Two-stage Stochastic Mixed-integer Linear Programming Research Articles

Resilience-oriented planning for active distribution systems: A probabilistic approach considering regional weather profiles

Recent experiences with increasing climatic variability highlighted the importance of enhanced power distribution system resilience. Conventional methods of power distribution system planning usually focus on persistentinvestments associated with predicted faulty events, aiming to enhance the overall system reliability. Improving distribution system resilience in response to extreme weather events involves reduction of impact of the events having varying probability of occurrence. The probability of an extreme event varies significantly by location due to a combination of geographical, climatological, and meteorological factors. Thus, instead of requiring persistent investments, the solutions for resilience-oriented system upgrades should be guided by the potential effects and the probability of occurrence of extreme weather events in a specific area. To achieve this objective, the present paper proposes a two-stage stochastic mixed-integer linear programming (MILP) model based on regional weather profiles for the long-term resilience planning of distribution systems against extreme weather events. In the planning stage, investment decisions are made regarding overhead distribution line hardening and the distributed generation (DG) installation. During restoration process in the operational stage, dynamic microgrid formation is carried out for advanced distribution grid operations to minimize the cost corresponding to loss of load. The inclusion of regional weather profiles into planning objectives provide added flexibility to the network operators for analysing the trade-off between investment cost and load loss cost in resilient distribution system planning strategies.

Integrating pre and post-disaster activities for designing an equitable humanitarian relief supply chain

The design of an efficient humanitarian relief supply chain (HRSC) for disaster management depends on the integration of pre- and post-disaster decisions, which reduces the suffering of the victims and saves lives. The use of strategies such as quantity flexibility contracts in the pre-disaster phase and evacuation management in the post-disaster phase significantly increase the efficiency of HRSCs for disaster management. Therefore, in this paper, for the first time, a comprehensive two-stage stochastic mixed-integer linear programming model is presented for integrating the pre- and post-disaster activities for possible disasters such as floods and earthquakes by considering quantity flexibility contract and equitable relief goods distribution. In addition, the proposed model determines the location of warehouses and inventory levels in these warehouses in the pre-disaster phase. It also deals with public donations and budget planning, planning vehicles and helicopters, locating hospitals and shelters, and optimizing the inventory flow between echelons in the post-disaster phase. In the proposed model, the quantity flexibility contract is applied to decrease the inventory level in the pre-disaster phase, and reduce the supply risk in the post-disaster phase. The efficiency and performance of the proposed model are evaluated using the data related to a possible earthquake in Karaj (one of the metropolises of Iran).

Stochastic service network design: The value of fixed routes

This study introduces a challenging service network design problem with stochastic demands and fixed routes. In this problem, the routes of some service vehicles are fixed during the whole planning horizon, while the remaining vehicles are flexible in that their routes can adapt to different realizations of customer demands each day. The problem is to determine which vehicles should be set as fixed and how their fixed routes should be designed. To solve this problem, we formulate a two-stage stochastic mixed-integer linear program. The fixed routes are designed in the first stage before customer demands are realized, and the routes for flexible vehicles are designed in the second stage after customer demands are observed. A learning-based multiple scenario approach is developed as the solution method. Numerical experiments on real-world operational data from a logistics company show that fixing the routes of some service vehicles may reduce the operational cost by an average of approximately 6.37% compared with that in the case of no fixed routes.

Traffic signal control under stochastic traffic demand and vehicle turning via decentralized decomposition approaches

Traffic congestion is a global pressing issue but can be mitigated via effective traffic signal control schemes. In this paper, based on a cell transmission model we coordinate the control of traffic signals at multiple intersections to maximize vehicle throughput on corridors or road networks, under stochastic traffic demand and vehicle turning. We formulate a two-stage stochastic mixed-integer linear program using finite samples of the uncertain parameter, and combine Benders decomposition with the alternating direction method of multipliers to develop spatially-temporally distributed algorithms for optimizing the problem. We test instances of traffic signal control on corridors and grid networks, generated based on synthetic and real-world traffic data. Our results show that (i) considering traffic uncertainty can significantly improve the signal control quality and (ii) decentralized decomposition approaches can quickly find high-quality signal plans for multiple intersections in complex road networks, and fully utilize the computation and communication technologies in smart-transportation infrastructures.

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.

A Two-Stage Stochastic Fuzzy Mixed-Integer Linear Programming Approach for Water Resource Allocation under Uncertainty in Ajabshir Qaleh Chay Dam

Due to the dry climate and unsuitable distribution of rainfall in Iran, sustainable agriculture depends on the proper use of water resources. In this study, the optimal allocation of water at Ajabshir Qaleh Chay Dam in agricultural sector is investigated using an interval parameter two-stage stochastic mixed-integer linear programming approach. Indeed, interval parameters two-stage stochastic programming (ITSP) with fuzzy variables is developed based on mixed-integer programming for a water resource allocation model to retrieve the water shortage of agricultural products and to achieve the optimal allocation of Ajabshir Qaleh Chay Dam water through its river canals between different products under uncertainty conditions. In this developed method, called extended ITSP (EITSP), a number of alternatives are used to compensate for the difference between the amount of promised water allocation targets and the actual allocated water in the optimal allocation of water. Then a new solving approach based on Huang Algorithm, fuzzy chance constrained programming and Zimmermann fuzzy programming will be presented to solve the problems. Furthermore, using a case study in this dam, the results are obtained for the developed approaches to clarify the described methods and to compare these results with each other. Finally, comparing the total system profits of the models shows that in the fuzzy model, the profit and system certainty increase simultaneously. Therefore, due to the lack of water resources in the agricultural sector and the uncertainty, the agricultural authorities of Ajabshir can decrease the unsustainability of water resources using the optimal model while increasing the cost-effectiveness of the farmers.

STOCHOS: Stochastic opportunistic maintenance scheduling for offshore wind farms

Despite the promising outlook, the numerous economic and environmental benefits of offshore wind energy are still compromised by its high Operations and Maintenance (O&M) expenditures. On one hand, offshore-specific challenges such as site remoteness, harsh weather, transportation requirements, and production losses, significantly inflate the O&M costs relative to land-based wind farms. On the other hand, the uncertainties in weather conditions, asset degradation, and electricity prices largely constrain the farm operator’s ability to identify the time windows at which maintenance is possible, let alone optimal. In response, we propose STOCHOS, short for the stochastic holistic opportunistic scheduler—a maintenance scheduling approach tailored to address the unique challenges and uncertainties in offshore wind farms. Given probabilistic forecasts of key environmental and operational parameters, STOCHOS optimally schedules the offshore maintenance tasks by harnessing the opportunities that arise due to favorable weather conditions, on-site maintenance resources, and maximal operating revenues. STOCHOS is formulated as a two-stage stochastic mixed-integer linear program, which we solve using a scenario-based rolling horizon algorithm that aligns with the industrial practice. Tested on real-world data from the U.S. North Atlantic where several offshore wind farms are in-development, STOCHOS demonstrates considerable improvements relative to prevalent maintenance benchmarks, across various O&M metrics, including total cost, downtime, resource utilization, and maintenance interruptions.

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.

AWS IoT analytics platform for microgrid operation management

Microgrid (MG) represents a promising opportunity for integrating renewable energy systems with the electric power grid. However, numerous complexities need to be addressed in the process. The electrical grid is complex, vulnerable, and centralized. Thus, the integration is challenging owing to the stochastic nature of renewable energy generation, which affects the possibility of reliable forecasting. The wastage due to poor estimation of clean energy generation discourages new investments in this area. However, recent advancements in big data technologies enable processing a large amount of data captured from multiple sources in real-time. It opens the possibility of improving the operational optimization of MGs and the performance of forecasting models. The overall MG problem is formulated using a two-stage stochastic mixed-integer linear programming problem with recourse. Amazon Web Services (AWS) IoT analytics platform inputs data in real-time and runs a sophisticated wind generation forecast analysis. The stochastic model is solved using the Sample Average Approximation (SAA) algorithm. The innovative methodology leads to significant improvements in the total average operating cost by integrating AWS IoT Analytics compared to traditional methods that use historical data. Computations are performed for different power-grid settings, including a power-outage, and different power generators units capacities with total operation average cost savings of 7.6% and 5.9%, respectively. Sensitivity analysis showed that the SSA algorithm could solve all the instances by providing high-quality solutions. The AWS IoT strategy outperformed at 7.7 % and 3.6% for optimality gap and CPU time, respectively. We examined an actual case in Peru for an agricultural application to assess the performance of a stochastic optimization model with a real-time IoT wind generation forecast strategy. The results revealed the following capabilities of our novel framework: (1) it can realize higher cost savings from the MG operating systems; (2) improve real-time renewable energy forecasting; (3) facilitate robust decision-making under conditions of uncertainty.

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.

Stochastic program for disassembly lot-sizing under uncertain component refurbishing lead times

Planning disassembly operations for a given demand in components is challenging in practice because the quality of recovered components is very uncertain, and thus the duration of refurbishing operations is unpredictable. In this paper, we address the capacitated disassembly lot-sizing problems under uncertain refurbishing durations. More precisely, we consider a two-level disassembly system with a single type of end-of-life product, a dynamic demand, and stochastic refurbishing lead times for all components. To deal with the static decision frameworks, this problem is modeled as a two-stage stochastic Mixed-Integer Linear Program (MILP), where the objective is to minimize the expected total cost. To alleviate the scalability issues, we propose a reformulation of the inventory constraint that significantly reduces the number of scenarios. In addition, to solve large scale problems, we couple this reformulation with Monte–Carlo sampling. We provide a rolling horizon approach to deal with the static decision framework, where disassembly decisions are updated when new information unfolds. Experimental results show the effectiveness of the proposed models and the convergence of the resulting Sample Average Approximation (SAA) estimator.

Resilience-oriented operation of power systems: Hierarchical partitioning-based approach

As an achievement of innovations resulting from partitioning mechanisms, these mechanisms can contribute to the more flexible operation of power systems in local communities. The ever-increasing frequency and severity of unexpected real-time failures have created challenges for partitioned-based power system operators, affecting each partition’s resiliency. With this in mind, this paper presents an adaptive local operation strategy (ALOS) for resilient scheduling of the renewable-dominated partitioned-based power systems under normal and islanding modes in a decentralized manner. The main objective of the developed ALOS lies in reaching an affordable preparedness level in each partition to deal with unscheduled islanding mode, which can occur subsequent to real-time failures at common lines between adjacent partitions on transmission level. To this end, a set of resilience-target constraints is presented to prepare sufficient spinning reserve capacity in each partition to ensure continuity of supply during islanding mode. The proposed strategy is formulated as a two-stage stochastic mixed-integer linear program (MILP), and the nested formation algorithm is employed to execute it in a hierarchical fashion based on the privacy-preserving protocols. Besides, the tri-state compressed air energy storage (CAES) system is also included in the proposed strategy to mitigate the negative consequences caused by real-time failures and uncertain sources. Numerical results conducted on the IEEE 30-bus test system reveal that the proposed ALOS can enhance the resilience of each partition in responding to unscheduled islanding mode by efficiently utilizing all available capacities on the generation side. Furthermore, the DIgSILENT PowerFactory is used to identify the worst possible series of events and to evaluate the effectiveness of the proposed resilience-promoting proactive strategy in dealing with these events.

Design of Sustainable Renewable-Based Utility Plants in the Face of Uncertainty

This work presents the formulation of a two-stage stochastic mixed-integer linear programming (MILP) model to include uncertainty in the design of renewable-based utility plants. The model is based on a superstructure that integrates technologies to process biomass, waste, solar radiation, and wind and considers uncertainty in availability of the renewable resources and on the utility demands. The uncertain parameter space is calculated based on a monthly probability density function for each uncertain parameter and discretized into different levels. It is shown that as uncertainty is considered in the model formulation, design flexibility improves with respect to the deterministic-based designs, although the flexibility is achieved at the expense of higher underused facilities and therefore unused investment cost.

Stochastic pre-event preparation for enhancing resilience of distribution systems

Extreme weather events are the common causes for power supply interruptions and power outages in electrical distribution systems. Improving the distribution system and enhancing its resilience is becoming crucial due to the increased frequency of extreme weather events. Preparation and allocation of multiple flexible resources, such as mobile resources, fuel resources, and labor resources before extreme weather events can mitigate the effects of extreme weather events and enhance the resilience of power distribution systems. In this paper, a two-stage stochastic mixed-integer linear programming (SMILP) is proposed to optimize the preparation and resource allocation process for upcoming extreme weather events, which leads to faster and more efficient post-event restoration. The objective of the proposed two-stage SMILP is to maximize the served load and minimize the operating cost of flexible resources. The first stage in the optimization problem selects the amounts and locations of different resources. The second stage considers the operational constraints of the distribution system and repair crew scheduling constraints. The proposed stochastic pre-event preparation model is solved by a scenario decomposition method, Progressive Hedging (PH), to ease the computational complexity introduced by a large number of scenarios. Furthermore, to show the impact of solar photovoltaic (PV) generation on system resilience, three types of PV systems are considered during a power outage and the resilience improvements with different PV penetration levels are compared. Numerical results from simulations on a large-scale (more than 10,000 nodes) distribution feeder have been used to validate the effectiveness and scalability of the proposed method.

Role of vanadium redox flow batteries in the energy management system of isolated microgrids

The small size of isolated microgrids (IMGs), high intermittency and fluctuations of renewable generations, and low inertia of distributed energy resources make the frequency an important factor in the operation of IMGs that should be managed considering technical and economic constraints. The fast response time and long lifecycle of vanadium redox flow batteries (VRFBs) make them a promising candidate for energy and frequency management of IMGs. Accordingly, this paper proposes a novel energy management system (EMS) for IMGs which aims to optimize energy and reserve scheduling of VRFBs in coordination with droop controlled distributed generators. The objective of the EMS is to minimize the total operation cost of the IMG while dynamic and static frequency securities are preserved. In the proposed EMS, the hierarchical control structure of IMGs is precisely formulated and the VRFB is incorporated in the EMS to provide both energy and reserve in primary and secondary control levels. Moreover, to get accurate results, the nonlinear characteristics of VRFBs are considered in the developed model and linearized by an efficient approach. The proposed EMS is formulated as a two-stage stochastic mixed-integer linear programming problem (MILP) that guarantees the global optimal solution. Case studies on a test IMG indicate that optimal coordination of VRFBs in energy and reserve scheduling can reduce the operation cost more than 11.67%.

A Benders-Decomposition and Meta-Heuristic Algorithm for a Bi- Objective Stochastic Reliable Capacitated Facility Location Problem Not Dealing with Benders Feasibility-Cut Stage

This paper addresses a bi-objective two-stage stochastic mixed-integer linear programming model for a stochastic reliable capacitated facility location in which the optimum numbers, locations and as well as shipment quantity of the product between the network nodes for all scenarios should be determined. Unlike most of previous relevant works, multiple levels of capacities available to the manufacturers in different scenarios are permitted in this study. The proposed objectives of the model include: the minimization of expected sum of installation, production, transportation under uncertainty of parameters, such as transportation and production and disruption of facilities, as well as minimizing expected standard deviation of network costs for whole scenarios. Since one of the most important reasons for researchers' reluctance to apply Benders-decomposition algorithm in facility-location concept is the time-consuming nature of its feasibility-cut stage, one of the most outstanding innovation in this paper is to add a strengthening redundant constraint to the proposed model in order to eliminate the mechanism related to feasibility cuts in master problem. to the best of our knowledge, it is the first time that this technique, not being involved in keeping master-problem feasibility, is used to solve a reliable capacitated facility location problem. In this approach, in terms of time-consuming the Benders algorithm is able to powerfully compete with metaheuristic algorithms, but with an exact solution. To prove advantage of this algorithm satisfying both ultimate solution optimality and appropriate running time compared to metaheuristic algorithms at the same time, one metaheuristic algorithm, namely Imperialist Competitive Algorithm (ICA), is presented. Usefulness and practicality of the proposed model and solution method demonstrated through a case example in different class with variable size.

Managing access to primary care clinics using scheduling templates.

An important challenge confronting healthcare is the effective management of access to primary care. Appointment scheduling policies/templates can help strike an effective balance between the lead-time to an appointment (a.k.a. indirect waiting time, measuring the difference between a patient's desired and actual appointment dates) and waiting times at the clinic on the day of the appointment (a.k.a. direct waiting time). We propose methods for identifying effective appointment scheduling templates using a two-stage stochastic mixed-integer linear program model. The model embeds simulation for accurate evaluation of direct waiting times and uses sample average approximation method for computational efficiency. The model accounts for patients' no-show behaviors, provider availability, overbooking, demand uncertainty, and overtime constraints. The model allows the scheduling templates to be potentially updated at regular intervals while minimizing the patient expected waiting times and balancing provider utilization. Proposed methods are validated using data from the U.S. Department of Veterans Affairs (VA) primary care clinics.

A Stochastic Planning Model for Improving Resilience of Distribution System Considering Master-Slave Distributed Generators and Network Reconfiguration

The recent experiences of extreme weather events highlight the significance of boosting the resilience of distribution systems. In this situation, the resilience of distribution systems planning leads to an efficient solution for protecting the system from these events via line hardening and the installation of distributed generators (DGs). For this aim, this study presents a new two-stage stochastic mixed-integer linear programming model (SMILP) to hedge against natural disaster uncertainty. The first stage involves making investment decisions about line hardening and DG installation. Then, in the second stage, the dynamic microgrids are created according to a master-slave concept with the ability of integrating distributed generators to minimize the cost of loss of load in each uncertain outage scenario. In particular, this paper presents an approach to select the line damage scenarios for the SMILP. In addition, the operational strategies such as load control capability, microgrid formation and network reconfiguration are integrated into the distribution system plans for resilience improvement in both planning and emergency response steps. The simulation results for an IEEE 33-bus test system demonstrate the effectiveness of the proposed model in improving disaster-induced the resilience of distribution systems.