1. Lecture 8 Transmission Lines, Transformers, Per Unit Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS

2. 1 Announcements Start reading Chapter 3. HW 2 is due now. HW 3 is 4.32, 4.41, 5.1, 5.14. Due September 22 in class. “Energy Tour” opportunity on Oct 1 from 9am to 9pm. Visit a coal power plant, a coal mine, a wind farm and a bio-diesel processing plant. Sponsored by Students for Environmental Concerns. Cost isn’t finalized, but should be between $10 and $20. Contact Rebecca Marcotte at marcott1@illinois.edu for more information or to sign up. marcott1@illinois.edu

3. 2 V, I Relationships, cont’d

4. 3 Equation for Voltage

5. 4 Real Hyperbolic Functions For real x the cosh and sinh functions have the following form:

6. 5 Complex Hyperbolic Functions For x =  + j  the cosh and sinh functions have the following form

7. 6 Determining Line Voltage

8. 7 Determining Line Voltage, cont’d

9. 8 Determining Line Current

10. 9 Transmission Line Example

11. 10 Transmission Line Example, cont’d

12. 11 Transmission Line Example, cont’d

13. 12 Lossless Transmission Lines

14. 13 Lossless Transmission Lines If P > SIL then line consumes vars; otherwise line generates vars.

15. 14 Transmission Matrix Model Oftentimes we’re only interested in the terminal characteristics of the transmission line. Therefore we can model it as a “black box”. VSVS VRVR ++ -- ISIS IRIR Transmission Line

16. 15 Transmission Matrix Model, cont’d

17. 16 Equivalent Circuit Model Next we’ll use the T matrix values to derive the parameters Z' and Y'.

18. 17 Equivalent Circuit Parameters

19. 18 Equivalent circuit parameters

20. 19 Simplified Parameters

21. 20 Simplified Parameters

22. 21 Medium Length Line Approximations

23. 22 Three Line Models

24. 23 Power Transfer in Short Lines Often we'd like to know the maximum power that could be transferred through a short transmission line V1V1 V2V2 ++ -- I1I1 I1I1 Transmission Line with Impedance Z S 12 S 21

25. 24 Power Transfer in Lossless Lines

26. 25 Limits Affecting Max. Power Transfer Thermal limits – limit is due to heating of conductor and hence depends heavily on ambient conditions. – For many lines, sagging is the limiting constraint. – Newer conductors limit can limit sag. For example, in 2004 ORNL working with 3M announced lines with a core consisting of ceramic Nextel fibers. These lines can operate at 200 degrees C. – Trees grow, and will eventually hit lines if they are planted under the line.

27. 26 Other Limits Affecting Power Transfer Angle limits – while the maximum power transfer occurs when line angle difference is 90 degrees, actual limit is substantially less due to multiple lines in the system Voltage stability limits – as power transfers increases, reactive losses increase as I 2 X. As reactive power increases the voltage falls, resulting in a potentially cascading voltage collapse.

28. 27 Transformers Overview Power systems are characterized by many different voltage levels, ranging from 765 kV down to 240/120 volts. Transformers are used to transfer power between different voltage levels. The ability to inexpensively change voltage levels is a key advantage of ac systems over dc systems. In this section we’ll development models for the transformer and discuss various ways of connecting three phase transformers.

29. 28 Transmission to Distribution Transfomer

30. 29 Transmission Level Transformer

31. 30 Ideal Transformer First we review the voltage/current relationships for an ideal transformer – no real power losses – magnetic core has infinite permeability – no leakage flux We’ll define the “primary” side of the transformer as the side that usually takes power, and the secondary as the side that usually delivers power. – primary is usually the side with the higher voltage, but may be the low voltage side on a generator step-up transformer.

32. 31 Ideal Transformer Relationships

33. 32 Current Relationships

34. 33 Current/Voltage Relationships

35. 34 Impedance Transformation Example Example: Calculate the primary voltage and current for an impedance load on the secondary

36. 35 Real Transformers Real transformers – have losses – have leakage flux – have finite permeability of magnetic core 1.Real power losses – resistance in windings (i 2 R) – core losses due to eddy currents and hysteresis

37. 36 Transformer Core losses Eddy currents arise because of changing flux in core. Eddy currents are reduced by laminating the core Hysteresis losses are proportional to area of BH curve and the frequency These losses are reduced by using material with a thin BH curve

38. 37 Effect of Leakage Flux

39. 38 Effect of Finite Core Permeability

40. 39 Transformer Equivalent Circuit Using the previous relationships, we can derive an equivalent circuit model for the real transformer

41. 40 Simplified Equivalent Circuit

42. 41 Calculation of Model Parameters The parameters of the model are determined based upon – nameplate data: gives the rated voltages and power – open circuit test: rated voltage is applied to primary with secondary open; measure the primary current and losses (the test may also be done applying the voltage to the secondary, calculating the values, then referring the values back to the primary side). – short circuit test: with secondary shorted, apply voltage to primary to get rated current to flow; measure voltage and losses.

43. 42 Transformer Example Example: A single phase, 100 MVA, 200/80 kV transformer has the following test data: open circuit: 20 amps, with 10 kW losses short circuit: 30 kV, with 500 kW losses Determine the model parameters.

44. 43 Transformer Example, cont’d

45. 44 Residential Distribution Transformers Single phase transformers are commonly used in residential distribution systems. Most distribution systems are 4 wire, with a multi-grounded, common neutral.

46. 45 Per Unit Calculations A key problem in analyzing power systems is the large number of transformers. – It would be very difficult to continually have to refer impedances to the different sides of the transformers This problem is avoided by a normalization of all variables. This normalization is known as per unit analysis.

47. 46 Per Unit Conversion Procedure, 1  1. Pick a 1  VA base for the entire system, S B 2. Pick a voltage base for each different voltage level, V B. Voltage bases are related by transformer turns ratios. Voltages are line to neutral. 3. Calculate the impedance base, Z B = (V B ) 2 /S B 4. Calculate the current base, I B = V B /Z B 5. Convert actual values to per unit Note, per unit conversion on affects magnitudes, not the angles. Also, per unit quantities no longer have units (i.e., a voltage is 1.0 p.u., not 1 p.u. volts)

48. 47 Per Unit Solution Procedure 1. Convert to per unit (p.u.) (many problems are already in per unit) 2. Solve 3. Convert back to actual as necessary

49. 48 Per Unit Example Solve for the current, load voltage and load power in the circuit shown below using per unit analysis with an S B of 100 MVA, and voltage bases of 8 kV, 80 kV and 16 kV. Original Circuit

50. 49 Per Unit Example, cont’d Same circuit, with values expressed in per unit.

51. 50 Per Unit Example, cont’d

52. 51 Per Unit Example, cont’d To convert back to actual values just multiply the per unit values by their per unit base