Chance Constraints Research Articles

Low-carbon and economic optimization of a source-load-storage system based on Stackelberg game and chance constraints

As energy demand grows and environmental pollution increases, low-carbon development has become a key focus in energy systems. To address the conflicting interests of the source-load-storage system, while also considering environmental benefits, this paper proposes an optimization model for the low-carbon economy of the source-load-storage system based on Stackelberg game theory and opportunity constraints. First, to ensure low carbon emissions and environmental protection, the carbon emissions of each entity in the source-load-storage system are constrained by a reward-penalty laddering carbon trading mechanism. Additionally, a demand response strategy is introduced on the user side, which accounts for both price and carbon compensation incentives. Next, recognizing the autonomy of the entities in the source-load-storage system, a decision-making model is developed based on the Stackelberg game, with the Power Management Operator as the leader, and the Power Generation Operator, Energy Storage Operator, and User as the followers. This model also outlines the low-carbon interaction mechanisms among the various entities of the source-load-storage system. Finally, the model is solved by combining an improved particle swarm algorithm with the Gurobi optimization tool. Simulation results effectively validate the proposed model and method, showing that the source-load-storage system can rationally adjust its strategy within the low-carbon framework while balancing both economic and environmental considerations.

A Two-Stage Fault Reconfiguration Strategy for Distribution Networks with High Penetration of Distributed Generators

In distribution networks with high penetration of distributed generators (DGs), traditional fault reconfiguration strategies often fail to achieve maximum load recovery and encounter operational stability challenges. This paper proposes a novel two-stage fault reconfiguration strategy that addresses both the fault ride-through capability and output uncertainty of DGs. The first stage introduces a rapid power restoration reconfiguration model that integrates network reconfiguration with fault ride-through, enabling DGs to provide power support to the distribution network during faults, thereby significantly improving the recovery rate of lost loads. An AdaBoost-enhanced decision tree algorithm is utilized to accelerate the computational process. The second stage proposes a post-recovery optimal reconfiguration model that uses fuzzy mathematics theory and the transformation of chance constraints to quantify the uncertainty of both generation and load, thereby improving the system’s static voltage stability index. Case studies using the IEEE 69-bus system and a real-world distribution network validate the effectiveness of the proposed strategy. This two-stage strategy facilitates short-term rapid load power restoration and enhances long-term operational stability, improving both the resilience and reliability of distribution networks with high DG penetration. The findings of this research contribute to enhancing the fault tolerance and operational efficiency of modern power systems, which is essential for integrating higher levels of renewable energy.

Uncertain Variational Inequalities Based on Chance Constraints

Uncertain Variational Inequalities Based on Chance Constraints

Optimal scheduling of electricity–hydrogen–ammonia coupled integrated energy system based on uncertain renewable generations

Power-to-ammonia (P2A) technology presents an effective strategy for facilitating the low-carbon transition of integrated energy system (IES), effectively addressing the stochastic and intermittent nature of renewable energy generation. In response to the challenges posed by the increased penetration of renewable energy, this paper develops an optimization scheduling model of IES based on chance constraints within an electricity–hydrogen–ammonia coupled architecture. The model integrates P2A technology, hydrogen blending in gas turbine (GT) and gas boiler (GB), and coal–ammonia co-combustion, while further modeling the dynamic characteristics of the electrolyzer. To account for renewable energy uncertainties, the probability reserve is formulated using chance-constrained programming (CCP). The model is solved using the sequence operation theory, which converts the CCP into a deterministic equivalent model to balance the expected and stochastic outputs of renewable generation. Case studies on four scenarios demonstrate that the proposed model achieves a 5.82% reduction in carbon emissions and a renewable energy utilization rate of 98.3%, outperforming traditional scheduling approaches. Additionally, the optimal system performance is achieved when hydrogen blending ratios reach 16% for GT and 18% for GB. These results underscore the model's effectiveness in enhancing low-carbon operation, maximizing renewable energy utilization, and improving the economic efficiency of IES.

Vessel Routing Chance-constrained Multi-objective Optimization by LNG and Diesel Hybrid Energy Vessels for Marine Debris Collection

Abstract Marine debris causes great harm to the ecosystem and human health, and it is urgent to clean up marine debris. In this paper, the method of logistics network is adopted to optimize the vessel route for debris collection. In order to reduce carbon emissions, we use LNG hybrid energy vessels. Since debris weight is difficult to predict, we treat it as a random variable. Considering the relationship among the three factors: speed, travel time, and carbon emissions, a multi-objective chance constraint model is established, and then transformed into a deterministic model by distribution function. We also design an AM-PHALNS algorithm using the AMOSA algorithm as a framework. Finally, through the case analysis of waste collection in the East China Sea, it is verified that the AM-PHALNS algorithm is superior to the NSGAII algorithm, saving 13.55% of travel time, 18.52% of total carbon emissions, and 20.25% of running time on average.

Integer programming approaches for distributionally robust chance constraints with adjustable risks

Integer programming approaches for distributionally robust chance constraints with adjustable risks

Multi-source shared operation optimization strategy for multi-virtual power plants based on distributionally robust chance constraint

Multi-source shared operation optimization strategy for multi-virtual power plants based on distributionally robust chance constraint

Electric Bus Scheduling Problem with Time Windows and Stochastic Travel Times

This work develops a scheduling tool for electric buses that accounts for daily disruptions while minimizing the operational costs. The contribution of this study lies in the development of electric bus schedules that consider many factors, such as multiple depots, multiple charging stations, and stochastic travel times, providing schedules resilient to extreme conditions. The developed model is a mixed-integer linear program (MILP) with chance constraints. The main decision variables are the assignment of electric vehicles to scheduled trips and charging events to ensure the improved operation of daily services under uncertain conditions. Numerical experiments and a sensitivity analysis based on the variation in travel times are conducted, demonstrating the performance of our solution approach. The results from these experiments indicate that the variant of the model with the chance constraint produces schedules with lower operational costs compared to the case where the chance constraints are not introduced.

Research on optimization of waste sorting and transportation network in smart cities based on garbage volume prediction

As urban populations and economies grow, the amount of municipal solid waste is expected to increase. To address this, waste sorting and collection have become crucial for implementing effective waste classification policies and promoting smart city development. Given the uncertainties in waste volume and the environmental impact of wet waste transportation, this paper presents optimization models for both dry and wet garbage vehicle routing. The goal is to minimize total costs, carbon footprint, and secondary pollution. A random chance constraint is introduced based on decision-makers' preferences, and stochastic theory is used to convert this constraint into a probability density equivalent, improving solution efficiency. To tackle the complexity of waste collection and transportation, an improved genetic algorithm (GA-LS) combining genetic algorithms and a two-layer local search method is proposed. This enhances the exploration ability of the solution space. The model is validated through actual case studies and sensitivity analysis, demonstrating that it can generate near-optimal solutions within an acceptable time frame. The optimization reduces costs, minimizes environmental pollution, and improves residents' satisfaction. Furthermore, it provides a theoretical basis for determining the final wet waste treatment site and optimal transportation routes. Future research will integrate generative deep learning models to improve waste volume prediction accuracy, supporting more efficient and cost-effective smart city decisions.

Exponential cone approach to joint chance constraints in stochastic model predictive control

Stochastic model predictive control addresses uncertainties by incorporating the probabilistic description of the disturbances into joint chance constraints. Yet, the classic methods for handling this class of constraints are often computationally inefficient and overly conservative. To overcome this, we propose to replace the nonconvex inverse cumulative distribution function of the standard normal distribution in the deterministic counterpart of these constraints with a highly accurate, exponential cone-representable approximation. This allows the constraints to be formulated as exponential cone functions, and the problem is solved as an exponential cone optimization with risk allocation as decision variables. The main advantage of the proposed approach is that the optimization problem is efficiently solved with off-the-shelf software, and with reduced conservativeness. Moreover, it applies to any problem with linear joint chance constraints subject to normally distributed disturbances. We validate our method with numerical examples of stochastic model predictive control applications.

Data-Driven Optimal Dispatch of Integrated Energy in Zero-Carbon Building System Considering Occupant Comfort and Uncertainty

Zero-carbon emissions in building systems play a critical role in promoting energy transition and mitigating climate change, while optimal energy dispatch in highly electrified building systems is essential to achieve this goal. To address this issue, we develop an integrated energy system model for zero-carbon buildings. The carbon capture and carbon processing devices are incorporated in the system, while the uncertainty of renewable energy sources is handled by a robust optimization approach. Occupant comfort is also expressed using chance constraints to minimize energy costs. Furthermore, we propose a data-driven approach to solve the optimization problem, where uncertain parameters are clustered into subsets to construct uncertainty sets. Numerical results demonstrate that the total energy cost can be reduced by 18.87% in summer and 27.22% in winter when relaxing the occupant comfort constraints, and comparative analysis shows that the proposed approach can achieve a balance between conservativeness and computational complexity.

A Proximal Difference-of-Convex Algorithm for Sample Average Approximation of Chance Constrained Programming

Chance constrained programming (CCP) refers to a type of optimization problem with uncertain constraints that are satisfied with at least a prescribed probability level. In this work, we study the sample average approximation (SAA) method for chance constraints, which is an important approach to CCP in the data-driven setting where only a sample of multiple realizations of the random vector in the constraints is available. The SAA method approximates the underlying distribution with an empirical distribution over the available sample. Assuming that the functions in the chance constraints are all convex, we reformulate the SAA of chance constraints into a difference-of-convex (DC) form. Additionally, by assuming the objective function is also a DC function, we obtain a DC constrained DC program. To solve this reformulation, we propose a proximal DC algorithm and show that the subproblems of the algorithm are suitable for off-the-shelf solvers in some scenarios. Moreover, we not only prove the subsequential and sequential convergence of the proposed algorithm, but also derive the iteration complexity for finding an approximate Karush-Kuhn-Tucker point. To support and complement our theoretical development, we show via numerical experiments that our proposed approach is competitive with a host of existing approaches. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: P. Wang and L. Balzano received financial support from the National Science Foundation [CAREER Award CCF-1845076], the Army Research Office Young Investigator Program [Award W911NF1910027], and the Department of Energy [Award DE-SC0022186]. R. Jiang received financial support from the Major Program of the National Natural Science Foundation of China [Grants 72394360, 72394364] and the Natural Science Foundation of Shanghai [Grant 22ZR1405100]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0648 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0648 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Unscented Hypersonic Trajectory Optimization with a Heating-Rate Chance Constraint

The aerothermal load on an aerospace vehicle is one of the most critical conditions during hypersonic flight. To maximize performance, a typical hypersonic vehicle rides the maximum allowable value of the heating-rate constraint during a portion of its flight. Because atmospheric density has high uncertainty, guiding a hypersonic vehicle along a deterministic optimal trajectory will violate the maximum heating-rate limit with an unacceptable probability. To address this problem, we pose the maximum heating rate on the vehicle as a chance constraint in a tychastic trajectory optimization problem (from Tyche, the Greek goddess of chance). To generate a tractable problem formulation, the chance constraint is mapped to a constraint on a deviation measure of the heating rate. The resulting tychastic problem is transcribed to a constrained unscented trajectory optimization problem and solved by a guess-free, spectral algorithm. The validity of the entire approach is independently verified via a Monte Carlo analysis. Sample numerical results demonstrate how risk can be reduced from unsafe (i.e., 70% risk) to safe (i.e., near 0% risk) operations.

Emergent structures of attention on social media are driven by amplification and triad transitivity.

As they evolve, social networks tend to form transitive triads more often than random chance and structural constraints would suggest. However, the mechanisms by which triads in these networks become transitive are largely unexplored. We leverage a unique combination of data and methods to demonstrate a causal link between amplification and triad transitivity in a directed social network. Additionally, we develop the concept of the "attention broker," an extension of the previously theorized tertius iungens (or "third who joins"). We use an innovative technique to identify time-bounded Twitter/X following events, and then use difference-in-differences to show that attention brokers cause triad transitivity by amplifying content. Attention brokers intervene in the evolution of any sociotechnical system where individuals can amplify content while referencing its originator.

A Real-Life Demonstration of Secondary Frequency Reserve Provision with Electric Water Heaters

Residential electric water heaters have the potential to significantly contribute to the balancing of the grid by providing frequency services. However, this entails a large-scale, challenging control problem subject to several uncertainties. In this paper, we perform the first real-life validation of secondary frequency reserve provision with a cluster of residential thermal loads in a near-commercial setting. We adopt an aggregate-and-dispatch control approach, which combines a scalable optimization step enabled by a reduced-order model with a real-time dispatch step. To handle the uncertainty related to service activation, we incorporate chance constraints in the optimization model and reformulate it as a robust problem. We validate the control approach under the assumption of perfect merit order knowledge in different stages, with a cluster of up to 600 electric water heaters, and show that this pool is able to effectively provide reserves, and that the integration of the chance constraints is beneficial for performance.

Optimal Power Procurement for Green Cellular Wireless Networks Under Uncertainty and Chance Constraints.

Given the increasing global emphasis on sustainable energy usage and the rising energy demands of cellular wireless networks, this work seeks an optimal short-term, continuous-time power-procurement schedule to minimize operating expenditure and the carbon footprint of cellular wireless networks equipped with energy-storage capacity, and hybrid energy systems comprising uncertain renewable energy sources. Despite the stochastic nature of wireless fading channels, the network operator must ensure a certain quality-of-service (QoS) constraint with high probability. This probabilistic constraint prevents using the dynamic programming principle to solve the stochastic optimal control problem. This work introduces a novel time-continuous Lagrangian relaxation approach tailored for real-time, near-optimal energy procurement in cellular networks, overcoming tractability problems associated with the probabilistic QoS constraint. The numerical solution procedure includes an efficient upwind finite-difference solver for the Hamilton-Jacobi-Bellman equation corresponding to the relaxed problem, and an effective combination of the limited memory bundle method (LMBM) for handling nonsmooth optimization and the stochastic subgradient method (SSM) to navigate the stochasticity of the dual problem. Numerical results, based on the German power system and daily cellular traffic data, demonstrate the computational efficiency of the proposed numerical approach, providing a near-optimal policy in a practical timeframe.

Rolling optimization control method for hydro-photovoltaic-storage microgrid based on stochastic chance constraints

Hydro-photovoltaic-storage (HPS) microgrid has gradually become an important measure to optimize the energy structure and ensure the reliability of regional power supply. However, due to the strong randomness and spatiotemporal correlations of hydropower and photovoltaic (PV) output, traditional deterministic optimization methods are difficult to support the accurate regulation and reliable operation of microgrid with a high proportion of renewable energy integration. On this basis, a rolling optimization control method for HPS microgrid based on stochastic chance constraints is proposed. A novel multivariate scenario reduction method considering hydro-PV correlations is presented to characterize the uncertainty of renewable energy output, and a day-ahead stochastic optimal scheduling model based on chance-constrained programming is constructed. Combined with stochastic model predictive control strategies, the day-ahead scheduling plan can be adjusted at multiple time scales, both intraday power compensation and real-time adjustments, to suppress the intraday power fluctuations induced by day-ahead scenario errors and reduce the influence of the uncertainty of hydro-PV power output on microgrid operation. Experimental results show that compared with the traditional deterministic scheduling method, the proposed method can effectively improve the stability and economy of HPS microgrid operation under complex uncertain conditions.

Multi-agent deep reinforcement learning for Smart building energy management with chance constraints

Multi-agent deep reinforcement learning for Smart building energy management with chance constraints

Low-carbon dispatch optimization of wind-solar-thermal-storage multi-energy system based on stochastic chance constraints and carbon trading mechanism

To improve the low-carbon economic performance of renewable energy-dominated power systems, a multi-energy coordinated optimization dispatch model for wind, solar, thermal, and storage systems considering uncertainties on both the supply and demand sides is proposed. This paper comprehensively considers the economic costs of thermal power unit operation, wind and solar power curtailment, energy storage operation, carbon trading and spinning reserve. The model incorporates a penalizing carbon trading mechanism and uses a stochastic chance-constrained approach to handle fluctuations in wind and solar power generation as well as uncertainties in load forecasting. The study, based on the IEEE 30-bus system, is solved using a stochastic simulation particle swarm optimization algorithm. Results show that after introducing the carbon trading mechanism, the system's carbon emissions were reduced by 8.35%, wind and solar curtailment penalties were reduced by 65.48%, and overall costs decreased by 14.94%. Additionally, the chance-constrained model effectively reduced the system's reserve capacity requirements, with reserve capacity decreasing by 31.84%, leading to a further reduction of 26.83% in overall costs. In the scenario of combined wind-solar-thermal-storage output, the wind and solar curtailment rate dropped to 7.37%, and carbon emissions decreased to 6474.69 tons. Through the "energy shifting" function, the energy storage system provided effective support during peak loads, further optimizing the dispatch outcomes.

Combined Long-Term Collision Avoidance and Stochastic Station-Keeping in Geostationary Earth Orbit

To limit the spread of space debris, space situational awareness (SSA) delineates guidelines to preserve current space assets. Developing effective collision avoidance maneuver (CAM) strategies is emerging as a global top priority among the considered countermeasures to debris-generating events. Despite most encounters happening over very short time frames, some conjunctions occur over a longer time window, such as in geostationary Earth orbit (GEO), where the involved objects may have small relative velocities. Besides, external perturbations, particularly the geopotential, lunisolar, and solar radiation pressure ones, exert forces on the spacecraft, causing it to deviate from its designated slot and potentially endanger neighboring satellites. This issue is compounded when considering state uncertainty. The presented work, therefore, introduces convex optimization approaches for long-term CAM and tailored stochastic station-leeping (SK) policy regarding longitude and latitude in this regime. The formulation enables continuous CAM and chance-constrained SK, ensuring satellite adherence to an assigned GEO slot with a given probability. Two kinds of chance constraints are devised: the first one does not consider the correlation between longitude and latitude, but the latter does.