1. ECE 576 – Power System Dynamics and Stability Prof. Tom Overbye Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign overbye@illinois.edu 1 Lecture 9: Synchronous Machine Models, Integral Manifolds and Reduced-Order Models

2. Announcements Read Chapter 5 and Appendix A; then Chapters 4 and 6 Homework 2 is posted on the web; it is due on Tuesday Feb 16 Midterm exam is moved to March 31 (in class) because of the ECE 431 field trip on March 17 We will have extra classes on Mondays Feb 15 and March 7 from 1:30 to 3pm in 5070 ECEB – No class on Tuesday Feb 23, Tuesday March 1 and Thursday March 3 2

3. Previous Machine Equations 3 We will next focus on these stator flux equations

4. Chapter 5, Single Machine, Infinite Bus System (SMIB) Book introduces new variables by combining machine values with line values Usually infinite bus angle,  vs, is zero 4

5. “Transient Speed” Mechanical time constant A small parameter Introduce New Constants 5 We are ignoring the exciter and governor for now; they will be covered in much more detail later

6. Stator Flux Differential Equations 6

7. Introduce an exact integral manifold (for any sized ε): Special Case of Zero Resistance Without resistance this is just an oscillator 7

8. Reduced Order Models Before going further, we will consider a formal approach to reduce the model complexity – Reduced order models Idea is to approximate the behavior of fast dynamics without having to explicitly solve the differential equations – Essentially all models have fast dynamics that not explicitly modeled Goal is a more easily solved model (i.e., a reduced order model) without significant loss in accuracy 8

9. Manifolds Hard to precisely define, but "you know one when you see one" – Smooth surfaces – In one dimensions a manifold is a curve without any kinks or self-intersections (line, circle, parabola, but not a figure 8) 9

10. Two-Dimensional Manifolds 10 Images from book and mathworld.wolfram.com

11. Suppose we could find z = h(x) Integral Manifolds 11 Desire is to express z as an algebraic function of x, eliminating dz/dt

12. Integral Manifolds 12 Replace z by h(x) If the initial conditions satisfy h, so z 0 = h(x 0 ) then the reduced equation is exact Chain rule of differentiation

13. Integral Manifold Example Assume two differential equations with z considered "fast" relative to x 13 It is easy to see that z=1 is the equilibrium value of z

14. For this simple system we can get the exact solution 14 Integral Manifold Example

15. Solve for Equilibrium (Steady- state ) Values 15

16. Solve for Remaining Constants Use the initial conditions and derivatives at t=0 to solve for the remaining constants 16

17. Solve for Remaining Constants 17

18. Solution Trajectory in x-z Space Below image shows some of the solution trajectories of this set of equations in the x z space 18 z rapidly decays to 1.0

19. Candidate Manifold Function Consider a function of the form z = h(x) = mx + c We would then have 19 One equation and two unknowns: One solution is m=0, c=1

20. Candidate Manifold Function With the manifold z = 1 we have an exact solution if z = 1.0 since dz/dt = -10z+10 is always zero With this approximation then we simplify as 20 This is exact only if z 0 = 1.0 Exact solution

21. Linear Function, Full Coupling Now consider the linear function 21

22. Linear Function, Full Coupling Which has an equilibrium point at the origin, eigenvalues 1 = -2.3 (the slow mode) and 2 = -8.7 (the fast mode), and a solution of the form Using x 0 = 10 and z 0 = 10, the solution is 22

23. Solution Trajectory in x-z Space 23

24. Linear Function, Full Coupling Same function but change the initial condition to x 0 = 0 and z 0 = 10 Solving for the constants gives In general 24

25. Solution Trajectories in x-z Space 25

26. Candidate Manifold Function Trajectories appear to be heading to origin along a single axis Again consider a candidate manifold function z = h(x) = mx + c Again solve for m dx/dt 26

27. Candidate Manifold Function 27

28. Candidate Manifold Function The two solutions correspond to the two modes The one we've observed is z = -1.3x The other is z = -7.7x To observe this mode select x 0 = 1 and z 0 = -7.7 Zeroing out c 1 and c 3 is clearly a special case 28

29. Eliminating the Fast Mode Going with z = -1.3x we just have the equation This is a simpler model, with the application determining whether it is too simple 29

30. Formal Two Time-Scale Analysis This can be more formalized by introducing a parameter  30 In the previous example we had  = 0.1

31. Formal Two Time-Scale Analysis Using the previous process to get an expression for dx/dt we have For c(  ) = 0 31 Result is complex for larger values since system has complex eigenvaues

32. Formal Two Time-Scale Analysis If z is infinitely fast (  = 0) then z = -z, h(  1 To compute for small  use a power series 32

33. Formal Two Time-Scale Analysis Solving for the coefficients 33

34. Applying to the Previous Example Note the slow mode eigenvalue approximation has changed from -2.3 to -2.2 34

35. General two-time-scale linear system To generalize assume 35 Expression for z is the equilibrium manifold; D must be nonsingular

36. Application to Nonlinear Systems For machine models this needs to be extended to nonlinear systems 36 In general solution is difficult, but there are special cases similar to the stator transient problem

37. Example 37

38. Example Solving we get 38

39. An exact integral manifold (for any sized ε): Special Case of Zero Resistance Without resistance this is just an oscillator 39

40. Elimination of Stator Transients If we assume the stator flux equations are much faster than the remaining equations, then letting  go to zero creates an integral manifold with 40

41. Impact on Studies 41 Image Source: P. Kundur, Power System Stability and Control, EPRI, McGraw-Hill, 1994 Stator transients are not considered in transient stability

42. 3 fast dynamic states, now eliminated 7 not so fast dynamic states 8 algebraic states Machine Variable Summary 42 We'll get to the exciter and governor shortly

43. Direct Axis Equations 43

44. Quadrature Axis Equations 44

45. Swing Equations 45 These are equivalent to the more traditional swing expressions