1. ECE 576 – Power System Dynamics and Stability
Lecture 8: Synchronous Machine Modeling
Prof. Tom Overbye
Dept. of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign

2. Announcements Read Chapter 5 and Appendix A
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 8, 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

3. Sinusoidal Steady-State
Here we consider the application to balanced, sinusoidal conditions

4. Transforming to dq0

5. Simplifying Using d Recall that Hence
These algebraic equations can be written as complex equations,
The conclusion is if we know d, then we can easily relate the phase to the dq values!

6. Summary So Far
The model as developed so far has been derived using the following assumptions
The stator has three coils in a balanced configuration, spaced 120 electrical degrees apart
Rotor has four coils in a balanced configuration located 90 electrical degrees apart
Relationship between the flux linkages and currents must reflect a conservative coupling field
The relationships between the flux linkages and currents must be independent of qshaft when expressed in the dq0 coordinate system

7. Two Main Types of Synchronous Machines
Round Rotor
Air-gap is constant, used with higher speed machines
Salient Rotor (often called Salient Pole)
Air-gap varies circumferentially
Used with many pole, slower machines such as hydro
Narrowest part of gap in the d-axis and the widest along the q-axis

8. Assuming a Linear Magnetic Circuit
If the flux linkages are assumed to be a linear function of the currents then we can write
The rotor self- inductance matrix Lrr is independent of qshaft

9. Inductive Dependence on Shaft Angle
L12 = + maximum
L12 = - maximum

10. Stator Inductances
The self inductance for each stator winding has a portion that is due to the leakage flux which does not cross the air gap, Lls
The other portion of the self inductance is due to flux crossing the air gap and can be modeled for phase a as
Mutual inductance between the stator windings is modeled as
The offset angle is either 2p/3 or -2p/3

11. Conversion to dq0 for Angle Independence

12. Conversion to dq0 for Angle Independence
For a round rotor machine LB is small and hence Lmd is close to Lmq. For a salient pole machine Lmd is substantially
larger

13. Convert to Normalized at f = ws
Convert to per unit, and assume frequency of ws
Then define new per unit reactance variables

14. Example Xd/Xq Ratios for a WECC Case

15. Normalized Equations

16. Key Simulation Parameters
The key parameters that occur in most models can then be defined the following transient values
These values will be used in all the synchronous machine
models
In a salient rotor machine Xmq is small so Xq = X'q; also X1q is small so T'q0 is small

17. Example X'q/Xq Ratios for a WECC Case
About 75% are Clearly Salient Pole Machines!

18. Key Simulation Parameters
And the subtransient parameters
These values will be used in the subtransient machine models. It is common to assume X"d = X"q

19. Internal Variables
Define the following variables, which are quite important in subsequent models
Hence E'q and E'd are scaled flux linkages and Efd is the scaled field voltage

20. Dynamic Model Development
In developing the dynamic model not all of the currents and fluxes are independent
In the book formulation only seven out of fourteen are independent
Approach is to eliminate the rotor currents, retaining the terminal currents (Id, Iq, I0) for matching the network boundary conditions

21. Rotor Currents
Use new variables to solve for the rotor currents

22. Rotor Currents

23. Final Complete Model

24. Final Complete Model
TFW is the friction and windage component that we'll consider later

25. Final Complete Model

26. Single-Machine Steady-State
Previous derivation was done assuming a linear magnetic circuit
We'll consider the nonlinear magnetic circuit (section 3.5) but will first do the steady-state condition (3.6)
In steady-state the speed is constant (equal to ws), d is constant, and all the derivatives are zero
Initial values are determined from the terminal conditions: voltage magnitude, voltage angle, real and reactive power injection

27. Single-Machine Steady-State
The key variable we need to determine the initial conditions is actually d, which doesn't appear explicitly in these equations!

28. Field Current The field current, Ifd, is defined in steady-state as
However, what is usually used in transient stability simulations for the field current is
So the value of Xmd is not needed

29. Determining d without Saturation
In order to get the initial values for the variables we need to determine d
We'll eventually consider two approaches: the simple one when there is no saturation, and then later a general approach for models with saturation
To derive the simple approach we have

30. Determining d without Saturation
Recalling the relation between d and the stator values
We then combine the equations for Vd and Vq and get

31. Determining d without Saturation
Then in terms of the terminal values

32. D-q Reference Frame
Machine voltage and current are “transformed” into the d-q reference frame using the rotor angle, 
Terminal voltage in network (power flow) reference frame are VS = Vt = Vr +jVi

33. A Steady-State Example
Assume a generator is supplying 1.0 pu real power at 0.95 pf lagging into an infinite bus at 1.0 pu voltage through the below network. Generator pu values are Rs=0, Xd=2.1, Xq=2.0, X'd=0.3, X'q=0.5

34. A Steady-State Example, cont.
First determine the current out of the generator from the initial conditions, then the terminal voltage

35. A Steady-State Example, cont.
We can then get the initial angle and initial dq values Or

36. A Steady-State Example, cont.
The initial state variable are determined by solving with the differential equations equal to zero.

37. PowerWorld Two-Axis Model
Numerous models exist for synchronous machines, some of which we'll cover in-depth. The following is a relatively simple model that represents the field winding and one damper winding; it also includes the generator swing eq.
For a salient pole machine, with Xq=X'q, then E'd would rapidly decay to zero

38. PowerWorld Solution of 11.10

39. Nonlinear Magnetic Circuits
Nonlinear magnetic models are needed because magnetic materials tend to saturate; that is, increasingly large amounts of current are needed to increase the flux density
Linear

40. Saturation

41. Saturation Models Many different models exist to represent saturation
There is a tradeoff between accuracy and complexity
Book presents the details of fully considering saturation in Section 3.5
One simple approach is to replace
With

42. Saturation Models In steady-state this becomes
Hence saturation increases the required Efd to get a desired flux
Saturation is usually modeled using a quadratic function, with the value of Se specified at two points (often at 1.0 flux and 1.2 flux)
A and B are determined from the two data points

43. Saturation Example
If Se = 0.1 when the flux is 1.0 and 0.5 when the flux is 1.2, what are the values of A and B using the

44. Implementing Saturation Models
When implementing saturation models in code, it is important to recognize that the function is meant to be positive, so negative values are not allowed
In large cases one is almost guaranteed to have special cases, sometimes caused by user typos
What to do if Se(1.2) < Se(1.0)?
What to do if Se(1.0) = 0 and Se(1.2) <> 0
What to do if Se(1.0) = Se(1.2) <> 0
Exponential saturation models have been used
We'll cover other common saturation approaches in Chapter 5

45. 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

46. 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)

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

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

49. Integral Manifolds Replace z by h(x) Chain rule of differentiation
If the initial conditions satisfy h, so z0 = h(x0) then the reduced equation is exact

50. Integral Manifold Example
Assume two differential equations with z considered "fast" relative to x

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

52. Solve for Equilibrium (Steady-state ) Values

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

54. Solve for Remaining Constants