1. Bernie Lesieutre UW-Madison
Dynamic Phasors Single Phase Induction Motors point-on-wave disturbances
Bernie Lesieutre
UW-Madison
WECC MVWG, January 2018

2. Modeling Need
We need single-phase motor models for positive sequence simulations.
Three types have been developed as part of load modeling efforts:
Performance model (static)
Dynamic phasor model
Point-on-wave model – detailed, first principles model

3. What Are Dynamic Phasors?
Start with simple example. Consider the basic RL circuit; given a sinusoidal voltage, solve for sinusoidal current.
Given
Solve for
Direct Substitution yields
Tediously solve for

4. What Are Dynamic Phasors?
Transition to phasor notation.
Where is our traditional phasor.

5. What Are Dynamic Phasors?
Solve using phasor analysis. Use Direct Substitution yields Trivially solve for current phasor And time domain current:

6. What Are Dynamic Phasors?
Allow phasors to vary slowly in time Use Direct Substitution yields Which can be simulated. N.B. Assumes phasor vary slowly relative underlying frequency. i.e. not intended to capture sub-cycle phenomenon.

7. What Are Dynamic Phasors?
In practice, we can model multiple harmonics using dynamic phasors:
So far in our work, we’ve only used 1st harmonics for electrical variables, and 0th harmonic for mechanical. More detail is possible.

8. Dynamic phasors - example
The information is contained in the envelope and phase.
The dyanamic phasor model is similar to detailed model, but “averaged” in some sense over a cycle.

9. Single Phase Compressor Simulation Model
Point-on-Wave model in stationary reference frame.

10. Dynamic phasors - conceptually
Represent a waveform as a short-timescale complex Fourier series with time-varying coefficients.
offset term
fundamental term
harmonic terms
For the purpose of implementation in a positive sequence simulator, we ignore offset and harmonic terms.
What are the implications of this?

11. Dynamic phasors - conceptually
Most of the time we expect the fundamental terms to dominate the characteristic. The offset should be small, and we neglect harmonics.
However, we find that this fundamental-terms model does not necessarily represent subcycle phenomena.

12. Back to dynamic Phasors…
How might we represent these effects in a dynamic phasor model? Assuming a voltage-driven model, what should the driving terms look like?

13. Time-windowed voltage phasors
Disturbance at voltage crest
Disturbance at voltage zero
Analyze this with a one-cycle window, sliding Fourier Transform

14. Time-windowed voltage phasors
Disturbance at voltage crest
Disturbance at voltage zero
What to do with this information?

15. Adjustments to simulation/model
Issues
We would like to keep the model suitable for positive sequence simulators.
We don’t want to expand it to include additional terms (harmonics), if we don’t have to.

16. Adjustments to simulation/model
Possible approaches to insert braking torque
Direct in torque: for zero-crossing events, apply short braking torque. (how much?)
Direct in voltage: for zero-crossing events, apply short phase angle pulse to applied voltage.
Fit the model parameters to match voltage crest data.

17. Voltage Phase Implementation
For zero crossing events, include negative phase angle pulse

18. Laboratory Tests and Simulations
Air Conditioner Tests at BPA Facility
Test Point-on-Wave Response
Trane Compressor
Dynamic Phasor Simulations with voltage angle pulse.
Voltage dip to 30-60%% nominal
Recovery voltage at 85% nominal.
(lower value to ensure stalling)
Find fault duration to result in a stall
Voltage ramp test for model fitting.

19. Voltage Ramp Test

20. Current and Power for 55%-85%-5cycle

21. Current and Power for 55%-85%-6cycle

22. Fault Regions, Instantaneous Voltage Dip
Zero Crossing
Peak

23. Needed Continuing Work
More on accommodating fast transients – possibly adding other harmonic terms to the model
Parameter Fitting
Minor adjustments to reference frame

24. Zero Crossing, with ramp
Instantaneous
One cycle ramp