The day the current disappeared / APJ (Johan) Rens

The day the current disappeared

Johan.Rens@nwu.ac.za 3 May 2019

Contents

• Finding current: What did Maxwell miss?

• Why the Polish knight wanted to climb another mountain…

• That is why the current went missing!

• Until sanity was restored by the IEEE 1459-2010

• Application in impedance-constrained networks

A brief history of nearly every electron

• Electricity as a matter of physics since BC

• Found application during 1800’s: “electrical engineering” – using this resource to

do “something”, mostly to better life for mankind

• Moving electrically charged electrons and protons = current is flowing

• Mostly done in an alternating manner (ac electricity)

• How well it is done governed by the constant law of misery in electrical circuits:

Ohm

On average, the voltage remains shockingly

i(t)

v(t)

_source

v(t)

load

v(t)

_loss

That is why that single-phase generator remained so s

i(t)

v(t)

_source

v(t)

load

v(t)

_loss

v(t)

_b

v(t)

_c

v(t)

_a

i(t)

_a

i(t)

_b

i(t)

_c

p(t)

_a

p(t)

_c

p(t)

_b

p(t)

_3-phase

And if those voltages are no longer symmetrical and those currents

unbalanced?

The time-independent feature of three-phase energy transfer is lost……

It get worse when the load is linear (and supply voltage

non-sinusoidal)….

• What does non-linear loading mean?

• More or less the modern way of consuming current • Such as in a LED lamp, a laptop charger and and and • Nowadays, MW’s and kV’s

• The energy transfer is impacted • We are used to:

Not anymore….

2 2 2

S > P + Q

LED lamp

Now this is where the current got lost!

•

Distortion power?

•

What is that?

•

Something missing in power theory?

•

Budeanu defined the concept of distortion power – as “distortion” is the cause

•

Many contributions to power theory followed

•

Modern power phenomena required new definition

•

Engineers have to design, specify and operate power system in which energy

phenomena has physical intrepetation

•

Why is the term “distortion power” then still in use?

•

Discussion became intense………

Sixth International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions Milano, October 13-15, 2003

And then the Polish knight arrived…

But what is the problem?

• Steinmetz (1892): Ratio of active to apparent power decrease when waveform becomes more distorted such as electric arc (lighting application).

• Impact of distortion and unbalance new phenomena • Power factor reduction a concern

-• Unbalance in loading, asymmetry in supply voltages, AND distortion in voltage and/or current contributes to the degradation of power factor (the effiency in the transfer of real energy)

• Classical power theory can only deal with perfectly sinusoidal voltages, perfectly symmetrical between phases and perfectly linear loads that withdraw perfectly balanced currents

• Budeanu (1927) described S2 > P2+Q2 when waveforms are non-sinusoidal

 IEEE-1459-2010 attempted to further practical formulations universally acceptable for engineers to deal with modern power systems

Lyon 1920 Depenbrock 1977-2003

Bucholz 1922 Kusters and Moore 1980

Budeanu 1927 Page 1980 Fryze 1932 Nomowiesjki 1981 Goodhue 1933 Akagi-Nabae 1983….. Quade 1937 Filipski 1984 Nudelcu 1963 Sawicki 1986 Sharon 1972 Czarnecki 1987-….. Shepherd-Zakikhani 1972 Enslin 1988 Emanuel 1974 Tenti 1990 Harashima 1976 Ferrero 1991-….. Willems 1992-….. IEEE, Emanuel 1996-…..

Effective values: nonsinusoidal waveform conditions (1)

• A non-sinusoidal, single-phase, time-dependent voltage v(t)_1ϕwith fixed and repetitive

period T is applied to a load - represented as a finite series of harmonic components: h=1, 5, 7 • Single phase distortion component of v(t)_1ϕ can be isolated as v_H(t)_1ϕ:

• The IEEE 1459-2000 document further practical guidelines on power definitions •The time-domain or the frequency domain can be used for power definitions • Focus will be on frequency domain power definitions in this presentation

( )

₁

( )

₁

(

1

)

(

5

)

(

7

)

1

1

1

1sin

sin 5

sin 7

5

7

N h h

v t

_φ

v t

_φ

ω θ

t

ω θ

t

ω θ

t

=

=

∑

=

−

+

−

+

−

Effective values: nonsinusoidal waveform conditions (2)

• i(t)_1ϕ when v(t)_1ϕ is applied to frequency dependent impedance load:

( )

( )

(

)

[

]

,1 1 1 1 1 1 2 sin 1, 5, 7 N N h h h h h i t i t I h t h φ φ φ

ω

β

= = = = + =

∑

∑

• Distortion components i_H(t)_1ϕ:

( )

₁ 1 1

2

sin(

)

N H h h h

i

t

_φ

I

h t

ω

β

≠

=

∑

+

Active Power: nonsinusoidal waveform conditions

• The effective or RMS values: 1 2,1 1 2,1

1 1 ; N N h h h h V_{φ φ} I _{φ φ} = = =

∑

V =

∑

I

• The single phase power: 1

( ) ( )

_{1 1}

( )

₁

( )

₁

1 1

( )

N N h h h h

p

_φ

t

v t

_φ

i t

_φ

v t

_φ

i t

_φ = =







=

_{= }

_

_



∑



∑



( )

₁

Re

2

( )

₁

Re

2

( )

₁ h h h

p t

_φ

=

_



v

t

_φ



_

_



i

t

_φ



_

• Classical power theory formulates Time-dependent Active Power (per harmonic order h):

• The Time-dependent Total Active Power of a circuit under distorted waveform conditions:

( )

₁

( )

₁ 1 N h h p t _φ p t _φ = =

∑

• Total (or Joint) Average Active Power requires integration over a period T:

( )

(

)

1 ₁ ,1 ,1 1 1 cos N h h h h h T P p t dt V I T φ _φ φ φ

α

β

= =

∫

=

∑

−

Reactive Power: nonsinusoidal waveform conditions

• Classical power theory formulates Time-dependent Reactive Power (per harmonic order h):

• The Time-dependent Total Reactive Power of a circuit under distorted waveform conditions:

( )

( )

1 N h h q t q t = =

∑

• Total (or Joint) Average Reactive Power requires integration over a period T: Does it make sense?

( ) ( )

( )

( )

1 _{1 1} 1 1 1 1

( )

N N h h h h

p t

v t

i t

v t

i t

φ _{φ φ} φ φ = =

=

=

∑

∑

( )

₁ ,1

(

1

)

,1

(

1

)

1 1 2 2 2 ,1 ,1 ,1 1 1 , 1( )

2

sin

*

2

sin

N N h h h h h h N N N h B h k h h h k h k H B B

p t

V

h t

I

h t

P

Q

P

Q

D

φ φ φ φ φ φ

ω

α

ω

β

= = = = = ≠

=

+

+













=

_

_

+

_

_

_{+ }

_









_

_

=

+

+

∑

∑

∑

∑

∑

V

I

Total Active Power Total Reactive Power (Budeanu’s Reactive Power) Budeanu’s Distortion Power

What is wrong with Q

_B

– the Budeanu Reactive Power?

• Physical nature of reactive power follows from the application of field theory (Maxwell’s equations)

• Reactive Power not to contribute to real energy transfer

• Physical nature of reactive power - energy accumulation in electric and magnetic fields of reactive components in the load and source

• Results in oscillatory exchange of energy between these reactive components

• Similar explanation assigned to harmonic reactive power Q_h at each harmonic order h

• Is Q_B (Joint/Total Reactive Power) a useful concept? •Let’s investigate……

( )

( )

,1 ,1

(

)

1 1 1 1 sin N B _h h h h h h h T T Q q t q t V I T T φ φ

α

β

∞ = = =

∫

=

∫

∑

=

∑

−

Application of Q

_B

in power factor correction (1)

v(t)_1ϕ 10 ohm 100 mH i(t) • PF = 0.3 based on Q_B

Application of Q

_B

in power factor correction (2)

v(t)_1ϕ 10 ohm 100 mH 64 μF i(t) 2 2 2 2 1 1 5 5 7 7 11 11 B

Q

C

V

ω

V

ω

V

ω

V

ω

=

+

+

+

Application of Q

_B

in power factor correction (3)

• Q_B completely compensated by capacitor, but- power factor of compensated circuit did

not change significantly.

• Apparent power is less in compensated circuit but unnecessary loading remains (difference between apparent power and real power).

Remark

• Budeanu’s reactive power (Q_B) not useful for power factor compensation.

• Power factor correction results in “distortion power” (D_B) to increase significantly due to increased interaction between uneven harmonic voltage and current components.

v(t)_{1ϕ 1 ohm} i(t) Time V ol tage 1.676 1.676 − V_ac j 600 0 j Time C u rre n t 1.676 1.676 − I_{a c} j 600 0 j

S

1φ

1.194 VA

P

1φ

1.194 Watt

Q

B

0 VAr

D

B

0 VAr

• D_B has zero value: waveforms perfectly sinusoidal? • Both voltage and current are distorted!

Summary: Power theory in a modern

power system

• It must, as far as possible consist of a generalisation of the classic single-frequency power theory that has by now been universally accepted.

• It must be as amenable to conventional measurement techniques as possible and require the minimum of sophistication in instrumentation.

• It’s different defined components must be relatable to physically observable or ascribable phenomena and not to hypothetical or abstract mathematical

definition.

• It must present a suitable basis for quantifiable measurement, control, tariff systems and design.

• It must cater for every conceivable practical situation and never violate circuit laws, regardless of which domain it is transformed into.

• It must be useful to the engineer who has to apply these definitions in design, specification and operation of the power system.

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2. The IEEE 1459 made easy

What is a modern Power System? (1)

-400 -300 -200 -100 0 100 200 300 400 P h as e V ol ta ges (V )

Line-neutral voltages in a 400 V power system

-80 -60 -40 -20 0 20 40 60 80 P ha se c ur re nt s (A )

Three phase line-currents in a 400 V power system

Condition 3: Unbalance in loading… -1000 -800 -600 -400 -200 0 200 400 600 800 1000 C u rr en t (A )

-60000 -40000 -20000 0 20000 40000 60000 Li n e-L in e V ol ta ges ( V )

Asymmetrical voltages at PCC feeding Arc Furnace

• A Three-phase power system under non-sinusoidal waveform conditions, unbalanced loading and asymmetrical supply voltages is considered.

• Sinusoidal, balanced and single-phase power system operation is easily introduced as a simplification

Define nonsinusoidal line-neutral voltages and line currents:

, , 0 0 , , 0 0 , , 0 ( ) 2 sin( ) ( ) 2 sin( 120 ) ( ) 2 sin( 120 ) a a h a h h b b h a h h c c h c h h v t V h t v t V h t h v t V h t h

ω α

ω α

ω α

∞ ≠ ∞ ≠ ∞ ≠ = + = + − = + +

∑

∑

∑

0 , , 0 0 0 , , 0 0 0 , , 0

( )

2

sin(

)

( )

2

sin(

120

)

( )

2

sin(

120

)

a a a h a h h b b b h a h h c c c h c h h

i t

I

I

h t

i t

I

I

h t

h

i t

I

I

h t

h

ω

β

ω

β

ω

β

∞ ≠ ∞ ≠ ∞ ≠

=

+

+

=

+

+

−

=

+

+

+

∑

∑

∑

• DC components in voltage, (V_a0, V_b0 and V_c0) should always be zero.

• DC values in line currents (I_a0, I_b0 and I_c0) could be nonzero depending on nature of load.

• The RMS line-neutral voltage V_a and RMS line current I_a (similarly for phase b and c) are related to harmonic components by:

2 2 2 2 2 2 2 ,1 , ,1 , 2 2 2 2 2 2 2 2 2 ,1 , ,1 , 2 2

(

)

(

)

a a a h a aH aH a h h h a a a h a aH aH a h h h

V

V

V

V

V

V

V

I

I

I

I

I

I

I

∞ ∞ ≠ ≠ ∞ ∞ ≠ ≠

=

+

=

+

=

=

+

=

+

=

∑

∑

∑

∑

• Fundamental frequency and the nonfundamental frequency (harmonic frequencies grouped)

☝

What is the “effect” of three-phase voltages and three-phase currents in terms of the three-phase power system?

“Effective” voltage and “effective” current to represent the state of the

three-phase power system:

2 2 2 1 2 2 2 1 e e eH e e eH

V

V

V

I

I

I

=

+

=

+

2 2 2 2 2 2

3

3

a b c e a b c e

V

V

V

V

I

I

I

I

+

+

=

+

+

=

☝

If an artificial neutral point in a 3-wire three-phase system is not used to find the line-neutral voltage values, the effective three-phase voltage can be

calculated from the RMS phase-phase voltage values as:

2 2 2

9

ab bc ca e

V

V

V

V

=

+

+

• The fundamental frequency

componens of the effective voltage

and current in a 3-wire three-phase

power system: 2 2 2 ,1 ,1 ,1 ,1 2 2 2 ,1 ,1 ,1 ,1

9

3

ab bc ca e a b c e

V

V

V

V

I

I

I

I

+

+

=

+

+

=

• Non-fundamental frequency components of the effective voltage and current in a 3-wire three-phase power system:

2 2 2 2 2 2

9

3

abH bcH caH eH aH bH cH eH

V

V

V

V

I

I

I

I

+

+

=

+

+

=

(

)

(

)

2 2 2 , , , 1 2 2 2 , , , 1

3

3

a h b h c h h eH a h b h c h h eH

V

V

V

V

I

I

I

I

∞ ≠ ∞ ≠

+

+

=

+

+

=

∑

∑

• Non-fundamental frequency components

relates to the harmonic components (line-neutral voltages assumed):

• Unbalanced condition in a

4-wire three-phase power system

requries reformulation of the

effective voltage and current :

(

2 2 2

)

2 2 2 2 2 2 2

1

3

18

3

e a b c ab bc ca a b c n e

V

V

V

V

V

V

V

I

I

I

I

I





=

_

+

+

+

+

+

_

+

+

+

=

• The fundamental frequency components of

the effective voltage and

current in a 4-wire

three-phase power system:

(

2 2 2

)

2 2 2 ,1 ,1 ,1 ,1 ,1 ,1 ,1 2 2 2 2 ,1 ,1 ,1 ,1 ,1

1

3

18

3

e a b c ab bc ca a b c n e

V

V

V

V

V

V

V

I

I

I

I

I





=

_

+

+

+

+

+

_

+

+

+

=

• The non-fundamental frequency components of the effective voltage and

current in a 4-wire

three-phase power system:

(

2 2 2

)

2 2 2 2 2 2 2

1

3

18

3

eH aH bH cH abH bcH caH aH bH cH nH eH

V

V

V

V

V

V

V

I

I

I

I

I





=

_

+

+

+

+

+

_

+

+

+

=

☞

Implementation straightforward with modern digital instrumentation

• Arithmetic apparent power S_A: 2 2 2 2 2 2 2 2 2 a a Ba Ba b b Bb Bb c c Bc Bc A a b c

S

P

Q

D

S

P

Q

D

S

P

Q

D

S

S

S

S

=

+

+

=

+

+

=

+

+

=

+

+

• Active, Budeanu’s reactive and

distortion power is summated over all three phases:

• Vector apparent power S_v:

• The Budeanu power definition for single-phase power systems are applied per phase ,3 a b c B Ba Bb Bc B Ba Bb Bc

P

P

P

P

Q

Q

Q

Q

D

D

D

D

φ Σ

=

+ +

=

+

+

=

+

+

2 2 2 ,3 V B B S = P_{Σ φ} + Q + D

And the effective apparent power

is…..

The three-phase or system effective apparent power S_e can be written in terms of the contribution of all harmonic components:

2 2 2

,1

e e eN

S

=

S

+

S

The components in the system effective apparent power S_e can be grouped in the

fundamental and nonfundamental frequency voltage and current components:

The non-fundamental frequency apparent power S_eN consist of 3 “distortion” components:

And the “distortion” powers are…..

V_e,1I_eH: The current distortion power, D_eI

V_eHI_e,1: The voltage distortion power, D_eV

V_eHI_eH: The harmonic distortion power, D_eH

An effective harmonic apparent power S_eHis defined:

2 2 2 ,3 eH H eH

S

=

P

_φ

+

D

(

) (

2

)

2

(

)

₂ 2 ,1 ,1 2 2 2 eN e eH eH e eH eH eI eV eH

S

V I

V I

V I

D

D

D

=

+

+

=

+

+

The Total Three-Phase Active Power

The Joint (Total) Harmonic Active Power of three-phase power system:

(

)

3 1 3 1 Re N H , ah ah bh bh ch ch h N h, h P P φ φ ∗ ∗ ∗ ≠ ≠ = + + =

∑

∑

V I V I V I

The Joint Active Power of three-phase power system:

( )

( )

(

)

3 _{3 3} 1 3 1 Re N ah ah bh bh ch ch h N h, h P t t P φ _{φ φ} φ ∗ ∗ ∗ = =   =   = + + =

∑

∑

v ,i V I V I V I

Quantification on the level of distortion is done with three-phase effective values: ,1 , ,1 eH e e e H e e

V

VTHD

V

I

ITHD

I

=

=

☞

Voltage total harmonic distortion factor: VTHD_e

☞

Current total harmonic distortion factor: ITHD_e

A shortcut to S_eN:

(

)

2

1

eN e e e e e

S

=

S

VTHD

+

ITHD

+

VTHD ITHD

And to D_eI; D_eV; D_eH (The “distortion” powers):

1 1 1 eI e e eV e e eH e e e

D

S ITHD

D

S VTHD

D

S ITHD VTHD

=

=

=

To do what with?

Harmonic pollution (S_eN/S_e1):

Unbalance pollution:

☝

The positive sequence voltage (V₁+_{) and current (I}

1+) in the three-phase fundamental frequency components have to be found…..

Then calculate:

The “unbalance pollution”:

☝

“Unbalance pollution” includes both the effect on loading unbalance and voltage asymmetry

☞

The positive sequence components in the three-phase fundamental frequency voltages and currents:



The 50 Hz positive sequence active power:



The 50 Hz positive sequence reactive power:



The 50 Hz positive sequence apparent power: And power factor?

Various options exist….

Different apparent powers have been formulated: • Arithmetic apparent power, S_A

• Vector apparent power, S_v

• Three-phase effective apparent power, S_e • Positive sequence apparent power, S₁+

1 1 P PF S + + + = A A P PF S = V V P PF S = Different apparent power factors have been formulated:

• Arithmetic apparent power factor • Vector apparent power factor

• Three-phase effective apparent power factor • Positive sequence apparent power factor

If the waveforms are sinusoidal, the loading in perfect balance and the supply voltages in perfect symmetry, then:

A V e

PF = PF = PF = PF+

In a practical power system with distorted waveforms, unbalanced loading and asymmetrical supply voltages:

e A V

PF < PF < PF

• PF_e reflects the impact of harmonics and asymmetrical waveforms the best.

☞

The smallest numerical value – regulatory application

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3. Case Study

Application of IEEE 1459-2000 - practical power system

R_s + ωL_s 6-pulse controller Heating load R_L + ωL_L 11kV/400 V 1 MVA 0.01+j0.043 p.e. V and I measurements PCC

Voltage and Current waveforms

-400 -300 -200 -100 0 100 200 300 400 Vol ta ge (V) Line-neutral voltages Va Vb Vc -800 -600 -400 -200 0 200 400 600 800 C u rr en t (A ) Line currents Ia Ib Ic • Line-neutral voltages • Line Currents

The IEEE 1459 power definitions require translating time-domain waveforms to the frequency domain by means of the Fourier transform:

The level of distortion requires quantification: • ITHD_e= 48.3%

• The non-fundamental frequency components of the effective voltage and

current:

(

2 2 2

)

2 2 2 2 2 2 2

1

3

18

17.2 V

3

145.8 A

eH aH bH cH abH bcH caH aH bH cH nH eH

V

V

V

V

V

V

V

I

I

I

I

I





=

_

+

+

+

+

+

_

=

+

+

+

=

=

• The fundamental frequency components of

the effective voltage and

current:

(

2 2 2

)

2 2 2 ,1 ,1 ,1 ,1 ,1 ,1 ,1 2 2 2 2 ,1 ,1 ,1 ,1 ,1

1

3

18

227.6 V

3

301.5 A

e a b c ab bc ca a b c n e

V

V

V

V

V

V

V

I

I

I

I

I





=

_

+

+

+

+

+

_

=

+

+

+

=

=

The effective voltage and current :

The RMS line-neutral voltages per phase:

2 , 1 229.2 V 227.7 V 227.8 V a a h h b c V V V V ∞ = = = = =

∑

2 , 1 315.4 A 342.9 A 345.8 A 8.1 A a a h h b c n I I I I I ∞ = = = = = =

∑

The RMS line currents per phase:

The powers

1

7.5 kVar

eH e e e

D

=

S ITHD VTHD

=

Current distortion power:

Voltage distortion power: Harmonic distortion power:

Effective apparent power: Arithmetic apparent power:

Vector apparent power:

_S

_V

₌

_{199.7 KVA}

Joint (three-phase) active power:

(

)

3 3 1 1 Re 387 W N N H , ah ah bh bh ch ch h, h h P _φ ∗ ∗ ∗ P _φ ≠ ≠ =

∑

V I +V I +V I =

∑

= −

( )

( )

(

)

3 _{3 3} 3 1 1 Re 92 9 kW N N ah ah bh bh ch ch h, h h P_φ t _φ t _φ ∗ ∗ ∗ P _φ . = =   = _v ,i _ =

∑

V I +V I +V I =

∑

=

Joint harmonic active power:

1 229.4 KVA e S = 199.7 KVA A S = 1

99.9 kVar

eI e e

D

=

S ITHD

=

1

15.5 kVar

eV e e

D

=

S VTHD

=

And the power factors….

The arithmetic power factor: 0.465 p.u. The vector power factor: 0.465 p.u. The effective power factor: 0.405 p.u

( ) (

) (

2

) (

2

)

2 1 % *100 3.6% eN e e e e e S ITHD VTHD ITHD VTHD S = + + =

And the “pollution” factors….

Harmonic pollution: Unbalance pollution: 1 21

( )

1 2 1 1 9.1 KVA (%) *100 4.4% u e u e S S S S UnbalancePollution S + = − = ∴ = = Unbalance factors: 2 1 2 1 0.4% 3% UB UB V V V I I I = = = =

☞

The three-phase effective power factor is numerically the smallest if unbalance and/or waveform distortion exist.

☞

The contribution to apparent power by both voltage asymmetry and unbalance in loading, was shown to be significant even with “low” values in V_UB and I_UB.

☞

The three-phase effective distortion index for voltage (VTHD_e) and current (ITHD_e) furthers straightforward calculation of distortion powers.

☞

It is possible to isolate the contribution of harmonics to useles power by means of voltage distortion power (D_V), current distortion power (D_I) and harmonic distortion power (D_H).

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It is a helpful reflection on the impact of nonsinusoidal waveforms,

unbalanced loading and asymmetrical supply voltages from the point of view from operating such power system.

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Conclusion on Power Theory

• The classical power theory explains phenomena in a power system with a sound physical explanation but is inadequate when waveforms are non-sinusoidal,

loading is unbalanced and voltage waveforms are asymmetrical.

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Conform to Electrical Network Laws

• Numerous approaches to power theory exists which are formulated in either the time- or frequency domain.

• New contributions are forthcoming as inadequacies are better understood. • In general, a power theory has to:

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Explain Physical Phenomena

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Be Measurable

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Enables Compensation