Risk-averse Formulation Research Articles

Bilevel optimization with a multi-objective lower-level problem: risk-neutral and risk-averse formulations

In this work, we propose different formulations and gradient-based algorithms for deterministic and stochastic bilevel problems with conflicting objectives in the lower level. Such problems have received little attention in the deterministic case and have never been studied from a stochastic approximation viewpoint despite the recent advances in stochastic methods for single-level, bilevel, and multi-objective optimization. To solve bilevel problems with a multi-objective lower level, different approaches can be considered depending on the interpretation of the lower-level optimality. An optimistic formulation that was previously introduced for the deterministic case consists of minimizing the upper-level function over all non-dominated lower-level solutions. In this paper, we develop new risk-neutral and risk-averse formulations, address their main computational challenges, and develop the corresponding deterministic and stochastic gradient-based algorithms.

A scalable framework for multi-objective PDE-constrained design of building insulation under uncertainty

This paper introduces a scalable computational framework for optimal design under high-dimensional uncertainty, with application to thermal insulation components. The thermal and mechanical behaviors are described by continuum multi-phase models of porous materials governed by partial differential equations (PDEs), and the design parameter, material porosity, is an uncertain and spatially correlated field. After finite element discretization, these factors lead to a high-dimensional PDE-constrained optimization problem. The framework employs a risk-averse formulation that accounts for both the mean and variance of the design objectives. It incorporates two regularization techniques, the ℓ0-norm and phase field functionals, implemented using continuation numerical schemes to promote spatial sparsity in the design parameters. To ensure efficiency, the framework utilizes a second-order Taylor approximation for the mean and variance and exploits the low-rank structure of the preconditioned Hessian of the design objective. This results in computational costs that are determined by the rank of preconditioned Hessian, remaining independent of the number of uncertain parameters. The accuracy, scalability with respect to the parameter dimension, and sparsity-promoting abilities of the framework are assessed through numerical examples involving various building insulation components.

Risk-averse stochastic optimal control: An efficiently computable statistical upper bound

In this paper, we discuss an application of the Stochastic Dual Dynamic Programming (SDDP) type algorithm to nested risk-averse formulations of Stochastic Optimal Control (SOC) problems. We propose a construction of a statistical upper bound for the optimal value of risk-averse SOC problems. This outlines an approach to a solution of a long standing problem in that area of research. The bound holds for a large class of convex and monotone conditional risk mappings. Finally, we show the validity of the statistical upper bound to solve a real-life stochastic hydro-thermal planning problem.

Optimal pricing for electricity retailers based on data-driven consumers’ price-response

In the present work, we tackle the problem of finding the optimal price tariff to be set by a risk-averse electric retailer participating in the pool and whose customers are price sensitive. We assume that the retailer has access to a sufficiently large smart-meter dataset from which it can statistically characterize the relationship between the tariff price and the demand load of its clients. Three different models are analyzed to predict the aggregated load as a function of the electricity prices and other parameters, as humidity or temperature. More specifically, we train linear regression (predictive) models to forecast the resulting demand load as a function of the retail price. Then, we will insert this model in a quadratic optimization problem which evaluates the optimal price to be offered. This optimization problem accounts for different sources of uncertainty including consumer’s response, pool prices and renewable source availability, and relies on a stochastic and risk-averse formulation. In particular, one important contribution of this work is to base the scenario generation and reduction procedure on the statistical properties of the resulting predictive model. This allows us to properly quantify (data-driven) not only the expected value but the level of uncertainty associated with the main problem parameters. Moreover, we consider both standard forward-based contracts and the recently introduced power purchase agreement contracts as risk-hedging tools for the retailer. The results are promising as profits are found for the retailer with highly competitive prices and some possible improvements are shown if richer datasets could be available in the future. A realistic case study and multiple sensitivity analyses have been performed to characterize the risk-aversion behavior of the retailer considering price-sensitive consumers. It has been assumed that the energy procurement of the retailer can be satisfied from the pool and different types of contracts. The obtained results reveal that the risk-aversion degree of the retailer strongly influences contracting decisions, whereas the price sensitiveness of consumers has a higher impact on the selling price offered.

Risk Averse Model Predictive Control of Bioreactors

Uncertainty is inherent in bioprocess modelling and control. Typically, uncertainties are handled using either the stochastic approach or the robust approach. Recently, the risk-averse approach, i.e., an interpolation between the stochastic and worst-case robust approach is gaining popularity. Risk-averse formulations are very useful in avoiding conservative solutions while still handling high-effect, low-probability events. In the bioreactor case considered in this paper, one such high effect low probability event is wash off caused by high feed rate or low inlet substrate concentration. A risk-averse, risk-constrained model predictive control formulation is proposed in this paper. The dynamic optimisation problem to be solved at every measurement instance is formulated using AV@R type risk objective. Similarly, probabilistic chance constraints are approximated by an AV@R-type risk constraints. The problem is then solved using the conic duality of the risk measure and an epigraphical decomposition of the nested multistage problem.

Risk-averse multi-stage stochastic optimization for surveillance and operations planning of a forest insect infestation

We derive a nested risk measure for a maximization problem and implement it in a scenario-based formulation of a multi-stage stochastic mixed-integer programming problem. We apply the risk-averse formulation to the surveillance and control of a non-native forest insect, the emerald ash borer (EAB), a wood-boring beetle native to Asia and recently discovered in North America. Spreading across the eastern United States and Canada, EAB has killed millions of ash trees and cost homeowners and local governments billions of dollars. We present a mean-Conditional Value-at-Risk (CVaR), multi-stage, stochastic mixed-integer programming model to optimize a manager’s decisions about surveillance and control of EAB. The objective is to maximize the benefits of healthy ash trees in forests and urban environments under a fixed budget. Combining the risk-neutral objective with a risk measure allows for a trade-off between the weighted expected benefits from ash trees and the expected risks associated with experiencing extremely damaging scenarios. We define scenario dominance cuts (sdc) for the maximization problem and under the decision-dependent uncertainty. We then solve the model using the sdc cutting plane algorithm for varying risk parameters. Computational results demonstrate that scenario dominance cuts significantly improve the solution performance relative to that of CPLEX. Our CVaR risk-averse approach also raises the objective value of the least-benefit scenarios compared to the risk-neutral model. Results show a shift in the optimal strategy from applying less expensive insecticide treatment to more costly tree removal as the manager becomes more risk-averse. We also find that risk-averse managers survey more often to reduce the risk of experiencing adverse outcomes.

An integrated optimization framework for combined heat and power units, distributed generation and plug-in electric vehicles

This paper provides a six-level integrated optimization framework for a distribution system that transacts energy with upward electricity market and downward active microgrids in day-ahead and real-time horizons. The proposed method uses a risk-averse formulation and the distribution system utilizes multiple combined heating and power units, distributed generation, plug-in electric vehicles parking lots, and electric and thermal storage units. Demand response program alternatives are also utilized by the distribution system. A three-stage uncertainty modeling is proposed to model six sources of uncertainties that are consist of energy resource power generations, loads and prices, active microgrids contributions and contingencies. Two case studies evaluate the proposed algorithm for the 123-bus test system that multiple 33-bus microgrid systems are transacting energy and ancillary services with the main grid. Further, different sensitivity analyses are performed to evaluate the effect of energy and ancillary services prices on the simulation results.

Implementation of RTO in a large hydrogen network considering uncertainty

This paper describes the problems associated with the implementation of a real-time optimization (RTO) decision support tool, for the operation of a large scale hydrogen network of an oil refinery. In addition, a formulation which takes into account the stochastic uncertainty of hydrogen demand, due to hydrocarbons quality change, is described and further studied, focusing on its utility in the decision-making process of operators. An integrated robust data reconciliation, and economic optimization, considering plant-wide uncertain parameters is presented and discussed. Moreover, stochastic uncertainty in hydrogen demand is assessed for its inclusion within the RTO framework. A novel approach of the decisions stages at hydrogen producers and consumers is proposed, which supports the formulation of the problem as a two-stage stochastic non-linear program. Representative results are presented and discussed, aimed at assessing the potential impact in the hydrogen management policies. For this purpose, the value of the stochastic solution, perfect information, and expectation of the expected value are analyzed. Complementarily, a risk-averse formulation is presented (value-at-risk and conditional-value-at-risk) and its results compared against the formulation without risk considerations. Finally, some attention is given to future directions of this decision support tool, based on these work contributions, including the importance of the decision makers’ participation in the analysis of the potential impact of risk-averse results.

Risk-averse formulations and methods for a virtual power plant

In this paper, we address the optimal operation of a virtual power plant using stochastic programming. We consider one risk-neutral and two risk-averse formulations that rely on the conditional value at risk. To handle large-scale problems, we implement two decomposition methods with variants using single- and multiple-cuts. We propose the utilization of wind ensembles obtained from the European Centre for Medium Range Weather Forecasts (ECMWF) to quantify the uncertainty of the wind forecast. We present detailed results relative to the computational performance of the risk-averse formulations, the decomposition methods, and risk management and sensitivities analysis as a function of the number of scenarios and risk parameters. The implementation of the two decomposition methods relies on the parallel solution of subproblems, which turns out to be paramount for computational efficiency. The results show that one of the two decomposition methods is the most efficient.

An Arbitrage-Free Nelson-Siegel Term Structure Model with Stochastic Volatility for the Determination of Currency Risk Premia

This paper uses a risk-averse formulation of the uncovered interest rate parity to determine exchange rates through interest rate differentials, and ultimately extract currency risk premia. The method proposed consists of developing an affine Arbitrage-Free class of dynamic Nelson-Siegel term structure models with stochastic volatility to obtain the domestic and foreign discount rate variations, which in turn are used to derive a representation of exchange rate depreciations. No-arbitrage restrictions allow to endogenously capturing currency risk premia. Empirical findings suggest that estimated currency risk premia are able to account for the forward premium puzzle and their properties are examined.

Evaluating extraordinarily large requests in a revenue management environment

More and more companies in various industry sectors run automated revenue management (RM) systems to control their sales processes. Customer requests are assumed to be relatively small and a single acceptance or rejection decision has a minimal impact in the total revenue. But how should a company deal with arbitrary large requests, which can have fundamental revenue and capacity impacts? This article proposes evaluation methods to derive the least acceptable prices for the buyer or seller in a network RM environment. Purely resource sale-type requests are extended to contracts, which include return options for unsold resources. We propose deterministic and stochastic models to evaluate both request types. In addition, different business and economical situations require different evaluation approaches. Therefore, we additionally derive risk-averse formulations to enable decision makers to compare expected profit with underlying risks.

Risk-averse feasible policies for large-scale multistage stochastic linear programs

We consider risk-averse formulations of stochastic linear programs having a structure that is common in real-life applications. Specifically, the optimization problem corresponds to controlling over a certain horizon a system whose dynamics is given by a transition equation depending affinely on an interstage dependent stochastic process. We put in place a rolling-horizon time consistent policy. For each time step, a risk-averse problem with constraints that are deterministic for the current time step and uncertain for future times is solved. To each uncertain constraint corresponds both a chance and a Conditional Value-at-Risk constraint. We show that the resulting risk-averse problems are numerically tractable, being at worst conic quadratic programs. For the particular case in which uncertainty appears only on the right-hand side of the constraints, such risk-averse problems are linear programs. We show how to write dynamic programming equations for these problems and define robust recourse functions that can be approximated recursively by cutting planes. The methodology is assessed and favourably compared with Stochastic Dual Dynamic Programming on a real size water-resource planning problem.

SDDP for multistage stochastic linear programs based on spectral risk measures

We consider risk-averse formulations of multistage stochastic linear programs. For these formulations, based on convex combinations of spectral risk measures, risk-averse dynamic programming equations can be written. As a result, the Stochastic Dual Dynamic Programming (SDDP) algorithm can be used to obtain approximations of the corresponding risk-averse recourse functions. This allows us to define a risk-averse nonanticipative feasible policy for the stochastic linear program. Formulas for the cuts that approximate the recourse functions are given. In particular, we show that some cut coefficients have analytic formulas.

Sampling-Based Decomposition Methods for Multistage Stochastic Programs Based on Extended Polyhedral Risk Measures

We define a risk-averse nonanticipative feasible policy for multistage stochastic programs and propose a methodology to implement it. The approach is based on dynamic programming equations written for a risk-averse formulation of the problem. This formulation relies on a new class of multiperiod risk functionals called extended polyhedral risk measures. Dual representations of such risk functionals are given and used to derive conditions of coherence. In the one-period case, conditions for convexity and consistency with second order stochastic dominance are also provided. The risk-averse dynamic programming equations are specialized considering convex combinations of one-period extended polyhedral risk measures such as spectral risk measures. To implement the proposed policy, the approximation of the risk-averse recourse functions for stochastic linear programs is discussed. In this context, we detail a stochastic dual dynamic programming algorithm which converges to the optimal value of the risk-averse problem. (A corrected PDF is attached to the original pdf.)

The value of rolling-horizon policies for risk-averse hydro-thermal planning

We consider the optimal management of a hydro-thermal power system in the mid and long terms. From the optimization point of view, this amounts to a large-scale multistage stochastic linear program, often solved by combining sampling with decomposition algorithms, like stochastic dual dynamic programming. Such methodologies, however, may entail prohibitive computational time, especially when applied to a risk-averse formulation of the problem. We propose instead a risk-averse rolling-horizon policy that is nonanticipative, feasible, and time consistent. The policy is obtained by solving a sequence of risk-averse problems with deterministic constraints for the current time step and future chance and CVaR constraints. The considered hydro-thermal model takes into account losses resulting from run-of-river plants efficiencies as well as uncertain demand and streamflows. Constraints aim at satisfying demand while keeping reservoir levels above minzones almost surely. We show that if the problem uncertainty is represented by a periodic autoregressive stochastic process with lag one, then the probabilistic constraints can be computed explicitly. As a result, each one of the aforementioned risk-averse problems is a medium-size linear program, easy to solve. For a real-life power system we compare our approach with three alternative policies. Namely, a robust nonrolling-horizon policy and two risk-neutral policies obtained by stochastic dual dynamic programming, implemented in nonrolling- and rolling-horizon modes, respectively. Our numerical assessment confirms the superiority of the risk-averse rolling-horizon policy that yields comparable average indicators, but with reduced volatility and with substantially less computational effort.

Analysis of stochastic dual dynamic programming method

In this paper we discuss statistical properties and convergence of the Stochastic Dual Dynamic Programming (SDDP) method applied to multistage linear stochastic programming problems. We assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original (true) distribution is generated and consequently the SDDP algorithm is applied to the constructed Sample Average Approximation (SAA) problem. Then we proceed to analysis of the SDDP solutions of the SAA problem and their relations to solutions of the “true” problem. Finally we discuss an extension of the SDDP method to a risk averse formulation of multistage stochastic programs. We argue that the computational complexity of the corresponding SDDP algorithm is almost the same as in the risk neutral case.