Analysis of Power System Dynamics Subject to Stochastic Power Injections

We propose a framework to study the impact of stochastic active/reactive power injections on power system dynamics with a focus on time scales involving electromechanical phenomena. In this framework, the active/reactive power injections evolve according to a continuous-time Markov chain (CTMC), while the power system dynamics are described by the standard differential algebraic equation (DAE) model. The DAE model is linearized around a nominal set of active/reactive power injections, and the combination of the linearized DAE model and the CTMC forms a stochastic process known as a stochastic hybrid system (SHS). The extended generator of the SHS yields a system of ordinary differential equations that governs the evolution of the power system dynamic and algebraic state moments. We illustrate the application of the framework to the computation of long-term power system state statistics, and to short-term probabilistic dynamic performance/reliability assessment.

[1]  M. Collares-Pereira,et al.  Simple procedure for generating sequences of daily radiation values using a library of Markov transition matrices , 1988 .

[2]  Michael Milligan,et al.  Grid Impacts of Wind Power Variability: Recent Assessments from a Variety of Utilities in the United States; Preprint , 2006 .

[3]  Peter W. Sauer,et al.  Towards a wind farm reduced-order model , 2011 .

[4]  Sairaj V. Dhople,et al.  A Stochastic Hybrid Systems framework for analysis of Markov reward models , 2014, Reliab. Eng. Syst. Saf..

[5]  Alejandro D. Dominguez-Garcia,et al.  Models for Impact Assessment of Wind-Based Power Generation on Frequency Control , 2012 .

[6]  J. Driesen,et al.  Characterization of the Solar Power Impact in the Grid , 2007, 2007 International Conference on Clean Electrical Power.

[7]  Pulgar Painemal,et al.  Wind farm model for power system stability analysis , 2010 .

[8]  Chika O. Nwankpa,et al.  Stochastic model for power system planning studies , 1991 .

[9]  Catherine Rosenberg,et al.  Markovian models for home electricity consumption , 2011, GreenNets '11.

[10]  Takashi Hikihara,et al.  Application of hybrid system theory to power system stability analysis , 2005 .

[11]  H.R. Pota,et al.  Decentralized control of power systems via robust control of uncertain Markov jump parameter systems , 2005, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[12]  Ian A. Hiskens,et al.  Power system modeling for inverse problems , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[13]  L.C.P. da Silva,et al.  Investigation of the relationship between ill-conditioned power flow and voltage collapse , 2000 .

[14]  Yu Christine Chen,et al.  A Method to Study the Effect of Renewable Resource Variability on Power System Dynamics , 2012, IEEE Transactions on Power Systems.

[15]  Mark H. Davis.  Markov Models and Optimization , 1995 .

[16]  S. M. Shahidehpour,et al.  Stochastic Approach to Small Disturbance Stability Analysis , 1992, IEEE Power Engineering Review.

[17]  Z.Q. Liu,et al.  Research on the relationship of the singular point for load flow Jacobian matrix and the critical point of voltage collapse , 2005, IEEE Power Engineering Society General Meeting, 2005.

[18]  J. Hespanha.  A model for stochastic hybrid systems with application to communication networks , 2005 .

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

[20]  Johanna L. Mathieu,et al.  Modeling and Control of Aggregated Heterogeneous Thermostatically Controlled Loads for Ancillary Services , 2011 .

[21]  Felix F. Wu,et al.  Dynamic security regions of power systems , 1982 .

[22]  M. Pai,et al.  Power system steady-state stability and the load-flow Jacobian , 1990 .

[23]  M. L. Crow,et al.  Structure-Preserved Power System Transient Stability Using Stochastic Energy Functions , 2012, IEEE Transactions on Power Systems.

[24]  William E. Boyson,et al.  Photovoltaic array performance model. , 2004 .

[25]  Sridevi Parise,et al.  Clustering Markov States into Equivalence Classes using SVD and Heuristic Search Algorithms , 2003, AISTATS.

[26]  C. O. Nwankpa,et al.  Stochastic analysis and simulation of grid-connected wind energy conversion system , 2000 .

[27]  D.M. Falcao,et al.  Probabilistic Wind Farms Generation Model for Reliability Studies Applied to Brazilian Sites , 2006, IEEE Transactions on Power Systems.

[28]  Roy Billinton,et al.  Reliability evaluation of power systems , 1984 .

[29]  Y. Manichaikul,et al.  Physically Based Industrial Electric Load , 1979, IEEE Transactions on Power Apparatus and Systems.

[30]  Munther A. Dahleh,et al.  Model reduction of irreducible Markov chains , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[31]  A. D. Dominguez-Garcia,et al.  Reliability modeling of cyber-physical electric power systems: A system-theoretic framework , 2012, 2012 IEEE Power and Energy Society General Meeting.

[32]  P. Billingsley,et al.  Probability and Measure , 1980 .

[33]  R. Malhamé,et al.  Electric load model synthesis by diffusion approximation of a high-order hybrid-state stochastic system , 1985 .

[34]  M. A. Pai,et al.  Hybrid systems view of power system modelling , 2000, 2000 IEEE International Symposium on Circuits and Systems. Emerging Technologies for the 21st Century. Proceedings (IEEE Cat No.00CH36353).

[35]  Ian A. Hiskens,et al.  Trajectory Sensitivity Analysis of Hybrid Systems , 2000 .

[36]  C. Chong,et al.  Stochastic hybrid state systems for electric load modeling , 1983, The 22nd IEEE Conference on Decision and Control.

[37]  J. Hespanha.  Modelling and analysis of stochastic hybrid systems , 2006 .

[38]  Roy Jensen,et al.  Reliability Modeling in Electric Power Systems , 1979 .

[39]  A. Debs,et al.  Statistical synthesis of power system functional load models , 1979, 1979 18th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[40]  Sigifredo Gonzalez,et al.  Performance Model for Grid-Connected Photovoltaic Inverters , 2007 .

[41]  A. Megretski,et al.  Model reduction of discrete-time Markov jump linear systems , 2006, 2006 American Control Conference.

[42]  Carmen Lucia Tancredo Borges,et al.  A Comparison of Load Models for Composite Reliability Evaluation by Nonsequential Monte Carlo Simulation , 2010, IEEE Transactions on Power Systems.

[43]  Li Li,et al.  Decentralized robust control of uncertain Markov jump parameter systems via output feedback , 2006, 2006 American Control Conference.

[44]  Roland Malhamé,et al.  A jump-driven Markovian electric load model , 1990, Advances in Applied Probability.

[45]  D. Hill,et al.  Numerical Simulation for Stochastic Transient Stability Assessment , 2012, IEEE Transactions on Power Systems.

[46]  W. G. Watson,et al.  Reliability Modeling in Electric Power Systems , 1979 .

[47]  Dominik Szynal,et al.  ON MARKOV-TYPE INEQUALITIES , 2010 .

[48]  Felix F. Wu,et al.  Probabilistic dynamic security assessment of power systems-I: Basic model , 1983 .

[49]  S. Kumagai,et al.  Steady-State Security Regions of Power Systems , 1982 .

[50]  G. Papaefthymiou,et al.  MCMC for Wind Power Simulation , 2008, IEEE Transactions on Energy Conversion.

[51]  Kenneth A. Loparo,et al.  A probabilistic mechanism for small disturbance instabilities in electric power systems , 1985 .

[52]  M. Crow,et al.  Numerical simulation of Stochastic Differential Algebraic Equations for power system transient stability with random loads , 2011, 2011 IEEE Power and Energy Society General Meeting.

[53]  Sean P. Meyn,et al.  A simulation-based method for aggregating Markov chains , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[54]  P. Kundur,et al.  Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions , 2004, IEEE Transactions on Power Systems.

[55]  C.D. Vournas,et al.  Hybrid systems modeling for power systems , 2004, IEEE Circuits and Systems Magazine.