ECE 576 POWER SYSTEM DYNAMICS AND STABILITY Lecture 40 Structure Preserving Energy Function Angle and Voltage Stability Professor M.A. Pai Department of Electrical and Computer Engineering © 2000 University of Illinois Board of Trustees, All Rights Reserved

Angle Stability Model With classical model for machines we have ~ ~ Trans. Network m Machines n Physical buses ~ Model With classical model for machines we have and 2n network equations swing equations (m)

Trans Network 1 2 m m+1 m+(m+1) - + m+(m+2) m+2 - + Trans. Network m+m (m+n) - +

Trans Network (contd) Let rotor angles, velocities and bus angles with respect to syn. ref. frame be Define Internal voltages Network voltages

Trans Network (contd) Augmented includes transient reactances. Swing eqns. Actually the R.H. side is i i+m

Network Equations Let load be represented as i Network equations Swing equations plus (1) and (2) constitute a differential-algebraic model (DAE) Assume for all i so that there is no frequency dependent load. i

Objective To construct the first integral of motion for the system ie is the first integral or energy function. Introduction of C.O.I. facilitates constructing V. Define

Swing Equations Swing equations become Where Steps to get the first integral 1. Multiply (1) by and sum over m equations

Swing Equations (contd) 2. Multiply real power flow equations (slide 4) by and sum over all n physical buses. 3. Divide reactive power equation (slide 4) by Multiply (2) by and sum over n physical buses

Swing Equations (contd) (1), (2) and (3) of previous slide and (3) of slide # 7 give Problem term! Consider simpler case constant. Then,

Swing Equations (contd) The first integral is V=V+K where K= constant. K is chosen so that V=0 at s.e.p. Thus, Thus

Swing Equations (contd) since It can be shown that Define a vector int. nodes physical buses = injected power at nodes

Swing Equations (contd) Then This is a compact expression relates to reactive power in network.

Voltage Stability – Use of Energy Functions As the system gets loaded Q reserves in the system are not sufficient locally or globally. Hence there is voltage depression and if not controlled may lead to voltage instability or collapse. Two possibilities System gets progressively loaded till the point of voltage collapse. System presently is stable but close to the point of voltage collapse. A small disturbance will then lead to voltage collapse.

What is the Role of System and Load Dynamics? View point 1 Buses View point 1 Since voltage stability is a slow phenomenon both S and L are considered static. Hence voltage stability is a static problem. Techniques 1. Det. J. (J=load flow Jacobian) 2. Min. sing. value of J 3. Condition number of J 4. Continuation method to find saddle node bifurcation 5. Use of sensitivities from OPF

View Point 2 Since voltage profile is controlled by excitation dynamics, modeling of system dynamics is important. Finding both saddle node and Hopf bifurcation by eigenvalue monitoring. Sensitivity of voltage collapse due to injected Q at GEN, SVC, TAP-changer dynamics, etc.

View Point 4 All of the above! There is consensus that some sort of dynamics does play a part. Hence, an energy function approach is desirable (Demarco, Overbye) Single machine case ~ SEP – high voltage solution UEP – low voltage solution

View Point 3 Load dynamics are crucial in determining voltage collapse. Load modeling in a dynamic sense (D.E.’s for motors, etc) Fast changing or cyclic loads.

Voltage Energy Margin For a slowly varying load Hence (energy at UEP)-(energy at SEP) voltage energy margin (VEM) For a slowly varying load Hence Hence derived from SPM (transient stability). and assume UEPV values. Hence,

Voltage Energy Margin (contd) As in angle stability we need efficient algorithms to find UEP solutions. VEM a Voltage collapse point If VEM>0, system is voltage stable a Voltage collapse point

Conclusion A unified picture is now possible for both angle and voltage stability using same energy function. Effects of dynamic modeling on system and load size are needed to be studied for both angle and voltage stability. For angle stability with s.p. models, controlling u.e.p. is not defined easily. However PEBS method works. Sensitivity analysis of voltage energy margin may provide clues to preventing voltage collapse.

Coherency, synchrony, slow coherency Podmore, R. ‘Identification of coherent generators for dynamic equivalents’ IEEE Trans. Power Appar. Syst. Vol PAS-97 (July/August 1978) pp1344-1354. Pai, M A and Adgaonkar, R P. ‘Identification of coherent generators using weighted eigenvectors’ Proc. IEEE PES Winter Meeting New York, USA A79 022-5 (February 1979). Slow coherency Chow, JH (Ed.). ‘Time scale modeling of dynamic networks with applications to power systems’ Lecture notes in control and information sciences Vol 46, SpringerVerlag,USA(1982). Synchrony 4. G.Ramaswamy, G.C. Verghese, Luis Rouco, C. Vialas, C.L. Demarco. ‘Synchrony, aggression and multi-area eigenvalues analysis’ IEEE Trans. Power Systems 1995.

Linearizing Swing Equations Assume faulted period of the order of few cycles. Assume as step input=accelerating power at Put (1) in state space form

Linearizing Swing Equations (contd) Solution of post-fault system Mi are right eigenvectors of A ki determined from values of X(t) at t = tcl and left eigenvectors of A. There are (n-1) complex pairs of eigenvalues and one negative real eigenvalue pil and qil represent amplitudes of complex modes.

Coherency and Synchrony Two generators i and j are classified as coherent if where is a small number. dij is the norm of the row vectors corresponding to δi and δj in the right eigenvector matrix M. Synchrony Uses the angle between the two row vectors as the criterion.