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We consider empirical approximations of two-stage stochastic mixed-integer linear programs and derive central
limit theorems for the objectives and optimal values. The limit theorems are based on empirical process theory
and the functional delta method. We also show how these limit theorems can be used to derive confidence intervals
for optimal values via a certain modification of the bootstrapping method.
Bei der Produktions- und Handelsplanung treffen Energieversorgungsunternehmen eine Reihe von Entscheidungen unter unsicheren Randbedingungen. Ein Optimierungsmodell für einen mittelfristigen Planungshorizont muss diese Unsicherheiten berücksichtigen, etwa durch Einbeziehung von statistischen Modellen für die zufallsbehafteten Eingangsdaten. Dadurch ist es prinzipiell möglich, Risikobetrachtungen direkt in die Optimierung zu integrieren. Wir demonstrieren in dieser Arbeit die Möglichkeit, spezielle dynamische Risikomaße, so genannte polyedrische Risikomaße, in die Zielfunktion der Optimierung mit aufzunehmen. Im Gegensatz zu vielen anderen Ansätzen wird dadurch die Komplexität des Problems nicht wesentlich erhöht. Das vorgestellte Modell stellt ein Werkzeug zur Entscheidungsunterstützung für kleinere Marktteilnehmer hinsichtlich der Beschaffungsplanung dar. Dabei werden insbesondere konkrete mittelfristig bindende Bezugsverträge mit der Möglichkeit verglichen, die Versorgung in erster Linie auf der Basis von Spot- und Futuremarkt zu planen.
We present an applied mathematical model with stochastic input data for mean-risk optimization of electricity portfolios containing electricity futures as well as several components to satisfy a stochastic electricity demand: electricity spot market, two different types of supply contracts offered by a large power producer, and a combined heat and power production facility with limited capacity. Stochasticity enters the model via uncertain electricity demand, heat demand, spot prices, and future prices. The model is set up as a decision support system for a municipal power utility (price taker) and considers a medium term optimization horizon of one year in hourly discretization. The objective is to maximize the expected overall revenue and, simultaneously, to minimize risk in terms of multiperiod risk measures. Such risk measures take into account intermediate cash values in order to avoid uncertainty and liquidity problems at any time. We compare the effect of different multiperiod risk measures taken from the class of polyhedral risk measures which was suggested in our earlier work.
We present a mathematical model with stochastic input data for mean-risk optimization of electricity portfolios containing
several physical components and energy derivative products. The model is designed for a medium term optimization horizon
of one year in hourly discretization. With the objective of maximization of the mean book value of the portfolio at the end of
optimization horizon simultaneously several risk measures are taken into account. We present numerical results for a largescale
realistic problem adapted to a municipal utility and study the effects of varying weighting of risk on the book value of
the portfolio during the whole time horizon.