Robust Cournot-Bertrand equilibria on power networks

Mather, Jonathan; Munsing, Eric

doi:10.23919/acc.2017.7963367

Cited by 6 publications

(5 citation statements)

References 38 publications

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“…Thus, we do not expect uniqueness of solutions for the setting considered in this paper. great importance for practice; see, e.g., [4,32,45]. In the following, we consider arobustification of the WMP (6).…”

Section: An Equivalent Variational Inequalitymentioning

confidence: 99%

“…However, the stochastic approach to LCPs is rather mature (see, e.g., [10][11][12]44]) compared to the field of robust LCPs or robust market equilibrium problems, which are still in their infancies. The only studies we are aware of are [7,40,43,[54][55][56] on robust LCPs, [39,45] on robust market equilibrium modeling, and the very recent and related paper [18] on robust bidding strategies in auctions. In this paper, we contribute to the second of the three mentioned fields and consider robustified market equilibrium models with transmission and generation investments.…”

Section: Introductionmentioning

confidence: 99%

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$$\Gamma $$-Robust electricity market equilibrium models with transmission and generation investments

2021

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We consider uncertain robust electricity market equilibrium problems including transmission and generation investments. Electricity market equilibrium modeling has a long tradition but is, in most of the cases, applied in a deterministic setting in which all data of the model are known. Whereas there exist some literature on stochastic equilibrium problems, the field of robust equilibrium models is still in its infancy. We contribute to this new field of research by considering $$\Gamma $$ Γ -robust electricity market equilibrium models on lossless DC networks with transmission and generation investments. We state the nominal market equilibrium problem as a mixed complementarity problem as well as its variational inequality and welfare optimization counterparts. For the latter, we then derive a $$\Gamma $$ Γ -robust formulation and show that it is indeed the counterpart of a market equilibrium problem with robustified player problems. Finally, we present two case studies to gain insights into the general effects of robustification on electricity market models. In particular, our case studies reveal that the transmission system operator tends to act more risk-neutral in the robust setting, whereas generating firms clearly behave more risk-averse.

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Section: An Equivalent Variational Inequalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

$$\Gamma $$-Robust electricity market equilibrium models with transmission and generation investments

2021

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“…The same concept has also been studied in Xie and Shanbhag (2014, 2016), where the authors consider strict robustness for uncertain LCPs for the case of different uncertainty sets like box or ellipsoidal uncertainties and focus on questions of tractability of the respective robust counterparts. The first and, to the best of our knowledge, only application of the results in Xie and Shanbhag (2014, 2016) is given in Mather and Munsing (2017), where the general theory is applied to Cournot–Bertrand equilibria on power networks. A related study of robustified market equilibrium problems can be found in the recent paper by Kramer et al.…”

Section: Introductionmentioning

confidence: 99%

Γ‐robust linear complementarity problems with ellipsoidal uncertainty sets

Krebs

Müller

Schmidt

2021

Int Trans Operational Res

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We study uncertain linear complementarity problems (LCPs), that is, problems in which the LCP vector q or the LCP matrix M may contain uncertain parameters. To this end, we use the concept of Γ‐robust optimization applied to the gap function formulation of the LCP. Thus, this work builds upon Krebs and Schmidt (2020). There, we studied Γ‐robustified LCPs for ℓ1‐ and box‐uncertainty sets, whereas we now focus on ellipsoidal uncertainty sets. For uncertainty in q or M, we derive conditions for the tractability of the robust counterparts. For these counterparts, we also give conditions for the existence and uniqueness of their solutions. Finally, a case study for the uncertain traffic equilibrium problem is considered, which illustrates the effects of the values of Γ on the feasibility and quality of the respective robustified solutions.

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“…In particular, these papers focus on tractability of the corresponding robust counterparts. The results are applied to the case of Cournot-Bertrand equilibria in power networks in [22]; see also [18] for a related study of Nash-Cournot and perfect competition equilibria in comparable settings.…”

Section: Introductionmentioning

confidence: 99%

Affinely Adjustable Robust Linear Complementarity Problems

Biefel,

Liers,

Rolfes

et al. 2020

Preprint

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Linear complementarity problems are a powerful tool for modeling many practically relevant situations such as market equilibria. They also connect many sub-areas of mathematics like game theory, optimization, and matrix theory. Despite their close relation to optimization, the protection of LCPs against uncertainties-especially in the sense of robust optimization-is still in its infancy. During the last years, robust LCPs have only been studied using the notions of strict and Γ-robustness. Unfortunately, both concepts lead to the problem that the existence of robust solutions cannot be guaranteed. In this paper, we consider affinely adjustable robust LCPs. In the latter, a part of the LCP solution is allowed to adjust via a function that is affine in the uncertainty. We show that this notion of robustness allows to establish strong characterizations of solutions for the cases of uncertain matrix and vector, separately, from which existence results can be derived. For an uncertain LCP vector, we additionally provide sufficient conditions on the LCP matrix for the uniqueness of a solution. Moreover, based on characterizations of the affinely adjustable robust solutions, we derive a mixed-integer programming formulation that allows to solve the corresponding robust counterpart. If the LCP matrix is uncertain, characterizations of solutions are developed for every nominal matrix, i.e., these characterizations are, in particular, independent of the definiteness of the nominal matrix. Robust solutions are also shown to be unique for positive definite LCP matrix but both uniqueness and mixedinteger programming formulations still remain open problems if the nominal LCP matrix is not positive definite.

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Robust Cournot-Bertrand equilibria on power networks

Cited by 6 publications

References 38 publications

$$\Gamma $$-Robust electricity market equilibrium models with transmission and generation investments

$$\Gamma $$-Robust electricity market equilibrium models with transmission and generation investments

Γ‐robust linear complementarity problems with ellipsoidal uncertainty sets

Affinely Adjustable Robust Linear Complementarity Problems

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