Andris Möller

Optimization of a continuous distillation process under random inflow rate

Abstract The paper deals with a continuous distillation process under stochastic rate of inflows collected in a feed tank. The aim of analysis is to find a robust control of extracting feed from the tank over a certain time horizon such that—without knowledge of future realizations of the inflow rate—some level constraints in the feed tank will be met with high probability. This approach relies on formulating and numerically treating probabilistic constraints. The inflow rate is considered as a stochastic process for which two basically different model assumptions are made: the first model assumes a Gaussian process, and thus reflects the superposition of many independent elementary inflows; the second model treats maybe the simplest case of a single elementary inflow profile, namely rectangular inflows with fixed rate and duration but stochastic starting time. Numerical results illustrating both assumptions are presented, and advantages over the simple anticipation of nominal inflow profiles are highlighted.

A Gradient Formula for Linear Chance Constraints Under Gaussian Distribution

We provide an explicit gradient formula for linear chance constraints under a (possibly singular) multivariate Gaussian distribution. This formula allows one to reduce the calculus of gradients to the calculus of values of the same type of chance constraints (in smaller dimension and with different distribution parameters). This is an important aspect for the numerical solution of stochastic optimization problems because existing efficient codes for, e.g., calculating singular Gaussian distributions or regular Gaussian probabilities of polyhedra can be employed to calculate gradients at the same time. Moreover, the precision of gradients can be controlled by that of function values, which is a great advantage over using finite difference approximations. Finally, higher order derivatives are easily derived explicitly. The use of the obtained formula is illustrated for an example of a transportation network with stochastic demands.

On probabilistic constraints induced by rectangular sets and multivariate normal distributions

In this paper, we consider optimization problems under probabilistic constraints which are defined by two-sided inequalities for the underlying normally distributed random vector. As a main step for an algorithmic solution of such problems, we prove a derivative formula for (normal) probabilities of rectangles as functions of their lower or upper bounds. This formula allows to reduce the calculus of such derivatives to the calculus of (normal) probabilities of rectangles themselves thus generalizing a similar well-known statement for multivariate normal distribution functions. As an application, we consider a problem from water reservoir management. One of the outcomes of the problem solution is that the (still frequently encountered) use of simple individual probabilistic constraints can completely fail. By contrast, the (more difficult) use of joint probabilistic constraints, which heavily depends on the derivative formula mentioned before, yields very reasonable and robust solutions over the whole time horizon considered.

Airline network revenue management by multistage stochastic programming

A multistage stochastic programming approach to airline network revenue management is presented. The objective is to determine seat protection levels for all itineraries, fare classes, points of sale of the airline network and all dcps of the booking horizon such that the expected revenue is maximized. While the passenger demand and cancelation rate processes are the stochastic inputs of the model, the stochastic protection level process represents its output and allows to control the booking process. The stochastic passenger demand and cancelation rate processes are approximated by a finite number of tree structured scenarios. The scenario tree is generated from historical data using a stability-based recursive scenario reduction scheme. Numerical results for a small hub-and-spoke network are reported.

Stochastic Optimization for Operating Chemical Processes under Uncertainty

Mathematical optimization techniques are on their way to becoming a standard tool in chemical process engineering. While such approaches are usually based on deterministic models, uncertainties such as external disturbances play a significant role in many real-life applications. The present article gives an introduction to practical issues of process operation and to basic mathematical concepts required for the explicit treatment of uncertainties by stochastic optimization.

On joint probabilistic constraints with Gaussian coefficient matrix

The paper deals with joint probabilistic constraints defined by a Gaussian coefficient matrix. It is shown how to explicitly reduce the computation of values and gradients of the underlying probability function to that of Gaussian distribution functions. This allows us to employ existing efficient algorithms for calculating this latter class of functions in order to solve probabilistically constrained optimization problems of the indicated type. Results are illustrated by an example from energy production.

Primal and Dual Methods for Unit Commitment in a Hydro-Thermal Power System

The unit commitment prob lem in a power generation system com prising thermal and pumped storage hy dro units is addressed A large scale mixed integer optimization model for unit commitment in a real power system is de veloped and solved by primal and dual ap proaches Both solution methods employ state of the art algorithms and software Results of test runs are reported

Solving the Unit Commitment Problem in Power Generation by Primal and Dual Methods

The unit commitment problem in power plant operation planning is addressed. For a real power system comprising coal- and gas-fired thermal and pumped-storage hydro plants a large-scale mixed integer optimization model for unit commitment is developed. Then primal and dual approaches to solving the optimization problem are presented and results of test runs are reported.

Probabilistic constraints via SQP solver: application to a renewable energy management problem

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.

Optimal Control of a Continuous Distillation Process under Probabilistic Constraints

A continuous distillation process with random inflow rate is considered. The aim is to find a control (feed rate, heat supply, reflux rate) which is optimal with respect to energy consumption and which is robust at the same time with respect to the stochastic level constraints in the feed tank. The solution approach is based on the formulation of probabilistic constraints. An overall model including the dynamics of the distillation process and probabilistic constraints under different assumptions on the randomness of inflow is developed and numerical results are presented.