Nonlinear Programming Formulation Research Articles

Control allocation optimization for an over-actuated tandem tiltwing eVTOL aircraft considering aerodynamic interactions

This paper describes control allocation strategies for a tandem tiltwing electric vertical take-off and landing (eVTOL) aircraft with distributed propulsion. The control mappings are constructed as local surrogate models that relate the control effectors to the aircraft's aeropropulsive forces and moments across the full flight envelope, including interactional aerodynamics effects. Control allocation optimization problems are formulated based on force and moment commands for specific flight conditions. Specifically, a nonlinear programming formulation based on nonlinear control mapping is proposed for control allocation optimization, and the problem is solved using a sparse nonlinear optimizer. The solutions obtained from different formulations such as the minimization of force and moment residuals and the minimization of control effort are investigated. Furthermore, these solutions are compared with the solutions obtained using linear control mapping-based formulations. Results presented for the cases of forward flight, low speed flight & hover, and flight in the transition corridor suggest that considering aerodynamic interactions while formulating and solving the control allocation problem is necessary for higher levels of accuracy compared to linear approaches. One-propeller-out scenarios and moment and force generation cases are also investigated. Finally, control allocation for steady and accelerated operation in the transition corridor is discussed. The contribution of the methods and results in the paper is a nonlinear control allocation methodology for the entire flight envelope of an over-actuated tandem tiltwing eVTOL aircraft.

A two-phase sequential algorithm for global optimization of the standard quadratic programming problem

AbstractWe introduce a new sequential algorithm for the Standard Quadratic Programming Problem (StQP), which exploits a formulation of StQP as a Linear Program with Linear Complementarity Constraints (LPLCC). The algorithm is finite and guarantees at least in theory a $$\delta $$ δ -approximate global minimum for an arbitrary small $$\delta $$ δ , which is a global minimum in practice. The sequential algorithm has two phases. In Phase 1, Stationary Points (SP) with strictly decreasing objective function values are computed. Phase 2 is designed for giving a certificate of global optimality for the last SP computed in Phase 1. Two different Nonlinear Programming Formulations for LPLCC are proposed for each one of these phases, which are solved by efficient enumerative algorithms. New procedures for computing a lower bound for StQP are also proposed, which are easy to implement and give tight bounds in general. Computational experiments with a number of test problems from known sources indicate that the two-phase sequential algorithm is, in general, efficient in practice. Furthermore, the algorithm seems to be an efficient way to study the copositivity of a matrix by exploiting an StQP with this matrix.

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.

Collaborative Scheduling Optimization of Container Port Berths and Cranes under Low-Carbon Environment

Motivated by the need for a green and low-carbon economy, we explore the co-scheduling optimization of berths and cranes. Our aim is to balance the carbon tax and operating costs of ports under uncertain conditions, proposing an innovative nonlinear mixed-integer programming formulation. To address this optimization challenge, we have developed an enhanced version of the adaptive spiral flying dung beetle algorithm (ASFDBO). In order to evaluate the performance of the ASFDBO algorithm, we performed a benchmark function test and a convergence analysis with other recognized metaheuristics. In addition, we verified the practical applicability of the ASFDBO algorithm in different test scenarios. Through numerical experiments, we analyze the feasibility and effectiveness of the algorithm’s scheduling solutions and improvement strategies. Results indicate that our collaborative scheduling optimization, which considers both carbon and production costs, achieves feasible solutions and reduces carbon expenses. Finally, we investigate the impact of different carbon tax rates on the joint scheduling optimization of berths and quay cranes, and the results show that a reasonable carbon tax policy can effectively reduce the carbon emissions of ports.

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.

Multi-Objective Battery Coordination in Distribution Networks to Simultaneously Minimize CO2 Emissions and Energy Losses

The techno–environmental analysis of distributed energy resources in electrical distribution networks is a complex optimization task due to the non-convexities of its nonlinear programming formulation. This research employs convex optimization to address this issue while minimizing the expected carbon dioxide emissions and daily energy losses of a distribution grid via the optimal dispatch of battery energy storage units (BESUs) and renewable energy units (REUs). The exact non-convex model is approximated via semi-definite programming in the complex variable domain. The optimal Pareto front is constructed using a weighting-based optimization approach. Numerical results using an IEEE 69-bus grid confirm the effectiveness of our proposal when considering unitary and variable power factor operation for the BESUs and the REUs. All numerical simulations were carried out using MATLAB software (version 2022b), a convex disciplined tool (CVX), and the semi-definite programming solvers SEDEUMI and SDPT3.

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.

Fueling the seaport of the future: Investments in low‐carbon energy technologies for operational resilience in seaport multi‐energy systems

AbstractThe ability to withstand and recover from disruptions is essential for seaport energy systems, and in light of the growing push for decarbonization, incorporating clean energy sources has become increasingly imperative to ensure resilience. This paper proposes a resilience enhancement planning strategy for a seaport multi‐energy system that integrates various energy modalities and sources, including heating, cooling, hydrogen, solar, and wind power. The planning strategy aims to ensure the reliable operation of the system during contingency events, such as power outages, equipment failures, or extreme weather incidents. The proposed optimization model is designed as a mixed‐integer nonlinear programming formulation, in which McCormick inequalities and other linearization techniques are utilized to tackle the model nonlinearities. The model allocates fuel cell electric trucks (FCETs), renewable energy sources, hydrogen refueling stations, and remote control switches such that the system resilience is enhanced while incorporating natural‐gas‐powered combined cooling, heating, and power system to minimize the operation and unserved demand costs. The model considers various factors such as the availability of renewable energy sources, the demand for heating, cooling, electricity, and hydrogen, the operation of remote control switches to help system reconfiguration, the travel behaviour of FCETs, and the power output of FCETs via vehicle‐to‐grid interface. The numerical results demonstrate that the proposed strategy can significantly improve the resilience of the seaport multi‐energy system and reduce the risk of service disruptions during contingency scenarios.

A novel framework for the day-ahead market clearing process featuring the participation of distribution system operators and a hybrid pricing mechanism

The electricity market liberalization propelled by the increased operational flexibility observed in distribution systems urges the development of mathematical models for competitive electricity markets accounting for the participation of distribution systems operators. Most existing approaches model the interaction between the agents and the market as a bilevel programming problem, wherein the market is characterized at the lower level. Hence, the upper-level agent’s objective function is favored over minimizing the costs of supplying the system’s demand. In this paper, a new bilevel formulation is proposed. In the upper level, the market operator determines the optimal energy price that ensures the demand supply and maximizes the overall social welfare. In the lower level, the generation companies and distribution systems operators determine the amount of energy injected/extracted into/from the transmission system to maximize profits/minimize operational costs. A new pricing method is proposed in this paper as an alternative to the transmission system’s dual variables, which are inaccessible under the proposed framework. The resulting nonlinear mixed-integer bilevel programming formulation is transformed into an equivalent single-level mixed-integer linear program. Numerical results illustrate the proposed approach’s effectiveness in maintaining the minimum operational costs while distributing the payoff amongst the agents.

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.

Optimum Scheduling of a Multi-Machine Flexible Manufacturing System Considering Job and Tool Transfer Times without Tool Delay

In order to minimize makespan (Cmax) without causing tool delay with the fewest copies of each tool type, this study investigates the concurrent scheduling of automated guided vehicles (AGVs), machines (MCs), tool transporter (TT), and tools in a multi-machine flexible manufacturing system (FMS). The tools are housed in a central tool magazine (CTM), accessible to and utilized by several machines. AGVs and the tool transporter (TT) move jobs and tools between machines. Since it involves allocating tool copies and AGVs to job operations, sequencing job operations on machines, and related trip operations, such as AGVs’ and TT’s empty trip and loaded trip times, this simultaneous scheduling problem is highly complicated. This issue is resolved using the symbiotic organisms search algorithm (SOSA), based on the symbiotic interaction strategies that organisms adapt to survive in the ecosystem. This study proposes a mixed nonlinear integer programming formulation to address this problem. Verification is performed using an industrial problem from a manufacturing organization. The results show that employing two copies for two tool types out of 22 tool kinds and one copy for the remaining tool types results in no tool delay, which causes a reduction in the Cmax as well as cost. The industries that can benefit directly from this study are consumer electronics manufacturers, original equipment manufacturers, automobile manufacturers, and textile machine producers. The results demonstrate that the SOSA provides promising results compared to the flower pollination algorithm (FPA).

Enabling Efficient Spatio-Temporal GPU Sharing for Network Function Virtualization

By leveraging standard IT virtualization technology and Commercial-Off-The-Shelf (COTS) servers, Network Function Virtualization (NFV) decouples network functions from proprietary hardware devices for flexible service provisioning. But the potential of NFV is significantly limited by its performance inefficiency. With the unparalleled advantages of multi-core parallelism and high memory bandwidth, Graphics Processing Units (GPUs) are regarded as a promising way to accelerate Virtualized Network Functions (VNF). However, the special architecture of GPU brings new challenges to task scheduling and resource allocation. To this end, we propose a <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</b> PU <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</b> riented <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</b> patio- <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</b> emporal sharing framework for NFV called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Gost</i> , aiming for GPU based VNF performance promotion. The execution order and GPU resource allocation (i.e., the number of threads) are considered in task scheduling to minimize the end-to-end latency for VNF flows. First, we formulate the task scheduling problem into a nonlinear programming form, and then transform it into an equivalent Integer Linear Programming (ILP) form.The problem is proved as NP-hard. We customize the classical list scheduling algorithm and propose a List Scheduling based Spatio-Temporal GPU sharing strategy (LSSTG), whose achievable worst-case performance is also formally analyzed. We practically implement <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Gost</i> prototype, based on which extensive experiments verify the high performance efficiency of LSSTG compared to state-of-the-art in terms of latency and throughput.

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.