Using stochastic dual dynamic programming in problems with multiple near‐optimal solutions

Stochastic dual dynamic programming (SDDP) is one of the few algorithmic solutions available to optimize large-scale water resources systems while explicitly considering uncertainty. This paper explores the consequences of, and proposes a solution to, the existence of multiple near-optimal solutions (MNOS) when using SDDP for mid or long-term river basin management. These issues arise when the optimization problem cannot be properly parametrized due to poorly defined and/or unavailable data sets. This work shows that when MNOS exists, (1) SDDP explores more than one solution trajectory in the same run, suggesting different decisions in distinct simulation years even for the same point in the state-space, and (2) SDDP is shown to be very sensitive to even minimal variations of the problem setting, e.g., initial conditions—we call this “algorithmic chaos.” Results that exhibit such sensitivity are difficult to interpret. This work proposes a reoptimization method, which simulates system decisions by periodically applying cuts from one given year from the SDDP run. Simulation results obtained through this reoptimization approach are steady state solutions, meaning that their probability distributions are stable from year to year.

[1]  A. Tilmant,et al.  Assessing marginal water values in multipurpose multireservoir systems via stochastic programming , 2008 .

[2]  A. Tilmant,et al.  Restoring a flow regime through the coordinated operation of a multireservoir system: The case of the Zambezi River basin , 2010 .

[3]  A. Gjelsvik,et al.  Long- and Medium-term Operations Planning and Stochastic Modelling in Hydro-dominated Power Systems Based on Stochastic Dual Dynamic Programming , 2010 .

[4]  C. S. Buchanan,et al.  Nested Benders decomposition and dynamic programming for reservoir optimisation , 1999, J. Oper. Res. Soc..

[5]  Jery R. Stedinger,et al.  Water Resources Systems Planning And Management , 2006 .

[6]  M. Pereira.  Optimal stochastic operations scheduling of large hydroelectric systems , 1989 .

[7]  Keith Beven,et al.  Prophecy, reality and uncertainty in distributed hydrological modelling , 1993 .

[8]  Birger Mo,et al.  Integrated risk management of hydro power scheduling and contract management , 2001 .

[9]  Ximing Cai,et al.  Finding multiple optimal solutions to optimal load distribution problem in hydropower plant , 2012 .

[10]  Tito Homem-de-Mello,et al.  Sampling strategies and stopping criteria for stochastic dual dynamic programming: a case study in long-term hydrothermal scheduling , 2011 .

[11]  M. V. F. Pereira,et al.  Multi-stage stochastic optimization applied to energy planning , 1991, Math. Program..

[12]  David E. Rosenberg,et al.  Hydro-economic models: concepts, design, applications, and future prospects. , 2009 .

[13]  C. Perrin,et al.  Improvement of a parsimonious model for streamflow simulation , 2003 .

[14]  W. Kinzelbach,et al.  The cost of noncooperation in international river basins , 2012 .

[15]  L. A. Barroso,et al.  Assessment of parameter uncertainty in autoregressive streamflow models for stochastic long-term hydrothermal scheduling , 2012, 2012 IEEE Power and Energy Society General Meeting.

[16]  Pan Liu,et al.  A Bayesian model averaging method for the derivation of reservoir operating rules , 2015 .

[17]  Otto E. Rössler,et al.  Different Types of Chaos in Two Simple Differential Equations , 1976 .

[18]  Keith Beven,et al.  A manifesto for the equifinality thesis , 2006 .

[19]  Amaury Tilmant,et al.  A stochastic approach to analyze trade‐offs and risks associated with large‐scale water resources systems , 2007 .

[20]  Peter Bauer-Gottwein,et al.  Optimization of Multipurpose Reservoir Systems Using Power Market Models , 2015 .

[21]  Y. Mohamed,et al.  Optimal operation of a multipurpose multireservoir system in the Eastern Nile River Basin , 2010 .

[22]  S. Wallace,et al.  Stochastic Programming Models in Energy , 2003 .

[23]  Pan Liu,et al.  Deriving multiple near‐optimal solutions to deterministic reservoir operation problems , 2011 .

[24]  D. Whittington,et al.  Incentive compatibility and conflict resolution in international river basins: A case study of the Nile Basin , 2006 .

[25]  Tarjei Kristiansen.  Financial risk management in the electric power industry using stochastic optimization , 2004 .

[26]  Amaury Tilmant,et al.  Economic valuation of benefits and costs associated with the coordinated development and management of the Zambezi river basin , 2012 .

[27]  Keith Beven,et al.  Equifinality and the problem of robust calibration in nitrogen budget simulations , 1999 .

[28]  Sharon A. Johnson,et al.  Comparison of two approaches for implementing multireservoir operating policies derived using stochastic dynamic programming , 1993 .

[29]  Amaury Tilmant,et al.  Hydro-economic risk assessment in the eastern Nile River basin , 2014 .

[30]  Pan Liu,et al.  Deriving Near-Optimal Solutions to Deterministic Reservoir Operation Problems , 2010 .

[31]  D. E. Rheinheimer,et al.  Parameter uncertainty analysis of reservoir operating rules based on implicit stochastic optimization. , 2014 .

[32]  Leon S. Lasdon,et al.  A framework for sustainability analysis in water resources management and application to the Syr Darya Basin , 2002 .

[33]  A. Gjelsvik,et al.  Stochastic dual dynamic programming for seasonal scheduling in the Norwegian power system , 1992 .

[34]  D. Rosbjerg,et al.  The cost of ending groundwater overdraft on the North China Plain , 2015 .

[35]  Quentin Goor,et al.  Optimal Multipurpose-Multireservoir Operation Model with Variable Productivity of Hydropower Plants , 2011 .