Report

Review of AC Circuits Smith College, EGR 325 March 27, 2006 1 Objectives • Power calculations and terminology • Expand understanding of electrical power – from simple linear circuits to – a high voltage power system 2 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 3 ac Waveform v Vmax v Vmax sin t t 2 f f is thefrequencyof the waveform 4 How AC is Generated Stator N S Rotor Windings 5 How AC is Generated f v 2700 900 N Angle S 1800 3600 X 6 AC Phasor Representation v t v Vmax sin t Vmax Vrms V 2 2 f 377 rad / sec 7 V1 Reference V2 v1 v2 V1 V10 V2 V2 t v1 V1max sin t v2 V2 max sin (t ) 8 V1 Reference V2 v1 v2 V1 V10 V2 V2 t v1 V1max cost v2 V2 max cos(t ) 9 Phasors j V Vm e Vm v1 (t ) Re Ve jt 10 Representing Power 11 Power Calculations • P = VI • P = I2R • P = V2/R • S = VI • S = I2Z • S = V2/Z 12 Resistance Impedance • • • • Resistance in Capacitance in F Inductance in H Z = R + jX 13 Instantaneous Electric Power [p(t)] v(t ) Vmax sin(t ) v i i(t ) I max sin(t ) V t I (t ) v(t ) * i(t ) Vmax I max sin(t ) sin(t ) Vmax I max (t ) [cos( ) cos( 2t ) ] 2 Fixed average Zero average 14 Instantaneous vs. Average Power 1 1 p(t ) Vm I m cos( v i ) Vm I m cos( 2t v i ) 2 2 15 Instantaneous vs. Average Power • Instantaneous power is written as 1 1 p(t ) Vm I m cos( v i ) Vm I m cos( 2t v i ) 2 2 • The average of this expression is 1 P Vm I m cos( v i ) 2 16 Real & Reactive Power – Time Domain Vmax I max p(t ) [cos( vi ) cos( 2t vi ) ] 2 p(t ) P Q(t ) p Q(t) t t 17 Complex Power IMPORTANT is the power factor angle S V I V * I I I S V I V0 I VI * S V I cos j V I sin S P j Q Real Power Reactive Power 18 Example: Current Flow 19 Example: Power Flow 20 Power System Operations 21 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 22 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 23 Real Power Flow – Voltage Relation Voltage (pu) • In normal system operation, frequency/real-power dynamics are decoupled from voltage/reactive-power Power (pu) 24 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 25 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 26 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 27 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 28 Voltage (pu) Real Power Flow – Voltage Relation Power (pu) 29 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 30