1. Advanced Power Systems
Dr. Kar
U of Windsor

2. Dr. Kar 271 Essex Hall Office Hour: Thursday, 12:00-2:00 pm
GA: TBA
B20 Essex Hall TBA & TBA Office Hour: -----

3. Course Text Book: Grading Policy: Attendance (5%) Project (20%)
Electric Machinery Fundamentals by Stephen J. Chapman, 4th Edition, McGraw-Hill, 2005
Electric Motor Drives – Modeling, Analysis and Control by R. Krishnan Pren. Hall Inc., NJ, 2001
Power Electronics – Converters, Applications and Design by N. Mohan, J. Wiley & Son Inc., NJ, 2003
Power System Stability and Control by P. Kundur, McGraw Hill Inc., 1993
Research papers
Grading Policy:
Attendance (5%)
Project (20%)
Midterm Exam (30%)
Final Exam (45%)

4. Course Content
Working principles, construction, mathematical modeling, operating characteristics and control techniques for synchronous machines
Working principles, construction, mathematical modeling, operating characteristics and control techniques for induction motors
Introduction to power switching devices
Rectifiers and inverters
Variable frequency PWM-VSI drives for induction motors
Control of High Voltage Direct Current (HVDC) systems

5. Exam Dates
Midterm Exam:
Final Exam:

6. Term Projects
Group 1: Student 1 Student 2 Student 3
Project Title: Group 2: Student 1 Student 2 Student 3 Project Title: Group 3: Student 1 Student 2 Student 3

7. Synchronous Machines Construction Working principles
Mathematical modeling
Operating characteristics

8. CONSTRUCTION

9. Salient-Pole Synchronous Generator
Most hydraulic turbines have to turn at low speeds (between 50 and 300 r/min)
A large number of poles are required on the rotor
Turbine
Hydro (water)
D » 10 m
Non-uniform air-gap N S
d-axis
q-axis
Hydrogenerator

10. Salient-Pole Synchronous Generator
Stator
Salient-pole rotor

11. Cylindrical-Rotor Synchronous Generator
Stator
Cylindrical rotor

12. Damper Windings

13. The rotor of the generator is driven by a prime-mover
Operation Principle
The rotor of the generator is driven by a prime-mover
A dc current is flowing in the rotor winding which produces a rotating magnetic field within the machine
The rotating magnetic field induces a three-phase voltage in the stator winding of the generator

14. Electrical Frequency
Electrical frequency produced is locked or synchronized to the mechanical speed of rotation of a synchronous generator:
where fe = electrical frequency in Hz
P = number of poles
nm= mechanical speed of the rotor, in r/min

15. Direct & Quadrature Axes
d-axis
q-axis
Stator winding N
Uniform air-gap
Stator
Rotor winding
Rotor S
Turbogenerator

16. PU System
Per unit system, a system of dimensionless parameters, is used for computational convenience and for readily comparing the performance of a set of transformers or a set of electrical machines.
Where ‘actual quantity’ is a value in volts, amperes, ohms, etc.
[VA]base and [V]base are chosen first.

17. Classical Model of Synchronous Generator
The leakage reactance of the armature coils, Xl
Armature reaction or synchronous reactance, Xs
The resistance of the armature coils, Ra
If saliency is neglected, Xd = Xq = Xs
jXs
jXl Ra + Ia +
E d
Vt o
Equivalent circuit of a cylindrical-rotor synchronous machine

18. Phasor Diagram
q-axis E
IaXs d Vt
IaXl f
IaRa Ia
d-axis

19. The following are the parameters in per unit on machine rating of a 555 MVA, 24 kV, 0.9 p.f., 60 Hz, 3600 RPM generator
Lad=1.66 Laq=1.61 Ll=0.15 Ra=0.003
When the generator is delivering rated MVA at 0.9 p. f. (lag) and rated terminal voltage, compute the following:
(i) Internal angle δi in electrical degrees
(ii) Per unit values of ed, eq, id, iq, ifd
(iii) Air-gap torque Te in per unit and in Newton-meters

20. (b) Compute the internal angle δi and field current ifd using the following
equivalent circuit

21. Direct and Quadrature Axes
The direct (d) axis is centered magnetically in the center of the north pole
The quadrature axis (q) axis is 90o ahead of the d-axis
q: angle between the d-axis and the axis of phase a
Machine parameters in abc can then be converted into d/q frame using q
Mathematical equations for synchronous machines can be obtained from the d- and q-axis equivalent circuits
Advantage: machine parameters vary with rotor position w.r.t. stator, q, thus making analysis harder in the abc axis frame. Whereas, in the d/q reference frame, parameters are constant with time or q.
Disadvantage: only balanced systems can be analyzed using d/q-axis system

22. d- and q-Axis Equivalent Circuits
Ifd
Xfd
Rfd Xl
pyd
Ikd1
Imd
Vtd Ra Id
Xkd1
Xmd
Rkd1 yq
d-axis
vfd + -
pyfd
pykd1
- yd
pyq
Ikq1
Imq
Vtq Iq
Xkq1
Xmq
Rkq1
q-axis
pykq1
Imd=-Id+Ifd+Ikd1
Imq=-Iq+Ikq1

23. Small disturbances in a power system
Gradual changes in loads
Manual or automatic changes of excitation
Irregularities in prime-mover input, etc.
Importance of steady-state stability
Knowledge of steady-state stability provides valuable information about the dynamic characteristics of different power system components and assists in their design
- Power system planning
- Power system operation
- Post-disturbance analysis

24. Related Terms
Generator Modeling using the d- and q-axis equivalent circuits
Transmission System Modeling with a RL circuit
A Small Disturbance is a disturbance for which the set of equations describing the power system may be linearized for the purpose of analysis
Steady-State Stability is the ability to maintain synchronism when the system is subjected to small disturbances
Loss of synchronism is the usual symptom of loss of stability
Infinite Bus is a system with constant voltage and constant frequency, which is the rest of the power system
Eigen values and eigen vectors are used to identify system steady-state stability condition

25. The Flux Equations

26. Rearranged Flux Linkage equations

27. The Voltage Equations
……………..(1)

28. The Mechanical Equations
where
……………..(2)

29. Linearized Form of the Machine Model
……………..(3)

30. Terminal Voltage
The d- and q-axis components of the machine terminal voltage
can be described by the following equations:
………………………….(4)
where, Vt is the machine terminal voltage in per unit.
The linearized form of Vtd and Vtq are:
……………………….…(5)

31. Substituting ∆Vtd and ∆Vtq in the flux equations:
……..(6)

32. Rearranging the flux equations in a matrix form:
………………...…..(7)
where,

33. and…

34. Flux Linkage Equations (from the d- and q-axis equivalent circuits)
Linearized flux linkage equations:

35. and thus,
………………………………………...(8)

36. : from (8) : inserting (8) into (7) where, : system state matrix
………..(9)
: system state matrix

37. System to be Studied
Infinite Bus
Generator Vt It

38. System State Matrix and Eigen Values

39. Eigen Values Eigen values are the roots of the characteristic equation
Number of eigen values is equal to the order of the characteristic equation or number of state variables
Eigen values describe the system response ( ) to any disturbance

40. Analyzing the Eigen Values of the System State Matrix
Compute the eigen values of the system state matrix, A
The eigen values will give necessary information about the steady-state stability of the system
Stable System: If the real parts of ALL the eigen values are negative
Example:
A system with the above eigen values is on the verge of instability

41. Salient-pole synchronous generator
Machine Parameters
Salient-pole synchronous generator
3kVA, 220V, 4-pole, 60 Hz and 1800 r/min
Machine parameters
Per unit values
d-axis magnetizing reactance, Xmd
1.189
q-axis magnetizing reactance, Xmq
0.7164
Armature leakage reactance, Xl
0.100
Field circuit leakage reactance, Xfd
0.276
First d-axis damper circuit leakage reactance, Xkd1
0.181
First q-axis damper circuit leakage reactance, Xkq1
0.193
Armature winding resistance, Ra
0.0186
Field winding resistance, Rfd
0.0058
First d-axis damper winding resistance, Rkd1
0.062
First q-axis damper winding resistance, Rkq1
0.052