Advanced Power Systems

1. Advanced Power Systems Dr. Kar U of Windsor

2. Dr. Kar271 Essex HallEmail: nkar@uwindsor.ca Office Hour: Thursday, 12:00-2:00 pm http://www.uwindsor.ca/users/n/nkar/88-514.nsf GA: TBA B20 Essex Hall Email: TBA & TBA Office Hour: -----

3. Course Text Book: • 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 (---@uwindsor.ca)Student 2 (---@uwindsor.ca)Student 3 (---@uwindsor.ca) Project Title:Group 2:Student 1 (---@uwindsor.ca)Student 2 (---@uwindsor.ca)Student 3 (---@uwindsor.ca)Project Title: Group 3:Student 1 (---@uwindsor.ca)Student 2 (---@uwindsor.ca)Student 3 (---@uwindsor.ca)

7. Synchronous Machines • Construction • Working principles • Mathematical modeling • Operating characteristics

8. CONSTRUCTION

9. d-axis • Non-uniform air-gap • N • D»10m • q-axis • S • S • Turbine • N • Hydro (water) • 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 Hydrogenerator

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

11. Cylindrical-Rotor Synchronous Generator • Stator • Cylindrical rotor

12. Damper Windings

13. 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. d-axis q-axis Direct & Quadrature Axes 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 jXl Ra jXs + Ia + Vt 0o E d 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. yq Xl Xfd Ifd Ikd1 Imd + + Rfd pykd1 Rkd1 Xmd pyd - pyfd + vfd Ra Id Xkd1 - - d-axis Imd=-Id+Ifd+Ikd1 Imq=-Iq+Ikq1 Vtd - yd Xl Ikq1 Imq + pykq1 Rkq1 Xmq pyq - Xkq1 q-axis Ra Iq Vtq d- and q-Axis Equivalent Circuits

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 Modelingusing the d- and q-axis equivalent circuits • Transmission System Modelingwith 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. ……………..(2) where The Mechanical Equations

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

37. Vt Generator It Infinite Bus System to be Studied

38. Eigen Values: System State Matrix and Eigen Values System State Matrix:

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. Machine Parameters Salient-pole synchronous generator 3kVA, 220V, 4-pole, 60 Hz and 1800 r/min