Reference Frame Theory

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?

Transformation in Transformator

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

Symmetrical Components• It was invented by Fortescue• It decomposes unbalance abc phase variables into three

symmetrical component• Representative symmetrical components

Symmetrical Components(2)• The transformation matrix is

Where

and the inverse

ExampleFault

Va1 Va2

Va0

Vb0

Vc2 Vc0

Vb1

Vb2

Vc1

We will begin other kind of transformations

You will appreciate the benefit of using them later on

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

The inverse

axis axis

axis 2

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.

Park’s Transformation (2)

Park’s Transformation (3)• A vector in space can be seen from several coordinate

• We can change from one coordinate to another coordinate

v

Park’s Transformation (4)

Park’s Transformation (5)

The abc to dqo transformation

The abc to dqo transformation (2)

The abc to dqo transformation (3)

The abc to dqo transformation (3)

Commonly Used Reference Frame

Transformation Between Reference Frame

Application to Voltage Equation

Application to Voltage Equation(2)

Application to Flux Linkage

Application to Flux Linkage (2)

Application to Flux Linkage (3)

Application to Flux Linkage (4)

Application to Flux Linkage (5)

Transformation of Balanced Set

Transformation of Balanced Set (2)

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.