Instantaneous Reactive Power Theory and its Applications to Active

Power Filtering

Arun JayendranB060110EE

Conventional Power Theory• Traditional power theory for 1 phase, sinusoidal systems is well

established• But these concepts fail to explain power components under non linear

load conditions• These theories assumes reactive power arises due to oscillation of power

between source and load

Relationship between v, i and p in a capacitor

Courtesy : www.wikipedia.org

Power under non-sinusoidal conditions • Power definitions by Budeanu [1927] (in frequency domain) can

be applied only for steady state analysis, limited to periodic waveforms of voltage and current

• Power definitions by Fryze [1930] (in time domain) is based on rms values of voltages and currents too not valid under transient phenomena

Need for a new power theory• Developments in the field of semiconductor technology

have led to an explosion in electronic and power electronic devices in use today• Power electronic converters behave as a non linear load

and represent a significant amount of power compared with other traditional linear loads.• The speed response of these converters and the way they

introduce harmonic components demands the evolution of time domain techniques to analyze energy flow in non linear circuit at instantaneous levels

About P-Q Theory…• In 1983 Hirofumi Akagi introduced p-q theory through his

paper “The Generalized theory of Instantaneous Reactive Power in Three Phase Circuits” • Most widely used for Non Linear load compensation for

APF.• Introduction of instantaneous imaginary power makes p-q

formulation a formal theory of electrical power in three phase system.• Compensates harmonic power optimally in

balanced/unbalanced and sinusoidal supply voltage systems, but not so good with non sinusoidal voltages.

Basis of P-Q Theory• Based on instantaneous powers defined in time

domain.• Can be applied to 3 phase systems with or with out

neutral.• It is valid not only in the steady state, but also in the

transient state.• P-Q theory considers 3 phase system as a unit , not as a

superposition or sum of three single phase circuits• Clarke transformation of voltages and current from the

abc to 0 coordinates.• Defines instantaneous power on 0 coordinates

The Clarke Transformation

[𝜈0𝜈𝛼𝜈𝛽

]=√ 23 [1/√2 1 ⁄ √2 1 ⁄ √21 −1 /2 −1/20 √3 /2 −√3 /2][𝜈a𝜈b𝜈c ]

[ i0i𝛼i𝛽 ]=√ 23 [1 /√2 1 ⁄ √2 1 ⁄ √ 21 −1/2 −1/20 √3/2 −√3/2] [ iaibic ]

are zero sequence voltage, α axis ,β axis voltages respectively

are zero sequence current, α axis ,β axis currents respectively

(1.a)

(1.b)

Clarke Transformation• Transforms voltages, currents from a-b-c coordinates to mutually

perpendicular set of α-β-0 axis

1200

a

b

c

αβ

0

900

ω = 0

• α-β-0 is a stationary frame

Inverse Clarke Transformation

are zero sequence voltage, α axis ,β axis voltages respectively

are zero sequence current, α axis ,β axis currents respectively

(2.a)

(2.b)

The Instantaneous Powers of the P-Q Theory

p = vα.iα + vβiβ

p is the instantaneous real power

q =vα i⋅ β − vβ i⋅ α

q is the instantaneous imaginary power

p0= v0 i⋅ 0

p0 is the instantaneous zero-sequence power

(3)

(4)

(5)

The P-Q Theory in 3Phase, 3 Wire Systems• In axis, voltage vector, (6) current vector, (7) • Complex power,

(8)

• The same can be expressed in matrix form as follows:

• (9)

Physical meaning of p and q

• For a three phase system with or with out neutral conductor in steady state or during transients, the three phase instantaneous active powerdescribes the total instantaneous energy flow per second between two subsystems

Definition of p (unit: Watt)

Definition of q (unit: Volt Ampere Imaginary)• The imaginary power q is proportional to the quantity of energy

that is being exchanged between the phases of the system. It does not contribute to energy transfer* between source and load at any time

Physical meaning of p and q•

p

qab

c

vavb vc

p: instantaneous total energy flow per time unitq: energy exchanged between phases without transferring energy

Note on Instantaneous imaginary power q• Conventional power theory defined reactive power as a

component of the instantaneous (active ) power, which has an average value equal to zero.

•

• The imaginary power means a sum of products of instantaneous three phase voltage and current portions that does not contribute to energy transfer between two subsystems at any time

• Eqn. (10) is similar to that implemented in some instruments for measuring reactive power. But instead of phasors, here instantaneous values are used

(10)

Components of p , q and p0

•

Courtesy: [5]

, • - Mean value of the instantaneous real power. • transferredfrom the power source to the load, in a

balanced way through the a-b-c coordinates • only desired power component to be supplied by the

power source• due to fundamental active current

• - Alternating value of the instantaneous real power. • exchanged between the power source and the load,

through the a-b-c coordinates. • since does not involve any energy transference from the

power source to load, it must be compensated.• due to harmonic currents

, • - Mean value of the instantaneous imaginary power. • exchangedbetween system phases and does not imply

transfer of energy between power source and load• choice of compensation of depends on reactive power

compensation • due to fundamental reactive current

• -Alternating value of the instantaneous imaginary power. • exchangedbetween system phases and does not imply

transfer of energy between power source and load• since is not necessary, it must be compensated.• due to harmonic currents

Block Diagram of Shunt APF(Active Power Filter)

Courtesy:[5]

P-Q theory Application : Shunt APF Harmonic Compensation

Shunt Current compensation based on p-q theory

[𝜈0𝜈𝛼𝜈𝛽

]=√ 23 [1/√2 1 ⁄ √2 1 ⁄ √21 −1 /2 −1/20 √3 /2 −√3 /2][𝜈a𝜈b𝜈 c ]

[ i0i𝛼i𝛽 ]=√ 23 [1 /√2 1 ⁄ √2 1 ⁄ √ 21 −1/2 −1/20 √3/2 −√3/2] [ iaibic ]

p = vαiα + vβiβ

q =vαiβ − vβiα

[ ica∗icb∗ica∗]=√ 23 [1/√2 1 ⁄ √2 1 ⁄ √21 −1/2 −1/20 √3/2 −√3/2] [ ic 0∗ica∗i cb∗][ ic𝛼∗ic 𝛽

∗ ]= 1𝜈𝛼

2+𝜈𝛽2 [𝜈𝛼−𝜈𝛽

𝜈𝛽𝜈𝛼 ][ pc∗qc∗ ]Ica*

Icb*

Icc*

Cour

tesy

: [4]

Block Diagram of Shunt APF compensation based on P-Q theory

Courtesy:[5]

Simulation results for constant instantaneous supply power strategy

Cour

tesy

: [5]

Simulink Model – Load Section

Simulink Model – Control Section

Simulation Video

Waveforms – Load Current

Waveforms – Compensation current

Waveform – Source current (compensated)

FFT of Load and Source currents

THD= 33.09%V1 = 22.16V5= 32(% of V1)V7= 2 (% of V1)V11= 7 (% of V1)

THD= 1.61%V1 = 22.16V5= 1.5(% of V1)V7= 0.2 (% of V1)V11= 0.1 (% of V1)

Developments in power theory• Original definitions of PQ theory were applicable only for

balanced 3 phase systems with or with out neutral wire• Later on P-Q theory for single phase and N Phase systems

were developed.• Some researchers argue that P-Q theory is not a complete

power theory and it doesn’t assign correct physical meaning to its power terms• Research is still going for developing unified power theory

electrical engineering applicable under any generic voltage and current conditions.• Some of these theories are : P-Q-r theory, CPC theory, Cross

vector theory, Hilbert space based power theory etc.

Conclusions• Instantaneous reactive power theory (P-Q theory) is one among

the modern power theories which is used in instantaneous evaluation of real and imaginary powers.

• It was introduced at first as a compensation theory, but later on formal definitions were given.

• It introduces the term instantaneous imaginary power q • Widely used in power conditioners for harmonic/reactive power

compensation.• Optimal for balanced/unbalanced,sinusoidal 3 Phase systems.• Latest reformulation of P-Q theory is P-Q-r theory.

References• [1] Mr. Suresh Kumar K.S &Dr. S. Ashok, “FACTS Controllers and

Applications”,Nalanda Digital Library at National Institute of Technology Calicut.,2003

• [2] Zainal Salam, Tan Perng Cheng and Awang Jusoh,“Harmonics Mitigation Using Active Power Filter: A Technological Review”, ELEKTRIKA, VOL. 8, NO.2, 2006, 17 26.‐

• [3] A. M. Massoud, S. J. Finney, and B. W. Williams, “Review of Harmonic Current Extraction Techniques for an Active Power Filter”, 2004 11th International Conference on Harmonics and Quality of Power.

• [4] Hirofumi Akagi, Mauricio Aredes, E.H. Watanabe, “Instantaneous Power Theory and Applications in Power Conditioning”, page no.53-220.

• [5] João Afonso, Carlos Couto, Júlio Martins, “Active Filters with Control Based on the p-q Theory”, IEEE Industrial Electronics Society Newsletter vol. 47, no 3, Sept. 2000, ISSN: 0746-1240, pp. 5-10.

• [6] Arun Jayendran, Sreeram V, Subin V Sivadas “Comparative Simulation Study of Harmonic Extraction Schemes in Active Power Filtering Applications”, published by Academy Publishers, Finland, in IJRTE( International Journal for Recent Trends in Engineering)

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