Abstract
This paper aims at illustrating the efficient solution of nonlinear optimization problems with joint probabilistic constraints under multivariate Gaussian distributions. The numerical solution approach is based on Sequential Quadratic Programming (SQP) and is applied to a renewable energy management problem. We consider a coupled system of hydro and wind power production used in order to satisfy some local demand of energy and to sell/buy excessive or missing energy on a day-ahead and intraday market, respectively. A short term planning horizon of 2 days is considered and only wind power is assumed to be random. In the first part of the paper, we develop an appropriate optimization problem involving a probabilistic constraint reflecting demand satisfaction. Major attention will be payed to formulate this probabilistic constraint not directly in terms of random wind energy produced but rather in terms of random wind speed, in order to benefit from a large data base for identifying an appropriate distribution of the random parameter. The second part presents some details on integrating Genz’ code for Gaussian probabilities of rectangles into the environment of the SQP solver SNOPT. The procedure is validated by means of a simplified optimization problem which by its convex structure allows to estimate the gap between the numerical and theoretical optimal values, respectively. In the last part, numerical results are presented and discussed for the original (nonconvex) optimization problem.
Similar content being viewed by others
References
Aksoy H, Toprak ZF, Aytek A, Ünal NE (2004) Stochastic generation of hourly mean wind speed data. Renew Energy 29:2111–2131
Andrieu L, Henrion R, Römisch W (2010) A model for dynamic chance constraints in hydro power reservoir management. Eur J Oper Res 207:579–589
Arnold T, Henrion R, Möller A, Vigerske S (2014) A mixed-integer stochastic nonlinear optimization problem with joint probabilistic constraints. Pacific J Optim 10:5–20
Ben-Tal A, Nemirovski A (2002) Robust solutions of linear programming problems contaminated with uncertain data. Math Program 88:411–424
Calafiore GC, Campi MC (2006) The scenario approach to robust control design. IEEE Trans Automat Control 51:742–753
de Doncker E, Genz A, Ciobanu M (1999) Parallel computation of multivariate normal probabilities. Comput Sci Stat 31:89–93
Dentcheva D, Martinez G (2013) Regularization methods for optimization problems with probabilistic constraints. Math Program 138:223–251
Ermoliev YM, Ermolieva TY, Macdonald GJ, Norkin VI (2000) Stochastic optimization of insurance portfolios for managing exposure to catastrophic risk. Ann Oper Res 99:207–225
Genz A (1992) Numerical computation of multivariate normal probabilities. J Comput Gr Stat 1:141–149
Genz A, Bretz F (2009) Computation of multivariate normal and t probabilities. Lecture notes in statistics, vol 195. Springer, Dordrecht
Gill PE, Murray W, Saunders MA (1997) SNOPT: An SQP algorithm for large-scale constraint optimization, Numerical Analysis Report 97–1. Department of Mathmatics, University of California, San Diego
Henrion R (2012) Gradient estimates for Gaussian distribution functions: application to probabilistically constrained optimization problems. Numer Algebra Control Optim 2:655–668
Henrion R, Möller A (2012) A gradient formula for linear chance constraints under Gaussian distribution. Math Oper Res 37:475–488
Luedtke J, Ahmed S (2008) A sample approximation approach for optimization with probabilistic constraints. SIAM J Optim 19:674–699
Nemirovski A, Shapiro A (2006) Convex approximations of chance constrained programs. SIAM J Optim 17:969–996
Olieman N, van Putten B (2010) Estimation method of multivariate exponential probabilities based on a simplex coordinates transform. J Stat Comput Simul 80:355–361
Pagnoncelli B, Ahmed S, Shapiro A (2009) Sample average approximation method for chance constrained programming: theory and applications. J Optim Theory Appl 142:399–416
Prékopa A (2003) Probabilistic programming, Stochastic programming Ruszczyński A, Shapiro A (eds), Handbooks in operations research and management science, vol 10. Elsevier, Amsterdam
Prékopa A, Szántai T (1978) Flood control reservoir system design using stochastic programming. Math Program Study 9:138–151
Prékopa A (1995) Stochastic Programming. Kluwer, Dordrecht
Royset JO, Polak E (2007) Extensions of stochastic optimization results to problems with system failure probability functions. J Optim Theory Appl 133:1–18
Shapiro A, Dentcheva D, Ruszczyński A (2009) Lectures on stochastic programming. MPS-SIAM series on optimization p 9
Szántai T (1996) Evaluation of a special multivariate Gamma distribution. Math Program Study 27:1–16
Tamura J (2012) Calculation method of losses and efficiency of wind generators. In: Muyeen M (ed) Wind energy conversion systems. Technology and trends, Springer, Berlin, pp 25–51
Van Ackooij W, Henrion R (2014) Gradient formulae for nonlinear probabilistic constraints with Gaussian and Gaussian-like distributions. SIAM J Optim 24:1864–1889
Van Ackooij W, Henrion R, Möller A, Zorgati R (2010) On probabilistic constraints induced by rectangular sets and multivariate normal distributions. Math Methods Oper Res 71:535–549
Van Ackooij W, Henrion R, Möller A, Zorgati R (2011) On joint probabilistic constraints with Gaussian coefficient matrix. Oper Res Lett 39:99–102
Van Ackooij W, Zorgati R, Henrion R, Möller A (2014) Joint chance constrained programming for hydro reservoir management. Optim Eng 15:509–531
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.
Rights and permissions
About this article
Cite this article
Bremer, I., Henrion, R. & Möller, A. Probabilistic constraints via SQP solver: application to a renewable energy management problem. Comput Manag Sci 12, 435–459 (2015). https://doi.org/10.1007/s10287-015-0228-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10287-015-0228-z
Keywords
- Probabilistic constraints
- Renewable energies
- Multivariate Gaussian probability
- SQP with low precision data