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Capacity expansion using stochastic programming with

Decision Dependent Probabilities

Lars Hellemo

Asgeir Tomasgard

Paul I. Barton

ICSP XIII 2013

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Decision dependent distribution

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Outline

Decision dependent uncertainty in stochastic programs - taxonomy

Decision dependent probabilities

Capacity expansion application

Some numerical results

Original motivation:

1) How can we reformulate DDP problems to recourse models?

2) Can we use a generalization of Generalized Benders to solve them?

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Decision dependent probabilities

The underlying problem is stochastic with decision dependent uncertainty

Formulated as large scale non-convex stochastic program

Ax0 £ b

ps(y)s

å =1

Wsxs = Ts

T x0 - Hs, "s

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Real world versus problem I Take a problem where a dam in a river is to be built, and the dimensions to be determined.

The risk of a dam break has to be balanced against the extra cost of further reinforcing the dam. Depending on how it is built, the dam will resist different amounts of inflow. Inflows vary stochastically from year to year.

If you think there is a deterministic link between the inflow and the risk of breakage, this is exogenous uncertainty. The stochastic inflow is not influenced by the way the dam is build, rather the dam's resistance to various inflows is (deterministically) .

However, the relationship between the dam dimensions and the probability of a break based on stochastic inflows can be difficult to include in an optimization model. The modeller can choose to model this with endogenous uncertainty .

Here the probability of breakage depends on how strong we build the dam, possibly hiding some of the underlying stochastic phenomena.

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Real world versus problem II

Consider uncertainty about the properties of an oil reservoir. For example, where to drill wells, when (and if) to drill the wells

The reserves in the reservoir are deterministic (in the short term...). But we do not know what they are.....How should we model?

Our decision to drill a well does normally not change the size of the reservoir as such (that is deterministic!), but it will provide more information about the true state of the reservoir. This information, is however not revealed unless we pay the substantial cost of drilling the well (endogenous uncertainty in the information process of the problem ).

Some drilling approaches may also jeopardize the reservoir itself by introducing leaks between layers in the ground, something that could leave parts of the reservoir unrecoverable (endogenous uncertainty in the real world and in the problem).

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Mapping of real world to model

Deterministic

Exogenous

Endogenous

Real world

Deterministic

Exogenous

Endogenous

Problem Model class

Non-linear (non-

convex)

MIP

LP

Example: Reformulation to Stochastic programs with recourse

in discrete time and with discrete distributions.

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T. Jonsbråten, R. Wets, and D. Woodruff. A class of stochastic programs with decision dependent random

elements. Annals of Operations Research, 82:83, 106, 1998.

J. Dupačová. Optimization under exogenous and endogenous uncertainty. L. Lukáš (Ed.), Proc. of MME06,

University of West Bohemia in Pilsen (2006)

Taxonomy suggested extension

Decision

Dependent

Dynamics &

Distribution

(Type 3)

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Modify probability distribution

Adjusting scenario probabilities Many earlier approaches based on MILP and distribution

selection with binary y

We want to study y continuous and more general formulations

|( )p y

K. Viswanath, S. Peeta, and S. Salman. Investing in the Links of a Stochastic Network to Minimize Expected

Shortest Path. Length. Purdue University Economics Working Papers, 2004.

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Other relevant litterature

G. Pflug. Optimization of stochastic models: the interface between simulation and optimization. Kluwer Academic Pub, 1996.

G. C. Pflug. On-line optimization of simulated Markovian processes. Mathematics of Operations Research, 15(3):381{395, August 1990.

R. Rubinstein and A. Shapiro. Discrete event systems: Sensitivity analysis and stochastic optimization by the score function method. 1993.

Z. Artstein and R. Wets. Stability results for stochastic programs and sensors, allowing for discontinuous objective functions. SIAM Journal on Optimization, 4:537, 1994.

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Formulation 1: Scaling

In the special case where the original distribution is

uniform, this gives the function

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Formulation 2: Convex combination

Example of mixture distribution of normal distributions

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Discreteization of distribution function

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Form. 3: Optimization over parameters in the distribution

f (x | a,b) = abxa-1(1- xa)b-1

F(x | a,b) =1- (1- xa)b

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Kumaraswamy distribution

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Kumaraswamy distribution

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Form 4: Approximation of distribution + parameter opt.

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Standard distribition

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Standard distribution

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Example: Change of parameters

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Branch-and-bound

To guarantee convergence for these non-convex non-linear models: Continuous branch-and-bound

Branching strategy is essential

In addition integer B&B depending on the model.

Gams versions 23.6.2 and 23.7.2 with Baron using CPLEX in combination with CONOPT or MINOS.

GBD inspired decomposition at first seemed to work well in many cases . Until we tried strategic branching....

Baron using intelligent branching outperformed the GBD inspired decomposition

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Test Case, application from energy

Inspired by Louveaux and Smeers (1988)

Invest in energy producing technologies (i)

Serve demand in modes (j) approximating a load curve

Technology/activity (y) available to modify probability. This is our addition

F. V. Louveaux and Y. Smeers. Optimal investments for electricity generation: A stochastic model and a test

problem. In Y. Ermoliev and R. J.-B. Wets, editors, Numerical Techniques for Stochastic Optimization,

pages 445-454. Springer-Verlag, 1988.

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Base model

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Scalable subset of scenarios uniform distribution

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Convex combination of distributions

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Computational results

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Probabilities from solution of Gams implementation of model using 100 scenarios and Kumaraswamy distribution

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Probabilities from optimal solution of Gams implementation of model using 100 scenarios and approximation of normal distribution

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Research

Structural properties

Decomposition

New formulations

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