Standard Mixed Integer Linear Programming Research Articles

Towards sustainable smart cities: Integration of home energy management system for efficient energy utilization

Smart Homes (SHs) play a crucial role in the broader context of smart cities, contributing to the overall efficiency, sustainability, and quality of life for residents. This work presents a computationally efficient mathematical model for Home Energy Management Systems (HEMS) to activate the Demand Side Management (DSM) while minimizing the energy consumption costs of SHs. The proposed model is represented as a standard Mixed-Integer Linear Programming (MILP) optimization problem. The proposed HEMS provides the optimal scheduling of home appliances and plugging time for charging Electric Vehicles (EV) at home. The main objective of this optimization problem is to minimize the electricity bills while improving the load profile of the distribution system in the context of smart cities. In this study, load management for a typical household has been studied considering several electricity tariffs in Portugal. The simulation results confirm that in the presence of the proposed HEMS, the electricity bill will be effectively reduced while consumer behavior is changed. The cost savings that can be attained through only shifting loads is 5.7 %, while it can reach 67.65 % by installing a hybrid energy system at the studied smart home. The weekly cost reduction percentage due to charging the EV at home is around 16.24 % compared to charging the EV using public charging stations regardless of the traveling time and waiting at the public charging stations.

A cut-and-solve algorithm for virtual machine consolidation problem

The virtual machine consolidation problem attempts to determine which servers to be activated, how to allocate virtual machines to the activated servers, and how to migrate virtual machines among servers such that the summation of activated, allocation, and migration costs is minimized subject to the resource constraints of the servers and other practical constraints. In this paper, we first propose a new mixed integer linear programming formulation for the virtual machine consolidation problem. We show that compared with existing formulations, the proposed formulation is much more compact in terms of smaller numbers of variables or constraints, which makes it suitable for solving large-scale problems. We then develop a cut-and-solve algorithm, a tree search algorithm to efficiently solve the virtual machine consolidation problem to optimality. The proposed cut-and-solve algorithm is based on a novel relaxation of the virtual machine consolidation problem that provides a stronger lower bound than the natural continuous relaxation of the virtual machine consolidation problem, making a smaller search tree. By extensive computational experiments, we show that (i) the proposed formulation significantly outperforms existing formulations in terms of solution efficiency; (ii) compared with standard mixed integer linear programming solvers, the proposed cut-and-solve algorithm is much more efficient; and (iii) compared with an existing heuristic algorithm, the proposed cut-and-solve algorithm is better able to balance workload distribution across servers and achieve higher resource utilization.

Forward and Line-Based Cycle Bases for Periodic Timetabling

The optimization of periodic timetables is an indispensable planning task in public transport. Although the periodic event scheduling problem (PESP) provides an elegant mathematical formulation of the periodic timetabling problem that led to many insights for primal heuristics, it is notoriously hard to solve to optimality. One reason is that for the standard mixed-integer linear programming formulations, linear programming relaxations are weak, and the integer variables are of pure technical nature and in general do not correlate with the objective value. While the first problem has been addressed by developing several families of cutting planes, we focus on the second aspect. We discuss integral forward cycle bases as a concept to compute improved dual bounds for PESP instances. To this end, we develop the theory of forward cycle bases on general digraphs. Specifically for the application of timetabling, we devise a generic procedure to construct line-based event-activity networks and give a simple recipe for an integral forward cycle basis on such networks. Finally, we analyze the 16 railway instances of the benchmark library PESPlib, match them to the line-based structure, and use forward cycle bases to compute better dual bounds for 14 out of the 16 instances.

Optimal controlled-islanding with AC power flow constraints: A benders-decomposition approach

The controlled-islanding process aims to prevent large devastating blackouts through an optimal splitting of a power network into smaller islands followed by corrective control actions. Generally, the controlled-islanding problem is cast as a mixed-integer non-linear programming (MINLP) model, which can be prohibitively cumbersome, particularly when it accommodates AC power flow constraints. This paper exploits the Benders decomposition to enhance the mathematical tractability of the problem. The AC power flow constraints are converted to equality constraints using proper auxiliary variables. These new constraints are applied to form a modified Jacobian matrix. The modified equations are embedded in the Benders subproblem involving slack variables in a linear iterative-based optimization structure. Hence, the obtained solution is totally free of any approximation or linearization error. The objective function of the controlled-islanding problem is a combination of total load shedding and generation curtailment of the network. This objective function implicitly mitigates the stability concerns in the islands, which can threaten their steady operation. As opposed to a unique value, adaptive lower bound are considered for the steady-state frequency deviation of islands. This adaptive setting prevents an incomplete frequency recovery after the islanding, considering the reserve capacity of each island. The proposed method yields more practically sound results than a standard mixed-integer linear programming (MILP) model with DC or linearized AC power flow constraints as voltage and apparent power quantities are retained in the permissible ranges. The effectiveness of the proposed algorithm is examined via the IEEE 30-bus test system and a 76-bus case as a part of an actual power system. The time-domain simulation results are also provided to support the accuracy of the frequency stability model captured in the optimization problem.

An MILP based optimization model for reservoir flood control operation considering spillway gate scheduling

Reservoir flood control operation is an important and effective measure for avoiding or reducing flood disasters. Considering that reservoir operation is a control process that essentially manages the spillway gates to increase or decrease the amount of released water, this paper focuses on studying the reservoir flood control operation which considers spillway gate scheduling. Minimizing the peak outflow from the reservoir is adopted as the optimization objective to protect downstream areas from flood disasters. In addition to the conventional reservoir constraints, the operation constraints of individual spillway gates and the influence of the reservoir forebay water level and opening degree on the gates’ discharge capacity are also well considered to obtain precise scheduling. The nonlinear factors, especially the objective function in the form of maximin, the three-dimensional water discharge curves, and the rules for the opening and closing sequence of the spillway gates, are linearized through several strategies. The original nonlinear and non-convex model is then converted into a standard mixed-integer linear programming (MILP) formulation. The results for the real-world case study indicate that: 1) the proposed model is believed to be computationally efficient with a calculation time of about 30 min; 2) for the flood events with return periods of 10 years and 100 years, the flood peaks are reduced from 4160 m3/s to 3681 m3/s and from 11007 m3/s to 9293 m3/s, representing decreases of 11.5 % and 15.6 %, respectively. The outflow curves become more stable, demonstrating that the flood control pressure on the downstream had been relieved after optimization; 3) the proposed model can produce more realistic and executable flood control operation scheduling compared with the conventional optimization model without considering spillway gate scheduling. These research findings can provide an effective reference for both the operators of reservoirs and researchers in the field of reservoir flood control operation.

Generating optimal robust continuous piecewise linear regression with outliers through combinatorial Benders decomposition

Using piecewise linear (PWL) functions to model discrete data has applications for example in healthcare, engineering and pattern recognition. Recently, mixed-integer linear programming (MILP) approaches have been used to optimally fit continuous PWL functions. We extend these formulations to allow for outliers. The resulting MILP models rely on binary variables and big- constructs to model logical implications. The combinatorial Benders decomposition (CBD) approach removes the dependency on the big- constraints by separating the MILP model into a master problem of the complicating binary variables and a linear sub problem over the continuous variables, which feeds combinatorial solution information into the master problem. We use the CBD approach to decompose the proposed MILP model and solve for optimal PWL functions. Computational results show that vast speedups can be found using this robust approach, with problem-specific improvements including smart initialization, strong cut generation and special branching approaches leading to even faster solve times, up to more than 12,000 times faster than the standard MILP approach.

Pool trading model within a local energy community considering flexible loads, photovoltaic generation and energy storage systems

This paper presents a pool trading model within a local energy community considering home energy management systems (HEMSs) and other consumers. A transparent mechanism for market clearing is proposed to incentivise active prosumers to trade their surplus energy within a rule-based pool market in the local energy community. A price-based demand response program (PBDRP) is considered to increase the consumers’ willingness to modify their consumption. The mathematical optimization problem is a standard mixed-integer linear programming (MILP) problem to allow for rapid assessment of the trading market for real energy communities which have a considerable number of consumers. This allows for novel energy trading strategies amongst different clients in the model and for the integration of a pool energy trading model at the level of the local energy community. The objective function of the energy community is to minimize the overall bills of all participants while fulfilling their demands. Two different scenarios have been evaluated, independent and integrated operation modes, to show the impacts of coordination amongst different end-users. Results show that through cooperation, end-users in the local energy community market can reduce the total electricity bill. This is shown in a 16.63% cost reduction in the independent operation and a 21.38% reduction in the integrated case. Revenues for active consumers under coordination increased compared to independent operation of the HEMS.

A MILP approach to DRAM access worst-case analysis

The Dynamic Random Access Memory (DRAM) is among the major points of contention in multi-core systems. We consider a challenging optimization problem arising in worst-case performance analysis of systems architectures: computing the worst-case delay (WCD) experienced when accessing the DRAM due to the interference of contending requests. The WCD is a crucial input for micro-architectural design of systems with reliable end-to-end performance guarantees, which is required in many applications, such as when strict real-time requirements must be imposed. The problem can be modeled as a mixed integer linear program (MILP), for which standard MILP software struggles to solve even small instances. Using a combination of upper and lower scenario bounding, we show how to solve realistic instances in a matter of few minutes. A novel ingredient of our approach, with respect to other WCD analysis techniques, is the possibility of computing the exact WCD rather than an upper bound, as well as providing the corresponding scenario, which represents crucial information for future memory design improvements.

A two-stage joint operation and planning model for sizing and siting of electrical energy storage devices considering demand response programs

This study describes a computationally efficient model for the optimal sizing and siting of Electrical Energy Storage Devices (EESDs) in Smart Grids (SG), accounting for the presence of time-varying electricity tariffs due to Demand Response Program (DRP) participation. The joint planning and operation problem for optimal siting and sizing of the EESD is proposed in a two-stage optimization problem. In this regard, the long-term decision variables deal were the size and location of the EESDs and have been considered at the master level while the operating point of the generation units and EESDs is determined by the slave stage of the model utilizing a standard mixed-integer linear programming model. To examine the effectiveness of the model in the slave sub-problem, the operation model is solved for different working days of different seasons. Binary Particle Swarm Optimization (BPSO) and Binary Genetic Algorithm (BGA) have been used at the master level to propose different scenarios for investment in the planning stage. The slave problem optimizes the model in terms of the short-term horizon (day-ahead). Additionally, the slave problem determines the optimal schedule for an SG considering the presence of EESD (with sizes and locations provided by the upper level). The electricity price fluctuates throughout the day, according to a Time-of-Use (ToU) DRP pricing scheme. Moreover, the impacts of DRPs have been addressed in the slave stage. The proposed model is examined on a modified IEEE 24-Bus test system.

A Shrinking Horizon Model Predictive Controller for Daily Scheduling of Home Energy Management Systems

In this paper, the model predictive control (MPC) strategy is utilized in smart homes to handle the optimal operation of controllable electrical loads of residential end-users. In the proposed model, active consumers reduce their daily electricity bills by installing photovoltaic (PV) panels and battery electrical energy storage (BEES) units. The optimal control strategy will be determined by the home energy management system (HEMS), benefiting from the meteorological and electricity market data stream during the operation horizon. In this case, the optimal scheduling of home appliances is managed using the shrinking horizon MPC (SH-MPC) and the main objective is to minimize the electricity cost. To this end, the HEMS is augmented by the SH-MPC, while maintaining the desired operation time slots of controllable loads for each day. The HEMS is cast as a standard mixed-integer linear programming (MILP) model that is incorporated into the SH-MPC framework. The functionality of the proposed method is investigated under different scenarios applied to a benchmark system while both time-of-use (TOU) and real-time pricing (RTP) mechanisms have been adopted in this study. The problem is solved using six case studies. In this regard, the impact of the TOU tariff was assessed in Scenarios 1&#x2013;3 while Scenarios 4&#x2013;6 evaluate the problem with the RTP mechanism. By adopting the TOU tariff and without any load shifting program, the cost is <inline-formula> <tex-math notation="LaTeX">$\$ $ </tex-math></inline-formula>1.2274 while by using the load shifting program without the PV and BEES system, the cost would reduce to <inline-formula> <tex-math notation="LaTeX">$\$ $ </tex-math></inline-formula>0.8709. Furthermore, by using the SH-MPC model, PV system and the BEES system, the cost would reduce to <inline-formula> <tex-math notation="LaTeX">$\$ $ </tex-math></inline-formula>-0.282713 with the TOU tariff. This issue shows that the prosumer would be able to make a profit. By adopting the RTP tariff and without any load shifting program, the cost would be <inline-formula> <tex-math notation="LaTeX">$\$ $ </tex-math></inline-formula>1.22093 without any PV and BEES systems. By using the SH-MPC model, the cost would reduce to <inline-formula> <tex-math notation="LaTeX">$\$ $ </tex-math></inline-formula>1.08383. Besides, by adopting the SH-MPC, and the PV and BEES systems, the cost would reduce to <inline-formula> <tex-math notation="LaTeX">$\$ $ </tex-math></inline-formula>0.05251 with the RTP tariff, showing the significant role of load shifting programs, local power generation, and storage systems.

Short-term optimal scheduling of cascade hydropower plants shaving peak load for multiple power grids

Cascade hydropower plants which have good regulation performance and are managed by the dispatching center of regional power grids are usually required to simultaneously shave the peak load for multiple provincial power grids, which is an important way of relieving the growing peak shaving pressure on power grids in central and eastern China. This paper establishes an optimization model for the short-term generation scheduling of cascade hydropower plants in regional power grids. In this model, minimizing the peak-valley load difference of multiple power grids is adopted as the objective function. In addition to conventional hydraulic constraints, the operation constraints of individual hydropower units, electrical constraints and head effect on the power generation are all considered to obtain precise scheduling. The original nonlinear and non-convex model is converted into a standard mixed-integer linear programming (MILP) formulation through several linearization strategies. The results for the real-world case study indicate that: 1) the proposed model is computationally efficient with a calculation time of 326 s; 2) the peak-valley differences of the Shanghai Power Grid and Zhejiang Power Grid decreased by 9.71% and 2.29%, respectively; 3) compared with actual operation, the proposed model shows better performance in peak shaving for multiple provincial power grids.

Transactive energy framework in multi-carrier energy hubs: A fully decentralized model

This paper investigates a fully decentralized model for electricity trading within a transactive energy market. The proposed model presents a peer-to-peer (P2P) trading framework between the clients. The model is incorporated for industrial, commercial, and residential energy hubs to serve their associated demands in a least-cost paradigm. The alternating direction method of multipliers (ADMM) is implemented to address the decentralized power flow in this study. The optimal operation of the energy hubs is modeled as a standard mixed-integer linear programming (MILP) optimization problem. The corresponding decision variables of the energy hubs operation are transferred to the peer-to-peer (P2P) market, and ADMM is applied to ensure the minimum data exchange and address the data privacy issue. Two different scenarios have been studied in this paper to show the effectiveness of the electricity trading model between peers, called integrated and coordinated operation modes. In the integration mode, there is no P2P energy trading while in the coordinated framework, the P2P transactive energy market is taken into account. The proposed model is simulated on the modified IEEE 33-bus distribution network. The obtained results confirm that the coordinated model can efficiently handle the P2P transactive energy trading for different energy hubs, addressing the minimum data exchange issue, and achieving the least-cost operation of the energy hubs in the system. The obtained results show that the total operating cost of the hubs in the coordinated model is lower than that of the integrated model by $590.319, i.e. 11.75 % saving in the costs. In this regard, the contributions of the industrial, commercial, and residential hubs in the total cost using the integrated model are $3441.895, $596.600, and $988.789, respectively. On the other hand, these energy hubs contribute to the total operating cost in the coordinated model by $2932.645, $590.155, and $914.165 respectively. The highest decrease relates to the industrial hub by 14.8 % while the smallest decrease relates to the residential hub by 1 %. Furthermore, the load demand in the integrated and coordinated models is mitigated by 13 % and 17 %, respectively. These results indicate that the presented framework could effectively and significantly reduce the total load demand which in turn leads to reducing the total cost and power losses.

A Single-Level Rule-Based Model Predictive Control Approach for Energy Management of Grid-Connected Microgrids

A single-level rule-based model predictive control (RBMPC) scheme is presented for optimizing the energy management of a grid-connected microgrid composed of local production units, renewable energy sources, local loads, and several types of energy storage systems (ESSs). The single-level controller uses two different models that yield different descriptions of the microgrid and use different sampling times. The model with a smaller sampling time provides a more detailed description of the microgrid, in order to keep track of the fast dynamics, while the model with a higher sampling time provides a less detailed description and is used for making long-term predictions when it is not needed anymore to track the fast dynamics. Moreover, we propose a novel RBMPC method that assigns the value to the binary decision variables in the hybrid microgrid model, e.g., ON or OFF status of the generators and charging or discharging mode of ESSs, through if–then–else rules, which rely on the price of electricity and the local net imbalance. The standard method of applying model predictive control (MPC) to a hybrid model results in a mixed-integer linear programming (MILP) problem. Our proposed rule-based method is able to convert the standard MILP problem into a linear one. We compare our approach through simulations to the MILP approach and show that our method yields almost no loss in performance while providing a significant reduction in the computation time.

A matheuristic for the vehicle routing problem with drones and its variants

In this work, we are interested in studying the Vehicle Routing Problem with Drones (VRPD). Given a fleet of trucks, where each truck carries a given number of drones, the objective consists in designing feasible routes and drone operations such that all customers are served and minimal makespan is achieved. We formulate the VRPD as a Mixed Integer Linear Program (MILP), which can be solved by any standard MILP solver. Moreover, with the aim of improving the performance of solvers, we introduce several sets of valid inequalities (VIEQ). Due to limited performance of the solvers in addressing large instances, we propose a matheuristic approach that effectively exploits the problem structure of the VRPD. Integral to this approach, we propose the Drone Assignment and Scheduling Problem (DASP) that, given an existing routing of trucks, looks for an optimal assignment and schedule of drones such that the makespan is minimized. In this context, we propose two MILP formulations for the DASP. In order to evaluate the performance of a state-of-the-art solver in tackling the MILP formulation of the VRPD, the benefit of the proposed VIEQs, and the performance of the matheuristic, we carried out extensive computational experiments. According to the numerical results, the use of drones can significantly reduce the makespan and the proposed VIEQ as well as the matheuristic approach have a significant contribution in solving the VRPD effectively.

Short-term optimal operation of hydro-wind-solar hybrid system with improved generative adversarial networks

The high penetration of variable renewable energy sources (RESs) has greatly increased the difficulty in power system scheduling and operation. To fully utilize the complementary characteristics of various RESs, a stochastic optimization model considering the strong regulation capacity of cascade hydropower stations and the uncertainty of wind and photovoltaic (PV) power is presented. Based on the improved generative adversarial networks, the spatial and temporal correlation characteristics between wind farms and PV plants are accurately captured via measured data. Due to the nonlinear features of the hydroelectric plants, linearization methods are adopted to reformulate the original model into a standard mixed integer linear programming (MILP) formulation. Then, the model is solved with a proposed two-stage approach, in which a heuristic algorithm is used to solve the first-stage unit commitment optimization. The cascade hydraulic connection and time delay of the water flow are established in the second stage to exploit the considerably controllable adjustment capability of hydropower generation. A renewable energy base in southwest China is chosen as a detailed case study. The simulation results reveal the potential of the large-scale application of only a hydro-wind-solar hybrid system to satisfy the power transmission demand with the guidance of the coordinated operation strategy, and the performance of the hybrid system can be further enhanced with high-quality scenarios from the proposed deep neural network.

Optimizing resource utilization in NFV dynamic systems: New exact and heuristic approaches

Network Function Virtualization (NFV) orchestration and management have attracted a lot of attention in recent years as it provides new opportunities regarding performance and deployment. In particular, several models have attempted to capture the behavior of such systems under various restricted assumptions. However, previously proposed mathematical models can only handle problems of fairly small size. This paper proposes a Mixed Integer Linear Programming (MILP) model for the resource utilization problem in an NFV dynamic context with several enhancements regarding the state of the art. We include the utilization of flow constraints to ensure the order of functions in a service chain. By systematic generation of Flow Cover inequalities, significant improvements in processing time are obtained with a standard MILP solver to compute exact optimal solutions. We also propose three efficient heuristics (two MILP-based heuristics) to find high-quality feasible solutions for large-scale systems within reduced execution time. We also carry out a set of experiments to evaluate the proposed algorithms and provide valuable guidelines for the efficient design of such systems. The results show that our approach is capable of handling large size instances of the NFV deployment problem involving up to 200 nodes and 100 demands.

Optimization-Based Scheduling Method for Agile Earth-Observing Satellite Constellation

This Paper proposes a two-step binary linear programming formulation for task scheduling of a constellation of low-Earth-orbit satellites and demonstrates its applicability and scalability to obtain high-quality solutions using a standard mixed-integer linear programming solver. In this instance, the goal of satellite constellation task scheduling is to allocate each task for the satellites and to determine the task starting times in order to maximize the overall mission performance metric. The scheduling problem is formulated to find the solution by first finding a set of candidate communication time intervals for each satellite/ground-station pair as one of the key constraints and time tabling the observation task to acquire the user-requested data, with the incorporation of key constraints for satellite constellation operation. Numerical experiments are designed for investigating the trends, sensitivity, and characteristics of scheduling outputs based on multiple representative instances. The performance of the scheduling solutions by the proposed two-step binary linear programming method exhibits significant improvement of up to 35% in the number of assignments and the sum of profits over the general greedy algorithm.

Optimization of wireless sensor networks deployment with coverage and connectivity constraints

Wireless sensor networks have been widely deployed in the last decades to provide various services, like environmental monitoring or object tracking. Such a network is composed of a set of sensor nodes which are used to sense and transmit collected information to a base station. To achieve this goal, two properties have to be guaranteed: (i) the sensor nodes must be placed such that the whole environment of interest (represented by a set of targets) is covered, and (ii) every sensor node can transmit its data to the base station (through other sensor nodes). In this paper, we consider the Minimum Connected k-Coverage (MCkC) problem, where a positive integer $$k \ge 1$$ defines the coverage multiplicity of the targets. We propose two mathematical programming formulations for the MCkC problem on square grid graphs and random graphs. We compare them to a recent model proposed by Rebai et al. (Comput Oper Res 59:11–21, 2015). We use a standard mixed integer linear programming solver to solve several instances with different formulations. In our results, we point out the quality of the LP-bound of each formulation as well as the total CPU time or the proportion of solved instances to optimality within a given CPU time.

A two-stage stochastic programming approach for identifying optimal postponement strategies in supply chains with uncertain demand

In the course of globalization, applying mass-customization strategies has led to a high diversity of variants in many economic sectors. Thus, customer demands are often less predictable, and handling increasing inventory stocks as well as avoiding shortfalls have become particularly important. All these complexity drivers result in higher supply chain risks. Postponement strategies have been proposed as a suitable approach to address these problems. Although the concept of postponement and its impact on the supply chain are theoretically well discussed, optimally configuring the entire production and distribution activities is still challenging. We present a two-stage stochastic mixed-integer linear programming model, which comprises an integrated production and distribution planning approach, and considers postponement concepts. In comparison to earlier approaches that examine postponement strategies, our model supports the decision maker under demand uncertainty and considers lead times, penalty costs for shortfalls, as well as inventory-keeping decisions over a tactical planning horizon. This allows an integrated investigation of both form and logistics postponement concepts. Moreover, we consider the decision maker’s risk attitude identifying non-dominated profitable and risk-averse strategies. We illustrate the benefits of the model by using a case study from the apparel industry, and present the results of a sensitivity analysis with respect to varying demand uncertainty and demand correlations as well as different preferences regarding risk aversion. Furthermore, we carry out performance and quality benchmarks and compare the results of a standard mixed-integer linear programming solver, a parallel nested Benders approach and a sample average approximation technique.

Non-discriminatory train dispatching in a rail transport market with multiple competing and collaborative train operating companies

Train dispatching is vital for the punctuality of train services, which is critical for a train operating company (TOC) to maintain its competitiveness. Due to the introduction of competition in the railway transport market, the issue of discrimination is attracting more and more attention. This paper focuses on delivering non-discriminatory train dispatching solutions while multiple TOCs are competing in a rail transport market, and investigating impacting factors of the inequity of train dispatching solutions. A mixed integer linear programming (MILP) model is first proposed, in which the inequity of competitors (i.e., trains and TOCs) is formalized by a set of constraints. In order to provide a more flexible framework, a model is further reformulated where the inequity of competitors is formalized as the maximum individual deviation of competitors’ delay cost from average delay cost in the objective function. Complex infrastructure capacity constraints are considered and modelled through a big M-based approach. The proposed models are solved by a standard MILP solver. A set of comprehensive experiments is conducted on a real-world dataset adapted from the Dutch railway network to test the efficiency, effectiveness, and applicability of the proposed models, as well as determine the trade-off between train delays and delay equity.