Stochastic Optimization Model Research Articles

Modeling and analysis of heterogeneous traffic flow with mixed regular and connected human-driven vehicles considering driver stochasticity

This paper presents a novel microscopic modeling framework to explore the impact of driver stochasticity and other related factors on heterogeneous traffic flows, consisting of both regular human-driven vehicles (RHDVs) and connected human-driven vehicles (CHDVs). We apply the stochastic optimal velocity car-following model for RHDVs, while an extended version of the stochastic continuous car-following model, taking into account optimal velocity guidance and driver compliance, is devised for CHDVs. The stability of CHDV car-following model is examined theoretically and numerically. Simulation experiments are conducted to analyze the stochastic heterogeneous traffic flow properties (i.e. stability and safety) with the proposed microscopic modeling framework under diverse parameter settings including CHDV penetration rate, driver compliance, and driver stochasticity. Results show that traffic flow stability and safety improve with the increase in CHDV penetration rate and driver compliance but deteriorate as the driver’s stochasticity strength increases. CHDVs significantly enhance traffic stability when the CHDV penetration rate or driver compliance exceeds a certain threshold, typically ranging between 0.5 and 0.7. Additionally, there is an interaction between driver stochasticity and CHDV penetration rate in their effect on traffic flow stability, where the contribution of stochasticity to traffic oscillations first increases and then decreases as the penetration rate rises. When the penetration rate is between 0 and 0.4, the traffic risk in emergency situations can be significantly reduced as the penetration rate increases. These findings may guide the management and control strategies of heterogeneous traffic flow.

Nash Bargaining-Based Coordinated Frequency-Constrained Dispatch for Distribution Networks and Microgrids

As the penetration of distributed renewable energy continues to increase in distribution networks, the traditional scheduling model that the inertia and primary frequency support of distribution networks are completely dependent on the transmission grid will place enormous regulatory pressure on the transmission grid and hinder the active regulation capabilities of distribution networks. To address this issue, this paper proposes a coordinated optimization method for distribution networks and microgrid clusters considering frequency constraints. First, the confidence interval of disturbances was determined based on historical forecast deviation data. On this basis, a second-order cone programming model for distribution networks with embedded frequency security constraints was established. Then, microgrids performed economic dispatch considering the reserves requirement to provide inertia and primary frequency support, and a stochastic optimization model with conditional value-at-risk was built to address uncertainties. Finally, a cooperative game between the distribution network and microgrid clusters was established, determining the reserve capacity provided by each microgrid and the corresponding prices through Nash bargaining. The model was further transformed into two sub-problems, which were solved in a distributed manner using the ADMM algorithm. The effectiveness of the proposed method in enhancing the operational security and economic efficiency of the distribution networks is validated through simulation analysis.

Stochastic Decision-Making Optimization Model for Large Electricity Self-Producers Using Natural Gas in Industrial Processes: An Approach Considering a Regret Cost Function

In the context of high energy costs and energy transition, the optimal use of energy resources for industrial consumption is of fundamental importance. This paper presents a decision-making structure for large consumers with flexibility to manage electricity or natural gas consumption to satisfy the demands of industrial processes. The proposed modelling energy system structure relates monthly medium and hourly short-term decisions to which these agents are subjected, represented by two connected optimization models. In the medium term, the decision occurs under uncertain conditions of energy and natural gas market prices, as well as hydropower generation (self-production). The monthly decision is represented by a risk-constrained optimization model. In the short term, hourly optimization considers the operational flexibility of energy and/or natural gas consumption, subject to the strategy defined in the medium term and mathematically connected by a regret cost function. The model application of a real case of a Brazilian aluminum producer indicates a measured energy cost reduction of USD 3.98 millions over a six-month analysis period.

Optimization of renewable energy-based integrated energy systems: A three-stage stochastic robust model

The integration of renewable energy into an integrated energy system (REIES) represents a promising approach to achieving clean and low-carbon energy consumption. However, the inherent uncertainty and variability of renewable energy sources and load demands present significant challenges to the operational efficiency and stability of REIES. To address these challenges, we propose a three-stage stochastic robust optimization (TRSRO) model. This model accounts for market electricity prices, source-load scenario probabilities, and output uncertainties to optimize system configurations in REIES. It also includes the optimization of power exchanges with the regional grid and strategic operational scheduling. Initial scenarios of uncertainty variables are generated using the spectral normalized conditional Wasserstein generative adversarial network (SNCWGAN), forming polyhedral uncertainty sets. Comprehensive norm constraints are then applied to capture probability distribution uncertainties within these sets. To improve the efficiency of solving the model, we introduce a two-layer column constraint generation algorithm, considering variable alternate iterations (TLCCG-VAI). The results demonstrate that the proposed method can achieve a 12.16 % reduction in total cost compared to the baseline method. Furthermore, the TLCCG-VAI algorithm reduces runtime by 82.97 % while maintaining the same level of solution accuracy.

Scenario-based stochastic optimization on the variability of solar and wind for component sizing of integrated energy systems

The inherent intermittency and variability of renewable energy sources present significant challenges to the optimal design and implementation of integrated energy systems (IES). This paper introduces a novel stochastic optimization model that integrates advanced scenario generation and clustering algorithm for renewable energy sources within a multi-objective, bi-level optimization framework. Specifically, the clearness index is employed to represent the stochastic distribution of solar radiation intensity by beta distribution, while wind speed uncertainty is modeled seasonally using the Weibull distribution. Monte Carlo sampling with synchronous back substitution is applied for scenario generation and reduction of solar radiation and wind speed. To address the multi-objective evaluation, the analytic hierarchy process is utilized, and the joint optimization is achieved by combining a region contraction algorithm with stochastic programming. The proposed methodology is validated on an IES featuring various heating devices, incorporating uncertainties in both wind and solar energy. The results indicate that the absorption heat pump-based scheme achieves superior energy-saving performance, achieving an energy rate of 0.4724. Additionally, the compression heat pump-based scheme exhibits excellent economic efficiency and environmental sustainability, with a cost of energy of 0.3639 and a renewable fraction of 0.5536.

Stochastic optimal allocation of grid-side independent energy storage considering energy storage participating in multi-market trading operation

The integration of large-scale intermittent renewable energy generation into the power grid imposes challenges to the secure and economic operation of the system, and energy storage (ES) can effectively mitigate this problem as a flexible resource. However, the conventional ES allocation is mostly planned to meet the regulation demands of individual entities, which is likely to result in low utilization of ES and difficult to recover the investment cost. Therefore, a two-stage stochastic optimal allocation model for grid-side independent ES (IES) considering ES participating in the operation of multi-market trading, such as peak-valley arbitrage, frequency regulation, and leasing, is proposed in this paper to improve the comprehensive benefits and utilization rate of ES. The first stage aims to allocate IES and develop a systematic scheduling plan based on the forecast of wind power output and load demand, while the second stage responds to the uncertainty of wind power output by re-dispatching generating units and invoking ES power leased by wind farms. Then, a two-layer loop iterative solution algorithm based on the Benders decomposition is formed to effectively solve the proposed model. Finally, the approach developed in this paper is applied to a modified IEEE RTS-79 test system, and the results verify that it is both feasible and effective.

Stochastic optimal scheduling of a combined wind-photovoltaic-CSP-fire system accounting for electrical heat conversion

Aiming at the consumption problem caused by the increasing scale of wind power and photovoltaic (PV) grid-connected, a multi-energy co-generation system is constructed with wind power, PV, concentrated solar power (CSP), and thermal power; in addition, in order to reduce the impact of the prediction errors of wind power, PV, and loads on the system’s economic operation, the photovoltaic and thermal power plants are used to provide the system's backup capacity together, and the opportunity constraint model of the reliability of the backup capacity is established, so as to satisfy the system reliability constraints at a certain higher confidence level; finally, a sampling-based deterministic transformation method is introduced to simplify the model. The model is simplified by introducing a sampling-based deterministic transformation method of opportunity constraint; finally, a stochastic optimal dispatch model of the combined wind-photovoltaic-CSP-fire system, which accounts for the conversion of electricity and heat, is constructed with the objective of minimising the integrated operating cost of the combined system, and the effectiveness of the proposed model is analysed by simulation verification.

A two-stage stochastic programming framework for blood product inventory management with ABO substitution and lateral transshipment

This work proposes a two-stage stochastic optimization model for managing the inventory of a two-echelon blood supply chain comprised of a blood center and a network of hospitals and transfusion points. The proposed model identifies optimal distribution quantities across the supply chain, considering demand stochasticity and product perishability. Furthermore, the model utilizes redistribution policies, such as lateral transshipment and product substitution based on ABO compatibility, as corrective actions to minimize outdates (i.e., expired product) and shortages across the network. Using a computational experiment that includes a network with one regional blood center and twenty hospitals, we show that inventory management policies that include both ABO substitution and transshipment can significantly decrease the expected total network costs. Similarly, we see corresponding reductions in the expected number of outdated units and emergency requests to external networks for additional supply to satisfy unmet demand.

Multi-objective optimization for a green forward-reverse meat supply chain network design under uncertainty: Utilizing waste and by-products

The essential requirement for food stands as a pivotal human need, exerting significant ecological impact from production to consumption. Meat, a key dietary component, offers essential nutrients vital for human health. This paper presents a bi-objective two-stage stochastic optimization model for a green forward-reverse meat supply chain network design, addressing both economic and environmental concerns throughout the chain. In the forward flow, the supply chain manages various meat products consist of fresh, processed, and frozen meat products, ensuring their eco-efficient production and distribution. Meanwhile, in the reverse flow, waste and by-products generated during the production process are repurposed and reused. The study promotes environmental sustainability by repurposing waste, utilizing by-products, and minimizing carbon emissions. The proposed model is solved using popular exact ε-constraint method for smaller instances and Non-dominated Sorting Genetic Algorithm II (NSGA-II), Multi-Objective Particle Swarm Optimization (MOPSO), and Strength Pareto Evolutionary Algorithm 2 (SPEA2) meta-heuristic algorithms are employed for larger instances. SPEA2 outperforms both MOPSO and NASG-II, demonstrating less average gap that is 0.38% and 0.76%, respectively. Additionally, the findings reveals that an average 2.5% reduction in environmental impacts associated with an average 5% decrease in profit. The noteworthy outcomes of the research empower managers to navigate the implications of fluctuating demand effectively. Moreover, it is crucial to underscore the effects of conversion rates, particularly those associated with the manufacturing process. An excessively high conversion rate can negatively impact profitability and worsen environmental issues. Conversely, lowering the conversion rate can enhance profitability and mitigate environmental impacts.

Stochastic optimal control model for COVID-19: mask wearing and active screening/testing

Stochastic optimal control model for COVID-19: mask wearing and active screening/testing

A multi-objective stochastic optimization model for combined heat and power virtual power plant considering carbon recycling and utilizing

In order to give full play to the energy supply potential of distributed energy resources, this paper studies the scheduling optimization of CHP-VPP. First, the CHP unit and various distributed energy sources are aggregated into VPP. Carbon recycling and utilizing are realized through carbon capture and power-to-gas devices. At the same time, carbon storage and hydrogen storage devices are added to decouple carbon capture and P2G procedures. Then, the risk of VPP real-time scheduling is quantified through uncertainty scenario generation and CVaR. Finally, with the goals of operating cost, carbon emission, and operation risk, a multi-objective stochastic scheduling optimization model of VPP is constructed, and the subjective and objective ensemble weighting method is used to solve the problem. The example results show that the proposed method can boost the wastage of wind and photovoltaic power, and also lower the carbon emissions of VPPs.

Decentralized coordinated optimal dispatching of multi-area interconnected electricity-gas-heat system based on inertia Bregman ADMM

The coordinated interconnection of multi-area interconnected electricity-gas-heat systems (MIEGHS) has become a typical feature of the modern energy internet. While enhancing system operational efficiency, it also introduces complexity and uncertainty into system operation. In consideration of these factors, a decentralized coordinated robust optimization scheduling framework for MIEGHS is proposed to facilitate coordinated optimization. Firstly, a two-stage robust stochastic optimal dispatching model for a single-area integrated electricity-gas-heat system (IEGHS) is established, considering the uncertainty of wind power output and load. Secondly, a decentralized self-regulation control model for MIEGHS is proposed based on the principle of consistency constraints. Next, considering both the timeliness and convergence of the traditional alternating direction method of multipliers (ADMM) in solving problems, an improved ADMM method based on inertial Bregman distance is presented for the optimized solution of the proposed model. Finally, the effectiveness and superiority of the presented model and methods are verified by three interconnected IEGHS.

Impact of power outages depends on who loses it: Equity-informed grid resilience planning via stochastic optimization

This research presents a novel approach for enhancing power grid resilience with a focus on social equity in light of increasing natural disasters. Utilizing a two-stage stochastic optimization model for flood mitigation investments, we optimize substation hardening and power flow decisions to minimize the weighted combination of expected load shed and expected equity metric. Our method uniquely approximates community power outages stemming from transmission grid disruptions, sidestepping the complexities of the distribution grid, and it incorporates two equity metrics (affected population and duration of loss) to address the uneven postdisaster well-being loss across communities. Our approach’s novelty lies in integrating this equity-informed resilience model with realistic flood scenario generation and utilizing a large-scale synthetic but a realistic power grid of Texas. The findings highlight the importance of the composite objective function in altering power flow decisions to prioritize electricity provision and save communities in disadvantaged areas even without investing in substation hardening. The results also quantify the equity and load shed benefits of substation hardening as a function of the investment budget with a parameterized analysis. With an attention to equity, power outages increase in nonvulnerable communities — a trade-off made to mitigate well-being loss in the most vulnerable areas. We further explore a justice model inspired by the government’s Justice40 initiative but find it less effective than our equity-informed models at preventing well-being loss. Our research, enriched by a comprehensive sensitivity analysis, offers valuable insights for policymakers, grid operators, and utilities aiming for a more resilient and equitable power grid.

Stochastic Techno-Economic Optimization of Hybrid Energy System with Photovoltaic, Wind, and Hydrokinetic Resources Integrated with Electric and Thermal Storage Using Improved Fire Hawk Optimization

In this paper, a stochastic techno-economic optimization framework is proposed for three different hybrid energy systems that encompass photovoltaic (PV), wind turbine (WT), and hydrokinetic (HKT) energy sources, battery storage, combined heat and power generation, and thermal energy storage (Case I: PV–BA–CHP–TES, Case II: WT–BA–CHP–TES, and Case III: HKT–BA–CHP–TES), with the inclusion of electric and thermal storage using the 2m + 1 point estimate method (2m + 1 PEM) utilizing real data obtained from the city of Espoo, Finland. The objective function is defined as planning cost minimization. A new meta-heuristic optimization algorithm named improved fire hawk optimization (IFHO) based on the golden sine strategy is applied to find the optimal decision variables. The framework aims to determine the best configuration of the hybrid system, focusing on achieving the optimal size for resources and storage units to ensure efficient electricity and heat supply simultaneously with the lowest planning cost in different cases. Also, the impacts of the stochastic model incorporating the generation and load uncertainties using the 2m + 1 PEM are evaluated for different case results compared with the deterministic model without uncertainty. The results demonstrated that Case III obtained the best system configuration with the lowest planning cost in deterministic and stochastic models and. This case is capable of simply meeting the electrical and thermal load with the contribution of the energy resources, as well as the CHP and TESs. Also, the IFHO superiority is proved compared with the conventional FHO, and particle swarm optimization (PSO) achieves the lowest planning cost in all cases. Moreover, incorporating the stochastic optimization model, the planning costs of cases I–III are increased by 4.28%, 3.75%, and 3.57%, respectively, compared with the deterministic model. Therefore, the stochastic model is a reliable model due to its incorporating the existence of uncertainties in comparison with the deterministic model, which is based on uncertain data.

Alleviating bus bunching via modular vehicles

The notorious phenomenon of bus bunching prevailing in uncontrolled bus systems produces irregular headways and downgrades the level of service by increasing passengers’ expected waiting time. Modular autonomous vehicles (MAVs), due to their ability to split and merge en route, have the potential to help both late and early buses recover from schedule deviation while providing continuous service. In this paper, we propose a novel bus bunching alleviation strategy for MAV-aided transit systems. We first consider a soft vehicle capacity constraint and establish a continuum approximation (CA) model (Model I) to capture the system dynamics intertwined with the MAV splitting and merging operations, and then establish an infinite-horizon stochastic optimization model to determine the optimal splitting and merging strategy. To capture the reality that passengers may fail to board an overcrowded bus, we propose a second model (Model II) by extending Model I to accommodate a hard vehicle capacity constraint. Based on the characteristics of the problem, we develop a customized deep Q-network (DQN) algorithm with multiple relay buffers and a penalized ruin state applicable for both models to optimize the strategy for each MAV. Numerical results show that the strategy obtained via the DQN algorithm is an effective bunch-proof strategy and has a better performance than the myopic strategy for MAV-aided systems and the two-way-looking strategy for conventional bus systems. Sensitivity analyses are also conducted to examine the effectiveness and benefits of the proposed strategy across different operation scenarios.

Managing oversaturation in BRT corridors: A new approach of timetabling for resilience enhancement using a tailored integer L-shaped algorithm

Bus rapid transit (BRT) is a high-capacity public transport system that typically operates along urban transit corridors with dense travel demand. Maintaining the efficiency and stability of the BRT is paramount for daily transport operations. Owing to the difficulty in ensuring an exclusive right-of-way along the entire route, stochastic congestion events may occur resulting from road segments without dedicated BRT lanes. This may lead to volatility in travel time and resulting in passenger stranding, determined as common-case disruptions in this study. These common-case disruptions frequently occur in the daily operation of oversaturated BRT routes. To manage and mitigate their negative impacts, a novel timetabling problem for enhancing the resilience of a BRT system was proposed to assess the ability of the system to withstand and recover from these disruptions. We formulated the problem as a two-stage stochastic mixed-integer optimization model and designed an exact algorithm based on a tailored integer L-shaped method. We then analyzed the structural properties of our model and developed several acceleration techniques to further improve the efficiency of the algorithm. The computational results show that the proposed algorithm outperforms the commercial solver in large-scale instances and can provide near-optimal solutions when the commercial solver is invalid. Besides, external comparisons with dynamic programming also demonstrate the superiority of the proposed algorithm in solution efficiency. Compared with the benchmark timetabling problem, which aims to reduce passenger waiting time, the proposed method can efficiently reduce the duration of the oversaturation period by 35.3%.

Stochastic Optimal Strategies and Management of Electric Vehicles and Microgrids

This study combines the Nash–Cournot competition model and the stochastic optimization model to examine the impact of electric vehicle (EV) quantity fluctuations on microgrid operations, aiming to optimize energy usage in a competitive electricity market. Integrating distributed energy resources and bidirectional charging, microgrids offer a novel approach for energy optimization, aiding in renewable energy generation, peak demand management, and emission reduction. Empirical evidence highlights benefits in Taiwan’s electricity market and net-zero emissions target by 2050, with a case study demonstrating enhanced local renewable energy generation due to EVs and microgrid integration. As the number of EVs increases, electricity sales from microgrids decline, but electricity purchases remain stable. The degree of electricity liberalization also influences the supply and demand dynamics of the electricity market. Microgrids selling electricity only to the main grid increases total power consumption by 65.55 million MWh, reducing the market share of the state-owned utility (Taipower). Conversely, allowing retailers to purchase from microgrids increases total consumption by 30.87 million MWh with a slight market share decrease for Taipower. This study contributes to providing an adaptable and flexible general model for future studies to modify and expand based on different scenarios and variables to shape energy and environmental policies.

Comparing resilience strategies for a multistage green supply chain to mitigate disruptions: A two-stage stochastic optimization model

Supply chain disruptions, notably exemplified by the COVID-19 pandemic, have substantial impacts, requiring effective resilience strategies. This study introduces a comprehensive multi-stage, multi-period green supply chain design model, incorporating six resilience strategies to tackle downstream and upstream disruptions. A rigorous two-stage stochastic optimization approach is employed to manage parameter uncertainties and disruptions. The objectives include minimizing disruption costs and CO2 emissions, for a chain that operates under cap-and-trade emission regulation. The findings, derived from a numerical experiment and sensitivity analysis, offer the optimal supply chain structure and effective resilience strategies to mitigate disruptions. The demand structure for essential and non-essential products during disruptions are explored, emphasizing proactive resource allocation. In particular, the impact of facility capacity reductions underscores the importance of capacity management. Sensitivity analyses reflect the trade-offs involved in capacity management strategies, emphasizing the critical need to maintain an optimal level of capacity to prevent service level degradation. Additionally, examination of carbon abatement regulations reveals the intricate balance between environmental responsibility and economic efficiency. In summary, this study addresses supply chain disruptions and offers actionable insights for supply chain managers and policy makers. The research highlights the importance of specific and data-driven resilience strategies to optimize supply chain performance in face of disruptions.

Cashflow-driven investment beyond expectations

ABSTRACT This paper presents a computationally tractable optimization model for cashflow-driven investment where the aim is to find asset portfolios whose future payouts cover given liability payments as well as possible. While current industry solutions are largely based on expected future cash flows, we use a stochastic optimization model that seeks portfolios that give the best possible match across time as well as scenarios. Cashflow matching across scenarios is controlled by risk aversion while the timing is driven by illiquidity of money markets. When illiquidity increases, the hedging strategy quickly shifts towards portfolios suggested by deterministic models, but significant uncertainty remains. The model can incorporate hundreds of quoted instruments in the construction of optimal hedging strategies. The hedging strategies are able to employ any statistical connections between the liabilities and publicly quoted assets. The model is solved with simple Monte Carlo approximations and off-the-shelf convex optimization software. Besides optimal hedging strategies, we find the least cost of hedging which provides a market-consistent valuation based on the current quotes and the liquidity factors as well as the views and risk preferences of the investor/regulator. The approach is illustrated by pricing and hedging defined benefit pension liabilities which depend on uncertain longevity developments and the consumer price index. The hedging strategies are constructed from 128 publicly quoted instruments including index-linked bonds and equities. We find that the optimized hedging strategies achieve lower risk at a lower cost than the strategies obtained by matching expected cashflows. The hedging-based liability valuations are robust with respect to model parameters and the additional risk reduction achieved by optimization does not add much to the overall hedging cost.

Coordinated optimization of configuration and operation of a photovoltaic integrated building cooling system with electricity and ice storages under source-load uncertainties

Solar photovoltaic integration into buildings provides an effective and attractive method to decrease fossil energy consumption and reduce greenhouse gases emission. This study proposes a flexible photovoltaic integrated building cooling system with electricity and ice storages to address the challenges posed by the instability of solar energy and the integration and coordination of multiple components. Uncertainties in solar irradiance, ambient temperature, and cooling load are characterized using a scenario analysis method that combines Latin Hypercube Sampling and the K-Means++ algorithm. Then, aiming to achieve the minimal total cost, including investment, operational, and CO2 emission costs, a two-layer stochastic optimization model is developed to simultaneously determine the optimal capacities and operational strategies for each device. Genetic algorithm and mixed integer linear programming are, respectively, used to solve the two-layer optimization problem. The comparisons between optimum schemes with and without CO2 emission cost demonstrates that when the base chiller’s capacities are 1179 kW and 1081 kW in cases 1 and 2 respectively, it can independently meet 72.37 % and 61.27 % of the cooling load demand throughout the cooling season. In addition, the optimal capacity of photovoltaics, considering CO2 emission, increases by 109.7 %, and its penetration rate reaches 18.3 %, which is 9.7 % higher than the optimal scheme not considering CO2 emission. However, the annualized investment cost increases by 52.3 %. The developed methods provide effective tools for the system design and operation management of solar energy integrated building cooling system.