RA/TA Kahraman Yumak

IEEE Standard 1459-2010 Single Phase Power Definitions

September 12, 2012 Electrical Engineering Department

Outline

1. Single Phase Power Definitions Under Sinusoidal Conditions

2. Single Phase Power Definitions Under Nonsinusoidal Conditions

3. Numerical Study

4. References

Slide 1 / 9

1. Single Phase Power Definitions Under Sinusoidal Conditions

Slide 2 / 9

𝑣 = 2𝑉 sin 𝜔𝑡 and 𝑖 = 2𝐼 sin 𝜔𝑡 − 𝜃 (1)

The instantaneous power 𝑝, consists of instantaneous active power and instantaneous reactive power.

𝑝 = 𝑣𝑖 = 𝑝𝑎 + 𝑝𝑞 (2)

Instantaneous active power 𝑝𝑎 is the rate of unidirectional flow of the energy from the source to the load. Its steady state rate of flow is not negative. Consists of active 𝑃 and intrinsic power −𝑃𝑐𝑜𝑠 2𝜔𝑡 . Intrinsic power is always present . This oscillating component does not cause power loss.

𝑝𝑎 = 𝑉𝐼𝑐𝑜𝑠𝜃 1 − 𝑐𝑜𝑠 2𝜔𝑡 = 𝑃 1 − 𝑐𝑜𝑠 2𝜔𝑡 (3)

𝑃 =1

𝑘𝑇 𝑝𝑑𝑡

𝜏+𝑘𝑇

𝜏

=1

𝑘𝑇 𝑝𝑎𝑑𝑡

𝜏+𝑘𝑇

𝜏

= 𝑉𝐼 cos 𝜃 (4)

Let the voltage and current:

The well-known and universally accepted concept.

Active power 𝑃 ;

SIN

GLE P

HA

SE PO

WER D

EFINITIO

NS U

ND

ER SIN

USO

IDA

L CO

ND

ITION

S

Slide 3 / 9

Instantaneous reactive power 𝑝𝑞

oscillates between the source and load where the net transfer of energy to the load is nil. These power oscillations cause power loss in the conductors.

𝑝𝑞 = −𝑉𝐼𝑠𝑖𝑛𝜃𝑠𝑖𝑛 2𝜔𝑡 = −𝑄𝑠𝑖𝑛 2𝜔𝑡 (5)

Reactive power Q ; due to the phase shift between voltage and current

𝑄 = 𝑉𝐼𝑠𝑖𝑛𝜃 (6)

The apparent power S ; is the product of the rms voltage and the rms current. Maximum active power that can be transmitted through the same line while keeping load rms voltage and rms current are constant.

𝑆 = 𝑉𝐼 = 𝑃2 + 𝑄2 (7)

Power factor: the ratio between the energy transmitted to the load over the max. energy that could be transmitted provided the line losses are kept same

𝑃𝐹 =𝑃

𝑆

2. Single Phase Power Definitions Under Nonsinusoidal Conditions

Slide 4 / 9

Let the voltage and current: the power system frequency components 𝑣1, 𝑖1and the remaining terms; harmonic components 𝑣𝐻 and 𝑖𝐻.

𝑣 = 𝑣1 + 𝑣𝐻 and 𝑖 = 𝑖1 + 𝑖𝐻 (8)

𝑣1 = 2𝑉1 sin 𝜔𝑡 − 𝛼1 (9)

𝑖1 = 2𝐼1 sin 𝜔𝑡 − 𝛽1 (10)

𝑣𝐻 = 𝑉0 + 2 𝑉ℎ sin ℎ𝜔𝑡 − 𝛼ℎ

ℎ≠1

(11)

𝑖𝐻 = 𝐼0 + 2 𝐼ℎ sin ℎ𝜔𝑡 − 𝛽ℎ

ℎ≠1

(12)

where

SIN

GLE P

HA

SE PO

WER D

EFINITIO

NS U

ND

ER SIN

USO

IDA

L CO

ND

ITION

S

Slide 5 / 9

Voltage and current is divided into two components, fundamental and harmonic parts. rms values are calculated.

𝑉2 =1

𝑘𝑇 𝑣2𝑑𝑡 = 𝑉1

2 +

𝜏+𝑘𝑇

𝜏

𝑉𝐻2 (13)

𝐼2 =1

𝑘𝑇 𝑖2𝑑𝑡 = 𝐼1

2 +

𝜏+𝑘𝑇

𝜏

𝐼𝐻2 (14)

where

𝑉𝐻2 = 𝑉0

2 + 𝑉ℎ2 =

ℎ≠1

𝑉2 − 𝑉12 (15)

𝐼𝐻2 = 𝐼0

2 + 𝐼ℎ2 =

ℎ≠1

𝐼2 − 𝐼12 (16)

𝑇𝐻𝐷𝑉 =𝑉𝐻

𝑉1=

𝑉

𝑉1

2

− 1 (17)

𝑇𝐻𝐷𝐼 =𝐼𝐻𝐼1

= 𝐼

𝐼1

2

− 1 (18)

Total harmonic distortion (THD) for voltage and current is defined

IEEE’S PO

WER D

ECO

MP

OSITIO

N

Slide 6 / 9

Active power 𝑃;

𝑃 =1

𝑘𝑇 𝑝𝑑𝑡

𝜏+𝑘𝑇

𝜏

= 𝑃1 + 𝑃𝐻 (19)

𝑃1 = 𝑉1𝐼1 cos 𝜃1 (20)

𝑃𝐻 = 𝑃 − 𝑃1 = 𝑉ℎ𝐼ℎ cos 𝜃ℎ

ℎ≠1

(21)

Only fundamental reactive power definition is given and no explanation is made. Distortion powers for individually voltage, current and harmonics are defined by using THD. But there is not any physical interpretation and also a definition for total distortion power. Reactive power is related to energy oscillations. Distortion powers are related to waveform distortions.

Fundamental reactive power:

𝑄1 = 𝑉1𝐼1 sin 𝜃1 (22)

Fundamental apparent power:

𝑆11 = 𝑉1𝐼1 = 𝑃112 + 𝑄11

2 (23)

IEEE’S PO

WER D

ECO

MP

OSITIO

N

Slide 7 / 9

Current distortion power:

Voltage distortion power:

Harmonic apparent power:

Harmonic distortion power:

Finally, apparent power becomes as;

Nonfundamental apparent power:

Nonactive power:

𝐷𝐼 = 𝑉1𝐼𝐻 = 𝑆1 𝑇𝐻𝐷𝐼 (24)

𝐷𝑉 = 𝑉𝐻𝐼1 = 𝑆1 𝑇𝐻𝐷𝑉 (25)

𝑆𝐻 = 𝑉𝐻𝐼𝐻 = 𝑆1 𝑇𝐻𝐷𝐼 𝑇𝐻𝐷𝑉 (26)

𝐷𝐻 = 𝑆𝐻

2 − 𝑃𝐻2 (27)

𝑆2 = 𝑉𝐼 2 = 𝑆12 + 𝐷𝐼

2 + 𝐷𝑉2 + 𝑆𝐻

2 (28)

𝑆𝑁2 = 𝑆2 − 𝑆1

2 = 𝐷𝐼2 + 𝐷𝑉

2 + 𝑆𝐻2 (29)

𝑁 = 𝑆2 − 𝑃2 (30)

IEEE’S PO

WER D

ECO

MP

OSITIO

N

Slide 7 / 9

Fundamental Power Factor (Displacement Power Factor):

𝑃𝐹1 =

𝑃1

𝑆1 (31)

Power Factor : Line utilization

𝑃𝐹 =𝑃

𝑆 (32)

max. utilization of the line is obtained when 𝑆 = 𝑃

Harmonic Pollution : Harmonic injection produced by consumer

𝐻𝑃 =

𝑆𝑁

𝑆1 (33)

3. Numerical Study

Slide 8 / 9

𝑉1 100 𝐼1 100

𝑉3 8 𝐼3 20

𝑉5 15 𝐼5 15

𝑉7 5 𝐼7 10

𝑉 101.56 𝐼 103.56

𝑉ℎ 17.72 𝐼ℎ 26.926

𝛼1 0° 𝛽1 30°

𝛼3 70° 𝛽3 165°

𝛼5 -141° 𝛽5 -234°

𝛼7 -142° 𝛽7 -234°

Table 1. RMS Values and Phase Angles

1 1 3 3 5 5 7 7

1 1 3 3 5 5 7 7

2 sin 2 sin 3 2 sin 5 2 sin 7

2 sin 2 sin 3 2 sin 5 2 sin 7

v t V t V t V t V t

i t I t I t I t I t (34)

𝑆 10517.55

𝑆11 10000

𝑆𝐻 477.13

𝑆𝑁 3256.88

𝑃 8632.54

𝑃11 8660

𝑃𝐻 -27.46

𝑄11 5000

𝐷𝐼 2692.58

𝐷𝑉 1772

𝐷𝐻 476.34

𝑁 6008.17

𝑇𝐻𝐷𝑉 0.177

𝑇𝐻𝐷𝐼 0.269

𝑃𝐹1 0.866

𝑃𝐹 0.821

𝐻𝑃 0.3257

Table 2. IEEE’s Power Definitions

6. References

Slide 9 / 9

1. IEEE Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions, IEEE Std. 1459-2010, Feb. 2010.

2. E. Emanuel, “Power Definitions and the Physical Mechanism of Power Flow”, John Wiley & Sons Ltd., UK, 2010

THANK YOU.

September 12, 2012 Electrical Engineering Department