Chance Constraints Research Articles

Water–Food Nexus System Management under Uncertainty through an Inexact Fuzzy Chance Constraint Programming Method

This study discusses the planning of a regional-scale water–food nexus (WFN) system using an inexact fuzzy chance constraint programming (IFCCP) method. The IFCCP approach can handle uncertainties expressed as interval and fuzzy parameters, as well as the preferences of decision makers. An inexact fuzzy chance constraint programming-based water–food nexus (IFCCP-WFN) model has been developed for the City of Jinan with the consideration of various restrictions related to water and land availability, as well as food and vegetable demands. Solutions for the planting areas for different crops in different periods have been generated under the different preferences of decision makers. The water resource availability would be the priority factor affecting the WFN system under demanding conditions, in which wheat cultivation would be dominated by this factor under fuzzy confidence levels of 0.2 and 0.5, and the planting area of corn would be determined by this factor under high fuzzy confidence levels (e.g., 0.8). In addition, the reliability of irrigation would decrease with increasing fuzzy confidence levels under demanding conditions, limiting the planting areas for crops and leading to a decreasing trend of the system benefit. Adequate water resources would be available for irrigation under optimistic conditions, implying no significant contributions to the planting schemes. Nevertheless, increasing food loss rates would result in more planting areas to satisfy food requirements and thus a greater system benefit under advantageous conditions. Compared with the developed IFCCP-WFN model, the interval-linear-programming-based water–food nexus (ILP-WFN) model can merely reflect the lower and upper bounds of uncertain parameters and neglects the inherent distributional information within the fuzzy parameters. Thus, the ILP-WFN model is unable to reveal the inherent impacts of the fuzzy parameters on the resulting planting strategies.

Stackelberg-Nash asymmetric bargaining-based scheduling optimization and revenue-allocation for multi-operator regional integrated energy system considering competition-cooperation relationship and source-load uncertainties

The integration of source-grid-load-storage greatly improves the energy managing flexibility of regional integrated energy systems (RIES). Considering the conflicting interests of multi-operator, this paper divides RIES into 1) energy supply center (ESC); 2) load aggregator (LA); 3) energy storage aggregator (ESA), and establishes a multi-operator two-layer optimal scheduling model by combining 1) the Stackelbelberg-Nash asymmetric bargaining game based on the genetic algorithm; and 2) the alternating direction method of multipliers algorithm (GA-ADMM). Firstly, considering the absence of historical data, trapezoidal fuzzy parameters are introduced to construct fuzzy chance constraints based on credibility theory to simulate the source-load uncertainties. Secondly, the uncertainty of demand response behavior, real-time carbon credits trading within RIES, and cooperation-competition behavior of followers (LA and ESA) are considered to simulate the asymmetric relationship between followers and leaders (ESC) based on GA. Then, based on the ADMM algorithm, a Nash asymmetric bargaining game considering energy dependency and energy supply-reception status is proposed to realize the revenue allocation between followers. Finally, simulations are implemented to verify the validity and reliability of the proposed model. Results show that: 1) Cooperation improves the energy-managing flexibility of followers, reduces their dependence on ESC, and thus enhances their game ability. 2) The real-time carbon credits trading enables zero-carbon energy interactions and natural gas replenishment within RIES. 3) Increases in the price of various energy and carbon credits can enhance cooperation between followers.

Fourth-party logistics network design with demand surge: A greedy scenario-reduction and scenario-price based decomposition algorithm

The sharing economy has offered new viable approaches for firms to tackle demand surges. In this paper, we study a novel network design problem for the fourth-party logistics network (4PLN) to cope with unpredictable demand surges, in which temporary renting of logistics resources from a sharing market is available to supplement network capacity. A chance-constrained stochastic programming model is established to minimize the overall cost for 4PLN under service level targets. To deal with the difficulties brought by the evaluation of service level, we reformulate a mixed-integer linear programming (MILP) model, that can be solved straightly, based on the sample average approximation method. Then, to address the enormous challenge posed by the coupling of the basic NP-hard network design problem and the large number of demand scenarios in the MILP version, the Scenario-Price based Decomposition Algorithm (P-DA) is designed based on the key idea of decomposing the above-coupled factors. Further, to mitigate the performance deterioration brought on by large system scale and/or sample size, we expand our base algorithm to the Greedy Scenario-Reduction and Scenario-Price based Decomposition Algorithm (GR&P-DA) through the fast processing of chance constraints by introducing a greedy method. Through computational experiments, we demonstrate the effectiveness of our proposed model and algorithms, and we analyze the impact of various model parameters, including demand level, surge magnitude, surge probability, and rental price of third-party logistics resources, on 4PLN design. Furthermore, our comparative analysis reveals that renting logistics resources to address demand surges can significantly reduce costs and discovers that deploying resources in advance does not always provide an advantage, a temporary “case-by-case” planning approach for renting will give a more cost-saving scheme when surge probability is at a low level as it avoids idle resources by not being stuck with a conservative strategy.

Computationally Tractable Gaussian Process-based Stochastic Predictive Control Using Backoffs

Current stochastic nonlinear model predictive control (SNMPC) hinges on the lack of high-fidelity models that describe the system behavior and the lack of tractable solution methods that handle chance constraints. Motivated by this, a model-and data-driven predictive control approach using Gaussian processes (GP-MDPC) is synthesized in this paper. It exploits GPs to learn the unknown dynamics and apply Taylor expansion for uncertainty propagation through probabilistic modeling. A backoff approximation method is explored to reformulate the chance constraints into tractable expressions. Finally, a finite-horizon stochastic optimal control problem (FH-SOCP) is formulated.

A Two-Step Approach to Wasserstein Distributionally Robust Chance- and Security-Constrained Dispatch

This paper considers a security constrained dispatch problem involving generation and line contingencies in the presence of the renewable generation. The uncertainty due to renewables is modeled using joint chance-constraint and the mismatch caused by contingencies and renewables are handled using reserves. We consider a distributionally robust approach to solve the chance-constrained program. We assume that samples of the uncertainty are available. Using them, we construct a set of distributions, termed ambiguity set, containing all distributions that are close to the empirical distribution under the Wasserstein metric. The chance constraint is imposed for all distributions in the ambiguity set to form the distributionally robust optimization problem. This problem is nonconvex and computationally heavy to solve exactly. We adopt a two-step approach to find an approximate solution. In the first step, we construct a polyhedral set in the space of uncertainty that contains enough mass under all distributions in the ambiguity set. This set is constructed by solving several two-dimensional distributionally robust problems. In the second step, we solve a linear robust optimization problem where the uncertain constraint is imposed for all uncertainty values lying in the polyhedral set. We demonstrate the scalability and robustness of our method using numerical experiments.

A Distributionally Robust Optimization Approach to Two-Sided Chance-Constrained Stochastic Model Predictive Control With Unknown Noise Distribution

In this work, we propose a distributionally robust stochastic model predictive control (DR-SMPC) algorithm to address the problem of multiple two-sided chance constrained discrete-time linear systems corrupted by additive noise. The prevalent mechanism to cope with two-sided chance constraints is the so-called risk allocation approach, which conservatively approximates the two-sided chance constraints with two single chance constraints by applying Bool's inequality. In this proposed DR-SMPC framework, an exact second-order cone approach is adopted to abstract the multiple two-sided chance constraints by considering the first and second moments of the noise. With the proposed DR-SMPC algorithm, the worst-case probability of violating safety constraints is guaranteed to be within a pre-specified maximum value. By flexibly adjusting this pre-specified maximum probability, the feasible region of the initial state can be increased for the SMPC problem. The recursive feasibility and convergence of the proposed DR-SMPC are rigorously established by introducing a binary initialization strategy for the nominal state. A simulation study of a single spring and double mass system was conducted to demonstrate the effectiveness of the proposed DR-SMPC algorithm.

An Analytical Polytope Approximation Aggregation of Electric Vehicles Considering Uncertainty for the Day-Ahead Distribution Network Dispatching

The integration of large-scale heterogeneous electric vehicles (EVs) into the distribution network increases the system management complexity significantly. Due to the flexible charging/discharging operation and shiftable energy consumption during the parking time, EVs possess a great dispatching potential for the distribution network. This paper develops an analytical polytope approximation (APA) aggregation model to efficiently depict the dispatchable regions of large-scale EVs with uncertainties and proposes a bilevel cooperative optimization approach for EV aggregators to participate in the distribution network day-ahead optimal scheduling. Uncertainties of EVs are modeled by chance constraints with different confidence levels to accommodate various regulatory requirements. The proposed approach can obtain large-scale EVs dispatchable capabilities with minimum flexibility loss and realize the regulation command disaggregation without violations. Numerical case studies are conducted on IEEE 33-bus and 141-bus distribution networks based on real data sets, which show that the proposed approach outperforms other state-of-art methods in terms of low operation flexibility conservatism, high system economics, and computational efficiency.

Optimizing Chance Constraint Multiple-Objective Fractional Mathematical Programming Problem Involving Dependent Random Variable

This manuscript suggests a methodology to solve chance constraint multiple-objective linear fractional mathematical programming problem in which the parameters are dependent random variables to each other. The proposed problem is formulated by taking few of the parameters as continuous dependent random variables. The proposed model cannot be solved directly by using existing methodology. Thus in order to solve the proposed model, an equivalent deterministic model is derived. The procedure to solve the proposed model is accomplished in two main steps. Initially, the proposed multiple-objective chance constraint linear fractional mathematical problem is transformed to deterministic equivalent multiple-objective linear fractional mathematical programming by the help of chance constrained method. In the second step, multiple-objective functions, which consist of fractional functions is solved by using lexicographic programming approach. Finally, an example is mentioned to illustrate the methodology.

Optimal Spatial–Temporal Scheduling for EVs With Stochastic Behaviors and Possible Inauthentic Information

This paper studies the spatial-temporal scheduling for electric vehicles with stochastic charging behaviors. A new spatial-temporal scheduling model is proposed to achieve both the spatial coordination and temporal coordination for EV users and is modeled as a non-convex optimization problem which can be solved by the coordinate descent method. Further, a new billing method of charging cost is proposed based on the Vickrey-Clarke-Groves to avoid possible inauthentic information from users. Specially, the individual rationality and incentive compatibility of the method are theoretically proved. Moreover, a stochastic optimization problem is proposed to describe the stochastic charging behavior of users, which can fit the real application better than the deterministic one. Consequently, an efficient charging strategy is obtained via chance constraint method. Simulation results and comparison studies show the effectiveness of both the model and the billing method.

Model-Based Chance-Constrained Reinforcement Learning via Separated Proportional-Integral Lagrangian.

Safety is essential for reinforcement learning (RL) applied in the real world. Adding chance constraints (or probabilistic constraints) is a suitable way to enhance RL safety under uncertainty. Existing chance-constrained RL methods, such as the penalty methods and the Lagrangian methods, either exhibit periodic oscillations or learn an overconservative or unsafe policy. In this article, we address these shortcomings by proposing a separated proportional-integral Lagrangian (SPIL) algorithm. We first review the constrained policy optimization process from a feedback control perspective, which regards the penalty weight as the control input and the safe probability as the control output. Based on this, the penalty method is formulated as a proportional controller, and the Lagrangian method is formulated as an integral controller. We then unify them and present a proportional-integral Lagrangian method to get both their merits with an integral separation technique to limit the integral value to a reasonable range. To accelerate training, the gradient of safe probability is computed in a model-based manner. The convergence of the overall algorithm is analyzed. We demonstrate that our method can reduce the oscillations and conservatism of RL policy in a car-following simulation. To prove its practicality, we also apply our method to a real-world mobile robot navigation task, where our robot successfully avoids a moving obstacle with highly uncertain or even aggressive behaviors.

Solving constrained matrix games with fuzzy random linear constraints

In real-world games, players may face an uncertain environment where fuzziness and randomness coexist. The main difficulty in dealing with games involving fuzziness and randomness arises when comparing the payoffs. The purpose of this paper is to introduce a new approach to deal with constrained matrix games where the entries of the constraint matrices are LR-fuzzy random variables. Our methodology is based on constructing a new matrix game using the chance constraint method adapted to the probability-possibility measures. First, a specific type of saddle point is defined as an equilibrium solution. Then, conditions for the existence of the proposed solution are established. Further, a technique based on second-order programming for computing the saddle point is presented. Finally, a numerical illustration of the approach is provided.

Chance-Constrained Multiple-Choice Knapsack Problem: Model, Algorithms, and Applications.

The multiple-choice knapsack problem (MCKP) is a classic NP-hard combinatorial optimization problem. Motivated by several significant real-world applications, this work investigates a novel variant of MCKP called the chance-constrained MCKP (CCMCKP), where item weights are random variables. In particular, we focus on the practical scenario of CCMCKP, in which the probability distributions of random weights are unknown and only sample data is available. We first present the problem formulation of CCMCKP and then establish the two benchmark sets. The first set contains synthetic instances, while the second set is designed to simulate a real-world application scenario of a telecommunication company. To solve CCMCKP, we propose a data-driven adaptive local search (DDALS) algorithm. Compared to existing stochastic optimization and distributionally robust optimization methods, the main novelty of DDALS lies in its data-driven solution evaluation approach, which does not make any assumptions about the underlying distributions and is highly effective even when faced with a high intensity of the chance constraint and a limited amount of sample data. Experimental results demonstrate the superiority of DDALS over the baselines on both the benchmarks. Finally, DDALS can serve as the baseline for future research, and the benchmark sets are open-sourced to further promote research on this challenging problem.

Stochastic Data-Driven Predictive Control: Regularization, Estimation, and Constraint Tightening

Data-driven predictive control methods based on the Willems’ fundamental lemma have shown great success in recent years. These approaches use receding horizon predictive control with nonparametric data-driven predictors instead of model-based predictors. This study addresses three problems of applying such algorithms under unbounded stochastic uncertainties: 1) tuning-free regularizer design, 2) initial condition estimation, and 3) reliable constraint satisfaction, by using stochastic prediction error quantification. The regularizer is designed by leveraging the expected output cost. An initial condition estimator is proposed by filtering the measurements with the one-step-ahead stochastic data-driven prediction. A novel constraint-tightening method, using second-order cone constraints, is presented to ensure high-probability chance constraint satisfaction. Numerical results demonstrate that the proposed methods lead to satisfactory control performance in terms of both control cost and constraint satisfaction, with significantly improved initial condition estimation.

Sample-Adaptive Robust Economic Dispatch With Statistically Feasible Guarantees

The high penetration of renewable energy brings significant uncertainty to the power grids. Taking economic dispatch (ED) as an example, the inaccurate prediction of renewable energy generations dramatically increases the dispatch cost and risks the power grid's reliable operation. The accurate distribution knowledge of the renewable generations enables modeling the ED as stochastic programming with joint chance constraints, which various classical methods can tackle. However, in practice, such distribution knowledge is inaccessible, and we can only observe samples from some unknown distribution. This makes conducting effective ED solely based on the observed samples challenging. It is particularly true when we need to handle the joint chance constraints. To tackle these challenges, we introduce the notions of statistical feasibility and statistically feasible ED to guarantee the satisfaction of the joint chance constraints. Specifically, we first propose a sample-adaptive robust optimization (RO) to decouple the joint constraints. We then identify that the inaccurate uncertainty set leads to RO's conservativeness, and then reconstruct the constraint-specific uncertainty sets. We design the corresponding sample-adaptive reconstruction-based RO (ReconRO) based on the reconstructed uncertainty sets to further enhance the ED's effectiveness.

A distributionally robust energy management of microgrid problem with ambiguous chance constraints and its tractable approximation method

As the penetration of intermittent renewable energy increases in microgrid systems, flexible power generation resources cause the power imbalance problem. To improve the absorption of renewable energy and to deal with its uncertainty more effectively, this paper proposes a distributionally robust approximate model for microgrid with ambiguous chance constraints (DR-CCP). First, the distributionally robust model with ambiguous chance constraints is a semi-infinite chance constrained planning problem, which is computationally difficult and inefficient, so the use of Chernoff's inequality to derive a safe tractable approximation form for the ambiguous chance constraint on the basis of a probability distribution for ambiguous sets including bounded perturbations with mean zero, transforms it into a mixed integer linear programming problem that can be directly solved using CPLEX to solve it. Then, the optimal value of the proposed model is used to approximate equivalent conditional value at risk (CVaR), and the relationship with CVaR is established. The optimal solution of our model is employed to construct multiple sets of comparison models, enhancing the richness of numerical experiments. Finally, to validate the feasibility and effectiveness of our proposed model, a series of diverse tests are performed on the IEEE 33-bus power distribution system.

Hybrid algorithm for industrial micro-grid demand response with assembly process based on distributionally robust chance constraint

Hybrid algorithm for industrial micro-grid demand response with assembly process based on distributionally robust chance constraint

A Two-stage Robust Scheduling Optimization of an Energy Hub with Multiple Chance Constraints

A Two-stage Robust Scheduling Optimization of an Energy Hub with Multiple Chance Constraints

Multiple Joint Chance Constraints Approximation for Uncertainty Modeling in Dispatch Problems

Multiple Joint Chance Constraints Approximation for Uncertainty Modeling in Dispatch Problems

Stochastic Model Predictive Control With Minimal Constraint Violation Probability for Time-Variant Chance Constraints

Stochastic Model Predictive Control With Minimal Constraint Violation Probability for Time-Variant Chance Constraints

Multi-Stage Decision Rules for Power Generation & Storage Investments With Performance Guarantees

We develop multi-stage linear decision rules (LDRs) for dynamic power system generation and energy storage investment planning under uncertainty and propose their chance-constrained optimization with performance guarantees. First, the optimized LDRs guarantee operational and carbon policy feasibility of the resulting dynamic investment plan even when the planning uncertainty distribution is ambiguous. Second, the optimized LDRs internalize the tolerance of the system planner towards the stochasticity (variance) of uncertain investment outcomes. They can eventually produce a quasi-deterministic investment plan, which is insensitive to uncertainty (as in deterministic planning) but robust to its realizations (as in stochastic planning). Last, we certify the performance of the optimized LDRs with the bound on their sub-optimality due to their linear functional form. Using this bound, we guarantee that the preference of LDRs over less restrictive – yet poorly scalable – scenario-based optimization does not lead to financial losses exceeding this bound. We use a testbed of the U.S. Southeast power system to reveal the trade-offs between the cost, stochasticity, and feasibility of LDR-based investments. We also conclude that the LDR sub-optimality depends on the amount of uncertainty and the tightness of chance constraints on operational, investment and policy variables.