Lecture 2 Complex Power, Reactive Compensation, Three Phase Professor Tom Overbye Department of Electrical and Computer Engineering ECE 476 POWER SYSTEM ANALYSIS

1 Reading and Homework For lectures 2 through 3 please be reading Chapters 1 and 2 HW 1 is 2.7, 12, 21, 26; due Thursday 9/4

2 Vertical Monopolies Within a particular geographic market, the electric utility had an exclusive franchise Generation Transmission Distribution Customer Service In return for this exclusive franchise, the utility had the obligation to serve all existing and future customers at rates determined jointly by utility and regulators It was a “cost plus” business

3 Vertical Monopolies Within its service territory each utility was the only game in town Neighboring utilities functioned more as colleagues than competitors Utilities gradually interconnected their systems so by 1970 transmission lines crisscrossed North America, with voltages up to 765 kV Economies of scale keep resulted in decreasing rates, so most every one was happy

4 Current Midwest Electric Grid

5 History, cont’d ’s 1970’s brought inflation, increased fossil-fuel prices, calls for conservation and growing environmental concerns Increasing rates replaced decreasing ones As a result, U.S. Congress passed Public Utilities Regulator Policies Act (PURPA) in 1978, which mandated utilities must purchase power from independent generators located in their service territory (modified 2005) PURPA introduced some competition

6 History, cont’d – 1990’s & 2000’s Major opening of industry to competition occurred as a result of National Energy Policy Act of 1992 This act mandated that utilities provide “nondiscriminatory” access to the high voltage transmission Goal was to set up true competition in generation Result over the last few years has been a dramatic restructuring of electric utility industry (for better or worse!) Energy Bill 2005 repealed PUHCA; modified PURPA

7 State Variation in Electric Rates

8 The Goal: Customer Choice

9 The Result for California in 2000/1 OFF

10 The California-Enron Effect Source : RI AK electricity restructuring delayed restructuring no activity suspended restructuring WA OR NV CA ID MT WY UT AZ CO NM TX OK KS NE SD ND MN IA WI MO IL IN OH KY TN MS LA AL GA FL SC NC W VA PA NY VT ME MI NH MA CT NJ DE MD AR HI DC

11 August 14 th, 2003 Blackout

Illinois Electricity Crisis Two main electric utilities in Illinois are ComEd and Ameren Restructuring law had frozen electricity prices for ten years, with rate decreases for many. Prices rose on January 1, 2007 as price freeze ended; price increases were especially high for electric heating customers who had previously enjoyed rates as low as 2.5 cents/kWh Current average residential rate (in cents/kWh) is 10.4 in IL, 8.74 IN, 11.1 WI, 7.94 MO, 9.96 IA, CT, 6.09 ID, in CA, US average

13 Review of Phasors Goal of phasor analysis is to simplify the analysis of constant frequency ac systems v(t) = V max cos(  t +  v ) i(t) = I max cos(  t +  I ) Root Mean Square (RMS) voltage of sinusoid

14 Phasor Representation

15 Phasor Representation, cont’d (Note: Some texts use “boldface” type for complex numbers, or “bars on the top”)

16 Advantages of Phasor Analysis (Note: Z is a complex number but not a phasor)

17 RL Circuit Example

18 Complex Power

19 Complex Power, cont’d

20 Complex Power (Note: S is a complex number but not a phasor)

21 Complex Power, cont’d

22 Conservation of Power At every node (bus) in the system – Sum of real power into node must equal zero – Sum of reactive power into node must equal zero This is a direct consequence of Kirchhoff’s current law, which states that the total current into each node must equal zero. – Conservation of power follows since S = VI*

23 Conversation of Power Example Earlier we found I = 20  -6.9  amps

24 Power Consumption in Devices

25 Example First solve basic circuit

26 Example, cont’d Now add additional reactive power load and resolve

27 Power System Notation Power system components are usually shown as “one-line diagrams.” Previous circuit redrawn Arrows are used to show loads Generators are shown as circles Transmission lines are shown as a single line

28 Reactive Compensation Key idea of reactive compensation is to supply reactive power locally. In the previous example this can be done by adding a 16 Mvar capacitor at the load Compensated circuit is identical to first example with just real power load

29 Reactive Compensation, cont’d Reactive compensation decreased the line flow from 564 Amps to 400 Amps. This has advantages – Lines losses, which are equal to I 2 R decrease – Lower current allows utility to use small wires, or alternatively, supply more load over the same wires – Voltage drop on the line is less Reactive compensation is used extensively by utilities Capacitors can be used to “correct” a load’s power factor to an arbitrary value.

30 Power Factor Correction Example

31 Distribution System Capacitors

32 Balanced 3 Phase (  ) Systems A balanced 3 phase (  ) system has – three voltage sources with equal magnitude, but with an angle shift of 120  – equal loads on each phase – equal impedance on the lines connecting the generators to the loads Bulk power systems are almost exclusively 3  Single phase is used primarily only in low voltage, low power settings, such as residential and some commercial

33 Balanced 3  -- No Neutral Current

34 Advantages of 3  Power Can transmit more power for same amount of wire (twice as much as single phase) Torque produced by 3  machines is constrant Three phase machines use less material for same power rating Three phase machines start more easily than single phase machines

35 Three Phase - Wye Connection There are two ways to connect 3  systems – Wye (Y) – Delta (  )

36 Wye Connection Line Voltages V an V cn V bn V ab V ca V bc -V bn Line to line voltages are also balanced (α = 0 in this case)

37 Wye Connection, cont’d Define voltage/current across/through device to be phase voltage/current Define voltage/current across/through lines to be line voltage/current

38 Delta Connection I ca IcIc I ab I bc IaIa IbIb

39 Three Phase Example Assume a  -connected load is supplied from a 3  13.8 kV (L-L) source with Z = 100  20 

40 Three Phase Example, cont’d

41 Delta-Wye Transformation

42 Delta-Wye Transformation Proof

43 Delta-Wye Transformation, cont’d

44 Three Phase Transmission Line