1. Reference Frame Theory

2. Background: Linear Transformation We have seen that transformation can simplify the problem in power system. Choosing appropriate transformation can significantly reduce the problem. Can you mention the example of the transformation you have ever used?

3. Transformation in Transformator You can always express all variables & parameters in secondary of the transformator into corresponding primer or vice versa.

4. Symmetrical Components It was invented by Fortescue It decomposes unbalance abc phase variables into three symmetrical component Representative symmetrical components

5. Symmetrical Components(2) The transformation matrix is Where and the inverse

6. Example Fault V a1 V a2 V a0 V b0 V c2 V c0 V b1 V b2 V c1

7. We will begin other kind of transformations You will appreciate the benefit of using them later on

8. Clarke’s Transformation It transforms abc quantity into stationary and coincides with phase a-axis and leads the by The equation with The inverse

9. Park’s Transformation It was proposed by Robert H. Park from MIT There are wide ranges of application from machine simulation, control, drives, etc. This transform is one of the most important application for power engineers.

10. Park’s Transformation (2)

11. Park’s Transformation (3) A vector in space can be seen from several coordinate We can change from one coordinate to another coordinate

12. Park’s Transformation (4)

13. Park’s Transformation (5)

14. The abc to dqo transformation

15. The abc to dqo transformation (2)

16. The abc to dqo transformation (3)

18. Commonly Used Reference Frame

19. Transformation Between Reference Frame

20. Application to Voltage Equation

21. Application to Voltage Equation(2)

22. Application to Flux Linkage

23. Application to Flux Linkage (2)

24. Application to Flux Linkage (3)

25. Application to Flux Linkage (4)

26. Application to Flux Linkage (5)

27. Transformation of Balanced Set

28. Transformation of Balanced Set (2)

29. References Power System Stability lecture notes by Dr. Naebboon Hoochareon. Dynamic Simulations of Electric Machinery: Using MATLAB/SIMULINK by Prof. Chee-Mun Ong Analysis of Electric Machinery and Drive Systems by Prof. Paul C. Krause et.al.