1 Review of AC Circuits Smith College, EGR 325 March 27, 2006

2 Objectives Power calculations and terminology Expand understanding of electrical power –from simple linear circuits to –a high voltage power system

3 Overview Basic Circuits Sinusoidal waveform representation Root mean square Phase shift Phasors Complex numbers Complex impedance Electric Power Complex: real & reactive power Power factor and power factor correction

4 ac Waveform

5 How AC is Generated Stator Windings N S Rotor

6 Angle v X N S ff How AC is Generated

7  t v AC Phasor Representation

8 V1V1 V2V2  t v 1 v 2  Reference

9 V1V1 V2V2  t v 1 v 2 

10 Phasors

11 Representing Power

12 Power Calculations P = VI P = I 2 R P = V 2 /R S = VI S = I 2 Z S = V 2 /Z

13 Resistance  Impedance Resistance in  Capacitance in F Inductance in H Z = R + jX

14 Instantaneous Electric Power [p(t)] Fixed average Zero average V I

15 Instantaneous vs. Average Power

16 Instantaneous vs. Average Power Instantaneous power is written as The average of this expression is

17 Real & Reactive Power – Time Domain  t Q(t)   tt p

18 Complex Power V I IMPORTANT  is the power factor angle Real Power Reactive Power

19 Example: Current Flow

20 Example: Power Flow

21 Power System Operations

22 Operating Challenges Load is stochastic and is not controlled Power flows cannot be directed or controlled Electricity cannot be stored Everything happens in real-time Generation can be controlled

23 Power System Variables Generators produce complex power –S = P + jQ –Real power, P, able to perform useful work –Reactive power, Q, supports the system electromagnetically Single system frequency, f Voltage profile, V

24 Real Power Flow – Voltage Relation Power (pu) Voltage (pu) In normal system operation, frequency/real-power dynamics are decoupled from voltage/reactive-power

25 Real Power and Frequency P and f dynamics are coupled –Demand > Supply: frequency will decrease (more energy drained from system than produced, acts like brakes on the turbines) –Supply > Demand: frequency will increase (more energy in the power system than consumed, acts like an accelerator so turbines spin faster) Generation-based frequency regulation –Generator inertia –Generator governors

26 Frequency Problems Imbalances in supply and demand beyond the capabilities of these generator controls –Load may be dropped, or “shed” by operators –Equipment protection may disconnect generators –Operators may disconnect regional tie lines

27 Reactive Power Analogy Voltage and reactive power allow real power to flow Reactive power –Energy stored in capacitance and inductance –Supports the electromagnetic fields along transmission lines –Cannot be transmitted long distances Analogy –Inflatable water pipes

28 Voltage Collapse The real power demanded is above the transfer capability of a transmission line Return to the water pipe analogy –Load draws too much power – dips into the stored reactive power – “collapses” the pipe Equations: P = V*I, I = V/Z –Load wants more power: Decrease apparent impedance (Z), to increase current draw (I), which allows increased P –But, if P at limit, result is to decrease V

29 Power (pu) Voltage (pu) Real Power Flow – Voltage Relation

30 Power System Response to Outages Power flows on the paths of least impedance As elements are removed (fail), the impedance changes and so power flows change  Instantaneously Human and computer monitoring of and reaction to problems is on a much slower timescale