scholarly journals A solution framework for linear PDE-constrained mixed-integer problems

2021 ◽  
Author(s):  
Fabian Gnegel ◽  
Armin Fügenschuh ◽  
Michael Hagel ◽  
Sven Leyffer ◽  
Marcus Stiemer
Keyword(s):  
Computation Time ◽  
Mixed Integer ◽  
Continuous Control ◽  
State Variables ◽  
Mixed Integer Linear Program ◽  
Numerical Solution Method ◽  
Linear Pde ◽  
Finite Set ◽  
Continuous Constraints ◽  
Naive Approach

AbstractWe present a general numerical solution method for control problems with state variables defined by a linear PDE over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs to derive constraints for a mixed-integer linear program (MILP) leads to systems that are too large to be solved with state-of-the-art solvers for MILPs, especially if we desire an accurate approximation of the state variables. Our framework comprises two techniques to mitigate the rise of computation times with increasing discretization level: First, the linear system is solved for a basis of the control space in a preprocessing step. Second, certain constraints are just imposed on demand via the IBM ILOG CPLEX feature of a lazy constraint callback. These techniques are compared with an approach where the relations obtained by the discretization of the continuous constraints are directly included in the MILP. We demonstrate our approach on two examples: modeling of the spread of wildfire and the mitigation of water contamination. In both examples the computational results demonstrate that the solution time is significantly reduced by our methods. In particular, the dependence of the computation time on the size of the spatial discretization of the PDE is significantly reduced.

scholarly journals On mixed-integer optimal control with constrained total variation of the integer control

2020 ◽  
Author(s):  
Sebastian Sager ◽  
Clemens Zeile
Keyword(s):  
Optimal Control ◽  
Total Variation ◽  
Optimal Control Problems ◽  
Feasible Solution ◽  
Mixed Integer ◽  
Continuous Control ◽  
Mixed Integer Linear Program ◽  
Solution Quality ◽  
Frequent Switching ◽  
Control Trajectory

AbstractThe combinatorial integral approximation (CIA) decomposition suggests solving mixed-integer optimal control problems by solving one continuous nonlinear control problem and one mixed-integer linear program (MILP). Unrealistic frequent switching can be avoided by adding a constraint on the total variation to the MILP. Within this work, we present a fast heuristic way to solve this CIA problem and investigate in which situations optimality of the constructed feasible solution is guaranteed. In the second part of this article, we show tight bounds on the integrality gap between a relaxed continuous control trajectory and an integer feasible one in the case of two controls. Finally, we present numerical experiments to highlight the proposed algorithm’s advantages in terms of run time and solution quality.

scholarly journals Train-Scheduling Optimization Model for Railway Networks with Multiplatform Stations

2019 ◽  
Vol 12 (1) ◽  
pp. 257
Author(s):  
Gianmarco Garrisi ◽  
Cristina Cervelló-Pastor
Keyword(s):  
Mathematical Formulation ◽  
Computation Time ◽  
Mixed Integer ◽  
Initial Population ◽  
Mixed Integer Linear Program ◽  
Transit System ◽  
Train Scheduling ◽  
Railway Networks ◽  
Np Hard Problem ◽  
Problem Instances

This paper focuses on optimizing the schedule of trains on railway networks composed of busy complex stations. A mathematical formulation of this problem is provided as a Mixed Integer Linear Program (MILP). However, the creation of an optimal new timetable is an NP-hard problem; therefore, the MILP can be solved for easy cases, computation time being impractical for more complex examples. In these cases, a heuristic approach is provided that makes use of genetic algorithms to find a good solution jointly with heuristic techniques to generate an initial population. The algorithm was applied to a number of problem instances producing feasible, though not optimal, solutions in several seconds on a laptop, and compared to other proposals. Some improvements are suggested to obtain better results and further improve computation time. Rail transport is recognized as a sustainable and energy-efficient means of transport. Moreover, each freight train can take a large number of trucks off the roads, making them safer. Studies in this field can help to make railways more attractive to travelers by reducing operative cost, and increasing the number of services and their punctuality. To improve the transit system and service, it is necessary to build optimal train scheduling. There is an interest from the industry in automating the scheduling process. Fast computerized train scheduling, moreover, can be used to explore the effects of alternative draft timetables, operating policies, station layouts, and random delays or failures.

scholarly journals Learning-‘N-Flying: A Learning-Based, Decentralized Mission-Aware UAS Collision Avoidance Scheme

2021 ◽  
Vol 5 (4) ◽  
pp. 1-26
Author(s):  
Alëna Rodionova ◽  
Yash Vardhan Pant ◽  
Connor Kurtz ◽  
Kuk Jang ◽  
Houssam Abbas ◽  
...  
Keyword(s):  
Collision Avoidance ◽  
Computation Time ◽  
Unmanned Aircraft ◽  
Mixed Integer ◽  
Mixed Integer Linear Program ◽  
Worst Case ◽  
Online Computation ◽  
Multiple Case ◽  
Aircraft System ◽  
The City

Urban Air Mobility, the scenario where hundreds of manned and Unmanned Aircraft Systems (UASs) carry out a wide variety of missions (e.g., moving humans and goods within the city), is gaining acceptance as a transportation solution of the future. One of the key requirements for this to happen is safely managing the air traffic in these urban airspaces. Due to the expected density of the airspace, this requires fast autonomous solutions that can be deployed online. We propose Learning-‘N-Flying (LNF), a multi-UAS Collision Avoidance (CA) framework. It is decentralized, works on the fly, and allows autonomous Unmanned Aircraft System (UAS)s managed by different operators to safely carry out complex missions, represented using Signal Temporal Logic, in a shared airspace. We initially formulate the problem of predictive collision avoidance for two UASs as a mixed-integer linear program, and show that it is intractable to solve online. Instead, we first develop Learning-to-Fly (L2F) by combining (1) learning-based decision-making and (2) decentralized convex optimization-based control. LNF extends L2F to cases where there are more than two UASs on a collision path. Through extensive simulations, we show that our method can run online (computation time in the order of milliseconds) and under certain assumptions has failure rates of less than 1% in the worst case, improving to near 0% in more relaxed operations. We show the applicability of our scheme to a wide variety of settings through multiple case studies.

scholarly journals Optimal Coordinated Bidding of a Profit Maximizing, Risk-Averse EV Aggregator in Three-Settlement Markets Under Uncertainty

Energies ◽  
2019 ◽  
Vol 12 (9) ◽  
pp. 1755 ◽  
Author(s):  
Yelena Vardanyan ◽  
Henrik Madsen
Keyword(s):  
Real Time ◽  
Value At Risk ◽  
Risk Measure ◽  
Convex Combination ◽  
Computation Time ◽  
Mixed Integer ◽  
Conditional Value At Risk ◽  
Two Stage ◽  
Mixed Integer Linear Program ◽  
Market Prices

This paper develops a two-stage stochastic and dynamically updated multi-period mixed integer linear program (SD-MILP) for optimal coordinated bidding of an electric vehicle (EV) aggregator to maximize its profit from participating in competitive day-ahead, intra-day and real-time markets. The hourly conditional value at risk (T-CVaR) is applied to model the risk of trading in different markets. The objective of two-stage SD-MILP is modeled as a convex combination of the expected profit and the T-CVaR hourly risk measure. When day-ahead, intra-day and real-time market prices and fleet mobility are uncertain, the proposed two-stage SD-MILP model yields optimal EV charging/discharging plans for day-ahead, intra-day and real-time markets at per device level. The degradation costs of EV batteries are precisely modeled. To reflect the continuous clearing nature of the intra-day and real-time markets, rolling planning is applied, which allows re-forecasting and re-dispatching. The proposed two-stage SD-MILP is used to derive a bidding curve of an aggregator managing 1000 EVs. Furthermore, the model statistics and computation time are recorded while simulating the developed algorithm with 5000 EVs.

scholarly journals Investment Incentives in Competitive Electricity Markets

2018 ◽  
Vol 8 (10) ◽  
pp. 1978 ◽  
Author(s):  
Jaber Valinejad ◽  
Taghi Barforoshi ◽  
Mousa Marzband ◽  
Edris Pouresmaeil ◽  
Radu Godina ◽  
...  
Keyword(s):  
Electricity Markets ◽  
Electricity Market ◽  
Planning Horizon ◽  
Optimization Method ◽  
Design Criterion ◽  
Mixed Integer ◽  
Investment Incentives ◽  
Mixed Integer Linear Program ◽  
Interconnected Power System ◽  
The Impact

This paper presents the analysis of a novel framework of study and the impact of different market design criterion for the generation expansion planning (GEP) in competitive electricity market incentives, under variable uncertainties in a single year horizon. As investment incentives conventionally consist of firm contracts and capacity payments, in this study, the electricity generation investment problem is considered from a strategic generation company (GENCO) ′ s perspective, modelled as a bi-level optimization method. The first-level includes decision steps related to investment incentives to maximize the total profit in the planning horizon. The second-level includes optimization steps focusing on maximizing social welfare when the electricity market is regulated for the current horizon. In addition, variable uncertainties, on offering and investment, are modelled using set of different scenarios. The bi-level optimization problem is then converted to a single-level problem and then represented as a mixed integer linear program (MILP) after linearization. The efficiency of the proposed framework is assessed on the MAZANDARAN regional electric company (MREC) transmission network, integral to IRAN interconnected power system for both elastic and inelastic demands. Simulations show the significance of optimizing the firm contract and the capacity payment that encourages the generation investment for peak technology and improves long-term stability of electricity markets.

Optimizing Location and Relocation of Response Units in Guarding Critical Facilities

2005 ◽  
Vol 1923 (1) ◽  
pp. 127-136 ◽  
Author(s):  
Aamod Sathe ◽  
Elise Miller-Hooks
Keyword(s):  
Transportation Network ◽  
Computational Effort ◽  
Optimal Location ◽  
Mixed Integer ◽  
Mixed Integer Linear Program ◽  
Public And Private ◽  
Emergency Operations ◽  
Response Unit ◽  
Critical Facilities ◽  
Military Units

The ability to locate military units or equipment, police forces, and first responders optimally and to relocate idle units quickly in response to changing conditions is crucial to a country's ability to guard its critical facilities. Such facilities include vital components of the transportation infrastructure, government and monumental buildings, locations of large gatherings, emergency operations centers, and public and private utilities and communications facilities. In this paper, the problem of making optimal location and relocation decisions for a fixed fleet of response units in a transportation network, where travel conditions are uncertain, is addressed. A mixed integer linear program with multiple objectives (maximize secondary coverage and minimize cost) is presented. Because exact solution of such problems may require considerable computational effort, a metaheuristic based on the principles of genetic algorithms is proposed. The heuristic seeks the set of Pareto-optimal location and relocation decisions for each network state. All facilities of concern must be covered by at least one response unit. If the state of the network changes so that coverage is lost (e.g., travel times increase or a response unit is no longer available), one or more of the response units must be relocated. These relocation decisions are also addressed.

scholarly journals Locating Multiple Size and Multiple Type of Charging Station for Battery Electricity Vehicles

2018 ◽  
Vol 10 (9) ◽  
pp. 3267 ◽  
Author(s):  
Shaohua Cui ◽  
Hui Zhao ◽  
Huijie Wen ◽  
Cuiping Zhang
Keyword(s):  
Travel Time ◽  
Electric Vehicles ◽  
Sensitivity Analyses ◽  
Mixed Integer ◽  
Range Limit ◽  
Mixed Integer Linear Program ◽  
Charging Station ◽  
Different Types ◽  
Initial Charge ◽  
Charging Stations

As environmental and energy issues have attracted more and more attention from the public, research on electric vehicles has become extensive and in-depth. As driving range limit is one of the key factors restricting the development of electric vehicles, the energy supply of electric vehicles mainly relies on the building of charging stations, battery swapping stations, and wireless charging lanes. Actually, the latter two kinds of infrastructure are seldom employed due to their immature technology, relatively large construction costs, and difficulty in standardization. Currently, charging stations are widely used since, in the real world, there are different types of charging station with various levels which could be suitable for the needs of network users. In the past, the study of the location charging stations for battery electric vehicles did not take the different sizes and different types into consideration. In fact, it is of great significance to set charging stations with multiple sizes and multiple types to meet the needs of network users. In the paper, we define the model as a location problem in a capacitated network with an agent technique using multiple sizes and multiple types and formulate the model as a 0–1 mixed integer linear program (MILP) to minimize the total trip travel time of all agents. Finally, we demonstrate the model through numerical examples on two networks and make sensitivity analyses on total budget, initial quantity, and the anxious range of agents accordingly. The results show that as the initial charge increases or the budget increases, travel time for all agents can be reduced; a reduction in range anxiety can increase travel time for all agents.

scholarly journals Quantized-Feedback-Based Adaptive Event-Triggered Control of a Class of Uncertain Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1603
Author(s):  
Yun Ho Choi ◽  
Sung Jin Yoo
Keyword(s):  
Nonlinear Systems ◽  
Total Sample ◽  
Control Law ◽  
Continuous Control ◽  
State Variables ◽  
Event Time ◽  
Quantized Feedback ◽  
Event Triggering ◽  
Sample Data ◽  
Event Triggered

A quantized-feedback-based adaptive event-triggered tracking problem is investigated for strict-feedback nonlinear systems with unknown nonlinearities and external disturbances. All state variables are quantized through a uniform quantizer and the quantized states are only measurable for the control design. An approximation-based adaptive event-triggered control strategy using quantized states is presented. Compared with the existing recursive quantized feedback control results, the primary contributions of the proposed strategy are (1) to derive a quantized-states-based function approximation mechanism for compensating for unknown and unmatched nonlinearities and (2) to design a quantized-states-based event triggering law for the intermittent update of the control signal. A Lyapunov-based stability analysis is provided to conclude that closed-loop signals are uniformly ultimately bounded and there exists a minimum inter-event time for excluding Zeno behavior. In simulation results, it is shown that the proposed quantized-feedback-based event-triggered control law can be implemented with less than 10% of the total sample data of the existing quantized-feedback continuous control law.

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