XYLENE POWER LTD.

ELECTRICITY POWER FACTOR

By C. Rhodes

POWER FACTOR:
In the absence of harmonics, power factor is the cosine of the phase angle between the alternating voltage and current waveforms. In an electricity transmission/distribution system power factor correction is used to shift the voltage and current waveforms into phase with each other. Power factor correction maximizes the amount of power that can be delivered to a load via a transmission/distribution system that is subject to specific voltage and current constraints and hence maximizes the power transfer capability.

Practical implementation of power factor correction in a transmission network with time varying reactive parameters is a complex process involving insertion of reactive power. There are potential severe complications related to unintended tripping of nuclear generation safety shutdown systems if the transmission network has to be reconfigured while under high load. A better strategy is to use the electricity rate structure to encourage all generators and loads to operate at close to unity power factor to minimize power factor correction requirements within the transmission/distribution system.

Over reliance on power factor correction within a transmission/distribution system generally leads to system instability and unreliability. Reliability is enhanced by keeping all grid customers at unity power factor and by designing the transmission/distribution system to minimize reliance on power factor correction to keep within safe operating limits.

Power Factor is a measure of reduction in electricity system power delivery capability caused by a grid customer having a finite behind the meter reactance at the line frequency. However, if the grid customer has a behind the meter power inverter that generates voltage and/or current harmonics, the situation becomes more complex. It is necessary to measure received and transmitted power to properly determine transmission/distribution usage. This web page reviews basic electrical engineering concepts relating to power factor and its effect on power transmission capability and then shows how transmission/distribution usage can be determined from the measured value of:
[(Etb - Eta) / (Erb - Era)],
as defined in the section titled Electricity Metering. This methodology allows for arbitrary current and voltage waveforms that may be introduced by power inverters and similar high frequency switching devices.

MAXIMUM TRANSMITTED POWER:
The maximum instantaneous power Pmax that can be transmitted via a two wire circuit is given by:
Pmax = Vo Io
where:
Vo = maximum voltage
Io = maximum current

Apply the trigonometric identity:
(sin WT)^2 + (cos WT)^2 = 1
to get:
Pmax = Vo Io [(sin WT)^2 + (cos WT)^2]
Note that for all values of WT the instantaneous power never exceeds Pmax.

At the metering point in an AC system with no harmonics:
V = Vo sin(WT)
where:
V = instantaneous voltage
W = angular frequency
T = time

The corresponding instantaneous current I is given by:
I = Io sin(WT - B)
where B is the phase angle by which the current lags the voltage due to inductance (reactance)
Expanding this relationship using the trigonometric identity:
sin(WT - B) = sin(WT)cos(-B) + cos(WT)sin(-B)
= sin(WT)cos(B) - cos(WT)sin(B)
gives:
I = Io [sin(WT)cos(B) - cos(WT)sin(B)]

The corresponding instantaneous power P is given by:
P = V I
or
P = Vo Io [(sin(WT))^2 cos(B) - sin(WT)cos(WT)sin(B)]

From time T = 0 to time T = 2 Pi / W the net energy E that is transmitted is:
E = Integral from T = 0 to T = 2 Pi / W of P dT
or
E = Integral from T = 0 to T = 2 Pi / W of Vo Io [(sin(WT))^2 cos(B) - sin(WT)cos(WT)sin(B)]dT
= Integral from A = 0 to A = 2 Pi of (Vo Io / W) [(sin(A))^2 cos(B) - sin(A)cos(A)sin(B)]dA
or
E = (Vo Io / W) Pi cos(B)

The average power Pave transmitted between T = 0 and T = 2 Pi / W is:
Pave = E / (2 Pi / W)
= (Vo Io / W) Pi cos(B) / (2 Pi / W)
or
Pave = (Vo Io / 2) cos(B)

When cos(B) = 1, the power transmission capability of a single phase AC circuit takes its maximum value of (Vo Io / 2). The actual power transmission capability is determined by the phase angle B which determines the size of the factor cos(B). Hence:
cos(B) = power factor
This result is well known and can be found in most introductory electrical engineering textbooks.

For an individual load cos(B) can take any value in the range 0 to 1, so that the transmission capacity that must be assigned to that load is:
[(Vo Io)/ 2] = Pave / cos(B)
This power transmission capacity has units of kVA.

Recall that:
Pave = [(Erb - Era) - (Etb - Eta)] / (Tb - Ta)
where:
Ta = time at beginning of measurement interval
Tb = time at end of measurement interval
Erb - Era = energy received during the measurement interval
Etb - Eta = energy transmitted during the measurement interval

Hence the transmission capacity used to deliver the net received power is:
Pave / cos(B) = [(Erb - Era) - (Etb - Eta)] / [(Tb - Ta) cos(B)]

This expression can be rearranged to give:
Pave / cos(B) = [(Erb-Era)/(Tb-Ta)]{1 - [(Etb-Eta)/(Erb-Era)]}/ cos(B)
or
Pave / cos(B) = [-(Etb-Eta)/(Tb-Ta)]{1 - [(Erb-Era)/(Etb-Eta)]}/ cos(B)

In order to properly allocate transmission/distribution costs it is necessary to find an expression for:
{1 - [(Erb-Era)/(Etb-Eta)]}/ cos(B) in terms of only (Etb-Eta)/(Erb - Era).

DETERMINATION OF (Erb - Era) AND (Etb - Eta) AS FUNCTIONS OF B:
Recall that for the period from time Ta = 0 to time Tb = 2 Pi / W:
[(Erb - Era) - (Etb - Eta)] = Integral from A = 0 to A = 2 Pi of (Vo Io / W)[(sin(A))^2 cos(B) - sin(A)cos(A)sin(B)]dA
= Integral from A = 0 to A = 2 Pi of (Vo Io / W)[(sin(A) (sin(A) cos(B) - cos(A)sin(B))]dA
= Integral from A = 0 to A = A1 of (Vo Io / W)[(sin(A) (sin(A) cos(B) - cos(A)sin(B))]dA
+ Integral from A = A1 to A = Pi of (Vo Io / W)[(sin(A) (sin(A) cos(B) - cos(A)sin(B))]dA
+ Integral from A = Pi to A = Pi+A1 of (Vo Io / W)[(sin(A) (sin(A) cos(B) - cos(A)sin(B))]dA
+ Integral from A = Pi+A1 to A = 2 Pi of (Vo Io / W)[(sin(A) (sin(A) cos(B) - cos(A)sin(B))]dA
Where angle A1 is given by:
sin(A1) cos(B) - cos(A1) sin(B) = 0
or
tan(A1) = tan(B)
or
A1 = B

Consider the case of: (Erb - Era) > (Etb - Eta). Then in the above 4 line integration the 1st and 3rd lines are negative and contribute to transmitted power (Etb - Eta). The 2nd and 4th lines are positive and contribute to received power (Erb - Era).

A table of integrals gives:
Integral (sin(A))^2 dA = [(A / 2) - (sin(2A)) / 4]
and
Integral -sin(A)cos(A)dA = -(sin(A))^2 / 2

FIND (Erb - Era):
Integral from A = B to A = Pi of (Vo Io / W)[(sin(A) (sin(A) cos(B) - cos(A)sin(B))]dA
= (Vo Io / W)cos(B)[(Pi / 2) - (B / 2) + (sin(2B)) / 4]+(Vo Io / W)sin(B)[(sin(B))^2 / 2]
and
Integral from A = Pi+B to A = 2 Pi of (Vo Io / W)[(sin(A) (sin(A) cos(B) - cos(A)sin(B))]dA
=(Vo Io / W)cos(B)[(2 Pi/2) - ((B+Pi)/2) + (sin(2B))/4] + (Vo Io / W)sin(B)[(sin(Pi + B))^2 /2]
Thus:
(Erb - Era) = (Vo Io / W)cos(B)[(Pi) - (B) + (sin(2B)) / 2]+(Vo Io / W)[(sin(B))^3]

FIND (Etb - Eta):
Integral from A = 0 to A = B of (-Vo Io / W)[(sin(A) (sin(A) cos(B) - cos(A)sin(B))]dA
= (-Vo Io / W) cos(B)[(B / 2) - (sin(2B)) / 4] + (Vo Io / W)sin(B)[(sin(B))^2 / 2]
and
Integral from A = Pi to A = Pi+B of (-Vo Io / W)[(sin(A) (sin(A) cos(B) - cos(A)sin(B))]dA
= (-Vo Io / W) cos(B)[(B / 2) - (sin(2B)) / 4]+(Vo Io / W)sin(B)[((sin(B))^2 / 2]
Thus:
(Etb - Eta) = (-Vo Io / W)cos(B)[ (B) - (sin(2B)) / 2]+(Vo Io / W)[(sin(B))^3]

FIND [(Erb - Era) - (Etb - Eta)]:
[(Erb - Era) - (Etb - Eta)] = (Vo Io / W)cos(B)(Pi)
Hence:
Pave = [(Erb - Era) - (Etb - Eta)]/ (2 Pi / W)
= (Vo Io / W)cos(B)(Pi) / (2 Pi / W)
= (Vo Io / 2)cos(B)
as expected.

FIND [(Etb - Eta) / (Erb - Era)]:
[(Etb - Eta) / (Erb - Era)] = {-cos(B)[B - (sin(2B))/2]+[(sin(B))^3]}/{cos(B)[Pi - B + (sin(2B))/2]+(sin(B))^3}
= {-B cos(B) + sin(B)(cos(B))^2+(sin(B))^3}/{(Pi - B)cos(B) + sin(B)(cos(B))^2+(sin(B))^3}
= {-B cos(B) + sin(B)}/{(Pi - B)cos(B) + sin(B)}
or
[(Etb - Eta) / (Erb - Era)] = {-B + tan(B)}/{(Pi - B) + tan(B)}

FIND F(B):
F(B) = {1 - [(Etb-Eta)/(Erb-Era)]}/ cos(B)
= {1 - {-B + tan(B)}/{(Pi - B) + tan(B)}}/cos(B)
= Pi /{(Pi - B) + tan(B)}cos(B)
= Pi / {(Pi - B)cos(B) + sin(B)}

Thus for B in the range 0 < B < (Pi / 2) the functions:
[(Etb - Eta) / (Erb - Era)] = {-B + tan(B)}/{(Pi - B) + tan(B)}
and
F(B) = Pi / {(Pi - B)cos(B) + sin(B)}
are easily tabulated.

Thus we can make a lookup table which for 0 < B < (Pi/2) tabulates:
B,
[(Etb - Eta) / (Erb - Era)]= {-B + tan(B)}/{(Pi - B) + tan(B)}
and
{1 - [(Etb-Eta)/(Erb-Era)]}/ F = Pi / {(Pi - B)cos(B) + sin(B)}.

The calculation procedure is to first measure (Etb - Eta) and (Erb - Era).
Then calculate [(Etb - Eta) / (Erb - Era)] and use the lookup table to find the corresponding value of B and hence F(B) = [Pi / {(Pi - B)cos(B) + sin(B)}].

Since the input data is just [(Etb - Eta) / (Erb - Era)], this procedure will work for determining power factor for 1, 2 or 3 phases.

If (Etb - Eta) > (Erb - Era), (Erb - Era) and (Etb - Eta) interchange roles, so for:
(Etb - Eta) > (Erb - Era):
[(Erb - Era) / (Etb - Eta)] = {-B + tan(B)}/{(Pi - B) + tan(B)}

SUMMARY:
The transmission capacity in kVA used by a load customer is given by:
[(Erb - Era) /(Tb-Ta)][F(B)]
where the value of the parameter:
F(B) = [ Pi / {(Pi - B)cos(B) + sin(B)}]
is obtained from a table of:
B, [(Etb-Eta) / (Erb-Era)] = {-B + tan(B)}/{(Pi-B) + tan(B)}, F(B)=[Pi/{(Pi-B)cos(B)+sin(B)}]
using as input data the measured value of:
[(Etb - Eta) / (Erb - Era)].

Similarly,the transmission capacity in kVA used by an operating generator is given by:
[(Etb - Eta) /(Tb-Ta)][F(B)]
where the value of the parameter:
F(B)=[ Pi / {(Pi - B)cos(B) + sin(B)}]
is obtained from a table of:
B, [(Erb-Era) / (Etb-Eta)] = {-B + tan(B)}/{(Pi-B) + tan(B)}, F(B)=[Pi/{(Pi-B)cos(B)+sin(B)}]
using as input data the measured value of:
[(Erb - Era) / (Etb - Eta)].

Note that in the presence of harmonics B does not mean the phase difference between the voltage and current waveforms. Instead B is a function of:
[(Etb-Eta)/(Erb-Era)].
However, if there are no harmonics present then B equals the phase angle between the voltage and the current waveforms.

For a load customer (Etb-Eta) < (Erb-Era) and numerical tabulation gives:
TABLE OF B (RADIANS), [(Etb-Eta)/(Erb-Era)], F(B):
B = 0.00000000 [(Etb-Eta)/(Erb-Era)] = 0.00000000 F(B) = 1.00000000
B = 0.01745327 [(Etb-Eta)/(Erb-Era)] = 0.00000056 F(B) = 1.00015176
B = 0.03490655 [(Etb-Eta)/(Erb-Era)] = 0.00000451 F(B) = 1.00060503
B = 0.05235983 [(Etb-Eta)/(Erb-Era)] = 0.00001524 F(B) = 1.00135708
B = 0.06981311 [(Etb-Eta)/(Erb-Era)] = 0.00003617 F(B) = 1.00240563
B = 0.08726638 [(Etb-Eta)/(Erb-Era)] = 0.00007072 F(B) = 1.00374884
B = 0.10471967 [(Etb-Eta)/(Erb-Era)] = 0.00012236 F(B) = 1.00538523
B = 0.12217294 [(Etb-Eta)/(Erb-Era)] = 0.00019461 F(B) = 1.00731374
B = 0.13962622 [(Etb-Eta)/(Erb-Era)] = 0.00029100 F(B) = 1.00953369
B = 0.15707950 [(Etb-Eta)/(Erb-Era)] = 0.00041516 F(B) = 1.01204477
B = 0.17453278 [(Etb-Eta)/(Erb-Era)] = 0.00057073 F(B) = 1.01484704
B = 0.19198606 [(Etb-Eta)/(Erb-Era)] = 0.00076148 F(B) = 1.01794093
B = 0.20943933 [(Etb-Eta)/(Erb-Era)] = 0.00099120 F(B) = 1.02132721
B = 0.22689261 [(Etb-Eta)/(Erb-Era)] = 0.00126380 F(B) = 1.02500702
B = 0.24434589 [(Etb-Eta)/(Erb-Era)] = 0.00158327 F(B) = 1.02898183
B = 0.26179917 [(Etb-Eta)/(Erb-Era)] = 0.00195371 F(B) = 1.03325348
B = 0.27925244 [(Etb-Eta)/(Erb-Era)] = 0.00237932 F(B) = 1.03782416
B = 0.29670572 [(Etb-Eta)/(Erb-Era)] = 0.00286442 F(B) = 1.04269638
B = 0.31415900 [(Etb-Eta)/(Erb-Era)] = 0.00341345 F(B) = 1.04787302
B = 0.33161228 [(Etb-Eta)/(Erb-Era)] = 0.00403100 F(B) = 1.05335730
B = 0.34906556 [(Etb-Eta)/(Erb-Era)] = 0.00472180 F(B) = 1.05915282
B = 0.36651883 [(Etb-Eta)/(Erb-Era)] = 0.00549072 F(B) = 1.06526350
B = 0.38397211 [(Etb-Eta)/(Erb-Era)] = 0.00634282 F(B) = 1.07169365
B = 0.40142539 [(Etb-Eta)/(Erb-Era)] = 0.00728330 F(B) = 1.07844793
B = 0.41887867 [(Etb-Eta)/(Erb-Era)] = 0.00831758 F(B) = 1.08553138
B = 0.43633194 [(Etb-Eta)/(Erb-Era)] = 0.00945125 F(B) = 1.09294943
B = 0.45378522 [(Etb-Eta)/(Erb-Era)] = 0.01069012 F(B) = 1.10070788
B = 0.47123850 [(Etb-Eta)/(Erb-Era)] = 0.01204023 F(B) = 1.10881295
B = 0.48869178 [(Etb-Eta)/(Erb-Era)] = 0.01350782 F(B) = 1.11727124
B = 0.50614506 [(Etb-Eta)/(Erb-Era)] = 0.01509942 F(B) = 1.12608981
B = 0.52359833 [(Etb-Eta)/(Erb-Era)] = 0.01682178 F(B) = 1.13527612
B = 0.54105161 [(Etb-Eta)/(Erb-Era)] = 0.01868195 F(B) = 1.14483809
B = 0.55850489 [(Etb-Eta)/(Erb-Era)] = 0.02068727 F(B) = 1.15478408
B = 0.57595817 [(Etb-Eta)/(Erb-Era)] = 0.02284536 F(B) = 1.16512295
B = 0.59341144 [(Etb-Eta)/(Erb-Era)] = 0.02516420 F(B) = 1.17586403
B = 0.61086472 [(Etb-Eta)/(Erb-Era)] = 0.02765209 F(B) = 1.18701718
B = 0.62831800 [(Etb-Eta)/(Erb-Era)] = 0.03031770 F(B) = 1.19859277
B = 0.64577128 [(Etb-Eta)/(Erb-Era)] = 0.03317007 F(B) = 1.21060172
B = 0.66322456 [(Etb-Eta)/(Erb-Era)] = 0.03621866 F(B) = 1.22305554
B = 0.68067783 [(Etb-Eta)/(Erb-Era)] = 0.03947333 F(B) = 1.23596630
B = 0.69813111 [(Etb-Eta)/(Erb-Era)] = 0.04294441 F(B) = 1.24934672
B = 0.71558439 [(Etb-Eta)/(Erb-Era)] = 0.04664270 F(B) = 1.26321014
B = 0.73303767 [(Etb-Eta)/(Erb-Era)] = 0.05057948 F(B) = 1.27757061
B = 0.75049094 [(Etb-Eta)/(Erb-Era)] = 0.05476659 F(B) = 1.29244283
B = 0.76794422 [(Etb-Eta)/(Erb-Era)] = 0.05921640 F(B) = 1.30784229
B = 0.78539750 [(Etb-Eta)/(Erb-Era)] = 0.06394188 F(B) = 1.32378521
B = 0.80285078 [(Etb-Eta)/(Erb-Era)] = 0.06895662 F(B) = 1.34028864
B = 0.82030406 [(Etb-Eta)/(Erb-Era)] = 0.07427487 F(B) = 1.35737047
B = 0.83775733 [(Etb-Eta)/(Erb-Era)] = 0.07991158 F(B) = 1.37504948
B = 0.85521061 [(Etb-Eta)/(Erb-Era)] = 0.08588243 F(B) = 1.39334537
B = 0.87266389 [(Etb-Eta)/(Erb-Era)] = 0.09220387 F(B) = 1.41227882
B = 0.89011717 [(Etb-Eta)/(Erb-Era)] = 0.09889319 F(B) = 1.43187155
B = 0.90757044 [(Etb-Eta)/(Erb-Era)] = 0.10596855 F(B) = 1.45214636
B = 0.92502372 [(Etb-Eta)/(Erb-Era)] = 0.11344901 F(B) = 1.47312718
B = 0.94247700 [(Etb-Eta)/(Erb-Era)] = 0.12135464 F(B) = 1.49483913
B = 0.95993028 [(Etb-Eta)/(Erb-Era)] = 0.12970652 F(B) = 1.51730863
B = 0.97738356 [(Etb-Eta)/(Erb-Era)] = 0.13852683 F(B) = 1.54056339
B = 0.99483683 [(Etb-Eta)/(Erb-Era)] = 0.14783894 F(B) = 1.56463254
B = 1.01229011 [(Etb-Eta)/(Erb-Era)] = 0.15766742 F(B) = 1.58954671
B = 1.02974339 [(Etb-Eta)/(Erb-Era)] = 0.16803818 F(B) = 1.61533808
B = 1.04719667 [(Etb-Eta)/(Erb-Era)] = 0.17897850 F(B) = 1.64204048
B = 1.06464994 [(Etb-Eta)/(Erb-Era)] = 0.19051714 F(B) = 1.66968952
B = 1.08210322 [(Etb-Eta)/(Erb-Era)] = 0.20268445 F(B) = 1.69832264
B = 1.09955650 [(Etb-Eta)/(Erb-Era)] = 0.21551241 F(B) = 1.72797925
B = 1.11700978 [(Etb-Eta)/(Erb-Era)] = 0.22903481 F(B) = 1.75870084
B = 1.13446306 [(Etb-Eta)/(Erb-Era)] = 0.24328730 F(B) = 1.79053110
B = 1.15191633 [(Etb-Eta)/(Erb-Era)] = 0.25830757 F(B) = 1.82351608
B = 1.16936961 [(Etb-Eta)/(Erb-Era)] = 0.27413542 F(B) = 1.85770427
B = 1.18682289 [(Etb-Eta)/(Erb-Era)] = 0.29081295 F(B) = 1.89314685
B = 1.20427617 [(Etb-Eta)/(Erb-Era)] = 0.30838467 F(B) = 1.92989774
B = 1.22172944 [(Etb-Eta)/(Erb-Era)] = 0.32689770 F(B) = 1.96801389
B = 1.23918272 [(Etb-Eta)/(Erb-Era)] = 0.34640191 F(B) = 2.00755540
B = 1.25663600 [(Etb-Eta)/(Erb-Era)] = 0.36695013 F(B) = 2.04858571
B = 1.27408928 [(Etb-Eta)/(Erb-Era)] = 0.38859835 F(B) = 2.09117191
B = 1.29154256 [(Etb-Eta)/(Erb-Era)] = 0.41140592 F(B) = 2.13538486
B = 1.30899583 [(Etb-Eta)/(Erb-Era)] = 0.43543580 F(B) = 2.18129955
B = 1.32644911 [(Etb-Eta)/(Erb-Era)] = 0.46075481 F(B) = 2.22899530
B = 1.34390239 [(Etb-Eta)/(Erb-Era)] = 0.48743389 F(B) = 2.27855608
B = 1.36135567 [(Etb-Eta)/(Erb-Era)] = 0.51554841 F(B) = 2.33007084
B = 1.37880894 [(Etb-Eta)/(Erb-Era)] = 0.54517849 F(B) = 2.38363386
B = 1.39626222 [(Etb-Eta)/(Erb-Era)] = 0.57640934 F(B) = 2.43934508
B = 1.41371550 [(Etb-Eta)/(Erb-Era)] = 0.60933161 F(B) = 2.49731056
B = 1.43116878 [(Etb-Eta)/(Erb-Era)] = 0.64404185 F(B) = 2.55764286
B = 1.44862206 [(Etb-Eta)/(Erb-Era)] = 0.68064289 F(B) = 2.62046156
B = 1.46607533 [(Etb-Eta)/(Erb-Era)] = 0.71924435 F(B) = 2.68589374
B = 1.48352861 [(Etb-Eta)/(Erb-Era)] = 0.75996315 F(B) = 2.75407455
B = 1.50098189 [(Etb-Eta)/(Erb-Era)] = 0.80292408 F(B) = 2.82514781
B = 1.51843517 [(Etb-Eta)/(Erb-Era)] = 0.84826039 F(B) = 2.89926666
B = 1.53588844 [(Etb-Eta)/(Erb-Era)] = 0.89611450 F(B) = 2.97659428
B = 1.55334172 [(Etb-Eta)/(Erb-Era)] = 0.94663867 F(B) = 3.05730466
B = 1.57079633 [(Etb-Eta)/(Erb-Era)] = 1.00000000 F(B) = 3.14159265

FOR A NET GENERATOR:
(Etb-Eta) > (Erb-Era)
and the table is identical except that the numbers in the second column represent:
[(Erb-Era)/(Etb-Eta)]

This web page last updated September 11, 2010.