Optimal Hedging Rule for Reservoir Refill
Publication: Journal of Water Resources Planning and Management
Volume 142, Issue 11
Abstract
In this study, an optimal reservoir refill hedging rule (RHR) is developed under hydrologic uncertainty using a two-stage model. Based on the probability distribution of the maximum refill water availability at the end of refill season, three possible cases exist: unfilled without flood damage, complete filling without flood damage, and complete filling with flood damage. These cases are characterized based on relationships among storage capacity, expected storage buffer, and maximum safe excess discharge. Karush–Kuhn–Tucker (KKT) conditions for the two-stage model show that the optimal refill operation equates the expected marginal loss of conservation benefit from not complete filling (i.e., ending storage of refill period less than storage capacity) and the expected marginal flood damage from levee overtopping downstream, unless constrained by capacity constraints. A RHR curve, which is analogous to water supply hedging and flood hedging rules, is drawn and shows the trade-off between the two objectives. The optimal refill hedging release decision shows a linear relationship with expected current water availability for a wide range of water conservation benefit functions (linear, concave, or convex). Several operational results are derived. A large downstream flood conveyance capacity and remaining storage capacity allow for a smaller current release and greater storage of water. Relative economic drivers are important; a greater economic potential for flood damage drives a greater release of water in the current stage, and vice versa. Below a critical forecast uncertainty value, improving forecasts reduces the volume of water released, although the opposite effect occurs above this critical value. Finally, the Danjiangkou Reservoir case shows that the RHR, combined with a rolling horizon decision approach, performs better than current rule curves and leads to a gradual dynamic refilling based on forecast information, indicating its potential for practical use.
Get full access to this article
View all available purchase options and get full access to this article.
References
Bazaraa, M. S., Sherali, H. D., and Shetty, C. M. (2006). “The Fritz John and Karush-Kuhn-Tucker optimality conditions.” Chapter 4, Nonlinear programming: Theory and algorithms, Wiley, Canada, 163–236.
Boettle, M., Rybski, D., and Kropp, J. P. (2013). “How changing sea level extremes and protection measures alter coastal flood damages.” Water Resour. Res., 49(3), 1199–1210.
Bower, B. T., Hufschmidt, M. M., and Reedy, W. W. (1962). “Operating procedures: Their role in the design of water-resource systems by simulation analyses.” Design of water resource systems, Harvard University Press, Cambridge, MA, 443–458.
Chand, S., Hsu, V. N., and Sethi, S. (2002). “Forecast, solution, and rolling horizons in operations management problems: A classified bibliography.” Manuf. Serv. Oper. Manage., 4(1), 25–43.
Chen, L., McPhee, J., and Yeh, W. W. (2007). “A diversified multiobjective GA for optimizing reservoir rule curves.” Adv. Water Resour., 30(5), 1082–1093.
Clark, E. J. (1956). “Impounding reservoirs.” J. Am. Water Works Assoc., 8(4), 349–354.
Datta, B., and Burges, S. J. (1984). “Short-term, single, multiple-purpose reservoir operation: Importance of loss functions and forecast errors.” Water Resour. Res., 20(9), 1167–1176.
Ding, W., Zhang, C., Peng, Y., Zeng, R., Zhou, H., and Cai, X. (2015). “An analytical framework for flood water conservation considering forecast uncertainty and acceptable risk.” Water Resour. Res., 51(6), 4702–4726.
Draper, A. J., and Lund, J. R. (2004). “Optimal hedging and carryover storage value.” J. Water Res. Plann. Manage., 83–87.
Eum, H., and Simonovic, S. P. (2010). “Integrated reservoir management system for adaptation to climate change: The Nakdong River Basin in Korea.” Water Resour. Manage., 24(13), 3397–3417.
Faber, B. A., and Stedinger, J. R. (2001). “Reservoir optimization using sampling SDP with ensemble streamflow prediction (ESP) forecasts.” J. Hydrol., 249(1), 113–133.
Fisher, I. (1930). The theory of interest as determined by impatience to spend income and opportunity to spend it, MacMillan, New York.
Geoffrion, A. M. (1976). “The purpose of mathematical programming is insight, not numbers.” Interfaces, 7(1), 81–92.
Hsu, N., and Wei, C. (2007). “A multipurpose reservoir real-time operation model for flood control during typhoon invasion.” J. Hydrol., 336(3), 282–293.
Hu, Z. P., Feng, S. Y., and Yu, F. Q. (1991). “Strategy of flood control water level for Danjiangkou reservoir.” Water Resour. Hydropower Eng., 11, 1–7 (in Chinese).
Kolesar, P., and Serio, J. (2011). “Breaking the deadlock: Improving water-release policies on the Delaware river through operations research.” Interfaces, 41(1), 18–34.
Le Ngo, L., Madsen, H., and Rosbjerg, D. (2007). “Simulation and optimisation modelling approach for operation of the Hoa Binh reservoir, Vietnam.” J. Hydrol., 336(3), 269–281.
Li, X., Guo, S., Liu, P., and Chen, G. (2010). “Dynamic control of flood limited water level for reservoir operation by considering inflow uncertainty.” J. Hydrol., 391(1-2), 124–132.
Li, Y., Guo, S., Guo, J., Wang, Y., Li, T., and Chen, J. (2014). “Deriving the optimal refill rule for multi-purpose reservoir considering flood control risk.” J. Hydroenviron. Res., 8(3), 248–259.
Liu, P., et al. (2015). “Optimal design of seasonal flood limited water levels and its application for the Three Gorges reservoir.” J. Hydrol., 527, 1045–1053.
Liu, X., Guo, S., Liu, P., Chen, L., and Li, X. (2011). “Deriving optimal refill rules for multi-purpose reservoir operation.” Water Resour. Manage., 25(2), 431–448.
Loucks, D. P., Van Beek, E., Stedinger, J. R., Dijkman, J. P., and Villars, M. T. (2005). Water resources systems planning and management: An introduction to methods, models and applications, UNESCO, Paris.
Lund, J. R., and Guzman, J. (1999). “Derived operating rules for reservoirs in series or in parallel.” J. Water Res. Plann. Manage., 143–153.
Marsaglia, G., and Bray, T. A. (1964). “A convenient method for generating normal variables.” Siam Rev., 6(3), 260–264.
Marsaglia, G., MacLaren, M. D., and Bray, T. A. (1964). “A fast procedure for generating normal random variables.” Commun. ACM, 7(1), 4–10.
Maurer, E. P., and Lettenmaier, D. P. (2004). “Potential effects of long-lead hydrologic predictability on Missouri River main-stem reservoirs*.” J. Clim., 17(1), 174–186.
Milly, P., et al. (2007). “Stationarity is dead.” Ground Water News Views, 4(1), 6–8.
Paredes, J., and Lund, J. R. (2006). “Refill and drawdown rules for parallel reservoirs: Quantity and quality.” Water Resour. Manage., 20(3), 359–376.
Sankarasubramanian, A., Lall, U., Devineni, N., and Espinueva, S. (2009). “The role of monthly updated climate forecasts in improving intraseasonal water allocation.” J. Appl. Meteorol. Clim., 48(7), 1464–1482.
Valeriano, S., et al. (2010). “Decision support for dam release during floods using a distributed biosphere hydrological model driven by quantitative precipitation forecasts.” Water Resour. Res., 46(10), W10544.
Wang, Y., Guo, S., Yang, G., Hong, X., and Hu, T. (2014). “Optimal early refill rules for Danjiangkou reservoir.” Water Sci. Eng., 7(4), 403–419.
Williams, O. (1997). “Engineering and design: Hydrologic engineering requirements for reservoirs.”, Corps of Engineers, Washington, DC.
Xu, W., Zhang, C., Peng, Y., Fu, G., and Zhou, H. (2014a). “A two stage Bayesian stochastic optimization model for cascaded hydropower systems considering varying uncertainty of flow forecasts.” Water Resour. Res., 50(12), 9267–9286.
Xu, W., Zhao, J., Zhao, T., and Wang, Z. (2014b). “Adaptive reservoir operation model incorporating nonstationary inflow prediction.” J. Water Res. Plann. Manage., 04014099.
You, J., and Cai, X. (2008a). “Determining forecast and decision horizons for reservoir operations under hedging policies.” Water Resour. Res., 44(11), W11430.
You, J. Y., and Cai, X. (2008b). “Hedging rule for reservoir operations. I. A theoretical analysis.” Water Resour. Res., 44(1), W01415.
You, J. Y., and Cai, X. (2008c). “Hedging rule for reservoir operations. II. A numerical model.” Water Resour. Res., 44(1), W01416.
Yun, R., and Singh, V. P. (2008). “Multiple duration limited water level and dynamic limited water level for flood control, with implications on water supply.” J. Hydrol., 354(1), 160–170.
Zhang, C., Zhu, X., Fu, G., Zhou, H., and Wang, H. (2014). “The impacts of climate change on water diversion strategies for a water deficit reservoir.” J. Hydroinf., 16(4), 872–889.
Zhao, J., Cai, X., and Wang, Z. (2011a). “Optimality conditions for a two-stage reservoir operation problem.” Water Resour. Res., 47(8), W08503.
Zhao, T., Cai, X., Lei, X., and Wang, H. (2012a). “Improved dynamic programming for reservoir operation optimization with a concave objective function.” J. Water Res. Plann. Manage., 590–596.
Zhao, T., Cai, X., and Yang, D. (2011b). “Effect of streamflow forecast uncertainty on real-time reservoir operation.” Adv. Water Resour., 34(4), 495–504.
Zhao, T., Yang, D., Cai, X., Zhao, J., and Wang, H. (2012b). “Identifying effective forecast horizon for real-time reservoir operation under a limited inflow forecast.” Water Resour. Res., 48(1).
Zhao, T., and Zhao, J. (2014). “Forecast-skill-based simulation of streamflow forecasts.” Adv. Water Resour., 71, 55–64.
Zhao, T., Zhao, J., Lund, J. R., and Yang, D. (2014). “Optimal hedging rules for reservoir flood operation from forecast uncertainties.” J. Water Res. Plann. Manage., 04014041.
Zhao, T., Zhao, J., Yang, D., and Wang, H. (2013). “Generalized martingale model of the uncertainty evolution of streamflow forecasts.” Adv. Water Resour., 57, 41–51.
Zhu, X., Zhang, C., Yin, J., Zhou, H., and Jiang, Y. (2014). “Optimization of water diversion based on reservoir operating rules—A case study of the Biliu River Reservoir, China.” J. Hydrol. Eng., 411–421.
Information & Authors
Information
Published In
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Jun 16, 2015
Accepted: Apr 20, 2016
Published online: Jul 14, 2016
Published in print: Nov 1, 2016
Discussion open until: Dec 14, 2016
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.