Stochastic Linear Programming Model Research Articles

Resilience enhancement of distribution network under typhoon disaster based on two-stage stochastic programming

The reliability of power supply in distribution network is vulnerable to extreme weather events such as typhoon. Pre-event preparation can effectively mitigate the deterioration of system resilience. Therefore, we propose a distribution network resilience enhancement decision-making framework which is formulated as a two-stage stochastic mixed-integer linear programming (SMILP) model. The first stage invests coordinately in four strategies, including hardening lines, installing distributed generators (DG), allocating mobile emergency generators (MEG), and deploying switches, etc. And the goal at the first stage is to minimize the investment cost of resilience enhancement strategies. The objective at the second stage is to ensure the minimum expected recourse operation cost of the comprehensive strategies for all typical scenarios. The proposed model can combine distribution network long-term resilience planning with short-term post-event recovery. Furthermore, in order to address the uncertain problems of wind field, line damage and load fluctuations under typhoon disaster, this paper proposes the following solutions. For wind field uncertainty, a detailed wind field model considering time transition, sea-land transition and extreme value distribution is established to deal with wind speed prediction. For line damage uncertainty, an improved stress-strength interference model is set up. And for the load fluctuation uncertainty, scenario generation using load random multipliers is applied to address load uncertainty. The proposed framework is tested in IEEE 33-bus distribution system using historical data from 2018 super typhoon “Mangkhut” in China and demonstrates the SMILP model can significantly reduce the post-event expected recourse operation cost and meanwhile improve the distribution network resilience.

Profit-Oriented BESS Siting and Sizing in Deregulated Distribution Systems

Within the deregulation process of distribution systems, the distribution locational marginal price (DLMP) provides effective market signals for future unit investment. In that context, this paper proposes a two-stage stochastic bilevel programming (TS-SBP) model for investors to best allocate battery energy storage systems (BESSs). The first stage obtains the optimal siting and sizing of BESSs on a limited budget. The second stage, a bilevel BESS arbitrage model, maximizes the arbitrage revenue in the upper level and clears the distribution market in the lower level. Karush-Kuhn-Tucker (KKT) optimality conditions, strong duality theory, and the big-M method are utilized to transform the TS-SBP model into a tractable two-stage stochastic mixed-integer linear programming (TS-SMILP) model. A novel statistics-based scenario extraction algorithm is proposed to generate a series of typical operating scenarios. Then, scale reduction strategies for BESS candidate buses and inactive voltage constraints are proposed to reduce the scale of the TS-SMILP model. Finally, case studies on the IEEE 33-bus and 123-bus systems validate the effectiveness of the DLMP in incentivizing BESS planning and the efficiency of the two proposed scale reduction strategies.

Multi-objective two-stage stochastic optimization model for post-disaster waste management

Abstract Post-disaster waste management is one of the most crucial tasks in the recovery phase of the disaster cycle, and it was created to assist affected communities in returning to a stable state following a disaster. To develop an efficient post-disaster waste management strategy, this study presents a multi-objective two-stage stochastic mixed integer linear programming model for post-disaster waste management. The proposed mathematical model was developed based on a mixed strategy of on-site and off-site waste separation in the supply chain. This study aims to minimize not only the total cost and the environmental impact to provide waste flow decisions and choose collection and separation sites, recycling sites, landfill sites, and incineration sites throughout the supply chain under the uncertain situation. To solve a multi-objective problem, a normalized weighted sum method is used to find the solution. A numerical case based on realistic data is presented to validate and verify the proposed model. Based on the numerical example, the results demonstrated that the implementation of the mixed strategy for waste separation with the consideration of uncertain situations can reduce the total cost, balance the environmental impact, and determine the unexpected situation in the post-disaster waste supply chain efficiently.

Multi-Objective Bus Timetable Coordination Considering Travel Time Uncertainty

This paper proposes a timetable coordination method for transfer problems in a bus transit system. With a given bus network, a stochastic mixed-integer linear programming (MILP) model has been formulated to obtain coordinated bus timetables with the objective of minimizing a weighted sum of the average value of total waiting time and its average absolute deviation value, allowing for random bus travel time. The vital decision variable is the terminal departure offset time of each target bus trip within a certain off-peak period. The robust MILP model can also be used to solve the first-bus transfer problem with the introduction of several new linear constraints. A solution method based on the Monte Carlo simulation has been developed to solve the MILP model. Numerical experiments have been conducted for different scenarios. The results indicate that bus timetables coordinated by the developed model are capable of substantially reducing waiting time for transfer and non-transfer passengers. In addition, the feasibility of simplifying a common sub-route into a single transfer stop in a timetable coordination problem has been explored based on numerical experiments.

A stochastic linear programming model for maximizing generation and firm output at a reliability in long-term hydropower reservoir operation

The firm output, usually determined in the design stage of a hydroplant to serve as a threshold to measure system reliability, can be regarded as an unknown parameter to be explored to its maximum in long-term hydropower reservoir operation (LHRO). An unknown firm output to be ensured at certain reliability, however, will make the problem much more nonlinear and then complicate the modeling. This work presents a stochastic linear programming (SLP-1) model that can explicitly incorporate reliability in ensuring an unknown firm output to be maximized by using a probability variable to represent a decision at a state and introducing binary variables to decide whether the decision will ensure the unknown firm output. The present SLP-1 is improved on 1) another previous SLP model (SLP-0) that must have a firm output prespecified at certain reliability, and compared with 2) the stochastic dynamic programming (SDP) model that can only manage to estimate the firm output at a reliability with trial and error. Case studies show the superiority of the present SLP to the SDP that can hardly make the reliability any closer to what the SLP can achieve, especially in ensuring a high firm output, with gaps to desired reliability ranging up to 42.33% for Xiaowan and 31.8% for Nuozhadu. Indeed, the SLP-1 will encounter the dimensional difficulty that needs further efforts to overcome when applied to cascaded reservoirs.

Improved managerial decisions for fuzzy stochastic bi-objective linear programming models: A Trade-off ratio-based autonomized approach

Improved managerial decisions for fuzzy stochastic bi-objective linear programming models: A Trade-off ratio-based autonomized approach

Improving distribution system resilience by undergrounding lines and deploying mobile generators

To improve the resilience of electric distribution systems, this paper proposes a stochastic multi-period mixed-integer linear programming model that determines where to underground new distribution lines and how to coordinate mobile generators in order to serve critical loads during extreme events. The proposed model represents the service restoration process using the linearized DistFlow approximation of the AC power flow equations as well as binary variables for the undergrounding decisions of the lines, the configurations of switches, and the locations of mobile generators during each time period. The model also enforces a radial configuration of the distribution network and considers the transportation times needed to deploy the mobile generators. It is shown that long-term line undergrounding decisions which are cognizant of short-term mobile generator deployments yield superior results relative to undergrounding decisions made without considering mobile generators. Using an extended version of the IEEE 123-bus test system, numerical simulations show that combining the ability to underground distribution lines with the deployment of mobile generators can significantly improve the resilience of the power supply to critical loads. • Resilience can be improved via mobile generators and underground lines. • Stochastic optimization for siting underground lines and operating mobile generators. • Show benefits from jointly considering undergrounding and mobile generator decisions.

A novel separate chance-constrained programming model to design a sustainable medical ventilator supply chain network during the Covid-19 pandemic

<p style='text-indent:20px;'>Providing new models or designing sustainable networks in recent studies represents a growing trend. However, there is still a gap in the simultaneous modeling of the three dimensions of sustainability in the electronic medical device supply chain (SC). In this paper, a novel hybrid chance-constrained programming and cost function model is presented for a green and sustainable closed-loop medical ventilator SC network design. To bring the problem closer to reality, a wide range of parameters including all cost parameters, demands, the upper bound of the released <inline-formula><tex-math id="M1">\begin{document}$ co_2 $\end{document}</tex-math></inline-formula>, and the minimum percentage of the units of product to be disposed and collected from a customer and to be dismantled and shipped from DCs are modeled as uncertain along with the normal probability distribution. The problem was first formulated into the framework of a bi-objective stochastic mixed-integer linear programming (MILP) model; then, it was reformulated into a tri-objective deterministic mixed-integer nonlinear programming (MINLP) one. In order to model the environmental sustainability dimension, in addition to handling the total greenhouse gas emissions, the total waste products were also controlled. The efficiency and applicability of the proposed model were tested in an Iranian medical ventilator production and distribution network. For sensitivity analyses, the effect of some critical parameters on the values of the objective functions was carefully examined. Finally, valuable managerial insights into the challenges of companies during the COVID-19 pandemic were presented. Numerical results showed that with the increase in the number of customers in the COVID-19 crisis, social responsibility could improve cost mean by up to 8%.</p>

A Stochastic Modeling Approach for Interplanetary Supply Chain Planning

The unique and complex challenges of sustainable interplanetary travel necessitate a novel space logistics paradigm, which is hereafter referred to as the interplanetary supply chain. To lay the groundwork for this anticipated paradigm shift, this work aims to introduce principles and methodologies of supply chain (SC) planning to the field of space logistics. To this end, a stochastic multistage mixed integer linear programming model is proposed that explores the vision of a permanent human presence on Mars from a logistics standpoint. The model optimizes material flows, propellant production, and in-space infrastructure allocations to identify the optimal SC network design. Notably, demand uncertainty at the Mars base is considered, which makes it the first model to investigate the effects of stochasticity in space logistics planning. Several case studies are investigated, which incorporate data on Earth–Mars transfer windows and flight trajectories obtained through the National Aeronautics and Space Administration’s trajectory browser. The results reveal numerous interdependencies and, thus, highlight the need for holistic SC modeling approaches that go beyond the isolated mission planning of the extant literature. For instance, the findings show frequent interactions among spacecraft as they regularly engage in division of labor regarding cargo and propellant transportation. Moreover, complex propellant supply networks that involve strategic in-space infrastructure allocation and propellant production on celestial bodies are established. Overall, the developed model provides valuable insights into the planning of a sustainable interplanetary travel, offers a new SC perspective, and lays the foundation for future research in this area.

Optimizing the aluminum supply chain network subject to the uncertainty of carbon emissions trading market

Aluminum production is characterized by high electricity consumption; hence its decarbonization hinges on upstream low-carbon electricity generation. As a result, the supply chain of aluminum product manufacturing can be strongly affected by the uncertainty of carbon emissions price via electricity generation. This paper investigates these impacts on the strategic design and the tactic operations of the aluminum supply chain by constructing a stochastic mixed-integer linear programming (SMILP) model, in which carbon price uncertainty and demand uncertainty are both accounted for. The supply chain case in the study covers a full process from upstream power generation to end-use consumption. A variety of stochastic carbon price scenarios are designed and assessed. Sensitivity analysis with respect to demand fluctuation is also performed. The results show that carbon price uncertainty significantly affects the configuration of the whole system. The main driving forces are the supplier selection and configuration alteration triggered by the carbon price variation, whereas other metrics such as carbon emissions costs and the responses to demand fluctuations display complicated patterns of change. This study provides a useful analytical framework that can be applied to the planning problem for other industries facing the same challenge.

A chance-constraint optimization model for a multi-echelon multi-product closed-loop supply chain considering brand diversity: An accelerated Benders decomposition algorithm

The closed-loop supply chain (CLSC) network design has become one of the most critical issues due to the importance of resource optimization. Moreover, increasing competition in commercial markets leads to the diversity of a brand’s product portfolio to meet the customer’s demand. Hence, this paper develops a multi-period, multi-brand stochastic mixed-integer linear programming (MILP) model with direct and indirect distribution for the proposed CLSC network. Because of the uncertain nature of demand, the uncertainties for new and second-hand product demand are considered. Besides, a chance-constraint optimization (CCO) approach is applied to deal with uncertainty. Moreover, the accelerated Benders decomposition (BD) algorithm is designed to solve the proposed model. Several test problems are created and used to solve the accelerated BD compared with the conventional BD algorithm to investigate this model. Finally, the results are compared and described analytically, and some future research is suggested.

A strategy to enhance the distribution systems recoverability via the simultaneous coordination of planning actions and operational resources

This paper presents a planning and operational strategy to improve the recoverability of distribution systems (DSs) to deal with a set of possible line fault scenarios. The strategy simultaneously optimizes the allocation of dispatchable distributed generation (DG) units while coordinating a dynamic restoration process based on a radial topology reconfiguration, an islanding operation, a demand response program, and the pre-positioning and dispatch of mobile emergency storage units. The uncertainty and variability associated with solar irradiation and demand are captured via a multi-period formulation based on a stochastic mixed-integer linear programming model. The objective function of this model minimizes the investment cost of new dispatchable DG units and the amount of energy shedding within the system. Simulations are performed on adapted 33-node and 53-node test systems to validate the proposed strategy under four different test conditions, numerical results reveal the advantages of simultaneously solving the planning and operational stages to improve the recoverability of the system.

Optimisation models for the day-ahead energy and reserve self-scheduling of a hybrid wind–battery virtual power plant

This work presents a suite of two optimisation models for the short-term self-scheduling and redispatch of a virtual power plant (VPP) composed of a wind farm and a Li-ion battery, that participates in the day-ahead energy and secondary regulation reserve markets of the Spanish power system. First, a two-stage stochastic mixed-integer linear programming model is used to obtain the VPP’s generation and reserve schedule and the opportunity cost of the energy stored in the battery. The model has an hourly resolution and a look-ahead period beyond the markets’ scheduling horizon and considers the hourly battery degradation costs as a function of both the depth of discharge and the discharge rate. Different strategies are evaluated to forecast the real-time use of the committed secondary regulation reserves. Second, a deterministic MILP model is used to determine the redispatch of the VPP using as input the generation and reserve schedule and the VPP’s storage opportunity cost provided by the former model and is executed on an hourly rolling basis. The results obtained show that the proposed models are effective for the short-term scheduling and redispatch of the VPP used with a low computational time, making them tractable and practical for their daily use.

A Benders decomposition approach for solving a two-stage local energy market problem under uncertainty

New consumer-centric electricity market schemes have emerged as an alternative to manage the energy balancing locally in distribution networks with high distributed energy resources (DERs) penetration. Peer-to-peer (P2P) energy trading has become an attractive scheme for reducing user and distribution system operators (DSO) costs through the energy surplus exchange between the energy community users, where the uncertainty of DERs and the consumption are challenging issues that require suitable managing for a proper market operation. This article presents a novel two-stage stochastic mixed-integer linear programming model to address the scheduling day-ahead problem of an energy community operating under a P2P energy trading scheme. The formulation proposed minimizes the expected community cost considering the network limitations, ensuring fair trading in the local energy market (LEM), and allowing the prosumers to act as buyers or sellers depending on their load consumption and self-generation. In order to reduce the shared information by consumers/prosumers that trade energy, the proposed model is decomposed into a master problem (MP) and subproblems (SPs) to manage the network constraints and the market requirements, respectively. With this purpose, the Benders decomposition approach is implemented using the recently introduced Strengthened Benders cuts to address the binary variables related to the market and battery operation in the SPs. The model and the algorithm are tested in the 69-bus radial distribution system, considering from 3 to 39 agents trading energy to measure the model scalability. In this analysis, the algorithm convergence shows that the proposed methodology reduces the LEM’s shared information without increasing the energy community cost.

Analytics with stochastic optimisation: experimental results of demand uncertainty in process industries

This study reports the test results of a two-stage stochastic linear programming (SLP) model with recourse using a user-friendly generic decision support system (DSS) in a North American steel company. This model has the flexibility to configure multiple material facilities, activities and storage areas in a multi-period and multi-scenario environment. The value of stochastic solution (VSS) with a real-world example has a potential benefit of US$ 24.61 million. Experiments were designed according to the potential joint probability distribution scenarios and the magnitude of demand variability. Overall, 144 SLP optimisation model instances were solved across four industries, namely, steel, aluminium, polymer and pharmaceuticals. The academic contribution of this research is two-fold: first, the potential contribution to profit in a steel company using an SLP model; and second, the optimisation empirical experiments confirm a pattern that the VSS and expected value of perfect information (EVPI) increase with the increase in demand variability. This study has implications for practicing managers seeking business solutions with prescriptive analytics using stochastic optimisation-based DSS. This study will attract more industry attention to business solutions, and the prescriptive analytics discipline will garner more scholarly and industry attention.

A two-stage stochastic programming model for seaport berth and channel planning with uncertainties in ship arrival and handling times

The limited availability of berths and channels is generally the bottleneck restricting the capacity of a seaport and thus resulting in traffic congestion. Optimizing the operations of the berths and channels has been recognized as a more economic avenue for mitigating seaport traffic congestion compared with channel dredging and berth expanding that needs significant capital and time costs. This paper presents a two-stage stochastic mixed integer linear programming model for the seaport berth and channel planning, aiming to minimize the expected total weighted completion times of ships under uncertain ship arrival times and ship handling durations. The first stage decides the berth allocation of ships under uncertainty. In the second stage, the channel planning, including the selection of lanes, assignment of tugboats, and sequencing of ships, is determined after the uncertainty has been realized. To effectively solve the model, we propose two tailored decomposition methods, that is, the stage decomposition method and the decomposition-based heuristic algorithm (DHA). Then, a lower bound of the problem is derived to evaluate the quality of the solution. Numerical experiments on Tianjin Port of China show the satisfactory performance of these two proposed methods. Especially, the DHA is able to obtain near-optimal solutions with the average optimality gap less than 3% within four-hour computational time for the instances up to 500 scenarios and 190 ship movements. Some managerial insights are obtained to guide the operations of the port.

Optimal location and operation of waste-to-energy plants when future waste composition is uncertain

In many countries, waste management is increasingly geared towards a circular economy, aiming for a sustainable society with less waste generation, fewer landfills, and a higher rate of recycling. Waste-to-Energy (WtE) plants, which convert waste into heat and energy, can contribute to the circular economy by utilizing types of waste that cannot be recycled. Due to the varying quality of sorting and socio-economic conditions in individual regions, the waste composition differs between regions and has an uncertain future development. Waste composition significantly affects the operation of WtE plants due to differences in energy potential. This paper supports strategic capacity planning for waste energy recovery by introducing a two-stage stochastic mixed-integer linear programming model that captures waste composition uncertainty through scenarios of possible future development. The results of the model provide insights into the economics of operation and identify important factors in the sustainability of the waste handling system. The model is demonstrated on an instance with six scenarios for waste management in the Czech Republic for the year 2030. The solution of the proposed model is to build 14 new WtE plants with a total capacity of 1970 kt in addition to the four existing plants with a capacity of 831 kt. The annual energy recovery capacity is expected to increase almost four times to satisfy EU directives that restrict waste landfilling.

A stochastic programming model for the aircraft sequencing and scheduling problem considering flight duration uncertainties

AbstractThis study presents a stochastic mixed-integer linear programming model for the aircraft sequencing and scheduling problem. The proposed model aims to minimise the average fuel consumption per aircraft in the Terminal Manoeuvring Area while considering uncertain flight durations for each flight. The tabu search algorithm was selected to solve the problem. The stochastic solution and deterministic solution results were compared to show the benefits of the stochastic solution. The average sample approximation technique was applied to this problem, and enhancement rates of the average fuel consumption per aircraft were 8.78% and 9.11% comparing the deterministic approach

The use of parametric programming and simulation-optimisation approaches for stochastic inventory control in the food markets under fuzzy deterioration rate

In this study, we address the stochastic inventory control problem of green vegetables deteriorating in a fruit and vegetable wholesaler. The problem is to re-order green vegetables periodically to meet demand of the green markets. The aim is to decide the order quantity and the order-up-to level for each period minimising the total inventory cost. In the fruit and vegetable wholesaler, a staff separates the fresh and non-fresh products to sustain freshness in the green vegetable crates. The behaviour of the wholesaler staff on selecting the non-fresh green vegetables is personal reasoning that fuzzy logic can help to deal with the uncertainty. Accordingly, we introduce a stochastic integer linear programming model accounting for the uncertain demand. We represent the deterioration rate indicated in the model by fuzzy numbers and transform the resulting fuzzy model into an α-parametric programming model. As a solution method, we propose a simulated annealing based simulation-optimisation algorithm. The models and the heuristic algorithm are applied to a case study that reflects the real settings of a food market in Turkey. The numerical analysis is conducted on the case study to understand the effect of the fuzziness on the solution quality and time under varying cost parameters and deterioration rates. As a result, the numerical analyses demonstrate that the proposed models allow us better to estimate the total costs and the waste quantity. The heuristic algorithm yields near-optimal solutions. The solutions approach approximately 3% difference on average for the medium and long term decisions. The heuristic algorithm results in significantly shorter computation times compared to the proposed model.

Operation of a Technical Virtual Power Plant Considering Diverse Distributed Energy Resources

Virtual power plants (VPPs) have emerged as a way to coordinate and control the growing number of distributed energy resources (DERs) within power systems. Typically, VPP models have focused on financial or commercial outcomes and have not considered the technical constraints of the distribution system. The objective of this article is the development of a technical VPP (TVPP) operational model to optimize the scheduling of a diverse set of DERs operating in a day-ahead energy market, considering grid management constraints. The effects on network congestion, voltage profiles, and power losses are presented and analyzed. In addition, the thermal comfort of the consumers is considered and the tradeoffs between comfort, cost, and technical constraints are presented. The model quantifies and allocates the benefits of the DER operation to the owners in a fair and efficient manner using the Vickrey–Clarke–Grove mechanism. This article develops a stochastic mixed-integer linear programming model and various case studies are thoroughly examined on the IEEE 119 bus test system. Results indicate that electric vehicles provide the largest marginal contribution to the TVPP, closely followed by solar photovoltaic (PV) units. Also, the results show that the operations of the TVPP improve financial metrics and increase consumer engagement while improving numerous technical operational metrics. The proposed TVPP model is shown to improve the ability of the system to incorporate DERs, including those from commercial buildings.