106 km2, with a mean annual runoff at the dam site of 4.51 × 1011 m3. The management objectives of the TGR include flood control, power generation and improved navigation. With a flood storage capacity of 2.215 × 1011 m3/yr, the
Conclusions
Two alternative methods, linear regression and Bayesian simulation, were used to analyze the uncertainty associated with the reservoir operating rule parameters. A case study of the Three Gorges Reservoir revealed that the confidence interval of an operation decision could be estimated based on the operating rule parameter uncertainty analysis methods described, which is a potential benefit when evaluating and selecting the most realistic confidence interval for specific parameters. Overall,
Acknowledgments
This study was supported by the Program for New Century Excellent Talents in University (NCET-11-0401), the National Natural Science Foundation of China (51190094) and Non-Profit Industry Financial Program of Ministry of Water Resources (201201051). The authors also thank the editor, four anonymous reviewers and Ximing Cai for comments and suggestions, which helped improve the quality of this paper.
References (57)
- et al.
Equifinality, data assimilation, and uncertainty estimation in mechanistic modelling of complex environmental systems using the glue methodology
J. Hydrology
(2001) - et al.
Generalized likelihood uncertainty estimation (GLUE) using adaptive Markov Chain Monte Carlo sampling
Adv. Water Res.
(2008) - et al.
Evaluation of stochastic reservoir operation optimization models
Adv. Water Res.
(2009) - et al.
A diversified multi-objective GA for optimizing reservoir rule curves
Adv. Water Res.
(2007) - et al.
Evaluation of the subjective factors of the GLUE method and comparison with the formal Bayesian method in uncertainty assessment of hydrological models
J. Hydrology
(2010) - et al.
Dynamic control of flood limited water level for reservoir operation by considering inflow uncertainty
J. Hydrology
(2010) - et al.
Simulation and optimisation modelling approach for operation of the Hoa Binh reservoir Vietnam
J. Hydrology
(2007) - et al.
Bayesian MCMC flood frequency analysis with historical information
J. Hydrology
(2005) - et al.
Comparing uncertainty analysis techniques for a SWAT application to the Chaohe Basin in China
J. Hydrology
(2008) - et al.
Sobols sensitivity analysis for a distributed hydrological model of Yichun River Basin, China
J. Hydrology
(2013)
The future of distributed models: model calibration and uncertainty in prediction
Hydrological Processes
Derivation of monthly reservoir release policies
Water Res. Res.
Intelligent control for modelling of real-time reservoir operation
Hydrological Processes
Understanding the Metropolis-Hastings algorithm
Am Statistician
Evaluation of Bayesian estimation of a hidden continuous-time Markov Chain model with application to threshold violation in water-quality indicators
J. Environ. Inf.
Joint Bayesian model selection and parameter estimation of the generalized extreme value model with covariates using birth-death Markov chain Monte Carlo
Water Resour. Res.
Bayesian Data Analysis
A reservoir flood forecasting and control system in China
Hydrological Sci. J.
An adaptive metropolis algorithm
Bernoulli
Fuzzy constrained optimization of eco-friendly reservoir operation using self-adaptive genetic algorithm: a case study of a cascade reservoir system in the Yalong River, China
Ecohydrology
Optimal operation of multi-reservoir systems: state-of-the-art review
J. Water Resour. Plan. Manage.
Deriving reservoir refill operating rules by using the proposed DPNS model
Water Resour. Manage.
Derivation of aggregation-based joint operating rules curves for cascade hydropower reservoirs
Water Resour. Manage.
Deriving multiple near-optimal solutions to deterministic reservoir operation problems
Water Resour. Res.
Uncertainty in hydrologic modeling: Toward an integrated data assimilation framework
Water Resour. Res.
Water resources systems planning and management: an introduction to methods, models, and applications
Cited by (91)
Water-energy-environment nexus under different urbanization patterns: A sensitivity-based framework for identifying key feedbacks
2023, Journal of Cleaner ProductionA parallel approximate evaluation-based model for multi-objective operation optimization of reservoir group
2023, Swarm and Evolutionary ComputationDNN-SSDP for hydropower system operation using small state sets
2022, Journal of HydrologyThe complementary management of large-scale hydro-photovoltaic hybrid power systems reinforces resilience to climate change
2022, Journal of HydrologyCitation Excerpt :In this study, three management evaluation indicators were used: annual power generation (economy), the generation assurance rate (power-supply reliability), and the water shortage index (WSI) (water-supply reliability). Power generation is a critical operational economic indicator, while the power generation assurance rate must be high enough to ensure reliability in the electricity supply (Liu et al., 2014; Yang et al., 2021). Additionally, the WSI is a proper quantitative indicator that can comprehensively describe the matching degree of water supply and demand, and it is ideal to supply water as close to downstream comprehensive water requirements as possible (Ming et al., 2019; Si et al., 2019).