A risk-averse optimization model for trading wind energy in a market environment under uncertainty

In this paper, a stochastic programming approach is proposed for trading wind energy in a market environment under uncertainty. Uncertainty in the energy market prices is the main cause of high volatility of profits achieved by power producers. The volatile and intermittent nature of wind energy represents another source of uncertainty. Hence, each uncertain parameter is modeled by scenarios, where each scenario represents a plausible realization of the uncertain parameters with an associated occurrence probability. Also, an appropriate risk measurement is considered. The proposed approach is applied on a realistic case study, based on a wind farm in Portugal. Finally, conclusions are duly drawn. (C) 2011 Elsevier Ltd. All rights reserved.

Convergence analysis of sampling-based decomposition methods for risk-averse multistage stochastic convex programs

We consider a class of sampling-based decomposition methods to solve risk-averse multistage stochastic convex programs. We prove a formula for the computation of the cuts necessary to build the outer linearizations of the recourse functions. This formula can be used to obtain an efficient implementation of Stochastic Dual Dynamic Programming applied to convex nonlinear problems. We prove the almost sure convergence of these decomposition methods when the relatively complete recourse assumption holds. We also prove the almost sure convergence of these algorithms when applied to risk-averse multistage stochastic linear programs that do not satisfy the relatively complete recourse assumption. The analysis is first done assuming the underlying stochastic process is interstage independent and discrete, with a finite set of possible realizations at each stage. We then indicate two ways of extending the methods and convergence analysis to the case when the process is interstage dependent.

Multistep stochastic mirror descent for risk-averse convex stochastic programs based on extended polyhedral risk measures

We consider risk-averse convex stochastic programs expressed in terms of extended polyhedral risk measures. We derive computable con dence intervals on the optimal value of such stochastic programs using the Robust Stochastic Approximation and the Stochastic Mirror Descent (SMD) algorithms. When the objective functions are uniformly convex, we also propose a multistep extension of the Stochastic Mirror Descent algorithm and obtain con dence intervals on both the optimal values and optimal solutions. Numerical simulations show that our con dence intervals are much less conservative and are quicker to compute than previously obtained con dence intervals for SMD and that the multistep Stochastic Mirror Descent algorithm can obtain a good approximate solution much quicker than its nonmultistep counterpart. Our con dence intervals are also more reliable than asymptotic con dence intervals when the sample size is not much larger than the problem size.

Dynamic stackelberg game with risk-averse players: optimal risk-sharing under asymmetric information

The objective of this paper is to clarify the interactive nature of the leader-follower relationship when both players are endogenously risk-averse. The analysis is placed in the context of a dynamic closed-loop Stackelberg game with private information. The case of a risk-neutral leader, very often discussed in the literature, is only a borderline possibility in the present study. Each player in the game is characterized by a risk-averse type which is unknown to his opponent. The goal of the leader is to implement an optimal incentive compatible risk-sharing contract. The proposed approach provides a qualitative analysis of adaptive risk behavior profiles for asymmetrically informed players in the context of dynamic strategic interactions modelled as incentive Stackelberg games.

Avoiding the dark side of positive online consumer reviews: enhancing reviews' usefulness for high risk-averse travelers

The influence of positive online consumer reviews on a traveler's decision making remains unclear. To better understand the perceived usefulness of online reviews, this study conducts two experiments using positive and negative online consumer reviews. Study results suggest that high risk-averse travelers find negative online reviews more useful than positive reviews. For positive online reviews, high-risk averse travelers feel expert reviewers' postings, travel product pictures, and well-known brand names enhance usefulness of the positive online reviews. These findings offer interesting implications for both marketing theory and practice.

Price stabilization and the risk-averse firm

An analytically simple and tractable approach to firm-level welfare analysis of complete and partial mean-preserving price stabilization for producers with general risk-averse preferences facing a stochastic technology is developed. Necessary and sufficient conditions for price stabilization to be welfare enhancing are derived under different assumptions of the producer's preferences and the producer's technology. Existing stabilization results for the risk-averse firm are shown to be corollaries of these more general results.

Risk management for mathematical optimization under uncertainty

We present a general multistage stochastic mixed 0-1 problem where the uncertainty appears everywhere in the objective function, constraints matrix and right-hand-side. The uncertainty is represented by a scenario tree that can be a symmetric or a nonsymmetric one. The stochastic model is converted in a mixed 0-1 Deterministic Equivalent Model in compact representation. Due to the difficulty of the problem, the solution offered by the stochastic model has been traditionally obtained by optimizing the objective function expected value (i.e., mean) over the scenarios, usually, along a time horizon. This approach (so named risk neutral) has the inconvenience of providing a solution that ignores the variance of the objective value of the scenarios and, so, the occurrence of scenarios with an objective value below the expected one. Alternatively, we present several approaches for risk averse management, namely, a scenario immunization strategy, the optimization of the well known Value-at-Risk (VaR) and several variants of the Conditional Value-at-Risk strategies, the optimization of the expected mean minus the weighted probability of having a "bad" scenario to occur for the given solution provided by the model, the optimization of the objective function expected value subject to stochastic dominance constraints (SDC) for a set of profiles given by the pairs of threshold objective values and either bounds on the probability of not reaching the thresholds or the expected shortfall over them, and the optimization of a mixture of the VaR and SDC strategies.

Risk optimization of a steel frame communications tower subject to tornado winds

The structural engineering community in Brazil faces new challenges with the recent occurrence of high intensity tornados. Satellite surveillance data shows that the area covering the south-east of Brazil, Uruguay and some of Argentina is one of the world most tornado-prone areas, second only to the infamous tornado alley in central United States. The design of structures subject to tornado winds is a typical example of decision making in the presence of uncertainty. Structural design involves finding a good balance between the competing goals of safety and economy. This paper presents a methodology to find the optimum balance between these goals in the presence of uncertainty. In this paper, reliability-based risk optimization is used to find the optimal safety coefficient that minimizes the total expected cost of a steel frame communications tower, subject to extreme storm and tornado wind loads. The technique is not new, but it is applied to a practical problem of increasing interest to Brazilian structural engineers. The problem is formulated in the partial safety factor format used in current design codes, with all additional partial factor introduced to serve as optimization variable. The expected cost of failure (or risk) is defined as the product of a. limit state exceedance probability by a limit state exceedance cost. These costs include costs of repairing, rebuilding, and paying compensation for injury and loss of life. The total expected failure cost is the sum of individual expected costs over all failure modes. The steel frame communications, tower subject of this study has become very common in Brazil due to increasing mobile phone coverage. The study shows that optimum reliability is strongly dependent on the cost (or consequences) of failure. Since failure consequences depend oil actual tower location, it turn,,; out that different optimum designs should be used in different locations. Failure consequences are also different for the different parties involved in the design, construction and operation of the tower. Hence, it is important that risk is well understood by the parties involved, so that proper contracts call be made. The investigation shows that when non-structural terms dominate design costs (e.g, in residential or office buildings) it is not too costly to over-design; this observation is in agreement with the observed practice for non-optimized structural systems. In this situation, is much easier to loose money by under-design. When by under-design. When structural material cost is a significant part of design cost (e.g. concrete dam or bridge), one is likely to lose significantmoney by over-design. In this situation, a cost-risk-benefit optimization analysis is highly recommended. Finally, the study also shows that under time-varying loads like tornados, the optimum reliability is strongly dependent on the selected design life.

RISK-BASED MULTIOBJECTIVE PATH PLANNING AND DESIGN OPTIMIZATION FOR UNMANNED AERIAL VEHICLES

Safe operation of unmanned aerial vehicles (UAVs) over populated areas requires reducing the risk posed by a UAV if it crashed during its operation. We considered several types of UAV risk-based path planning problems and developed techniques for estimating the risk to third parties on the ground. The path planning problem requires making trade-offs between risk and flight time. Four optimization approaches for solving the problem were tested; a network-based approach that used a greedy algorithm to improve the original solution generated the best solutions with the least computational effort. Additionally, an approach for solving a combined design and path planning problems was developed and tested. This approach was extended to solve robust risk-based path planning problem in which uncertainty about wind conditions would affect the risk posed by a UAV.

Multi-level, Multi-stage and Stochastic Optimization Models for Energy Conservation in Buildings for Federal, State and Local Agencies

Energy Conservation Measure (ECM) project selection is made difficult given real-world constraints, limited resources to implement savings retrofits, various suppliers in the market and project financing alternatives. Many of these energy efficient retrofit projects should be viewed as a series of investments with annual returns for these traditionally risk-averse agencies. Given a list of ECMs available, federal, state and local agencies must determine how to implement projects at lowest costs. The most common methods of implementation planning are suboptimal relative to cost. Federal, state and local agencies can obtain greater returns on their energy conservation investment over traditional methods, regardless of the implementing organization. This dissertation outlines several approaches to improve the traditional energy conservations models. Any public buildings in regions with similar energy conservation goals in the United States or internationally can also benefit greatly from this research. Additionally, many private owners of buildings are under mandates to conserve energy e.g., Local Law 85 of the New York City Energy Conservation Code requires any building, public or private, to meet the most current energy code for any alteration or renovation. Thus, both public and private stakeholders can benefit from this research. The research in this dissertation advances and presents models that decision-makers can use to optimize the selection of ECM projects with respect to the total cost of implementation. A practical application of a two-level mathematical program with equilibrium constraints (MPEC) improves the current best practice for agencies concerned with making the most cost-effective selection leveraging energy services companies or utilities. The two-level model maximizes savings to the agency and profit to the energy services companies (Chapter 2). An additional model presented leverages a single congressional appropriation to implement ECM projects (Chapter 3). Returns from implemented ECM projects are used to fund additional ECM projects. In these cases, fluctuations in energy costs and uncertainty in the estimated savings severely influence ECM project selection and the amount of the appropriation requested. A risk aversion method proposed imposes a minimum on the number of “of projects completed in each stage. A comparative method using Conditional Value at Risk is analyzed. Time consistency was addressed in this chapter. This work demonstrates how a risk-based, stochastic, multi-stage model with binary decision variables at each stage provides a much more accurate estimate for planning than the agency’s traditional approach and deterministic models. Finally, in Chapter 4, a rolling-horizon model allows for subadditivity and superadditivity of the energy savings to simulate interactive effects between ECM projects. The approach makes use of inequalities (McCormick, 1976) to re-express constraints that involve the product of binary variables with an exact linearization (related to the convex hull of those constraints). This model additionally shows the benefits of learning between stages while remaining consistent with the single congressional appropriations framework.

Background risk in generalized expected utility theory

In this paper, it is shown that, for a wide range of risk-averse generalized expected utility preferences, independent risks are complementary, contrary to the results for expected utility preferences satisfying conditions such as proper and standard risk aversion.

Learning about Risk and Return: A Simple Model of Bubbles and Crashes

This paper demonstrates that an asset pricing model with least-squares learning can lead to bubbles and crashes as endogenous responses to the fundamentals driving asset prices. When agents are risk-averse they need to make forecasts of the conditional variance of a stock’s return. Recursive updating of both the conditional variance and the expected return implies several mechanisms through which learning impacts stock prices. Extended periods of excess volatility, bubbles and crashes arise with a frequency that depends on the extent to which past data is discounted. A central role is played by changes over time in agents’ estimates of risk.

Inequality and Risk-Taking Behaviour

This paper investigates social influences on attitudes to risk and offers an evolutionary explanation of risk-taking by young low-ranked males. Becker, Murphy and Werning (2005) found that individuals about to participate in a status tournament may take fair gambles even though they are risk averse in both wealth and status. Here their model is generalised by use of the insight of Hopkins and Kornienko (2010) that in a tournament or status competition one can consider equality in terms of the status or rewards available as well as in initial endowments. While Becker et al. found that risk-taking is increasing in the equality of initial endowments, it is found here that it is increasing in the inequality of rewards in the tournament. Further, it is shown that the poorest will be risk loving if the lowest level of status awarded is sufficiently low. Thus, the disadvantaged in society rationally engage in risky behavior when social rewards are sufficiently unequal. Finally, as greater inequality in terms of social status induces gambling, it can cause greater inequality of wealth.

Entrepreneurial risk, investment and innovation

In this paper I develop a general equilibrium model with risk averse entrepreneurialfirms and with public firms. The model predicts that an increase in uncertainty reducesthe propensity of entrepreneurial firms to innovate, while it does not affect thepropensity of public firms to innovate. Furthermore, it predicts that the negativeeffect of uncertainty on innovation is stronger for the less diversified entrepreneurialfirms, and is stronger in the absence of financing frictions in the economy. In thesecond part of the paper I test these predictions on a dataset of small and mediumItalian manufacturing firms.

A simple model of multiple equilibria based on risk

This paper shows how risk may aggravate fluctuations in economies with imperfect insurance and multiple assets. A two period job matching model is studied, in which risk averse agents act both as workers and as entrepreneurs. They choose between two types of investment: one type is riskless, while the other is a risky activity that creates jobs.Equilibrium is unique under full insurance. If investment is fully insured but unemployment risk is uninsured, then precautionary saving behavior dampens output fluctuations. However, if both investment and employment are uninsured, then an increase in unemployment gives agents an incentive to shift investment away from the risky asset, further increasing unemployment. This positive feedback may lead to multiple Pareto ranked equilibria. An overlapping generations version of the model may exhibit poverty traps or persistent multiplicity. Greater insurance is doubly beneficial in this context since it can both prevent multiplicity and promote risky investment.