Joint Chance Constraints Research Articles

Chance-constrained games with mixture distributions

In this paper, we consider an n-player non-cooperative game where the random payoff function of each player is defined by its expected value and her strategy set is defined by a joint chance constraint. The random constraint vectors are independent. We consider the case when the probability distribution of each random constraint vector belongs to a subset of elliptical distributions as well as the case when it is a finite mixture of the probability distributions from the subset. We propose a convex reformulation of the joint chance constraint of each player and derive the bounds for players’ confidence levels and the weights used in the mixture distributions. Under mild conditions on the players’ payoff functions, we show that there exists a Nash equilibrium of the game when the players’ confidence levels and the weights used in the mixture distributions are within the derived bounds. As an application of these games, we consider the competition between two investment firms on the same set of portfolios. We use a best response algorithm to compute the Nash equilibria of the randomly generated games of different sizes.

Chance constrained unit commitment approximation under stochastic wind energy

With the increasing integration of renewable energy into power networks, the scheduling and operation of power systems become more challenging, in particular because of the intermittency of wind speed. Thus, appropriate optimization tools are needed to efficiently manage the operation of modern power systems. This paper considers the unit commitment problem of power systems under high wind energy penetration. To mitigate the potential negative effects of inaccurate wind energy forecast, a chance constrained model is employed to guarantee that supply and demand for energy be balanced at each time period over a finite planning horizon. As joint chance constraint is known to be difficult to handle analytically, two approximation approaches, namely quantile- and p-efficiency-based, are proposed and benchmarked against the well-known scenario-based approximation. The results indicate that the proposed schemes provide feasible solutions and tight lower bounds of the true operation cost within a much shorter time frame than the scenario-based counterpart.

Lagrangian relaxation based heuristics for a chance-constrained optimization model of a hybrid solar-battery storage system

We develop a stochastic optimization model for scheduling a hybrid solar-battery storage system. Solar power in excess of the promise can be used to charge the battery, while power short of the promise is met by discharging the battery. We ensure reliable operations by using a joint chance constraint. Models with a few hundred scenarios are relatively tractable; for larger models, we demonstrate how a Lagrangian relaxation scheme provides improved results. To further accelerate the Lagrangian scheme, we embed the progressive hedging algorithm within the subgradient iterations of the Lagrangian relaxation. We investigate several enhancements of the progressive hedging algorithm, and find bundling of scenarios results in the best bounds. Finally, we provide a generalization for how our analysis extends to a microgrid with multiple batteries and photovoltaic generators.

Chance-constrained optimal power flow based on a linearized network model

In this study, the focus is put on improving the solution process for a chance-constrained alternating current (AC) optimal power flow model. Firstly, a novel chance-constrained optimal power flow model is proposed and introduced based on a linearized network model with explicit bounds on voltage magnitude, reactive power, and apparent power flow, which can achieve a desirable computation performance from the perspective of modeling. In particular, the linearization imposed on the apparent power flow will induce joint chance constraints, making the deterministic transformation of the chance constraints challenging to perform. Thus, this paper adopts an improved Boole’s inequality to address this issue. In a further step, as a substitute to the analytical reformulation method and the Monte Carlo simulation method, the three-point estimation and the Cornish-Fisher series expansion are combined to efficiently conduct uncertainty evaluation on the AC power flow recovered solution, while ensuring all chance constraints in the stochastic scenarios are satisfied. If any violation probability exceeds the given value, the corresponding constraint bounds will be tightened, and an updated deterministic linearly-constrained model needs to be solved again. This process is repeated until all the convergence conditions are reached. Case studies on two test systems verified the characteristics/advantages of the proposed chance-constrained optimal power flow modeling and solution approach.

Energy and reserve dispatch with distributionally robust joint chance constraints

We develop a two-stage stochastic program for energy and reserve dispatch of a joint power and gas system with a high penetration of renewables. Data-driven distributionally robust chance constraints ensure that there is no load shedding and renewable spillage with high probability. We solve this problem efficiently using conditional value-at-risk approximations and linear decision rules. Out-of-sample experiments show that this model dominates the corresponding stochastic program without chance constraints that models the effects of load shedding and renewable spillage explicitly.

Games with distributionally robust joint chance constraints

This paper studies an n-player non-cooperative game where each player has expected-value payoff function and chance-constrained strategy set. We consider the case where the row vectors defining the constraints are independent random vectors whose probability distributions are not completely known and belong to a certain distributional uncertainty set. The chance-constrained strategy sets are defined using a distributionally robust framework. We consider one density based uncertainty set and four two-moments based uncertainty sets. One of the considered uncertainty sets is based on a nonnegative support. Under the standard assumptions on the players’ payoff functions, we show that there exists a Nash equilibrium of a distributionally robust chance-constrained game for each uncertainty set. As an application, we study Cournot competition in electricity market and perform the numerical experiments for the case of two electricity firms.

Constrained Q-Learning for Batch Process Optimization

Chemical process optimization and control often require satisfaction of constraints for safe operation. Reinforcement learning (RL) has been shown to be a powerful control technique that can handle nonlinear stochastic optimal control problems. Despite this promise, RL has yet to see significant translation to industrial practice due to its inability to satisfy state constraints. This work aims to address this challenge. We propose an “oracle”-assisted constrained Q-learning algorithm that guarantees the satisfaction of joint chance constraints with high probability, which is required for safety critical tasks. To that end, constraint tightening (backoffs) are introduced, which are adjusted using Broyden’s method, hence making the backoffs self-tuned. This results in a general methodology that can be integrated into approximate dynamic programming-based algorithms to guarantee constraint satisfaction with high probability. Finally, a case study is presented to compare the performance of the proposed approach with that of model predictive control (MPC). The superior performance of the proposed algorithm, in terms of constraint handling, signifies a step toward the use of RL in real world optimization and control of systems, where constraints are critical in ensuring safety.

Data-driven distributionally robust chance-constrained optimization with Wasserstein metric

We study distributionally robust chance-constrained programming (DRCCP) optimization problems with data-driven Wasserstein ambiguity sets. The proposed algorithmic and reformulation framework applies to all types of distributionally robust chance-constrained optimization problems subjected to individual as well as joint chance constraints, with random right-hand side and technology vector, and under two types of uncertainties, called uncertain probabilities and continuum of realizations. For the uncertain probabilities (UP) case, we provide new mixed-integer linear programming reformulations for DRCCP problems. For the continuum of realizations case with random right-hand side, we propose an exact mixed-integer second-order cone programming (MISOCP) reformulation and a linear programming (LP) outer approximation. For the continuum of realizations (CR) case with random technology vector, we propose two MISOCP and LP outer approximations. We show that all proposed relaxations become exact reformulations when the decision variables are binary or bounded general integers. For DRCCP with individual chance constraint and random right-hand side under both the UP and CR cases, we also propose linear programming reformulations which need the ex-ante derivation of the worst-case value-at-risk via the solution of a finite series of linear programs determined via a bisection-type procedure. We evaluate the scalability and tightness of the proposed MISOCP and (MI)LP formulations on a distributionally robust chance-constrained knapsack problem.

Green Traffic Off-Loading Over Uncertain Shared Spectrums With End-to-End QoS Guarantee

The fast proliferation of Internet of Things (IoT) exacerbates the current spectrum shortage. As a promising solution, licensed spectrum sharing has received great attentions recently. However, due to the unpredictable activities of incumbent users, the availability of the shared spectrum is uncertain, making the quality of service (QoS) provisioning based on it a difficult task, especially under multi-hop scenarios. In this paper, considering the energy-efficiency issue, we propose an end-to-end (E2E) QoS guaranteed green traffic off-loading scheme over the uncertain shared spectrums in multi-hop networks. Specifically, we formulate it into a two-step problem, an off-loaded traffic maximization (OTM) problem and an energy-efficiency improvement (EEI) problem. The former one is to determine the admitted sessions. To achieve the E2E QoS guarantee under the uncertain spectrum supply, we develop a concept of equivalent link capacity, and construct a risk-averse joint chance constraint. Since the formulated problem is intractable, we first develop a data-driven conservative transformation to make the joint chance constraint a set of deterministic individual ones. Then, we design a convex-hull-based and a substitution-based linearization approach to make all the constraints tractable. For the second step, the EEI problem with the same constraints as the OTM problem is designed to refine the resource scheduling for the admitted sessions to optimize the energy-efficiency. To deal with the fractional-form energy-efficiency objective, a Dinkelbach method based iterative algorithm is further developed. Numerical results have shown the effectiveness of the proposed scheme.

Joint chance constrained input shaping

This paper addresses the problem of robust input shaping for rest to rest maneuvers of linear systems with parametric uncertainties. A stochastic optimization problem with a quadratic cost function is posed which probabilistically penalizes excursions of terminal time state values from desired values due to uncertainties. This quadratic cost represented by a hyper-sphere, is approximated by a hyper-polygon to permit a convex problem formulation, where the joint chance constraints are represented using statistics of the uncertain terminal states. Polynomial Chaos is used as an uncertainty quantification tool to estimate the first two moments of the stochastic state variables necessary for the implementation of the chance constraints. The solution to the optimization problem yields the desired input shaper. Several analytical methods of dealing with the joint chance constraints are investigated and compared on illustrative benchmark examples. The framework presented permits the users to trade-off performance for robustness to any desired level.

Cost-Based attraction recommendation for tour operators under stochastic demand

Attraction recommendation is a key functionality offered by tour operators. The main stakeholders of attraction recommendations include tourists and tour operators. The former use recommendations to make travel decisions, and the latter manage recommendations for their own benefits. Most existing attraction recommendation methods focus on providing recommendations that best match tourists’ preferences, yet overlook the benefits of tour operators. To address this gap, we conduct a two-phase study that focuses on cost-based attraction recommendations under stochastic tourist demand from the perspective of tour operators. In the first phase, we obtain preliminary recommendation solutions that best match tourists’ topic preferences. Then, with the consideration of cost factors, a stochastic programming model with a joint chance constraint is proposed to refine the preliminary recommendation solutions in the second phase, and a tractable model based upon Sample Average Approximation (SAA) method is further presented. To assess the performance of the proposed method, comprehensive experiments are conducted with both simulated instances and real-world data. The results indicate that the proposed optimization model can significantly reduce tour operators’ recommendation cost while maintaining a high service level and tourist satisfaction.

Two‐stage chance‐constrained unit commitment based on optimal wind power consumption point considering battery energy storage

Wind power generation has developed rapidly in recent decades due to low carbon emissions. However, the significant uncertainty makes it uneconomic or even unreliable to be consumed, which hinders the development of wind energy. To guarantee the minimum wind utilisation level without jeopardising system reliability and cost-effectiveness, this study proposes a concept of optimal wind power consumption point. Based on that, a two-stage chance-constrained unit commitment model is presented to co-optimise the day-ahead energy and reserve schedules, which achieves a reasonable trade-off between robustness and costs. The battery energy storage is also investigated to enhance system flexibility and promote wind consumption. The joint chance constraint is dealt with through a sample average approximation method in bilinear forms. The resulting large-scale mixed-integer programming is decomposed into the master and subproblem formulations and then solved iteratively by the developed bilinear Benders decomposition (BBD) method. To achieve computational tractability, several techniques are used to enhance the convergence property of BBD and accelerate the solution process, with a novel optimality-check-only bilinear Benders decomposition method proposed. Case studies on six-bus, IEEE 118-bus and 236-bus systems demonstrate the effectiveness of the proposed model and algorithm.

DC Optimal Power Flow With Joint Chance Constraints

Managing uncertainty and variability in power injections has become a major concern for power system operators due to increasing levels of fluctuating renewable energy connected to the grid. This work addresses this uncertainty via a joint chance-constrained formulation of the DC optimal power flow (OPF) problem, which satisfies all the constraints jointly with a pre-determined probability. The few existing approaches for solving joint chance-constrained OPF problems are typically either computationally intractable for large-scale problems or give overly conservative solutions that satisfy the constraints far more often than required, resulting in excessively costly operation. This paper proposes an algorithm for solving joint chance-constrained DC OPF problems by adopting an S$\ell _1$QP-type trust-region algorithm. This algorithm uses a sample-based approach that avoids making strong assumptions on the distribution of the uncertainties, scales favorably to large problems, and can be tuned to obtain less conservative results. We illustrate the performance of our method using several IEEE test cases. The results demonstrate the proposed algorithm's advantages in computational times and limited conservativeness of the solutions relative to other joint chance-constrained DC OPF algorithms.

Compacting multistage stochastic programming models through a new implicit extensive framework

Deterministic equivalent models reformulate optimization problems from a computational perspective. Nonetheless, these models become computationally intractable quickly when the number of stages increase. In this context, a framework to reduce the size of scenario tree and multistage stochastic optimization problems is proposed. Scenario trees are generated using the Knuth transformation for a more compact representation. Moreover, the optimization model is described by using an implicit extensive form approach. The framework is tested in an asset-liability management multistage stochastic model with joint chance constraints, making it possible to acquire the optimal solution for large instances without any relaxation or decomposition mechanism.

Sustainable supplier selection and order allocation: Distributionally robust goal programming model and tractable approximation

A challenge of planning real-life sustainable supplier selection and order allocation (SS/OA) problems for purchasing companies is to gather the extensive data and exact distributions of input data. Motivated by this challenge, this paper addresses imprecise probability distributions of uncertain per unit cost, CO2 emissions, demand, supply capacity and the minimum order quality and characterizes these distribution uncertainty by ambiguity sets including the true distributions. Moreover, the purchasing company decisions to be optimized simultaneously include four conflicting objectives regarding cost, CO2 emissions, society and suppliers’ comprehensive value while considering risk measures incurred by cost and CO2 emissions to achieve company sustainability. To help decision makers formulate practicable policy, a novel distributionally robust sustainable SS/OA goal programming model is developed with expected constraints and joint chance constraints. The proposed optimization model can balance multiple conflicting objectives, and effectively solve our sustainable SS/OA problem. More importantly, we structure ambiguous distributions sets, and thus derive the computationally tractable approximation form of the proposed practical model. Finally, we illustrate our optimization method through a case study about a steel company, conduct a thorough inquiry into the effect of uncertainty and summarize the management implications of the results.

Chemical Process Design Taking into Account Joint Chance Constraints

A method for solving the problems of chemical process design taking into account the incompleteness and inaccuracy of the initial information presented in the form of one-stage optimization problems with joint chance constraints is proposed. Joint chance constraints guarantee the required probability level of performance, unlike individual chance constraints, which provide only an estimate of the level of performance. However, the calculation of joint chance constraints is much more complicated than that of individual chance constraints. This work proposes a method for solving the problems of chemical process design with allowance for joint chance constraints, which eliminates the operation of the direct calculation of constraints. The method involves converting a stochastic programming problem into a sequence of deterministic nonlinear programming problems and can significantly reduce the time it takes to solve the problem. The effectiveness of the proposed approach is illustrated by model examples.

Resource and Network Management for Satellite Communications Systems: A Chance-Constrained Approach

In this paper, we propose an optimization framework for the resource and network management in large-scale satellite communications (SATCOM) systems with stochastic and time-varying communication requests. Because further expansions of SATCOM systems are expected in the near future, it is of practical importance to develop management methodologies for efficiently and robustly operating large-scale SATCOM systems. To address this problem, in this paper, we formulate a SATCOM network optimization problem within the framework of the chance-constrained model predictive control. In this problem, we require that a joint chance constraint on the bandwidth loss rate is satisfied with a user-specified performance and probability. Because the joint chance constraint is nonlinear in design variables, we relax the constraint to a linear deterministic one for reformulating the original problem into a relaxed deterministic problem. We numerically verify that the proposed method outperforms a baseline methodology and therefore allows us to efficiently manage large-scale SATCOM systems.

Automatic Exploration Process Adjustment for Safe Reinforcement Learning with Joint Chance Constraint Satisfaction

In reinforcement learning (RL) algorithms, exploratory control inputs are used during learning to acquire knowledge for decision making and control, while the true dynamics of a controlled object is unknown. However, this exploring property sometimes causes undesired situations by violating constraints regarding the state of the controlled object. In this paper, we propose an automatic exploration process adjustment method for safe RL in continuous state and action spaces utilizing a linear nominal model of the controlled object. Specifically, our proposed method automatically selects whether the exploratory input is used or not at each time depending on the state and its predicted value as well as adjusts the variance-covariance matrix used in the Gaussian policy for exploration. We also show that our exploration process adjustment method theoretically guarantees the satisfaction of the constraints with the pre-specified probability, that is, the satisfaction of a joint chance constraint at every time. Finally, we illustrate the validity and the effectiveness of our method through numerical simulation.

A Decomposition Algorithm for the Two-Stage Chance-Constrained Operating Room Scheduling Problem

The required time for surgical interventions in operating rooms (OR) may vary significantly from the predicted values depending on the type of operations being performed, the surgical team, and the patient. These deviations diminish the efficient utilization of OR resources and result in the disruption of projected surgery start times. This paper proposes a two-stage chance-constrained model to solve the OR scheduling problem under uncertainty. The goal is to minimize the costs associated with OR opening and overtime as well as reduce patient waiting times. The risk of OR overtime is controlled using chance constraints. Numerical experiments show that the proposed model provides a better trade-off between minimizing costs and reducing solution variability when compared to two existing models in the literature. It is also shown that the three models converge as the overtime probability threshold approaches one. Moreover, it is observed that the individual chance constraints result in the opening of fewer rooms, lower waiting times, and shorter solution times when compared to that of joint chance constraints. A decomposition algorithm is applied that solves large test instances of the OR scheduling problem, that of which is known to be NP-hard. Strong valid inequalities are derived in order to accelerate the convergence speed. The proposed approach outperformed both a commercial solver and a basic decomposition algorithm after solving all test instances with up to 89 surgeries and 20 ORs in less than 48 minutes.

Constrained Reinforcement Learning for Dynamic Optimization under Uncertainty

Dynamic real-time optimization (DRTO) is a challenging task due to the fact that optimal operating conditions must be computed in real time. The main bottleneck in the industrial application of DRTO is the presence of uncertainty. Many stochastic systems present the following obstacles: 1) plant-model mismatch, 2) process disturbances, 3) risks in violation of process constraints. To accommodate these difficulties, we present a constrained reinforcement learning (RL) based approach. RL naturally handles the process uncertainty by computing an optimal feedback policy. However, no state constraints can be introduced intuitively. To address this problem, we present a chance-constrained RL methodology. We use chance constraints to guarantee the probabilistic satisfaction of process constraints, which is accomplished by introducing backoffs, such that the optimal policy and backoffs are computed simultaneously. Backoffs are adjusted using the empirical cumulative distribution function to guarantee the satisfaction of a joint chance constraint. The advantage and performance of this strategy are illustrated through a stochastic dynamic bioprocess optimization problem, to produce sustainable high-value bioproducts.