Mixed-integer Nonlinear Programming Formulation Research Articles

Model predictive control for optimal dispatch of chillers and thermal energy storage tank in airports

Cost of energy consumption is one of the biggest operational cost for airports, and it is increasing from time to time as airports expand to support growing number of passengers. Various factors affect the energy consumption including efficiency of airport Heating Ventilation and Air conditioning (HVAC) systems, which in turn depends on the efficiency of individual subsystems. In this paper, we present an optimal scheduling method for the central plant system at Dallas Fort Worth airport, involving chillers, pumps, and a thermal energy storage (TES) system. A model predictive control (MPC) problem is formulated to minimize both energy and demand charge costs while satisfying the cooling needs of the airport. The proposed Mixed-Integer Nonlinear Programming (MINLP) formulation includes performance curve based models for chillers and pumps and a simplified state of charge model for TES. The formulation also includes predictions of cooling load and chilled water return temperature. Simulation results for a month in summer show savings around 10% compared to the baseline. Initial recommendations based on insights from simulation results to the manual operation procedures resulted in significant savings. Field test results show a 7% chiller efficiency improvement.

Multi-objective mammography unit location–allocation problem: A case study

This work addresses the Multi-Objective Mammography Unit Location–allocation Problem (MOMULAP), aiming to meet three objectives: maximize mammography screening coverage, minimize the total distance traveled weighted by the number of users, and maximize equity in access to mammography screening. We introduce a mixed-integer nonlinear programming (MINLP) formulation to represent the MOMULAP and algorithms based on the Non-dominated Sorting Genetic Algorithm II (NSGA-II) and Strength Pareto Evolutionary Algorithm (SPEA2) for treating it. The algorithms were tested with data from seven Brazilian states. In these states, the number of cities ranges from 139 to 853, equipment from 23 to 347 units, and estimated annual demand for screenings from 96,592 to 1,739,085. The solutions provided by this work allow health managers to choose the appropriate location and allocation of the mammography units, considering different objectives.

Modeling and optimization of steady flow of natural gas and hydrogen mixtures in pipeline networks

We extend the canonical problems of simulation and optimization of steady-state gas flows in pipeline networks with compressors to the transport of mixtures of highly heterogeneous gases injected throughout a network. Our study is motivated by proposed projects to blend hydrogen generated using clean energy into existing natural gas pipeline systems as part of efforts to reduce the reliance of energy systems on fossil fuels. Flow in a pipe is related to endpoint pressures by a basic Weymouth equation model, with an ideal gas equation of state, where the wave speed depends on the hydrogen concentration. At vertices, in addition to mass balance, we also consider mixing of incoming flows of varying hydrogen concentrations. The problems of interest are the heterogeneous gas flow simulation (HGFS), which determines system pressures and flows given fixed boundary conditions and compressor settings, as well as the heterogeneous gas flow optimization (HGFO), which extremizes an objective by determining optimal boundary conditions and compressor settings. We examine conditions for uniqueness of solutions to the HGFS, as well as compare and contrast mixed-integer and continuous nonlinear programming formulations for the HGFO. We develop computational methods to solve both problems, and examine their performance using four test networks of increasing complexity.

Discrete optimal designs for distributed energy systems with nonconvex multiphase optimal power flow

The optimal selection, sizing, and location of small-scale technologies within a grid-connected distributed energy system (DES) can contribute to reducing carbon emissions, consumer costs, and network imbalances. There is a significant lack of studies on how DES designs, especially those with electrified heating systems, impact unbalanced low-voltage distribution networks to which most DES are connected. This is the first study to present an optimisation framework for obtaining discrete technology sizing and selection for grid-connected DES design, while simultaneously considering multiphase optimal power flow (MOPF) constraints to accurately represent unbalanced low-voltage distribution networks. An algorithm is developed to solve the resulting Mixed-Integer Nonlinear Programming (MINLP) formulation. It employs a decomposition based on Mixed-Integer Linear Programming (MILP) and Nonlinear Programming (NLP) and uses integer cuts and complementarity reformulations to obtain discrete designs that are also feasible with respect to the network constraints. A heuristic modification to the original algorithm is also proposed to improve computational speed. Improved formulations for selecting feasible combinations of air source heat pumps (ASHPs) and hot water storage tanks are also presented. Two networks of varying size are used to test the optimisation methods. Designs with electrified heating (ASHPs and tanks) are compared to those with conventional gas boilers. The algorithms outperform one of the existing state-of-the-art commercial deterministic MINLP solvers, which fails to find any solutions in two instances within specified time limits. While feasible solutions were obtained for all cases, convergence was not achieved for all, especially for those involving the larger network. Where converged, the algorithm with the heuristic modification has achieved results up to 70% faster than the original algorithm. Results for case studies suggest that including ASHPs can support up to 16% higher renewable generation capacity compared to gas boilers, albeit with higher ASHP investment costs, as local generation and consumption minimises network violations associated with excess power export. The results also show the importance of including nonlinear power flow constraints in DES design problems. The optimisation framework and results can be used to inform stakeholders such as policymakers and network operators, to increase renewable energy capacity and aid the decarbonisation of domestic heating systems.

Equitable supply in intermittently operated rural water networks in emerging economies

Abstract Many operators of water distribution networks (WDNs) are unable to meet the increasing demand for water. Utility operators in such situations resort to rationing the supply as a partial solution to this problem; this, in turn, may lead to disproportionate allocation of water or inequity in supply. In this study, we propose a mixed integer non-linear program formulation and an efficient solution approach to minimize the inequity in supply, subject to hydraulic constraints and additional constraints on hours of supply and valve operation. Further, we show that the schedule can be obtained using a data-driven approach based on flow and level measurements, which eliminates the modelling effort and uncertainty associated with the use of hydraulic models. We demonstrate the proposed approaches through simulations of a real WDN, and experiments conducted on a topologically similar laboratory-scale network.

Computer aided molecular design coupled to deep learning techniques as a less-expensive approach to design organic photoredox catalysts

To date, the prediction of electronic absorption spectra of organic molecules by computational chemistry methods is still limited, with differences of even 100 nm between the theoretical and experimental values, often at relatively high computational cost. In this work, we present a computationally cheaper approach for the design of new molecules that may be candidates for organic photoredox catalysts based on the prediction of their absorbance in the visible region of the electromagnetic spectrum. This problem is important to promote the development and use of solar energy in chemical processes, as an exemplary application. The design of new chromophoric molecules is approached as a two-step procedure. In the first step, a Mixed-Integer Nonlinear Programming (MINLP) model was formulated for the In-silico design of organic chromophores. A quantitative structure–activity relationship model was developed for the prediction of the difference in energy from the HOMO to the LUMO orbitals of organic dyes. This molecular design approach produced 4 molecules, where 3 have not been previously reported. In the second step, a large chemical compound database was leveraged to develop a deep learning model aimed at forecasting the absorption of visible light by such and related compounds. We have taken advantage of the deep learning model to verify that the light absorption between this model and the results of the MINLP formulation match with reasonable accuracy. For 3 of the 4 molecules, both results appear similar, allowing to verify that the molecules could in fact offer the desired absorbance at the target wavelength.

An MINLP formulation to identify thermodynamically-efficient distillation configurations

Designing configurations for separation of multicomponent mixtures by distillation is challenging because of (i) combinatorial explosion of the choice set, and (ii) nonconvex nature of the governing equations. This work proposes a novel Mixed Integer Nonlinear Program (MINLP) that is formulated to identify configurations that minimize the total exergy loss/maximize thermodynamic efficiency of the separation process. The formulation in its default form has several nonlinear nonconvex equations. We propose a model reformulation via a simple variable elimination technique to reduce the system of nonlinear equations to a single equation, which we refer to as the exergy constraint. We describe the properties of exergy constraints and exploit them in deriving additional valid cuts for the problem. Finally, we use the model for a case study concerning the recovery of Natural Gas Liquids (NGLs) from shale gas.

A novel mathematical model to incorporate carbon trading and other emission reduction techniques within energy planning models

A dependable and sustainable energy supply is crucial as energy consumption continues to rise due to population growth, economic development, and improved living standards. The use of fossil fuels leads to CO2 emissions and are subject to volatility in prices. Capital-intensive technologies to reduce emissions are challenging to implement on a practical scale, and economic instruments are likely to play a role in future energy systems by encouraging adoption of these technologies. Carbon trading is an emerging economic instrument that enables entities (plants, governments, etc.) to exchange emission rights, allowing economic and environmental aspects to be balanced. This study introduces a scalable carbon trading modelling approach, integrated into previously developed DECO2 open-source energy planning framework. Direct and indirect optimisation approaches are proposed, both consisting of superstructure-based mixed-integer nonlinear programming formulations. Carbon price is a variable in the direct optimisation or a parameter in the indirect optimisation approach. While the direct optimisation approach involves more non-linearity, it is shown to result in solutions with greater decarbonisation, higher profits, and lower costs, compared to the indirect optimisation results. A novel feature of this multi-period model not considered in previous works is the simultaneous emissions trading across time periods and among entities (power plants and government). This enables efficient and coordinated emission allowances trading among various entities and timeframes. Various new costs and revenue streams are added into the energy planning framework; therefore, profits can also be predicted, along with predictions of electricity prices. New energy resources (nuclear and wind) and carbon capture utilisation and storage are also introduced to the modified DECO2 model. The models are tested on the Pakistan's power sector. Minimisation of emissions using direct optimisation showed that the carbon trading increased profits significantly in the second, third, and fourth planning periods (4.74, 3.86, and 3.55 times, respectively), but in the first period, profits were slightly higher without carbon trading (1.06 times more). Minimisation of budget using indirect optimisation showed higher profits in case of no carbon trading for all the periods. Between 2021 and 2040, hydropower is expected to grow the most (by a minimum of 3.14 times and a maximum of 15.87 times), followed by solar (with an expected increase between 2.54 and 3.26 times) and wind generation (which may increase by 2.35 to 2.66 times). Deployment of emission reduction technologies is significantly lower when carbon trading is implemented as compared to when it is not, due to increased pressure on CO2-intensive generation. Results show that incorporating carbon trading into an energy market leads to both financial (increased profits) and environmental (lower emissions) sustainability, and that using direct optimisation approach increases benefits of carbon markets.

Facility location problems on graphs with non-convex neighborhoods

In this paper, we deal with some variants of classical facility location problems on graphs in which both the customers and the facilities belong to a non-necessarily convex neighborhood. Therefore, a point in each neighborhood representing the customer/facility has to be determined, and the customers have to be assigned to the facilities depending on a criterion. In particular, the p-median, the p-center, and the p-maximal covering versions of this problem on graphs are analyzed. An important difference with respect to their classical versions is that the lengths of the arcs depend on the location of the points chosen in the neighborhoods. Therefore, the lengths are not part of the input but part of the decision process. Assuming that the neighborhoods are Mixed-Integer Second Order Cone representable, different mixed-integer non-linear programming formulations are proposed for each one of the considered problems. Moreover, solution procedures providing bounds and a preprocessing phase are developed to reduce the number of variables and constraints of the proposed formulations. The results of an extensive computational experience are reported.

Energy Minimization of Dynamic Multi-UAV Communication Network for Cooperative MultiHop Data Gathering

This study proposes a minimum-energy operational planning problem for multiple unmanned aerial vehicles (UAVs) conducting a cooperative multi-hop data-gathering mission. A multi-UAV ad-hoc network collects data from multiple sources and transmits them to a ground base station. The proposed mixed-integer nonlinear programming (MINLP) formulation simultaneously determines cooperative task allocation, data gathering schedule, dynamic aerial data flow, and UAV trajectories to minimize operational energy consumption. A hierarchical two-phase solution procedure that decomposes the original problem into a series of less complex problems and solves them efficiently is developed. The effectiveness of the proposed formulation and the solution procedure is demonstrated through numerical experiments and comparison with benchmarking operational schemes.

A multi-objective multi-criteria decision-making approach for sustainable vaccine distribution, inventory control (R,s,S policy), and waste management considering risk assessment, a case study

Vaccines play a vital role in monitoring and preventing the growth of contagious diseases. As of December 2019, With the prevalence of corona worldwide, all communities, particularly developing countries, came across irresistible challenges in providing a proper vaccine for citizens. Moreover, accurate waste control is of great importance, particularly in the context of infectious medical waste. To address mentioned difficulties regarding vaccine inventory ordering & waste management, this paper develops a sustainable multi-objective mixed-integer non-linear programming (MINLP) formulation. In order to examine real goals, all three proposed objective functions are regulated in compliance with the obtained result of the risk assessment. besides, for exploring the applicability of the proposed network, a real case study in one of the central cities in Iran, Kashan province, is considered. Two test problems have also been defined, encompassing small and medium size solutions. The LP-metric and goal programming approaches are utilized to solve the model. According to one of the sensitivity analysis results, extending the review period will be conducive to proliferation in the amount of order quantity and subsequently, imposing more costs on the chain at a particular time and getting the opportunity from countries to capitalize monetary values in other public affairs.

Dynamic optimization of formation and pace in a cycling team time trial considering effects of drafting and cyclists’ fatigue and recovery

This paper proposes a dynamic formation optimization approach to cycling team time trials, taking into account road elevation, drafting effects, and fatigue and recovery of the cyclists. Dynamic models of fatigue and recovery are those proposed in authors’ earlier work and calibrated using human subject data. The formation optimization problem is first solved for a team of two cyclists, with insights then used to formulate the problem for the more complex and general n-cyclist case. A mixed-integer nonlinear program formulation is proposed to determine the optimal position rotation strategy in addition to optimal power and velocity trajectory of the platoon. Simulation results show improved time compared to a baseline strategy that is popular among cycling teams.

Efficient Integration of Photovoltaic Solar Generators in Monopolar DC Networks through a Convex Mixed-Integer Optimization Model

The problem regarding the optimal siting and sizing of photovoltaic (PV) generation units in electrical distribution networks with monopolar direct current (DC) operation technology was addressed in this research by proposing a two-stage convex optimization (TSCO) approach. In the first stage, the exact mixed-integer nonlinear programming (MINLP) formulation was relaxed via mixed-integer linear programming, defining the nodes where the PV generation units must be placed. In the second stage, the optimal power flow problem associated with PV sizing was solved by approximating the exact nonlinear component of the MINLP model into a second-order cone programming equivalent. The main contribution of this research is the use of two approximations to efficiently solve the studied problem, by taking advantage of convex optimization models. The numerical results in the monopolar DC version of the IEEE 33-bus grid demonstrate the effectiveness of the proposed approach when compared to multiple combinatorial optimization methods. Two evaluations were conducted, to confirm the efficiency of the proposed optimization model. The first evaluation considered the IEEE 33-bus grid without current limitations in all distribution branches, to later compare it to different metaheuristic approaches (discrete versions of the Chu and Beasley genetic algorithm, the vortex search algorithm, and the generalized normal distribution optimizer); the second simulation included the thermal current limits in the model’s optimization. The numerical results showed that when the maximum point power tracking was not regarded as a decision-making criterion, the expected annual investment and operating costs exhibited better performances, i.e., additional reductions of about USD 100,000 in the simulation cases compared to the scenarios involving maximum power point tracking.

New Mixed-Integer Nonlinear Programming Formulations for the Unit Commitment Problems with Ramping Constraints

Mixed-Integer Formulations for Power Production Problems The unit commitment problem is a complex mixed-integer nonlinear program that originates in the field of power production. Although it arises in a monopolistic system, there is still great attention to this problem even in a free-market regime, where it constitutes only a subproblem of larger ones. Historically, it was usually solved by Lagrangian relaxation methods. However, the advances achieved by commercial solvers of mixed-integer (linear and convex) programming problems have made such approaches an attractive option. T. Bacci, A. Frangioni, C. Gentile, and K. Tavlaridis-Gyparakis present the first mixed-integer nonlinear programming formulation with a polynomial number of both variables and constraints that describes the convex hull of the feasible solutions of the unit commitment problem with a single thermal generation unit, comprising all typical constraints and convex power generation costs. Proving that the formulation is exact requires a new result about the convex envelope of specially structured functions that can have independent interest. This new formulation for a single power generation unit is used to derive three new formulations for the general unit commitment problem whose effectiveness has been tested against the state-of-art formulation.

Hybrid Machine Learning-Mathematical Programming Approach for Optimizing Gas Production and Water Management in Shale Gas Fields

This paper presents a novel mathematical programming approach that simultaneously incorporates a mixed-integer nonlinear programming formulation with machine learning models to determine the operating conditions, gas production, and optimal water management for the completion phase in shale gas fields. The dataset for the development of an artificial neural network model has been collected from the Eagle Ford Texas formation. The total cumulative gas production and flowback water generated in shale gas wells are selected as output variables. The mathematical optimization model considers machine learning models for each well, mass balances, treatment, storage, reuse, and disposal options as well as well location selection and associated costs and revenues from the sale of the shale gas produced. A case study has been used to illustrate the benefits of the proposed approach. The compromise solution offers a water consumption per produced energy of 6.71 L/GJ in addition to the fact that 27% of the total fracture water required can be obtained by reusing the flowback water with attractive economic indicators.

Optimization-Based Selection of Hydrants and Valves Control in Water Distribution Networks for Fire Incidents Management

In England and Wales, water utilities reduce hydraulic pressure to a minimum regulatory threshold in order to reduce leakage and avoid financial penalties. However, utilities are not legally bound to guarantee specific flow rates from fire hydrants, thus posing a risk for firefighting. We formulate a biobjective mixed-integer nonlinear program (MINLP) to simultaneously determine control valve settings and the location of fire hydrants to be utilized in a water distribution network during urban fire incidents. The goal is to provide the required flow rate from the fire hydrants while minimizing 1) the distance of the utilized fire hydrants from the fire location and 2) the impact on customer supply. As the solution is required in real-time, we propose an optimization-based heuristic, which relies on iteratively solving a NLP approximation and relaxation of the MINLP formulation. Furthermore, we assess the quality of the heuristic solutions for the presented study case by calculating global optimality bounds. The proposed heuristic is applied to an operational water distribution network.

Dynamic coal blending under sustainability concerns: a case study from the Indian steel industry

PurposeIncreasing scarcity of natural resources and the adverse effects of unsustainable practices call for more and more efficient management strategies in the energy industry. The quality of the coke plays a significant role in the quality and durability of the output steel which is produced using the energy from the coal. This paper aims to investigate the dynamic coal blending problem under overall cost and coke quality constraints in the steel industry within a periodic cycle of operations.Design/methodology/approachConsidering the variability of the natural properties over a periodic cycle, this study proposes a multi-period mixed-integer non-linear programming formulation to optimize the total blending costs while taking various coke quality constraints into account. Besides, this study applies factorial design to investigate about the significant effect of coal proportions as well as improvement into the overall cost of blending.FindingsIn this case study, utilizing real data from a coal blending facility in India, through a factorial design, the authors obtain optimal desirable levels of coal proportions and their criticality levels towards the total cost of blending (TCB) or objective function. This analysis reflects the role of the coke quality constraints in the objective function value while characterizing the price of sustainability for the case study among other critical insights.Originality/valueObjective function (or TCB) includes basic coal cost, movement cost and environmental costs during the coal and coke processing at a coke-oven and blast furnace of steel industry. The price of sustainability provides managerial insights on that sacrifices the industry has to make in order to become more “sustainable”.

Style: A new optimization model for Synthesis of uTility sYstems with steam LEvel placement

Moving industrial production to a more sustainable basis requires a step change in the efficiency of site steam and cogeneration systems, coupled with a switch away from fossil fuels to renewable energy sources. This paper presents the first step in the development of an optimization framework capable of identifying a roadmap to move utility systems to a more sustainable basis. Previous approaches to the design and optimization of steam and cogeneration systems have simplified the problem to the extent that many important practical issues have been neglected, restricting the scope of the options included. The use of grossly simplified models has been prompted by the mathematical difficulties of optimizing such complex energy systems. To overcome limitations in previous work and provide a sound basis for future developments, this paper proposes a new superstructure-based optimization model for the optimization of utility systems, accounting for optimum steam level placement. The latter is important for improving systems efficiency and reducing energy consumption. The optimization problem involves the selection of more realistic operating conditions of the steam mains (superheat conditions and pressure). The model accounts for water preheating, as well as superheating and de-superheating for process steam generation and use. Hot oil circuits are also included as hot utility option to overcome potential steam temperature and/or pressure limitations at high temperatures, and with it provide more flexibility in the framework. The general problem requires making several continuous and discrete decisions, where non-linearities and non-convexities from underlying physics and binary decisions exacerbate the complex nature of the problem, yielding a nonconvex Mixed Integer Non-Linear Programming (MINLP) formulation. However, MINLP formulations can become computationally intractable to solve. Thus, to guarantee tractability and fast conversion, the STYLE model develops a novel successive Mixed Integer Linear Programming formulation. The STYLE methodology is applied to two case studies to illustrate the advantages of the synthesis method and the benefits of optimizing steam levels for the reduction of overall energy consumption at industrial sites. The proposed approach addresses major shortcomings inherent in previous research and provides a foundation for future work to explore the next generation of sustainable utility systems.

Real-time optimization of train regulation and passenger flow control for urban rail transit network under frequent disturbances

Urban rail transit networks face frequent delays and oversaturation resulting from unexpected disturbances and high demands during rush hours, which inevitably leads to unstable train operations, poor service experiences, and potential accidents on crowded platforms. This paper explores the train regulation and passenger flow control strategies for large-scale urban rail transit networks. Specifically, we develop a mixed-integer nonlinear programming (MINLP) formulation to improve the punctuality and regularity in train operations, reduce the passenger waiting time, and alleviate the passenger flow burden of platforms, where the dynamic coupling interactions among network-wide passengers, trains and stations are systematically considered. To reduce the computational complexity and satisfy real-time requirements, we particularly devise a real-time optimization approach based on iterative nonlinear programming (INP) combined with Lagrangian relaxation (LR) under the rolling horizon (RH) framework, which effectively disposes of the intractable nonconvexity and constructs the desirable properties of decomposability and parallelism. The original problem is transformed into a sequence of line-level subproblems that can be solved quickly, so as to circumvent the computational burden. Moreover, to verify the effectiveness of our approach, we apply it to address numerical examples on a simplified network and the real-world Beijing Subway network. The realistic case studies illustrate that the proposed approach can reduce the train deviation from the original timetable, the passenger waiting time, and the passenger accumulation risk on the platform, by around 84.92%, 30.34% and 61.42% in comparison with the common rule-based strategy. Additionally, the computation time for the realistic large-scale experiment is acceptable for a real-time implementation.

A two-stage Genetic Algorithm for joint coordination of spare parts inventory and planned maintenance under uncertain failures

The main problem of spare parts management is to maintain the minimal requirement of stock keeping units kept. This research proposes a joint optimization model of multi-item multi-period spare parts inventory management and planned maintenance under uncertain failures in order to balance inventory cost and spare parts availability. This paper presents a Mixed Integer Non-linear Programming formulation of the inventory optimization model under a minimum and maximum inventory policy with stock review intervals. Some studies in the literature have considered aggregating spare parts inventory management as they assume that it will reduce the ordering cost. We consider both independent and aggregate spare parts inventory policies and then combine the formulation with the predictive maintenance interval, which is a replacement action due to uncertain failures under predefined distribution. Furthermore, a novel two-stage Genetic Algorithm is proposed as a sim-heuristic approach to deal with the non-linearity, combinations, and stochasticity of the problem and solve large-scale instances. In the end, we perform a computational study on some instances and a real-world case study to demonstrate the proposed approach’s effectiveness and efficiency. The computational study shows that the independent policy results in lower cost than the aggregate policy, and the proposed Genetic Algorithm performs efficiently for large-scale problems.