Power in Resistive and Reactive AC Circuits

7/28/2019 Power in Resistive and Reactive AC Circuits

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Power in resistive and reactive AC circuits

Consider a circuit for a single-phase AC power system, where a 120 volt, 60 Hz AC voltagesource is delivering power to a resistive load: (Figurebelow)

Ac source drives a purely resistive load.

In this example, the current to the load would be 2 amps, RMS. The power dissipated at the loadwould be 240 watts. Because this load is purely resistive (no reactance), the current is in phase

with the voltage, and calculations look similar to that in an equivalent DC circuit. If we were to

plot the voltage, current, and power waveforms for this circuit, it would look like Figurebelow.
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Current is in phase with voltage in a resistive circuit.

Note that the waveform for power is always positive, never negative for this resistive circuit.

This means that power is always being dissipated by the resistive load, and never returned to thesource as it is with reactive loads. If the source were a mechanical generator, it would take 240

watts worth of mechanical energy (about 1/3 horsepower) to turn the shaft.

Also note that the waveform for power is not at the same frequency as the voltage or current!

Rather, its frequency is double that of either the voltage or current waveforms. This differentfrequency prohibits our expression of power in an AC circuit using the same complex

(rectangular or polar) notation as used for voltage, current, and impedance, because this form of

mathematical symbolism implies unchanging phase relationships. When frequencies are not thesame, phase relationships constantly change.

As strange as it may seem, the best way to proceed with AC power calculations is to usescalar

notation, and to handle any relevant phase relationships with trigonometry.

For comparison, let's consider a simple AC circuit with a purely reactive load in Figurebelow.

AC circuit with a purely reactive (inductive) load.
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Power is not dissipated in a purely reactive load. Though it is alternately absorbed from and

returned to the source.

Note that the power alternates equally between cycles of positive and negative. (Figure above)

This means that power is being alternately absorbed from and returned to the source. If the

source were a mechanical generator, it would take (practically) no net mechanical energy to turn

the shaft, because no power would be used by the load. The generator shaft would be easy tospin, and the inductor would not become warm as a resistor would.

Now, let's consider an AC circuit with a load consisting of both inductance and resistance in

Figurebelow.

AC circuit with both reactance and resistance.
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At a frequency of 60 Hz, the 160 millihenrys of inductance gives us 60.319 of inductive

reactance. This reactance combines with the 60 of resistance to form a total load impedance of60 + j60.319 , or 85.078 45.152o. If we're not concerned with phase angles (which we'renot at this point), we may calculate current in the circuit by taking the polar magnitude of the

voltage source (120 volts) and dividing it by the polar magnitude of the impedance (85.078 ).

With a power supply voltage of 120 volts RMS, our load current is 1.410 amps. This is the figurean RMS ammeter would indicate if connected in series with the resistor and inductor.

We already know that reactive components dissipate zero power, as they equally absorb power

from, and return power to, the rest of the circuit. Therefore, any inductive reactance in this load

will likewise dissipate zero power. The only thing left to dissipate power here is the resistiveportion of the load impedance. If we look at the waveform plot of voltage, current, and total

power for this circuit, we see how this combination works in Figurebelow.

A combined resistive/reactive circuit dissipates more power than it returns to the source. Thereactance dissipates no power; though, the resistor does.


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As with any reactive circuit, the power alternates between positive and negative instantaneous

values over time. In a purely reactive circuit that alternation between positive and negative

power is equally divided, resulting in a net power dissipation of zero. However, in circuits withmixed resistance and reactance like this one, the power waveform will still alternate between

positive and negative, but the amount of positive power will exceed the amount of negative

power. In other words, the combined inductive/resistive load will consume more power than itreturns back to the source.

Looking at the waveform plot for power, it should be evident that the wave spends more time on

the positive side of the center line than on the negative, indicating that there is more power

absorbed by the load than there is returned to the circuit. What little returning of power thatoccurs is due to the reactance; the imbalance of positive versus negative power is due to the

resistance as it dissipates energy outside of the circuit (usually in the form of heat). If the source

were a mechanical generator, the amount of mechanical energy needed to turn the shaft would bethe amount of power averaged between the positive and negative power cycles.

Mathematically representing power in an AC circuit is a challenge, because the power wave isn'tat the same frequency as voltage or current. Furthermore, the phase angle for power means

something quite different from the phase angle for either voltage or current. Whereas the anglefor voltage or current represents a relativeshift in timingbetween two waves, the phase angle for

power represents a ratio between power dissipated and power returned. Because of this way in

which AC power differs from AC voltage or current, it is actually easier to arrive at figures forpower by calculating withscalarquantities of voltage, current, resistance, and reactance than it

is to try to derive it from vector, orcomplex quantities of voltage, current, and impedance that

we've worked with so far.

REVIEW:

In a purely resistive circuit, all circuit power is dissipated by the resistor(s). Voltage andcurrent are in phase with each other.

In a purely reactive circuit, no circuit power is dissipated by the load(s). Rather, power is

alternately absorbed from and returned to the AC source. Voltage and current are 90 o out

of phase with each other.

In a circuit consisting of resistance and reactance mixed, there will be more power

dissipated by the load(s) than returned, but some power will definitely be dissipated and

some will merely be absorbed and returned. Voltage and current in such a circuit will be

out of phase by a value somewhere between 0o and 90o.

We know that reactive loads such as inductors and capacitors dissipate zero power, yet the fact

that they drop voltage and draw current gives the deceptive impression that they actually do

dissipate power. This phantom power is called reactive power, and it is measured in a unitcalled Volt-Amps-Reactive (VAR), rather than watts. The mathematical symbol for reactive

power is (unfortunately) the capital letter Q. The actual amount of power being used, or

dissipated, in a circuit is called true power, and it is measured in watts (symbolized by the capital


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letter P, as always). The combination of reactive power and true power is called apparent power,

and it is the product of a circuit's voltage and current, without reference to phase angle. Apparent

power is measured in the unit ofVolt-Amps (VA) and is symbolized by the capital letter S.

As a rule, true power is a function of a circuit's dissipative elements, usually resistances (R).

Reactive power is a function of a circuit's reactance (X). Apparent power is a function of acircuit's total impedance (Z). Since we're dealing with scalar quantities for power calculation,

any complex starting quantities such as voltage, current, and impedance must be represented bytheirpolar magnitudes, not by real or imaginary rectangular components. For instance, if I'm

calculating true power from current and resistance, I must use the polar magnitude for current,

and not merely the real or imaginary portion of the current. If I'm calculating apparent powerfrom voltage and impedance, both of these formerly complex quantities must be reduced to their

polar magnitudes for the scalar arithmetic.

There are several power equations relating the three types of power to resistance, reactance, and

impedance (all using scalar quantities):

Please note that there are two equations each for the calculation of true and reactive power.There are three equations available for the calculation of apparent power, P=IE being useful only

for that purpose. Examine the following circuits and see how these three types of powerinterrelate for: a purely resistive load in Figurebelow, a purely reactive load in Figurebelow,and a resistive/reactive load in Figurebelow.

Resistive load only:
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True power, reactive power, and apparent power for a purely resistive load.

Reactive load only:

True power, reactive power, and apparent power for a purely reactive load.

Resistive/reactive load:


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True power, reactive power, and apparent power for a resistive/reactive load.

These three types of power -- true, reactive, and apparent -- relate to one another in trigonometric

form. We call this thepower triangle: (Figurebelow).


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Power triangle relating appearant power to true power and reactive power.

Using the laws of trigonometry, we can solve for the length of any side (amount of any type of

power), given the lengths of the other two sides, or the length of one side and an angle.

REVIEW: Power dissipated by a load is referred to as true power. True power is symbolized by the

letter P and is measured in the unit of Watts (W).

Power merely absorbed and returned in load due to its reactive properties is referred to as

reactive power. Reactive power is symbolized by the letter Q and is measured in the unit

of Volt-Amps-Reactive (VAR).

Total power in an AC circuit, both dissipated and absorbed/returned is referred to as

apparent power. Apparent power is symbolized by the letter S and is measured in the unit

of Volt-Amps (VA).

These three types of power are trigonometrically related to one another. In a righttriangle, P = adjacent length, Q = opposite length, and S = hypotenuse length. The

opposite angle is equal to the circuit's impedance (Z) phase angle.

Calculating power factor

As was mentioned before, the angle of this power triangle graphically indicates the ratio

between the amount of dissipated (orconsumed) power and the amount of absorbed/returnedpower. It also happens to be the same angle as that of the circuit's impedance in polar form.

When expressed as a fraction, this ratio between true power and apparent power is called the

power factorfor this circuit. Because true power and apparent power form the adjacent andhypotenuse sides of a right triangle, respectively, the power factor ratio is also equal to the cosineof that phase angle. Using values from the last example circuit:

It should be noted that power factor, like all ratio measurements, is a unitless quantity.


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For the purely resistive circuit, the power factor is 1 (perfect), because the reactive power equals

zero. Here, the power triangle would look like a horizontal line, because the opposite (reactive

power) side would have zero length.

For the purely inductive circuit, the power factor is zero, because true power equals zero. Here,

the power triangle would look like a vertical line, because the adjacent (true power) side wouldhave zero length.

The same could be said for a purely capacitive circuit. If there are no dissipative (resistive)components in the circuit, then the true power must be equal to zero, making any power in the

circuit purely reactive. The power triangle for a purely capacitive circuit would again be a

vertical line (pointing down instead of up as it was for the purely inductive circuit).

Power factor can be an important aspect to consider in an AC circuit, because any power factorless than 1 means that the circuit's wiring has to carry more current than what would be

necessary with zero reactance in the circuit to deliver the same amount of (true) power to the

resistive load. If our last example circuit had been purely resistive, we would have been able todeliver a full 169.256 watts to the load with the same 1.410 amps of current, rather than the mere

119.365 watts that it is presently dissipating with that same current quantity. The poor power

factor makes for an inefficient power delivery system.

Poor power factor can be corrected, paradoxically, by adding another load to the circuit drawingan equal and opposite amount of reactive power, to cancel out the effects of the load's inductive

reactance. Inductive reactance can only be canceled by capacitive reactance, so we have to add a

capacitorin parallel to our example circuit as the additional load. The effect of these twoopposing reactances in parallel is to bring the circuit's total impedance equal to its total resistance

(to make the impedance phase angle equal, or at least closer, to zero).

Since we know that the (uncorrected) reactive power is 119.998 VAR (inductive), we need to

calculate the correct capacitor size to produce the same quantity of (capacitive) reactive power.Since this capacitor will be directly in parallel with the source (of known voltage), we'll use the

power formula which starts from voltage and reactance:


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Let's use a rounded capacitor value of 22 F and see what happens to our circuit: (Figurebelow)

Parallel capacitor corrects lagging power factor of inductive load. V2 and node numbers: 0, 1,

2, and 3 are SPICE related, and may be ignored for the moment.


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The power factor for the circuit, overall, has been substantially improved. The main current has

been decreased from 1.41 amps to 994.7 milliamps, while the power dissipated at the loadresistor remains unchanged at 119.365 watts. The power factor is much closer to being 1:

Since the impedance angle is still a positive number, we know that the circuit, overall, is stillmore inductive than it is capacitive. If our power factor correction efforts had been perfectly on-

target, we would have arrived at an impedance angle of exactly zero, or purely resistive. If we

had added too large of a capacitor in parallel, we would have ended up with an impedance anglethat was negative, indicating that the circuit was more capacitive than inductive.

A SPICE simulation of the circuit of (Figure above) shows total voltage and total current arenearly in phase. The SPICE circuit file has a zero volt voltage-source (V2) in series with the

capacitor so that the capacitor current may be measured. The start time of 200 msec ( instead of0) in the transient analysis statement allows the DC conditions to stabilize before collecting data.

See SPICE listing pf.cir power factor.

pf.cir power factorV1 1 0 sin(0 170 60)C1 1 3 22uF


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v2 3 0 0L1 1 2 160mHR1 2 0 60# resolution stop start.tran 1m 200m 160m.end

The Nutmeg plot of the various currents with respect to the applied voltage V total is shown in

(Figurebelow). The reference is Vtotal, to which all other measurements are compared. This is

because the applied voltage, Vtotal, appears across the parallel branches of the circuit. There is nosingle current common to all components. We can compare those currents to V total.

Zero phase angle due to in-phase Vtotaland Itotal . The lagging IL with respect to Vtotal is corrected

by a leading IC .

Note that the total current (Itotal) is in phase with the applied voltage (V total), indicating a phase

angle of near zero. This is no coincidence. Note that the lagging current, IL of the inductor wouldhave caused the total current to have a lagging phase somewhere between (Itotal) and IL. However,

the leading capacitor current, IC, compensates for the lagging inductor current. The result is a

total current phase-angle somewhere between the inductor and capacitor currents. Moreover, thattotal current (Itotal) was forced to be in-phase with the total applied voltage (V total), by the

calculation of an appropriate capacitor value.

Since the total voltage and current are in phase, the product of these two waveforms, power, will

always be positive throughout a 60 Hz cycle, real power as in Figureabove. Had the phase-anglenot been corrected to zero (PF=1), the product would have been negative where positive portions

of one waveform overlapped negative portions of the other as in Figure above. Negative power is

fed back to the generator. It cannont be sold; though, it does waste power in the resistance ofelectric lines between load and generator. The parallel capacitor corrects this problem.
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Note that reduction of line losses applies to the lines from the generator to the point where the

power factor correction capacitor is applied. In other words, there is still circulating current

between the capacitor and the inductive load. This is not normally a problem because the powerfactor correction is applied close to the offending load, like an induction motor.

It should be noted that too much capacitance in an AC circuit will result in a low power factorjust as well as too much inductance. You must be careful not to over-correct when adding

capacitance to an AC circuit. You must also be very careful to use the proper capacitors for thejob (rated adequately for power system voltages and the occasional voltage spike from lightning

strikes, for continuous AC service, and capable of handling the expected levels of current).

If a circuit is predominantly inductive, we say that its power factor is lagging(because thecurrent wave for the circuit lags behind the applied voltage wave). Conversely, if a circuit is

predominantly capacitive, we say that its power factor is leading. Thus, our example circuit

started out with a power factor of 0.705 lagging, and was corrected to a power factor of 0.999

lagging.

REVIEW:

Poor power factor in an AC circuit may be corrected, or re-established at a value close

to 1, by adding a parallel reactance opposite the effect of the load's reactance. If the load's

reactance is inductive in nature (which is almost always will be), parallel capacitance iswhat is needed to correct poor power factor.

AC power

From Wikipedia, the free encyclopedia

Jump to: navigation, search

This article deals with power in AC systems. See Mains electricity for information on

utility-supplied AC power.
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Usually hidden from the unaided eye, the blinking of (non-incandescent) lighting

powered by AC mains is revealed in this motion-blurred long exposure of city lights.

Light is emitted twice each cycle.

Powerin an electric circuit is the rate of flow of energy past a given point of the circuit. In

alternating currentcircuits, energy storage elements such asinductance and capacitance mayresult in periodic reversals of the direction of energy flow. The portion of power that, averagedover a complete cycle of the AC waveform, results in net transfer of energy in one direction is

known as real power. The portion of power due to stored energy, which returns to the source in

each cycle, is known as reactive power.

Contents[hide]

1 Real, reactive, and apparentpowers

2 Power factor

3 Reactive power

4 Reactive power control

5 Unbalanced polyphase systems

6 Basic calculations using realnumbers

7 Multiple frequency systems

8 See also

9 References

[edit] Real, reactive, and apparent powers

In a simple alternating current (AC) circuit consisting of a source and a linear load, both the

current and voltage are sinusoidal. If the load is purely resistive, the two quantities reverse their

polarity at the same time. At every instant the product of voltage and current is positive,indicating that the direction of energy flow does not reverse. In this case, only real power is

transferred.

If the loads are purely reactive, then the voltage and current are 90 degrees out of phase. For half

of each cycle, the product of voltage and current is positive, but on the other half of the cycle, theproduct is negative, indicating that on average, exactly as much energy flows toward the load as

flows back. There is no net energy flow over one cycle. In this case, only reactive energy flows

there is no net transfer of energy to the load.
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Practical loads have resistance, inductance, and capacitance, so both real and reactive power will

flow to real loads. Power engineers measure apparent power as the magnitude of the vector sum

of real and reactive power. Apparent power is the product of the root-mean-square of voltage andcurrent.

Engineers care about apparent power, because even though the current associated with reactivepower does no work at the load, it heats the wires, wasting energy. Conductors, transformers and

generators must be sized to carry the total current, not just the current that does useful work.

Another consequence is that adding the apparent power for two loads will not accurately give the

total apparent power unless they have the same displacement between current and voltage (the

samepower factor).

Conventionally, capacitors are considered to generate reactive power and inductors to consumeit. If a capacitor and an inductor are placed in parallel, then the currents flowing through the

inductor and the capacitor tend to cancel rather than add. This is the fundamental mechanism for

controlling the power factor in electric power transmission; capacitors (or inductors) are insertedin a circuit to partially cancel reactive power 'consumed' by the load.

The complex power is the vector sum of real and reactive power. The apparent

power is the magnitude of the complex power.

Real power (P)

Reactive power (Q)

Complex power (S)

Apparent Power (|S|)

Phase of Current ()

Engineers use the following terms to describe energy flow in a system (and assign each of them adifferent unit to differentiate between them):

Real power (P) or active power[1]: watt [W] Reactive power (Q): volt-ampere reactive [var]

Complex power (S): volt-ampere [VA]
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Apparent Power (|S|), that is, the absolute value of complex power S: volt-ampere [VA]

Phase of Voltage Relative to Current (), the angle of difference (indegrees) between voltage and current; Current lagging Voltage (Quadrant IVector), Current leading voltage (Quadrant IV Vector)

In the diagram,Pis the real power, Q is the reactive power (in this case positive), Sis the

complex power and the length ofSis the apparent power.

Reactive power does not transfer energy, so it is represented as the imaginary axis of the vector

diagram. Real power moves energy, so it is the real axis.

The unit for all forms of power is the watt (symbol: W), but this unit is generally reserved for

real power. Apparent power is conventionally expressed involt-amperes(VA) since it is theproduct ofrmsvoltage and rms current. The unit for reactive power is expressed as VAR, which

stands forvolt-amperes reactive. Since reactive power transfers no net energy to the load, it is

sometimes called "wattless" power. It does, however, serve an important function in electricalgrids and its lack has been cited as a significant factor in theNortheast Blackout of 2003.[2]

Understanding the relationship among these three quantities lies at the heart of understanding

power engineering. The mathematical relationship among them can be represented by vectors or

expressed using complex numbers, S=P+ jQ (where j is the imaginary unit).

[edit] Power factorMain article: Power factor

The ratio between real power and apparent power in a circuit is called thepower factor. It's a

practical measure of the efficiency of a power distribution system. For two systems transmitting

the same amount of real power, the system with the lower power factor will have highercirculating currents due to energy that returns to the source from energy storage in the load.

These higher currents produce higher losses and reduce overall transmission efficiency. A lower

power factor circuit will have a higher apparent power and higher losses for the same amount ofreal power.

The power factor is one when the voltage and current are inphase. It is zero when the current

leads or lags the voltage by 90 degrees. Power factors are usually stated as "leading" or "lagging"

to show the sign of the phase angle of current with respect to voltage.

Purely capacitive circuits cause reactive power with the current waveform leading the voltagewaveform by 90 degrees, while purely inductive circuits consume reactive power with the

current waveform lagging the voltage waveform by 90 degrees. The result of this is that

capacitive and inductive circuit elements tend to cancel each other out.
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Where the waveforms are purely sinusoidal, the power factor is the cosine of the phase angle ()

between the current and voltage sinusoid waveforms. Equipment data sheets and nameplates

often will abbreviate power factor as " " for this reason.

Example: The real power is 700 W and the phase angle between voltage and current is 45.6.

The power factor is cos(45.6) = 0.700. The apparent power is then: 700 W / cos(45.6) = 1000VA.[3]

[edit] Reactive powerMain article: Volt-ampere reactive

Reactive power flow is needed in an alternating-current transmission system to support the

transfer of real power over the network. In alternating current circuits, energy is stored

temporarily in inductive and capacitive elements, which can result in the periodic reversal of thedirection of energy flow. The portion of power flow remaining, after being averaged over a

complete AC waveform, is the real power; that is, energy that can be used to do work (for

example overcome friction in a motor, or heat an element). On the other hand, the portion ofpower flow that is temporarily stored in the form of magnetic or electric fields, due to inductiveand capacitive network elements, and then returned to source, is known as reactive power.

AC connected devices that store energy in the form of a magnetic field include inductive devices

called reactors, which consist of a large coil of wire. When a voltage is initially placed across the

coil, a magnetic field builds up, and it takes a period of time for the current to reach full value.This causes the current to lag behind the voltage in phase; hence, these devices are said to

absorb reactive power.

A capacitor is an AC device that stores energy in the form of an electric field. When current is

driven through the capacitor, it takes a period of time for a charge to build up to produce the fullvoltage difference. On an AC network, the voltage across a capacitor is constantly changing

the capacitor will oppose this change, causing the voltage to lag behind the current. In other

words, the current leads the voltage in phase; hence, these devices are said togenerate reactivepower.

Energy stored in capacitive or inductive elements of the network give rise to reactive power

flow. Reactive power flow strongly influences the voltage levels across the network. Voltagelevels and reactive power flow must be carefully controlled to allow a power system to be

operated within acceptable limits.

[edit] Reactive power control

Transmission connected generators are generally required to support reactive power flow. For

example on the Great Britain transmission system generators are required by the Grid Code

Requirements to supply their rated power between the limits of 0.85 power factor lagging and

0.90 power factor leading at the designated terminals. The system operator will performswitching actions to maintain a secure and economical voltage profile while maintaining a

reactive power balance equation:
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Generator_MVARs + System_gain + Shunt_capacitors = MVAR_Demand + Reactive_losses +

Shunt_reactors

The System gain is an important source of reactive power in the above power balance equation,which is generated by the capacitive nature of the transmission network itself. By making

decisive switching actions in the early morning before the demand increases, the system gain canbe maximized early on, helping to secure the system for the whole day.

To balance the equation some pre-fault reactive generator use will be required. Other sources ofreactive power that will also be used include shunt capacitors, shunt reactors, Static VAR

Compensators and voltage control circuits.

[edit] Unbalanced polyphase systems

While real power and reactive power are well defined in any system, the definition of apparent

power for unbalanced polyphase systems is considered to be one of the most controversial topics

in power engineering. Originally, apparent power arose merely as a figure of merit. Majordelineations of the concept are attributed to Stanley'sPhenomena of Retardation in the Induction

Coil(1888) and Steinmetz's Theoretical Elements of Engineering(1915). However, with the

development ofthree phasepower distribution, it became clear that the definition of apparent

power and the power factor could not be applied to unbalancedpolyphase systems. In 1920, a"Special Joint Committee of the AIEE and the National Electric Light Association" met to

resolve the issue. They considered two definitions:

that is, the quotient of the sums of the real powers for each phase over the sum of the apparent

power for each phase.

that is, the quotient of the sums of the real powers for each phase over the magnitude of the sumof the complex powers for each phase.

The 1920 committee found no consensus and the topic continued to dominate discussions. In

1930 another committee formed and once again failed to resolve the question. The transcripts oftheir discussions are the lengthiest and most controversial ever published by the AIEE (Emanuel,1993). Further resolution of this debate did not come until the late 1990s.
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[edit] Basic calculations using real numbers

A perfect resistor stores no energy, so current and voltage are in phase. Therefore there is noreactive power and . Therefore for a perfect resistor

For a perfect capacitor or inductor there is no net power transfer, so all power is reactive.

Therefore for a perfect capacitor or inductor:

WhereXis the reactanceof the capacitor or inductor.

If X is defined as being positive for an inductor and negative for a capacitor then we can removethe modulus signs from Q and X and get

Instantaneous power is defined as:

where v(t) and i(t) are the time varying voltage and current waveforms.

This definition is useful because it applies to all waveforms, whether they are sinusoidal or not.This is particularly useful in power electronics, where nonsinusoidal waveforms are common.

In general, we are interested in the real power averaged over a period of time, whether it is a low

frequency line cycle or a high frequency power converter switching period. The simplest way to

get that result is to take the integral of the instantaneous calculation over the desired period.
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This method of calculating the average power gives the real power regardless of harmonic

content of the waveform. In practical applications, this would be done in the digital domain,where the calculation becomes trivial when compared to the use of rms and phase to determine

real power.

[edit] Multiple frequency systems

Since an RMS value can be calculated for any waveform, apparent power can be calculated from

this.

For real power it would at first appear that we would have to calculate loads of product terms andaverage all of them. However if we look at one of these product terms in more detail we come toa very interesting result.

however the time average of a function of the form cos(t+k) is zero provided that is nonzero.

Therefore the only product terms that have a nonzero average are those where the frequency ofvoltage and current match. In other words it is possible to calculate real (average) power by

simply treating each frequency separately and adding up the answers.

Furthermore, if we assume the voltage of the mains supply is a single frequency (which it usually

is), this shows that harmonic currentsare a bad thing. They will increase the rms current (sincethere will be non-zero terms added) and therefore apparent power, but they will have no effect on

the real power transferred. Hence, harmonic currents will reduce the power factor.

Harmonic currents can be reduced by a filter placed at the input of the device. Typically this willconsist of either just a capacitor (relying on parasitic resistance and inductance in the supply) or a

capacitor-inductor network. An activepower factor correctioncircuit at the input would

generally reduce the harmonic currents further and maintain the power factor closer to unity.

_[Contents]

Search
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Copyright 2005 jsd

* Contents1 Power Grid Principles

2 Routine Feedback Loops

2.1 Voltage Stability

2.2 Phase : Load Sharing

2.3 Phase Locking

2.4 Frequency : Total Power Regulation

2.5 Buying and Selling Power

2.6 Timekeeping

3 Things I Dont Understand

4 Non-Routine Feedback : Breakers etc.

5 Remarks on the Blackout

A Basic Notions and Jargon

B References

1 Power Grid Principles

Lets talk about the physics and engineering of the North American electric power grid. Some

useful notions and jargon can be found in appendix A.

As discussed in section 2, it is important to realize that there are several different feedback loopsinvolved. They involve different physics, and have different roles to play in stabilizing (or

destabilizing) the grid. These loops include

Voltage Stability (section 2.1). Timescale: fast (a few cycles or less). Role:keeping generator output in equilibrium with the load. Stability: could bepositive or negative.

Phase (section 2.2 and section 2.3). Timescale: fast. Role: allowing onegenerator to share the load with other generators.

Frequency (section 2.4). Timescale: slow (seconds and longer). Role:regulation of total power generation.

Long-term frequency (section 2.6). Timescale: hours and longer. Role:Timekeeping.
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2 Routine Feedback Loops

2.1 Voltage Stability

Lets start by considering the phenomenon the industry calls voltage stability or its opposite,

voltage collapse. Voltage stability plays the role of keeping generator output in equilibriumwith the load.

Consider the diagram in figure 1. It shows a generator (which includes some nonzero output

impedance) driving a transmission network in series with the useful load.

_________________| || output Z

----- impedance Z--- |___________----- (A) |

--- Z| Z| |__________| (B) || Z load| Z impedance|________________________________________|

(... generator ...) (transmission) (load)

Figure 1: Power Grid as Thvenin Voltage Divider

We would like this circuit to be stable. Specifically, we would like the voltage at points (A) and

(B) to be well-behaved with respect to changes in the load impedance. For an ordinary voltage-divider made of resistors, stability would be guaranteed ... but the power grid is much morecomplicated.

A necessary condition for stability is that a reduction in voltage should result in a reduction in

current. There are plenty of loads on the power grid that do not behave this way.

Perhaps the most obvious examples are the regulated power supplies incomputers and other electronic equipment. If the computer is drawing 40amps from its 5V power supply, that is 200 watts, and the regulator isdesigned to make sure that changes in the line voltage have no effect on

this. Therefore, by conservation of energy, if the line voltage drops, thecurrent that the power supply draws from the line goes up ... which makes anegative contribution to the stability of the power grid.

Similar effects arise naturally in more primitive equipment. The saturabletransformers that are used throughout the power grid serve as voltageregulators, which means that even a purely resistive load in your house canmake a negative contribution to stability, because if the generator voltagedrops, the transformer serving your house will come out of saturation a little
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bit, keeping your voltage relatively constant at the expense of drawing morecurrent from the grid.

As another example, if you decrease the line voltage on a synchronous motor(clock motor), the rotation rate will not change and the load will not change,so again by conservation of energy the line current must go up. Non-

synchronous motors behave more-or-less the same, although the argument isa little more complicated, because the motor might slow down a little bit,reducing somewhat the energy delivered to the useful load. On the otherhand, when the motor slows down, its power factor probably gets worse,adding a reactive component to the current drawn from the grid.

On top of all that, keep in mind that under conditions when lowering the voltage at point (B)

results in unchanged power flow and therefore increase current flow past point (B), the power

flow past point (A) actually goes up, because of increasedI2R losses in the transmissionimpedance. Similarly whenever the load factor gets worse it increases theI2R losses.

As a result of all this, the generator could well be looking into a load that has a negative small-signal impedance. Actually the analysis is easier if we consider the Norton equivalent, namely

figure 2, in which we speak of conductances instead of impedances. We care about the small-signal conductance, defined as G := I/V(notI/V). Many people find zero conductance easier to

think about than infinite impedance. Infinities are always annoying.

_________________ _________ _________| | | || | | |

constant- ^^^ output G G Gcurrent ^ ^ conductance G G Gsource ^^^ | | |

|_________________|_________|_________|

(... generator ...) (transmission) (load)

Figure 2: Power Grid as Norton Current Divider

In some sense the easiest thing to think about is a zero-conductance load, which carries the sameamount of current even if you vary the voltage. A slightly positive or slightly negative

conductance isnt much different from zero. The worst case is when the load conductance

becomes so strongly negative that the parallel combination of the load and everything else

becomes zero. In that case, the circuit becomes unstable: you have a constant-current sourcelooking into zero conductance, which cant be good. We see that a necessary condition for

stability is to keep the generators output conductance sufficiently large to overcome whatevernegative terms the rest of the circuit might contribute. (To say the same thing the other way: thegenerator needs to have a small output impedance.)

At the next level of detail, note that theI2R terms are nonlinear, so a large voltage swing at point

(A) can lead to instability in cases where a smaller swing would not. According to reference 1,

the First Energy system for years considered five-percent voltage swings to be the largest
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allowable, but at some point they decided to cut corners and allow ten-percent voltage swings,

and this contributed to voltage collapse that was part of the 14-August-2003 blackout scenario.

Also the reactiveI2R terms make a nonlinear contribution, so it is a good idea to keep them small(using capacitors or whatever). According toreference 1, First Energy didnt do a very good job

of that, either.

Of course the idea of negative conductance only applies to the small-signal conductance. In

contrast, if you cut the customers voltage all the way to zero, then surely the current flow andpower flow go to zero as well. So at some point, for abnormally large voltage swings, the system

is stable. To say the same thing another way, rolling blackouts make more sense than brownouts.

2.2 Phase : Load Sharing

Phase is used for balancing the short-term load-sharing between generators.

By way of analogy, suppose you have two horses harnessed to the same wagon. You want themto share the load. There is enough stiffness in the harness so that if one horse tries to get ahead of

the other, it will take more than its share of the load. The horse will feel this instantly and will be

retarded by the extra load.

The rotational version of this is shown in figure 3. There are three disks rotating coaxially CCW.The two outer disks (yellow) are generators. The middle disk (green) is the load. If the turbine

driving the left-hand generator is a little friskier than the turbine driving the right-hand generator,

the left one will become slightly advanced in phase and will take more of the load. Power istransferred to the load by rubber bands shown in red. You can see that the rubber band attached

to the left disk has two advantages: is more stretched and has better leverage than the other one.

Figure 3: Phase : Load Sharing between Generators

As you can see, this short-term load balancing is achieved by basic physics. No signaling

network is required; the power lines themselves carry all the information that is needed. No

governors or servos are involved. No software, no nuthin, just physics.

2.3 Phase Locking

The same mechanism ensures that the two generators will remain synchronous. The phase signal

that tells one or the other to pick up the slack involves a boundedamount of phase, so the long-

term frequency is exactly the same for both generators.
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This is of course related to the phase sensitivity discussed in section 2.2, but it is not a trivial

corollary. It didnt have to be that way. You could imagine other load-balancing schemes, such

as two generators each coupled to the common load by a fluid clutch, so that they would sharethe load but not be synchronous.

Also, if the load mismatch becomes extreme, one generator could slip 2 of phase relative to theother. In the industry this is called pole slip and it is a Bad Thing for a number of reasons,

including the fact that it puts large, rapidly-changing mechanical stresses on the generators.

The possibility of pole slip drives home the point that phase locking shouldnt be taken for

granted.

2.4 Frequency : Total Power Regulation

Lets consider again what happens if the magnitude of the load increases. Conservation of energytells us there are really only two possible responses: either reduce the load somehow, or increase

the amount of power generated.

We saw in section 2.1 that reducing the load is not as simple as it sounds. And in any case

customers more-or-less expect to be able to draw greater or lesser amounts of power at theirdiscretion, and they expect the grid to accommodate this.

The grid is rigged up so that when the load increases, it causes all the generators to slow down.

This is not just a phase shift with a bounded amount of phase as mentioned in section 2.2, but

rather a full-blown frequency change involving an unbounded amount of phase.

This frequency change is detected (typically using the finest technology from the year 1788,

namely a flyball governor) and that leads to the opening of the throttle on the turbine driving thegenerator, so that power production comes into equilibrium with the increased load.

Now here comes a slightly sneaky bit: you could imaging putting a PID(proportional/integral/differential) controller on each throttle to drive the frequency error right to

zero, but thats not whats done! There is intentionally a system-wide (specifically, island-wide)

frequency error, and that is used as a signal to all producers to increase their power output by a

certain percentage of their capacity. This is called frequency droop. There is a system-wideagreement as to how each producer should respond to a given amount of droop. Without such an

agreement, whoever had the least-droopy generator would wind up picking up all the variations

in load for the entire island.

Again, this functionality is not entirely independent of the phase sensitivity discussed in section2.2, but its not a direct corollary either. You could have had one without the other. Frequency

droop depends on additional functionality in the throttle servos, and on agreed-upon semantics.

The alternative would have been to use non-droopy servos ... plus some nice out-of-bandsignaling scheme to respond to changes in overall load.
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2.5 Buying and Selling Power

The folks who own the generators can look at the volts and amps (or, analogously, the stretch

and angle of the rubber bands infigure 3) and see how much power each one is generating. Then,depending on who wants to buy and who wants to sell, they can apply an upward or downward

bias to the throttle settings on the turbines that drive the generators. This happens on a muuuuchslower timescale than the load-shifting mentioned in section 2.2 and the total power regulationmentioned in section 2.4.

2.6 Timekeeping

Customers like to use the 60Hz power as a reference for timekeeping. The classic electric clock

uses a synchronous motor.

This is possible because of yet another feedback loop, a very slow feedback loop implemented asfollows: Grid operators count cycles. Every so often they compare this to official time signals,

originally the Naval Observatory but nowadays just GPS. When the discrepancy starts gettingtoo big, due mainly to the droop mentioned in itemsection 2.4, somebody sends out an AGC

(automatic generator control) signal that tells everybody to bias their throttles a little bit to makeup for lost time. There are also manual procedures for doing this. In any case, this happens on a

much slower timescale than load-shifting (section 2.2 and total power regulation (section 2.4).

The clock error can be as much as ten seconds either way before they necessarily do anythingabout it. However, the long-term frequency and timekeeping should be excellent. The phase error

is bounded, so the frequency error goes to zero if you wait long enough. Unless there is a

blackout :-(.

3 Things I Dont Understand Consider the case of two transmission lines running in parallel, both bringing

power from subsystem A to subsystem B. I dont have a clear understandingof how it is decided which line carries what percentage of the power. (Thisstands in contrast to load-sharing involving two generators, as discussed insection 2.2. Generators can change their phase, and generators havethrottles, whereas transmission lines dont.) We know there are big step-up/step-down transformers at the input/output of these lines; perhaps theturns-ratio of these transformers is adjustable.

Reference 1 talks a lot about "zone-3 impedance relays". Evidently they aresome kind of fancy circuit breakers, but I dont know what problem they aresupposed to solve (not counting problems that are better solved by othermeans). In particular, I dont know what variables factor into the decision totrip such a breaker.

On page 88 the report says ... many regions in the United States and Canada haveeliminated the use of zone 3 relays on 230-kV and greater lines which raises the

question, if they can be eliminated, why were they installed in the first place?
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I dont understand the interplay between voltage stability (section 2.1) andphase-based load sharing (section 2.2). We have two feedback loops withsimilar timescales, trying to solve problems that are neither fully identical norfully independent. This seems like a recipe for chaos.

4 Non-Routine Feedback : Breakers etc.

Circuit breakers are important and necessary ... but they are not supposed to trip during routine,

normal operations.

Also we should not fall into the trap of thinking that a circuit breaker is the solution to aproblem, because typically there are multiple ways of solving any given problem, including more

than one way to employ circuit breakers.

For example, consider a long transmission line. There are two basic ways in which it could

become overloaded: (a) there could be a short to ground (e.g. due to contact between a tree and a

sagging wire) or (b) it could be that the load at the downstream end is drawing too much powerfrom the generators at the upstream end. These two cases are readily distinguishable: In case (a),

there is a big discrepancy between the power entering upstream and power leaving downstream,while in case (b) there is not. It is important to make this distinction and respond accordingly. In

case (a), you need to trip the breakers at both ends of the line. In case (b), in my opinion tripping

those two breakers is generally not what you want, and should be done only as a last resort. Thefirst resort should be to decrease the net demand imposed by the downstream subsystem, perhaps

by increasing generation on the downstream side, or buying power through another link, or by

shedding some percentage of the load (by blacking out selected customers).

This is important, because on 14 August 2003, the early stages of the blackout involved

transmission lines tripping off line. (It was only in the later stages that generators tripped off.)Only one of these was due to a fault to ground; the rest were due to overloads. Every time a link

tripped off, it increased the load on remaining links, leading to further overloads ... and you caneasily see how a runway situation could result.

Similar remarks apply to the breakers that trip a generator off line in response to overload

conditions. Overload could be manifested as overcurrent, undervoltage, or underfrequency. (One

also needs to respond to overvoltage and/or overfrequency, but those are not typical of anoverload situation, and are presumably less common.) When faced with an overload, the first

response should be to throttle up other nearby sources and/or to buy power from more distant

sources. The second response should be to shed some fraction of the load, by blacking out

selected customers. Either of these first two responses leaves the remaining customers in bettershape. The last resort is to trip the overloaded generator off line. This is better than allowing it to

be damaged by the overload, but still it is a bad thing because it increases the load on the rest of

the system, which is a recipe for a runway situation.

You dont need to be a genius to see that the system was badly designed. It needed more actions

that reduced real load, and fewer actions that reduced generation and transmission capacity.
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5 Remarks on the Blackout

Reference 1 has remarkably little to say about why the 14 August 2003 blackout lasted so long.There is a long discussion of how the blackout started, and a long discussion of how it spread,

but almost nothing about how long it lasted, or why.

The report has only a handful of references to black-start capability, and most of those are

merely a passing mention of generating plants that completely lack the capability.

I have seen many reports (e.g. reference 2) of subsystems that could not return to normal

operation with stable voltage and frequency until they could tie into a larger, already-functioning

grid. This seems like grounds for expecting larger and longer-lasting blackouts in the future.Long Island is literally and physically an island, and I dont see why it wouldnt have been

possible to make it electrically an island, bring up the on-island generating capacity, and have

most customers service restored within minutes. The switchgear needed to accomplish this

presumably exists, and all that is lacking is the control network and planning necessary to use the

switchgear effectively.

Similar remarks apply to Manhattan. I am particularly unimpressed by Con Edisons policy of

not accepting any responsibility for the blackout (reference 3). We agree that the blackout didnt

startin Con Ed territory, but Con Ed remains responsible for the fact that it spread into theirterritory. If theyd played their cards right, it wouldnt have. They also remain responsible for the

fact that customers were without service for periods ranging from 12 to 29 hours. Why couldnt

they have restored service within 29 minutes? Why did it take 29 hours?

The importance of black-start capability had been documented in industry standards for years

before the 2003 blackout. Alas the industry has a track record of not adhering to its own

standards as discussed in reference 1.

More generally, readingreference 1creates the strong impression that the system containedmany features that serve to minimize the number of small outages ... even though the same

features increase the risk of large, long-lasting outages. I have no idea whether this is

intentional ... I guess such a strategy would make sense if the penalty for a small outage was the

same as the penalty for a huge outage.

One also gets the impression that many of the feedback loops and system protection mechanisms

were both myopic and disproportionate. That is, when faced with a moderately bad situation, the

mechanisms acted in such a way as to make the local situation moderately better while making

the overall situation dramatically worse. This includes responding to a 10% overload on a majortransmission cable by tripping off the entire cable, instead of shedding 10% or more of the

downstream load. This happened repeatedly as the blackout was spreading.

It is sometimes argued that it is impossible to put intelligent (non-myopic) control systems intothe power grid. This argument has some partial factual basis, because electrical power is nearly

100% perishable on a very short timescale. That is, there is very, very little energy storage

capacity in the grid. Therefore it is true that phase-based load sharing (section 2.2) and voltage
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stability (section 2.1) must rely on basic physics, because there is no time for anything else.

Similarly, because electricity travels at essentially the speed of light, you cant send timely

notification of certain problems; the problem would arrive at least as soon as the notification. Onthe other hand, not all problems fit this description. According to reference 1, in many cases the

blackout was spread by the action of zone-3 impedance relays, which act after a one- or two-

second delay [ibid., page 88]. You can gather a lot of information from faraway places and doquite a lot of computing in one second.

Reference 1 said Some of NERCs planning and operational requirements and standards were

sufficiently ambiguous that FE could interpret them to include practices that were inadequate for

reliable system operation. Things do not appear to have improved much in this regard. On 1April 2005 NERC promulgated a black-start standard (reference 4), which requires only that the

black-start plan meet the expectations of the regional system restoration plan. That strikes me

as non-quantitative and remarkably open to interpretation.

I havent made an exhaustive study, but I havent seen anything that looks like an enforceable

reliability standard, or a tangible reward for reliability, or a significant penalty for unreliability.

If I had to guess, Id say theres a high probability of another huge blackout within the next ten

years. Id estimate at 10% chanceper yearfor each of the next several years.

Former Secretary of Energy Bill Richardson said the United States was a major superpowerwith a third-world electrical grid.

A Basic Notions and Jargon

To make sense ofreference 5, one needs to know some jargon and some background concepts:

What NERC calls an "Interconnect" is basically what civilians would call a grid.It does not mean an interconnection point.

Various pieces of the system operate as islands. Everything within a givenisland is synchronous, i.e operating at the same frequency, but differentislands are free to have different frequencies. Indeed frequency mismatchcan be used to map out the boundaries of islands.

Normal operation is the limiting case of three huge islands (east, west, andTexas). (The pros dont like to apply the word island to these huge areas, butthe logic is the same.) In emergencies, the system is supposed to break upinto much smaller islands, to prevent problems from spreading. On 14-

August-2003, not nearly enough islanding occurred.

There are tie-lines between east, west, and Texas, but they are allasynchronous DC links.

What NERC calls a "Control Area" is something that has power-generationcapability, a connection to the grid, and the proper servo performance to playby the rules described in section 2. The smallest possible Control Area is a
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single generator (with servos), but in general a Control Area typically hasmultiple generators and other complexities.

B References1.

Final Report on the August 14th Blackout in the United States and Canada

http://www.google.com/url?

sa=Ustart=1q=https://reports.energy.gov/e=8092

2.

Richard Kessel, Blackout 2003: How Did It Happen and Why?

http://energycommerce.house.gov/108/Hearings/09042003hearing1062/Kess

el1683.htm

3.

Youre on your own - Con Ed http://www.nydailynews.com/news/v-

pfriendly/story/110527p-99864c.html

4.

NERC Standard EOP-007-0 Establish, Maintain, and Document a Regional

Blackstart Capability Plan

ftp://www.nerc.com/pub/sys/all_updl/standards/rs/EOP-007-0.pdf

5.

NERC Operating Manualftp://www.nerc.com/pub/sys/all_updl/oc/opman/opman.pdf
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