List of notations Preface to the second edition Preface to the first edition 1. Stochastic programming models 2. Two-stage problems 3. Multistage problems 4. Optimization models with probabilistic constraints 5. Statistical inference 6. Risk averse optimization 7. Background material 8. Bibliographical remarks Bibliography Index.

Lectures on Stochastic Programming: Modeling and Theory

This paper presents a security-constrained unit commitment (SCUC) algorithm which takes into account the intermittency and volatility of wind power generation. The UC problem is solved in the master problem with the forecasted intermittent wind power generation. Next, possible scenarios are simulated for representing the wind power volatility. The initial dispatch is checked in the subproblem and generation redispatch is considered for satisfying the hourly volatility of wind power in simulated scenarios. If the redispatch fails to mitigate violations, Benders cuts are created and added to the master problem to revise the commitment solution. The iterative process between the commitment problem and the feasibility check subproblem will continue until simulated wind power scenarios can be accommodated by redispatch. Numerical simulations indicate the effectiveness of the proposed SCUC algorithm for managing the security of power system operation by taking into account the intermittency and volatility of wind power generation.

Security-Constrained Unit Commitment With Volatile Wind Power Generation

Portfolio and risk management problems of power utilities may be modeled by multistage stochastic programs. These models use a set of scenarios and corresponding probabilities to model the multivariate random data process (electrical load, stream flows to hydro units, and fuel and electricity prices). For most practical problems the optimization problem that contains all possible scenarios is too large. Due to computational complexity and to time limitations this program is often approximated by a model involving a (much) smaller number of scenarios. The proposed reduction algorithms determine a subset of the initial scenario set and assign new probabilities to the preserved scenarios. The scenario tree construction algorithms successively reduce the number of nodes of a fan of individual scenarios by modifying the tree structure and by bundling similar scenarios. Numerical experience is reported for constructing scenario trees for the load and spot market prices entering a stochastic portfolio management model of a German utility.

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Scenario reduction and scenario tree construction for power management problems

This paper considers stochastic programming models for decision-making under uncertainty in the context of electricity markets. It provides a brief overview of modeling and solution techniques within a mathematical programming framework. Tutorial as well as recent references are provided. This paper provides the guidelines for discussion in a panel session organized by the authors on "Decision Making under Uncertainty in Electricity Markets", scheduled for the IEEE PES 2006 General Meeting.

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Decision Making Under Uncertainty in Electricity Markets

Model predictive control (MPC) has demonstrated exceptional success for the high-performance control of complex systems [1], [2]. The conceptual simplicity of MPC as well as its ability to effectively cope with the complex dynamics of systems with multiple inputs and outputs, input and state/output constraints, and conflicting control objectives have made it an attractive multivariable constrained control approach [1]. MPC (a.k.a. receding-horizon control) solves an open-loop constrained optimal control problem (OCP) repeatedly in a receding-horizon manner [3]. The OCP is solved over a finite sequence of control actions {u0,u1,f,uN- 1} at every sampling time instant that the current state of the system is measured. The first element of the sequence of optimal control actions is applied to the system, and the computations are then repeated at the next sampling time. Thus, MPC replaces a feedback control law p(m), which can have formidable offline computation, with the repeated solution of an open-loop OCP [2]. In fact, repeated solution of the OCP confers an "implicit" feedback action to MPC to cope with system uncertainties and disturbances. Alternatively, explicit MPC approaches circumvent the need to solve an OCP online by deriving relationships for the optimal control actions in terms of an "explicit" function of the state and reference vectors. However, explicit MPC is not typically intended to replace standard MPC but, rather, to extend its area of application [4]-[6].

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Stochastic Model Predictive Control: An Overview and Perspectives for Future Research

We consider convex stochastic programs with an (approximate) initial probability distribution P having finite support supp P, i.e., finitely many scenarios. The behaviour of such stochastic programs is stable with respect to perturbations of P measured in terms of a Fortet-Mourier probability metric. The problem of optimal scenario reduction consists in determining a probability measure that is supported by a subset of supp P of prescribed cardinality and is closest to P in terms of such a probability metric. Two new versions of forward and backward type algorithms are presented for computing such optimally reduced probability measures approximately. Compared to earlier versions, the computational performance (accuracy, running time) of the new algorithms has been improved considerably. Numerical experience is reported for different instances of scenario trees with computable optimal lower bounds. The test examples also include a ternary scenario tree representing the weekly electrical load process in a power management model.

Scenario Reduction Algorithms in Stochastic Programming

An important issue for solving multistage stochastic programs consists in the approximate representation of the (multivariate) stochastic input process in the form of a scenario tree. In this paper, we develop (stability) theory-based heuristics for generating scenario trees out of an initial set of scenarios. They are based on forward or backward algorithms for tree generation consisting of recursive scenario reduction and bundling steps. Conditions are established implying closeness of optimal values of the original process and its tree approximation, respectively, by relying on a recent stability result in Heitsch, Romisch and Strugarek (SIAM J Optim 17:511–525, 2006) for multistage stochastic programs. Numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management.

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Scenario tree modeling for multistage stochastic programs

A framework for the reduction of scenario trees as inputs of (linear) multistage stochastic programs is provided such that optimal values and approximate solution sets remain close to each other. The argument is based on upper bounds of the L r -distance and the filtration distance, and on quantitative stability results for multistage stochastic programs. The important difference from scenario reduction in two-stage models consists in incorporating the filtration distance. An algorithm is presented for selecting and removing nodes of a scenario tree such that a prescribed error tolerance is met. Some numerical experience is reported.

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Scenario tree reduction for multistage stochastic programs

http://www.math.hu-berlin.de/%7Eheitsch/HR07orl.pdf

Holger Heitsch

Papers

Scenario Reduction Algorithms in Stochastic Programming

Scenario reduction and scenario tree construction for power management problems

Scenario tree modeling for multistage stochastic programs

Scenario tree reduction for multistage stochastic programs

A note on scenario reduction for two-stage stochastic programs