A Sufficient Condition for Small-Signal Stability and Construction of Robust Stability Region

The small-signal stability is an integral part of the power system security analysis. The introduction of renewable source related uncertainties is making the stability assessment difficult as the equilibrium point is varying rapidly. This paper focuses on the Differential Algebraic Equation (DAE) formulation of power systems and bridges the gap between the conventional reduced system and the original one using logarithmic norm. We propose a sufficient condition for stability using Bilinear Matrix Inequality and its inner approximation as Linear Matrix Inequality. Another contribution is the construction of robust stability regions in state-space in contrast to most existing approaches trying same in the parameter space. Performance evaluation of the sufficiency condition and its inner approximation has been given with the necessary and sufficient condition for a small-scale test case. The paper provides a necessary base to develop tractable construction techniques for the robust stability region of power systems.

[1]  Thanh Long Vu,et al.  Contraction Analysis of Nonlinear DAE Systems , 2017, IEEE Transactions on Automatic Control.

[2]  F. Lewis,et al.  Optimal control for singular systems , 1985, 1985 24th IEEE Conference on Decision and Control.

[3]  Hsiao-Dong Chiang,et al.  Network-Preserving Sensitivity-Based Generation Rescheduling for Suppressing Power System Oscillations , 2017, IEEE Transactions on Power Systems.

[4]  Line A. Roald,et al.  Towards an AC Optimal Power Flow Algorithm with Robust Feasibility Guarantees , 2018, 2018 Power Systems Computation Conference (PSCC).

[5]  Ronald L. Smith.  Some interlacing properties of the Schur complement of a Hermitian matrix , 1992 .

[6]  Karen L. Butler-Purry,et al.  Multi-Time Step Service Restoration for Advanced Distribution Systems and Microgrids , 2018, IEEE Transactions on Smart Grid.

[7]  Hung D. Nguyen,et al.  Constructing Convex Inner Approximations of Steady-State Security Regions , 2019, IEEE Transactions on Power Systems.

[8]  V. Ajjarapu,et al.  Critical Eigenvalues Tracing for Power System Analysis via Continuation of Invariant Subspaces and Projected Arnoldi Method , 2007, IEEE Transactions on Power Systems.

[9]  F. L. Pagola,et al.  On Sensitivities, Residues and Participations. Applications to Oscillatory Stability Analysis and Control , 1989, IEEE Power Engineering Review.

[10]  Krishnamurthy Dvijotham,et al.  A Framework for Robust Long-Term Voltage Stability of Distribution Systems , 2019, IEEE Transactions on Smart Grid.

[11]  Rama K. Yedavalli,et al.  Robust Stability Region of a Microgrid Under Parametric Uncertainty Using Bialternate Sum Matrix Approach , 2018, IEEE Transactions on Power Systems.

[12]  O. Toker,et al.  On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback , 1995, Proceedings of 1995 American Control Conference - ACC'95.

[13]  Chi-Tsong Chen,et al.  Linear System Theory and Design , 1995 .

[14]  Peter B. Luh,et al.  Formal Analysis of Networked Microgrids Dynamics , 2018, IEEE Transactions on Power Systems.

[15]  P. Kundur,et al.  Power system stability and control , 1994 .

[16]  P. N. Paraskevopoulos,et al.  Modern Control Engineering , 2001 .

[17]  Zahra Aminzarey,et al.  Contraction methods for nonlinear systems: A brief introduction and some open problems , 2014, 53rd IEEE Conference on Decision and Control.

[18]  Qifeng Li.  A Tight SDP Relaxation for MIQCQP Problems in Power Systems Based on Disjunctive Programming , 2015, 1509.05141.

[19]  M. A. Pai,et al.  Applications of interval matrices in power system stabilizer design , 1997 .

[20]  Peng Zhang,et al.  A Geršgorin theory for robust microgrid stability analysis , 2016, 2016 IEEE Power and Energy Society General Meeting (PESGM).

[21]  M. Vidyasagar,et al.  Nonlinear systems analysis (2nd ed.) , 1993 .

[22]  J.H. Chow,et al.  Power system stabilizers as undergraduate control design projects , 2004, IEEE Transactions on Power Systems.

[23]  Hoay Beng Gooi,et al.  Ellipsoidal Prediction Regions for Multivariate Uncertainty Characterization , 2017, IEEE Transactions on Power Systems.

[24]  Johan Löfberg,et al.  Automatic robust convex programming , 2012, Optim. Methods Softw..

[25]  V. Ajjarapu,et al.  Sensitivity-Based Efficient Identification of Oscillatory Stability Margin and Damping Margin Using Continuation of Invariant Subspaces , 2011, IEEE Transactions on Power Systems.

[26]  Hsiao-Dong Chiang,et al.  Investigation of an Effective Strategy for Computing Small-Signal Security Margins , 2018, IEEE Transactions on Power Systems.

[27]  Zhengchun Du,et al.  A novel method for computing small-signal stability boundaries of large-scale power systems , 2013, IEEE Transactions on Power Systems.

[28]  Hansheng Wu,et al.  Stability and robust stabilization of nonlinear descriptor systems with uncertainties , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.