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Optimization of conditional valueat-risk
... With λ := [λ T 1 , . . . , λ T I ] T denoting the Lagrange multiplier vector associated with constraints (23c), the approximated augmented Lagrangian of (23) is given by (26) [cf. (19) ...
... Based on the approximated augmented Lagrangian in (26), problem (23) can be then tackled by solving the decomposed subproblems given by (27) and (28), which update the primal and dual variables at BS i , respectively, ∀i. ...
... CVaR has been widely used in various real-world applications, especially in the finance area, to account not only for the expected cost of the resource allocation actions, but also for their "risks" [22], [26]- [28]. In the present context, recall that the transaction costf i (P i , s i ) in (5) is a function associated with the decision variable P i and the random state ...
The integration of renewable energy sources (RES) has facilitated efficient and sustainable resource allocation for wireless communication systems. In this paper, a novel framework is introduced to develop coordinated multicell beamforming (CMBF) design for wireless cellular networks powered by a smart microgrid, where the BSs are equipped with RES harvesting devices and can perform two-way (i.e., buying/selling) energy trading with the main grid. To this end, new models are put forth to account for the stochastic RES harvesting, two-way energy trading, and conditional value-at-risk (CVaR) based energy transaction cost. Capitalizing on these models, we propose a distributed CMBF solution to minimize the grid-wide transaction cost subject to user quality-of-service (QoS) constraints. Specifically, relying on state-of-the-art optimization tools, we show that the relevant task can be formulated as a convex problem that is well suited for development of a distributed solver. To cope with stochastic availability of the RES, the stochastic alternating direction method of multipliers (ADMM) is then leveraged to develop a novel distributed CMBF scheme. It is established that the proposed scheme is guaranteed to yield the optimal CMBF solution, with only local channel state information available at each BS and limited information exchange among the BSs. Numerical results are provided to corroborate the merits of the proposed scheme.
... A generator with renewable units can use risk measures such as the value-at-risk (VaR) and conditional value-at-risk (CVaR) to limit the risk of generation shortage. Let ∆ j,h (p j,h , p j,h ) denote a function that captures the penalty for generation shortage in time slot h for generator j with renewable units [28]. It is defined as ...
... is a random variable, since the actual generation p ren j,h is a stochastic process. Under a given confidence level β j ∈ (0, 1) and the offered generation level p ren j,h in time slot h, the VaR for generator j is defined as the minimum threshold cost α j,h , for which the probability of generation shortage of generator j being less than α j,h is at least β j [28]. That is, ...
... The CVaR is an alternative risk measure, which is convex and can be optimized using sampling techniques. The CVaR for generator j with renewable units in time slot h is defined as the expected value of the generation shortage cost ∆ j,h (p ren j,h , p ren j,h ) when only the costs that are greater than or equal to VaR βj j,h (p ren j,h ) are considered [28]. That is, ...
The uncertainties of the renewable generation units and the proliferation of price-responsive loads make it a challenge for independent system operators (ISOs) to manage the energy trading market in the future power systems. A centralized energy market is not practical for the ISOs due to the high computational burden and violating the privacy of different entities, i.e., load aggregators and generators. In this paper, we propose a day-ahead decentralized energy trading algorithm for a grid with generation uncertainty. To address the privacy issues, the ISO determines some control signals using the Lagrange relaxation technique to motivate the entities towards an operating point that jointly optimize the cost of load aggregators and profit of the generators, as well as the risk of the generation shortage of the renewable resources. More, specifically, we deploy the concept of conditional-value-at-risk (CVaR) to minimize the risk of renewable generation shortage. The performance of the proposed algorithm is evaluated on an IEEE 30-bus test system. Results show that the proposed decentralized algorithm converges to the solution of the ISO's centralized problem in 45 iterations. It also benefits both the load aggregators by reducing their cost by 18% and the generators by increasing their profit by 17.1%.
... The notion of a risk measure [23], [26] is of key significance in the context of DRO. In fact, under very general conditions DRO functionals (e.g., used as problem objectives/constraints) arise as coherent risk measures [23,Chapters 6 and 7], since every such risk measure can be represented (in duality) as a supremum of expectations over a set of distributions [23, Section 6.3], i.e., a coherent risk measure ρ(·) on an integrable random cost ξ can be written as ρ(ξ) = sup Q∈U E Q [ξ], where U is a corresponding set of probability distributions. ...
... The selection of U ϵ and ϵ regulates the degree of relaxation, i.e., ϵ → 0 (or when ϵ is sufficiently small) reduces to an expectation (ergodic setting), while ϵ → ∞ yields the essential supremum over the fading distribution, ensuring deterministic feasibility (initial instantaneous setting). The ball U ϵ is tacitly selected herein by capitalizing on the dual representation of the Conditional Value-at-Risk (CVaR) [23], [26], the latter thus being adopted as a coherent measure of power policy fluctuation risk. ...
... In this work, we exploit the CVaR to facilitate tackling the DRO (3). Specifically, the CVaR is defined for an integrable random cost ξ(·) as [23], [26] ...
Modern wireless communication systems necessitate the development of cost-effective resource allocation strategies, while ensuring maximal system performance. While commonly realizable via efficient waterfilling schemes, ergodic-optimal policies often exhibit instantaneous resource constraint fluctuations as a result of fading variability, violating prescribed specifications possibly within unacceptable margins, inducing further operational challenges and/or costs. On the other extent, short-term-optimal policies -- commonly based on deterministic waterfilling-- while strictly maintaining operational specifications, are not only impractical and computationally demanding, but also suboptimal in a long-term sense. To address these challenges, we introduce a novel distributionally robust version of a classical point-to-point interference-free multi-terminal constrained stochastic resource allocation problem, by leveraging the Conditional Value-at-Risk (CVaR) as a coherent measure of power policy fluctuation risk. We derive closed-form dual-parameterized expressions for the CVaR-optimal resource policy, along with corresponding optimal CVaR quantile levels by capitalizing on (sampling) the underlying fading distribution. We subsequently develop two dual-domain schemes -- one model-based and one model-free -- to iteratively determine a globally-optimal resource policy. Our numerical simulations confirm the remarkable effectiveness of the proposed approach, also revealing an almost-constant character of the CVaR-optimal policy and at rather minimal ergodic rate optimality loss.
... The notion of a risk measure [23], [26] is of key significance in the context of DRO. In fact, under very general conditions DRO functionals (e.g., used as problem objectives/constraints) arise as coherent risk measures [23,Chapters 6 and 7], since every such risk measure can be represented (in duality) as a supremum of expectations over a set of distributions [23, Section 6.3], i.e., a coherent risk measure ρ(·) on an integrable random cost ξ can be written as ρ(ξ) = sup Q∈U E Q [ξ], where U is a corresponding set of probability distributions. ...
... The selection of U ϵ and ϵ regulates the degree of relaxation, i.e., ϵ → 0 (or when ϵ is sufficiently small) reduces to an expectation (ergodic setting), while ϵ → ∞ yields the essential supremum over the fading distribution, ensuring deterministic feasibility (initial instantaneous setting). The ball U ϵ is tacitly selected herein by capitalizing on the dual representation of the Conditional Value-at-Risk (CVaR) [23], [26], the latter thus being adopted as a coherent measure of power policy fluctuation risk. ...
... In this work, we exploit the CVaR to facilitate tackling the DRO (3). Specifically, the CVaR is defined for an integrable random cost ξ(·) as [23], [26] ...
Modern wireless communication systems necessitate the development of cost-effective resource allocation strategies, while ensuring maximal system performance. While commonly realizable via efficient waterfilling schemes, ergodic-optimal policies often exhibit instantaneous resource constraint fluctuations as a result of fading variability, violating prescribed specifications possibly within unacceptable margins, inducing further operational challenges and/or costs. On the other extent, short-term-optimal policies-commonly based on deterministic waterfilling-, while strictly maintaining operational specifications, are not only impractical and computationally demanding, but also suboptimal in a long-term sense. To address these challenges, we introduce a novel distributionally robust version of a classical point-to-point interference-free multi-terminal constrained stochastic resource allocation problem, by leveraging the Conditional Value-at-Risk (CVaR) as a coherent measure of power policy fluctuation risk. We derive closed-form dual-parameterized expressions for the CVaR-optimal resource policy, along with corresponding optimal CVaR quantile levels by capitalizing on (sampling) the underlying fading distribution. We subsequently develop two dual-domain schemes-one model-based and one model-free-to iteratively determine a globally-optimal resource policy. Our numerical simulations confirm the remarkable effectiveness of the proposed approach, also revealing an almost-constant character of the CVaR-optimal policy and at rather minimal ergodic rate optimality loss.
... Optimizing average performance, as is typical in standard stochastic optimal control, often fails to yield effective policies for decision making in stochastic environments where deviations from expected outcomes carry significant risk; e.g. in financial markets [1], safe robotics and autonomous systems [2], and healthcare [3]. As such, incorporating risk measures become vital in such decision making problems for balancing the performance with resilience to rare events. ...
... Risk-aware approaches, on the other hand, offer a (probabilistic) compromise by building on available stochastic priors to manage both risk and performance, simultaneously. Consequently, there have been significant efforts [5]- [8] in developing risk-aware decision making frameworks using tools like Conditional Value at Risk (CVaR) [1], Markov Risk Measure [9], and Entropic Value at The authors are with Harvard University; emails: talebi@seas.harvard.edu and nali@seas.harvard.edu. ...
... capturing the uncertainty in g(X t , U t ) relative to that past information-see Figure 1. 1 To account for the long-term risk behavior, especially in non-stationary systems with heavytailed noise, we define the ergodic-risk criterion C ∞ as the limit of the normalized cumulative uncertainty: ...
In stochastic systems, risk-sensitive control balances performance with resilience to less likely events. Although existing methods rely on finite-horizon risk criteria, this paper introduces \textit{limiting-risk criteria} that capture long-term cumulative risks through probabilistic limiting theorems. Extending the Linear Quadratic Regulation (LQR) framework, we incorporate constraints on these limiting-risk criteria derived from the asymptotic behavior of cumulative costs, accounting for extreme deviations. Using tailored Functional Central Limit Theorems (FCLT), we demonstrate that the time-correlated terms in the limiting-risk criteria converge under strong ergodicity, and establish conditions for convergence in non-stationary settings while characterizing the distribution and providing explicit formulations for the limiting variance of the risk functional. The FCLT is developed by applying ergodic theory for Markov chains and obtaining \textit{uniform ergodicity} of the controlled process. For quadratic risk functionals on linear dynamics, in addition to internal stability, the uniform ergodicity requires the (possibly heavy-tailed) dynamic noise to have a finite fourth moment. This offers a clear path to quantifying long-term uncertainty. We also propose a primal-dual constrained policy optimization method that optimizes the average performance while ensuring limiting-risk constraints are satisfied. Our framework offers a practical, theoretically guaranteed approach for long-term risk-sensitive control, backed by convergence guarantees and validations through simulations.
... Risk-aware techniques take into account highly-unlikely yet probable worst-case events, and, as a consequence, hedge against these tail outcomes to an extent proportional to their probabilities. Robustness to stochasticity naturally emerges from such methods, as demonstrated in the risk-sensitive optimization literature [16], [17]. In more related work [10], the authors reformulate the SSP problem as Difference Convex Programs (DCPs) with the decision variable set as a dynamic risk functional and for Markov state transitions. ...
... From (16) and by Lemma 1, we can thus write: ...
... COST STATISTICS COMPARISONS (σ C k = 0.5) As additional support for our results, we notice here that, consistent with the known VaR-CVaR relationship established in the literature[16], for both values of α, CVaRα[L] > VaRα[L]. ...
With the pervasiveness of Stochastic Shortest-Path (SSP) problems in high-risk industries, such as last-mile autonomous delivery and supply chain management, robust planning algorithms are crucial for ensuring successful task completion while mitigating hazardous outcomes. Mainstream chance-constrained incremental sampling techniques for solving SSP problems tend to be overly conservative and typically do not consider the likelihood of undesirable tail events. We propose an alternative risk-aware approach inspired by the asymptotically-optimal Rapidly-Exploring Random Trees (RRT*) planning algorithm, which selects nodes along path segments with minimal Conditional Value-at-Risk (CVaR). Our motivation rests on the step-wise coherence of the CVaR risk measure and the optimal substructure of the SSP problem. Thus, optimizing with respect to the CVaR at each sampling iteration necessarily leads to an optimal path in the limit of the sample size. We validate our approach via numerical path planning experiments in a two-dimensional grid world with obstacles and stochastic path-segment lengths. Our simulation results show that incorporating risk into the tree growth process yields paths with lengths that are significantly less sensitive to variations in the noise parameter, or equivalently, paths that are more robust to environmental uncertainty. Algorithmic analyses reveal similar query time and memory space complexity to the baseline RRT* procedure, with only a marginal increase in processing time. This increase is offset by significantly lower noise sensitivity and reduced planner failure rates.
... In this paper, the conditional value-at-risk (CVaR) is used to measure the violation rate of a chance-constraint. As demonstrated in [18], enforcing a CVaR ≤ 0 ensures that the corresponding chance constraint is satisfied. ...
... where (19) is the same as the objective function in (12a), and (20) is the complementary slackness condition based on (12e). Therefore, the two partial derivatives in (18) can be calculated as: ...
... We first intuitively analyze the expected outcomes of Algorithm (1) at each iteration T . Given that the partial derivative terms in (18) are always non-negative, the direction in which ϵ is updated, indicated by the sign of ∂L(ϵ) ∂ϵ , depends on relative magnitudes of the expected versus actual cost and CVaR values. If Cost exp t > Cost act t , indicating that the DSO is overly conservative by planning for adverse scenarios that never materialize, then the value of ϵ should be decreased. ...
Information asymmetry between the Distribution System Operator (DSO) and Distributed Energy Resource Aggregators (DERAs) obstructs designing effective incentives for voltage regulation. To capture this effect, we employ a Stackelberg game-theoretic framework, where the DSO seeks to overcome the information asymmetry and refine its incentive strategies by learning from DERA behavior over multiple iterations. We introduce a model-based online learning algorithm for the DSO, aimed at inferring the relationship between incentives and DERA responses. Given the uncertain nature of these responses, we also propose a distributionally robust incentive design model to control the probability of voltage regulation failure and then reformulate it into a convex problem. This model allows the DSO to periodically revise distribution assumptions on uncertain parameters in the decision model of the DERA. Finally, we present a gradient-based method that permits the DSO to adaptively modify its conservativeness level, measured by the size of a Wasserstein metric-based ambiguity set, according to historical voltage regulation performance. The effectiveness of our proposed method is demonstrated through numerical experiments.
... Generally speaking, the proper objective for a risk-aware operator penalizes the highest overall operation costs. Therefore, minimizing the Conditional Value-at-Risk (CVaR) (a widely-used metric for quantifying risk [7]) of the overall operation costs is used as the training objective. Using such an objective, a bilevel program is built for parameter estimation, where the upper level optimizes over the forecast model parameter, and the lower level solves the DA and RT operation problems for each sample. ...
... The goal is to find an optimal solution of x to minimize CVaR β . [7] shows that such a goal can be achieved via solving an optimization program, [7]. The optimal solution x * minimizes the CVaR β , and α * gives the corresponding Var β . ...
... The goal is to find an optimal solution of x to minimize CVaR β . [7] shows that such a goal can be achieved via solving an optimization program, [7]. The optimal solution x * minimizes the CVaR β , and α * gives the corresponding Var β . ...
This paper develops a risk-aware net demand forecasting product for virtual power plants, which helps reduce the risk of high operation costs. At the training phase, a bilevel program for parameter estimation is formulated, where the upper level optimizes over the forecast model parameter to minimize the conditional value-at-risk (a risk metric) of operation costs. The lower level solves the operation problems given the forecast. Leveraging the specific structure of the operation problem, we show that the bilevel program is equivalent to a convex program when the forecast model is linear. Numerical results show that our approach effectively reduces the risk of high costs compared to the forecasting approach developed for risk-neutral decision makers.
... To address the challenges posed by the two types of uncertainties, we proposed a two-stage algorithm. The first stage involves constructing ambiguity tubes and deriving distributionally robust bounds for dynamic uncertainties using the WDR optimization method, which includes risk function formulations by Conditional Value-at-Risk (CVaR) [25] and convex reformulations. In the second stage, we derive a nominal MPC system from a stochastic MPC system, leveraging the ambiguity tubes for dynamic uncertainties and WDR optimization reformulations for static uncertainties. ...
... where µ k (t) := B ⊤ * k RB I − α(t)e ⊤ ⊤ /V 0 is introduced to simplify the formulation, d G/V k (t) and u G/V k (t) are auxiliary optimization variables. Following a standard CVaR procedure [25], we can transform the uncertain constraints (26c) and (26d) into risk functions, leading to the following proposition: Proposition 1. The risk function J G risk and J V risk for (26c) and (26d) are respectively, ...
... (25b), (26a), (26b), (25d), (28b) where µ 1 and µ 2 are the weighting factors between the cost functions and risk functions, and y(t) is defined in the formulation (25). All auxiliary variables are collected in ...
The integration of various power sources, including renewables and electric vehicles, into smart grids is expanding, introducing uncertainties that can result in issues like voltage imbalances, load fluctuations, and power losses. These challenges negatively impact the reliability and stability of online scheduling in smart grids. Existing research often addresses uncertainties affecting current states but overlooks those that impact future states, such as the unpredictable charging patterns of electric vehicles. To distinguish between these, we term them static uncertainties and dynamic uncertainties, respectively. This paper introduces WDR-MPC, a novel approach that stands for two-stage Wasserstein-based Distributionally Robust (WDR) optimization within a Model Predictive Control (MPC) framework, aimed at effectively managing both types of uncertainties in smart grids. The dynamic uncertainties are first reformulated into ambiguity tubes and then the distributionally robust bounds of both dynamic and static uncertainties can be established using WDR optimization. By employing ambiguity tubes and WDR optimization, the stochastic MPC system is converted into a nominal one. Moreover, we develop a convex reformulation method to speed up WDR computation during the two-stage optimization. The distinctive contribution of this paper lies in its holistic approach to both static and dynamic uncertainties in smart grids. Comprehensive experiment results on IEEE 38-bus and 94-bus systems reveal the method’s superior performance and the potential to enhance grid stability and reliability.
... To address the challenges posed by the two types of uncertainties, we proposed a two-stage algorithm. The first stage involves constructing ambiguity tubes and deriving distributionally robust bounds for dynamic uncertainties using the WDR optimization method, which includes risk function formulations by Conditional Value-at-Risk (CVaR) [25] and convex reformulations. In the second stage, we derive a nominal MPC system from a stochastic MPC system, leveraging the ambiguity tubes for dynamic uncertainties and WDR optimization reformulations for static uncertainties. ...
... where µ k (t) := B ⊤ * k RB I − α(t)e ⊤ ⊤ /V 0 is introduced to simplify the formulation, d G/V k (t) and u G/V k (t) are auxiliary optimization variables. Following a standard CVaR procedure [25], we can transform the uncertain constraints (26c) and (26d) into risk functions, leading to the following proposition: Proposition 1. The risk function J G risk and J V risk for (26c) and (26d) are respectively, ...
... (25b), (26a), (26b), (25d), (28b) where µ 1 and µ 2 are the weighting factors between the cost functions and risk functions, and y(t) is defined in the formulation (25). All auxiliary variables are collected in ...
The integration of various power sources, including renewables and electric vehicles, into smart grids is expanding, introducing uncertainties that can result in issues like voltage imbalances, load fluctuations, and power losses. These challenges negatively impact the reliability and stability of online scheduling in smart grids. Existing research often addresses uncertainties affecting current states but overlooks those that impact future states, such as the unpredictable charging patterns of electric vehicles. To distinguish between these, we term them static uncertainties and dynamic uncertainties, respectively. This paper introduces WDR-MPC, a novel approach that stands for two-stage Wasserstein-based Distributionally Robust (WDR) optimization within a Model Predictive Control (MPC) framework, aimed at effectively managing both types of uncertainties in smart grids. The dynamic uncertainties are first reformulated into ambiguity tubes and then the distributionally robust bounds of both dynamic and static uncertainties can be established using WDR optimization. By employing ambiguity tubes and WDR optimization, the stochastic MPC system is converted into a nominal one. Moreover, we develop a convex reformulation method to speed up WDR computation during the two-stage optimization. The distinctive contribution of this paper lies in its holistic approach to both static and dynamic uncertainties in smart grids. Comprehensive experiment results on IEEE 38-bus and 94-bus systems reveal the method's superior performance and the potential to enhance grid stability and reliability.
... where H T j (y) denotes the j th row of the matrix H(y); and g j (y) is the j th element of the vector g(y). Constraint (29) is an individual chance constraint that can be equivalently reformulated as a worst-case conditional value at risk (CVaR) constraint [20], [41], [42]. ...
... Remark: for a given measurable loss function L: R k ® R, probability distribution P on R k , and tolerance β Î(01), it is well known that [41]: ...
... We then use the CVaR definition introduced in [41]: (33) where (×) + = max(×0). We then require the CVaR constraint (33) to hold for a family of distributions defined directly from observed samples via the Wasserstein metric. ...
Moving away from fossil fuels towards renewable sources requires system operators to determine the capacity of distribution systems to safely accommodate green and distributed generation (DG). However, the DG capacity of a distribution system is often underestimated due to either overly conservative electrical demand and DG output uncertainty modelling or neglecting the recourse capability of the available components. To improve the accuracy of DG capacity assessment, this paper proposes a distributionally adjustable robust chance-constrained approach that utilises uncertainty information to reduce the conservativeness of conventional robust approaches. The proposed approach also enables fast-acting devices such as inverters to adjust to the real-time realisation of uncertainty using the adjustable robust counterpart methodology. To achieve a tractable formulation, we first define uncertain chance constraints through distributionally robust conditional value-at-risk (CVaR), which is then reformulated into convex quadratic constraints. We subsequently solve the resulting large-scale, yet convex, model in a distributed fashion using the alternating direction method of multipliers (ADMM). Through numerical simulations, we demonstrate that the proposed approach outperforms the adjustable robust and conventional distributionally robust approaches by up to 15% and 40%, respectively, in terms of total installed DG capacity.
... To minimize the risk that results in a severe loss, the development of advanced risk-aware control approaches becomes essential. The Conditional Value-at-Risk (CVaR) [17], This work was supported by JST, PRESTO Grant Number JPMJPR22C3, Japan. ...
... [18] is a risk measure that is defined as the conditional expectation of losses exceeding a certain threshold, thus quantifying the tail risk. CVaR was first used in the finance [17], [19], and now it is also used in the controls [20]- [24]. However, the computation of CVaR requires the exact knowledge of the uncertainty probability distribution, which may limit its practical use. ...
... However, the computation of CVaR requires the exact knowledge of the uncertainty probability distribution, which may limit its practical use. The worst-case CVaR [17], [19], [25], on the other hand, does not require the exact knowledge of the uncertainty probability distributions, but considers the maximum risk over a set of possible uncertainty distributions, which enhances its applicability in practice. Moreover, it is known that the computation of the worst-case CVaR can be expressed as a quadratic problem for common cases [18], [26]. ...
This paper proposes a risk-aware control approach to enforce safety for discrete-time nonlinear systems subject to stochastic uncertainties. We derive some useful results on the worst-case Conditional Value-at-Risk (CVaR) and define a discrete-time risk-aware control barrier function using the worst-case CVaR. On this basis, we present optimization-based control approaches that integrate the worst-case CVaR into the control barrier function, taking into account both safe set and tail risk considerations. In particular, three types of safe sets are discussed in detail: half-space, polytope, and ellipsoid. It is shown that control inputs for the half-space and polytopic safe sets can be obtained via quadratic programs, while control inputs for the ellipsoidal safe set can be computed via a semidefinite program. Through numerical examples of an inverted pendulum, we compare its performance with existing methods and demonstrate the effectiveness of our proposed controller.
... Secondly, [20] requires explicitly calculating the game payoff for each pair of players' actions, whereas we propose a single-shot semi-definite program (SDP). Finally, [20] uses the Value-at-Risk (VaR) as the risk metric, whereas we use Conditional VaR (CVaR) as a risk metric, which has much more advantages, notably convexity [21]. We note that game-theoretic approaches have also been used in other research fields not limited to control for optimal allocation of monitoring resources [22], [23] Other works which focus on optimal security allocation using a game-theoretic approach are [24]- [27]. ...
... where R Ω is a risk metric chosen by the defender. The subscript Ω denotes that the risk acts over the set Ω whose probabilistic description is known to the defender (for the results of this article to hold, it is sufficient that the defender can draw samples from the set Ω). ◁ CVaR is extensively used in the literature due to its numerous advantages [21]. Thus we choose the CVaR as a risk metric in Problem 2. Before we introduce the risk metric, we make the following assumptions that follow from [21]. ...
... The subscript Ω denotes that the risk acts over the set Ω whose probabilistic description is known to the defender (for the results of this article to hold, it is sufficient that the defender can draw samples from the set Ω). ◁ CVaR is extensively used in the literature due to its numerous advantages [21]. Thus we choose the CVaR as a risk metric in Problem 2. Before we introduce the risk metric, we make the following assumptions that follow from [21]. ...
This article considers the problem of risk-optimal allocation of security measures when the actuators of an uncertain control system are under attack. We consider an adversary injecting false data into the actuator channels. The attack impact is characterized by the maximum performance loss caused by a stealthy adversary with bounded energy. Since the impact is a random variable, due to system uncertainty, we use Conditional Value-at-Risk (CVaR) to characterize the risk associated with the attack. We then consider the problem of allocating security measures to the set of actuators to minimize the risk. We assume that there are only a limited number of security measures available. Under this constraint, we observe that the allocation problem is a mixed-integer optimization problem. Thus we use relaxation techniques to approximate the security allocation problem into a Semi-Definite Program (SDP). We also compare our allocation method (i) across different risk measures: the worst-case measure, the average (nominal) measure, and (ii) across different search algorithms: the exhaustive and the greedy search algorithms. We depict the efficacy of our approach through numerical examples.
... Popularity bias is a popular topic in the recommender systems literature. There have been previous works on handling popularity bias [3,18,24]. Many of these approaches require identifying apriori which items are in the "long tail" and which ones are in the short-tail of recommendation. ...
... Compared to those approaches, we do not need to apriori identify which items are in the long tail versus short tail. We compare two competitive baselines in this category including IPW [18] and CVaR [24] in Figure 3 and experiments. The results show our significant improvement in addressing the popularity bias problem. ...
... In this section, we introduce four other baselines IPW [18], CVar [24], Rerank [3] and MP that we compare against in this paper. IPW. ...
In many recommender problems, a handful of popular items (e.g. movies/TV shows, news etc.) can be dominant in recommendations for many users. However, we know that in a large catalog of items, users are likely interested in more than what is popular. The dominance of popular items may mean that users will not see items they would likely enjoy. In this paper, we propose a technique to overcome this problem using adversarial machine learning. We define a metric to translate user-level utility metric in terms of an advantage/disadvantage over items. We subsequently use that metric in an adversarial learning framework to systematically promote disadvantaged items. The resulting algorithm identifies semantically meaningful items that get promoted in the learning algorithm. In the empirical study, we evaluate the proposed technique on three publicly available datasets and four competitive baselines. The result shows that our proposed method not only improves the coverage, but also, surprisingly, improves the overall performance.
... b) Value-at-Risk optimization: The proposed percentile formulation is well-known in portfolio optimization. In concrete, problem (2) is known as a value-at-risk (VaR) problem in the context of risk analysis [24]. This connection is formalized latter in section II after we review some background on this field. ...
... This connection is formalized latter in section II after we review some background on this field. Here, we describe the algorithmic literature for this optimization class, which is rooted in economic theory [24]- [26]. From our perspective, there have been four main ideas to approach VaR problems like the one in (2): (1) conditional value-at-risk (CVaR) methods based on the seminal work of Rockafellar et al. [24]; (2) difference-of-convex (DC) approaches that decompose the percentile function as a difference of two convex functions [27,28]; (3) integer programming schemes that explore the combinatorial nature of the percentile objective [29]- [32] and (4) smoothing techniques that filter out local, erratic modes of the VaR objective [33,34]. ...
... Here, we describe the algorithmic literature for this optimization class, which is rooted in economic theory [24]- [26]. From our perspective, there have been four main ideas to approach VaR problems like the one in (2): (1) conditional value-at-risk (CVaR) methods based on the seminal work of Rockafellar et al. [24]; (2) difference-of-convex (DC) approaches that decompose the percentile function as a difference of two convex functions [27,28]; (3) integer programming schemes that explore the combinatorial nature of the percentile objective [29]- [32] and (4) smoothing techniques that filter out local, erratic modes of the VaR objective [33,34]. The CVaR approach is, perhaps, the most popular since, for convex losses, it leads to a convex problem that upper bounds the VaR objective [24]. ...
This paper consider considers the problem of locating a two dimensional target from range-measurements containing outliers. Assuming that the number of outlier is known, we formulate the problem of minimizing inlier losses while ignoring outliers. This leads to a combinatorial, non-convex, non-smooth problem involving the percentile function. Using the framework of risk analysis from Rockafellar et al., we start by interpreting this formulation as a Value-at-risk (VaR) problem from portfolio optimization. To the best of our knowledge, this is the first time that a localization problem was formulated using risk analysis theory. To study the VaR formulation, we start by designing a majorizer set that contains any solution of a general percentile problem. This set is useful because, when applied to a localization scenario in 2D, it allows to majorize the solution set in terms of singletons, circumferences, ellipses and hyperbolas. Using know parametrization of these curves, we propose a grid method for the original non-convex problem. So we reduce the task of optimizing the VaR objective to that of efficiently sampling the proposed majorizer set. We compare our algorithm with four benchmarks in target localization. Numerical simulations show that our method is fast while, on average, improving the accuracy of the best benchmarks by at least 100m in a 1 Km area.
... Eq. (18) implies that power sources with higher k g N values (corresponding to lower PoFs) contribute more to the total generation compared to the lower k g N values (or high PoFs). As a result, the system will be less dependent on the vulnerable generators with high PoFs. ...
... By definition, for a given probability level α, the α-VaR of a portfolio represents the minimum value β such that the portfolio's loss will not exceed β with a probability of α. In contrast, the α-CVaR is the expected loss conditional on the losses exceeding β [18]. ...
As power systems become more complex with the continuous integration of intelligent distributed energy resources (DERs), new risks and uncertainties arise. Consequently, to enhance system resiliency, it is essential to account for various uncertain events when implementing the optimization problem for the energy management system (EMS). This paper presents a preventive EMS considering the probability of failure (PoF) of each system component across different scenarios. A conditional-value-at-risk (CVaR)-based framework is proposed to integrate the uncertainties of the distribution network. Loads are classified into critical, semi-critical, and non-critical categories to prioritize essential loads during generation resource shortages. A proximal policy optimization (PPO)-based reinforcement learning (RL) agent is used to solve the formulated problem and generate the control decisions. The proposed framework is evaluated on a notional MVDC ship system and a modified IEEE 30-bus system, where the results demonstrate that the PPO agent can successfully optimize the objective function while maintaining the network and operational constraints. For validation, the RL-based method is benchmarked against a traditional optimization approach, further highlighting its effectiveness and robustness. This comparison shows that RL agents can offer more resiliency against future uncertain events compared to the traditional solution methods due to their adaptability and learning capacity.
... In such scenarios, designing risk-sensitive policies may be preferable to enhance the worst-case outcomes while reducing the average performance (3). A common risk-sensitive measure is the conditional valueat-risk (CVaR) [22], which is defined as the conditional mean ...
... The obtained scheme, whose steps are detailed in Algorithm 3, is referred to as multi-agent conservative centralized Q-learning (MA-CCQL). The optimized policy is given in (22). ...
Reinforcement learning (RL) has been widely adopted for controlling and optimizing complex engineering systems such as next-generation wireless networks. An important challenge in adopting RL is the need for direct access to the physical environment. This limitation is particularly severe in multi-agent systems, for which conventional multi-agent reinforcement learning (MARL) requires a large number of coordinated online interactions with the environment during training. When only offline data is available, a direct application of online MARL schemes would generally fail due to the epistemic uncertainty entailed by the lack of exploration during training. In this work, we propose an offline MARL scheme that integrates distributional RL and conservative Q-learning to address the environment’s inherent aleatoric uncertainty and the epistemic uncertainty arising from the use of offline data. We explore both independent and joint learning strategies. The proposed MARL scheme, referred to as multi-agent conservative quantile regression, addresses general risk-sensitive design criteria and is applied to the trajectory planning problem in drone networks, showcasing its advantages.
... function F α ) addresses the challenge of uncertainty in future market price/LMP predictions. Therein, we adopt VaR and CVaR concepts as risk measures, enabled by [36]. Specifically, VaR (and CVaR) are functions of all dispatch variables D = D 1 , . . . ...
... In addition, it may not offer a convex (and differentiable) VaR α (F ) formulation and hence may not be tractable to obtain the required profit optimality guarantees via off-the-shelf solvers. Alternatively, CVaR formulations presented in [36] allow a convex representation, wherein CVaR provides an estimated mean of the profit values which are lesser than the VaR value for the same confidence level used to estimate the VaR. For instance, the CVaR 0.95 is the average of the expected profit for plausible profit values lesser than the VaR 0.95 threshold [42], i.e., the mean of the 0.95-negative tail distribution of F (Λ, D) as defined below. ...
This work proposes a novel degradation-infused energy portfolio allocation (DI-EPA) framework for enabling the participation of battery energy storage systems in multi-service electricity markets. The proposed framework attempts to address the challenge of including the rainflow algorithm for cycle counting by directly developing a closed-form of marginal degradation as a function of dispatch decisions. Further, this closed-form degradation profile is embedded into an energy portfolio allocation (EPA) problem designed for making the optimal dispatch decisions for all the batteries together, in a shared economy manner. We term the entity taking these decisions as `facilitator' which works as a link between storage units and market operators. The proposed EPA formulation is quipped with a conditional-value-at-risk (CVaR)-based mechanism to bring risk-averseness against uncertainty in market prices. The proposed DI-EPA problem introduces fairness by dividing the profits into various units using the idea of marginal contribution. Simulation results regarding the accuracy of the closed-form of degradation, effectiveness of CVaR in handling uncertainty within the EPA problem, and fairness in the context of degradation awareness are discussed. Numerical results indicate that the DI-EPA framework improves the net profit of the storage units by considering the effect of degradation in optimal market participation.
... Based on VaR, CVaR is defined [57] as CVaR Fig. 1 shows an illustration of VaR and CVaR. Our goal is to find a function V : R n → R and a controller π : R n → R m that satisfy the Lyapunov conditions in (3). ...
... This inequality implies the satisfaction of the original chance constraint in (5). As shown in [57], the inequality (6) can be written equivalently as: ...
This article presents novel methods for synthesizing distributionally robust stabilizing neural controllers and certificates for control systems under model uncertainty. A key challenge in designing controllers with stability guarantees for uncertain systems is the accurate determination of and adaptation to shifts in model parametric uncertainty during online deployment. We tackle this with a novel distributionally robust formulation of the Lyapunov derivative chance constraint ensuring a monotonic decrease of the Lyapunov certificate. To avoid the computational complexity involved in dealing with the space of probability measures, we identify a sufficient condition in the form of deterministic convex constraints that ensures the Lyapunov derivative constraint is satisfied. We integrate this condition into a loss function for training a neural network-based controller and show that, for the resulting closed-loop system, the global asymptotic stability of its equilibrium can be certified with high confidence, even with Out-of-Distribution (OoD) model uncertainties. To demonstrate the efficacy and efficiency of the proposed methodology, we compare it with an uncertainty-agnostic baseline approach and several reinforcement learning approaches in two control problems in simulation
... CV aR(α = 0.1) expresses the expected value for the worst 10% of possible outcomes. [31] introduced a methodology for including the risk measure CVaR in the objective of a stochastic programming problem while keeping the problem convex. This allows a stochastic algorithm to fine-tune its risk aversion. ...
... Ω decides the risk-aversion of the algorithm with a higher Ω increasing the weighting of CVaR. Equations (19) -(20) constrain the CVaR following [31]. Additionally, the operational constraints from Equations (8) -(16) still apply. ...
Controlled charging of electric vehicles, EVs, is a major potential source of flexibility to facilitate the integration of variable renewable energy and reduce the need for stationary energy storage. To offer system services from EVs, fleet aggregators must address the uncertainty of individual driving and charging behaviour. This paper introduces a means of forecasting the service volume available from EVs by considering several EV batteries as one conceptual battery with aggregate power and energy boundaries. This avoids the impossible task of predicting individual driving behaviour by taking advantage of the law of large numbers. The forecastability of the boundaries is demonstrated in a multiple linear regression model which achieves an of 0.7 for a fleet of 1,000 EVs. A two-stage stochastic model predictive control algorithm is used to schedule reserve services on a day-ahead basis addressing risk trade-offs by including Conditional Value-at-Risk in the objective function. A case study with 1.2 million domestic EV charge records from Great Britain shows that increasing fleet size improves prediction accuracy, thereby increasing reserve revenues and decreasing effective charging costs. For fleet sizes of 400 or above, charging cost reductions plateau at 60\%, with an average of 1.8kW of reserve provided per vehicle.
... Section VI connects our results with Stromberg's method [13] for the LQS problem. b) Value-at-Risk (VaR) optimization: Formulation (4) also has a strong connection to portfolio optimization [17], [18], [19], since (4) can be interpreted as optimizing the Value-at-Risk (VaR) measure of an underlying stochastic risk problem. Further details on this interpretation for the setting of target localization can be found in [20]. ...
... where the set ∂φ T (θ ⋆ ) denotes the sub-differential of the convex function φ T at the point θ ⋆ . Because φ T is a pointwise maximum of convex functions, namely, φ T (θ) = max{φ m (θ); m ∈ T }, for all θ, we can use Corollary 4.3.2 in [28], together with (19), to conclude that ...
We consider the problem of robustly fitting a model to data that includes outliers by formulating a percentile optimization problem. This problem is non-smooth and non-convex, hence hard to solve. We derive properties that the minimizers of such problems must satisfy. These properties lead to methods that solve the percentile formulation both for general residuals and for convex residuals. The methods fit the model to subsets of the data, and then extract the solution of the percentile formulation from these partial fits. As illustrative simulations show, such methods endure higher outlier percentages, when compared with standard robust estimates. Additionally, the derived properties provide a broader and alternative theoretical validation for existing robust methods, whose validity was previously limited to specific forms of the residuals.
... , the following risk-adjusted cost consensus model with compromise limit in different opinion adjustment directions (ε-RDRO-DC) is given: [34] Suppose β∈(0,1) is a given confidence level and the function Ψ(x, ξ) is continuous everywhere about ζ. The mean of the β truncated tail distribution of the loss function μ(x, ξ) and it can be defined as: ...
... Obviously, CVaR indicates that the risk loss is not less than the expected value of VaR. Since the defining equation of CVaR is not easy to calculate under continuous random variables, Rockafellar et al. [34] give an approximate calculation method that CVaR can be equivalently transformed into . ...
... Based on VaR, CVaR is defined [51] as CVaR Our goal is to find a function V : R n → R and a controller π : R n → R m that satisfy the Lyapunov conditions in (3). However, the uncertainty in the dynamical system (4), which manifests itself in the termV (x, π(x)), poses a challenge for ensuring that the condition on the derivative of the Lyapunov function (3b) are satisfied. ...
... This inequality implies the satisfaction of the original chance constraint in (5). As shown in [51], CVaR can also be formulated as the following convex program: ...
This article presents novel methods for synthesizing distributionally robust stabilizing neural controllers and certificates for control systems under model uncertainty. A key challenge in designing controllers with stability guarantees for uncertain systems is the accurate determination of and adaptation to shifts in model parametric uncertainty during online deployment. We tackle this with a novel distributionally robust formulation of the Lyapunov derivative chance constraint ensuring a monotonic decrease of the Lyapunov certificate. To avoid the computational complexity involved in dealing with the space of probability measures, we identify a sufficient condition in the form of deterministic convex constraints that ensures the Lyapunov derivative constraint is satisfied. We integrate this condition into a loss function for training a neural network-based controller and show that, for the resulting closed-loop system, the global asymptotic stability of its equilibrium can be certified with high confidence, even with Out-of-Distribution (OoD) model uncertainties. To demonstrate the efficacy and efficiency of the proposed methodology, we compare it with an uncertainty-agnostic baseline approach and several reinforcement learning approaches in two control problems in simulation. Open-source implementations of the examples are available at https://github.com/KehanLong/DR Stabilizing Policy. INDEX TERMS Learning for control, Lyapunov methods, optimization under uncertainty, stability of nonlinear systems
... ξ is the random parameter associated to uncertain PV generation and must-run loads' consumption. For further information about CVaR β , readers are referred to [23]. ...
... where u w n is the decision vector under the scenario of w ∈ W . In addition, samples generated from the distribution of the uncertain parameter ξ can be used to approximate CVaR term in (8) and (9) as [23] min u,z ...
This paper addresses the operational challenges in distribution systems caused by the integration of fluctuating renewable energy sources and rising peak loads from widespread electrification. We propose a new distributed energy coordination framework that facilitates seamless interoperation between peer-to-peer (P2P) energy trading and the operation of distributed flexibility resources. This framework ensures network constraints are respected while promoting Energy Trading Consistency (ETC). Our approach utilizes a two-stage dynamic energy and flexibility sharing mechanism. It achieves energy efficiency through demand-management and P2P energy trading in the day-ahead (DA) stage, followed by adjustments of flexible resources and flexibility sharing through a receding-horizon feedback mechanism inspired by model predictive control (MPC) in the real-time (RT) stage. High-risk scenarios and computational overhead are managed by employing conditional value-at-risk (CVaR) measures, as well as approximation of constraints through dual robust reformulation and linear decision rules. To uphold the autonomy and privacy of prosumers, we propose distributed algorithms based on the fast consensus alternating direction method of multipliers (ADMM). Numerical results demonstrate that our framework achieves a 20% reduction in social cost while providing a coherent and interactive P2P energy trading environment, facilitating reliable network operation under high penetration of renewable energy sources without third-party intervention.
... In such a case, incorporating CVaR in the data rate constraint during beamforming optimization effectively characterizes the potential loss in data rate due to poor channel conditions, as CVaR captures the risk in the tail distribution of the data rate by defining the average potential loss that exceeds the Value-at-Risk (VaR), allowing the system to prepare for extreme conditions and enhancing transmission reliability. In general, the CVaR of a random variable Z is defined as [31] CVaR α (Z) := inf ...
... Proof. The α-percentile (Value at Risk) of a random variable, i.e., the value for which the likelihood of a random variable is less than or equal to it is at least α, is given by [31]: ...
This paper proposes a robust beamforming algorithm for massive multiple-input multiple-output (MIMO) low earth-orbit (LEO) satellite communications under uncertain channel conditions. Specifically, a risk-aware optimization problem is formulated to optimize the hybrid digital and analog precoding aiming at maximizing the energy efficiency of the LEO satellite while considering the required quality-of-service (QoS) by each ground user. The Conditional Value at Risk (CVaR) is used as a risk measure of the downlink data rate to capture the high dynamic and random channel variations due to satellite movement, achieving the required QoS under the worst-case scenario. A deep reinforcement learning (DRL) based framework is developed to solve the formulated stochastic problem over time slots. Considering the limited computation capabilities of the LEO satellite, the training process of the proposed learning algorithm is performed offline at a central terrestrial server. The trained models are then sent periodically to the LEO satellite through ground stations to provide online executions on the transmit precoding based on the current network state. Simulation results demonstrate the efficacy of the proposed approach in achieving the QoS requirements under uncertain wireless channel conditions.
... In this section, we will explain how the asset allocation problem (2), a nonlinear optimization problem due to the CVaRbased constraint, can be restated as a linear programming problem and why this formulation is useful in practice. The foundational concept was established by Rockafellar and Uryasev (2000), who demonstrated that solving the optimization problem (2) is equivalent to solving the following optimization problem: ...
... A comprehensive explanation (including all the formal proofs) regarding the derivation of the equivalent continuous formulation (5) and its subsequent discretization and linearization, (6) and (7), can be found in Rockafellar and Uryasev (2000). An in-depth analysis of these equivalent formulations-including practical examples, and a discussion of the general problem-setting incorporating transaction costs, value constraints, liquidity constraints, and limits on positionare provided in Krokhmal et al. (2001). ...
... This allows us to design controllers that guarantee that the expected value of the constraint-violating cases is small. CVaR is a relatively new risk measure that is defined as the conditional expectation of losses exceeding a certain threshold [17]. It is known that the (worst-case) CVaR is a coherent risk measure [4] and enjoys nice mathematical properties. ...
... Here δ ij denotes the Kronecker delta and E P [·] denotes the expectation with respect to P. The true underlying probability measure P is not known exactly, but it is known that P ∈ P. Definition 2.1 (Conditional Value-at-Risk [17], [27]): For a given measurable loss function L : R n → R, a probability distribution P on R n and a level ε ∈ (0, 1), the CVaR at ε with respect to P is defined as ...
This paper introduces the notions of stability, ultimate boundedness, and positive invariance for stochastic systems in view of risk. More specifically, those notions are defined in terms of the worst-case Conditional Value-at-Risk (CVaR), which quantifies the worst-case conditional expectation of losses exceeding a certain threshold over a set of possible uncertainties. Those notions allow us to focus our attention on the tail behavior of stochastic systems in the analysis of dynamical systems and the design of controllers. Furthermore, some event-triggered control strategies that guarantee ultimate boundedness and positive invariance with specified bounds are derived using the obtained results and illustrated using numerical examples.
... In the forward auction, we limit the risk of constraint violations, where we measure this risk via the conditional value at risk (CVaR) measure. CVaR, analyzed and popularized by [22], [23], has recently been advocated in power system planning, e.g., in [24]. For a random variable (think loss) X with smooth cumulative distribution function F X , CVaR at level δ ∈ [0, 1) equals the conditional mean of X over the (1 − δ)-tail of the distribution, i.e., ...
... Per [27, p. 15],L −1 =S ⊺ , and bothL andS are invertible. Thus, (22) admits the compact representation, as in [20, p. 206], ...
In this two-part paper, we consider the problem of profit-seeking distributed energy resource aggregators (DERAs) participating in the wholesale electricity market operated by an independent system operator (ISO). Part I proposes a market-based coordination mechanism in the form of a forward auction to allocate network access limits to DERAs. The proposed coordination mechanism results in decoupled DSO-DERA-ISO operations in the real-time electricity market, ensuring the distribution system reliability independent of DERAs' aggregation actions and wholesale market conditions. Welfare-maximizing market clearing problems under robust and risk-sensitive formulations are derived. Properties of the proposed auction, such as nonnegative surpluses of DSO and DERAs, along with empirical performance studies, are presented. Part II of the paper develops competitive profit-maximizing aggregation strategies for DERAs and their bidding strategies as virtual storage participants in the wholesale market.
... One can also use a safe, convex approximation of (4). The conditional value-at-risk (CVaR) [36] is typically used as it is the tightest convex approximation to (4): ...
Optimally combining frequency control with self-consumption can increase revenues from battery storage systems installed behind-the-meter. This work presents an optimized control strategy that allows a battery to be used simultaneously for self-consumption and primary frequency control. Therein, it addresses two stochastic problems: the delivery of primary frequency control with a battery and the use of the battery for self-consumption. We propose a linear recharging policy to regulate the state of charge of the battery while providing primary frequency control. Formulating this as a chance-constrained problem, we can ensure that the risk of battery constraint violations stays below a predefined probability. We use robust optimization as a safe approximation to the chance-constraints, which allows to make the risk of constraint violation arbitrarily low, while keeping the problem tractable and offering maximum reserve capacity. Simulations with real frequency measurements prove the effectiveness of the designed recharging strategy. We adopt a rule-based policy for self-consumption, which is optimized using stochastic programming. This policy allows to reserve more energy and power of the battery on moments when expected consumption or production is higher, while using other moments for recharging from primary frequency control. We show that optimally combining the two services increases value from batteries significantly.
... It is worth noting that PT differs from other risk measures such as Conditional Value at Risk (CVaR) [35] which evaluates the market risk based on the expected value of the risk at some future time. The underlying assumption in evaluating CVaR is that the risk is measured based on the conventional expectation of the future uncertain price, while in PT, the expectation is replace by subjective perception of the individuals which up to some extent introduces a notion of bounded rationality into the model. ...
In this paper, the problem of energy trading between smart grid prosumers, who can simultaneously consume and produce energy, and a grid power company is studied. The problem is formulated as a single-leader, multiple-follower Stackelberg game between the power company and multiple prosumers. In this game, the power company acts as a leader who determines the pricing strategy that maximizes its profits, while the prosumers act as followers who react by choosing the amount of energy to buy or sell so as to optimize their current and future profits. The proposed game accounts for each prosumer's subjective decision when faced with the uncertainty of profits, induced by the random future price. In particular, the framing effect, from the framework of prospect theory (PT), is used to account for each prosumer's valuation of its gains and losses with respect to an individual utility reference point. The reference point changes between prosumers and stems from their past experience and future aspirations of profits. The followers' noncooperative game is shown to admit a unique pure-strategy Nash equilibrium (NE) under classical game theory (CGT) which is obtained using a fully distributed algorithm. The results are extended to account for the case of PT using algorithmic solutions that can achieve an NE under certain conditions. Simulation results show that the total grid load varies significantly with the prosumers' reference point and their loss-aversion level. In addition, it is shown that the power company's profits considerably decrease when it fails to account for the prosumers' subjective perceptions under PT.
... Both [117] and [18] employ a two-stage stochastic programming approach to solve the profit maximization problem [117] (cost minimization problem [18]) for an aggregator. Additionally, a conditional-value-at-risk [118] term is appended in the objective of [18], in order to implement risk control. Distribution generation uncertainty is dealt with by utilizing deterministic forecast in [111], [117]. ...
Smart interactions among the smart grid, aggregators and EVs can bring various benefits to all parties involved, e.g., improved reliability and safety for the smart gird, increased profits for the aggregators, as well as enhanced self benefit for EV customers. This survey focus on viewing this smart interactions from an algorithmic perspective. In particular, important dominating factors for coordinated charging from three different perspectives are studied, in terms of smart grid oriented, aggregator oriented and customer oriented smart charging. Firstly, for smart grid oriented EV charging, we summarize various formulations proposed for load flattening, frequency regulation and voltage regulation, then explore the nature and substantial similarity among them. Secondly, for aggregator oriented EV charging, we categorize the algorithmic approaches proposed by research works sharing this perspective as direct and indirect coordinated control, and investigate these approaches in detail. Thirdly, for customer oriented EV charging, based on a commonly shared objective of reducing charging cost, we generalize different formulations proposed by studied research works. Moreover, various uncertainty issues, e.g., EV fleet uncertainty, electricity price uncertainty, regulation demand uncertainty, etc., have been discussed according to the three perspectives classified. At last, we discuss challenging issues that are commonly confronted during modeling the smart interactions, and outline some future research topics in this exciting area.
... It yields very high accuracy for typical applications. The approximate χ J ( , ,F) can be computed by solving a series of linear programs using linear relaxations (Rockafellar et al. 2000). Each valueL * ,j , j = 1, . . . ...
Using stochastic spanning tests without any distributional assumptions on returns, we show that the two classes of GDP-linked bonds, floaters and linkers, are not spanned by a broad benchmark set of stocks, bonds, and cash for a wide range of design specifications. Thus, they provide a new asset class with significant diversification benefits for investors, with proportional investments to these novel instruments estimated in the double digits and an increase in Sharpe ratios by up to 0.37 over the benchmark. The benefits depend on the market risk premium, but they persist for a wide range of premia estimates from existing literature and are robust to a randomized test. Using the generalized method of moments regressions, we document the finance and macro determinants of GDP-linked bond returns.
... Bao and Liu [25], as well as Bao [26], employed the difference between two successive utility functions as a reward. Risk-averse agents and ES/CVAR, the average loss exceeding the worst % of cases [269], are also adopted as a risk-adjusted performance measure [7,51]. Finally, in market-making and trade execution literature, utility-based rewards are common [177,248,291]. ...
Reinforcement Learning (RL) has experienced significant advancement over the past decade, prompting a growing interest in applications within finance. This survey critically evaluates 167 publications, exploring diverse RL applications and frameworks in finance. Financial markets, marked by their complexity, multi-agent nature, information asymmetry, and inherent randomness, serve as an intriguing test-bed for RL. Traditional finance offers certain solutions, and RL advances these with a more dynamic approach, incorporating machine learning methods, including transfer learning, meta-learning, and multi-agent solutions. This survey dissects key RL components through the lens of Quantitative Finance. We uncover emerging themes, propose areas for future research, and critique the strengths and weaknesses of existing methods.
... In many applications, CVaR [18] is preferred over VaR as it is a more coherent risk measure and does not encourage risktaking that could be obscured by the measure itself. Additionally, CVaR is less dependent on the underlying distribution and offers a more reliable measure for skewed distributions. ...
Timely and effective load shedding in power systems is critical for maintaining supply-demand balance and preventing cascading blackouts. To eliminate load shedding bias against specific regions in the system, optimization-based methods are uniquely positioned to help balance between economical and equity considerations. However, the resulting optimization problem involves complex constraints, which can be time-consuming to solve and thus cannot meet the real-time requirements of load shedding. To tackle this challenge, in this paper we present an efficient machine learning algorithm to enable millisecond-level computation for the optimization-based load shedding problem. Numerical studies on both a 3-bus toy example and a realistic RTS-GMLC system have demonstrated the validity and efficiency of the proposed algorithm for delivering equitable and real-time load shedding decisions.
... A. Derivation of (20) We first reformulate (18) as [42]: ...
The rise of advanced data technologies in electric power distribution systems enables operators to optimize operations but raises concerns about data security and consumer privacy. Resulting data protection mechanisms that alter or obfuscate datasets may invalidate the efficacy of data-driven decision-support tools and impact the value of these datasets to the decision-maker. This paper derives tools for distribution system operators to enrich data-driven operative decisions with information on data quality and, simultaneously, assess data usefulness in the context of this decision. To this end, we derive an AC optimal power flow model for radial distribution systems with data-informed stochastic parameters that internalize a data quality metric. We derive a tractable reformulation and discuss the marginal sensitivity of the optimal solution as a proxy for data value. Our model can capture clustered data provision, e.g., from resource aggregators, and internalize individual data quality information from each data provider. We use the IEEE 33-bus test system, examining scenarios with varying photovoltaic penetration, to demonstrate the application of our approach and discuss the relationship between data quality and its value.
... The mathematical properties of coherent risks make them compatible with standard stochastic optimization methods. In order to take advantage of such properties, we can modify VaR to derive another risk measure which is coherent, known as conditional value-atrisk (CVaR), first introduced by Rockafellar and Uryasev [33]. Considering the tail distribution described by VaR, CVaR is the conditional mean over it and is defined as ...
Markov games (MGs) and multi-agent reinforcement learning (MARL) are studied to model decision making in multi-agent systems. Traditionally, the objective in MG and MARL has been risk-neutral, i.e., agents are assumed to optimize a performance metric such as expected return, without taking into account subjective or cognitive preferences of themselves or of other agents. However, ignoring such preferences leads to inaccurate models of decision making in many real-world scenarios in finance, operations research, and behavioral economics. Therefore, when these preferences are present, it is necessary to incorporate a suitable measure of risk into the optimization objective of agents, which opens the door to risk-sensitive MG and MARL. In this paper, we systemically review the literature on risk sensitivity in MG and MARL that has been growing in recent years alongside other areas of reinforcement learning and game theory. We define and mathematically describe different risk measures used in MG and MARL and individually for each measure, discuss articles that incorporate it. Finally, we identify recent trends in theoretical and applied works in the field and discuss possible directions of future research.
... where γ is a decision variable and (·) + = max(·, 0). Based on the derivation in [37], ...
As the issue of short-circuit currents becomes increasingly critical in contemporary ever-expanding power grids, back-to-back DC transmission systems (BDTSs) have emerged as a promising scheme to address such a concern. However, the study on the practical and accurate modeling of BDTSs in typical power system decision-making problems remains profoundly limited. To bridge this research gap, we formulate a novel BDTS embedded scheduling problem while considering the comprehensive hydrogen model, which incorporates the technologies of hydrogen production, storage, and utilization. In addition, a well-defined Wasserstein-distance-based ambiguity set that harnesses the Gaussian-based nominal distribution is leveraged to capture the renewable output uncertainty. By developing equivalent reformulations of the nonlinear constraints inherent in BDTSs and distributionally robust chance constraints, the concerned model is eventually recast as a mixed-integer second-order cone programming problem. The effectiveness and superiority of the proposed scheduling model are assessed on three representative test systems.
... In addition to the basic EUM strategy, this work also considers the EUM strategy in combination with the conditional value at risk (CVaR), which is a risk metric for the tail risk of the model's objective function [18]. Depending on the considered confidence level γ, the CVaR is equal to the average profit of the 1-γ scenarios with the lowest profit. ...
This paper studies the use of conformal prediction (CP), an emerging probabilistic forecasting method, for day-ahead photovoltaic power predictions to enhance participation in electricity markets. First, machine learning models are used to construct point predictions. Thereafter, several variants of CP are implemented to quantify the uncertainty of those predictions by creating CP intervals and cumulative distribution functions. Optimal quantity bids for the electricity market are estimated using several bidding strategies under uncertainty, namely: trust-the-forecast, worst-case, Newsvendor and expected utility maximization (EUM). Results show that CP in combination with k-nearest neighbors and/or Mondrian binning outperforms its corresponding linear quantile regressors. Using CP in combination with certain bidding strategies can yield high profit with minimal energy imbalance. In concrete, using conformal predictive systems with k-nearest neighbors and Mondrian binning after random forest regression yields the best profit and imbalance regardless of the decision-making strategy. Combining this uncertainty quantification method with the EUM strategy with conditional value at risk (CVaR) can yield up to 93% of the potential profit with minimal energy imbalance.
... CVaR is a risk assessment measure that quantifies the tail risk of a portfolio at a certain confidence level. CVaR is similar to VaR except that it provides a relatively conservative measure of loss [4]. This study uses VaR and CVaR to estimate currency exchange portfolio risks. ...
The recent currency fluctuations, which have created uncertainty in currency markets about the movement of exchange rates, have spurred renewed interest in portfolio dependence modelling and risk modelling. Currency exchange portfolio risks can be measured using Value at Risk (VaR) and Conditional Value at Risk (CVaR) based on Monte Carlo simulation. However, this method of risk estimation involves considerable challenges owing to the complexity of modelling the joint multivariate distribution of the assets in the portfolio. Therefore, copula functions, such as the t-Student and Clay-ton copulas, have been proposed to measure the dependence structure of the return of a currency exchange portfolio. This study proposes the use of the copula-VaR and copula-CVaR approaches to make strategic choices in currency management when evaluating the risks of equal-and mixed-weighted portfolios of the returns of investment in five foreign currencies in Malaysia: the United States dollar, United Kingdom pound sterling, European Union euro, Japanese yen, and Singapore dollar. The generalised autoregressive conditional heteroscedas-ticity (GARCH)-copula models are also evaluated. We find that the marginal distribution of the returns series of the currency exchange rates can be modelled using the Glosten-Jagannathan-Runkle (GJR)-GARCH model with the t-Student distribution, and the dependence structure of the currency exchange portfolio can be depicted by the t-Student copula. The best investment performance tends to the Singapore dollar.
... For example, the extended filtered historical simulation can be used to generate a forwardlooking sample of multivariate returns. Then, the currency exposure can be optimized with respect to the expected shortfall by formulating the optimization problem in the linear programming format, such as presented in Rockafellar and Uryasev (2000). ...
... We leverage the quantile representation to compute risk measures which map a real-valued distribution to a real number and quantify the probability of occurrence of an event away from the expectation [26]. Some well-known risk measures used in risk-sensitive RL are variance, Value at Risk (VaR) and CVaR [27]. In this work, we use the CVaR which represents the expected return we should experience in the worst α% of cases defined as: ...
In complex real-world decision problems, ensuring safety and addressing uncertainties are crucial aspects. In this work, we present an uncertainty-aware Reinforcement Learning agent designed for risk-sensitive applications in continuous action spaces. Our method quantifies and leverages both epistemic and aleatoric uncertainties to enhance agent's learning and to incorporate risk assessment into decision-making processes. We conduct numerical experiments to evaluate our work on a modified version of Lunar Lander with variable and risky landing conditions. We show that our method outperforms both Deep Deterministic Policy Gradient (DDPG) and TD3 algorithms by reducing collisions and having significant faster training. In addition, it enables the trained agent to learn a risk-sensitive policy that balances performance and risk based on a specific level of sensitivity to risk required for the task.
... where (x) + = max(x, 0). α ∈ (0, 1) is the confidence level, and Q π θ c (s t , a t ) is equal to the average of the worst-case αfraction of losses under optimal conditions [50]. Observe s t and generate action from current policy π θt ([a [1] t , a [2] t ] | s t ) 3: Transmit the packets based on action a Reconstruct the trajectory based on received packets by (1) 5: Predict the trajectory based on action a [2] t by (2) 6: ...
In this paper, we establish a task-oriented cross-system design framework to minimize the required packet rate for timely and accurate modeling of a real-world robotic arm in the Metaverse, where sensing, communication, prediction, control, and rendering are considered. To optimize a scheduling policy and prediction horizons, we design a Constraint Proximal Policy Optimization (C-PPO) algorithm by integrating domain knowledge from relevant systems into the advanced reinforcement learning algorithm, Proximal Policy Optimization (PPO). Specifically, the Jacobian matrix for analyzing the motion of the robotic arm is included in the state of the C-PPO algorithm, and the Conditional Value-at-Risk (CVaR) of the state-value function characterizing the long-term modeling error is adopted in the constraint. Besides, the policy is represented by a two-branch neural network determining the scheduling policy and the prediction horizons, respectively. To evaluate our algorithm, we build a prototype including a real-world robotic arm and its digital model in the Metaverse. The experimental results indicate that domain knowledge helps to reduce the convergence time and the required packet rate by up to 50%, and the cross-system design framework outperforms a baseline framework in terms of the required packet rate and the tail distribution of the modeling error.
... Using convex reformulation of CVaR (Rockafellar and Uryasev 2000), we derive a tractable reformulation of computing CVaR β as a convex program given by CVaR β à min X n ,z n 1 ,α α + 1 (n 1)β X n 1 ià1 z i s:t: z i X i α, z i 0 ∀i à 1, : : : , n 1 X n 2 U s (W n ), z n 1 2 R n 1 , α 2 R: ...
Modeling and decision making for queueing systems have been one of fundamental topics in operations research. For these problems, uncertainty models are established by estimation of key parameters such as expected interarrival and service times. In practice, however, their distributions are unknown, and decision makers are only given historical records of waiting times, which contain relevant but indirect information on the uncertainties. Their complex temporal dependence on the queueing dynamics and the absence of distributional information on the model primitives render estimation of queueing systems remarkably challenging. In the paper “Robust Queue Inference from Waiting Times” by Chaithanya Bandi, Eojin Han, and Alexej Proskynitopoulos, a new inference framework based on robust optimization is proposed to estimate unknown service times from waiting time observations. This new framework allows data-driven, distribution-free estimation on unknown model primitives by solving tractable optimization problems.
... where α CVaR is the confidence level parameter, while g w and ζ are the auxiliary variables used to calculate the CVaR. The detailed derivation and proof process of this calculation method are shown in detail in [26]. ...
To facilitate wind energy use and avoid low returns, or even losses in extreme cases, this paper proposes an integrated risk measurement and control approach to jointly manage multiple statistical properties of the expected profit distribution for a wind storage system. First, a risk-averse stochastic decision-making framework and multi-type risk measurements, including the conditional value at Risk (CVaR), value at risk (VaR) and shortfall probability (SP), are described in detail. To satisfy the various needs of multi-type risk-averse decision makers, integrated risk measurement and control approaches are then proposed by jointly considering the expected, boundary and probability values of the extreme results. These are managed using CVaR, VaR and SP, respectively. Finally, the effectiveness of the proposed risk control strategy is verified by conducting case studies with realistic market data, and the results of different risk control strategies are analyzed in depth. The impacts of the risk parameters of the decision maker, the energy capacity of the battery storage and the price difference between the day-ahead and real-time markets on the expected profits and risks are investigated in detail.
... and a 0 � (a 10 , a 20 , : : : , a n0 ) ⊤ , which can be computed using an MILP similar to the one in Theorem 1. Thus, the associated value-at-risk, VaR α (Φ 0 (X)) � min{y ∈ R + | P(Φ 0 (X) > y) ≤ 1 � α}, can be estimated via Theorem 4. According to Rockafellar and Uryasev (2000), given the samples Φ 0 (x 1 ), : : : , Φ 0 (x N ), the conditional valueat-risk of the worst-case losses can be approximated as CVaR α (Φ 0 (X)) ≈ min{y ∈ R + |Φ 0 (x j ) + {N(1 � α)} �1 P N j�1 z j ≤ z j + y, z j ≥ 0, j � 1, : : : , N}: ...
Robust Risk Quantification via Shock Propagation in Financial Networks
Despite the significance of risk contagion in financial networks, uncertainties arise in interbank network structures because of limited information. To address this, proposed is a robust optimization approach to estimate worst-case default probabilities and capital requirements for a specific group of banks (e.g., systemically important financial institutions). By applying this tool, we analyze the impact of different incomplete network information structures and gain regulatory insights into gathering actionable network information.
... While expected utility optimization dates back to the 1950s, if not earlier, the optimization of CVaR was not studied until the early 2000s [50], [51], [52]. This functional has been studied primarily by the operations research and financial engineering communities. ...
The standard approach to risk-averse control is to use the exponential utility (EU) functional, which has been studied for several decades. Like other risk-averse utility functionals, EU encodes risk aversion through an increasing convex mapping
of objective costs to subjective costs. An objective cost is a realization
y
of a random variable
Y
. In contrast, a subjective cost is a realization
of a random variable
that has been transformed to measure preferences about the outcomes. For EU, the transformation is
, and under certain conditions, the quantity
can be approximated by a linear combination of the mean and variance of
Y
. More recently, there has been growing interest in risk-averse control using the conditional value-at-risk (CVaR) functional. In contrast to the EU functional, the CVaR of a random variable
Y
concerns a fraction of its possible realizations. If
Y
is a continuous random variable with finite
, then the CVaR of
Y
at level
is the expectation of
Y
in the
worst cases. Here, we study the applications of risk-averse functionals to controller synthesis and safety analysis through the development of numerical examples, with an emphasis on EU and CVaR. Our contribution is to examine the decision-theoretic, mathematical, and computational tradeoffs that arise when using EU and CVaR for optimal control and safety analysis. We are hopeful that this work will advance the interpretability of risk-averse control technology and elucidate its potential benefits.
In the context of high energy costs and energy transition, the optimal use of energy resources for industrial consumption is of fundamental importance. This paper presents a decision-making structure for large consumers with flexibility to manage electricity or natural gas consumption to satisfy the demands of industrial processes. The proposed modelling energy system structure relates monthly medium and hourly short-term decisions to which these agents are subjected, represented by two connected optimization models. In the medium term, the decision occurs under uncertain conditions of energy and natural gas market prices, as well as hydropower generation (self-production). The monthly decision is represented by a risk-constrained optimization model. In the short term, hourly optimization considers the operational flexibility of energy and/or natural gas consumption, subject to the strategy defined in the medium term and mathematically connected by a regret cost function. The model application of a real case of a Brazilian aluminum producer indicates a measured energy cost reduction of USD 3.98 millions over a six-month analysis period.
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