Stochastic Mixed-integer Linear Programming Research Articles

Impact of government subsidies on a cellulosic biofuel sector with diverse risk preferences toward feedstock uncertainty

Investment risk and uncertainty about the availability of biomass feedstock hinders the development of a mature cellulosic biofuel sector. The Biomass Crop Assistance Program (BCAP) is a federal program designed to subsidize farmers to establish, produce and deliver biomass feedstock to biorefineries. This study evaluated the impacts of BCAP on the optimal biofuel supply chain decisions considering feedstock yield uncertainty and associated investment risk given diverse risk preferences of the biofuel sector. The expected cost for a risk-neutral biofuel sector was minimized using a two-stage stochastic mixed integer linear program, whereas the Conditional Value-at-Risk of the supply chain was optimized for a risk-averse sector. Ex-ante analysis of a switchgrass-based biofuel sector in west Tennessee indicates BCAP payments could lower expected cost and investment risk for both risk-averse and risk-neutral biofuel sectors. However, the cost saving and risk reduction resulting from BCAP incentives for the risk-averse biofuel sector were higher than the risk-neutral biofuel sector. In addition, BCAP payments may drive more cropland to be converted for switchgrass, which potentially mitigates water-induced soil erosion and reduces greenhouse gas emissions associated with net carbon sequestration, but may also create unintended consequence of competition for land between food and fuel use.

A risk-based optimal self-scheduling of smart energy hub in the day-ahead and regulation markets

Utilizing multi-carrier energies such as wind energy, electric vehicle (EV) and battery banks are a significant step toward a cleaner production. Hence, This paper proposes a stochastic-based decision-making framework for the efficient short-term management of smart energy hub (EH) in restructured power systems with high penetration of renewable energy. The electrical and natural gas carriers are the input of smart EH, while the electricity and heat demands are considered as the outputs. The EH, including battery storage system (BSS) and EV fleet, is managed in the regulation market and day-ahead (DA) horizons. The energy hub operator makes optimal decisions regarding the natural gas network and energy supply for the thermal and electricity customers. An optimal self-scheduling model is developed to take into account the day ahead (DA) and regulation markets (RM) and decisions regarding generations of electrical and thermal devices as well as EV aggregator decisions. The primary goal of the proposed framework is minimizing the cost of procuring electricity and heat energy carriers in DA and regulation markets, including upward/downward regulations via a stochastic mixed-integer linear programming (MILP) approach. In order to get the uncertainty around the exact outcomes of RM prices, wind generation, and EV patterns, the model also takes into account the conditional value at risk (CVaR) term. The proposed formulation is examined by applying to a smart EH. Results show the effectiveness and usefulness of the proposed framework in managing the smart EHs efficiently.

Optimal self-scheduling of home energy management system in the presence of photovoltaic power generation and batteries

Today, the fact that consumers are becoming more active in electrical power systems, along with the development in electronic and control devices, makes the design of Home Energy Management Systems (HEMSs) an expedient approach to mitigate their costs. The added costs incurred by consumers are mainly paying for the peak-load demand and the system’s operation and maintenance. Thus, developing and utilizing an efficient HEMS would provide an opportunity both to the end-users and system operators to reduce their costs. Accordingly, this paper proposes an effective HEMS design for the self-scheduling of assets of a residential end-user. The suggested model considers the existence of a dynamic pricing scheme such as Real-Time Pricing (RTP), Time-of-Use (TOU), and Inclining Block Rate (IBR), which are effective Demand Response Programs (DRPs) put in place to alleviate the energy bill of consumers and incentivize demand-side participation in power systems. In this respect, the self-scheduling problem is modeled using a stochastic Mixed-Integer Linear Programming (MILP) framework, which allows optimal determination of the status of the home appliances throughout the day, obtaining the global optimal solution with a fast convergence rate. It is noted that the consumer is equipped with self-generation assets through a Photovoltaic (PV) panel and a battery. This system would make the consumers have energy arbitrage and transact energy with the utility grid. Consequently, the proposed model is demonstrated by determining the best operation schedule for different case studies, highlighting the impact each different DRP has on designing and utilizing the HEMS system for best results.

Aircraft sequencing and scheduling in TMAs under wind direction uncertainties

ABSTRACTAircraft sequencing and scheduling within terminal airspaces has become more complicated due to increased air traffic demand and airspace complexity. A stochastic mixed-integer linear programming model is proposed to handle aircraft sequencing and scheduling problems using the simulated annealing algorithm. The proposed model allows for proper aircraft sequencing considering wind direction uncertainties, which are critical in the decision-making process. The proposed model aims to minimise total aircraft delay for a runway airport serving mixed operations. To test the stochastic model, an appropriate number of scenarios were generated for different air traffic demand rates. The results indicate that the stochastic model reduces the total aircraft delay considerably when compared with the deterministic approach.

Optimization of CCUS Supply Chains for Some European Countries under the Uncertainty

This paper develops a two-stage stochastic mixed integer linear programming model to optimize Carbon Capture, Utilization and Storage (CCUS) supply chains in Italy, Germany and the UK. Few works are present in the literature about this topic, thus this paper overcomes this limitation considering carbon supply chains producing different products. The objective of the numerical models is to minimize expected total costs, under the uncertainties of the production costs of carbon-dioxide-based compounds. Once carbon dioxide emissions that should be avoided are fixed, according to environmental protection requirements for each country, the optimal design of these supply chains is obtained finding the distribution of carbon dioxide captured between utilization and storage sections, the amount of different carbon-based products and the best connection between each element inside the system. The expected total costs for the CCUS supply chain of Italy, Germany and the UK are, respectively, 77.3, 98.0 and 1.05 billion€/year (1004, 613 and 164 €/ton CO2 captured). A comparison with the respective deterministic model, analyzed elsewhere, is considered through the evaluation of the Expected Value of Perfect Information (EVPI) and the Value of Stochastic Solution (VSS). The former is 1.29 billion€/year, 0.18 million€/year and 8.31 billion€/year, respectively, for the CCUS of Italy, the UK and Germany. VSS on the other hand is equal to 1.56 billion€/year, 0 €/year and 0.1 billion€/year, respectively, for the frameworks of Italy, the UK and Germany. The results show that the uncertain production cost in the stochastic model does not have a significant effect on the results; thus, in this case, there are few advantages in solving a stochastic model instead of the deterministic one.

A stochastic network design problem for hazardous waste management

Hazardous waste management is of paramount importance due to the potential threats posed to the environment and local residents. The design of a hazardous waste management system involves several important decisions, i.e., the determination of the locations and sizes of treatment, recycling and disposal facilities, and organizing the transportation of hazardous waste among different facilities. In this paper, we proposed a novel stochastic bi-objective mixed integer linear program (MILP) to support these decisions in order to reduce the population exposure to risk while simultaneously maintaining a high cost efficiency of the transportation and treatment of hazardous waste. Moreover, considering the inherent uncertainty within the planning horizon, the cost, demand and affected population are defined as stochastic parameters. A sample average approximation based goal programming (SAA-GP) approach is used to solve the mathematical model. The proposed model and solution method are validated through numerical experiments whose results show that uncertainty may not only affect the objective value but also lead to different strategic decisions in the network design of a hazardous waste management system. In this regard, the strategic decisions obtained by the stochastic model is more robust to the change of external environment. Finally, the model is applied in a real-world case study of healthcare waste management in Wuhan, China, in order to show its applicability.

Distribution System Resilience Enhancement via Mobile Emergency Generators

Natural calamities always have been a serious threat to energy systems. In this regard, this paper constitutes a stochastic mixed integer linear programming (SMILP) model to enhance the resilience of power distribution systems to deal with disastrous events. In particular, the proposed model is developed to enhance both survivability and restoration capability of distribution systems. In this regard, to increase the preparedness of the power distribution system, the system operator reconfigures the network by utilizing remote-control switches (RCSs), manual switches (MSs), and distributed generations (DGs) before the natural calamity hits. The proposed model contemplates the traveling time of crew teams to the MS sites in the transportation system in order to achieve the best switching sequence. In addition, it pre-positions the crews and mobile emergency generators (MEGs) in staging locations to hasten the likely post-disturbance operations. To do so, likely post-event operations are included in the model using the scenario generation technique. To validate the performance of the developed model a distribution system is employed. The results of simulations confirm the effectiveness of the proposed approach in declining the interruption of electric energy for customers.

Stochastic Operation of a Solar-Powered Smart Home: Capturing Thermal Load Uncertainties

This study develops a mixed-integer linear programming (MILP) model for the optimal and stochastic operation scheduling of smart buildings. The aim of this study is to match the electricity demand with the intermittent solar-based renewable resources profile and to minimize the energy cost. The main contribution of the proposed model addresses uncertainties of the thermal load in smart buildings by considering detailed types of loads such as hot water, heating, and ventilation loads. In smart grids, buildings are no longer passive consumers. They are controllable loads, which can be used for demand-side energy management. Smart homes, as a domain of Internet of Things (IoT), enable energy systems of the buildings to operate as an active load in smart grids. The proposed formulation is cast as a stochastic MILP model for a 24-h horizon in order to minimize the total energy cost. In this study, Monte Carlo simulation technique is used to generate 1000 random scenarios for two environmental factors: the outdoor temperature, and solar radiation. Therefore in the proposed model, the thermal load, the output power of the photovoltaic panel, solar collector power generation, and electricity load become stochastic parameters. The proposed model results in an energy cost-saving of 20%, and a decrease of the peak electricity demand from 7.6 KWh to 4.2 KWh.

A stochastic approach for integrated production and distribution planning in dairy supply chains

This work addresses production and distribution planning for a real-world dairy supply chain. The planning model accounts for the production and distribution of Cheese, Yogurt, Powdered Milk and UHT milk products across a two-echelon Supply Chain. This task is undermined by the inherent variability of raw materials and finished products demand. The integrated production and distribution planning methodology introduced is based on a two-stage stochastic mixed integer linear programming formulation. In real-world settings the number of scenarios grows substantially; thus, a scenario reduction strategy based on clustering techniques is given. A decomposition and solving strategy is also introduced and applied to a real-world case study. This study showed that the value of the stochastic solution might rise to 21.1% above the deterministic solution. This indicates the importance of considering uncertainty for dairy production and distribution. Besides, different-size instances are tested to study the scalability of the solution approach.

Markovian-based stochastic operation optimization of multiple distributed energy systems with renewables in a local energy community

Local energy communities (LECs), as locally and collectively organized multi-energy systems, are expected to play an important role in energy transition, since they enable deployment of sustainable energy technologies and consumer engagement, bringing various benefits to the community users and contributing to the overall energy and climate objectives. An LEC may consist of multiple distributed energy systems (DESs), which, interconnected through local grid and heating network, can share power and thermal energy with no costs for community's users. This paper focuses on stochastic daily operation optimization of multiple DESs with renewables in an LEC. The problem is to find the optimized operation strategies of energy devices in each DES, and decide the amount of electrical and thermal energy to be shared among DESs with the objective to minimize the total expected net energy and CO2 emission cost of the LEC, while meeting given day-ahead demand of community's users. The problem is challenging because of the intermittent and uncertain nature of renewable generation and the coupling of energy devices and energy processes intra and inter DESs. To address these issues, a stochastic mixed-integer linear programming model is established with uncertain renewable generation modeled by a Markovian process to avoid the difficulties and drawbacks associated with scenario-based methods. The problem is solved by using branch-and-cut. Numerical testing results show that the total expected cost of the LEC is reduced by the integrated management of the DESs as compared to the costs attained under other operation modes where there are no interconnections among DESs, demonstrating the potential benefits that can be achieved with LECs through the optimized management of local energy resources aiming to foster efficient use of the available energy. Results also highlight the benefits of the stochastic approach as compared with the deterministic one.

A hierarchical energy management system for islanded multi-microgrid clusters considering frequency security constraints

With the widespread development of microgrids (MGs) in future smart distribution networks, a number of neighboring MGs can be connected and form a multi-microgrid (MMG) cluster. In this regard, the energy management of a MMG is challenging due to more complex components and higher degrees of uncertainty in a small region of power system. Likewise, in the islanded MMG (IMMG) clusters, due to the low-inertia and high intermittent energy delivery of renewable resources, the frequency security should be considered in the energy management. To address this issue, this paper proposes an energy management system (EMS) in which hierarchical control structure of IMMG clusters is precisely modeled. The proposed EMS aims to minimize total operation cost of IMMG cluster while sufficient primary and secondary reserves are scheduled to preserve frequency security in a predefined range. Besides, the proposed EMS provides optimal strategies for MGs to exchange energy and reserves during scheduling horizon. To consider operational uncertainties, the proposed EMS is formulated as a two-stage stochastic mixed-integer linear programming problem that guaranties the global optimal solution. The obtained results verify that through the proposed EMS, total operation cost of the IMMG cluster is minimized while the frequency can be cost-effectively preserved within a pre-defined secure range.

The value of physical distribution flexibility in serving dense and uncertain urban markets

Abstract Urban last-mile distribution in emerging economies suffers from unique levels of operational complexity, largely due to the prevailing dominance of the highly fragmented traditional retail channel. To support companies in strategically designing efficient urban distribution networks in such uncertain market environments, we propose a large-scale stochastic mixed integer linear programming model that incorporates three commonly deployed measures of physical distribution flexibility. Being able to flexibly adjust transportation capacities, facility capacities, and demand allocations may enable companies to serve urban markets more efficiently, especially in the presence of demand uncertainty. The model supports strategic network design decisions by optimizing the number, type, and capacity of distribution facilities as well as the composition of corresponding vehicle fleets. We apply the model to two large-scale, real-world case studies based on real data from Coca-Cola Femsa’s last-mile operations in Bogota and Cali, Colombia. These case studies are representative for many last-mile distribution problems in emerging market megacities. Our numerical results demonstrate how considering stochasticity and incorporating physical distribution flexibility in the strategic design of urban last-mile distribution networks jointly and independently affect the resulting network design and improve its commercial and operational performance. The stochastic design approach outperforms the deterministic design approach on every instance in terms of expected cost and performance risk. In most cases, physical distribution flexibility improves performance. However, we show that in some cases, if strategic decisions are based on deterministic assumptions, allowing for operational flexibility may result in a deterioration in network performance.

Optimal Asset Planning for Prosumers Considering Energy Storage and Photovoltaic (PV) Units: A Stochastic Approach

In the distribution system, customers have increasingly use renewable energy sources and battery energy storage systems (BESS), transforming traditional loads into active prosumers. Therefore, methodologies are needed to provide prosumers with tools to optimize their investments and increase business opportunities. In this paper, a stochastic mixed integer linear programming (MILP) formulation is proposed to solve for optimal sizes of prosumer assets, considering the use of a BESS and photovoltaic (PV) units. The objective is to minimize the total cost of the system, which is defined as the combination of a solar PV system investment, BESS investment, maintenance costs of assets, and the cost of electricity supplied by the grid. The developed method defines the optimal size of PV units, the power/energy capacities of the BESS, and the optimal value for initial energy stored in the BESS. Both deterministic and stochastic approaches were explored. For each approach, the proposed model was tested for three cases, providing a varying combination of the use of grid power, PV units, and BESS. The optimal values from each case were compared, showing that there is potential to achieve more economic plans for prosumers when PV and BESS technologies are taken into account.

Relating single-scenario facets to the convex hull of the extensive form of a stochastic single-node flow polytope

We study when a facet-defining inequality for a deterministic, single-scenario subproblem is also facet-defining for the extensive form of a two-stage stochastic mixed-integer linear program (SMIP). To answer this question, we introduce a novel stochastic variant of the well-known single-node flow (SNF) polytope, and present necessary and sufficient conditions for single-scenario facet-defining inequalities to be facet-defining for the extensive form. We further demonstrate that our stochastic SNF polytope is a relaxation of a broad subclass of SMIPs, illustrating its generality.

A sensitivity analysis of stochastic programming for reverse logistic of herbs agro-industry: a case study of herbs logistic in Indonesia

This paper analyses the sensitivity of reverse logistic formulation of herbs agro- industry based on fuzzy stochastic mixed integer linear programming. A case study from real world problem of herbs logistic in Indonesia is provided in order to respond stochastic challenges in the reverse logistic system. For implementation purpose of this current progress, some related historical and hypothetical data were deployed. The model was then used to test how far this fuzzy quantitative modelling is capable to solve the problem within available data ranges with consideration on possibility in each data occurrence. A GRG non-linear was used as model solution to solve the fuzzy stochastic modelling with implementation using Excel solver. The fuzzy quantitative modelling result with a case study in herbs logistic in Indonesia is concluded with verification and validation on current model formulation for decision making purposes in herbs reverse logistic.

Multi-stage stochastic framework for energy management of virtual power plants considering electric vehicles and demand response programs

Nowadays, the importance of the coordinated operation of power system assets has caused severe challenges that can be addressed through some recently emerged concepts, such as microgrids (MGs) and virtual power plants (VPPs). In this paper, a hierarchical model is proposed for simultaneous modeling of an MG scheduling and VPP energy management problems. Given the stochastic nature of the scheduling inputs, power production and also, load demand uncertainties are modeled using a scenario-based method. Therefore, the final model is presented as a stochastic mixed-integer linear programming (MILP) model. It is possible to cover the fluctuations at the minimum cost using demand response (DR) programs and electric vehicles (EVs). In addition, the possibility of power transaction between the adjacent MGs is analyzed in scheduling model, preventing unreal power exchanges. The simulation results indicate that the coordinated operation of MGs can improve load restoration, and reduce the load supply costs in each MG. It is also observed that the profit of the VPP would be higher in the case with the time-of-use (TOU) tariff compared to the cases with the real-time pricing (RTP) and critical peak pricing (CPP) mechanisms.

Resilience-uncertainty nexus in building energy management integrated with solar system and battery storage

Application of local energy resources in the buildings encounters such systems with both uncertainties in the produced power and failures (events) of the resources. The energy planning excluding uncertainty and events may be economically efficient but it is not resilient and practical. This paper provides a platform for energy management in the building under a set of realistic failures and uncertainties. The energy resilience is improved and the energy cost is decreased at the same time while all events and uncertainties are modeled. A typical building integrated with solar-battery-grid is studied. The resilience index is to minimize the loss of demanded service following all possible events. Three objective functions are defined and compared including (i) resilience cost, (ii) energy cost, (ii) resilience and energy costs together. The stochastic mixed integer linear programming is implemented to minimize the objective functions. The results validate that the developed toolkit can cope with all events and uncertainties with minimum operating cost and maximum resilience. The optimal tradeoff between operating cost, resilience, and uncertainty is presented. The results demonstrate that the uncertainty and resilience increase the planning cost by 5.3% and 13.8%, respectively, and both of them together increase the cost by 19.5%.

A Benders’ Decomposition Approach for Renewable Generation Investment in Distribution Systems

A model suitable to obtain where and when renewable energy sources (RES) should be allocated as part of generation planning in distribution systems is formulated. The proposed model starts from an existing two-stage stochastic mixed-integer linear programming (MILP) problem including investment and scenario-dependent operation decisions. The aim is to minimize photovoltaic and wind investment costs, operation costs, as well as total substation costs including the cost of the energy bought from substations and energy losses. A new Benders’ decomposition framework is used to decouple the problem between investment and operation decisions, where the latter can be further decomposed into a set of smaller problems per scenario and planning period. The model is applied to a 34-bus system and a comparison with a MILP model is presented to show the advantages of the model proposed.

Rolling Optimization of Mobile Energy Storage Fleets for Resilient Service Restoration

Mobile energy storage systems (MESSs) provide promising solutions to enhance distribution system resilience in terms of mobility and flexibility. This paper proposes a rolling integrated service restoration strategy to minimize the total system cost by coordinating the scheduling of MESS fleets, resource dispatching of microgrids and network reconfiguration of distribution systems. The integrated strategy takes into account damage and repair to both the roads in transportation networks and the branches in distribution systems. The uncertainties in load consumption and the status of roads and branches are modeled as scenario trees using Monte Carlo simulation method. The operation strategy of MESSs is modeled by a stochastic multi-layer time-space network technique. A rolling optimization framework is adopted to dynamically update system damage, and the coordinated scheduling at each time interval over the prediction horizon is formulated as a two-stage stochastic mixed-integer linear program with temporal-spatial and operation constraints. The proposed model is verified on two integrated test systems, one is with Sioux Falls transportation network and four 33-bus distribution systems, and the other is the Singapore transportation network-based test system connecting six 33-bus distribution systems. The results demonstrate the effectiveness of MESS mobility to enhance distribution system resilience due to the coordination of mobile and stationary resources.

Combining agent-based residential demand modeling with design optimization for integrated energy systems planning and operation

When optimizing the design of integrated energy systems, their energy demands are usually assumed as given input. However, the demand of energy systems may not always be available information or it can be uncertain, particularly for projects at the planning stage. Moreover, the overall energy demand is dependent on the behavior of many individuals, which might change as a result of technical, economic or policy interventions. Therefore, this study proposes a holistic approach to combine the demand modeling with design and dispatch optimization for integrated energy systems. The approach can be decomposed into two stages. Firstly, at demand simulation stage, Agent Based Modeling is adopted to generate uncertain demand scenarios for a case study community and the energy-consuming activities for various types of households living in different kinds of apartments are simulated based on probability models and demographic information. Dozens of demand scenarios are obtained via iterative simulation, and k-means clustering approach is further applied to generate representative stochastic scenarios. Secondly, at system optimization stage, the uncertain demand scenarios are used as input of an established stochastic Mixed Integer Linear Programming model, by which the system design and dispatch strategy can be optimized simultaneously. The case study shows that the obtained optimal solutions can save 36% of annual total cost compared to the business-as-usual baseline scenario.