Particle Swarm Optimization for Computing Nash and Stackelberg Equilibria in Energy Markets

Interactions among stakeholders in deregulated markets lead to complex interdependent optimization problems. The present study is motivated by load control programs in energy markets and more precisely by using the power supply interruption as a tool for reducing consumers’ demand voluntarily, also known as voluntary load curtailment programs. The problem is formulated as a Stackelberg game, specifically, as a bilevel optimization problem that belongs to the mathematical programs with equilibrium constraints. In this game, a player that acts as leader determines the actions of the players that act as followers and play a Nash game among them through a subsidy program. The corresponding equilibria need to be found and the presence of nonconvex functions makes the use of metaheuristic algorithms attractive. An extension of particle swarm optimization is proposed for solving such problems based on the unified particle swarm optimization that is a variation of the plain particle swarm optimization algorithm. The proposed algorithm is tested by solving some examples of the formulated games in order to study its efficiency and the interactions between the stakeholders of the market.

[1]  Michael N. Vrahatis,et al.  Recent approaches to global optimization problems through Particle Swarm Optimization , 2002, Natural Computing.

[2]  Andreas Sumper,et al.  Experimental validation of a real time energy management system for microgrids in islanded mode using a local day-ahead electricity market and MINLP , 2013 .

[3]  Mousa Marzband,et al.  Adaptive load shedding scheme for frequency stability enhancement in microgrids , 2016 .

[4]  A. Huang,et al.  A game theoretic framework for a next-generation retail electricity market with high penetration of distributed residential electricity suppliers , 2014 .

[5]  Mark Karwan,et al.  Multilevel optimization: A mathematical programming perspective , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[6]  P. Cappers,et al.  Demand Response in U.S. Electricity Markets: Empirical Evidence , 2010 .

[7]  Alberto Leon-Garcia,et al.  Game-Theoretic Demand-Side Management With Storage Devices for the Future Smart Grid , 2014, IEEE Transactions on Smart Grid.

[8]  Mehdi Savaghebi,et al.  An Optimal Energy Management System for Islanded Microgrids Based on Multiperiod Artificial Bee Colony Combined With Markov Chain , 2017, IEEE Systems Journal.

[9]  Panos M. Pardalos,et al.  Energy Efficiency in Urban Electrical Grids through Consumer Networking , 2017 .

[10]  Michael N. Vrahatis,et al.  On the computation of all global minimizers through particle swarm optimization , 2004, IEEE Transactions on Evolutionary Computation.

[11]  Francisco Facchinei,et al.  Generalized Nash equilibrium problems , 2007, 4OR.

[12]  Mehdi Savaghebi,et al.  Distributed Smart Decision-Making for a Multimicrogrid System Based on a Hierarchical Interactive Architecture , 2016, IEEE Transactions on Energy Conversion.

[13]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[14]  David G. Luenberger,et al.  Linear and nonlinear programming , 1984 .

[15]  B. Hobbs,et al.  Complementarity Modeling in Energy Markets , 2012 .

[16]  Michael N. Vrahatis,et al.  Parameter selection and adaptation in Unified Particle Swarm Optimization , 2007, Math. Comput. Model..

[17]  Mousa Marzband,et al.  Optimal energy management for stand‐alone microgrids based on multi‐period imperialist competition algorithm considering uncertainties: experimental validation , 2016 .

[18]  M. N. Vrahatis,et al.  Computing Nash equilibria through computational intelligence methods , 2005 .

[19]  R. Walawalkar,et al.  Evolution and current status of demand response (DR) in electricity markets: Insights from PJM and NYISO , 2010 .

[20]  José Fortuny-Amat,et al.  A Representation and Economic Interpretation of a Two-Level Programming Problem , 1981 .

[21]  P. Harker Generalized Nash games and quasi-variational inequalities , 1991 .

[22]  Michael N. Vrahatis,et al.  Particle Swarm Optimization and Intelligence: Advances and Applications , 2010 .

[23]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[24]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[25]  T. Başar,et al.  A Stackelberg Network Game with a Large Number of Followers , 2002 .

[26]  M. Marzband,et al.  Distributed generation for economic benefit maximization through coalition formation–based game theory concept , 2017 .

[27]  James Kennedy,et al.  Particle swarm optimization , 2002, Proceedings of ICNN'95 - International Conference on Neural Networks.

[28]  George Papavassilopoulos Algorithms for static stackelberg games with linear costs and polyhedra constraints , 1982, 1982 21st IEEE Conference on Decision and Control.

[29]  Gerd Wachsmuth,et al.  On LICQ and the uniqueness of Lagrange multipliers , 2013, Oper. Res. Lett..

[30]  Ann van Ackere,et al.  A framework to evaluate security of supply in the electricity sector , 2017 .

[31]  A. Selvakumar,et al.  A New Particle Swarm Optimization Solution to Nonconvex Economic Dispatch Problems , 2007, IEEE Transactions on Power Systems.

[32]  Andreas Sumper,et al.  Real time experimental implementation of optimum energy management system in standalone Microgrid by using multi-layer ant colony optimization , 2016 .

[33]  Maurice Clerc,et al.  The particle swarm - explosion, stability, and convergence in a multidimensional complex space , 2002, IEEE Trans. Evol. Comput..

[34]  Andreas Sumper,et al.  Experimental Validation of a Real-Time Energy Management System Using Multi-Period Gravitational Search Algorithm for Microgrids in Islanded Mode , 2014 .

[35]  Cheng-Yan Kao,et al.  Applying Family Competition to Evolution Strategies for Constrained Optimization , 1997, Evolutionary Programming.

[36]  Mousa Marzband,et al.  Non-cooperative game theory based energy management systems for energy district in the retail market considering DER uncertainties , 2016 .