ECE 576 POWER SYSTEM DYNAMICS AND STABILITY Lecture 27 Reduced Order Multi-Machine Models Professor M.A. Pai Department of Electrical and Computer Engineering © 2000 University of Illinois Board of Trustees, All Rights Reserved

Simulation of 3-machine system All angles are relative to the angle of machine 1 No damping tf = 0.1 s tcl = 0.2 s

Simulation of 3-machine system tf = 0.1 s tcl = 0.395 s machine 2 machine 3 All angles are relative to the angle of machine 1 No damping

Simulation of 3-machine system tf = 0.1 s tcl = 0.2 s machine 2 machine 3 All angles are relative to the angle of machine 1 Damping is proportional to machine inertia

Simulation of 3-machine system tf = 0.1 s tcl = 0.2 s machine 2 machine 3 All angles are relative to the center of inertia Damping is proportional to machine inertia machine 1

Reduced Order Models Flux Decay Model (DAE) Classical Model (DAE Structure Preserving) (constant voltage behind transient reactance) Classical Model (DE only) (internal node model –network nodes eliminated) (1) is popular in small signal analysis, oscillations and their stabilization. (2) & (3) are popular in direct methods of stability analysis

Flux decay model Damper winding constant is small Use of Singular Perturbation (integral manifold) makes the equation algebraic Substitute in Torque Eqn and Stator Algebraic Eqn STATIC EXCITER

Flux Decay Model (Stator Eqns) Stator Equations (Assume ) These are in polar form. DYNAMIC CIRCUIT

Flux Decay Model (Generator Eqns) Generator Equations Network Equations: same as in previous DAE model Total number of equations 4m + 2m + 2n D.E Stator Network

Classical Model of Machine More assumptions… is large i.e. is a slow variable and remains constant at initial value . simplifies stator and torque equations. Generator Equations Will simplify later.

Classical Model Derivation Stator Equations Would like to have voltage behind .

Classical Model (contd) voltage behind transient reactance o No machine variables and no stator equations! = constant and computed as follows:

Classical Model (contd) Step 1: From the load flow, compute Step 2: Compute is kept constant. appears in torque equation (page 5)and we need to compute it in terms of state and network variables.

Classical Model (contd) Reactance does not consume real power. Hence, After expanding and simplifying (5)

Structure Preserving Multi-Machine Model (SPMM) 2 D.E’s at each machine plus algebraic equations at buses 1,…,n. (Note: at buses n+1,…, n+m, we know is constant and from D.E’s, therefore no algebraic eqn.

Structure Preserving Model At the generator buses i=1,…,m

Structure Preserving Model (contd) Hence, Now we can write the network equation in power balance form at all buses. At load bus Transmission Lines (Neglecting transmission line resistances)

Structure Preserving Model (contd) The total complex power in the network transmission lines from bus i is:

Structure Preserving Model (contd) The network power balance equations are:

Structure Preserving Model (contd) The D.E’s are: Plus the algebraic equations. Symbolically:

Numerical example (2 machine system) 1 2 ~ ~ 3

Numerical example (2 machine system) Flux decay model:

Numerical example (2 machine system)

Numerical example (2 machine system) Classical model: