This work presents a methodology to incorporate reliability constraints in the optimal power systems expansion planning problem. Besides LOLP and EPNS, traditionally used in power systems, this work proposes the use of the risk measures VaR (Value-at-Risk) and CVaR (Conditional Value-at-Risk), widely used in financial markets. The explicit consideration of reliability constraints in the planning problem can be an extremely hard task and, in order to minimize computational effort, this work applies the Benders decomposition technique splitting the expansion planning problem into an investment problem and two sub-problems to evaluate the system's operation cost and the reliability index. The operation sub-problem is solved by Stochastic Dual Dynamic Programming (SDDP) and the reliability sub-problem by Monte Carlo simulation. The proposed methodology is applied to the real problem of optimal expansion planning of the Bolivian power system.
Risk-averse model predictive control (MPC) offers a control framework that allows one to account for ambiguity in the knowledge of the underlying probability distribution and unifies stochastic and worst-case MPC. In this paper we study risk-averse MPC problems for constrained nonlinear Markovian switching systems using generic cost functions, and derive Lyapunov-type risk-averse stability conditions by leveraging the properties of risk-averse dynamic programming operators. We propose a controller design procedure to design risk-averse stabilizing terminal conditions for constrained nonlinear …
We use Markov risk measures to formulate a risk-averse version of the undiscounted total cost problem for a transient controlled Markov process. We derive risk-averse dynamic programming equations and we show that a randomized policy may be strictly better than deterministic policies, when risk measures are employed. We illustrate the results on an optimal stopping problem and an organ transplant problem.
In this paper, we consider a risk-averse control problem for diffusion processes, in which there is a partition of the admissible control strategy into two decision-making groups (namely, the {\it leader} and {\it follower}) with different cost functionals and risk-averse satisfactions. Our approach, based on a hierarchical optimization framework, requires that a certain level of risk-averse satisfaction be achieved for the {\it leader} as a priority over that of the {\it follower's} risk-averseness. In particular, we formulate such a risk-averse control problem involving a family of time-cons…
We consider the stochastic shortest path planning problem in MDPs, i.e., the problem of designing policies that ensure reaching a goal state from a given initial state with minimum accrued cost. In order to account for rare but important realizations of the system, we consider a nested dynamic coherent risk total cost functional rather than the conventional risk-neutral total expected cost. Under some assumptions, we show that optimal, stationary, Markovian policies exist and can be found via a special Bellman's equation. We propose a computational technique based on difference convex programs…
This paper features expectiles in dynamic and stochastic optimization. Expectiles are a family of risk functionals characterized as minimizers of optimization problems. For this reason, they enjoy various unique stability properties, which can be exploited in risk averse management, in stochastic optimization and in optimal control. The paper provides tight relates of expectiles to other risk functionals and addresses their properties in regression. Further, we extend expectiles to a dynamic framework. As such, they allow incorporating a risk averse aspect in continuous-time dynamic optimizati…
This paper studies risk in a stochastic auction which facilitates the integration of renewable generation in electricity markets. We model market participants who are risk averse and reflect their risk aversion through coherent risk measures. We uncover a closed form characterization of a risk-averse generator's optimal pre-commitment behaviour for a given real-time policy, both with and without risk trading.
In this paper, we consider the optimal controller design problem against data injection attacks on actuators for an uncertain control system. We consider attacks that aim at maximizing the attack impact while remaining stealthy in the finite horizon. To this end, we use the Conditional Value-at-Risk to characterize the risk associated with the impact of attacks. The worst-case attack impact is characterized using the recently proposed output-to-output $\ell_2$-gain (OOG). We formulate the design problem and observe that it is non-convex and hard to solve. Using the framework of scenario-based …
This paper investigates the use of risk measures and theories of choice for modeling risk-averse route choice and traffic network equilibrium with random travel times. We interpret the postulates of these theories in the context of routing, and we identify additive consistency as a plausible and relevant condition that allows to reduce risk-averse route choice to a standard shortest path problem. Within the classical theories of choice under risk, we show that the only preferences that satisfy this consistency property are the ones induced by the entropic risk measures.
The aim of this paper is to show that in some cases risk averse multistage stochastic programming problems can be reformulated in a form of risk neutral setting. This is achieved by a change of the reference probability measure making ``bad" (extreme) scenarios more frequent. As a numerical example we demonstrate advantages of such change-of-measure approach applied to the Brazilian Interconnected Power System operation planning problem.
We consider reinforcement learning with performance evaluated by a dynamic risk measure. We construct a projected risk-averse dynamic programming equation and study its properties. Then we propose risk-averse counterparts of the methods of temporal differences and we prove their convergence with probability one. We also perform an empirical study on a complex transportation problem.
Multistage risk-averse optimal control problems with nested conditional risk mappings are gaining popularity in various application domains. Risk-averse formulations interpolate between the classical expectation-based stochastic and minimax optimal control. This way, risk-averse problems aim at hedging against extreme low-probability events without being overly conservative. At the same time, risk-based constraints may be employed either as surrogates for chance (probabilistic) constraints or as a robustification of expectation-based constraints. Such multistage problems, however, have been id…
Two approaches to time consistency of risk averse multistage stochastic problems were discussed in the recent literature. In one approach certain properties of the cor-responding risk measure are postulated which imply its decomposability. The other approach deals directly with conditional optimality of solutions of the considered problem. The aim of this paper is to discuss a relation between these two approaches.
In this paper, we consider a risk-averse decision problem for controlled-diffusion processes, with dynamic risk measures, in which multiple risk-averse agents choose their decisions in such a way to minimize their individual accumulated risk-costs over a finite-time horizon. In particular, we introduce multi-structure dynamic risk measures induced from conditional $g$-expectations, where the latter are associated with the generator functionals of certain BSDEs that implicitly take into account the risk-cost functionals of the risk-averse agents. Here, we also require that such solutions of the…
The multi-armed bandit (MAB) is a classical online optimization model for the trade-off between exploration and exploitation. The traditional MAB is concerned with finding the arm that minimizes the mean cost. However, minimizing the mean does not take the risk of the problem into account. We now want to accommodate risk-averse decision makers. In this work, we introduce a coherent risk measure as the criterion to form a risk-averse MAB. In particular, we derive an index-based online sampling framework for the risk-averse MAB. We develop this framework in detail for three specific risk measure…
The Ohio State University（俄亥俄州立大学）；University of Michigan（密歇根大学）
Traditional reinforcement learning (RL) aims to maximize the expected total reward, while the risk of uncertain outcomes needs to be controlled to ensure reliable performance in a risk-averse setting. In this paper, we consider the problem of maximizing dynamic risk of a sequence of rewards in infinite-horizon Markov Decision Processes (MDPs). We adapt the Expected Conditional Risk Measures (ECRMs) to the infinite-horizon risk-averse MDP and prove its time consistency. Using a convex combination of expectation and conditional value-at-risk (CVaR) as a special one-step conditional risk measure,…
In this paper, we consider a risk-averse decision problem for controlled-diffusion processes, with dynamic risk measures, in which there are two risk-averse decision makers (i.e., {\it leader} and {\it follower}) with different risk-averse related responsibilities and information. Moreover, we assume that there are two objectives that these decision makers are expected to achieve. That is, the first objective being of {\it stochastic controllability} type that describes an acceptable risk-exposure set vis-á-vis some uncertain future payoff, and while the {\it second one} is making sure the sol…
We propose and investigate a discrete-time mean field game model involving risk-averse agents. The model under study is a coupled system of dynamic programming equations with a Kolmogorov equation. The agents' risk aversion is modeled by composite risk measures. The existence of a solution to the coupled system is obtained with a fixed point approach. The corresponding feedback control allows to construct an approximate Nash equilibrium for a related dynamic game with finitely many players.
We discuss risked competitive partial equilibrium in a setting in which agents are endowed with coherent risk measures. In contrast to socialplanning models, we show by example that risked equilibria are not unique, even when agents' objective functions are strictly concave. We also show that standard computational methods find only a subset of the equilibria, even with multiple starting points.
We study statistical properties of the optimal value and optimal solutions of the Sample Average Approximation of risk averse stochastic problems. Central Limit Theorem type results are derived for the optimal value and optimal solutions when the stochastic program is expressed in terms of a law invariant coherent risk measure. The obtained results are applied to hypotheses testing problems aiming at comparing the optimal values of several risk averse convex stochastic programs on the basis of samples of the underlying random vectors. We also consider non-asymptotic tests based on confidence i…
In this paper we present a framework for risk-averse model predictive control (MPC) of linear systems affected by multiplicative uncertainty. Our key innovation is to consider time-consistent, dynamic risk metrics as objective functions to be minimized. This framework is axiomatically justified in terms of time-consistency of risk preferences, is amenable to dynamic optimization, and is unifying in the sense that it captures a full range of risk assessments from risk-neutral to worst case. Within this framework, we propose and analyze an online risk-averse MPC algorithm that is provably stabil…
California Institute of Technology（加州理工学院）；NASA Jet Propulsion Laboratory（NASA喷气推进实验室）
We consider the problem of designing policies for Markov decision processes (MDPs) with dynamic coherent risk objectives and constraints. We begin by formulating the problem in a Lagrangian framework. Under the assumption that the risk objectives and constraints can be represented by a Markov risk transition mapping, we propose an optimization-based method to synthesize Markovian policies that lower-bound the constrained risk-averse problem. We demonstrate that the formulated optimization problems are in the form of difference convex programs (DCPs) and can be solved by the disciplined convex-…
In this paper, we consider a finite-horizon Markov decision process (MDP) for which the objective at each stage is to minimize a quantile-based risk measure (QBRM) of the sequence of future costs; we call the overall objective a dynamic quantile-based risk measure (DQBRM). In particular, we consider optimizing dynamic risk measures where the one-step risk measures are QBRMs, a class of risk measures that includes the popular value at risk (VaR) and the conditional value at risk (CVaR). Although there is considerable theoretical development of risk-averse MDPs in the literature, the computation…
Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology（决策与控制系统系，电气工程与计算机科学学院，皇家理工学院）；Chair of Information-oriented Control (ITR), Technical University of Munich（信息导向控制系（ITR），慕尼黑技术大学）
California Institute of Technology（加州理工学院）；NASA Jet Propulsion Laboratory（美国国家航空航天局喷气推进实验室）
We consider the problem of designing policies for partially observable Markov decision processes (POMDPs) with dynamic coherent risk objectives. Synthesizing risk-averse optimal policies for POMDPs requires infinite memory and thus undecidable. To overcome this difficulty, we propose a method based on bounded policy iteration for designing stochastic but finite state (memory) controllers, which takes advantage of standard convex optimization methods. Given a memory budget and optimality criterion, the proposed method modifies the stochastic finite state controller leading to sub-optimal soluti…
We introduce adaptive sampling methods for stochastic programs with deterministic constraints. First, we propose and analyze a variant of the stochastic projected gradient method where the sample size used to approximate the reduced gradient is determined on-the-fly and updated adaptively. This method is applicable to a broad class of expectation-based risk measures and leads to a significant reduction in the individual gradient evaluations used to estimate the objective function gradient. Numerical experiments with expected risk minimization and conditional value-at-risk minimization support …
Risk-averse optimal control problems have gained a lot of attention in the last decade, mostly due to their attractive mathematical properties and practical importance. They can be seen as an interpolation between stochastic and robust optimal control approaches, allowing the designer to trade-off performance for robustness and vice-versa. Due to their stochastic nature, risk-averse problems are of a very large scale, involving millions of decision variables, which poses a challenge in terms of efficient computation. In this work, we propose a splitting for general risk-averse problems and sho…
Storage is expected to be a critical source of firming in low-carbon grids. A common concern raised from ex-post assessments is that storage resources can fail to respond to strong price signals during times of scarcity. While commonly attributed to forecast error or failures in operations, we posit that this behavior can be explained from the perspective of risk-averse scheduling. Using a stochastic self-scheduling framework and real-world data harvested from the Australian National Electricity Market, we demonstrate that risk-averse storage resources tend to have a myopic operational perspec…
The paper deals with a risk averse dynamic programming problem with infinite horizon. First, the required assumptions are formulated to have the problem well defined. Then the Bellman equation is derived, which may be also seen as a standalone reinforcement learning problem. The fact that the Bellman operator is contraction is proved, guaranteeing convergence of various solution algorithms used for dynamic programming as well as reinforcement learning problems, which we demonstrate on the value iteration algorithm.
This paper considers a statistical signal processing problem involving agent based models of financial markets which at a micro-level are driven by socially aware and risk- averse trading agents. These agents trade (buy or sell) stocks by exploiting information about the decisions of previous agents (social learning) via an order book in addition to a private (noisy) signal they receive on the value of the stock. We are interested in the following: (1) Modelling the dynamics of these risk averse agents, (2) Sequential detection of a market shock based on the behaviour of these agents. Structur…
Bellman formulated a vague principle for optimization over time, which characterizes optimal policies by stating that a decision maker should not regret previous decisions retrospectively. This paper addresses time consistency in stochastic optimization. The problem is stated in generality first. The paper discusses time consistent decision-making by addressing risk measures which are recursive, nested, dynamically or time consistent and introduces stopping time risk measures. It turns out that the paradigm of time consistency is in conflict with various desirable, classical properties of gene…
This paper addresses risk awareness of stochastic optimization problems. Nested risk measures appear naturally in this context, as they allow beneficial reformulations for algorithmic treatments. The reformulations presented extend usual Hamilton-Jacobi-Bellman equations in dynamic optimization by involving risk awareness in the problem formulation. Nested risk measures are built on risk measures, which originate by conditioning on the history of a stochastic process. We derive martingale properties of these risk measures and use them to prove continuity. It is demonstrated that stochastic opt…
On the Global Convergence of Risk-Averse Natural Policy Gradient Methods with Expected Conditional Risk Measures
Department of Integrated Systems Engineering, The Ohio State University（俄亥俄州立大学整合系统工程系）；Department of Electrical Engineering and Computer Science, University of Michigan（密歇根大学电气工程与计算机科学系）
This paper studies the communication complexity of convex risk-averse optimization over a network. The problem generalizes the well-studied risk-neutral finite-sum distributed optimization problem and its importance stems from the need to handle risk in an uncertain environment. For algorithms in the literature, there exists a gap in communication complexities for solving risk-averse and risk-neutral problems. We propose two distributed algorithms, namely the distributed risk averse optimization (DRAO) method and the distributed risk averse optimization with sliding (DRAO-S) method, to close t…
We consider risk-averse convex stochastic programs expressed in terms of extended polyhedral risk measures. We derive computable confidence 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 confidence intervals on both the optimal values and optimal solutions. Numerical simulations show that our confidence intervals are much less conservative and are q…
In this paper we present a risk-averse model predictive control (MPC) scheme for the operation of islanded microgrids with very high share of renewable energy sources. The proposed scheme mitigates the effect of errors in the determination of the probability distribution of renewable infeed and load. This allows to use less complex and less accurate forecasting methods and to formulate low-dimensional scenario-based optimisation problems which are suitable for control applications. Additionally, the designer may trade performance for safety by interpolating between the conventional stochastic …
We consider two market designs for a network of prosumers, trading energy: (i) a centralized design which acts as a benchmark, and (ii) a peer-to-peer market design. High renewable energy penetration requires that the energy market design properly handles uncertainty. To that purpose, we consider risk neutral models for market designs (i), (ii), and their risk-averse interpretations in which prosumers are endowed with coherent risk measures reflecting heterogeneity in their risk attitudes. We characterize analytically risk-neutral and risk-averse equilibrium in terms of existence and uniquenes…
We consider optimal control problems for diffusion processes, where the objective functional is defined by a time-consistent dynamic risk measure. We focus on coherent risk measures defined by $g$-evaluations. For such problems, we construct a family of time and space perturbed systems with piecewise-constant control functions. We obtain a regularized optimal value function by a special mollification procedure. This allows us to establish a bound on the difference between the optimal value functions of the original problem and of the problem with piecewise-constant controls.
We develop a new approach to solving classification problems, which is bases on the theory of coherent measures of risk and risk sharing ideas. The proposed approach aims at designing a risk-averse classifier. The new approach allows for associating distinct risk functional to each classes. The risk may be measured by different (non-linear in probability) measures, We analyze the structure of the new classifier design problem and establish its theoretical relation to known risk-neutral design problems. In particular, we show that the risk-sharing classification problem is equivalent to an impl…
