American Institute of Mathematical Sciences

Advanced
July  2008, 4(3): 425-452. doi: 10.3934/jimo.2008.4.425

Dynamic oligopolistic competition on an electric power network with ramping costs and joint sales constraints

Reetabrata Mookherjee 1, , Benjamin F. Hobbs 2, , Terry L. Friesz 3, and  Matthew A. Rigdon 4,

1. 

Senior Scientist, Zilliant Inc, 3815 S Capital of Texas Highway, Suite 300, Austin, TX 78704, United States
2. 

Professor, Depts of Geography & Environmental Engineering & Applied Math. & Stat., The Johns Hopkins University, Baltimore, MD 21218, United States
3. 

Marcus Chaired Professor of Industrial Engineering, The Pennsylvania State University, University Park, PA 16802, United States
4. 

Doctoral Candidate, Dept of Industrial & Manufacturing Engineering, The Pennsylvania State University, University Park, PA 16802, United States

Received  July 2007 Revised  June 2008 Published  July 2008

Most previous Cournot-Nash models of competition among electricity generators have assumed a static perspective, resulting in finite dimensional variational and quasi-variational inequality formulations. However, these models' system costs and constraints fail to capture the dynamic nature of power networks. In this paper we propose a more general and complete model of Cournot-Nash competition on power networks that accounts for these features by including ($i$) explicit intra-day dynamics that describe the market's evolution from one Generalized Cournot-Nash Equilibrium to another for a 24 hour planning horizon, ($ii$) ramping constraints and costs for changing the power output of generators, and ($ iii $) joint constraints that include variables from other generating companies within the profit maximization problems for individual generators. These joint constraints yield a generalized Nash equilibrium problem which can be represented as a differential quasi-variational inequality (DQVI); such generalized Nash equlibrium problems can have multiple solutions. The resulting formulation poses computational challenges that can cause traditional algorithms for DVIs to fail. A restricted formulation is proposed that can be solved by an implicit fixed point algorithm. A numerical example is provided.
Keywords: differential quasi-variational inequalities, ramping costs., generalized Nash equilibrium.
Mathematics Subject Classification: Primary: 49N70, 58E35; Secondary: 49J1.
Citation: Reetabrata Mookherjee, Benjamin F. Hobbs, Terry L. Friesz, Matthew A. Rigdon. Dynamic oligopolistic competition on an electric power network with ramping costs and joint sales constraints. Journal of Industrial and Management Optimization, 2008, 4 (3) : 425-452. doi: 10.3934/jimo.2008.4.425
[1]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457
[2]

Yusuke Murase, Risei Kano, Nobuyuki Kenmochi. Elliptic Quasi-variational inequalities and applications. Conference Publications, 2009, 2009 (Special) : 583-591. doi: 10.3934/proc.2009.2009.583
[3]

Yurii Nesterov, Laura Scrimali. Solving strongly monotone variational and quasi-variational inequalities. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1383-1396. doi: 10.3934/dcds.2011.31.1383
[4]

Laura Scrimali. Mixed behavior network equilibria and quasi-variational inequalities. Journal of Industrial and Management Optimization, 2009, 5 (2) : 363-379. doi: 10.3934/jimo.2009.5.363
[5]

Yusuke Murase, Atsushi Kadoya, Nobuyuki Kenmochi. Optimal control problems for quasi-variational inequalities and its numerical approximation. Conference Publications, 2011, 2011 (Special) : 1101-1110. doi: 10.3934/proc.2011.2011.1101
[6]

Samir Adly, Tahar Haddad. On evolution quasi-variational inequalities and implicit state-dependent sweeping processes. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1791-1801. doi: 10.3934/dcdss.2020105
[7]

Haisen Zhang. Clarke directional derivatives of regularized gap functions for nonsmooth quasi-variational inequalities. Mathematical Control and Related Fields, 2014, 4 (3) : 365-379. doi: 10.3934/mcrf.2014.4.365
[8]

Bin Zhou, Hailin Sun. Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 521-535. doi: 10.3934/naco.2020049
[9]

Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091
[10]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51
[11]

Nobuyuki Kenmochi. Parabolic quasi-variational diffusion problems with gradient constraints. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 423-438. doi: 10.3934/dcdss.2013.6.423
[12]

Masao Fukushima. A class of gap functions for quasi-variational inequality problems. Journal of Industrial and Management Optimization, 2007, 3 (2) : 165-171. doi: 10.3934/jimo.2007.3.165
[13]

Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153
[14]

Savin Treanţă. On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces. Evolution Equations and Control Theory, 2022, 11 (3) : 827-836. doi: 10.3934/eect.2021027
[15]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022
[16]

Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123
[17]

Takeshi Fukao, Nobuyuki Kenmochi. Quasi-variational inequality approach to heat convection problems with temperature dependent velocity constraint. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2523-2538. doi: 10.3934/dcds.2015.35.2523
[18]

Barbara Panicucci, Massimo Pappalardo, Mauro Passacantando. On finite-dimensional generalized variational inequalities. Journal of Industrial and Management Optimization, 2006, 2 (1) : 43-53. doi: 10.3934/jimo.2006.2.43
[19]

Nguyen Thi Van Anh. On periodic solutions to a class of delay differential variational inequalities. Evolution Equations and Control Theory, 2022, 11 (4) : 1309-1329. doi: 10.3934/eect.2021045
[20]

O. Chadli, Z. Chbani, H. Riahi. Recession methods for equilibrium problems and applications to variational and hemivariational inequalities. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 185-196. doi: 10.3934/dcds.1999.5.185

2021 Impact Factor: 1.411

Tools

Metrics

  • PDF downloads (274)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy