Distributed Extremum Seeking for Constrained Networked Optimization and Its Application to Energy Consumption Control in Smart Grid

In this paper, a distributed extremum seeking scheme is proposed to find the solution of a nonmodel-based distributed optimization problem among networked agents. The agents are supposed to have measurements of the local cost functions and constraint functions. However, no explicit expressions on the cost functions, the constraint functions, or their gradients are available. The design of the distributed extremum seeking scheme is based on the saddle point dynamics. Stability analysis is conducted via using averaging analysis, Lyapunov stability analysis, and the concept of saddle point. It is shown that the solution to the distributed optimization problem is semiglobally practically asymptotically stable under the proposed extremum seeking law. An application of the proposed extremum seeking method to energy consumption control for the electricity consumers in smart grid is discussed. Simulation results on energy consumption control of a network of heating ventilation and air conditioning systems are provided to validate the proposed distributed extremum seeker.

[1]  Nicanor Quijano,et al.  Distributed extremum seeking for real-time resource allocation , 2013, 2013 American Control Conference.

[2]  Arye Nehorai,et al.  An Optimal and Distributed Demand Response Strategy With Electric Vehicles in the Smart Grid , 2014, IEEE Transactions on Smart Grid.

[3]  Angelia Nedic,et al.  Subgradient Methods for Saddle-Point Problems , 2009, J. Optimization Theory and Applications.

[4]  Lacra Pavel,et al.  An analytic framework for decentralized extremum seeking control , 2012, 2012 American Control Conference (ACC).

[5]  Jing Wang,et al.  Control approach to distributed optimization , 2010, 2010 48th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[6]  Na Li,et al.  Optimal demand response based on utility maximization in power networks , 2011, 2011 IEEE Power and Energy Society General Meeting.

[7]  Sonia Martínez,et al.  On Distributed Convex Optimization Under Inequality and Equality Constraints , 2010, IEEE Transactions on Automatic Control.

[8]  K. Johansson,et al.  Distributed positioning of autonomous mobile sensors with application to coverage control , 2011, Proceedings of the 2011 American Control Conference.

[9]  Ying Tan,et al.  On non-local stability properties of extremum seeking control , 2006, Autom..

[10]  Angelia Nedic,et al.  Distributed Optimization Over Time-Varying Directed Graphs , 2015, IEEE Trans. Autom. Control..

[11]  Miroslav Krstic,et al.  Multi-agent deployment in the plane using stochastic extremum seeking , 2010, 49th IEEE Conference on Decision and Control (CDC).

[12]  Jing Wang,et al.  A control perspective for centralized and distributed convex optimization , 2011, IEEE Conference on Decision and Control and European Control Conference.

[13]  A. Teel,et al.  Semi-global practical asymptotic stability and averaging , 1999 .

[14]  Fernando Paganini,et al.  Stability of primal-dual gradient dynamics and applications to network optimization , 2010, Autom..

[15]  Dirk Aeyels,et al.  Practical stability and stabilization , 2000, IEEE Trans. Autom. Control..

[16]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[17]  Ashish Cherukuri,et al.  Asymptotic stability of saddle points under the saddle-point dynamics , 2015, 2015 American Control Conference (ACC).

[18]  Guoqiang Hu,et al.  Distributed Energy Consumption Control via Real-Time Pricing Feedback in Smart Grid , 2014, IEEE Transactions on Control Systems Technology.

[19]  B. V. Dean,et al.  Studies in Linear and Non-Linear Programming. , 1959 .

[20]  Yeng Chai Soh,et al.  Distributed Extremum Seeking Control of networked large-scale systems under constraints , 2013, 52nd IEEE Conference on Decision and Control.

[21]  John S. Baras,et al.  Collaborative extremum seeking for welfare optimization , 2014, 53rd IEEE Conference on Decision and Control.

[22]  Anna Scaglione,et al.  Distributed Constrained Optimization by Consensus-Based Primal-Dual Perturbation Method , 2013, IEEE Transactions on Automatic Control.

[23]  T. Kose Solutions of Saddle Value Problems by Differential Equations , 1956 .

[24]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[25]  Miroslav Krstic,et al.  Stochastic Nash Equilibrium Seeking for Games with General Nonlinear Payoffs , 2011, SIAM J. Control. Optim..

[26]  Milos S. Stankovic,et al.  Distributed Seeking of Nash Equilibria With Applications to Mobile Sensor Networks , 2012, IEEE Transactions on Automatic Control.

[27]  Yan Zhang,et al.  Demand Response Management With Multiple Utility Companies: A Two-Level Game Approach , 2014, IEEE Transactions on Smart Grid.

[28]  Maojiao Ye,et al.  Distributed optimization for systems with time-varying quadratic objective functions , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[29]  Maojiao Ye,et al.  A distributed extremum seeking scheme for networked optimization , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[30]  Bahman Gharesifard,et al.  Distributed Continuous-Time Convex Optimization on Weight-Balanced Digraphs , 2012, IEEE Transactions on Automatic Control.

[31]  Milos S. Stankovic,et al.  Lie bracket approximation of extremum seeking systems , 2011, Autom..

[32]  Hans-Bernd Dürr,et al.  Saddle Point Seeking for Convex Optimization Problems , 2013, NOLCOS.

[33]  Na Li,et al.  Two Market Models for Demand Response in Power Networks , 2010, 2010 First IEEE International Conference on Smart Grid Communications.

[34]  Peng Yang,et al.  Stability and Convergence Properties of Dynamic Average Consensus Estimators , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[35]  Martin Guay,et al.  Extremum-seeking control of distributed systems using consensus estimation , 2014, 53rd IEEE Conference on Decision and Control.

[36]  Ioannis Lestas,et al.  On the convergence to saddle points of concave-convex functions, the gradient method and emergence of oscillations , 2014, 53rd IEEE Conference on Decision and Control.

[37]  Denis Dochain,et al.  A time-varying extremum-seeking control approach , 2013, ACC.

[38]  Miroslav Krstic,et al.  Stability of extremum seeking feedback for general nonlinear dynamic systems , 2000, Autom..

[39]  Anders Rantzer,et al.  Dynamic dual decomposition for distributed control , 2009, 2009 American Control Conference.

[40]  Hans-Bernd Dürr,et al.  Feedback design for multi-agent systems: A saddle point approach , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[41]  Deming Yuan,et al.  Distributed Primal-Dual Subgradient Method for Multiagent Optimization via Consensus Algorithms. , 2011, IEEE transactions on systems, man, and cybernetics. Part B, Cybernetics : a publication of the IEEE Systems, Man, and Cybernetics Society.

[42]  Lacra Pavel,et al.  A continuous-time decentralized optimization scheme with positivity constraints , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[43]  Miroslav Krstic,et al.  Nash Equilibrium Seeking in Noncooperative Games , 2012, IEEE Transactions on Automatic Control.

[44]  J. Neumann,et al.  SOLUTIONS OF GAMES BY DIFFERENTIAL EQUATIONS , 1950 .

[45]  Choon Yik Tang,et al.  Zero-gradient-sum algorithms for distributed convex optimization: The continuous-time case , 2011, Proceedings of the 2011 American Control Conference.

[46]  U. Ozguner,et al.  Nash solution by extremum seeking control approach , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[47]  Miroslav Krstic,et al.  Non-C2 Lie Bracket Averaging for Nonsmooth Extremum Seekers , 2014 .