On the multiplicity of solutions in generation capacity investment models with incomplete markets: a risk-averse stochastic equilibrium approach

Abada, Ibrahim; d'Aertrycke, Gauthier de Maere; Smeers, Yves

doi:10.1007/s10107-017-1185-9

Cited by 30 publications

(23 citation statements)

References 45 publications

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“…This can be viewed as a game where N firms compete in Cournot in a capacity market defined by an inverse demand function P (x) and subsequently make production decisions subject to a demand's capacity constraints while faced by random prices and costs, where P (x), the market price, is a decreasing function of total production, and C i (x i ) is the cost function of firm i. Capacity markets are utilized to price generation capacity in power markets [1,18]. Note that…”

Section: Comparisons Of Empirical and Theoretical Resultsmentioning

confidence: 99%

“…In the following theorem, we drive the upper bound for (1) i (η) and (2) i (η) defined in Theorem 3 and Corollary 3, respectively.…”

Section: Complexity Boundmentioning

confidence: 99%

“…Theorem 4 Set η a ∞ . Then we have the following bounds on the expression (1) i (η) and (2) i (η). a) Define n 0 = B 2 B 1 and n = B 1 (1 + n 0 ).…”

Section: Complexity Boundmentioning

confidence: 99%

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On Synchronous, Asynchronous, and Randomized Best-Response Schemes for Stochastic Nash Games

et al. 2020

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In this paper, we consider a stochastic Nash game in which each player minimizes a parameterized expectationvalued convex objective function. In deterministic regimes, proximal best-response (BR) schemes have been shown to be convergent under a suitable spectral property associated with the proximal BR map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization problems at each iteration. Instead, we propose an inexact generalization of this scheme in which an inexact solution to the BR problem is computed in an expected-value sense via a stochastic approximation (SA) scheme. On the basis of this framework, we present three inexact BR schemes: (i) First, we propose a synchronous inexact BR scheme where all players simultaneously update their strategies; (ii) Second, we extend this to a randomized setting where a subset of players is randomly chosen to update their strategies while the other players keep their strategies invariant; (iii) Third, we propose an asynchronous scheme, where each player chooses its update frequency while using outdated rival-specific data in updating its strategy. Under a suitable contractive property on the proximal BR map, we proceed to derive a.s. convergence of the iterates to the Nash equilibrium (NE) for (i) and (ii) and mean-convergence for (i)-(iii). In addition, we show that for (i)-(iii), the generated iterates converge to the unique equilibrium in mean at a linear rate with a prescribed constant rather than a sub-linear rate. Finally, we establish the overall iteration complexity of the scheme in terms of projected stochastic gradient (SG) steps for computing an −Nash equilibrium and note that in all settings, the iteration complexity is O(1/ 2(1+c)+δ ) where c = 0 in the context of (i) and represents the positive cost of randomization in (ii) and asynchronicity and delay in (iii). Notably, in the synchronous regime, we achieve a near-optimal rate from the standpoint of solving stochastic convex optimization problems by SA schemes. The schemes are further extended to settings where players solve two-stage stochastic Nash games with linear and quadratic recourse. Finally, preliminary numerics developed on a multi-portfolio investment problem and a two-stage capacity expansion game support the rate and complexity statements.

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Section: Comparisons Of Empirical and Theoretical Resultsmentioning

confidence: 99%

“…In the following theorem, we drive the upper bound for (1) i (η) and (2) i (η) defined in Theorem 3 and Corollary 3, respectively.…”

Section: Complexity Boundmentioning

confidence: 99%

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On Synchronous, Asynchronous, and Randomized Best-Response Schemes for Stochastic Nash Games

et al. 2020

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“…Notice that in order to compute the expected profit, all the market participants share the same probability functions. Equation (13) defines the profit of market participant m in scenario sc as the sum of the market incomes minus the generation costs (only thermal). Market incomes are computed by adding all the thermal, hydro, and RES generation that belong to such market agents, remunerated at the market price π t,i for all the nodes that comprise the scenario and taking into account the probability and the duration of each node.…”

Section: Mathematical Formulation Of the Market Equilibrium Modelmentioning

confidence: 99%

“…The interpretation of (41) is that in the risk-neutral equilibrium, the Lagrange multipliers of the Equation (13), formulated for all the market participants for every scenario, must coincide with the probability of that scenario. Then, Equations (32) and (33) become…”

Section: Impact Of Risk Aversion Levelmentioning

confidence: 99%

Impact of Risk Aversion on the Operation of Hydroelectric Reservoirs in the Presence of Renewable Energy Sources

et al. 2018

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Abstract:The increasing share of renewable energy sources, such as wind and solar generation, has a direct impact on the planning and operation of power systems. In addition, the consideration of risk criteria within the decision support tools used by market participants (generation companies, energy services companies, and arbitrageurs) is becoming a common activity given the increasing level of uncertainties faced by them. As a consequence, the behavior of market participants is affected by their level of risk aversion, and the application of equilibrium-based models is a common technique used in order to simulate their behavior. This paper presents a multi-stage market equilibrium model of risk-averse agents in order to analyze up to what extent the operation of hydro reservoirs can be affected by the risk-averse profile of market participants in a context of renewable energy source penetration and fuel price volatility.

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Preface

2017

Math. Program.

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On the multiplicity of solutions in generation capacity investment models with incomplete markets: a risk-averse stochastic equilibrium approach

Cited by 30 publications

References 45 publications

On Synchronous, Asynchronous, and Randomized Best-Response Schemes for Stochastic Nash Games

On Synchronous, Asynchronous, and Randomized Best-Response Schemes for Stochastic Nash Games

Impact of Risk Aversion on the Operation of Hydroelectric Reservoirs in the Presence of Renewable Energy Sources

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