A Unifying Energy-Based Approach to Stability of Power Grids With Market Dynamics

In this paper, a unifying energy-based approach is provided to the modeling and stability analysis of power systems coupled with market dynamics. We consider a standard model of the power network with a third-order model for the synchronous generators involving voltage dynamics. By applying the primal-dual gradient method to a social welfare optimization, a distributed dynamic pricing algorithm is obtained, which can be naturally formulated in port-Hamiltonian form. By interconnection with the physical model a closed-loop port-Hamiltonian system is obtained, whose properties are exploited to prove asymptotic stability to the set of optimal points. This result is extended to the case that also general nodal power constraints are included into the social welfare problem. Additionally, the case of line congestion and power transmission costs in acyclic networks is covered. Finally, a dynamic pricing algorithm is proposed that does not require knowledge about the power supply and demand.

[1]  Anuradha M. Annaswamy,et al.  The effect of a smart meter on congestion and stability in a power market , 2010, 49th IEEE Conference on Decision and Control (CDC).

[2]  Ufuk Topcu,et al.  Design and Stability of Load-Side Primary Frequency Control in Power Systems , 2013, IEEE Transactions on Automatic Control.

[3]  Claudio De Persis,et al.  A modular design of incremental Lyapunov functions for microgrid control with power sharing , 2015, 2016 European Control Conference (ECC).

[4]  W. Rudin.  Principles of mathematical analysis , 1964 .

[5]  Arjan van der Schaft,et al.  A port-Hamiltonian approach to optimal frequency regulation in power grids , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[6]  P. P. J. van den Bosch,et al.  On Constrained Steady-State Regulation: Dynamic KKT Controllers , 2009, IEEE Transactions on Automatic Control.

[7]  Claudio De Persis,et al.  An internal model approach to (optimal) frequency regulation in power grids with time-varying voltages , 2014, Autom..

[8]  Enrique Mallada,et al.  Distributed generator and load-side secondary frequency control in power networks , 2015, 2015 49th Annual Conference on Information Sciences and Systems (CISS).

[9]  Arjan van der Schaft,et al.  Port-Hamiltonian Systems Theory: An Introductory Overview , 2014, Found. Trends Syst. Control..

[10]  Peter W. Sauer,et al.  Power System Dynamics and Stability , 1997 .

[11]  Xuan Zhang,et al.  A real-time control framework for smart power networks: Design methodology and stability , 2015, Autom..

[12]  Enrique Mallada,et al.  Asymptotic convergence of constrained primal-dual dynamics , 2015, Syst. Control. Lett..

[13]  Enrique Mallada,et al.  Optimal Load-Side Control for Frequency Regulation in Smart Grids , 2014, IEEE Transactions on Automatic Control.

[14]  F. Alvarado.  The stability of power system markets , 1999 .

[15]  Arjan van der Schaft,et al.  Optimal power dispatch in networks of high-dimensional models of synchronous machines , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[16]  Babu Narayanan,et al.  POWER SYSTEM STABILITY AND CONTROL , 2015 .

[17]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

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

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

[20]  Na Li,et al.  Connecting Automatic Generation Control and Economic Dispatch From an Optimization View , 2014, IEEE Transactions on Control of Network Systems.

[21]  Nima Monshizadeh,et al.  Bregman Storage Functions for Microgrid Control , 2015, IEEE Transactions on Automatic Control.

[22]  C. De Persis,et al.  Port-Hamiltonian Formulation of the Gradient Method Applied to Smart Grids , 2015 .

[23]  F. Alvarado,et al.  Stability Analysis of Interconnected Power Systems Coupled with Market Dynamics , 2001, IEEE Power Engineering Review.

[24]  R. Rockafellar.  The multiplier method of Hestenes and Powell applied to convex programming , 1973 .

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

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

[27]  ASHISH CHERUKURI,et al.  Saddle-Point Dynamics: Conditions for Asymptotic Stability of Saddle Points , 2015, SIAM J. Control. Optim..

[28]  Selmer M. Johnson,et al.  Studies in Linear and Non-Linear Programming. , 1959 .

[29]  Lijun Chen,et al.  Reverse and forward engineering of frequency control in power networks , 2014, 53rd IEEE Conference on Decision and Control.

[30]  Samuel Karlin,et al.  Studies in Linear and Non-Linear Programming. , 1959 .

[31]  Xuan Zhang,et al.  Distributed dynamic feedback control for smart power networks with tree topology , 2014, 2014 American Control Conference.

[32]  Munther A. Dahleh,et al.  On the stability of wholesale electricity markets under real-time pricing , 2010, 49th IEEE Conference on Decision and Control (CDC).

[33]  Janusz Bialek,et al.  Power System Dynamics: Stability and Control , 2008 .

[34]  H. H. Happ,et al.  Power System Control and Stability , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[35]  Xuan Zhang,et al.  A real-time control framework for smart power networks with star topology , 2013, 2013 American Control Conference.

[36]  Enrique Mallada Steven H. Low.  Distributed Frequency-Preserving Optimal Load Control , 2014 .

[37]  Xuan Zhang,et al.  Achieving real-time economic dispatch in power networks via a saddle point design approach , 2015, 2015 IEEE Power & Energy Society General Meeting.

[38]  Anuradha M. Annaswamy,et al.  A hierarchical transactive control architecture for renewables integration in Smart Grids , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).