The Big-M method with the numerical infinite M

Cococcioni, Marco; Fiaschi, Lorenzo

doi:10.1007/s11590-020-01644-6

Cited by 61 publications

(29 citation statements)

References 19 publications

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“…Since P 0,agg,k is unknown, the big-M is used to limit the value of Ξ i,k i ∈ r j . It is worth mentioning that the big-M must be selected, large enough, but not too high to prevent the M value influencing the results of the optimization problem [37]. For instance, in this case, the values of all variables are in p.u., thus, an M value of 10 is adequate.…”

Section: Estimation Of Parameters Of the Emz-zip Load Modelmentioning

confidence: 99%

Modeling of Small Productive Processes for the Operation of a Microgrid

2021

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Small productive processes (SPPs) are promising drivers that promote the economic use of energy in microgrids (MGs). Both the complex nature of the SPPs and voltage variations make the operation of MGs challenging, since the quality of an energy management system’s (EMS) decisions depend on its characterization. The aim of this work is to propose a methodology for SPPs modeling, and to consider the influence of voltage on load consumption, which has general validity, and can be efficiently integrated into different MG EMS approaches. For this purpose, a novel extended multi-zone ZIP approach for the characterization of SPP loads and sensitivity to voltage changes is proposed. The associated framework herein presented was assessed using actual data collected from SPPs installed near the city of Arica, in northern Chile. The results showed that the proposed methodology was capable of representing the complex load behavior of the SPPs, properly considering the voltage influence. These results were compared with those obtained through common approaches found in the literature. The effectiveness of the proposed approach in representing SPP loads and their sensitivity to voltage changes was verified. The proposed scheme can be efficiently integrated into a wide range of EMS for MGs that include SPPs.

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Section: Estimation Of Parameters Of the Emz-zip Load Modelmentioning

confidence: 99%

Modeling of Small Productive Processes for the Operation of a Microgrid

2021

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“…Recent advances for multi-objective prioritized optimization, whether lexicographic [110][111][112] or Pareto-lexicographic [113][114][115], have developed programming tools and algorithms that have given new life to the study of lexicographic game theory [116,117]. As a consequence, the numerical study and solution of the practical problems mentioned above seem possible, possibly paving the way for a new approach to autonomous marine path planning.…”

Section: Opportunities and Way Aheadmentioning

confidence: 99%

Game Theory for Unmanned Vehicle Path Planning in the Marine Domain: State of the Art and New Possibilities

Cococcioni

Fiaschi

Lermusiaux

2021

JMSE

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Thanks to the advent of new technologies and higher real-time computational capabilities, the use of unmanned vehicles in the marine domain has received a significant boost in the last decade. Ocean and seabed sampling, missions in dangerous areas, and civilian security are only a few of the large number of applications which currently benefit from unmanned vehicles. One of the most actively studied topic is their full autonomy; i.e., the design of marine vehicles capable of pursuing a task while reacting to the changes of the environment without the intervention of humans, not even remotely. Environmental dynamicity may consist of variations of currents, the presence of unknown obstacles, and attacks from adversaries (e.g., pirates). To achieve autonomy in such highly dynamic uncertain conditions, many types of autonomous path planning problems need to be solved. There has thus been a commensurate number of approaches and methods to optimize this kind of path planning. This work focuses on game-theoretic approaches and provides a wide overview of the current state of the art, along with future directions.

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“…Recent literature has started to reconsider lexicographic problems, trying to solve the issues coming from both scalarization and preemptive approaches leveraging recent developments in numerical non-Archimedean computations [11,12,13]. Some results involve game theory [14,15,16], evolutionary optimization [17,18,19], linear programming tackled by Simplex-like methods [20,4,21,22]. Actually, the key idea is to scalarize the problem adopting infinite and infinitesimal weights.…”

Section: Introductionmentioning

confidence: 99%

A Non-Archimedean Interior Point Method for Solving Lexicographic Multi-Objective Quadratic Programming Problems

Fiaschi¹,

Cococcioni²

2021

Preprint

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This work presents a generalized implementation of the infeasible primal-dual Interior Point Method (IPM) achieved by the use of non-Archimedean values, i.e., infinite and infinitesimal numbers. The extended version, called here non-Archimedean IPM (NA-IPM), is able to solve a wider variety of linear and quadratic optimization problems than its standard counterpart. Among them, the lexicographic multi-objective one deserves particular attention, since NA-IPM overcomes the issues that standard techniques (such as scalarization or preemptive approach) have. In addition, infeasibility and unboundedness are no more corner cases to care of: by means of a mild embedding (addition of two variables and one constraint) NA-IPM implicitly and transparently manages their possible presence. To support the theoretical properties of NA-IPM, the manuscript also shows four linear and quadratic non-Archimedean programming test cases where the effectiveness of the algorithm is verified. This also stresses that NA-IPM is not just a mere symbolic or theoretical algorithm but actually a concrete numerical tool. Finally, the results of this study prove that non-Archimedean scientific computing is an extension of the standard one, suggesting it might have a huge number of practical applications in the near future in coping with both already existing and new real-world problems.

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The Big-M method with the numerical infinite M

Cited by 61 publications

References 19 publications

Modeling of Small Productive Processes for the Operation of a Microgrid

Modeling of Small Productive Processes for the Operation of a Microgrid

Game Theory for Unmanned Vehicle Path Planning in the Marine Domain: State of the Art and New Possibilities

A Non-Archimedean Interior Point Method for Solving Lexicographic Multi-Objective Quadratic Programming Problems

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