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

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

The aim of this paper is to illustrate the efficient solution of nonlinear optimization problems with joint probabilistic constraints by means of an SQP method. Here, the random vector is assumed to obey some multivariate Gaussian distribution. The numerical solution approach 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

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

The unit commitment problem in a power generation system comprising thermal and pumped-storage hydro units is adressed. A large-scale mixed-integer optimization model for unit commitment in a real power system is developed and solved by primal and dual approaches. Both solution methods employ state-of-the-art algorithms and software. Results of test runs are reported

Optimale Blockauswahl bei der Kraftwerkseinsatzplanung der VEAG

In der vorliegenden Arbeit beschreiben wir einen LP-basierten Branch-and-Bound- und einen Lagrange-Relaxations-Zugang für das Blockauswahlproblem in der Kraftwerkseinsatzplanung, wobei moderne Ansätze und Algorithmen für die entstehenden Teilprobleme zum Einsatz kommen. Für das zugrundeliegende Erzeugersystem aus thermischen Kraftwerken und Pumpspeicherwerken wurde ein gemischt-ganzzahliges lineares Optimierungsmodell entwickelt. Berichtet wird über Testrechnungen für dieses Modell in der mittelfristigen Planung zunächst mit Zeiträumen bis zu sechs Monaten

Design, Performance, and Calibration of the CMS Hadron-Outer Calorimeter

The CMS hadron calorimeter is a sampling calorimeter with brass absorber and plastic scintillator tiles with wavelength shifting fibres for carrying the light to the readout device. The barrel hadron calorimeter is complemented with an outer calorimeter to ensure high energy shower containment in the calorimeter. Fabrication, testing and calibration of the outer hadron calorimeter are carried out keeping in mind its importance in the energy measurement of jets in view of linearity and resolution. It will provide a net improvement in missing \et measurements at LHC energies. The outer hadron calorimeter will also be used for the muon trigger in coincidence with other muon chambers in CMS

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 seatprotection levels for all itineraries, fare classes, point of sales of the airlinenetwork and all data collection points of the booking horizon such that theexpected revenue is maximized. While the passenger demand and cance-lation rate processes are the stochastic inputs of the model, the stochasticprotection level process represents its output and allows to control the booking process. The stochastic passenger demand and cancelation rate processesare approximated by a finite number of tree structured scenarios. The scenario tree is generated from historical data using a stability-based recursivescenario reduction scheme. Numerical results for a small hub-and-spoke network are reported

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

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