Stochastic Robust Optimization Research Articles

Advancing Parameter Estimation in Differential Equations: A Hybrid Approach Integrating Levenberg–Marquardt and Luus–Jaakola Algorithms

This study presents an integrated approach that combines the Levenberg–Marquardt (LM) and Luus–Jaakola (LJ) algorithms to enhance parameter estimation for various applications. The LM algorithm, known for its precision in solving non-linear least squares problems, is effectively paired with the LJ algorithm, a robust stochastic optimization method, to improve accuracy and computational efficiency. This hybrid LM-LJ methodology is demonstrated through several case studies, including the optimization of MESH equations in distillation processes, modeling controlled diffusion in biopolymer films, and analyzing heat and mass transfer during the drying of cylindrical quince slices. By overcoming the convergence issues typical of gradient-based methods and performing global searches without initial parameter bounds, this approach effectively handles complex models and closely aligns simulation results with experimental data. These capabilities highlight the versatility of this approach in engineering and environmental modeling, significantly enhancing parameter estimation in systems governed by differential equations.

Low-carbon economic optimization strategy for industrial loads in parks considering source-load-price multivariate uncertainty

A low-carbon economic optimization method is proposed for industrial loads in parks considering multivariate uncertainty. Model of dynamic load node carbon intensity is proposed considering the uncertainty of clean energy and dynamic conventional units carbon intensity based on system currents. The robustness is improved based on multi-interval uncertainty set. The gap is filled by hybrid copula model with Generalized Autoregressive Conditional Heteroskedasticity (GARCH) prices in the uncertainty model. The GARCH is used to built the marginal distribution function of prices, and a hybrid copula model is proposed to obtain the joint probability density function based on the weights calculated by Euclidean distance. The LNCI probability distributions of the regulation potential of loads are proposed with the virtual energy storage and the virtual power model to improve the descriptive ability of uncertainty based on the expansion and contraction functions. Finally, the typical scenarios are extracted based on Monte Carlo and Wasserstein measures. The Column sum Constraint Generation algorithm and strong duality theory are used to get the robust-stochastic optimization. The method proposed in the paper has a positive effect on enterprises to rationalize their production schedules, reduce electricity and carbon emissions costs and increase the revenue from participating in grid regulation.

Optimal planning for hybrid renewable energy systems under limited information based on uncertainty quantification

Developing hybrid renewable energy systems (HRES) is crucial for ensuring affordable, sustainable, and reliable energy access in remote rural and island regions. Rich and detailed data play a pivotal role in the optimal design of HRES. However, remote areas often face data limitations due to inadequate metering infrastructure, which further increases uncertainties in HRES design. This paper proposes a planning optimization method that quantifies multiple uncertainties, requiring only limited information for the optimal HRES design. First, uncertainties in HRES planning are classified into different types based on intrinsic randomness, data availability, and impact level. Then, by integrating uncertainty quantification with credibility theory, knowledge-based scenario generation models are developed to leverage limited available information to generate extreme and typical operating conditions for a tailored HRES design. Finally, a fuzzy credibility-constrained stochastic-robust optimization model is proposed to optimize the design of HRES. A practical case facing significant data scarcity and gaps is used to validate the effectiveness of the proposed method. The proposed approach reduces the loss of load probability for HRES to 0.02% and lowers the levelized cost of energy to $0.16/kWh—33% lower than that of the deterministic method and 79% lower than that of the traditional stochastic method.

Scenario Generation Based on Ant Colony Optimization for Modelling Stochastic Variables in Power Systems

Uncertainty is an important subject in optimization problems due to the unpredictable nature of real variables in the power system area, which can condition the solution’s accuracy. The effective modelling of stochastic variables can contribute to the reduction in losses in the system under evaluation and facilitate the implementation of an effective response in advance. To model uncertainty variables, the most extended technique is the scenario generation (SG) method. This method evaluates possible combinations of complete curves. Classical scenario generation methods are founded on probability distributions or robust stochastic optimization. This paper proposes a novel approach for constructing scenarios using the Ant Colony Optimization (ACO) algorithm, referred to as ACO-SG. This methodology does not require a previous statistical study of uncertainty data to generate new scenarios. A historical dataset and the desired number of scenarios are the inputs inserted into the algorithm. In the case study, the algorithm used historical data from the Savona Campus Smart Polygeneration Microgrid of the University of Genoa. The approach was applied to generate scenarios of photovoltaic generation and building consumption. The low values of the Euclidean distance were used in order to check the validity of the scenarios. Moreover, the error deviation of the scenarios generated with the goal of daily power were 1.77% and 0.144% for the cases of PV generation and building consumption, respectively. The different results for both cases are explained by the characteristics of the specific cases. Despite these different results, both were significantly low, which indicates the capability of the algorithm to generate any kind of feature within curves and its adaptability to any case of SG.

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.

Budget allocation problem for projects with considering risks, robustness, resiliency, and sustainability requirements

Budget Allocation Problem (BAP) for projects is considered one of the critical risks that necessitate the completion of projects in the shortest possible duration. This aspect is significant in extensive projects such as national endeavors directly impacting people's livelihoods. This research focuses on introducing Sustainable, Robust, Resilient, and Risk-averse Budget Allocation for Projects (S3RBAP). A hybrid robust stochastic optimization approach was employed, incorporating Weighted VaR (WVaR) and the minimum function as risk criteria to ensure the robustness of the objective function. This model is designed to minimize the weighted expected value, WVaR, and the minimum progress function, thereby promoting the 3R and sustainability framework to tackle budget fluctuations and enhance project viability. The case study revolves around the construction of national projects in Iran. Ultimately, the project's required budget was allocated within the budgetary constraints. Sensitivity analysis found that the integration of 3R and sustainability led to a 13.5 % reduction in the progress function compared to scenarios without these principles. Furthermore, increasing the conservatism coefficient to 20 % decreased the progress function to −0.58 %. Reducing the resiliency coefficient had an adverse effect by disrupting the budget allocation process. The computational time is increased gradually with the escalation of problem complexity, while the progress function decreases due to the increased number of projects. The findings demonstrate that using an exponential function promotes risk aversion, whereas a sine function encourages risk-seeking behaviour within this research context.

A Two-Stage Hybrid Stochastic–Robust Coordination of Combined Energy Management and Self-Healing in Smart Distribution Networks Incorporating Multiple Microgrids

This paper presents a two-stage hybrid stochastic–robust coordination of energy management and self-healing in smart distribution networks with multiple microgrids. A multi-agent systems approach is first used for coupling energy management and self-healing strategies of microgrids, based on expert system rules. The second stage problem, a framework similar to that of the first stage, is then established for the smart distribution networks. Then, hybrid stochastic–robust optimization is used to model the uncertainties of demand, energy price, power generation of renewable energy sources, demand of electric vehicles, and accessibility of zone agents. Further, the grey wolf algorithm is used to solve the formulated optimization problem and achieve an optimal and reliable solution. The proposal is validated on a 69-bus distribution network consisting of three microgrids. The results validate that the proposal minimizes microgrids’ utilization indices, such as energy costs, energy losses, and network voltage drops, while simultaneously managing a flexible distribution network. It is also verified that the proposed multi-agent system design provides a high-speed and optimized self-healing solution for the network.

A viable supplier selection with order allocation by considering robustness, risk-averse and blockchain technology

The research introduces a novel approach called Viable Supplier Selection with Order Allocation considering Robustness and Risk-Aversion (VSSOARR), incorporating Blockchain Technology (BCT). The model utilizes a Robust Stochastic Optimization (RSO) technique with Weighted Value at Risk (WVaR) in the cost function. The objective function of this model combines the expected value and WVaR cost. The BCT platform is recommended to enhance agility in the face of disruptions. The research proposes sustainable requirements such as minimizing CO2 emissions and energy consumption management, ensuring minimum demand satisfaction, and incorporating facility capacity based on specific scenarios to enhance resiliency. The study involves various stakeholders, including suppliers and manufacturers. The main purpose of utilizing BCT is to facilitate swift transactions between components and support risk-averse decision-making. Integrating BCT as an agility instrument resulted in a 16.93% cost reduction for VSSOARR compared to scenarios where BCT was not considered. The research suggests incorporating viability with BCT positively impacts the supply chain’s overall performance. A sensitivity analysis was conducted on the conservatism coefficient, confidence level, agility rate, demand variation, and problem size.

Robust two‐level market coordinated transaction optimal model and benefit allocation strategy for a novel shared‐energy aggregator

AbstractShared‐energy storage (SES) can break the energy interaction barrier between the demand side and the supply side, which is becoming an option for improving the flexibility of energy systems. This study designed a novel structure for the shared‐energy aggregator (SEA), incorporating a microgrid (MG), prosumer and SES. Then, a two‐level market transaction optimization model for SEA, considering the robust stochastic optimization method, is proposed; namely, at the day‐ahead stage, a robust stochastic operation optimization model is constructed to achieve the dispatching plan from the demand side under the objective of achieving the minimum energy consumption cost for the prosumer, and at the real‐time market stage, a robust stochastic operation optimization model from the supply side is constructed to arrange the optimal dispatching strategy under the objective of the maximum aggregator benefits of MG and SES. Finally, an industrial park micro‐grid in Qinghai Province, China, is selected as an example for analysis.

Data-driven optimization for seismic-resilient power network planning

Many regions of the planet are exposed to seismic hazards that can cause devastating consequences on power systems. Due to these systems’ crucial role, the evaluation and planning for their safe and reliable operation are paramount. This paper develops a novel data-driven optimization framework to assess the power network’s seismic resilience and plan cost-effective investments for its enhancement. Under a robust optimization scheme, an earthquake attacker–defender model finds the worst-case realization of random earthquake network contingencies within an uncertainty set defined with a large number of scenarios generated by state-of-the-art engineering methods. Moreover, data-driven stochastic-robust optimization is employed in a two-stage seismic-resilient power network planning model, leveraging multiple seismic sources’ distributional information. Transmission line expansions and siting and sizing of battery energy storage systems are decided in the first stage, while the second stage decides operational variables. Experiments on a 281-node Chilean power system provide insights for seismic-resilient planning and demonstrate the efficiency of the proposed approach.

Net-zero, resilience, and agile closed-loop supply chain network design considering robustness and renewable energy.

The Net-zero, Resilience, and Agile Closed-Loop Supply Chain Network (NZRACLSCND) concept integrates net-zero, resiliency, and agility in a circular economy. Regarding net-zero, this research embeds renewable energy like solar energy and hybrid trucks to supply energy for facilities and transportation of goods and products between components. Applying redundancy, multi-source, and flexible capacity as resiliency strategies is suggested to cope with the demand disruption. Satisfaction demand level is utilized for the agile approach. This research proposes Robust Stochastic Optimization (RSO), including the weighted expected value and maximum CO2 for NZRACLSCND. This study locates and determines the flow of CLSC in the home appliance industry by considering NZRA, robustness, and risk against demand disruption. CO2 emission using the NZRA concept is 233.33% less than without considering NZRA concepts. In addition, the conservative coefficient, agile coefficient, decreased CO2 coefficient, and the model scale are analyzed. The results show that when the conservative coefficient increases, the risks of CO2 emission increase. In addition, when the agile coefficient increases, as a result, CO2 emission increases. Finally, when the decreased CO2 coefficient and the model scale increase, we can see that CO2 emission and cost are increased.

Day-ahead bidding and dispatch strategy of novel battery charging and swapping station under multiple uncertainties

The novel battery charging and swapping station (NBCSS) has great operational flexibility due to its integration of wind power, photovoltaic power, gas turbine and energy storage. This paper presents a day-ahead bidding and dispatch strategy of NBCSS under multiple uncertainties, which consists of battery demand, market price, renewable energy and load. Firstly, the optimized backup method is used to address the uncertainty of battery demand. Then, a two-stage stochastic robust optimization model is further applied to address the uncertainties of market price, renewable energy and load. The model’s goal is to minimize the total operational cost in the worst-case scenario, which is limited by the corresponding uncertainty set. Especially, to control the conservatism of the model, the total adjustable budget and temporal adjustable budget of uncertainties are considered together. Next, the two-stage model is solved efficiently based on the strong duality theory and column-and-constraint generation (C&CG) method. Finally, case studies are given to verify the effectiveness and performance of the proposed model. The results demonstrate that the NBCSS can be efficiently deployed, enabling market arbitrage and reducing the overall cost. Additionally, the conservatism of the proposed model can be adjusted according to the budgets of uncertainties.

Research on OPF Control of Three‐Phase Four‐Wire Low‐Voltage Distribution Network considering Uncertainty

As power systems become more complex and uncertain, low‐voltage distribution networks face numerous challenges, including three‐phase imbalances caused by asymmetrical loads and distributed energy resources. We propose a robust stochastic optimization (RSO)‐based optimal power flow (OPF) control method for three‐phase four‐wire low‐voltage distribution networks that consider uncertainty to address these issues. Using historical data and deep learning classification methods, the proposed method simulates optimal system behaviour without requiring communication infrastructure. The simulation results verify that the proposed method effectively controls the voltage and current amplitude while minimizing the operational cost and three‐phase imbalance within acceptable limits. The proposed method shows promise for managing uncertainties and optimizing performance in low‐voltage distribution networks.

Two-stage stochastic robust optimization scheduling of electric–thermal microgrid with solid electric thermal storage

During the periods of high heat load (HL) demand in winter, the increased heat output of combined heat and power units (CHPs) can significantly compress the electric power regulation range, thereby posing a threat to the safety of the power system. The solid electric thermal storage (SETS) can be employed as the regulating resource for both electric and thermal systems, expanding the dispatch space of microgrids to promote renewable energy consumption. In this paper, a two-stage stochastic robust optimization scheduling model of an electric–thermal microgrid with SETS is proposed, and the electric–thermal bi-directional regulation characteristics of SETS are considered. First, a SETS operation model based on the heat transfer characteristics of the homogeneous material-type thermal storage unit is established. Second, a regional thermal inertia model under a constant-flow variable-temperature system is established, which integrates HL at different locations to the heat source, avoiding the calculation of variation in water temperature, thus reducing the calculation difficulty. Finally, the two-stage stochastic robust optimization scheduling model of an electric–thermal microgrid with SETS is established. The model decouples the power and heat generation of CHP through the bi-directional regulation function of SETS. Case studies demonstrate the validity and effectiveness of the proposed scheduling model.

Aggregator-Network Coordinated Peer-to-Peer Multi-Energy Trading via Adaptive Robust Stochastic Optimization

Aggregator-Network Coordinated Peer-to-Peer Multi-Energy Trading via Adaptive Robust Stochastic Optimization

Optimization of intermodal routes for reefer containers under double uncertainty

Aiming at the influence of demand uncertainty and stochastic carbon trading price on the choice of intermodal transportation route for reefer containers, a hybrid robust stochastic optimization model with minimum economic cost is established, and a Monte Carlo sampling-based catastrophic adaptive genetic algorithm is designed to test the reasonableness and validity of the model. The results show that: influenced by the regret value constraint, the robust optimization of demand uncertainty will increase the total cost of reefer container multimodal transportation, while the increase of carbon trading price stochasticity doesn’t necessarily mean the increase of cost, and the reasonable setting of carbon trading price stochasticity and the trade-off between the maximum regret value and the economic cost, so that the cold-chain logistics and transportation enterprises can easily deal with the fluctuation of the demand of the freight transportation market, and also respond to the low-carbon requirements.

A data-driven robust optimization in viable supply chain network design by considering Open Innovation and Blockchain Technology

The research proposes a new method for Viable Supply Chain Network Design (VSCND) that incorporates Open Innovation (OI) and Blockchain Technology (BCT). A robust stochastic optimization and Conditional Value at Risk (CVaR) in the cost function is utilized to develop the model. This model's objective function incorporates the expected value, maximum cost, and CVaR cost. The OI and BCT platforms are suggested for an antifragile policy against disruption. In addition, CO2 emissions and energy consumption are proposed as sustainability requirements. Eventually, minimum demand satisfaction and resilience facilities by incorporating capacity based on specific scenarios are added to the model for an agile strategy. This research entails several parties, including vendors supplying the primary components and manufacturers creating products based on customer preferences. OI and BCT platforms aim to receive demand based on customer specifications and facilitate rapid transactions between components. Risk-averse decision-makers utilize a polyhedral Data-Driven Robust Optimization (DDRO) approach to manage uncertainty and flexible sets. Incorporating OI and BCT as antifragility instruments resulted in a 0.2% cost reduction for VSCNDOIBCT compared to when OI and BCT were not considered. This research suggests that integrating OI and BCT positively affects SC performance overall. The decreasing rate, DDRO coefficient, agility rate, demand variation, and problem size were subjected to a sensitivity analysis.

Hybrid robust-stochastic optimal scheduling for multi-objective home energy management with the consideration of uncertainties

Home energy management systems (HEMS) have transformed the traditional structure of electricity consumption on the customer side and facilitate real-time interaction between the customers and the grid. However, the HEMS scheduling is currently challenged by a variety of uncertainties, including photovoltaic (PV) output, electric vehicle (EV) charging/discharging behavior, and real-time pricing (RTP), which can seriously impact home equipment scheduling and critical objectives like total cost and users' comfort. Therefore, a hybrid robust-stochastic (HRS) optimization approach for multi-objective home energy management with the consideration of uncertainties and users' comfort has been proposed in this paper. Firstly, a detailed classification and modeling of the loads in the household are carried out to facilitate the exchange between the user-side resources and the grid in a benchmark based on RTP. Then the stochastic charging/discharging behavior of EV is modeled by stochastic optimization (SO) methods, and uncertainties in PV production and RTP are modeled by robust optimization (RO) methods, making full use of the flexibility of various uncertainty parameters. Meanwhile, the users' requirements for comfort are considered, and a multi-objective function for the economy and comfort of the HEMS is established. Finally, the effectiveness of the proposed method has been verified by case studies. The results show that the proposed HRS is effective to deal with different levels of the uncertainty parameters and ensures the users' comfort requirements in home energy system. Moreover, the users can make trade-offs between various levels of robust decisions according to their needs of economic and comfort level, thus obtaining a variety of electricity consumption decisions.

Robust planning for distributed energy storage systems considering location marginal prices of distribution networks

Energy storage plays an important role in integrating renewable energy sources and power systems, thus how to deploy growing distributed energy storage systems (DESSs) while meeting technical requirements of distribution networks is a challenging problem. This paper proposes an area-to-bus planning path with network constraints for DESSs under uncertainty. First, a distribution location marginal price (DLMP) formulation with maximum fluctuation boundaries of uncertainties is designed to select vulnerable areas exceeding voltage limits and higher line losses that occur in distribution networks. Different from simple multi-scenario power flow calculation and sensitivity analysis, DLMP with time and regional characteristics could be more intuitive to reflect line losses and voltage limits of distribution networks through price signals. After that, a two-stage stochastic robust optimization based planning method is developed to determine locations and capacities of DESSs in vulnerable areas. To make the uncertainty problem more tractable, stochastic scenarios are used to portray upper and lower boundaries of uncertainties, which avoids too-conservative decisions for robust optimization. Finally, numerical tests are implemented to testify the reasonability and validity of the proposed area-to-bus planning path under uncertainty. Compared with the DESSs planning framework without DLMP, the costs of DESSs are observably reduced with DLMP. With same budgets of uncertainty, investment costs of DESSs for the stochastic robust optimization with 30 and 50 scenarios are 3.91% and 4.45% lower than classical adaptive robust optimization (ARO).

A viable supply chain by considering vendor-managed-inventory with a consignment stock policy and learning approach

Inventory management challenges are optimizing inventory levels, minimizing stockouts, and streamlining the supply chain process for the vendor and the customer. The study tries to improve these challenges by proposing a new Viable Supply Chain (VSC) incorporating Vendor-Managed-Inventory (VMI) with a Consignment Stock (SC) policy for managing inventory between SC components. These strategies access the inventory level of the customer in the vendor, and if the items are sold, the customer pays the vendor for the inventory that was sold. The Robust Stochastic Optimization (RSO) with Conditional Value at Risk (CVaR) as risk criteria is utilized to solve the model. The objective function of this model combines the expected value, maximum, and CVaR of cost. To promote viability against disruption, sustainability requirements such as CO2 emissions and energy consumption are considered. An agile approach that includes a learning factor is suggested. In addition, a resiliency strategy by addressing order disruption requirements is proposed. The case study of this research is a small healthcare industry that prepares drug products. It is one of the largest industries with high sales and added value. The results show that the inclusion of VMI and CS in the main problem-solving approach resulted in a 14.8 % cost reduction compared to the scenario where VMI and CS were not considered. Finally, sensitivity analysis is performed on CVaR confidence, conservatism coefficient, learning rate, agility rate and scale of the main problem.