Mixed Integer Nonlinear Programming Model Research Articles

Optimising location, inventory and transportation in a sustainable closed-loop supply chain

Operations management researchers and practitioners have shown increasing interest in incorporating sustainability into supply chain (SC) design models. This means that sustainability must be considered in all aspects of the SC, including location, inventory and transportation (LIT) decisions. Hence the aim of this article is to propose an optimisation model that incorporates: (i) LIT decisions in an integrated manner; (ii) the three sustainability (3S) aspects, i.e. economic, environmental and social, into each named decisions; and (iii) a closed-loop supply chain (CLSC) structure. The proposed formulation is a multi-objective mixed integer non-linear programming (MO-MINLP) model whose objectives consider minimisation of economic and social costs (economic aspect) and carbon emissions (environmental aspect), and maximisation of the social impact of SC operations (social aspect). A transformation technique is applied to one of the objective functions, which results in an MO-MILP model solved by the lexicographic method. This article focuses on commodity industries where only one finished product is manufactured. Hence the 3S-LIT model is validated with a randomly generated dataset and against a recently published alternative model applied to the copper mining industry.

An inexact multi-objective mixed-integer nonlinear programming approach for water-soil-fertilizer management under uncertainty considering “footprint family-planetary boundary” assessment

Water, energy, and food (WEF) are interlinked and create a dynamic system that impacts both human well-being and ecology. Given the importance of ecology within the WEF system, water resources serve as the core issue. The allocation of water resources in irrigation districts is a challenging problem for the coordinated development of agricultural production, water resources, and the ecological environment. The integration of stochastic multi-objective programming, fuzzy credibility-constrained programming, and mixed integer programming offers a solution to this issue, with the construction of a fuzzy credibility-constrained stochastic multi-objective mixed-integer nonlinear programming model. The applicability and validity of this model were verified by applying it to the Kaikong Irrigation District (KID) of the Tarim River Basin in northwest China, with notable findings indicating that the optimized system reduces agricultural costs by 5.82%, increases irrigation water use efficiency by 1.80%, and reduces global warming potential by 6.45%. This study investigates the effects of diverse allocation strategies of water and land resources on the social, economic, and ecological subsystems and their interactions by downscaling four subprocesses PBs to the KID scale. The optimization model reveals that only the nitrogen footprint of Kuerle City surpasses the nitrogen boundary of the KID. The proposed solutions based on the model can encourage the green and ecologically-friendly development of agricultural production and can be applied to agricultural systems in arid regions with similar conditions.

Optimization of Fuel Supply Strategy for LNG Dual-fuel Ships

Fuel cost is an important part of voyage cost, and fuel replenishment strategy is an important decision affecting the operating cost and benefit of shipping companies. A mixed integer nonlinear programming model of fuel supply for LNG dual-fuel ships is established to expand the research scope of fuel supply in the shipping industry to minimize fuel cost. In this paper, the genetic algorithm is improved, the solving steps are designed, and the validity of the model and the algorithm are verified by example analysis. The results show that as long as the LNG price does not exceed 120% of the LSHFO price, the LNG dual-fuel ship can still reduce the fuel cost during the voyage. The lower the LNG price, the larger the LNG fuel tank capacity and the higher the proportion of fuel cost saved. When the fuel market fluctuates, the model can provide a scientific basis for shipping companies to determine the type of ship fuel, optimize the speed, and select the fuel port and fuel volume.

Multi-source coordinated stochastic restoration for SOP in distribution networks with a two-stage algorithm

After suffering from a grid blackout, distributed energy resources (DERs), such as local renewable energy and controllable distributed generators and energy storage can be used to restore loads enhancing the system’s resilience. In this study, a multi-source coordinated load restoration strategy was investigated for a distribution network with soft open points (SOPs). Here, the flexible regulation ability of the SOPs is fully utilized to improve the load restoration level while mitigating voltage deviations. Owing to the uncertainty, a scenario-based stochastic optimization approach was employed, and the load restoration problem was formulated as a mixed-integer nonlinear programming model. A computationally efficient solution algorithm was developed for the model using convex relaxation and linearization methods. The algorithm is organized into a two-stage structure, in which the energy storage system is dispatched in the first stage by solving a relaxed convex problem. In the second stage, an integer programming problem is calculated to acquire the outputs of both SOPs and power resources. A numerical test was conducted on both IEEE 33-bus and IEEE 123-bus systems to validate the effectiveness of the proposed strategy.

Distribution Network Expansion Planning Approach for Large Scale Electric Vehicles Accommodation

The increased power demand incurred by electric vehicles (EVs) has imposed new challenges on the distribution network. One significant responsibility of the distribution network operator (DNO) is to design and expand the distribution network periodically to accommodate the ever-increasing demand. The previous researches mainly use relaxation techniques to enhance the problem tractability, which would involve additional binary variables and incur computation burden to the optimization problem, or even worse, may lead to an intractable problem. This paper develops a mixed-integer non-linear programming (MINLP) model considering various network constraints for the distribution network expansion planning under a high penetration level of EVs. The stochastic characteristics of demands are addressed through set of scenarios. A distributed biased min-consensus algorithm based approach is proposed to solve the MINLP model. Finally, comparison tests are conducted on three different scales of distribution networks to validate the effectiveness of the proposed approach. Simulation results demonstrate that the computation time of the proposed approach is reduced by 42.46% on average for different systems compared with that of the traditional shortest path algorithm based approach.

Location and turbine parameter selection for offshore wind power maximization

Offshore wind turbines can capture more wind than onshore because of their larger structure and location. This higher yield even fails to reduce the high installation and maintenance cost of an offshore wind farm (OWF). Appropriate turbine parameters and installation site selection may maximize the power generation which is a way to trade off these costs. Knowing the wind thrust force, air density, and power coefficient beforehand can help select an appropriate site for turbine location. Once the site is selected, the optimal value of turbine variables such as height and radius can contribute to higher power yield. In this paper, a MINLP (Mixed Integer Non-Linear Programing) model is formulated with these important variables and the optimal values of these variables are determined to maximize the annual power production ([Formula: see text]) from offshore wind farm. The estimated power production, [Formula: see text], is calculated using two methods, mathematical programing method and simulation method. Computational result indicates that mathematical programing method is time consuming but more accurate whereas the accuracy of simulation method is proportional to the number of iterations. Although the result of a simulation can be improved to some extent, it cannot be as accurate as mathematical modeling for this study. These study results have great impact on the managerial decision and long range strategic and technical planning for maximizing power generation from an offshore wind farm.

Closed-loop scheduling optimization strategy based on particle swarm optimization with niche technology and soft sensor method of attributes-applied to gasoline blending process

Gasoline blending scheduling optimization can bring significant economic and efficient benefits to refineries. However, the optimization model is complex and difficult to build, which is a typical mixed integer nonlinear programming (MINLP) problem. Considering the large scale of the MINLP model, in order to improve the efficiency of the solution, the mixed integer linear programming - nonlinear programming (MILP-NLP) strategy is used to solve the problem. This paper uses the linear blending rules plus the blending effect correction to build the gasoline blending model, and a relaxed MILP model is constructed on this basis. The particle swarm optimization algorithm with niche technology (NPSO) is proposed to optimize the solution, and the high-precision soft-sensor method is used to calculate the deviation of gasoline attributes, the blending effect is dynamically corrected to ensure the accuracy of the blending effect and optimization results, thus forming a prediction-verification-reprediction closed-loop scheduling optimization strategy suitable for engineering applications. The optimization result of the MILP model provides a good initial point. By fixing the integer variables to the MILP-optimal value, the approximate MINLP optimal solution can be obtained through a NLP solution. The above solution strategy has been successfully applied to the actual gasoline production case of a refinery (3.5 million tons per year), and the results show that the strategy is effective and feasible. The optimization results based on the closed-loop scheduling optimization strategy have higher reliability. Compared with the standard particle swarm optimization algorithm, NPSO algorithm improves the optimization ability and efficiency to a certain extent, effectively reduces the blending cost while ensuring the convergence speed.

A novel economic scheduling of multi-product deterministic demand for co-production air separation system with liquid air energy storage

Air separation units (ASUs) integrated with liquid air energy storage (LAES) have the potential to balance grid demand and improve production profits. This study aims to enhance the demand response of a novel co-production air separation system (NCASS) by maximizing the energy arbitrage of a LAES system. We developed a mixed-integer nonlinear programming model to minimize operating costs by using a proxy model to represent the ASU production space and a black box model to represent the material and energy flow balance of LAES. The model is based on rigorous mathematical programming and, considers material and energy flow balance, analyzes the production and delivery processes and constraints for each product, and matches equipment start-up and shutdown time constraints. The case study shows that the model achieves a high level of matter and energy matching between the distillation process and the LAES process, maximizing the LAES peak-shaving capacity and improving system economics. During the 24 h schedule period, the total electricity consumption of NCASS increases by 18.94% during the valley electricity period and decreases by 20.59% during the flat and peak electricity period. The economic benefits increase by 9.51 k$, accounting for 14.29% of the total costs. This study contributes to the wider application of NCAS with LAES technology for sustainable energy management.

A MINLP model for the layout design of subsea oil gathering - transportation system in deep water oil field considering avoidance of subsea obstacles and pipe intersections

This paper focuses on the layout design of subsea oil gathering-transportation system in deep water oil field. In such a system, subsea manifolds are applied to gather and transport the produced fluids from subsea wells to floating processing terminals. A mixed integer nonlinear programming (MINLP) model is proposed, constrained by a series of operation and production requirements, aiming to minimize the total layout cost. Through the model, the pipe line network topology structure which reflects the allocations among the subsea wells, manifolds and processing terminals, the routes of pipes, as well as the size of the facilities could all be figured out. Two key contributions are made through this work. First, avoiding pipe intersections and subsea obstacles are integrated simultaneously, making the proposed model closer to practical situations. Second, a decomposition strategy based on Delaunay triangulation and gradient descent is constructed, achieving high quality initial solution and stable iteration process. The results of case studies indicate the validity, feasibility and stability of the proposed model and the solution method. Besides, the effect of manifold numbers on the layout optimization result is analyzed through the model, indicating its flexibility in the layout design analysis.

Queueing-based formulation model in a public transit network considering energy storage technology and demand charges

Battery Electric Buses (BEBs) are quickly emerging as a viable option for dramatically lowering transportation-related emissions. Finding optimal locations of charging stations for BEBs can facilitate the development of electric transit networks that provide continuous services. Using fast charging along with Energy Storage (ES) charger technologies has also garnered attention to mitigate the peak power demand. A consensus regarding two types of chargers may result in more investment in the electric transit network, which resulted in considering installing fast chargers in designing an electric transit network. While this study aims to consider two types of fast- and ES-chargers, along with demand charges and waiting times at terminal stops simultaneously, to minimize the total costs of the electric transit network. The queuing model integrated with the Mixed-Integer Non-Linear Programming (MINLP) model is proposed to locate the optimal placements of charging stations in the electric transit network. To validate the model, a benchmark Mumford0 network is applied. Results indicated that the proposed charging location model could efficiently determine optimal locations of fast- and ES-chargers across the electric transit network. Also, the results confirmed that considering ES chargers could significantly save 16.6 % of the total costs of the Mumford0 transit network and reduce the total demand charge imposed on the network by the fast charger. More interestingly, by purchasing more chargers, transit agencies can decide on an acceptable waiting time that would be comfortable for BEBs arriving at terminal stops.

Optimal Placement and Sizing of D-STATCOMs in Electrical Distribution Networks Using a Stochastic Mixed-Integer Convex Model

This paper addresses the problem regarding the optimal placement and sizing of distribution static synchronous compensators (D-STATCOMs) in electrical distribution networks via a stochastic mixed-integer convex (SMIC) model in the complex domain. The proposed model employs a convexification technique based on the relaxation of hyperbolic constraints, transforming the nonlinear mixed-integer programming model into a convex one. The stochastic nature of renewable energy and demand is taken into account in multiple scenarios with three different levels of generation and demand. The proposed SMIC model adds the power transfer losses of the D-STATOMs in order to size them adequately. Two objectives are contemplated in the model with the aim of minimizing the annual installation and operating costs, which makes it multi-objective. Three simulation cases demonstrate the effectiveness of the stochastic convex model compared to three solvers in the General Algebraic Modeling System. The results show that the proposed model achieves a global optimum, reducing the annual operating costs by 29.25, 60.89, and 52.54% for the modified IEEE 33-, 69-, and 85-bus test systems, respectively.

Efficient Reallocation of BESS in Monopolar DC Networks for Annual Operating Costs Minimization: A Combinatorial-Convex Approach

This article deals with the solution of a mixed-integer nonlinear programming (MINLP) problem related to the efficient reallocation of battery energy storage systems (BESS) in monopolar direct current (DC) grids through a master–slave optimization approach. The master stage solves the integer nature of the MINLP model, which is related to the nodes where the BESS will be located. In this stage, the discrete version of the vortex search algorithm is implemented. To determine the objective function value, a recursive convex approximation is implemented to solve the nonlinear component of the MINLP model (multi-period optimal power flow problem) in the slave stage. Two objective functions are considered performance indicators regarding the efficient reallocation of BESS in monopolar DC systems. The first objective function corresponds to the expected costs of the annual energy losses, and the second is associated with the annual expected energy generation costs. Numerical results for the DC version of the IEEE 33 bus grid confirm the effectiveness and robustness of the proposed master–slave optimization approach in comparison with the solution of the exact MINLP model in the General Algebraic Modeling System (GAMS) software. The proposed master–slave optimizer was programmed in the MATLAB software. The recursive convex solution of the multi-period optimal power flow problem was implemented in the convex discipline tool (CVX) with the SDPT3 and SEDUMI solvers. The numerical reductions achieved with respect to the benchmark case in terms of energy loss costs and energy purchasing costs were 7.2091% and 3.2105%, which surpassed the results reached by the GAMS software, with reductions of about 6.0316% and 1.5736%.

Dynamic zoning of the network using cooperative transmission and maintenance planning: A solution for sustainability of water distribution networks

Sustainability approaches in transmission and maintenance planning of network utilities with different maintenance zones, which involve increasing the reliability of the transfer while employing maintenance teams fairly, pose decision-making challenges. For these utilities, the best model to employ is the maximum-flow model, which considers network reliability and destination stocks. Therefore, the crucial point is to understand how zone owners can cooperate to maximize reliable transmission and increases maintenance income while considering fairness in allocating maintenance teams to maintenance centers over several periods. A Mixed-Integer Nonlinear Programming (MINLP) model is presented in this paper for maximizing the reliability in network utility issues, allocating maintenance teams, and exploring edge statuses. Cooperative Transmission and Maintenance (CTM) planning is a dynamic, multi-duration decision-making problem. The problem scale in this study is significantly enlarged to separate the lines into active, inactive, and in-repair sets, and consequently, a genetic algorithm is proposed. This study examines these approaches and analyses them (including collaborative outputs and managerial insights) in the context of three major water pipeline zones. Based on sensitivity analysis for 81 collaborative scenarios, the highest and lowest synergies were 83.61% and 36.7%, respectively. Finally, several collaborative income-sharing strategies based on game theory are investigated.

Risk-constrained energy management of an isolated multi-energy microgrid enhanced with hydrogen storage

Hydrogen energy has proven to be a key player in transitioning towards a carbon-neutral society. However, integrating it into existing multi-energy microgrids (MEMGs) is challenging due to uncertainties in hydrogen vehicle (HV) demand, gas and electricity demand, renewable energy output, and prices. To address this, a computationally feasible risk-constrained energy management model is developed using the information gap decision theory (IGDT) method. This model provides both robust and opportunistic strategies. The MEMG system includes electricity, hydrogen, and gas subsystems, and allows for conversions between power to hydrogen (P2H), hydrogen to power (H2P), hydrogen to gas (H2G), and gas to power (G2P). Liquified natural gas (LNG) is used as an external energy source to improve system security. The study also explores the use of hydrogen storage systems (HSS) and renewable energy for sustainability. While the problem is a mixed integer nonlinear programing model, it is converted into a mixed integer linear programing model to ensure global optimality. The proposed risk-based energy management model was tested on a case study and results showed a 2 % decrease in profit for the robust strategy and a 4 % increase in profit for the opportunistic strategy, with final profits of $4876.102 (robust) and $5171.624 (opportunistic).

Multi-Terminal Berth and Quay Crane Joint Scheduling in Container Ports Considering Carbon Cost

As container ports become increasingly important to the global supply chain, a growing number of ports are improving their competencies by consolidating multiple terminal resources internally. In addition, in the context of energy conservation and emission reduction, ports measure competitiveness not only in terms of terminal size, throughput and service level, but also in terms of low energy consumption and low pollution. Therefore, a nonlinear mixed-integer programming model considering the cost of carbon is developed for the multi-terminal berth and quay crane joint robust scheduling problem under uncertain environments to minimize the sum of expectation and variance of total cost under all randomly generated samples. The model considers the water depth and interference of quay cranes, etc. The expected vessel arrival time and the average operational efficiency plus relaxation are used as their actual values when scheduling. Finally, an improved adaptive genetic algorithm is developed by combining the simulated annealing mechanism, and numerical experiments are designed. The results show that the joint berth and quay crane scheduling with uncertainties and a multi-terminal coordination mechanism can effectively reduce the operating cost, including carbon costs and the vessel departure delay rate, and can improve resource utilization. Meanwhile, the scheduling with the multi-terminal coordination mechanism can obtain more significant improvement effects than the scheduling with uncertainties.

Optimizing Renewable Injection in Integrated Natural Gas Pipeline Networks Using a Multi-Period Programming Approach

In this paper, we propose an optimization model that considers two pathways for injecting renewable content into natural gas pipeline networks. The pathways include (1) power-to-hydrogen or PtH, where off-peak electricity is converted to hydrogen via electrolysis, and (2) power-to-methane, or PtM, where carbon dioxide from different source locations is converted into renewable methane (also known as synthetic natural gas, SNG). The above pathways result in green hydrogen and methane, which can be injected into an existing natural gas pipeline network. Based on these pathways, a multi-period network optimization model that integrates the design and operation of hydrogen from PtH and renewable methane is proposed. The multi-period model is a mixed-integer non-linear programming (MINLP) model that determines (1) the optimal concentration of hydrogen and carbon dioxide in the natural gas pipelines, (2) the optimal location of PtH and carbon dioxide units, while minimizing the overall system cost. We show, using a case study in Ontario, the optimal network structure for injecting renewable hydrogen and methane within an integrated natural gas network system provides a $12M cost reduction. The optimal concentration of hydrogen ranges from 0.2 vol % to a maximum limit of 15.1 vol % across the network, while reaching a 2.5 vol % at the distribution point. This is well below the maximum limit of 5 vol % specification. Furthermore, the optimizer realized a CO2 concentration ranging from 0.2 vol % to 0.7 vol %. This is well below the target of 1% specified in the model. The study is essential to understanding the practical implication of hydrogen penetration in natural gas systems in terms of constraints on hydrogen concentration and network system costs.

Optimal Integration of D-STATCOMs in Radial and Meshed Distribution Networks Using a MATLAB-GAMS Interface

This paper proposes an interconnection of the MATLAB and GAMS software interfaces, which were designed based on a master-slave methodology, to solve the mixed-integer nonlinear programming (MINLP) model problem associated with the problem regarding the optimal location and sizing of static distribution compensators (D-STATCOMs) in meshed and radial distribution networks, considering the problem of optimal reactive power flow compensation and the fact that the networks have commercial, industrial, and residential loads for a daily operation scenario. The objective of this study is to reduce the annual investment and operating costs associated with energy losses and the installation costs of D-STATCOMs. This objective function is based on the classical energy budget and the capacity constraints of the device. In the master stage, MATLAB software is used to program a discrete version of the sine-cosine algorithm (DSCA), which determines the locations where the D-STATCOMs will be installed. In the slave stage, using the BONMIN solver of the GAMS software and the known locations of the D-STATCOMs, the MINLP model representing the problem under study is solved to find the value of the objective function and the nominal power of the D-STATCOMs. To validate the effectiveness of the proposed master-slave optimizer, the 33-node IEEE test system with both radial and meshed topologies is used. With this test system, numerical comparisons were made with the exact solution of the MINLP model, using different solvers in the GAMS software, the genetic-convex strategy, and the discrete-continuous versions of the Chu and Beasley genetic algorithm and the salp swarm optimization algorithm. The numerical results show that DSCA-BONMIN achieves a global solution to the problem under study, making the proposed method an effective tool for decision-making in distribution companies.

Development of a MATLAB-GAMS Framework for Solving the Problem Regarding the Optimal Location and Sizing of PV Sources in Distribution Networks

This paper addresses the planning problem regarding the location and sizing of PV generators in distribution networks with a radial topology. This problem is mathematically modeled using a mixed integer nonlinear programming (MINLP) model, which seeks to reduce the total annual operating costs of the system for a planning horizon of 20 years. The objective function used in this paper comprises three elements: (i) the energy purchase costs at the substation node (i.e., the main supply node), (ii) the investment costs for the integration of PV generators, and (iii) the costs associated with the operation and maintenance of these devices. To solve this problem, the interconnection of MATLAB and GAMS software is proposed, while using a master–slave methodology, with which a high-quality solution to this problem is achieved. In the master stage, the MATLAB software is used as a tool to program a discrete version of the sine–cosine algorithm (DSCA), which determines the locations where the PV generators are to be installed. In the slave stage, using one of the solvers of the GAMS software (BONMIN) with the known locations of the PV generators, the MINLP model representing the problem to be studied is solved in order to find the value of the objective function and the nominal power of the PV generators. The numerical results achieved in the IEEE 33- and 69-node systems are compared with the mixed-integer conic programming model solution reported in the specialized literature, thus demonstrating the efficiency and robustness of the proposed optimization methodology.

Operational Reliability Assessment of Distribution Network With Energy Storage Systems

In this article, a novel approach that considers the time-varying load restoration capability is proposed for operational reliability assessment of distribution networks. To evaluate the operational reliability, two indices are firstly defined as the minimal load loss under the worst-case fault contingency in the upcoming time interval. To search for the optimal remedial actions for load restoration, a sequential network reconfiguration (NR) model is then proposed considering the reconfigurability of feeders and capability of emergent power support of energy storage systems (ESSs). Since this model is a complex nonlinear mixed integer programming model, an improved greedy searching-based algorithm is proposed to compute the optimal remedial actions and to further evaluate the risk of load loss. Comparative case studies are presented in this article to demonstrate that the load level, the expectation of the state-of-charge of ESS and the failure duration have the impacts on the system operational reliability. Besides, the proposed sequential NR model can be used to enhance the reliability of distribution networks by enabling an effective sequential load transfer strategy under emergent conditions. Unlike traditional reliability assessment, the proposed method provides the operators with the real-time awareness of the operational reliability level considering the real-time condition of distributed resource equipped feeders and worst-case emergency constraints.

An integrated approach for optimizing location-inventory and location-inventory-routing problem for perishable products

The grocery market, especially the perishable product market is under amazing developments in China, accounting for half of overall food expenditure. However, the supply chain of perishable products has encountered serious problems, including significant freight losses, high inventory and transportation costs, etc. To solve these problems, the current paper proposes an integrated approach for simultaneously optimizing the decisions involved in the supply chain, typically including location, inventory, and routing strategy. A three-echelon supply chain including suppliers, DCs and retailers is structured in this paper, with the impacts of whether allowing direct shipment from supplier to retailer being considered. Two typical problems are investigated: location-inventory problem (LIP) and location-inventory-routing problem (LIRP). Mixed integer nonlinear programming model for LIP is proposed while Mixed integer linear programming models for LIRP is proposed, with the objective of minimizing the overall costs of supply chain, which are composed of infrastructure construction costs, transportation costs, inventory costs, and freight losses. CPLEX is used for LIP model solution and a two-stage heuristic algorithm based on Simulated Annealing is designed to solve the LIRP model. The proposed mathematical models and solution method are then applied to a real supply chain network for fresh milk. The optimized results suggest that considering the direct shipment from supplier to retailer is beneficial in terms of reducing the overall supply chain costs by 31%, with 43% reductions in transportation costs, 69.4% reductions in freight losses, 99.4% increases in inventory costs. With the developments of e-retailers for perishable products, allowing the direct shipment from supplier to retailer will bring more benefits in the future.