System of variational inequalities with interconnected obstacles

Barkhudaryan, Rafayel; Gomes, Diogo A.; Shahgholian, Henrik; Salehi, Mohsen

doi:10.1080/00036811.2020.1757076

Cited by 4 publications

(7 citation statements)

References 28 publications

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“…When the dimensionality increases even further, the PDE-methods become computationally heavy and other ways of attacking the resulting optimal switching problem may be preferable, e.g. Monte Carlo-methods as in Aïd et al (2014), Barkhudaryan et al (2020). In the current setting, the algorithm for obtaining our strategy is run in only a few minutes on a standard laptop computer and is thus more than sufficiently quick for the purpose of the current paper.…”

Section: Discussionmentioning

confidence: 99%

“…Optimal switching is a relatively new and fast growing field of mathematics combining optimization, SDEs and partial differential equations (PDEs) (Barkhudaryan et al 2020;Biswas et al 2010;Djehiche et al 2010;El-Asri and Fakhouri 2017;El-Asri and Hamadéne 2009;Hamadéne and Morlais 2013;Hu and Tang 2010;Kharroubi 2016;Lundström et al 2014aLundström et al , b, 2019Martyr 2016a, b;Perninge 2018, Lundström and Olofsson 2021. However, a literature survey shows that, although the mathematical theory is well developed, applications of optimal switching to real life problems is a far less explored area.…”

Section: Literature Survey and Our Contributionmentioning

confidence: 99%

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Management strategies for run-of-river hydropower plants: an optimal switching approach

2021

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The mathematical theory for optimal switching is by now relatively well developed, but the number of concrete applications of this theoretical framework remains few. In this paper, we bridge parts of this gap by applying optimal switching theory to a conceptual production planning problem related to hydropower. In particular, we study two examples of small run-of-river hydropower plants and provide an outline of how optimal switching can be used to create fully automatic production schemes for these. Along the way, we create a new model for random flow of water based on stochastic differential equations and fit this model to historical data. We benchmark the performance of our model using actual flow data from a small river in Sweden and find that our production scheme lies close to the optimal, within 2 and 5 %, respectively, in a long term investigation of the two plants considered.

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Section: Discussionmentioning

confidence: 99%

Section: Literature Survey and Our Contributionmentioning

confidence: 99%

Management strategies for run-of-river hydropower plants: an optimal switching approach

2021

View full text Add to dashboard Cite

show abstract

“…When the dimensionality increases even further, the PDE-methods become computationally heavy and other ways of attacking the resulting optimal switching problem may be preferable, e.g. Monte Carlo-methods as in [2,5]. In the current setting, the algorithm for obtaining our strategy is run in only a few minutes on a standard laptop computer and is thus more than sufficient for the purpose of the current paper.…”

Section: Discussionmentioning

confidence: 99%

“…Optimal switching is a relatively new and fast growing field of mathematics combining optimization, SDEs and partial differential equations (PDEs) [5,6,15,17,18,23,25,27,28,29,30,31,32,38,39,40]. However, a literature survey shows that, although the mathematical theory is well developed, applications of optimal switching to real life problems is a far less explored area.…”

Section: Literature Survey and Our Contributionmentioning

confidence: 99%

Management strategies for run-of-river hydropower plants -an optimal switching approach

Lundström¹,

Olofsson²,

Önskog³

2020

Preprint

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The mathematical theory for optimal switching is by now relatively well developed, but the number of concrete applications of this theoretical framework remains few. In this paper, we bridge parts of this gap by applying optimal switching theory to a set of production planning problems related to hydropower plants. In particular, we study two different cases involving small run-of-river hydropower plants and show how optimal switching can be used to create fully automatic production schemes in these cases, with non-zero cost of switching between different states of production. Along the way of deriving these schemes, we also create a model for the random flow of water based on stochastic differential equations and fit this model to historical data. This stochastic flow model, which should be of independent interest, mimics the long term seasonal behaviour of the flow while still allowing for stochastic fluctuations and can incorporate a given forecast to damp the impact of such fluctuations in near time. We benchmark the performance of our model using actual flow data from a small river in Sweden and find that our production scheme lies close to the optimal, within 2 % and 5 %, respectively, in a long term investigation of the two plants considered.

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“…A more general reflection model puts us in the "oblique derivative" problem in which n(x) in (BC) should be replaced by ν(x), a vector field satisfying ν(x), n(x) > 0 on ∂Ω. The Dirichlet setting, which is studied in Barkhudaryan-Gomes-Shahgholian-Salehi [2], can be given a similar interpretation but then in the sense that the game "ends" when the process hits the boundary ∂Ω.…”

Section: Introductionmentioning

confidence: 99%

Systems of fully nonlinear parabolic obstacle problems with Neumann boundary conditions

Lundström¹,

Olofsson²

2021

Preprint

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We prove the existence of a unique viscosity solution to certain systems of fully nonlinear parabolic partial differential equations with interconnected obstacles in the setting of Neumann boundary conditions. The method of proof builds on the classical viscosity solution technique adapted to the setting of interconnected obstacles and construction of explicit viscosity sub-and supersolutions as bounds for Perron's method. Our motivation stems from so called optimal switching problems on bounded domains.

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System of variational inequalities with interconnected obstacles

Cited by 4 publications

References 28 publications

Management strategies for run-of-river hydropower plants: an optimal switching approach

Management strategies for run-of-river hydropower plants: an optimal switching approach

Management strategies for run-of-river hydropower plants -an optimal switching approach

Systems of fully nonlinear parabolic obstacle problems with Neumann boundary conditions

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