Mixed Integer Quadratic Programming Model Research Articles

Optimal household appliances scheduling for smart energy management considering inclining block rate tariff and net-metering system

Smart grids that integrate household renewable energy sources and share information with households can help create and maintain a smarter data-driven environment. Within this environment, flexible home energy management policies that minimize household energy costs can be adopted. This paper considers a smart home with a renewable energy source that favors satisfying its energy needs at minimum cost. This is achievable by smartly scheduling the use of its domestic appliances to match a given energy grid tariff. Focusing on the case of Egypt in which an inclining block rate (IBR) tariff is imposed, this paper fills a gap in the literature regarding the load scheduling models aiming to minimize energy cost at the household level whenever such a tariff exists. A new mixed integer quadratic programming (MIQP) model is formulated for this scheduling problem, considering the adopted net metering system with installed domestic photovoltaic (PV) systems in Egypt. The model generates the optimal household load schedule and the optimal amounts of energy to exchange with the grid while considering all the system and consumer utility constraints. To assess the applicability of the proposed model, a survey is conducted to identify the diversity and characteristics of using the electrical appliances by the Egyptian households. Based on the collected survey results, the effectiveness of the proposed MIQP model is investigated. Results confirm the effectiveness of the proposed model to minimize energy cost for different categories of the Egyptian households.

MPC-based framework incorporating pre-disaster and post-disaster actions and transportation network constraints for weather-resilient power distribution networks

Enhancing power system resilience to extreme weather requires an effective coordination of pre-disaster preventive response actions and post-disaster repair and restoration services. In this context, this article presents a two-stage framework to enhance resilience of power distribution networks to severe weather events. It considers operating and resource constraints in interdependent power and transportation networks based on official guidance from transport appraisal studies. It also incorporates sources of information available in a smart city context, such as energy measurements, traffic flows, and geographical information systems. In the first stage, a pre-disaster operational planning strategy is defined to prepare the grid for a high-risk outage scenario with the objective of minimizing the value of lost loads, formulated as a mixed-integer quadratic programming model. In the second stage, a post-disaster corrective strategy is implemented over a receding time horizon to compensate for deviations between the actions computed in the first stage and the actions required to minimize power outages, embedded into a model predictive control scheme. The effectiveness of the proposed framework is demonstrated on a real-world large-scale distribution network in the United Kingdom over a range of outage scenarios. Simulation results show that the proposed framework is capable of effectively minimizing first-stage operational costs and second-stage deviations between the projected and actual load demand supply. Numerical results obtained with the proposed framework indicate that the load energy unserved is at least twice smaller than with typical practices adopted by distribution network operators and computational times take two minutes or less. Therefore, it can be effectively used by distribution network operators to ensure an appropriate level of preparedness to power outages caused by extreme weather along with prompt restoration and repair services.

Optimal Solution for the Integrated Model of the Problems in Capacitated Vehicle Routing and Vehicle Scheduling at the Multi-door Depot

To have an efficient Supply Chain (SC), the coordination and integration of the activities in the SC are mandatory. Routing vehicles is an optimization problem in the SC. When the routed vehicles return to the depot and the doors at the depot are busy, the returned vehicles must wait to unload the accumulated shipments. Therefore, properly scheduling these vehicles to those doors at the depot to minimize the waiting time is an optimization problem in the SC. Therefore, in this study, routing vehicles to collect the shipments from suppliers and scheduling vehicles to doors at the depot, based on a first-come-first-serve basis, are simultaneously solved. Hence, the objective of this integrated vehicle routing and scheduling problem (VR&SP) is to minimize the total cost which contains the following components: vehicle travelling cost between suppliers, loading cost at the suppliers, vehicle waiting cost, unloading cost at the depot and vehicle operations cost. A Mixed Integer Quadratic Programming (MIQP) model is developed to solve the integrated VR&SP. The Branch and Bound algorithm method is employed to obtain the exact optimal solution to this MIQP using LINGO optimization software. The compatibility of the developed MIQP model is verified by the randomly generated small-scale instances of VR&SP. Therefore, it can be concluded that, this model solves the vehicle routing to suppliers and vehicle scheduling to doors at the depot simultaneously. Since VR&SP is a NP-hard problem, heuristics or meta-heuristic methods are recommended to solve the large-scale instances of VR&SP. Moreover, it can be recommended for further studies to amend this model by incorporating additional constraints to make it more applicable to real-world scenarios.

Optimal cyclic scheduling of ethylene plants in backup furnace mode with separation capacity constraints

Ethylene plants (EPs) are a critical output source of ethylene. The cyclic scheduling optimization is crucial to increase the profitability of an EP. The EP comprises the ethylene cracking furnace system (ECFS), quenching, compression, separation sections, etc., which culminates in a large-scale optimization problem. In addition, the performance-decaying and semi-continuous characteristics of the ECFS inevitably lead to fluctuations in product throughput, resulting in an unstable production process and product quality decline. To address this issue, we propose a mixed-integer quadratic programming (MIQP) model for the cyclic scheduling of the EP in backup furnace mode with separation capacity constraints. This novel model can reduce fluctuations in product throughput by running a backup furnace when one furnace is shut down for decoking. Besides, the separation capacity constraints ensure that high-quality products can be separated from the cracked gas downstream. Moreover, the reformulated normalized multiparametric disaggregation technique (RNMDT), which relies on convex envelopes, is employed to relax the proposed MIQP model into a mixed-integer linear programming (MILP) model by partitioning the domain of the variables in the quadratic terms. Subsequently, an optimization framework based on the RNMDT is established to enhance the solution efficiency of the original problem. Finally, a case study from the real world reveals the effectiveness of the proposed method, which can provide instructive optimal cyclic scheduling for the EP.

Flexible yard space allocation plan for new type of automated container terminal equipped with unilateral-cantilever rail-mounted gantry cranes

In this paper, new automated container terminals equipped with Unilateral-cantilever Rail-Mounted Gantry Cranes (URMGC) are considered as the research object. In the scenario, the automated guided vehicles can go deep among the container blocks arranged perpendicularly to the shoreline, which completely changes the yard’s operation mode and affects its operational efficiency. In addition, considering the shortage of storage space in the yard, a flexible space allocation strategy for these yards is studied, and a mixed integer quadratic programming model is established to minimize the horizontal transportation cost of containers. An improved particle swarm optimization algorithm is designed to compensate for the increased complexity of the proposed model. To accelerate the solution speed, the quantity and priority rules are developed to generate reasonable initial solutions, and two velocity updating strategies are adopted to accelerate convergence. Numerical experiments are conducted to validate effectiveness and efficiency of the proposed method, and comparisons with fixed space allocation strategy are presented. The results show that the strategy and solution adopted in this paper can reduce the driving distance of AGVs in such terminal configurations effectively and potentially increase the handling capacity of the yard.

Optimal Pole-Swapping in Bipolar DC Networks With Multiple CPLs Using an MIQP Model

This express brief presents a mixed-integer quadratic programming (MIQP) model to address the problem of optimal pole-swapping in asymmetric bipolar DC networks. The exact mixed-integer nonlinear programming model is convexified by using two stages. Using Taylor’s series expansion, the first stage corresponds to linearizing the hyperbolic relationship between voltage and powers in the constant power loads. The second stage applies the concept of the big-M number in order to obtain a linear equivalent for the product between binary and continuous variables. The proposed MIQP model is solved using the MATLAB programming environment, with the convex disciplined tool known as CVX and the Gurobi solver. Numerical results in the 21-and 85-bus grids demonstrate the effectiveness of the proposed MIQP formulation when compared to combinatorial optimization methods. The MIQP model ensures reductions of about 3.7609 and 49.7928kW in the total grid power losses for the 21-and 85-bus grids, respectively, when the neutral wire is assumed to be floating in all the nodes of the network except for the substation bus.

Risk-averse decision-making to maintain supply chain viability under propagated disruptions

In this paper, stochastic optimisation of CVaR is applied to maintain risk-averse viability and improve resilience of a supply chain under propagated disruptions. In order to establish the risk-averse boundaries on supply chain viability space, two stochastic optimisation models are developed with the two conflicting objectives: minimisation of Conditional Cost-at-Risk and maximisation of Conditional Service-at-Risk. Then, the risk-averse viable production trajectory between the two boundaries is selected using a stochastic mixed integer quadratic programming model. The proposed approach is applied to maintain the supply chain viability in the smartphone manufacturing and the results of computational experiments are provided. The findings indicate that when the decision-making is more risk-aversive, the size of the viability space between the two boundaries is greater. As a result, more room is available for selecting viable production trajectories under severe disruptions. Moreover, the larger is viability space, the higher is both worst-case and average resilience of the supply chain. Risk-neutral, single-objective decision-making may reduce the supply chain viability. A single-objective supply chain optimisation which moves production to the corresponding boundary of the viability space, should not be applied under severe disruption risks to avoid greater losses.

Co-optimization of integrated energy systems in the presence of renewable energy, electric vehicles, power-to-gas systems and energy storage systems with demand-side management

AbstractWith the widespread penetration of renewable energy sources and energy storage systems, the problem of energy management has received increasing attention. One of the systems that network owners consider today is the power-to-gas (P2G) system. This system causes surplus electricity generated from renewable energy resources or batteries in the network to be converted into gas and sold to the gas network. Two reasons for the existence of gas distributed generation resources and P2G systems cause the two power and gas networks to interact. Energy management and profit making considering these two networks, as a co-optimization of integrated energy systems, is a topic that has been discussed in this study to achieve the best optimal answer. Since the production of renewable energy resources and the purchase price of energy are uncertain, a scenario-based method has been chosen for modelling. Demand-side management is also one of the important problems in optimal operation of the electricity network, which can have a significant impact on reducing peak load and increasing profits. In this paper, a mixed-integer quadratic programming model for co-optimization of electric distribution and gas networks in the presence of distributed generation resources, P2G systems, storage facilities, electric vehicles and demand-side management is presented. The 33-bus distribution network is intended to analyse the proposed model. The results of different scenarios show the efficiency of the proposed model. Several key points are deduced from the obtained results: (i) demand-side management is able to reduce the peak load of the network, (ii) the presence of renewable resources and batteries can cause the network to convert excess electricity into gas and sell it to the gas network in the market and (iii) distributed generation can reduce the purchase of energy from the upstream network and cause a 36% reduction in the cost function.

Optimal layout model of feeder automation equipment oriented to the type of fault detection and local action

In feeder automation transformation there are difficulties in equipment and location selection. To help with this, an optimal layout model of feeder automation equipment oriented to the type of fault detection and local action is proposed. It analyzes the coordination relationship of the three most common types of automation equipment, i.e., fault indicator, over-current trip switch and non-voltage trip switch in the fault handling process, and the explicit expressions of power outage time caused by a fault on different layouts of the above three types of equipment are given. Given constraints of power supply reliability and the goal of minimizing the sum of equipment-related capital investment and power interruption cost, a mixed-integer quadratic programming model for optimal layout is established, in which the functional failure probability of equipment is linearized using the 3delta principle in statistics. Finally, the basic characteristics of the proposed model are illustrated by different scenarios on the IEEE RBTS-BUS6 system. It can not only take into account fault location and fault isolation to enhance user power consumption perception, but also can guide precise investment to improve the operational quality and efficiency of a power company.

Bi-objective unrelated parallel machines scheduling problem with worker allocation and sequence dependent setup times considering machine eligibility and precedence constraints

<p style='text-indent:20px;'>In today's competitive world, scheduling problems are one of the most important and vital issues. In this study, a bi-objective unrelated parallel machine scheduling problem with worker allocation, sequence dependent setup times, precedence constraints, and machine eligibility is presented. The objective functions are to minimize the costs of tardiness and hiring workers. In order to formulate the proposed problem, a mixed-integer quadratic programming model is presented. A strategy called repair is also proposed to implement the precedence constraints. Because the problem is NP-hard, two metaheuristic algorithms, a multi-objective tabu search (MOTS) and a multi-objective simulated annealing (MOSA), are presented to tackle the problem. Furthermore, a hybrid metaheuristic algorithm is also developed. Finally, computational experiments are carried out to evaluate different test problems, and analysis of variance is done to compare the performance of the proposed algorithms. The results show that MOTS is doing better in terms of objective values and mean ideal distance (MID) metric, while the proposed hybrid algorithm outperforms in most cases, considering other employed comparison metrics.</p>

An improved column generation algorithm for the disrupted flight recovery problem with discrete flight duration control and aircraft assignment constraints

The disrupted flight recovery problem is well-studied in the literature owing to its significant impact on airlines and passengers. In this work, we consider the disrupted flight recovery problem with two new realistic aspects, i.e., a new implementation for changing flight duration as a recovery option and considering the aircraft assignment constraints. Firstly, we develop a new mixed-integer quadratic programming model encapsulating a functional relationship between the reduced flight duration and the changes in fuel consumption using a piecewise function. Secondly, we propose an improved column generation approach to solve some scenarios based on actual flight data obtained from an airline. Lastly, the experimental results show that the improved column generation algorithm can obtain the optimal solution for all scenarios. Compared with not considering changing the flight duration, considering changing the flight duration can save about 24% of recovery costs on average. In addition, we analyze the effect of enforcing aircraft assignment constraints and different fuel prices on recovery costs. Based on the experimental results, we discuss how to support practitioners in choosing the appropriate combination of recovery options.

Multi-warehouse, multi-product inventory control model for agri-fresh products – A case study

Warehouses that operate under controlled micro-climates are necessary to reduce the deterioration of fresh produce, but they are also costly to operate. Developing an efficient operation regime for these premises while considering energy costs, demand, and harvesting requirements is crucial to managing the food supply chain sustainably and profitably. This paper proposes a method to develop an operation regime by solving a multi-period, multi-product, multi-warehouse inventory control optimization problem that originates from the management of agri-fresh products from an Australian company. We propose a mixed-integer quadratic programming model for this problem. A significant challenge is that each product has a specific deterioration rate over storage, which differs in each warehouse mode. Our model allows for two different operational modes for warehouses, each with different electricity consumption, affecting the time the products remain fresh. The objective function of the model minimizes the total inventory costs. The model provides the optimal inventory flow of each product, the optimal number of active warehouses, the optimal operational length of warehouses, and the optimal mode of warehouses in each period. We solve the model using Gurobi and integrate three other solution approaches based on model properties, which reduce the computational time by half. The model is tested using real data from the case study, an Australian agri-fresh company. The results indicate a reduction in total cost by 8% and quantity of product deterioration by 20%. We also investigate the impacts of variations in demand and supply caused by growing or shrinking markets in our case study under different scenarios and instance sizes of the problem and analyze the impact in solution times that are observed in practice.

Real-time optimization for train regulation and stop-skipping adjustment strategy of urban rail transit lines

This paper addresses the real-time train regulation problem by taking the stop-skipping strategy into account in high-frequency urban rail transit lines. During rush hours, if the train regulation is not adopted properly and rapidly to deal with the frequent disturbances, the delays may further spread and result in a large number of passengers stranded in the stations. The purpose of this paper is to explore the possibility of dynamic train regulation and stop-skipping adjustment strategy to cope with the disturbance management in a real-time manner. By considering the impact of dynamic passenger flows, a nonlinear programming model for train regulation problem is proposed with the objective of minimizing the total train deviation and enhancing the passenger service quality, which is further converted into a mixed-integer quadratic programming model for ease to solve. In addition, the constraints related to the train rolling-stock plan are taken into account to provide a feasible scheme. Based on a customized model predictive control (MPC) method, the proposed model can be solved in a real-time manner, laying a theoretical groundwork for implementation of the train regulation strategy. Computational results based on the real-world data of Beijing Yizhuang metro line illustrate the superiority of the train regulation strategy in comparison with the practical strategy. The robustness of the method is examined through various scenarios of uncertain passenger demands, and the adjustment performance with different prediction horizons is explored to illustrate the added value brought by the updated information of the proposed MPC method.

A combinatorial optimization approach to scenario filtering in portfolio selection

Recent studies stressed the fact that covariance matrices computed from empirical financial time series appear to contain a high amount of noise. This makes the classical Markowitz Mean–Variance Optimization model unable to correctly evaluate the performance associated to selected portfolios. Since the Markowitz model is still one of the most used practitioner-oriented tool, several filtering methods have been proposed in the literature to overcome the problem. Among them, the two most promising ones refer to the Random Matrix Theory and to the Power Mapping strategy. The basic idea of these methods is to transform the estimated correlation matrix before applying the Mean–Variance Optimization model. However, experimental analysis shows that these two strategies are not always effective when applied to real financial datasets.In this paper we propose a new filtering method based on Quadratic Programming. We develop a Mixed Integer Quadratic Programming model, which is able to filter those observations that may affect the performance of the selected portfolio. We discuss the properties of this new model and test it on some real financial datasets. We compare the out-of-sample performance of our portfolios with the one of the portfolios provided by the two above mentioned alternative filtering methods giving evidence that our method outperforms them. Although our model can be solved efficiently with standard optimization solvers, the computational burden increases for large datasets. To solve also these problems, we propose a heuristic procedure, which, on the basis of our empirical results, shows to be both efficient and effective.

Container Terminal Berth and Yard Collaborative Allocation for Import and Export Synchronous Operations by Computational Logistics

Container terminal operators have practical and clear requirements on improving operational efficiency and service level to address running challenges. The resource allocation and collaborative scheduling of quayside berth and storage yard are vital issues in the optimization of container terminal handling systems (CTHSs). The allocation of berth in quayside and blocks on storage yard directly affects the running efficiency of CTHS. Consequently, we focus on the berth and yard collaborative allocation for import and export synchronous operations (BYCA-IESOs) and propose the BYCA-IESO research model based on synchronous or asynchronous integrated scheduling strategy by computational logistics. The BYCA-IESO model is a mixed integer quadratic programming model to minimize multiple time consumption of calling liners and the total horizontal transferring distances of yard trailers at container terminals. Furthermore, we design other two allocation policies for comparison on the BYCA-IESO, which are both multistage scheduling strategy profiles. CPLEX 20.1.0 is applied to implement and solve the model and tactics of BYCA-IESO. Accordingly, the feasibility, validity, and relative merits of the models and strategies are obtained and verified by analyzing and comparing multiple numerical experimental results.

Strategy for Locating People to Reduce the Transmission of COVID-19 Using Different Interference Measures

COVID-19 is generally transmitted from person to person through small droplets of saliva emitted when talking, sneezing, coughing, or breathing. For this reason, social distancing and ventilation have been widely emphasized to control the pandemic. The spread of the virus has brought with it many challenges in locating people under distance constraints. The effects of wakes between turbines have been studied extensively in the literature on wind energy, and there are well-established interference models. Does this apply to the propagation functions of the virus? In this work, a parallel relationship between the two problems is proposed. A mixed-integer linear programming (MIP) model and a mixed-integer quadratic programming model (MIQP) are formulated to locate people to avoid the spread of COVID-19. Both models were constructed according to the distance constraints proposed by the World Health Organization and the interference functions representing the effects of wake between turbines. Extensive computational tests show that people should not be less than two meters apart, in agreement with the adapted Wells–Riley model, which indicates that 1.6 to 3.0 m (5.2 to 9.8 ft) is the safe social distance when considering the aerosol transmission of large droplets exhaled when speaking, while the distance can be up to 8.2 m (26 ft) if all the droplets in a calm air environment are taken into account.

An Approximate Mixed-Integer Convex Model to Reduce Annual Operating Costs in Radial Distribution Networks Using STATCOMs

The problem of optimal siting and sizing of distribution static compensators (STATCOMs) is addressed in this research from the point of view of exact mathematical optimization. The exact mixed-integer nonlinear programming model (MINLP) is decoupled into two convex optimization sub-problems, named the location problem and the sizing problem. The location problem is addressed by relaxing the exact MINLP model, assuming that all the voltages are equal to 1∠0∘, which allows obtaining a mixed-integer quadratic programming model as a function of the active and reactive power flows. The solution of this model provides the best set of nodes to locate all the STATCOMs. When all the nodes are selected, it solves the optimal reactive power problem through a second-order cone programming relaxation of the exact optimal power flow problem; the solution of the SOCP model provides the optimal sizes of the STATCOMs. Finally, it refines the exact objective function value due to the intrinsic non-convexities associated with the costs of the STATCOMs that were relaxed through the application of Taylor’s series expansion in the location and sizing stages. The numerical results in the IEEE 33- and 69-bus systems demonstrate the effectiveness and robustness of the proposed optimization problem when compared with large-scale MINLP solvers in GAMS and the discrete-continuous version of the vortex search algorithm (DCVSA) recently reported in the current literature. With respect to the benchmark cases of the test feeders, the proposed approach reaches the best reductions with 14.17% and 15.79% in the annual operative costs, which improves the solutions of the DCVSA, which are 13.71% and 15.30%, respectively.

Outer Approximation Method for the Unit Commitment Problem with Wind Curtailment and Pollutant Emission

This paper considers the fast and effective solving method for the unit commitment (UC) problem with wind curtailment and pollutant emission in power systems. Firstly, a suitable mixed-integer quadratic programming (MIQP) model of the corresponding UC problem is presented by some linearization techniques, which is difficult to solve directly. Then, the MIQP model is solved by the outer approximation method (OAM), which decomposes the MIQP into a mixed-integer linear programming (MILP) master problem and a nonlinear programming (NLP) subproblem for alternate iterative solving. Finally, simulation results for six systems with up to 100 thermal units and one wind unit in 24 periods are presented, which show the practicality of MIQP model and the effectiveness of OAM.

Optimizing decisions for a dual-channel retailer with service level requirements and demand uncertainties: A Wasserstein metric-based distributionally robust optimization approach

This study explores a dual-channel management problem of a retailer selling multiple products to customers through a traditional retail channel and an online channel to maximize expected profit. The prices and order quantities of both the online and the retail channels and the delivery times of the online channel are the decision variables. The demand for each product and each channel is assumed to be random and dependent on the prices of both channels and on the online delivery time. In addition, to ensure an adequate performance, service level requirements are considered and are modeled as joint chance constraints. Wasserstein uncertainty sets using the Wasserstein metric for demand probability distributions centered at the empirical distributions on the observed demands from the historical data are constructed in a data-driven approach. Accordingly, a data-driven distributionally robust joint chance constrained model is developed based on the data-driven Wasserstein uncertainty sets. A conservative CVaR approximation is used for the distributionally robust joint chance constraints. Through mathematical manipulations, the developed model is transformed into a bilinear program, which can be approximated by a mixed integer quadratic programming model using piecewise affine relaxations of the bilinear terms and can be solved efficiently. Numerical experiments are performed to illustrate the effectiveness and practicality of the proposed data-driven distributionally robust optimization approach to deal with demand uncertainties. The effects of the key parameters such as delivery time sensitivity, price sensitivity and customer channel preference are analyzed and managerial insights are provided. The results show that the decisions obtained by the proposed approach are robust to hedge against demand uncertainties. The proposed model and solution approach can provide effective decision supports for retailers selling products through an online channel and a traditional retail channel without reliable demand distribution information. Furthermore, compared with the L1-norm and the L2-norm, the L∞-norm is verified to perform better when used in the Wasserstein metric for constructing the Wasserstein uncertainty sets.

Distributed Energy Management of the Hybrid AC/DC Microgrid with High Penetration of Distributed Energy Resources Based on ADMM

This paper aims to investigate energy management of the hybrid AC/DC microgrid with the high penetration of distributed energy resources (DERs), such as electrical vehicles, heat pumps, and photovoltaics. In the previous studies, energy management of the hybrid microgrid is usually carried out by the system operator in a centralized manner, which suffers from the compromise of privacy information protection and the risk of single-point failure. Therefore, this paper proposes a distributed energy management scheme of the hybrid microgrid using the projection function-based alternating direction method of multipliers (P-ADMM), which allows each subgrid, i.e., AC subgrid and DC subgrid, to make day-ahead schedules independently with information exchanges while obtaining the optimal energy management solution. The energy management problem of the hybrid microgrid is formulated as a mixed-integer quadratic programming (MIQP) model, considering DER and energy storage system operation constraints, system operation constraints, and converter operation constraints. Then, the MIQP model is decomposed and distributed into smaller-scale QP models between subgrids using the P-ADMM algorithm, which can handle binary variables through projection functions. The numerical results conducted on the hybrid microgrid demonstrate that the proposed distributed scheme can effectively achieve optimal energy management for the hybrid AC/DC microgrid in a distributed manner.