Chance Constraints Research Articles

Studies on effective solar photovoltaic integration in distribution network with a blend of Monte Carlo simulation and artificial hummingbird algorithm

AbstractIn this paper, the two level stochastic optimisation approach has been suggested. In the lower level, the probability distribution functions (pdfs) for bus voltages and branch currents have been determined using the Monte Carlo simulation (MCS) to be employed in chance‐constrained probabilistic optimisation by taking into account solar radiation and power consumption uncertainties in the distribution networks (DNs). In the upper level, artificial hummingbird algorithm (AHA) handles the expected power loss minimisation subjected to chance constraints, which are related to bus voltages and branch currents, by optimising photovoltaic (PV) system capacities. This research examines the effect of uncertainties in PV system performing under diverse solar radiation and varying PV penetration level scenarios on expected power losses with stochastic DN limits. The stochastic optimisation approach has been compared with the deterministic method for observing the efficiency with optimal power usage. This research improves the knowledge base for optimal PV installation in DN by combining AHA with MCS and emphasising chance‐constrained methods. To indicate the efficacy of proposed strategy, the optimisation outcomes are tested utilising MCS under various uncertainty circumstances and DN parameters are assessed in terms of probabilities of exceeding limitations. The results are compared with the application of firefly algorithm (FA) using stochastic assessment and simulations. The simulation results show that the AHA technique outperforms the FA method in terms of effectively minimising power losses with less simulation time.

Low-Carbon Water–Rail–Road Multimodal Routing Problem with Hard Time Windows for Time-Sensitive Goods Under Uncertainty: A Chance-Constrained Programming Approach

In this study, a low-carbon freight routing problem for time-sensitive goods is investigated in the context of water–rail–road multimodal transportation. To enhance the on-time transportation of time-sensitive goods, hard time windows are employed to regulate both pickup and delivery services at the start and end of their transportation. The uncertainty of both the demand for time-sensitive goods and the capacity of the transportation network are modeled using L-R triangular fuzzy numbers in the routing process to make the advanced routing more feasible in the actual transportation. Based on the carbon tax policy, a fuzzy linear optimization model is established to address the proposed problem, and an equivalent chance-constrained programming formulation is then obtained to make the solution to the problem attainable. A numerical experiment is carried out to verify the feasibility of incorporating the carbon tax policy, uncertainty, and water–rail–road multimodal transportation to optimize the low-carbon freight routing problem for time-sensitive goods. Furthermore, a multi-objective optimization is used to reveal that lowering the transportation costs, reducing the carbon emissions, and avoiding the risk are in conflict with each in the routing. We also analyze the sensitivity of the optimization results concerning the confidence level of the chance constraints and the uncertainty degree of the uncertain demand and capacity. Based on the numerical experiment, we draw several conclusions to help the shipper, receiver, and multimodal transportation operator to organize efficient water–rail–road multimodal transportation for time-sensitive goods.

Distributionally robust single machine scheduling with release and due dates over Wasserstein balls

Single machine scheduling aims at determining the job sequence with the best desired performance, and provides the basic building block for more advanced scheduling problems. In the present study, a single machine scheduling model with uncertain processing time is considered by incorporating the job release time and due date. The job processing time follows unknown probability distribution, and can be estimated via the historical data. To model the uncertainty, the processing time distribution is defined over a Wasserstein ball ambiguity set, which covers all feasible probability distributions within the confidence radius of the empirical distribution, known as the center of the ball. Then a data-driven distributionally robust scheduling model is constructed with individual chance constraints. In particular, two equivalent reformulations are derived with respect to the ℓ1-norm and ℓ2-norm metrics of the Wasserstein ball, namely, a mixed-integer linear programming and a mixed-integer second order cone programming model, respectively. To accelerate the solving of large-scale instances, a tailored constraint generation algorithm is introduced. In the numerical analysis, the proposed distributionally robust scheduling approach is compared with the state-of-the-art methods in terms of out-of-sample performance.

Chance-constrained stochastic optimal control of epidemic models: A fourth moment method-based reformulation

This work proposes a methodology for the reformulation of chance-constrained stochastic optimal control problems that ensures reliable uncertainty management of epidemic outbreaks. Specifically, the chance constraints are reformulated in terms of the first four moments of the stochastic state variables through the so-called fourth moment method for reliability. Moreover, a spectral technique is employed to obtain surrogate models of the stochastic state variables, which enables the efficient computation of the required statistics. The practical implementation of the proposed approach is demonstrated via the optimal control of two different stochastic mathematical models of the COVID-19 transmission. The numerical experiments confirm that, unlike those reformulations based on the Chebyshev–Cantelli’s inequality, the proposed method does not exhibit the undesired outcomes that are typically observed when a higher precision is required for the risk level associated to the given chance constraints.

Covariance Control for Uncrewed Aircraft Systems Under Correlated Uncertainty

Uncrewed aircraft systems and advanced air mobility are associated with uncertain conditions, such as wind prediction errors, and they may deviate from dedicated airspaces referred to as corridors. However, further studies on these aspects are required. This study focuses on the covariance control for a stochastic discrete-time linear system under correlated uncertainties. A covariance control problem with chance constraints can be formulated as a convex-programming problem. Numerical simulations of uncrewed aircraft systems path planning can be used to compare two control laws: covariance control laws that consider correlated and noncorrelated noise. Numerical simulations were conducted to verify that the covariance control law considering correlated noise generates a trajectory that appropriately satisfies the constraints. Consequently, the effect of considering correlated noise in the covariance control law was clarified. In addition, covariance control has been demonstrated to generate appropriate trajectories for various state constraints. Moreover, numerical simulations indicated that covariance control facilitated the control of uncertainty, and the effectiveness of the covariance control was evaluated. The insights gained from the results of this study can inform enhanced uncertainty management for improving reliability and safety in real-world applications of uncrewed aircraft systems and advanced air mobility.

Signal-devices management and data-driven evidential constraints based robust dispatch strategy of virtual power plant

Ensuring the safety and reliability of energy systems is very important in power grid operation. However, inherent intricacies and uncertainties of modern-day power systems, such as the mismatch between signal frequency and equipment response speed and the stochasticity associated with renewable energy and load forecasts, create significant challenges for generation dispatch planning. This study proposes a robust optimization approach with constraints based on signal-devices management and data-driven evidential distribution constraints. The first stage of this approach is to decompose and recombine the net power demand according to the Hilbert-Huang transform and device capability. The signal-device management constraints are then established based on this. The second stage formulates an evidential constraint based on data-driven evidential distribution and chance constraints in a distributionally robust optimization framework that caters to the stochasticity associated with renewable energy and load forecasts. The proposed model is validated on a virtual power plant to assess the approach’s efficacy for different dispatching scenarios. Penalty-based sensitivity analysis provides further insights into the proposed method’s performance for varying levels of CO2 emissions. Simulation results demonstrate that system flexibility becomes increasingly crucial for maintaining system stability and security as the penetration of renewable energy grows. Compared with chance constraint, the proposed data-driven evidential constraint effectively enables the optimization framework to handle stochasticity after sacrificing 0.62% more economic loss and 0.7% more environmental loss. Excessively high penalty parameters for CO2 do not promote economic development, resulting in 21.08% and 52.51% more economic and environmental losses without obvious environmental protection.

Robust concave utility maximization over chance constraints

This paper first studies an expected utility problem with chance constraints and incomplete information on a decision maker’s utility function. The model maximizes the worst-case expected utility of random outcome over a set of concave functions within a novel ambiguity set, while the underlying probability distribution is known with the assumption of a discretization of possible realizations for the uncertainty parameter. To obtain computationally tractable formulations, we employ a discretization approach to derive a max–min chance-constrained approximation of this problem. This approximation is further reformulated as a mixed-integer program. We show that the discrete approximation converges to the true counterpart under mild assumptions. We also present a row generation algorithm for optimizing the max–min program. A computational study for a bin-packing problem and a multi-item newsvendor problem is conducted to demonstrate the benefit of the proposed framework and the computational efficiency of our algorithm. We find that the row generation algorithm can significantly reduce the computational time, and our robust policy could achieve a better out-of-sample performance when compared with the non-robust policy and the one without the chance constraints.

Optimal inventory decisions of infrequently needed spare parts for maintenance with inaccurate demand forecasts and limited storage space

Abstract In modern manufacturing workshops, maintaining adequate inventory for spare parts is an effective strategy to satisfy maintenance requirements. However, due to the stochastic nature of failures, spare parts often have characteristics of infrequent usage and inaccurate forecasted demand. Additionally, the multitude diversity of spare parts and limited storage space in warehouse poses challenges to inventory decisions regarding the types and quantity of spares. Thus, this paper investigates the inventory decision optimization problem of infrequently needed spare parts for maintenance within limited storage space and inaccurate forecasted demand. This paper constructed an integer programming model based on chance constraints to address the uncertainty of the storage space occupied by the spare parts. The main contributions of this paper are as follows. First, we investigate the inventory decision optimization problem within limited storage space and inaccurate forecasted demand, which are rarely considered in the optimization problem of the spare parts for maintenance. Second, we proposed a method for simplifying the computation of chance constraints based on a transformation of the Central Limit Theorem. Third, we proposed a heuristic iterate algorithm to search the optimal inventory decision by minimizing the overall downtime and storage costs. Using data from the workshop of an automotive enterprise, a case study is conducted by comparison with the Genetic Algorithm.

Designing a resilient supply chain network: A multi-objective data-driven distributionally robust optimization method

Uncertainty of supply chain disruption poses a significant challenge to decision-making in many real-life problems. In this paper, a new multi-objective data-driven distributionally robust optimization (MDDRO) model for the resilient supply chain network (RSCN) design problem under disruption scenario is developed, which includes minimizing the total cost, carbon emissions, and delivery time. In the context of supply chain disruption, it is important self-evidently that products can be supplied normally and the practical demand can be met. In handling the partial probability distribution information of the considered uncertain demand, this paper uses a data-driven Wasserstein-Moment ambiguity set (WMAS), which incorporate the Wasserstein metric and Moment information, and a robust counterpart to transform the developed model into a tractable approximation form. The finite-sample performance guarantee of the optimal solution of MDDRO model with Wasserstein-Moment ambiguous chance constraint is given. An accelerated Branch and cut algorithm (BCA) is constructed to solve the MDDRO model. Finally, through the investigation of a real-life case and sensitivity analysis of key parameters, some managerial insights are put forward.

A novel two-stage network data envelopment analysis model for kidney allocation problem under medical and logistical uncertainty: a real case study.

Organ transplantation is one of the most complicated and challenging treatments in healthcare systems. Despite the significant medical advancements, many patients die while waiting for organ transplants because of the noticeable differences between organ supply and demand. In the organ transplantation supply chain, organ allocation is the most significant decision during the organ transplantation procedure, and kidney is the most widely transplanted organ. This research presents a novel method for assessing the efficiency and ranking of qualified organ-patient pairs as decision-making units (DMUs) for kidney allocation problem in the existence of COVID-19 pandemic and uncertain medical and logistical data. To achieve this goal, two-stage network data envelopment analysis (DEA) and credibility-based chance constraint programming (CCP) are utilized to develop a novel two-stage fuzzy network data envelopment analysis (TSFNDEA) method. The main benefits of the developed method can be summarized as follows: considering internal structures in kidney allocation system, investigating both medical and logistical aspects of the problem, the capability of expanding to other network structures, and unique efficiency decomposition under uncertainty. Moreover, in order to evaluate the validity and applicability of the proposed approach, a validation algorithm utilizing a real case study and different confidence levels is used. Finally, the numerical results indicate that the developed approach outperforms the existing kidney allocation system.

Optimizing the economic dispatch of weakly-connected mini-grids under uncertainty using joint chance constraints

AbstractIn this paper, we deal with a renewable-powered mini-grid, connected to an unreliable main grid, in a Joint Chance Constrained (JCC) programming setting. In several rural areas in Africa with low energy access rates, grid-connected mini-grid system operators contend with four different types of uncertainties: forecasting errors of solar power and load; frequency and outages duration from the main-grid. These uncertainties pose new challenges to the classical power system’s operation tasks. Three alternatives to the JCC problem are presented. In particular, we present an Individual Chance Constraint (ICC), Expected-Value Model (EVM) and a so called regular model that ignores outages and forecasting uncertainties. The JCC model has the capability to guarantee a high probability of meeting the local demand throughout an outage event by keeping appropriate reserves for Diesel generation and battery discharge. In contrast, the easier to handle ICC model guarantees such probability only individually for different time steps, resulting in a much less robust dispatch. The even simpler EVM focuses solely on average values of random variables. We illustrate the four models through a comparison of outcomes attained from a real mini-grid in Lake Victoria, Tanzania. The results show the dispatch modifications for battery and Diesel reserve planning, with the JCC model providing the most robust results, albeit with a small increase in costs.

Fuzzy credibility chance-constrained multi-objective optimization for multiple transactions of electricity–gas–carbon under uncertainty

This paper proposes a fuzzy credibility chance-constrained multi-objective optimization model to optimize market transactions in the electricity–gas–carbon sectors under uncertainty. The model aims to maximize the profit of an integrated energy service provider by incorporating a ladder-type carbon trading mechanism, which adjusts carbon prices based on emission levels, and detailed multi-energy flow constraints. To effectively manage uncertainties in electricity, gas, and carbon markets, we derive credibility distributions for uncertain variables and introduce fuzzy credibility chance constraints—tools that assess the likelihood of meeting multiple transactions under uncertainty. The proposed model hedges the risks associated with multiple uncertainties while maximizing the credibility of expected costs, effectively balancing risk and cost. Through simulation analysis on an IEEE 33-node power network and a 32-node heat network, the proposed model achieved a 9.8% reduction in total system cost and an 8.5% reduction in total carbon emissions. Additionally, the model effectively determined credibility levels under different risk preferences, demonstrating its robustness in enhancing electricity–gas–carbon trading and promoting a low-carbon economy. This research provides a novel planning method for formulating trading strategies in multi-energy markets, with significant real-world implications for energy management and environmental sustainability.

Robust Optimization Research of Cyber-Physical Power System Considering Wind Power Uncertainty and Coupled Relationship.

To mitigate the impact of wind power uncertainty and power-communication coupling on the robustness of a new power system, a bi-level mixed-integer robust optimization strategy is proposed. Firstly, a coupled network model is constructed based on complex network theory, taking into account the coupled relationship of energy supply and control dependencies between the power and communication networks. Next, a bi-level mixed-integer robust optimization model is developed to improve power system resilience, incorporating constraints related to the coupling strength, electrical characteristics, and traffic characteristics of the information network. The upper-level model seeks to minimize load shedding by optimizing DC power flow using fuzzy chance constraints, thereby reducing the risk of power imbalances caused by random fluctuations in wind power generation. Furthermore, the deterministic power balance constraints are relaxed into inequality constraints that account for wind power forecasting errors through fuzzy variables. The lower-level model focuses on minimizing traffic load shedding by establishing a topology-function-constrained information network traffic model based on the maximum flow principle in graph theory, thereby improving the efficiency of network flow transmission. Finally, a modified IEEE 39-bus test system with intermittent wind power is used as a case study. Random attack simulations demonstrate that, under the highest link failure rate and wind power penetration, Model 2 outperforms Model 1 by reducing the load loss ratio by 23.6% and improving the node survival ratio by 5.3%.

Configuration and optimisation of a green closed-loop supply chain with delivery time and green investment considering government subsidy under meta-heuristics algorithms

Consumers' environment consciousness, together with the fierce rivalries among companies, has attracted increasing attention from scholars and industries to the configuration and optimisation of green closed-loop supply chain. However, current studies of closed-loop supply chains rarely consider both the uncertainty from the market and the quality from the recycled used products. Therefore, this article constructs a green closed-loop supply chain total profit model that considers government subsidies, delivery times, green investment and the above uncertainties. Fuzzy chance constraint method is used to solve the uncertain demand; genetic and particle swarm optimisation algorithms are adopted to calculate the model. This study finds (1) enterprises can formulate flexible and adaptable strategies based on different demands; (2) their delivery time decisions rely on both consumer demand and their logistics development levels; (3) enterprises should adjust their delivery times and green investment decisions according to consumers' different sensitivities; (4) enterprises can increase profits by recycling products within a certain range of greenness; (5) for improving recycling and remanufacturing rates, the government can provide secondary subsidies to enterprises for product recycling within the green range of 0.35 ∼ 0.40. This study contributes to the optimisation of strategies developed by enterprises under conditions of subsidies and multiple uncertainties.

Integrated aircraft routing and cargo routing problem for combination airlines

The combination airlines operate both passenger aircraft and freighter aircraft to meet passenger and cargo demand. At present, combination airlines employ a sequential approach to allocating their capacity for passenger and cargo demand. Nevertheless, implementing an integrated resource allocation procedure has the potential to improve overall resource allocation efficiency. In this paper, we introduce an integrated model to help combination airlines integrate their aircraft routing and cargo routing decisions to maximize the expected overall profits derived from both passenger and cargo demand. We considered the stochastic nature of passenger baggage and proposed a set of individual chance constraints to ensure the robustness of the integrated solution. We reformulate the chance constraints using piecewise linear approximation to ensure solution efficiency. In addition, we proposed a column-and-row generation based solution approach that removes the through-connection related constraints at the beginning of the solution process and then adds the columns and rows during the iterations as needed. We proved that the proposed column-and-row generation approach can obtain an optimal solution for the LP relaxation problem. The model and the solution approach were tested in a number of scenarios obtained from a major Chinese combination airline. The computational results show that the combination airline can improve their expected profits by integrating capacity allocation. The results also demonstrated that the proposed column-and-row generation solution approach can decrease the solution time of the integrated model. These findings indicate that the model and the solution method are useful and efficient tools for combination airlines when planning their aircraft and cargo routes.

Risk-averse distributionally robust optimization for construction waste reverse logistics with a joint chance constraint

The uncertainty of the amount of construction and demolition waste (CDW) generation affects the CDW reverse logistics network service level. We investigate a CDW reverse logistics network location-routing problem considering uncertainties. To minimize the total social cost, a two-stage risk-averse distributionally robust optimization model is developed, which aims to optimize the location and number of CDW disposal facilities and the CDW transportation scheme. We introduce the mean-conditional value at risk measure and a joint chance constraint into our model to consider the government’s risk aversion. The above model is approximated as a standard second-order cone programming model (SOCP) considering a special case. To exactly solve the SOCP, we design an outer approximation algorithm. Several performance tests and a case study are proposed, and sensitivity analysis provides helpful managerial insights.

Renewable energy-powered water desalination and treatment network under wind power and water demand uncertainty: A possibilistic chance-constrained programming

Given the scarcity of freshwater resources, the growing significance of desalination is undeniable. It holds immense potential, particularly in regions grappling with severe water shortages. However, desalination's Achilles heel lies in its voracious energy appetite, requiring roughly ten times more energy than wastewater treatment. Moreover, the prevalent use of fossil fuels in desalination plants poses concerning issues like environmental pollution, fossil fuel depletion, and rising costs. The present study has designed an integrated Water desalination and treatment Network that includes a number of desalination facilities, storage centers, wind farms, and wastewater treatment facilities. The water desalination and treatment network has been structured using a Mixed-Integer Linear Programming (MILP) model, considering uncertainties in wind power and water demand. Employing a chance constraint probabilistic programming approach, this model ensures robustness and balances conservatism with investment attractiveness. It aims to enhance resilience against fluctuations in wind energy and water demand within the water and energy supply chain network. The study applied this model to optimize the locations of desalination plants, treatment centers, and storage facilities. This integrated model ensures autonomy, eliminating the need for external water and energy sources while reliably meeting regional demands. In the context of the Makran coasts case study, our comprehensive mathematical model demonstrates an optimal allocation with 96.67 % attributed to fixed costs and only 3.33 % to variable costs. Moreover, this model precisely optimizes the locations of two desalination centers, two storage facilities, and ten water treatment centers, effectively managing the need for external water resources. Ultimately, through a rigorous sensitivity analysis, we unveiled that the chance constraint parameters have a significant impact on the variable costs.

The Value of Drilling—A Chance-Constrained Optimization Approach

Managing uncertainty is a core challenge in mine planning. Mine planners often represent various planning variables, such as equipment performance and geological parameters, as random variables due to inherent uncertainties. This paper looks at geological uncertainty and its impact on mine planning. Some traditional approaches to manage this uncertainty include using conditional simulations or mathematical programming in the planning process. Drilling additional holes, despite its cost, is a common method to reduce uncertainty using additional samples to reduce deposit variance. In this paper, we first outline an ore blending optimization model which uses chance-constrained programming to manage property limit risk when selecting the order of ore feed into a processing facility. In coal mining, in tactical planning horizons, the order of coal seam removal is usually predetermined, allowing a blending model to ensure optimal feed properties. Using chance-constrained programming allows us to blend the uncertainties from geological models to maximize plant output while adhering to property constraints. We use the chance-constrained blending model to determine the value of additional information from infill drilling. The model prioritizes drilling locations that reduce uncertainty and improve blending outcomes. A case study on a coking coal mine in Queensland, Australia, demonstrates the model’s application, highlighting significant improvements in blending by reducing the variance of high-quality blocks. The study concludes that targeting high-quality blocks for variance reduction can better accommodate lower-quality material, offering a more valuable approach than the traditional focus of reducing uncertainty in low-quality blocks. This approach provides insights for improving mine planning strategies and showcases the potential of chance constraints in optimizing ore blending under uncertainty.

Optimizing Supply Chain Efficiency using Innovative Goal Programming and Advanced Metaheuristic Techniques

This paper presents an optimization approach for supply chain management that incorporates goal programming (GP), dependent chance constraints (DCC), and the hunger games search algorithm (HGSA). The model acknowledges uncertainty by embedding uncertain parameters that promote resilience and efficiency. It focuses on minimizing costs while maximizing on-time deliveries and optimizing key decision variables such as production setups, quantities, inventory levels, and backorders. Extensive simulations and numerical results confirm the model’s effectiveness in providing robust solutions to dynamically changing supply chain problems when compared to conventional models. However, the integrated model introduces substantial computational complexity, which may pose challenges in large-scale real-world applications. Additionally, the model’s reliance on precise probabilistic and fuzzy parameters may limit its applicability in environments with insufficient or imprecise data. Despite these limitations, the proposed approach has the potential to significantly enhance supply chain resilience and efficiency, offering valuable insights for both academia and industry.

Joint chance‐constrained coordinated scheduling for electricity‐heat coupled systems considering hydrogen storage

AbstractStrong interdependence between power production and heat supply from combined power and heat units could restrict the wind power integration. Deploying hydrogen storage, typically formed by the electrolyser, hydrogen tank, and fuel cell, could be a promising measure to accommodate surplus wind power and facilitate the coordinated operation of power and heat. The authors consider coordinated scheduling under wind power uncertainty for an electricity‐heat coupled system with hydrogen storage and propose a distributionally robust joint chance‐constrained coordinated scheduling (DRJCC‐CS) model, where the waste heat from the electrolyser and fuel cell is incorporated. The authors design distributionally robust joint chance constraints to account for wind power uncertainty related constraints and derive their tractable inner and outer mixed‐integer convex approximations. Consequently, the proposed DRJCC‐CS model is casted into a mixed‐integer convex programme. Case studies are conducted to demonstrate the effectiveness of the proposed DRJCC‐CS method.