Mixed Integer Linear Programming Problem Research Articles

Resilience-Oriented Planning of Urban Distribution System Source–Network–Load–Storage in the Context of High-Penetrated Building-Integrated Resources

Building-integrated flexible resources can offer economical availability to accommodate high-penetrated renewable energy sources (RESs), which can be potentially coordinated to achieve cost-effective supply. This paper proposes a resilience-oriented planning model of urban distribution system source–network–load–storage in the context of high-penetrated building-integrated resources. In this model, source–network–load–storage resources are cost-optimally planned, including the lines, soft open point (SOP), building-integrated photovoltaics (BIPVs), building-integrated wind turbine (BIWT), building-integrated energy storage system (ESS), etc. To enhance fault recovery capability during extreme faults, fault scenarios are incorporated into the distribution system operation via coupled multiple recovery stages. The resilience-oriented planning is a thorny problem due to its source–network–load–storage couplings, normal-fault couplings, etc. The original resilience-oriented planning is reformulated as a mixed-integer linear programming (MILP) problem, which can then be solved with a two-stage method and evaluated via a multi-dimensional evaluation metrics. The proposed planning methodology is benchmarked over a Portugal 54-node urban distribution system to verify the superiority and effectiveness on the system economy and resilience levels. Case studies show that the proposed methodology can exploit the optimal synergies of different source–network–load–storage components and enhance system dispatchability.

A hybrid multi-objective solution approach for a reliable truck-drone routing problem integrated with pickup and delivery services

ABSTRACT Humanitarian logistics aims to reduce the arrival time of relief goods and optimize services in disaster-affected regions. This paper introduces a multi-objective hybrid truck-drone routing problem with pickup and delivery services, considering reliability (MHTDRP-PD-R) to enhance relief operations within humanitarian logistics. The model is formulated as a mixed-integer linear programming problem with the simultaneous goals of minimizing transportation time and maximizing reliability through routing and inventory decisions made by multiple trucks and drones. An adaptive large neighborhood search (ALNS) algorithm, coupled with heuristic technique, is developed to tackle the problem. The ALNS algorithm refines initial solutions through destroy and repair operators, integrated with multi-objective optimization methods to create a unified objective function. Small and large-scale test problems are then employed to evaluate solution quality using multi-objective optimization. The calculation results show the superiority of the developed ALNS integrated with the ε-constraint method regarding the objective value and solution time.

Interdependent Expansion Planning for Resilient Electricity and Natural Gas Networks

This study explores enhancing the resilience of electric and natural gas networks against extreme events like windstorms and wildfires by integrating parts of the electric power transmissions into the natural gas pipeline network, which is less vulnerable. We propose a novel integrated energy system planning strategy that can enhance the systems’ ability to respond to such events. Our strategy unfolds in two stages. Initially, we devise expansion strategies for the interdependent networks through a detailed tri-level planning model, including transmission, generation, and market dynamics within a deregulated electricity market setting, formulated as a mixed-integer linear programming (MILP) problem. Subsequently, we assess the impact of extreme events through worst-case scenarios, applying previously determined network configurations. Finally, the integrated expansion planning strategies are evaluated using real-world test systems.

Techno-economic evaluation of developing a zero-carbon highway service area energy system

ABSTRACT To further reduce CO2 emissions from highway service areas, this study proposes a novel zero-carbon highway service area energy system (S3). This system not only addresses energy supply challenges in remote locations but also expedites the reduction of CO2 emissions. A mixed-integer linear programming problem is formulated to optimize the configuration and operation of the proposed system. The system is compared with the conventional (S1) and low-carbon (S2) highway service area systems. Focused on northwest China, the results indicate that S3 can achieve zero CO2 emissions. However, the energy efficiency ratio (EER) of S1 is the highest at 84%, surpassing S3 by 2.4% (82%). In terms of economic analysis, the total cost and levelized cost of electricity (LCOE) of S1 are the highest at 1.62 MUSD and 0.14 USD/kWh, respectively. These values are 100% and 180% higher than those of S3, which are 0.81 MUSD and 0.05 USD/kWh.

Incorporating local uncertainty management into distribution system planning: An adaptive robust optimization approach

As the penetration of renewable energy sources (RES) significantly increases, their inherent uncertainty has brought about a paradigm shift in distribution system planning and operation. The existing planning methodologies usually consider the uncertainty during the investment and day-ahead stages while neglecting the interactive power fluctuations during the intraday stage, resulting in planning schemes with a risk of inadequate regional stability. In light of this, this paper proposes a novel adaptive robust distribution system planning approach that integrates local uncertainty management (LUM) to provide a comprehensive and refined consideration of uncertainty. First, we introduce the concept of LUM that can effectively address the deviations between predicted and actual power of RES and load. Second, we propose the adaptive robust planning model for the distribution system, in which the investment and day-ahead operation are considered in the first stage, while LUM is considered in the second stage to maintain the local stability of interactive power. The proposed model can actively manage uncertainties of RES and load demand, reaching the optimal balance between the investment cost, operational cost, and LUM cost. Finally, by employing the column-and-constraint generation (C&CG), the adaptive robust planning model is decoupled into a master min problem and a slaver max-min problem, after which the max-min problem is converted into a single mixed-integer linear programming (MILP) problem through the strong duality theory and big-M method. The effectiveness of the proposed model is demonstrated using a real-world distribution system located in Fujian Province, China. The simulation results indicate that the proposed model achieves a quantitative decrease of approximately 10% in total cost compared to deterministic planning. Sensitivity analysis is also conducted to reveal the trade-off between economic cost and robustness level.

Integrating Compressed CO2 Energy Storage in an Integrated Energy System

The integration of an energy storage system into an integrated energy system (IES) enhances renewable energy penetration while catering to diverse energy loads. In previous studies, the adoption of a battery energy storage (BES) system posed challenges related to installation capacity and capacity loss, impacting the technical and economic performance of the IES. To overcome these challenges, this study introduces a novel design incorporating a compressed CO2 energy storage (CCES) system into an IES. This integration mitigates the capacity loss issues associated with BES systems and offers advantages for configuring large-scale IESs. A mixed integer linear programming problem was formulated to optimize the configuration and operation of the IES. With an energy storage capacity of 267 MWh, the IES integrated with a CCES (IES–CCES) system incurred an investment cost of MUSD 161.9, slightly higher by MUSD 0.5 compared to the IES integrated with a BES (IES–BES) system. When not considering the capacity loss of the BES system, the annual operation cost of the IES–BES system was 0.5 MUSD lower than that of the IES–CCES system, amounting to MUSD 766.6. However, considering the capacity loss of the BES system, this study reveals that the operation cost of the IES–BES system surpassed that of the IES–CCES system beyond the sixth year. Over the 30-year lifespan of the IES, the total cost of the IES–CCES system was MUSD 4.4 lower than the minimum total cost of the IES–BES system.

Micro-grid tri-level EPEC based model for finding equilibria in the day-ahead and balancing markets

This paper presents a model that explores the strategic bidding equilibrium within a microgrid and its interactions with other strategic and non-strategic rivals in a joint energy and balancing market. The model utilizes a tri-level mathematical program with equilibrium constraints (MPEC) to represent the behavior of each strategic producer. Upper level maximizes the profit of each strategic producer, including microgrid and another strategic rival. The first lower-level problem involves maximizing social welfare through the day-ahead market clearing process, while the second lower-level problem deals with the balancing market clearing process. These objectives form the core of the proposed model. To simplify the tri-level problem, duality theory and the Karush-Kuhn-Tucker (KKT) optimally conditions are employed to transform it into a mixed integer linear programming (MILP) problem. By simultaneously solving all MPECs, an equilibrium problem with equilibrium constraints (EPEC) is formulated. The resulting EPEC is then addressed using a diagonalization algorithm and game theory to obtain a market Nash equilibrium, which constitutes the secondary objective of this paper. The effectiveness of the model is evaluated using the 6-bus test system as a case study. The results indicate that, at the equilibrium point, both the microgrid and rivals experience reduced profits compared to the initial state.

Evaluating the reliability of microgrids consisting of renewable energy sources using stochastic scheduling based on the data-driven uncertainty set

The paper addresses challenges arising from the unpredictable and probabilistic nature of wind speed and solar irradiance, leading to variable power production in renewable sources. Power imbalance issues in grids result from the difficulty in accurately estimating these variables. The study introduces an improved data-driven uncertainty set, utilizing a neural network trained with extensive historical data from a multi-microgrid system. This set efficiently identifies strong coupling and extracts characteristic scenarios, ensuring high reliability with a minimal number of scenes. The proposed probabilistic model, implemented in the coordination of pumped storage, wind, and solar power plants, incorporates energy limits and storage characteristics. A four-mode IEEE model for storage units allows quick response to system needs. Simulating storage unit states every hour, considering system load, output power, and energy limitations, optimizes power dispatch based on battery capacity. Additionally, the paper addresses the uncertainty in electric power demand in microgrids, proposing an improved data-driven uncertainty set obtained from a neural network. By eliminating unrealistic worst-case scenarios, the approach accurately captures system characteristics, maintaining high reliability while significantly reducing convergence time. The proposed uncertainty set is applied in a novel two-stage robust optimization dispatch model, transforming the optimization problem into a mixed-integer linear programming problem, and solving it using a column and constraint generation algorithm. Numerical case studies verify the feasibility and superiority of the proposed approach in terms of economy and convergence performance compared to existing robust optimization methods.

Linear optimization over the approximate solutions of a system of max-min equations

When a system of max-min equations is inconsistent, which is frequently encountered in modelling with fuzzy relations, the approximate solutions may be considered instead within an admissible error bound. This paper tackles the linear optimization problem over the approximate solutions with respect to the L∞ and the L1 residual error, respectively. It demonstrates that such an optimization problem can be reformulated as either a generalized set covering problem for the L∞ scenario or a mixed integer linear programming problem for the L1 scenario. As a result, it may be solved to optimality with the aid of a state-of-the-art optimization solver and hence does not demand a tailored solving method except for very large instances.

OPTIMAL INVENTORY POLICY USING ASSIGNMENT MODEL AS MIXED INTEGER LINEAR PROGRAMMING PROBLEM

This article discusses the assignment model as a mixed integer linear programming problem used in determining inventory policy for the multi-item case with backorder using the branch-and-bound method. The assignment model used must meet the specified constraints. The LINGO 18.0 application is used to solve mixed integer linear programming problems from the inventory problem assignment model. The optimal solution obtained from the assignment model uses the LINGO 18.0 application, which is based on the branch-and-bound method compared to the EOQ model with backorder. The results obtained from the comparison of these two models can be concluded to show that the inventory problem provides more optimal value using the assignment model as a mixed integer linear programming problem compared to using the EOQ model with backorder.

Analyzing flexibility options for microgrid management from economical operational and environmental perspectives

Power grids have undergone a massive transformation due to increasing number of renewable/non renewable distributed generators, storage technologies and electrical vehicles. This paper focuses on the flexible energy management of grid connected microgrid (MG) in order to analyze the effects of the flexibility options such as electric vehicle with demand response (EV-DR), battery storage system (BSS), hydrogen storage system (HSS) and dynamic line rating (DLR) in presence of renewable sources. In addition, microturbine (MT), fuel cell (FC), biomass (BIO), geothermal (GEO), and diesel engine (DIE) whose output is controllable are also considered as flexible sources. In this paper, optimum operation of grid-connected MG is modeled as a mixed-integer linear programming problem considering uncertainties of wind turbine (WT) and photovoltaic (PV) sources. A day-ahead forecasting of irradiance and wind speed are performed with Decision Tree Regression (DTR) and Long Short Term Memory (LSTM) algorithms for sensitive calculation of PV and WT output. The cost function merges operating costs including fuel, operation & maintains (O&M), depletion costs with emission cost of CO2, SO2, and NOx. Obtained numerical results show that total cost of the MG is reduced by 9.40% and emission cost is diminished by 5.59% with inclusion of all considered flexibility options. Further, load factor is improved from 0.7932 to 0.8236 with 100% flexibility condition for MG.

Coordinated optimization of source‐grid‐load‐storage for wind power grid‐connected and mobile energy storage characteristics of electric vehicles

AbstractThe rapid growth in the number of electric vehicles (EVs), driven by the ‘double‐carbon’ target, and the impact of uncontrolled charging and discharging behaviour and discharged battery losses severely limit electric vehicles’ low carbon characteristics. Existing research on systemic low‐carbon emissions and electric vehicle charging and discharging issues is usually determined by considering only carbon trading markets or charging and discharging management on the source side. In this regard, a coordinated and optimized operation model that considers the participation of electric vehicle clusters in deep peaking and the source network load and storage adjustable resources is proposed. The upper layer establishes a real‐time price‐based demand response mechanism for the load side with the minimum net load fluctuation as the objective function; the middle layer establishes a comprehensive operation mechanism for the source and storage side that includes an orderly charging and discharging peaking compensation mechanism for electric vehicles, and a deep peaking mechanism that takes into account clean emission, and constructs an optimal operation model with the minimum comprehensive operating cost as the objective function; the lower layer establishes a distribution network loss minimization model for the network side that takes into account the orderly charging and discharging of electric vehicle as the objective function. The optimal load model with the objective function of minimizing the distribution network loss is established at the lower level. Finally, the original problem is transformed into a mixed integer linear programming problem, and the model's effectiveness is verified by setting different scenarios. The model reduces the total cost by 22.22%, improves the wind power consumption rate by 19.55%, reduces the actual carbon emission by 16.66%, and reduces the distribution network loss by 13.91% compared to the basic model.

Planning Courses for Student Success at the American College of Greece

We are concerned with the personalized student course plan (PSCP) problem of optimizing the plan of courses students at the American College of Greece will need to take to complete their studies. We model the constraints set forth by the institution so that we guarantee the validity of all produced plans. We formulate several different objectives to optimize the resulting plan, including the fastest completion time, course difficulty balance, and maximization of the expected student grade point average given the student’s performance in past courses. All resulting problems are mixed-integer linear programming problems with a number of binary variables, that is, the max number of terms times the number of courses available for the student to take. The resulting mathematical programming problem is solvable in less than 10 seconds on a modern commercial off-the-shelf PC, whereas the manual process used to take more than one hour of advising time for every student and, as measured by the objectives set forth, resulted in suboptimal schedules. History: This paper was refereed.

An Optimization Approach to Berth Allocation Problems

The berth allocation problem determining the berthing time and position for incoming vessels in port operations has garnered increased attention within the global transportation network. This study focuses on the berth allocation problem with a continuous quay and dynamic vessel arrivals. With the overarching goal of enhancing service quality and optimizing berth utilization rates, this article proposes a mathematical programming model that minimizes the total waiting time of vessels and the overall completion time of vessel service. The formulated model is a mixed-integer linear programming problem that deterministic optimization techniques can globally solve. For large-scale problems, this study develops a genetic algorithm optimization approach to improve computational efficiency in reaching a near-optimal solution. Several numerical experiments are conducted to demonstrate the effectiveness and efficiency of the proposed approach.

Trapping Sets Search Using the Method of Mixed Integer Linear Programming with a Priori List of Variable Nodes

Purpose of research is to develop a new high-speed method for searching trappin sets in graph codes, ensuring the completeness of the search.Methods. There are two approaches to finding trappin sets. The first, based on the Monte Carlo method with a biased probability estimation using Importance Sampling, involves the use of a decoder. The advantage of this approach is its high performance. The disadvantages are the dependence on decoder parameters and channel characteristics and the finite probability of missing trappin sets. The second approach is based on the use of linear programming methods. The advantage of this approach is the completeness of the resulting list of trappin sets, due to its independence from the decoder parameters and channel characteristics. The disadvantage of this approach is its high computational complexity. In the article, within the framework of the second approach, a new method for searching trappin sets with less computational complexity is proposed. The method involves solving a mixed integer linear programming problem using an a priori list of code vertices participating in the shortest cycles in the code graph. Results. Using the proposed method, a search for trappin sets was performed for several low-density codes. For this purpose, the mathematical linear programming package IBM CPLEX version 12.8 was used, which was run on 32 threads of a 16-core AMD Ryzen 3950X processor with 32GB of RAM (DDR4). In the Margulis code (2640, 1320), using the proposed method, the trappin set TS(6,6) was found in a time of 0.53 s. The speedup provided by the method proposed in the paper compared to the Velazquez-Subramani method is 8252.415 times. Thanks to the high speed and completeness of the search, trappin sets were found for the first time TS(62,16) and TS(52,14) in the Margulis code (4896, 2474 ).Conclusion. The paper proposes a new method for searching trapping sets by solving a mixed integer linear programming problem with an a priori list of code. The method is fast and provides completeness of the search.

PORTFOLIO MODELS FOR OPTIMIZING DRAWDOWN DURATION

The drawdown duration, which measures the time elapsed since the portfolio obtained its maximum value, is an important criterion in active portfolio management for institutional investors. Although several optimization models exist for controlling portfolio drawdown magnitude (i.e. the percentage drop in portfolio value from its latest peak value), developing similar models for the drawdown duration has received minimal attention in the literature. Therefore, this paper develops a family of models for optimizing average, maximum and tail drawdown duration formulated as mixed-integer linear programming (MILP) problems, allowing the utilization of powerful solvers to identify optimal asset portfolios. We apply the developed models to real data on historical returns to compare their performance against traditional and drawdown-based portfolio selection models. The results indicate that the developed models lead to decrease in drawdown duration levels both in in-sample and out-of-sample tests. The constructed efficient frontiers also show a clear trade-off between minimizing drawdown duration and maximizing expected returns.

Trading strategies of energy storage participation in day-ahead joint market based on Stackelberg game

The goal of "carbon peak, carbon neutral" and the increasing expansion of new energy have helped to advance the development of energy storage. However, since the operating cost of energy storage is high, carbon emission trading and power market trading have emerged, effectively improving the efficiency. In this paper, a trading strategy and bidding framework of energy storage participation in the day-ahead joint market are studied. A market bidding model has been established in a framework based on the Stackelberg game. Finally, the "Day-Ahead and Intra-Day and Carbon Emission Trading (CET)" market clearing model has been constructed. It has been simplified to solve the equivalent mixed-integer linear programming (MILP) problem with equilibrium constraints through the use of the Karush–Kuhn–Tucker (KKT) optimality criterion and duality principle. The proposed model is validated through improved examples to obtain thermal unit output cuts of up to 32.2% during load trough periods, and up to 16.75% increase in clearing prices during peak load periods. The storage life is extended and the storage output variation is minimized.

Frequency Stability Constrained BESS Sizing Model for Microgrids

The grid integration of renewable energy sources necessitates using energy storage systems (ESSs) to provide more flexibility and controllability. This paper proposes a frequency stability-constrained battery energy storage system (BESS) sizing model for microgrids formulated as a mixed-integer linear programming (MILP) problem and decomposed using Benders decomposition. The optimal size of BESS is determined as a trade-off between minimizing the operating costs or maximizing the benefits and the high investment costs of BESS. Both the grid-connected and stand-alone operating modes are modeled for the microgrid along with the corresponding generation contingencies. The microgrid scheduling optimization model is built for cost-benefit analysis considering unit commitment and frequency stability (FS) constraints. The transient frequency dynamics are considered using an accurate time-domain model based on the discretized swing equation. The proposed method ensures frequency stability criteria, such as frequency nadir/overshoot, rate of change of frequency (RoCoF), and steady-state frequency, are within their allowable ranges following generation contingencies. Simulation results based on a medium-voltage microgrid test case with historical data of load, PV generation, and market price verify the effectiveness of the proposed method.

Sub‐optimal Internet of Thing devices deployment using branch and bound method

AbstractThe Internet of Thing (IoT) network deployments are widely investigated in 4G and 5G systems and will still be key technical systems to drive massive connectivity of 6G systems. In 6G, IoT systems will operate with 6G new technologies such as Integrated Sensing and Communications. The IoT systems of 6G will be a platform to collect information in real world and create new use cases and business models. As the IoT devices and cellular networks are getting smarter, the IoT ecosystem allows us to bridge between human life and digital life and accelerate the transition towards a hyper‐connected world. Optimal and scalable IoT network design has been investigated in many research groups but key challenges in this topic still remain. An IoT devices deployment problem is investigated to minimise the transmission and computation cost among network nodes. The IoT devices deployment problem is formulated as Mixed‐Integer Nonlinear Programming problem. After relaxing the constraints and transforming the problem to a mixed integer linear programming (MILP) problem, the authors propose a new branch and bound (BB) method with a machine learning function and solve the MILP problem as a sub‐optimal solution. In the numerical analysis, the authors evaluate both conventional BB method and the proposed BB method with weighting factors and compare the objective function values, the number of explored nodes, and computational time. The performances of the proposed BB method are significantly improved under the given simulation configuration. The author finds the optimal mapping of IoT devices to fusion nodes.

Online Optimization of Pickup and Delivery Problem Considering Feasibility

A pickup and delivery problem by multiple agents has many applications, such as food delivery service and disaster rescue. In this problem, there are cases where fuels must be considered (e.g., the case of using drones as agents). In addition, there are cases where demand forecasting should be considered (e.g., the case where a large number of orders are carried by a small number of agents). In this paper, we consider an online pickup and delivery problem considering fuel and demand forecasting. First, the pickup and delivery problem with fuel constraints is formulated. The information on demand forecasting is included in the cost function. Based on the orders, the agents’ paths (e.g., the paths from stores to customers) are calculated. We suppose that the target area is given by an undirected graph. Using a given graph, several constraints such as the moves and fuels of the agents are introduced. This problem is reduced to a mixed integer linear programming (MILP) problem. Next, in online optimization, the MILP problem is solved depending on the acceptance of orders. Owing to new orders, the calculated future paths may be changed. Finally, by using a numerical example, we present the effectiveness of the proposed method.