Decision Making Under Uncertainty

We introduce a number of stochastic programming models via examples and then proceed to derive one of the fundamental theorems in the field that brings to the fore the constrast between wait-and-see and here-and-now formulations.

A single thin oil-bearing stratum has constant thickness and is surrounded by a circular impermeable boundary. This circle C contains a domain D _o which is filled with oil, the rest is filled with water. The permeabilities for oil and for water are taken to be the same.

This stratum is pierced by several (vertical) wells one of which is a producer and intersects the oil domain D _o, the others intersect C outside D _o and are used for water injection.

We consider several problems concerning the control and optimization of the production by controlling the flow rates into and out of the various wells with a view to optimizing the net present value of oil produced or certain other relevant objective functions.

Following these deterministic optimization problems we allow for the considerable uncertainty in reservoir description by considering a stochastic scenario. The results are represented as probability distribution curves for the NPV and for total production. We also compute an efficient frontier analogous to the efficient frontier in portfolio management.

We present a dynamic multistage stochastic programming model for the cost-optimal generation of electric power in a hydro-thermal system under uncertainty in load, inflow to reservoirs and prices for fuel and delivery contracts. The stochastic load process is approximated by a scenario tree obtained by adapting a SARIMA model to historical data, using empirical means and variances of simulated scenarios to construct an initial tree, and reducing it by a scenario deletion procedure based on a suitable probability distance. Our model involves many mixed-integer variables and individual power unit constraints, but relatively few coupling constraints. Hence we employ stochastic Lagrangian relaxation that assigns stochastic multipliers to the coupling constraints. Solving the Lagrangian dual by a proximal bundle method leads to successive decomposition into single thermal and hydro unit subproblems that are solved by dynamic programming and a specialized descent algorithm, respectively. The optimal stochastic multipliers are used in Lagrangian heuristics to construct approximately optimal first stage decisions. Numerical results are presented for realistic data from a German power utility, with a time horizon of one week and scenario numbers ranging from 5 to 100. The corresponding optimization problems have up to 200,000 binary and 350,000 continuous variables, and more than 500,000 constraints.

Electricity producers participating in the Nordic wholesale-level market face significant uncertainty in inflow to reservoirs and prices in the spot and contract markets. Taking the view of a single risk-averse producer, we propose a stochastic programming model for the coordination of physical generation resources with hedging through the forward and option market. Numerical results are presented for a five-stage, 256 scenario model that has a two year horizon.

At the turn of the twentieth century, electric power was an exciting new technology. Capital flowed into highly leveraged entities that built power plants and wires in urban centers across the developed world. As time progressed electric utilities came under government regulation or control.

We consider wholesale electricity market pools in which generators must offer supply functions that are centrally dispatched. Each generator seeks a supply function to offer to the spot market, so as to maximise expected return. We give conditions under which a supply function exists that optimises return for every demand realisation. We also analyse the case in which the behaviour of the competition can be modelled by an appropriate probability distribution, and derive optimality conditions for the optimal supply-function offer in this case. The paper concludes with some remarks on applying the theory to the case where each generator must offer a limited number of prices in their stack.

Many economic equilibrium models have a structure that consists of econometrically estimated demand models and supply models that contain explicit representations of the supply technologies, known as process models. Econometric models measure the consequences of peoples’ decisions and are typically used to estimate demand because it is impossible to represent each individual decision and its consequences. Process modeling is an outgrowth of input-output analysis and linear programming and began with Markowitz [1955]. Here the technologies and possible decisions are modeled explicitly in an optimization model. The solution to the model consists of the decisions of optimizing firms and their consequences. Each modeling approach has had a long history and combining the two types of models into one economic equilibrium model is quite common. Examples are the energy-market models, PIES (Hogan [1975]), IFFS (Murphy, Conti, Sanders and Shaw [1988]), and NEMS (Energy Information Administration [1998]). For a summary of all three, see Murphy and Shaw [1995].