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We consider stochastic programs with risk measures in the objective and study
stability properties as well as decomposition structures. Thereby we place emphasis on dynamic
models, i.e., multistage stochastic programs with multiperiod risk measures. In this context, we
define the class of polyhedral risk measures such that stochastic programs with risk measures taken
from this class have favorable properties. Polyhedral risk measures are defined as optimal values of
certain linear stochastic programs where the arguments of the risk measure appear on the right-hand
side of the dynamic constraints. Dual representations for polyhedral risk measures are derived and
used to deduce criteria for convexity and coherence. As examples of polyhedral risk measures we
propose multiperiod extensions of the Conditional-Value-at-Risk.
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.
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 analyse stability aspects of linear multistage stochastic programs with polyhedral risk measures in the objective. In particular, we consider sensitivity of the optimal value with respect perturbations of the underlying stochastic input process. An existing stability result for multistage stochastic programs with expectation objective is carried forward to the case of polyhedral risk-averse objectives. Beside Lr-distances these results also involve filtration distances of the perturbations of the stochastic process. We discuss additional requirements for the
polyhedral risk measures such that the problem dependent filtration distances can be bounded by problem independent ones. Stability and such bounds are the basis for scenario tree approximation techniques used in practical problem solving.
The possibility of controlling risk in stochastic power optimization by incorporating special risk functionals, so-called polyhedral risk measures, into the objective is demonstrated. We present an exemplary optimization model for mean-risk optimization of an electricity portfolios of a price-taking retailer. Stochasticity enters the model via uncertain electricity demand, heat demand, spot prices, and future prices. The objective is to maximize the expected overall revenue and, simultaneously, to minimize risk in terms of multiperiod risk measures, i.e., risk measures that take into account intermediate cash values in order to avoid liquidity problems at any time. We compare the effect of different multiperiod polyhedral risk measures that had been suggested in our earlier work.
Stochastic Optimization of Electricity Portfolios: Scenario Tree Modeling and Risk Management
(2008)
We present recent developments in the field of stochastic programming with regard to application in power management. In particular we discuss issues of scenario tree modeling, i.e., appropriate discrete approximations of the underlying stochastic parameters. Moreover, we suggest risk avoidance strategies via the incorporation of
so-called polyhedral risk functionals into stochastic programs. This approach, motivated through tractability of the resulting problems, is a constructive framework providing particular flexibility with respect to the dynamic aspects of risk.
We compare different multiperiod risk measures taken from the class of polyhedral risk measures with respect to the effect
they show when used in the objective of a stochastic program. For this purpose, simulation results of a stochastic programming
model for optimizing the electricity portfolio of a German municipal power utility are presented and analyzed. This model
aims to minimize risk and expected overall cost simultaneously.
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.
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.
Stochastic optimization techniques are highly relevant for applications in electricity production and trading since, in particular after the deregulations of many electricity markets, there is a high number of uncertainty factors (e.g., demand, spot prices) to be considered that can be described reasonably by statistical models. Here, we want to highlight two aspects of this approach: scenario tree approximation and risk aversion. The former is a procedure to replace a general statistical model (probability distribution), which makes the optimization problem intractable, suitably by a finite discrete distribution (scenarios). This is typically an indispensable first step towards a solution of a stochastic optimization model. On the other hand, this is a highly sensitive concern, in particular if dynamic decision structures are involved (multistage stochastic programming). Then, the approximate distribution must exhibit tree structure. Moreover, it is of interest to get by with a moderate number of scenarios to have the resulting problem tractable. In any case, it has to be relied on suitably stability results to ensure that the obtained results are indeed related to the original (infinite dimensional) problem. These stability results involve probability distances and, for the multistage case, a filtration distance that evaluates the information increase over time. We present respective approximation schemes relying on Monte Carlo sampling and scenario reduction and combining techniques. The second topic of this talk is risk aversion. Namely, we present the approach of polyhedral risk measures which are given as (the optimal values of) certain simple stochastic programs. Well-known risk measures such as CVaR and expected polyhedral utility belong to this class and, moreover, multiperiod risk measures for multistage stochastic programs are suggested. For stochastic programs incorporating polyhedral risk measures it has been shown that numerical tractability as well as stability results known for classical (non-risk-averse) stochastic programs remain valid. In particular, the same scenario approximation methods can be used. Finally, we present illustrative numerical results from an electricity portfolio optimization model for a municipal power utility.