# American Institute of Mathematical Sciences

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

 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.
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 & Management Optimization, 2008, 4 (3) : 425-452. doi: 10.3934/jimo.2008.4.425
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