1. 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

2. 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

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

4. 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

5. 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

6. 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

7. 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

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

9. 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

10. 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.

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

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

13. 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.

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

15. 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.

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

17. 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)

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

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

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

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

22. Numerical example (2 machine system)
Flux decay model:

23. Numerical example (2 machine system)

24. Numerical example (2 machine system)
Classical model: