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 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 and i = 2I sin ωt θ (1) The instantaneous power p, consists of instantaneous active power and instantaneous reactive power. p = vi = p a + p q (2) p a = VIcosθ 1 cos 2ωt = P 1 cos 2ωt (3) Instantaneous active power p a 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 ; P = 1 kt τ+kt τ pdt = 1 kt τ+kt τ p a dt = VI cos θ (4) 2 / 9
Instantaneous reactive power p q 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. p q = 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. Power factor: PF = P S S = VI = P 2 + Q 2 (7) SINGLE PHASE POWER DEFINITIONS UNDER SINUSOIDAL CONDITIONS the ratio between the energy transmitted to the load over the max. energy that could be transmitted provided the line losses are kept same 3 / 9
2. Single Phase Power Definitions Under Nonsinusoidal Conditions Let the voltage and current: the power system frequency components v 1, i 1 and the remaining terms; harmonic components v H and i H. v = v 1 + v H and i = i 1 + i H (8) where v 1 = 2V 1 sin ωt α 1 (9) i 1 = 2I 1 sin ωt β 1 (10) v H = V 0 + 2 V h sin hωt α h h 1 i H = I 0 + 2 I h sin hωt β h h 1 (11) (12) 4 / 9
Voltage and current is divided into two components, fundamental and harmonic parts. rms values are calculated. where V 2 = 1 kt I 2 = 1 kt τ+kt τ τ+kt τ v 2 dt = V 2 2 1 + V H (13) i 2 dt = I 2 2 1 + I H (14) V 2 H = V 2 0 + V 2 h = V 2 2 V 1 (15) h 1 I 2 H = I 2 0 + I 2 h = I 2 2 I 1 (16) h 1 Total harmonic distortion (THD) for voltage and current is defined THD V = V H V 1 = THD I = I H I 1 = V V 1 I I 1 2 2 1 (17) 1 (18) SINGLE PHASE POWER DEFINITIONS UNDER SINUSOIDAL CONDITIONS 5 / 9
Active power P; P = 1 kt τ+kt pdt τ = P 1 + P H (19) P 1 = V 1 I 1 cos θ 1 (20) P H = P P 1 = h 1 V h I h cos θ h 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. (21) IEEE S POWER DECOMPOSITION Fundamental reactive power: Fundamental apparent power: Q 1 = V 1 I 1 sin θ 1 (22) S 11 = V 1 I 1 = P 2 2 11 + Q 11 (23) 6 / 9
Current distortion power: D I = V 1 I H = S 1 THD I (24) Voltage distortion power: D V = V H I 1 = S 1 THD V (25) Harmonic apparent power: S H = V H I H = S 1 THD I THD V (26) Harmonic distortion power: D H = S 2 2 H P H (27) IEEE S POWER DECOMPOSITION Finally, apparent power becomes as; Nonfundamental apparent power: S 2 = VI 2 = S 1 2 + D I 2 + D V 2 + S H 2 (28) S N 2 = S 2 S 1 2 = D I 2 + D V 2 + S H 2 (29) Nonactive power: N = S 2 P 2 (30) 7 / 9
Fundamental Power Factor (Displacement Power Factor): Power Factor : Line utilization PF 1 = P 1 S 1 (31) PF = P S max. utilization of the line is obtained when S = P Harmonic Pollution : Harmonic injection produced by consumer (32) HP = S N S 1 (33) IEEE S POWER DECOMPOSITION 7 / 9
3. Numerical Study 2 sin 2 sin 3 2 sin 1 1 3 3 5 5 5 2 sin 7 7 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 1 1 3 3 5 5 7 7 (34) Table 1. RMS Values and Phase Angles V 1 100 I 1 100 V 3 8 I 3 20 V 5 15 I 5 15 V 7 5 I 7 10 V 101.56 I 103.56 V h 17.72 I h 26.926 α 1 0 β 1 30 α 3 70 β 3 165 α 5-141 β 5-234 α 7-142 β 7-234 Table 2. IEEE s Power Definitions S 10517.55 S 11 10000 S H 477.13 S N 3256.88 P 8632.54 P 11 8660 P H -27.46 Q 11 5000 D I 2692.58 D V 1772 D H 476.34 N 6008.17 THD V 0.177 THD I 0.269 PF 1 0.866 PF 0.821 HP 0.3257 8 / 9
6. References 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 9 / 9
THANK YOU. September 12, 2012 Electrical Engineering Department