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IEEE Standard 1459-2010 Single Phase Power Definitions

RA/TA Kahraman Yumak

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 The well-known and universally accepted concept. Let the voltage and current: 𝑣 = 2𝑉 sin 𝜔𝑡

𝑖 = 2𝐼 sin 𝜔𝑡 − 𝜃

and

(1)

The instantaneous power 𝑝, consists of instantaneous active power and instantaneous reactive power. 𝑝 = 𝑣𝑖 = 𝑝𝑎 + 𝑝𝑞

𝑝𝑎 = 𝑉𝐼𝑐𝑜𝑠𝜃 1 − 𝑐𝑜𝑠 2𝜔𝑡

(2)

= 𝑃 1 − 𝑐𝑜𝑠 2𝜔𝑡

(3)

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. Active power 𝑃 ; 1 𝑃= 𝑘𝑇

𝜏+𝑘𝑇

𝜏

1 𝑝𝑑𝑡 = 𝑘𝑇

𝜏+𝑘𝑇

𝑝𝑎 𝑑𝑡 = 𝑉𝐼 cos 𝜃

(4)

𝜏

Slide 2/9

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 =

Power factor:

𝑃𝐹 =

𝑃2 + 𝑄2

𝑃 𝑆

(7)

SINGLE PHASE POWER DEFINITIONS UNDER SINUSOIDAL CONDITIONS

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.

the ratio between the energy transmitted to the load over the max. energy that could be transmitted provided the line losses are kept same Slide 3/9

2. Single Phase Power Definitions Under Nonsinusoidal Conditions

Let the voltage and current: the power system frequency components 𝑣1 , 𝑖1 and 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)

where

𝑣𝐻 = 𝑉0 + 2

𝑉ℎ sin ℎ𝜔𝑡 − 𝛼ℎ

(11)

𝐼ℎ sin ℎ𝜔𝑡 − 𝛽ℎ

(12)

ℎ≠1

𝑖𝐻 = 𝐼0 + 2 ℎ≠1

Slide 4/9

1 𝑉2 = 𝑘𝑇

𝜏+𝑘𝑇

1 𝐼2 = 𝑘𝑇

where

𝑣 2 𝑑𝑡 = 𝑉12 + 𝑉𝐻2

(13)

𝑖 2 𝑑𝑡 = 𝐼12 + 𝐼𝐻2

(14)

𝜏 𝜏+𝑘𝑇

𝜏

𝑉𝐻2 = 𝑉02 +

𝑉ℎ2 = 𝑉 2 − 𝑉12

(15)

𝐼ℎ2 = 𝐼 2 − 𝐼12

(16)

ℎ≠1

𝐼𝐻2 = 𝐼02 + ℎ≠1

Total harmonic distortion (THD) for voltage and current is defined 𝑉𝐻 𝑇𝐻𝐷𝑉 = = 𝑉1

𝑉 𝑉1

𝐼𝐻 𝑇𝐻𝐷𝐼 = = 𝐼1

𝐼 𝐼1

2

−1

(17)

−1

(18)

2

SINGLE PHASE POWER DEFINITIONS UNDER SINUSOIDAL CONDITIONS

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

Slide 5/9

1 𝑃= 𝑘𝑇

𝜏+𝑘𝑇 𝜏

𝑝𝑑𝑡 = 𝑃1 + 𝑃𝐻

𝑃1 = 𝑉1 𝐼1 cos 𝜃1 𝑃𝐻 = 𝑃 − 𝑃1 =

𝑉ℎ 𝐼ℎ cos 𝜃ℎ ℎ≠1

(19) (20) (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.

IEEE’S POWER DECOMPOSITION

Active power 𝑃;

Distortion powers are related to waveform distortions.

Fundamental reactive power: 𝑄1 = 𝑉1 𝐼1 sin 𝜃1

(22)

Fundamental apparent power: 𝑆11 = 𝑉1 𝐼1 =

2 2 𝑃11 + 𝑄11

(23)

Slide 6/9

𝐷𝐼 = 𝑉1 𝐼𝐻 = 𝑆1 𝑇𝐻𝐷𝐼

(24)

𝐷𝑉 = 𝑉𝐻 𝐼1 = 𝑆1 𝑇𝐻𝐷𝑉

(25)

𝑆𝐻 = 𝑉𝐻 𝐼𝐻 = 𝑆1 𝑇𝐻𝐷𝐼 𝑇𝐻𝐷𝑉

(26)

Voltage distortion power:

Harmonic apparent power:

Harmonic distortion power: 𝐷𝐻 =

𝑆𝐻2 − 𝑃𝐻2

(27)

IEEE’S POWER DECOMPOSITION

Current distortion power:

Finally, apparent power becomes as; 𝑆 2 = 𝑉𝐼

2

= 𝑆12 + 𝐷𝐼2 + 𝐷𝑉2 + 𝑆𝐻2

(28)

Nonfundamental apparent power: 𝑆𝑁2 = 𝑆 2 − 𝑆12 = 𝐷𝐼2 + 𝐷𝑉2 + 𝑆𝐻2

Nonactive power:

𝑁=

𝑆 2 − 𝑃2

(29)

(30) Slide 7/9

(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)

IEEE’S POWER DECOMPOSITION

Fundamental Power Factor (Displacement Power Factor): 𝑃1 𝑃𝐹1 = 𝑆1

Slide 7/9

3. Numerical Study v  t   2V1 sin t  1   2V3 sin  3t   3   2V5 sin  5t   5   2V7 sin  7t   7  i  t   2I1 sin t  1   2I 3 sin  3t  3   2I 5 sin  5t  5   2 I 7 sin  7t   7 

Table 1. RMS Values and Phase Angles 100 100 𝑉1 𝐼1 8 20 𝑉3 𝐼3 15 15 𝑉5 𝐼5 5 10 𝑉7 𝐼7 101.56 103.56 𝑉 𝐼 17.72 26.926 𝑉ℎ 𝐼ℎ 0° 30° 𝛼1 𝛽1 70° 165° 𝛼3 𝛽3 -141° -234° 𝛼5 𝛽5 -142° -234° 𝛼7 𝛽7

Table 2. IEEE’s Power Definitions 10517.55 𝑆 10000 𝑆11 477.13 𝑆𝐻 3256.88 𝑆𝑁 8632.54 𝑃 8660 𝑃11 -27.46 𝑃𝐻 5000 𝑄11 2692.58 𝐷𝐼 1772 𝐷𝑉 476.34 𝐷𝐻 6008.17 𝑁 𝑇𝐻𝐷𝑉 0.177 0.269 𝑇𝐻𝐷𝐼 0.866 𝑃𝐹1 0.821 𝑃𝐹 0.3257 𝐻𝑃

(34)

Slide 8/9

6. References

1.

IEEE Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions, IEEE Std. 14592010, Feb. 2010.

2.

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

Slide 9/9

THANK YOU.

September 12, 2012

Electrical Engineering Department