ECE 576 POWER SYSTEM DYNAMICS AND STABILITY Lecture 36 Energy Function for SMIB System Professor M.A. Pai Department of Electrical and Computer Engineering © 2000 University of Illinois Board of Trustees, All Rights Reserved

Energy Function for SMIB System Change of notation TEF is for post-fault system. ~ is angle relative to infinite bus. is relative rotor angle velocity.

TEF for SMIB System The right hand side of (1) can be written as ,where Multiplying (1) by , re-write since i.e i.e Hence, the energy function is

TEF for SMIB System (contd) is constant Equilibrium point is given by i.e. This is stable e.p. Verify by linearizing. Eigenvalues on axis. (Marginally Stable) With slight damping ,eigenvalues are in L.H.P. TEF is still constructed for undamped system.

TEF for SMIB System (contd) . Two unstable equilibrium points . (proved later) A change of coordinates so that then . becomes With this, the energy function is Where is the transient kinetic energy and , is the potential energy. and

Energy Function is equal to a constant E, which is the sum of the kinetic and potential energies. It remains constant once the fault is cleared since the system is conservative. evaluated at from the fault trajectory represents the total energy E present in the system at This energy must be absorbed by the system once the fault is cleared if the system is to be stable. The kinetic energy is always positive, and is the difference between E and

Potential Energy “well” Potential energy “well” or P.E. curve How is E computed?

Computation of E E is the value of total energy when fault is cleared. At is the post-fault s.e.p., both the and the is zero, since and at this point. Suppose, at the end of the faulted period the rotor angle is and the velocity is . Then This is the value of E.

u.e.p. ’s There are two other equilibrium points of are the other e.p. ’s

Nature of Equation Points Stable e.p Linearizing around Eigenvalues Marginally stable

Equation Values at u.e.p. ’s (unstable) Same for Hence Type 1 u.e.p.

Nature of . Relative maximum at and At are known from faulted trajectory a

P. E. “bowl” For the point goes up to with the K.E. it has acquired and then oscillates “within” the “well” with being nonlinear. With a small amount of damping This P.E. “well” is a powerful concept. If a b system is stable. system is unstable.

P. E. “bowl” (contd) If system goes unstable due to deceleration instead of acceleration then system is unstable if where are zero-dimensional potential energy boundary Physically ball rolling in a “well”. surfaces.

P. E. “well” and Linear Stability accelerating power Hence, Expand the right-hand side of (1) in Taylor series about an equilibrium point and retain only the linear term. Then

Summary for SMIB System If the equilibrium is unstable. If then it is an oscillatory system, and the oscillations around are bounded. Since there is always some positive damping, we may call it stable.

Summary for SMIB System (contd) is a stable equilibrium point and both and are unstable equilibrium points using this criterion (see slides 10 and 11). The energy function, Lyapunov function, and the PEBS are all equivalent in the case of a SMIB. For multimachine systems and nonconservative systems, each method gives only approximations to the true stability boundary.

Example Consider an SMIB system whose post-fault equation is given by The equilibrium points are given by Hence Linearizing around an equilibrium point “0” results in

State Space Form Define as the state variables. For eigenvalues of this matrix are It is a stable equilibrium point called the focus. For . the eigenvalues are Both are saddle points. Since there is only one eigenvalue in the right half plane, it is called a Type 1 u.e.p. Lecture 36 – Page 18

Example Construct the energy function. Verify the stability of the equilibrium points. The energy function is constructed for the undamped system, i.e., the coefficient of is set equal to zero. The energy function is

Example (contd) At is a stable equilibrium point. (See slide 15) At Hence, both and are unstable equilibrium points. Concept of ,PEBS, and Critical energy are important concepts for multimachine systems. If system goes unstable by acceleration, the critical u.e.p is . It is the only u.e.p for SMIB system. For multimachine systems, there are many u.e.p ’s. If Faulted trajectory goes unstable, the energy at critical u.e.p gives critical energy.