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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.
Quantitative stability of linear multistage stochastic programs is studied. It
is shown that the infima of such programs behave (locally) Lipschitz continuous
with respect to the sum of an Lr-distance and of a distance measure for the filtrations
of the original and approximate stochastic (input) processes. Various issues
of the result are discussed and an illustrative example is given. Consequences for
the reduction of scenario trees are also discussed.
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.
The paper provides a condition for differentiability as well as an equivalent criterion
for Lipschitz continuity of singular normal distributions. Such distributions are of interest,
for instance, in stochastic optimization problems with probabilistic constraints, where
a comparatively small (nondegenerate-) normally distributed random vector induces a large
number of linear inequality constraints (e.g. networks with stochastic demands). The criterion
for Lipschitz continuity is established for the class of quasi-concave distributions which
the singular normal distribution belongs to.
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.
Polyhedral discrepancies are relevant for the quantitative stability
of mixed-integer two-stage and chance constrained stochastic programs. We
study the problem of optimal scenario reduction for a discrete probability
distribution with respect to certain polyhedral discrepancies and develop
algorithms for determining the optimally reduced distribution approximately.
Encouraging numerical experience for optimal scenario reduction is provided.
Dynamic risk management in electricity portfolio optimization via polyhedral risk functionals
(2008)
We propose a methodology for combining risk management with optimal planning of power production and trading based on probabilistic knowledge about future uncertainties such as demands and spot prices. Typically, such a joint optimization of risk and (expected) revenue yields additional overall efficiency. Our approach is based on stochastic optimization (stochastic programming) with a risk functional as objective. The latter maps an uncertain cash flow to a real number. In particular, we employ so-called polyhedral risk functionals which, though being non-linear mappings, preserve linearity structures of optimization problems. Therefore, these are favorable to the numerical tractability of the optimization problems. The class of polyhedral risk functionals contains well-known risk functionals such as Average-Value-at-Risk and expected polyhedral utility. Moreover, it is also capable to model different dynamic risk mitigation strategies.
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 Lr-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.
Modeling several competitive leaders and followers acting in an electricity market
leads to coupled systems of mathematical programs with equilibrium constraints,
called equilibrium problems with equilibrium constraints (EPECs). We consider
a simplified model for competition in electricity markets under uncertainty of demand
in an electricity network
as a (stochastic) multi-leader-follower game. First order necessary conditions are
developed for the corresponding stochastic EPEC based on a result of Outrata.
For applying the general result an explicit representation of the co-derivative of
the normal cone mapping to a polyhedron is derived. Later the
co-derivative formula is used for verifying constraint qualifications and for identifying
$M$-stationary solutions of the stochastic EPEC if the demand is represented by a
finite number of scenarios.