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 zero

i(t)

v(t)

_source

v(t)

load

v(t)

_loss

Resistive component of loadReactive component: L, C or both

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 V ol ta ge time (ms) p(t) = v(t) * i(t)

= V_max*I_max * cos(wt +d)cos(wt +b) = 1

2 2Vrms 2Irmscos(d -b)+ VrmsIrmscos(d -b)cos 2(

[

wt +d)

]

+ VrmsIrmssin(d -b)sin 2éë

(

wt +d

)

ùû = V_RMSI_RMScos

(

d -b

)

{

1+ cos 2_éë

(

wt +d

)

_ùû

}

+ V_RMSI_RMSsin

(

d - b

)

sin 2_éë

(

wt +d

)

_ùû

That is why that single-phase generator remained so small….

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

The War of Currents winner : George Westinghouse; 1886

v

_a

(

t) = 2V

_LN

cos(

w

t +

d

)

v

_b

(

t) = 2V

_LN

cos(

w

t +

d

- 120

0

₎

v

_a

(

t) = 2V

_LN

cos(

w

t +

d

- 240

0

₎

i

_a

(

t) = 2I

_L

cos(

w

t +

b

)

i

_b

(

t) = 2I

_L

cos(

w

t +

b

- 120

0

₎

i

_c

(

t) = 2I

_L

cos(

w

t +

b

- 240

0

₎

The War of Currents winner : George Westinghouse; 1886

p_3j(t) = p_a(t) + p_b(t) + p_c(t)

= 3V_LNI_L cos(

d

-

b

) + V_LNI_L cos(

d

-

b

) + V_LNI_L[cos(

d

-

b

) + V_LNI_L cos(2

w

t +

d

+

b

)+ ... ....+ cos(2

w

t +

d

+

b

- 2400_{)+ cos(}

_d

_-

_b

_{) + V}

LNIL cos(2

w

t +

d

+

b

+ 240

0_)] = 3V_LNI_L cos(

d

-

b

) = P_3j

And if those voltages are no longer symmetrical and

those currents unbalanced?

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

It get worse when the load is non-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…. LED lamp Laptop charger

S= P + jQ

\ S

2

_=P

2

_+Q

2 2 2 2

S > P + Q

D

2

=S

2

- P

2

- Q

2

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

And then the Polish knight arrived…

Shouldn’t we rather abandon the

knighthood?

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

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 _ v t _

w 

t

w 

t

w 

t = =

v

_H



( )

t

= - + - + -1

=

2

V

h

sin(

hwt+ a

h

)

h¹1 N



=

1

5

sin 5

(

wt-

5

)

+

1

7

sin 7wt-

(

7

)

Effective values: nonsinusoidal waveform conditions (2)

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

( )

_H(t)_1ϕ

( )

:

(

)

[

]

,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   

w

b

= = = = + =





( )

₁ 1 1

2 sin(

)

N H h h h

i t

_

I

h t

w

b

¹

=



+

Active Power: nonsinusoidal waveform conditions

• The effective or RMS values: • The single phase power:

• 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:

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

2 2 1 ,1 1 ,1 1 1 ; N N h h h h V_{ } I _{ } = = =



V =



I

( ) ( )

( )

( )

1 _{1 1 1 1} 1 1

( )

N _h N _h h h

p t

_

v t

_

i t

_

v t

_

i t

_ = =







=

_{= }

_

_











( )

₁

Re 2

( )

₁

Re 2

( )

₁ h h h

p t

_

=

_ë

é

v

t

_

ù

_û

_ë

é

i

t

_

ù

_û

( )

₁

( )

₁ 1 N h h p t _ p t _ = =



( )

(

)

1 ₁ ,1 ,1 1 1 cos N h h h h h T P p t dt V I T  _   a b = =



=



-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:

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

☞

It does not make sense!

q t

( )

h

= Re 2v t

é

ë

( )

h

ù

û Im 2i t

é

ë

( )

h

ù

û

( )

( )

1 N h h q t q t = =



Q_B = 1 T _T



q t

( )

= 1 T q t

( )

h h=1 N



= T



V_h,1I_h,1 sin

(

a

_h -

b

_h

)

h=1 ¥



Summary of Budeanu’s definitions

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

( ) ( )

( )

( )

1 _{1 1} 1 1 1 1

( )

N N h h h h

p t

v t

i t

v t

i t

 _{ }   = =

=

=





( )

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

     

w

a

w

b

= = = = = ¹

=

+

+













=

_

_

+

_

_

_{+ }

_









_

_

=

+

+











V

I

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   a b ¥ = = =



=





=



-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

v t

( )

_1

=

100

h

sin 2

(

×π × 50 × h×t

)

;

hÎ 1,5,7,11

[

]

Application of Q

_B

in power factor correction (2)

v(t)_1ϕ 10 ohm

100 mH

64 μF

i(t)

• Calculate the capacitance in parallel to compensate Q_B

2 2 2 2 1 1 5 5 7 7 11 11 B

Q

C

V

w

V

w

V

w

V

w

=

+

+

+

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.

Does distortion power D

_B

make sense?

v(t)_{1ϕ 1 ohm} i(t) Time V ol ta ge 1.676 1.676 -V_{ac j} 600 0 j Time C ur re nt 1.676 1.676 -I_{ac j} 600 0 j

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

• D_B does not relate to degree of waveform distortion.

v t

( )

_1

=

1

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.

LPQI is part of

LPQI-VES is co-financed by

2. The IEEE 1459 made easy

What is a modern Power System? (1)

-400 -300 -200 -100 0 100 200 300

400 Line-neutral voltages in a 400 V power system

P h a s e V o lt a g e s ( V )

What is a modern Power System? (2)

-80 -60 -40 -20 0 20 40 60 80

Three phase line-currents in a 400 V power system

P h a s e c u rr e n ts ( A )

What is a modern Power System? (3)

Condition 3: Unbalance in loading…

-1000 -800 -600 -400 -200 0 200 400 600 800

1000 Load Currents withdrawn by Arc Furnace

C u rr e n t (A )

What is a modern Power System? (3)

-60000 -40000 -20000 0 20000 40000 60000

Asymmetrical voltages at PCC feeding Arc Furnace

L in e -L in e V o lt a g e s (V )

The IEEE 1459-2000: Voltage and Current (1)

• 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 w a w a w a ¥ ¹ ¥ ¹ ¥ ¹ = + = + -= + +







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

w

b

w

b

w

b

¥ ¹ ¥ ¹ ¥ ¹ = + + = + + -= + + +







The IEEE 1459-2000: Voltage and Current (2)

• 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:

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

¥ ¥ ¹ ¹ ¥ ¥ ¹ ¹

=

+

=

+

=

=

+

=

+

=









The IEEE 1459-2000: Voltage and Current (4)

☝

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:

“Effective” voltage and “effective” current in a 3-wire 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

+

+

=

+

+

=

Calculating the effective voltage and current (1)

☝

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:

• The fundamental frequency

componens of the effective voltage

and current in a 3-wire three-phase

power system: 2 2 2

9

ab bc ca e

V

V

V

V

=

+

+

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

+

+

=

+

+

=

Calculating the effective voltage and current (2)

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

• Non-fundamental frequency components relates to the harmonic components (line-neutral voltages assumed):

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

¥ ¹ ¥ ¹

+

+

=

+

+

=





Calculating the effective voltage and current (3)

• Unbalanced condition in a

4-wire three-phase power system

requries reformulation of the

effective voltage and current :

• The 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

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

V

V

V

V

V

V

V

I

I

I

I

I

é

ù

=

_ë

+

+

+

+

+

_û

+

+

+

=

(

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

é

ù

=

_ë

+

+

+

+

+

_û

+

+

+

=

Calculating the effective voltage and current (4)

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

current in a 4-wire

three-phase power system:

☞

Implementation straightforward with modern digital instrumentation

(

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

é

ù

=

_ë

+

+

+

+

+

_û

+

+

+

=

And power? Not all S is the same S…..

• Arithmetic apparent power S_A:

• 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 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

=

+

+

=

+

+

=

+

+

=

+

+

,3 a b c B Ba Bb Bc B Ba Bb Bc

P

P

P

P

Q

Q

Q

Q

D

D

D

D

 

=

+

+

=

+

+

=

+

+

2 2 2 ,3 V B B S = P_{ } + 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:

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:

S

_e2

_{= (V}

e

I

e

)

2

_{= (V}

e,1

I

e,1

)

2

_{+ (V}

e,1

I

eH

)

2

_{+ (V}

eH

I

e,1

)

2

_{+ (V}

eH

I

eH

)

2 2 2 2 ,1 e e eN

S

=

S

+

S

S

_eN2

_{= V}

e,1

I

eH

(

)

2

+ V

(

_eH

I

_e,1

)

2

+ V

(

_eH

I

_eH

)

2

= D

_eI2

_{+ D}

eV 2

_{+ D}

eH 2

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_eH is defined:

2 2 2 ,3 eH H eH

S

=

P

_

+

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:

The Joint 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      ¹ ¹ = + + =





V I V I V I

( )

( )

(

)

3 _{3 3} 1 3 1 Re N _{ah ah bh bh ch ch} h N h, h P t t P  _{ }     = = é ù = ë û = + + =





v ,i V I V I V I

And how much is that “distortion”?

Quantification on the level of distortion is done with three-phase effective values:

☞

Voltage total harmonic distortion factor: VTHD_e

☞

Current total harmonic distortion factor: ITHD_e

☞

The IEEE 1459-2000 write these symbols as THD_eV and THD_eI

,1 , ,1 eH e e e H e e

V

VTHD

V

I

ITHD

I

=

=

These distortion factors are useful!

A shortcut to S_eN:

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

To do what with?

(

)

2 1 eN e e e e e

S

=

S VTHD

+

ITHD

+

VTHD ITHD

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

D

S ITHD

D

S VTHD

D

S ITHD VTHD

=

=

=

To quantify the pollution….

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

S

_eN

S

_e1









2

= ITHD

(

_e

)

2

+ VTHD

(

_e

)

2

+ ITHD

(

_e

VTHD

_e

)

2

S

1 +

_{= 3V}

1 +

_I

1 +

S

_u1

= S

_e12

_{- S}

1 +

( )

2

UnbalancePollution(%) =

S

u1

S

_e1

*100

What powers are “useful”?

☞

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 answers to the same question!

P

₁+

= 3V

₁+

I

₁+

cos



₁+

Q

1 +

_{= 3V}

1 +

_I

1 +

_sin

_

1 + S₁+

( )

2 = P

( )

₁+ 2 _{+ Q} 1 +

( )

2

Power factor formulations

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+

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 1 1 P PF S + + + = A A P PF S = V V P PF S = PF_e = P,3 S_e = P_1,3 + P_{H ,3} S_e

Which power factor should be used?

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

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

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

☞

The smallest numerical value – regulatory application

☝

What power factor formulation does your instrument use?

A V e

PF = PF = PF = PF+

e A V

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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 Va Vb Vc Line-neutral voltages V o lt ag e ( V ) -800 -600 -400 -200 0 200 400 600 800 Ia Ib Ic Line currents C u rr e n t (A ) • Line-neutral voltages • Line Currents

Waveform analysis: Fourier transform

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%

Isolating the distortion components

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

current:

• The 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

é

ù

=

_ë

+

+

+

+

+

_û

=

+

+

+

=

=

(

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

é

ù

=

_ë

+

+

+

+

+

_û

=

+

+

+

=

=

The three-phase effective voltage and current

The effective voltage and current :

The RMS line-neutral voltages per phase:

The RMS line currents per phase:

The RMS neutral current:

=

2₁

₊

2

= 228.3 V

2

₌

1 2

₊

2

= 335 A

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 powers

Current distortion power: Voltage distortion power: Harmonic distortion power:

Effective apparent power: Arithmetic apparent power: Vector apparent power:

Joint (three-phase) active power: Joint harmonic active power:

1

7.5 kVar

eH e e e

D

=

S ITHD VTHD

=

199.7 KVA

V

S

=

(

)

3 3 1 1 Re N N 387 W H , ah ah bh bh ch ch h, h h P _    P _ ¹ ¹ =



V I +V I +V I =



=

-( )

( )

(

)

3 _{3 3} 3 1 1 Re N _{ah ah bh bh ch ch} N _h, 92 9 kW h h P_ t _ t _    P _ . = = é ù = _ëv ,i _û =



V I +V I +V I =



= 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

And the “pollution” factors….

Harmonic pollution: Unbalance pollution: Unbalance factors:

( )

(

) (

2

) (

2

)

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

( )

2 2 1 1 1 1 1 9.1 KVA (%) *100 4.4% u e u e S S S S UnbalancePollution S + = - = \ = = 2 1 2 1 0.4% 3% UB UB V V V I I I = = = =

Summary

☞

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).

☞

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

Conclusion

• 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.



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:



Explain Physical Phenomena



Be Measurable



Enables Compensation