AMPL Modeling Language Research Articles

Optimizing Vehicle Repairs Scheduling Using Mixed Integer Linear Programming: A Case Study in the Portuguese Automobile Sector

This study investigates the scheduling of mechanical repairs performed at a Portuguese firm in the automobile sector. The aim is to reduce the amount of time that vehicles spend inactive between interventions by developing a mathematical model that takes into account the available resources and mechanics, the necessary interventions, and the time required for each repair. To accomplish this, a mixed-integer linear programming (MILP) model was employed, incorporating various variables to schedule interventions, allocate resources, and determine start times for each vehicle. The problem was formulated using the AMPL modeling language, and real-world instances of the problem, derived from data provided by the company, were solved using the Gurobi solver. Results show that the developed model significantly improves the scheduling of the vehicles’ repairs at the firm, leading to a reduction of 67% on average in the downtime of the vehicles and allowing an automatic correct schedule of the mechanical interventions. Moreover, the comparison of the scheduling obtained from the developed model and the firm’s procedure shows that interventions on vehicles arriving at the repair shop are mostly repaired on the day of entry, allowing for quicker delivery to the customer.

A hybrid approach for optimization of electric power distributed networks with photovoltaic sources

• Reconfiguration is one of the most effective ways to reduce ohmic losses. • The hybrid methodology explores each set of problem variables with different techniques. • Particle swarm optimization considers distribution network reconfiguration and optimal power flow adjusts the value of PV inverters and maintenance network operating limits. • Photovoltaic systems can reduce the apparent power in the grid and reduce electrical losses. • The proposed solutions can avoid excess reactive power charges for PV purposes. This paper presents a hybrid approach, which associates a mathematical programming technique with a metaheuristic, to optimize the electricity distribution network with penetration of photovoltaic (PV) generation. This optimization considers the distribution network reconfiguration (DNR) and the adjustment of the power factor of photovoltaic (PV) inverters about the reactive power, controlling the voltage profile and minimizing network losses. The methodology uses particle swarm optimization (PSO) for DNR and an optimal power flow (OPF) to adjust the value of the PV inverters and maintenance of network operating limits, this being a great difference from other methodologies. The objective of the hybrid approach is to obtain the best result for the problem, taking advantage of the best characteristics of both techniques. The hybrid approach was compared with the PSO, highlighting its superiority in terms of the solutions found. The tests were performed using the AMPL Modeling Language in the 33-node system with the addition of PV generation. The approach considers the variations in the powers generated by the PV systems and the variations in the loads over 24 h. The results indicate that the combination of PV inverter control and DNR can reduce the active power losses by more than 37% in 24 h.

A Clustering Approach for the Optimal Siting of Recharging Stations in the Electric Vehicle Routing Problem with Time Windows

Transportation has been incorporating electric vehicles (EVs) progressively. EVs do not produce air or noise pollution, and they have high energy efficiency and low maintenance costs. In this context, the development of efficient techniques to overcome the vehicle routing problem becomes crucial with the proliferation of EVs. The vehicle routing problem concerns the freight capacity and battery autonomy limitations in different delivery-service scenarios, and the challenge of best locating recharging stations. This work proposes a mixed-integer linear programming model to solve the electric location routing problem with time windows (E-LRPTW) considering the state of charge, freight and battery capacities, and customer time windows in the decision model. A clustering strategy based on the k-means algorithm is proposed to divide the set of vertices (EVs) into small areas and define potential sites for recharging stations, while reducing the number of binary variables. The proposed model for E-LRPTW was implemented in Python and solved using mathematical modeling language AMPL together with CPLEX. Performed tests on instances with 5 and 10 clients showed a large reduction in the time required to find the solution (by about 60 times in one instance). It is concluded that the strategy of dividing customers by sectors has the potential to be applied and generate solutions for larger geographical areas and numbers of recharging stations, and determine recharging station locations as part of planning decisions in more realistic scenarios.

A Specialized Long-Term Distribution System Expansion Planning Method With the Integration of Distributed Energy Resources

The electrical distribution system (EDS) has undergone major changes in the last decade due to the increasing integration of distributed generation (DG), particularly renewable energy DG. Since renewable energy resources have uncertain generation, energy storage systems (ESSs) in the EDS can reduce the impact of those uncertainties. Besides, electric vehicles (EVs) have been increasing in recent years leveraged by environmental concerns, bringing new challenges to the operation and planning of the EDS. In this context, new approaches for the distribution system expansion planning (DSEP) problem should consider the distributed energy resources (DG units, ESSs, and EVs) and address environmental impacts. This paper proposes a mixed-integer linear programming model for the DSEP problem considering DG units, ESSs, and EV charging stations, thus incorporating the environmental impact and uncertainties associated with demand (conventional and EVs) and renewable generation. In contrast to other approaches, the proposed model includes the simultaneous optimization of investments in substations, circuits, and distributed energy resources, including environmental aspects (CO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> emissions). The optimization method was developed in the modeling language AMPL and solved via CPLEX. Tests carried out with a 24-node system illustrate its effectiveness as a valuable tool that can assist EDS planners in the integration of distributed energy resources.

Optimal Service Restoration in Active Distribution Networks Considering Microgrid Formation and Voltage Control Devices

This article presents a new mixed-integer linear programming model for the optimal restoration of active distribution networks during permanent fault events considering not only network reconfiguration but also the islanded operation of distributed generation, thereby giving rise to the formation of microgrids. To that end, the proposed approach accounts for both the black-start capacity of generators and the connection of multiple generators to the same microgrid. Moreover, voltage control devices such as substations’ on-load tap changers, voltage regulators, and capacitor banks are also considered in the formulation. The model is driven by the minimization of the out-of-service load after single or multiple faults in the network using the minimum number of switching operations and modifications in the statuses of the voltage control devices. The proposed model is implemented in the mathematical modeling language AMPL and solved with the CPLEX solver. Tests are conducted using a 53-node distribution system considering different fault locations. Numerical results show the relevant effect of microgrid formation and voltage control in the restoration process. Simulations also prove the effective performance of the proposed approach in terms of computation time.

A Hybrid Approach for Optimization of Distribution Networks with Photovoltaic Sources

This paper presents a hybrid approach, which associates a mathematical programming technique with a metaheuristic, to optimize the electricity distribution network with penetration of photovoltaic (PV) generation. This optimization considers the distribution network reconfiguration (DNR) and the adjustment of the power factor of photovoltaic (PV) inverters in relation to the reactive power, controlling the voltage profile and minimizing network losses. The methodology uses particle swarm optimization (PSO) for DNR and an optimal power flow (OPF) to adjust the value of the PV inverters and maintenance of network operating limits. The tests were performed using the AMPL Modeling Language in the 33-node system with the addition of PV generation. The approach considers the variations in the powers generated by the PV systems and the variations in the loads over a 24 h period. The results indicate that the combination of PV inverter control and DNR can reduce the active power losses by more than 37% in a 24-hour period.

State Estimation in Electric Power Systems Using an Approach Based on a Weighted Least Squares Non-Linear Programming Modeling

In this work, the state estimation problem of electric power systems is represented through a mathematical programming approach. Initially, a non-linear mathematical model based on the classical method of weighted least squares is proposed to solve the state estimation problem for comparative purposes. Due to the inherent limitations that this classical model presents when dealing with errors in the set of measurements, a new mathematical model is proposed that can be used within an iterative procedure to reduce the impact of measurement errors on the estimated results. Several tests on a didactic 5-bus power system and IEEE benchmark power test systems showed the effectiveness of the proposed approach which achieved better results than the proposed classical state estimation model. The non-linear programming models proposed in this paper are implemented in the mathematical modeling language AMPL. Additionally, to validate the results of the proposed methodologies, the power system operation points are compared with the results obtained using the Matpower simulation package. The results allowed concluding that the proposed mathematical models can be successfully applied to perform state estimation studies in power systems.

Problems on Shortest k-Node Cycles and Paths

The paper is devoted to the construction of mathematical models for problems on the shortest cycles and paths, that pass through a given number of nodes of a directed graph. Such cycles and paths are called k-node, where 1&lt;k &lt;n, n is the number of nodes in the graph. Section 1 formulates two problems for finding the shortest k-node cycle – a mixed Boolean and linear programming problem and a discrete programming problem. Both problems include constraints from the classical assignment problem, describing a one-time entry into a node and a one-time exit from a node for those nodes through which the cycle passes. The cycle connectivity in the first problem is ensured by modeling the flow problem, and in the second problem, it is ensured by using the A. Tucker constraints for the travelling salesman problem. Section 2 establishes a connection between the formulations of both problems from Section 1 and the travelling salesman problem and investigates the efficiency of their solution using modern versions of gurobi and cplex programs and the AMPL modeling language. Section 3 contains the formulation of the shortest k-node path problem, which is represented by a mixed Boolean and linear programming problem. With its help, the optimal routes were found for visiting the wine-making points of the Malopolskie Wine Route in the direction Lviv-Wroclaw-Lviv (Section 4). Here a map for the 20 most visited wine-making points of the Malopolskie Wine Route and a table of the distances between them and the distances from them to Lviv and Wroclaw, calculated using the Google Maps web service, are presented. The developed mathematical models of the problems of finding the shortest k-node paths and cycles and the developed software in the AMPL modeling language can be used for the design and arrangement of technical objects, optimization of the transportation of products, analysis and forecasting of economic processes, determination of optimal routes when planning passenger and freight traffic, optimal organization of the process of managing a set of transactions and queries during their implementation in network databases and other classes of applied optimization problems. Keywords: digraph, shortest path, Boolean variable, linear programming, Hamiltonian cycle, Hamiltonian path, travelling salesman problem, AMPL, gurobi, cplex.

A deterministic semi-infinite global optimization method to design slotless permanent magnet machines

PurposeIn this work, a method to design a slotless permanent magnet machine (SPMM) based on the joint use of an analytical model and deterministic global optimization algorithms is addressed. The purpose of this study is to propose to include torque ripples as an extra constraint in the optimization phase involving de facto the study of a semi-infinite optimization problem.Design/methodology/approachBased on the use of a well-known analytical model describing the electromagnetic behavior of an SPMM, this analytical model has been supplemented by the calculus of the dynamic torque and its ripples to carry out a more accurate optimized sizing method of such an electromechanical converter. As a consequence, the calculated torque depends on a continuous variable, namely, the rotor angular position, resulting in the definition of a semi-infinite optimization problem. The way to solve this kind of semi-infinite problem by discretizing the rotor angular position by using a deterministic global optimization solver, that is to say COUENNE, via the AMPL modeling language is addressed.FindingsIn this study, the proposed approach is validated on some numerical tests based on the minimization of the magnet volume. Efficient global optimal solutions with torque ripples about 5% (instead of 30%) can be so obtained.Research limitations/implicationsThe analytical model does not use results from the solution of two-dimensional field equations. A strong assumption is put forward to approximate the distribution of the magnetic flux density in the air gap of the SPMM.Originality/valueThe problem to design an SPMM can be efficiently formulated as a semi-infinite global optimization problem. This kind of optimization problems are hard to solve because they involve an infinity of constraints (coming from a constraint on the torque ripple). The authors show in this paper that by using analytical models, a discretization method and a deterministic global optimization code COUENNE, this problem is efficiently tackled. Some numerical results show that the deterministic global solution of the design can be reached even if the step of discretization is small.

Using interior point solvers for optimizing progressive lens models with spherical coordinates

Designing progressive lenses is a complex problem that has been previously solved by formulating an optimization model based on Cartesian coordinates. This work presents a new progressive lens model using spherical coordinates, and interior point solvers are used to solve this new optimization model. Although this results in a highly nonlinear, nonconvex, continuous optimization problem, the new spherical coordinates model exhibits better convexity properties compared to previous ones based on Cartesian coordinates. The real-world instances considered result in nonlinear optimization problems of about 900 variables and 15,000 constraints. Each constraint corresponds to a point on the grid that defines the lens surface. The number of variables depends on the precision of the B-spline basis used for representing the surface; and the number of constraints depends on the shape and quality of the design. We present our results for progressive lenses, which were obtained using the AMPL modeling language and the nonlinear interior point solvers IPOPT, LOQO and KNITRO. The computational results are reported, as well as some examples of real-world progressive lenses that were calculated using this new model. In terms of quality, the progressive lenses obtained by our model are competitive with those of previous models used for commercial eyeglasses.

An Optimization Model for Exercise Scheduling

The paper gives an optimization model for a special type of exercise session, circuit training. Circuit training involves a series of exercises performed in rotation with minimal rest. The goal of our model is to minimize the total circuit time while accomplishing a number of training goals. Our primary model is a linear integer program; additional constraints are added for muscle group and intensity requirements. The model is implemented and tested on algebraic modeling language AMPL. Our computational results show that the model can return an exercise schedule for a typical real-life data set within a few seconds.

Vehicle location models for Emergency Medical Services. An application for a Colombian company

Ambulance location, in a certain area of coverage, is a key element in reducing the waiting time of a potential patient and the initiation a healthcare service in Emergency Medical Services (EMS). The Operations Research area, developing and solving adequate mathematical models, helps to make good decisions in EMS. We propose two variants of a Mixed Integer Linear Programming model for locating heterogeneous fleet vehicles to support different types of services in emergency medical care, taking into account the operational requirements of a health care service company in Colombia. The proposed models do not exactly match any of those found in the literature. The models are solved with Gurobi solver and the modeling language AMPL and they are successfully validated with historical data of the company under study and some estimates based on external sources. These obtained results are compared using an adaptation of the concept of Preparedness taken from the literature for ambulance positioning, as regard to measure the readiness of the system to the expected demand. The new results show that the relevance of each model depends of the prioritization of services and/or areas that the company considers.

An MILP model for the static transmission expansion planning problem including HVAC/HVDC links, security constraints and power losses with a reduced search space

Abstract This paper presents a mixed-integer linear programming (MILP) model to solve the transmission network expansion planning (TNEP) problem, considering HVDC and HVAC proposals with security constraints and the system operation cost. The use of an MILP model guarantees convergence to optimality by using existing classical optimization methods. The operation cost is associated with the cost of transmission losses in the system. In this model, the HVDC lines can be either monopolar or bipolar type. Due to the size and complexity of the problem, a new method to reduce the search space solution is proposed, taking into account line losses, HVDC links, and contingencies. The set of HVDC lines in the proposal may use either line-commutated converters or voltage source converters. The model was implemented using the AMPL modeling language and solved using the CPLEX commercial solver. The Southern Brazilian electrical system was used to demonstrate the effectiveness of the proposed MILP model; tests were performed for cases with and without HVDC links, and with and without the system operation cost. The results show high quality solutions, and high performance when the search space reduction methodology (SSRM) is used. The computational time with the SSRM (the study case with HVDC and operation cost) was reduced by 85.9% with respect to the case that uses the complete search space, and also, the optimal solution was found. The importance of including the operation cost as an objective to be minimized is shown. The cost of losses is reduced by 55.7% when comparing the cases with and without the cost of operation included in the objective function and with HVDC lines as proposals.

An Integer Programming Model for the KenKen Problem

In this paper we consider modeling techniques for the mathematical puzzle KenKen. It is an interesting puzzle from modeling point of view since it has different kind of mathematical restrictions that are not trivial to express as linear constraints. We give an integer program for solving KenKen and and its implementation on modeling language AMPL. Our integer program uses an innovative way for converting product restrictions into linear constraints. It can be also used for teaching various integer programming techniques in an Operations Research course.

Interior-point solver for convex separable block-angular problems

Constraints matrices with block-angular structures are pervasive in optimization. Interior-point methods have shown to be competitive for these structured problems by exploiting the linear algebra. One of these approaches solves the normal equations using sparse Cholesky factorizations for the block constraints, and a preconditioned conjugate gradient (PCG) for the linking constraints. The preconditioner is based on a power series expansion which approximates the inverse of the matrix of the linking constraints system. In this work, we present an efficient solver based on this algorithm. Some of its features are as follows: it solves linearly constrained convex separable problems (linear, quadratic or nonlinear); both Newton and second-order predictor–corrector directions can be used, either with the Cholesky+PCG scheme or with a Cholesky factorization of normal equations; the preconditioner may include any number of terms of the power series; for any number of these terms, it estimates the spectral radius of the matrix in the power series (which is instrumental for the quality of the preconditioner). The solver has been hooked to the structure-conveying modelling language (SML) based on the popular AMPL modeling language. Computational results are reported for some large and/or difficult instances in the literature: (1) multicommodity flow problems; (2) minimum congestion problems; (3) statistical data protection problems using and distances (which are linear and quadratic problems, respectively), and the pseudo-Huber function, a nonlinear approximation to which improves the preconditioner. In the largest instances, of up to 25 millions of variables and 300,000 constraints, this approach is from 2 to 3 orders of magnitude faster than state-of-the-art linear and quadratic optimization solvers.

Transmission network expansion planning considering repowering and reconfiguration

Abstract Transmission expansion planning (TEP) is a classic problem in electric power systems. In current optimization models used to approach the TEP problem, new transmission lines and two-winding transformers are commonly used as the only candidate solutions. However, in practice, planners have resorted to non-conventional solutions such as network reconfiguration and/or repowering of existing network assets (lines or transformers). These types of non-conventional solutions are currently not included in the classic mathematical models of the TEP problem. This paper presents the modeling of necessary equations, using linear expressions, in order to include non-conventional candidate solutions in the disjunctive linear model of the TEP problem. The resulting model is a mixed integer linear programming problem, which guarantees convergence to the optimal solution by means of available classical optimization tools. The proposed model is implemented in the AMPL modeling language and is solved using CPLEX optimizer. The Garver test system, IEEE 24-busbar system, and a Colombian system are used to demonstrate that the utilization of non-conventional candidate solutions can reduce investment costs of the TEP problem.

A detailed MILP optimization model for combined cooling, heat and power system operation planning

A detailed optimization model is presented for planning the short-term operation of combined cooling, heat and power (CCHP) energy systems. The purpose is, given the design of a cogeneration system, to determine an operating schedule that minimizes the total operating and maintenance costs minus the revenue due to the electricity sold to the grid, while taking into account time-varying loads, tariffs and ambient conditions. The model considers the simultaneous use of different prime movers (generating electricity and heat), boilers, compression heat pumps and chillers, and absorption chillers to satisfy given electricity, heat and cooling demands. Heat and cooling load can be stored in storage tanks. Units can have one or two operative variables, highly nonlinear performance curves describing their off-design behavior, and limitations or penalizations affecting their start-up/shut-down operations. To exploit the effectiveness of state-of-the-art Mixed Integer Linear Program (MILP) solvers, the resulting Mixed Integer Nonlinear Programming (MINLP) model is converted into a MILP by appropriate piecewise linear approximation of the nonlinear performance curves. The model, written in the AMPL modeling language, has been tested on several plant test cases. The computational results are discussed in terms of the quality of the solutions, the linearization accuracy and the computational time.

Transmission network expansion planning for the Colombian electrical system: Connecting the Ituango hydroelectric power plant

Abstract The hydroelectric power plant HidroItuango represents a major expansion for the Colombian electrical system (with a total capacity of 2400 MW). This paper analyzes the possible interconnections and investments involved in connecting HidroItuango, in order to strengthen the Colombian national transmission system. A Mixed Binary Linear Programming (MBLP) model was used to solve the Multistage Transmission Network Expansion Planning (MTEP) problem of the Colombian electrical system, taking the N-1 safety criterion into account. The N-1 safety criterion indicates that the transmission system must be expanded so that the system will continue to operate properly if an outage in a system element (within a pre-defined set of contingencies) occurs. The use of a MBLP model guaranteed the convergence with existing classical optimization methods and the optimal solution for the MTEP using commercial solvers. Multiple scenarios for generation and demand were used to consider uncertainties within these parameters. The model was implemented using the algebraic modeling language AMPL and solved using the commercial solver CPLEX. The proposed model was then applied to the Colombian electrical system using the planning horizon of 2018–2025.

A Comparison of Optimal Control Methods for Minimum Fuel Cruise at Constant Altitude and Course with Fixed Arrival Time

Abstract The current lack of efficiency in the use of airspace, enforced by the economic crisis an the increasing competitiveness among air transport companies are drivers for a renewed interest for the application of optimization techniques to find answers to overcame current inefficiencies. The main objective of the present paper is to assess and compare different optimal aircraft trajectories techniques applied to the minimum fuel cruise problem at constant altitude and course with fixed arrival time, International Standard Atmosphere and without wind. Four trajectory optimization methods have been used: Hermite-Simpson, 5th degree Gauss-Lobatto and Radau pseudospectral collocation methods and the singular arc solution. Hermite-Simpson and 5th degree methods have been programmed in Ampl modeling language with an IPOPT solver and Radau pseudospectral method using gpops matlab tool with SNOPT solver. 5th degree Gauss-Lobatto collocation method gives the less fuel consumption solution followed by Radau pseudospectral, Hermite-Simpson and singular arc. In considering the program execution time, Hermite-Simpson collocation method is the fastest method followed by 5th degree and Radau pseudospectral. Also, taking into account the time for developing the program code the Radau pseudospectral is the most user friendly. Moreover, it has been observed that increasing the sample points in the Hermite- Simpson and 5th degree, the solution converge to the minimum fuel consumption solution. On the other hand, gpops does not show much sensitivity to the number of sample points.

Operation Planning of Electrical Distribution Systems Using a Mixed Integer Linear Model

This article presents a mixed integer linear programming model for the problem of operation planning of an electrical distribution system. The model defines the most appropriate adjustment of a set of control variables to minimize active energy losses of the system. The generation of active and reactive powers of distributed generators, the number of modules in operation of capacitor banks, and the number of tap steps for the voltage regulators are considered as the control variables. In the proposed formulation, the steady-state operation of the electrical distribution network is modeled mathematically, using linear expressions. The use of a mixed-integer linear model guarantees convergence to optimality using existing optimization software. The model was implemented in the mathematical modeling language AMPL and solved using the commercial solver CPLEX. A test system of 42 nodes was used to show the accuracy of the mathematical model as well as the efficiency of the proposed solution technique.