conservative power theory cpt

Conservative Power TheoryConservative Power TheoryA theoretical background to understand energy issue s of A theoretical background to understand energy issue s of

electrical networks under nonelectrical networks under non --sinusoidal conditionssinusoidal conditions

UNICAMP UNICAMP –– UNESP SorocabaUNESP SorocabaAugust 2012 August 2012

1

electrical networks under nonelectrical networks under non --sinusoidal conditionssinusoidal conditions

andand

to approach measurement, accountability and control to approach measurement, accountability and control problems in smart gridsproblems in smart grids

Paolo TentiDepartment of Information Engineering

University of Padova, Italy

Seminar OutlineSeminar Outline1. Motivation of work

2. Mathematical and physical foundations of the theor y• Mathematical operators and their properties• Instantaneous and average power & energy terms in p oly-

phase networks

3. Definition of current and power terms in single -phase

2

3. Definition of current and power terms in single -phase networks under non-sinusoidal conditions

4. Extension to poly-phase domain: 3-wires / 4-wires

5. Sequence components under non-sinusoidal conditions

6. Measurement & accountability issues

Describe physical phenomena energy transfer, energy storage, rate of utilization of power sources and distributio n

infrastructure, unwanted voltage and current terms, ….

Allow unambiguous measurement of quantities

1. Motivation of work1. Motivation of workWhy do we need to define power termsWhy do we need to define power terms

3

Allow unambiguous measurement of quantities load and source characterization, revenue metering, …

Compensation identify provisions which make the equipment or the

plant compliant with standards & regulations in ter ms of symmetry, purity of waveforms, power factor …

While the definition and meaning of instantaneous power and its average value ( active power ) are universally agreed, the situation is less clear wit h other popular power terms

What is/means reactive power ?

What is/means distortion power ?

1. Motivation of work1. Motivation of workFew basic questionsFew basic questions

4

What is/means distortion power ?

What is/means apparent power ?

These power terms are unambiguously defined when at least the voltage supply is sinusoidal, but are mat ter of controversial discussions (since nearly one century ) in case of distorted voltages and currents.

In the frequency domainIn the frequency domain BudeanuBudeanu (1927)(1927) SheperdSheperd & & ZakikhaniZakikhani (1971)(1971) CzarneckiCzarnecki (1984 …)(1984 …)

In the time domainIn the time domain FryzeFryze (1931)(1931)

1. Motivation of work1. Motivation of workMilestones of power theory historyMilestones of power theory history

55

KustersKusters & Moore (1975)& Moore (1975) DepenbrockDepenbrock (1993)(1993) AkagiAkagi & & NabaeNabae (1983)(1983)

No one of these theories was able to target all goa ls (characterization No one of these theories was able to target all goa ls (characterization of physical phenomena, load & line identification, compensation).of physical phenomena, load & line identification, compensation).

The timeThe time--domain theory presented here domain theory presented here tries to target all go als at the tries to target all goals at the same time.same time.

It represents an outcome of It represents an outcome of a longa long--standing cooperation between standing cooperation between UNIPD, UNICAMP and UNESP.UNIPD, UNICAMP and UNESP.

In modern scenarios (e.g., micro-grids) where: the grid is weak, frequency may change, voltages may be asymmetrical, distortion may affect voltages and currents,

are the usual definitions of reactive, unbalance an d distortion power still valid ?

1. Motivation of work1. Motivation of workNeed for a revision of power termsNeed for a revision of power terms

6

distortion power still valid ?

Which is the physical meaning of such terms ?

Are they useful for compensation ?

To which extent are power measurements affected by source non-ideality ?

It is possible to identify supply and load responsi bility on voltage distortion and asymmetry at a given net work port ?

Conservative Conservative Power TheoryPower Theory

• Definition of mathematical operators and their

2. 2. Mathematical and physical foundations

7

• Definition of mathematical operators and their properties

• Definition of instantaneous power and energy terms

• Conservative quantities

• Selection of voltage reference

• Definition of average power terms and their physica l meaning in real networks

Let T be the period of variables x and y, we define:

• Average value

• Time derivative

• Time integral

Mathematical operators for periodic Mathematical operators for periodic scalar quantitiesscalar quantities

( )∫==T

dttxT

xx0

1

dt

xdx=(

( )∫=t

dxx ττ

8

• Time integral

• Unbiased time integral

• Internal product

• Norm ( rms value )

• Orthogonality

( )∫=∫ dxx0

ττ

∫∫ −= xxx)

dtyxT

yxT

∫ ⋅=0

1,

xxxX ,==

0, =yx

Let x and y be vector quantities of size N, we define:

• Scalar product

• Magnitude

Mathematical operators for periodic Mathematical operators for periodic vector quantitiesvector quantities

n

N

nn yxyx ∑

==

1

o

∑=

==N

nnxxxx

1

2o

∑N

9

• Internal product

• Norm

• Orthogonality

• The vector norm is also called collective rms value

∑=

==N

nnn yxyxyx

1

,, o

∑∑==

===N

nn

N

nnn Xxxx

1

2

1

,X

0, =yx

The above operators have the following properties:

• Orthogonality

• Equivalences

Properties of mathematical operatorsProperties of mathematical operators(valid for scalar and vector quantities)(valid for scalar and vector quantities)

0,

0,

0,

0,

==

⇒==

xx

xx

xx

xx)

(

)

(

yxyxyxyx((

(( ,,,, −=−=

10

• For sinusoidal quantities

yxyxyx

yxyx

yxyx

yxyxyx

yxyx

yxyx

())(

))

())(

))

,,,

,,

,,

,,,

,,

,,

−=−=

−=

−=

⇒

−=−=−=−=

xxxX()

ωω 1===

ϕcos, XYyx = ϕω

senXYyx1

, =)

22

22222 2X

xxxx =+=+

ωω

(

)

Instantaneous power definitionsInstantaneous power definitions(for periodic variables)(for periodic variables)

Instantaneous (active) power: ∑∑ ==⋅=N

n

N

nn piuiup

Given the vectors of the N phase currents in and voltages un measured at a generic network port we define:

11

Instantaneous reactive energy(new definition ):

Instantaneous (active) power: ∑∑== n

nn

nn11

• Both quantities do not depend on the voltage refere nce• Both quantities are conservative in every real network

∑∑==

==⋅=N

nn

N

nnn wiuiuw

11

))

Conservation of instantaneous Conservation of instantaneous power and reactive energypower and reactive energy

For every real network π, let u and i be the vectors of the L branch voltages and currents, we claim that:

Branch voltages , their time derivative and unbiased integral are consistent with network π, i.e. they comply with KLV (Kirchhoff’s law for voltages)

12

Branch currents , their time derivative and unbiased integral are consistent with network π, i.e. they comply with KLC (Kirchhoff’s law for currents)

Thus, according to Tellegen’s Theorem allquantities shown here areconservative 0

0

0

=⋅=⋅=⋅=⋅

=⋅=⋅=⋅

iuiu

iuiu

iuiuiu

((

))

)(

()

Average power definitionsAverage power definitions(valid for periodic quantities)(valid for periodic quantities)

Reactive energy:

P

iuA

i,ui,uwW

i,upP

=λ

==

−===

==

IU

))

Active power:

Apparent power:

Power factor:

13

A=λPower factor:

• All quantities are defined in the time domain.• Reactive energy is a new definition , whose properties will be

analyzed in the following.• Active power and reactive energy are conservative

quantities which do not depend on the voltage refer ence.• Unlike P and W, apparent power A is non-conservative and

depends on the voltage reference . Skip voltage reference

Selection of voltage reference (1)Selection of voltage reference (1)

The equal sign is possible if:

1, ≤=⇒≤A

Piuiu λCauchy-Schwartz

inequality:

1, =⇒=⇒∝ λiuiuiu

14

We select the voltage reference so as to ensure uni ty power factor in case of symmetrical resistive load. This gives a physical meaning to the apparent power , which is the maximum active power that a supply line rated for Vrms Volts and Irms Amps can deliver to a (purely resistive and symmetrical) load .

Selection of voltage reference (2)Selection of voltage reference (2)NN--phase systems without neutral wirephase systems without neutral wire

Thus, the voltage reference must be selected to com ply

∑ ∑= =

=⇒=

=N

n

N

nnn ui

iRu

1 1

00

The proportionality condition between phase voltages and currents for symmetrical resistive load determines voltage reference

15

Thus, the voltage reference must be selected to com ply with the zero-sum condition:

( ) ∑∑∑===

=⇒=−⇒=N

nnref

N

nu

refn

N

nn measure

n

measureu

Nuuuu

111

100

44 344 21

This choice minimizes the norm of the voltage vector

Selection of voltage reference (3)Selection of voltage reference (3)NN--phase systems without neutral wirephase systems without neutral wire

R3

R2

L3

L2

i3

u21

u32

3

2

1

uSn

LOADLINESOURCEMeasurement of voltages and currents

16

R1 L1i1

u1

LLnRLnn∈ ∈ ∈ ∈ 1,2,3

∑=

≠=N

jjnnjn u

Nu

1

1

∑∑= =

≠=N

n

N

jjnnj u

Nu

1 1

22

2

1Nnuu

Nu

N

jjnnjn ÷=

−= ∑

=≠ 1,

1 2

1

22

Derivation of phase voltages

Selection of voltage reference (4)Selection of voltage reference (4)NN--phase systems with neutral wirephase systems with neutral wire

Nn,iRuuu nnoref ÷==⇒== 00

In case of symmetrical resistive load the proportio nality condition between phase voltages and currents holds only if the voltage reference is set to the neural wire.

17

Unity power factor may occur only if the neutral cu rrent is disregarded for apparent power computation (only ph ase currents are considered). Thus:

≠=

==== ∑∑∑∑

====

N

nn

N

nn

N

nn

N

nn IIUUPA

0

2

1

2

0

2

1

2, IUIU

Selection of voltage reference (5)Selection of voltage reference (5)NN--phase systems with neutral wirephase systems with neutral wire

R3

R2

R

L3

L2

L

3

2

1

LOADLINE

i1

i2

i3

SOURCE

Measurement of voltages and currents

18

R1 L1u30 u20 u10vSn

LLnRLnn∈ ∈ ∈ ∈ 0,1,2,3

i1

0

∑∑==

==N

nn

N

nn IU

1

2

1

2 IUCollective rms voltage and current

Homopolar voltage and current ∑

==

N

nn

z uN

u1

1

N

ii

Ni o

N

nn

z −== ∑=1

1

Resistor

uGi

iRu

=

=

Power terms in passive networksPower terms in passive networks

19

0,,

,22

===

===

iiRiuW

iRuGiuP

R

R

))

uGi =

L

ui

iLdt

diLu

)

(

=

==

Inductor


20

L

2,,0,, iLiiLiuW

L

uuiuP LL ====== )

)

22

1

2

1 22 LLLL

WiLiL ==Ε=⇒= εεInductorInductor

energyenergy

C

iu

uCdt

duCi

)

(

=

==

Capacitor


21

C

2,,0,, uCuCuiuWi

C

iiuP CC −=−=−====

)

)

CapacitorCapacitorenergyenergy 22

1

2

1 22 CCCC

WuCCu −==Ε=⇒= εε

Active and reactive power absorption of a Active and reactive power absorption of a linear passive network linear passive network ππππππππ

N resistors

M inductors

K capacitors

L=N+M+K ππππRemark: Whichever is the origin of reactive energy, including active and nonlinear loads, it can be compensated by reactive elements

22

Total active power and reactive energy

K capacitors

∑∑==

===N

nRR

L

lll to tn

PPiuP11

,

∑ ∑ ∑ ∑∑= = = ==

Ε−Ε=Ε−Ε=+==M

mC

K

kL

M

m

K

kCLCL

L

lll totto tkmkm

WWiuW1 1 1 11

)(2)(2,)

reactive elementswith proper energy storage capability



phase networks


23






3. Definition of current and power terms in single -phase networks under non -

sinusoidal conditions

24

• Orthogonal current decomposition into active, react ive and void terms

• Physical meaning of current terms• Apparent power decomposition into active, reactive and

void terms• Physical meaning of power terms• Application examples

sinusoidal conditions

Current terms

Orthogonal current decomposition in Orthogonal current decomposition in singlesingle --phase networksphase networks

(voltage and current measured at a generic network port)(voltage and current measured at a generic network port)

43421

vi

gsrsaravra iiiiiiiii ++++=++=

25

• ia active current• ir reactive current• iv void current

222222222

gsrsaravra iiiiiiiii ++++=++=

OrthogonalityOrthogonality : all terms in the above equations are orthogonal: all terms in the above equations are orthogonal

• isa scattered active current• isr scattered reactive current• ig generated current

Active current : the minimum current (i.e., with minimumrms value) needed to convey the active power P flowingthrough the port

u = port voltagePiu,



26

U = rms value of port voltage

Ge = equivalent conductanceuGu

U

Pu

u

iui ea ===

22

,

0,,

,, 2

===

====

uuGiuW

PUGuuGiuP

eaa

eeaa))

Active current conveys full active power and zero reactive energy

Reactive current : the minimum current needed to convey the reactive energy W flowing through the port

Be = equivalent reactivity



uBuW

uiu

i er))

)

)

)

)

===22

,

27

Be = equivalent reactivity

WUBuuBiuW

uuBiuP

eerr

err

====

===2,,

0,,)

)))

) Reactive current conveys full reactive energy and no active power

uBuU

uu

i er )

)

===22

0,, == uuBGii eera) Active and reactive current

are orthogonal

Void current : is the remaining current component

rav iiii −−=



Void current is not conveying active power or react ive energy

28

0,,,,

0,,,,

=−−=−−==

=−−=−−==

raravv

raravv

WWWiuiuiuiuW

PPPiuiuiuiuP))))

0,,

0,,

===

===

veverv

veveav

WBuiBii

PGuiGii)

Void current is orthogonal to active and reactive t erms

Void current is not conveying active power or react ive energy

gsrsav iiii ++=

The void current reflects the presence of scattered active , scattered reactive and load-generated harmonic terms



29

gsrsav

Scattered current terms :Account for different values of

equivalent admittance at different harmonics

Load-generated current harmonics :

Harmonic terms that exist in currents only, not in voltages

0,,, === gsrgsasrsa iiiiiiScattered and load-generated harmonic currents are orthogonal

Skip void current components

ForFor eacheach coco--existingexisting harmonicharmonic componentscomponents ofof voltagevoltageandand currentcurrent wewe definedefine::

Harmonic active current termsHarmonic active current terms

kkkkk uGuI

uP

uiu

i ==== ϕcos,

Scattered active currentScattered active current


30

kkkk

kkk

k

kk

k

kkak uGu

U

Iu

U

Pu

u

iui ==== ϕcos,

22

0, ==== ∑∈

haaKk

kha WPPPP

( )∑∈

−=−=Kk

kekahasa uGGiii 0,0 ==−= saahasa WPPP

∑∈

=Kk

akha ii Total harmonic active currentTotal harmonic active current

Scattered active currentScattered active current

Scattered reactive currentScattered reactive current

ForFor eacheach coco--existingexisting harmonicharmonic componentscomponents ofof voltagevoltageandand currentcurrent wewe definedefine::

HarmonicHarmonic reactivereactive currentcurrent termsterms

kkkkk uBuIk

uW

uiu

i)))

)

)

)

====ϕω sin,


31

kkkk

kkk

k

kk

k

kkrk uBu

U

Iku

U

Wu

u

iui

)))

)

)

)

====ϕω sin,

22

0, ==== ∑∈

hrrKk

khr PWWWW

( )∑∈

−=−=Kk

kekrhrsr uBBiii)

0,0 ==−= srhrsr PWWW

∑∈

=Kk

rkhr ii

Total harmonic reactive currentTotal harmonic reactive current

Scattered reactive currentScattered reactive current

aa IUiuP ==

IUiuQ ==

Apparent power decomposition Apparent power decomposition in in singlesingle --phase networksphase networks

222 VQPIUiuA ++===

Active power:

Reactive power:

32

Scattered active power:

Scattered reactive power:

Load-generated harmonic power:

rr IUiuQ ==

222gravv VSSIUiuV ++===

srsrr IUiuS ==

sasaa IUiuS ==

ggg IUiuV ==

Reactive power:

Void power:

Reactive PowerReactive Power

( )[ ]222 1 uTHDUUUU fhf +=+=

( )[ ]222 1 uTHDUUUU fhf)

))))

+=+=

ω=ff UU)

U and Û can be decomposed in fundamental and harmonic components(THD means total harmonic distortion)

Recalling that:

33

Note that, unlike reactive energy W, REACTIVE POWER QIS NOT CONSERVATIVE. In fact, it depends on line frequency and (local) voltage distortion.

Under sinusoidal conditions, the definition of Q coincides with the conventional one

( )[ ]( )[ ]2

2

1

1

uTHD

uTHDWW

U

UIUQ r

)

)

+

+=== ωWe have:

Void Power Terms Void Power Terms

Scattered active power:Scattered active power:

222grav VSSIUV ++==Void Power:Void Power:

2

2

222

kKk k

ksaa U

U

P

U

PUIUS ∑

∈

−==

34

Kk k UU∈

2

2

222

2

2

1

1k

Kk k

k

u

usrr U

U

W

U

WU

THD

THDIUS

)

))

)

)

∑∈

−

+

+== ω

Scattered reactive power:Scattered reactive power:

LoadLoad--generated harmonic power:generated harmonic power:

gg IUV =

Skip examples

Application ExamplesApplication Examples

Example Example ######## 11VoltageVoltage andand CurrentCurrent : : ResistiveResistive LoadLoad

0.5

1

1.5

[pu

] 0.5

1

1.5

[p

u]

u(t)

i(t)

u(t)

i(t)

35

Sinusoidal voltageSinusoidal voltage Non Non –– sinusoidal voltagesinusoidal voltage

0.1 0.11 0.12 0.13 0.14 0.15 0.16-1.5

-1

-0.5

0

u PC

C i P

CC

[pu

]

T ime [s]

0.1 0.11 0.12 0.13 0.14 0.15 0.16-1.5

-1

-0.5

0

u PC

C i

PC

C [

pu]

T ime [s]

i(t) i(t)

Current = i pu(t)/2

Application ExamplesApplication ExamplesExample Example ######## 11

ConservativeConservative Power Power TermsTerms: : ResistiveResistive LoadLoad

0

0.5

1

[pu]

p(t) = u(t)i(t) ϕUIcospP ==

0

0.5

1

[pu]

p(t) = u(t)i(t) pP =

36


This example shows the This example shows the correspondencescorrespondences between the between the CPTCPT theorytheory and and conventional theoryconventional theory

0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135

-0.5

0

T ime [s]

q(t) = ωωωω û(t)i(t) ϕUIsin(t)wωQ ==0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135

-0.5

0

T ime [s]

q(t) = ωωωω û(t)i(t) Q

Application ExamplesApplication ExamplesExample Example ######## 1 1 –– SingleSingle--phasephase

CurrentCurrent TermsTerms: : ResistiveResistive LoadLoad

0.1 0.11 0.12 0.13 0.14 0.15 0.16-1

-0.5

0

0.5

1

i PC

C [

pu]

-0.5

0

0.5

1

i a [p

u]

0.1 0.11 0.12 0.13 0.14 0.15 0.16-1

-0.5

0

0.5

1

i PC

C [

pu]

-1

-0.5

0

0.5

1

i a [pu

]Activecurrent

PCCcurrent

i PCC (t) = ia(t)

37

0.1 0.11 0.12 0.13 0.14 0.15 0.16-1

0.1 0.11 0.12 0.13 0.14 0.15 0.16-0.5

-0.25

0

0.25

0.5

i r [pu

]

0.1 0.11 0.12 0.13 0.14 0.15 0.16-0.5

-0.25

0

0.25

0.5

i v [p

u]

T ime [s]

0.1 0.11 0.12 0.13 0.14 0.15 0.16-1

0.1 0.11 0.12 0.13 0.14 0.15 0.16-0.5

-0.25

0

0.25

0.5

i r [pu

]

0.1 0.11 0.12 0.13 0.14 0.15 0.16-0.5

-0.25

0

0.25

0.5

i v [pu

]

T ime [s]


Reactivecurrent

Voidcurrent

current

i r (t)= 0

i v (t)= 0


VoltageVoltage andand CurrentCurrent : : OhmicOhmic--inductive Loadinductive Load

0

0.5

1

1.5

i PC

C [

pu] u(t) i(t)

0

0.5

1

1.5

i PC

C [

pu]

u(t) i(t)

38


0.3 0.31 0.32 0.33 0.34 0.35 0.36-1.5

-1

-0.5u PC

C i

Time [s]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1.5

-1

-0.5u PC

C i

T ime [s]

0.5

1

[pu]

p(t) = u(t)i(t) ϕUIcospP ==


ConservativeConservative Power Power TermsTerms: : OhmicOhmic--inductive Loadinductive Load

0.5

1

[pu]

p(t) = u(t)i(t) pP =

39

0.3 0.305 0.31 0.315 0.32 0.325 0.33 0.335

0

Time [s]

q(t) = ωωωω û(t)i(t) ϕUIsinwωQ ==


This example shows the This example shows the correspondencecorrespondence between between CPTCPTand and conventional conventional theory theory under sinusoidal conditionsunder sinusoidal conditions

0.3 0.305 0.31 0.315 0.32 0.325 0.33 0.335

0

T ime [s]

q(t) = ωωωω û(t)i(t) Q


CurrentCurrent TermsTerms: : OhmicOhmic--inductive Loadinductive Load

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.5

0

0.5

1

i PC

C [

pu]

-0.5

0

0.5

1

i a [pu

]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.5

0

0.5

1

i PC

C [

pu]

-0.5

0

0.5

1

i a [p

u]Activecurrent

PCCcurrent

40


0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.5

0

0.5

1

i r [pu

]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.5

-0.25

0

0.25

0.5

i v [pu

]

T ime [s]

i v (t)= 0

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.5

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.5

0

0.5

1

i r [pu

]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.5

-0.25

0

0.25

0.5

i v [p

u]

T ime [s]

current

Reactivecurrent

Voidcurrent

Physical meaning of void currentPhysical meaning of void currentExample Example ######## 22

VoidVoid CurrentCurrent TermsTerms: : OhmicOhmic--inductive Loadinductive Load

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.2

-0.1

0

0.1

0.2

i v [pu

]

-0.1

0

0.1

0.2

i sa [

pu]

Void current

Scattered active current

41

Non Non –– sinusoidal voltagesinusoidal voltage

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.2

-0.1

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.2

-0.1

0

0.1

0.2

i sr [

pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.2

-0.1

0

0.1

0.2

i g [pu

]

T ime [s]

Load-generated harmonic current

Scattered reactive current


VoltageVoltage andand CurrentCurrentDistortingDistorting LoadLoad

ConservativeConservative Power Power TermsTermsDistortingDistorting LoadLoad

0.5

1

[pu]

pP =wωQ =

-0.5

0

0.5

1

1.5

PC

C i P

CC

[pu

]

u(t)

i(t)

42


0.3 0.305 0.31 0.315 0.32 0.325 0.33 0.335

0

T ime [s]

p(t) = u(t)i(t)q(t) = ωωωω û(t)i(t)

0.3 0.305 0.31 0.315 0.32 0.325 0.33 0.335-1.5

-1

-0.5u P

T ime [s]

Physical meaning of void currentPhysical meaning of void currentVoidVoid CurrentCurrent TermsTerms: : OhmicOhmic--inductive Loadinductive Load

PCC current

Active current

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.5

0

0.5

1i P

CC [

pu]

-0.5

0

0.5

1

i a [pu

]

43


Void current

Reactive current

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.5

0

0.5

1

i r [pu

]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.5

0

0.5

1

i v [p

u]

T ime [s]

Physical meaning of void currentPhysical meaning of void currentVoidVoid CurrentCurrent TermsTerms: : OhmicOhmic--inductive Loadinductive Load

Void current

Scattered active current

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.8

-0.4

0

0.4

0.8

i v [pu

]

-0.4

0

0.4

0.8

i sa [

pu]

44


Load-generated harmonic current

Scattered reactive current

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.8

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.8

-0.4

0

0.4

0.8

i sr [

pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.8

-0.4

0

0.4

0.8

i g [pu

]

T ime [s]



phase networks


45






4. Extension to poly -phase domain: 3-wires / 4 -wires

46

• Orthogonal current decomposition into active, react ive, unbalance and void terms

• Physical meaning of current terms• Active power decomposition into active, reactive,

unbalance and void terms• Physical meaning of power terms

Orthogonal current decomposition Orthogonal current decomposition Extension to polyExtension to poly --phase: 3phase: 3 --wires / 4wires / 4 --wireswires

In poly-phase systems, the current components (active, reactive and void) can be defined for each phase:

NnuGuU

Pu

u

iui nnn

n

nn

n

nnan ÷==== 1,

,22

Active current

Piu

iuW

PiuP

aa

aa

≠=

==

==

IU

0,

,)

G = equivalent phase conductance

47

Reactive current

Void current

NnuBuU

Wu

u

iui nnn

n

nn

n

nnrn ÷==== 1,

,22

))

)

)

)

)

Nniiii rnannvn ÷=−−= 1,

Piu aa ≠=IU

Wiu

WiuW

iuP

rr

rr

rr

≠=

==

==

))

)

IU

,

0,

Gn= equivalent phase conductance

Bn = equivalent phase reactivity

0

0,,0,

>

====

v

vrvv iuWiuP

IU

)

Active and reactive current terms can also be defin ed collectively, i.e., by making reference to an equivalent balanced load absorbing the same active power and reactive energy of actual load:

Balanced Active currents: minimum collective currents needed to convey active power P


48

currents needed to convey active power P

Balanced Reactive currents: Balanced Reactive currents: minimum collective currents needed to convey reactive energy W

uGuP

uu

iui bba ===

22

,

U 0, === ba

ba

ba QPiuIU

uBuW

uu

iui bbr

))

)

)

)

)

===22

,

U 0, === br

br

br PWiu

))

IU

Gb= equivalent balanced conductance

Bb = equivalent balanced reactivity


UnbalancedUnbalanced ActiveActive currentscurrents

Unbalanced currentsUnbalanced currents account for the asymmetrical account for the asymmetrical behavior of the various phasesbehavior of the various phases

( ) NnuGGiiii nb

nuan

baa

ua ÷=−=⇒−= 1, 0=−= b

aaua PPP

49

UnbalancedUnbalanced ReactiveReactive currentscurrents

( ) nnanaaa

( ) NnuBBiiii nb

nurn

brr

ur ÷=−=⇒−= 1,

)

0=ua

aaa

W

0

0

=−=

=brr

ur

ur

WWW

P


Void currents: as for single-phase systems, they reflect the presence of scattered active, scattered reactive and generated terms.

gsr

sarav iiiiiii ++=−−=

50

0

0

=−−==−−=

rav

rav

WWWW

PPPP

Scattered current terms :Account for different values of

equivalent admittance at different harmonics

Load-generated harmonic current :

Harmonic terms that exist in currents only, not in voltages


Summary of current decomposition

i a active currents• i a

b balanced active currents

gsr

sa

ur

br

ua

bavra iiiiiiiiiii ++++++=++=

51

• i a balanced active currents• i a

u unbalanced active currents i r reactive currents

• i rb balanced reactive currents

• i ru unbalanced reactive currents

i v void currents• i a

s scattered active currents• i r

s scattered reactive currents• i g load-generated harmonic currents


43421321321

vra i

gsr

sa

i

ur

br

i

ua


Summary of current decompositionSummary of current decomposition

Each current component has a preciseEach current component has a precise

52

Moreover, all current terms defined in the above e quation Moreover, all current terms defined in the above e quation are are ORTHOGONALORTHOGONAL, thus:, thus:

22222222222

gsr

sa

ur

br

ua


Each current component has a preciseEach current component has a precisePHISICAL MEANING PHISICAL MEANING and is computed in the time domainand is computed in the time domain

Apparent power decomposition Apparent power decomposition in in polypoly --phase: 3phase: 3 --wires / 4wires / 4 --wireswires

2222 VNQPiuA +++=== IU

Active power:Active power: ba

ba iuP == IU

53

Reactive power:Reactive power:

UnbalanceUnbalance powerpower::

Void power:Void power:

br

br iuQ == IU

222gravv VSSiuV ++=== IU

22ra

u NNiuN +=== uIU

UnbalanceUnbalance Power TermsPower Terms

UnbalanceUnbalance powerpower::

Unbalance Active Power

22ra NNN +=

∑=

−===N

n n

nua

uaa

P

U

PiuN

12

2

2 UUIU

54

Unbalance Reactive Power

( )[ ]( )[ ] 2

2

122

2

1

1

UUIU ))

)

W

U

W

uTHD

uTHDN

N

n n

nurr −

+

+== ∑

=ω

=n nU1 U

Unbalance active and reactive power vanish if the load is balanced

Skip examples


Example Example ######## 1 1 : 3: 3--phase 3phase 3--wire wire –– BalancedBalanced load load (Resistive)

ConservativeConservative Power Power TermsTerms

1

p(t

) &

P;

w(t

) &

W [

pu]

VoltageVoltage andand CurrentCurrent

1

1.5

[p

u]

ua uc

ia iaib

ub

55

Sinusoidal voltageSinusoidal voltage

Current = i pu(t)/2

0.3 0.31 0.32 0.33 0.34 0.35 0.36

-0.5

0

0.5

p(t

) &

P;

w(t

) &

W [

pu]

T ime [s]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1.5

-1

-0.5

0

0.5

u P

CC

&

i

PC

C [

pu]

T ime [s]

ia iaib

0.3 0.31 0.32 0.33 0.34 0.35 0.36

-1-0.5

00.5

1

i b an [pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.50

0.51

i b r n [pu]


Balanced active currents

Balanced reactive currentsi r (t)= 0

Example Example ######## 1 1 : 3: 3--phase 3phase 3--wire wire –– BalancedBalanced load load (Resistive)

56

0.3 0.31 0.32 0.33 0.34 0.35 0.36

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.50

0.51

i u an [pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.50

0.51

i u r n [pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.50

0.51

i vn [pu]

T ime [s]

Void currentsi v (t)= 0

Unbalanced active currentsi ua (t)= 0

Unbalanced reactive currentsi u

r (t)= 0


0.5

1

1.5

CC

[pu

]

ubua uc

ia iai

VoltageVoltage andand CurrentCurrent

0.5

1

p(t)

& P

; w

(t)

& W

[pu

]

ConservativeConservative Power Power TermsTerms

Example Example ######## 1 1 : 3: 3--phase 3phase 3--wire wire –– UnbalancedUnbalanced load load (resistor connected between two phases)

57

Sinusoidal voltageSinusoidal voltage

Current = i pu(t)/2

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1.5

-1

-0.5

0

u P

CC

&

i

PC

T ime [s]

ib

0.3 0.31 0.32 0.33 0.34 0.35 0.36

-0.5

0

0.5

p(t)

& P

; w

(t)

& W

[pu

]T ime [s]


0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.50

0.51

i b an [pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.50

0.51

i b r n [pu]

Example Example ######## 1 1 : 3: 3--phase 3phase 3--wire wire –– UnbalancedUnbalanced load load (resistor connected between two phases)


Balanced reactive currentsi r (t)= 0

58

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.50

0.51

i u an [pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.50

0.51

i u r n [pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.50

0.51

i vn [pu]

T ime [s]

Void currents

Unbalanced active currents

Unbalanced reactive currents

i v (t)= 0


Example Example ######## 3 3 : 3: 3--phase 3phase 3--wirewireThreeThree--phase RL + Singlephase RL + Single--phase R loadphase R load

0.5

1

p(t)

& P

; w

(t)

& W

[pu

]

ConservativeConservative Power Power TermsTermsVoltageVoltage andand CurrentCurrent

0

0.5

1

1.5

&

i

PC

C [

pu]

ua uc

ia iaib

ua

59

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.5

0

p(t)

& P

; w

(t)

& W

[pu

]T ime [s]

Symmetrical nonSymmetrical non--sinusoidal voltagesinusoidal voltage

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1.5

-1

-0.5

0

u P

CC

&

i

T ime [s]

Application ExamplesApplication ExamplesExample Example ######## 3 3 : 3: 3--phase 3phase 3--wirewire

ThreeThree--phasephase RL + RL + SingleSingle--phasephase R loadR load

0.3 0.31 0.32 0.33 0.34 0.35 0.36-1

-0.50

0.51

i b an [pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.5

-0.250

0.250.5

i b r n [pu]

0.250.5


Balanced reactive currents

60

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.5

-0.250

0.25

i u an [pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.5

-0.250

0.250.5

i u r n [pu]

0.3 0.31 0.32 0.33 0.34 0.35 0.36-0.5

-0.250

0.250.5

i vn [pu]

T ime [s]

Void currents

Unbalanced active currents

Unbalanced reactive currents

Symmetrical nonSymmetrical non--sinusoidal voltagesinusoidal voltage

Orthogonal current Orthogonal current termsterms ::

43421321321

vra i

iii

i

ii

i

iiiiii gsrsaur

br

ua

bavra ++++++=++=

Each current component has Each current component has

Sharing of compensation dutiesSharing of compensation duties

61

Each current component has Each current component has a precise a precise PHYSICAL MEANINGPHYSICAL MEANING

Balanced Active currents Balanced Active currents convey active power Pconvey active power P

Balanced Reactive currents Balanced Reactive currents convey reactive power Q

Unbalanced Active and Reactive currents Unbalanced Active and Reactive currents account for account for asymmetrical behavior of the various phasesasymmetrical behavior of the various phases

Void currents Void currents reflect the presence of different behavior at reflect the presence of different behavior at different different frequencies and/or generated frequencies and/or generated current harmonicscurrent harmonics

i = i a + i r + i v = i ab + i r

b + i au + i r

u + i v

Reactive compensation

Unbalance compensation

requires controllable Harmonic

compensation

Sharing of compensation dutiesSharing of compensation duties

62

compensation requires controllable reactances (extended Steinmetz approach)

compensation requires high-

frequency response

Passive filters & Switching Power Compensators(SPC=APF+SPI)

Quasi-Stactionary

Compensators (SVC)

Stationary Compensators (reactive impedances)

&Quasi-Stationary

Compensators (SVC, Static VAR Compensators)

Effect of compensation on Effect of compensation on power termspower terms

Active power (balanced)Active power (balanced) ::

Reactive power (balanced)Reactive power (balanced) ::

Unbalance powerUnbalance power ::

baP IU=

brQ IU

)

=

22 uuuN IIUIU +==

0 →SVC

0 →SVC

Compensation

63

APPARENT APPARENT POWERPOWER

2222 VNQPA +++== IU

Unbalance powerUnbalance power ::

Void powerVoid power ::

22 ur

ua

uN IIUIU +==

X X X PA ba

SPC,SVC == → IU

0 →SVC

0 →SPCvV IU=



phase networks


64






5. 5. Sequence components under non -sinusoidal conditions

65

1. Problem statement2. Goal of decomposition3. Derivation of generalized symmetrical components in

the time domain (extension of Fortescue’s approach)4. Analysis of generalized symmetrical components in

the frequency domain5. Orthogonality of sequence components6. Application examples

Skip

Symmetrical components are very useful to simplify the analysis of three-phase networks under sinusoidal conditions

11. . Problem Problem statement (1)statement (1)

66

It is important to extend the definition and application of symmetrical components to non -sinusoidal periodic operation

11. . Problem Problem statement (2)statement (2) Given periodic three-phase variables fa(t), fb(t), fc(t) of

period T we define the following symmetry properties :

Homopolarity (zero symmetry)

( ) ( ) ( )tftftf cba ==

67

Positive (direct) symmetry

Negative (inverse) symmetry

( )

+=

+=3

2

3

Ttf

Ttftf cba

( )

−=

−=3

2

3

Ttf

Ttftf cba

cba

2. 2. Goal of decompositionGoal of decomposition

Given a set of generic three-phase variables:

we decompose them in the orthogonal form:

( )( )( )tf

tf

tf

f

c

b

a

=

68

where: f z are zero-sequence (homopolar) components f h are non-zero sequence (heteropolar) components f p are positive-sequence components f n are negative-sequence components f r are residual components

rnpzhz fffffff +++=+=

3. Derivation of generalized 3. Derivation of generalized symmetrical componentssymmetrical components

Zero sequence (homopolar) component

( ) ( ) ( ) ( ) ( )3

1

1

1tftftf

tftff cbazzz ++==

69

Heteropolar components

31

( )( )( )

( ) ( )( ) ( )( ) ( )tftf

tftf

tftf

tf

tf

tf

fffz

c

zb

za

hc

hb

ha

zh

−−−

==−=


Positive sequence component

( ) ( )

( ) ( ) ( ) ( )

−=

−==

++

++=

2,,

3

2

33

1

Ttftf

Ttftftftf

Ttf

Ttftftf

pppppp

hc

hb

ha

p

70

Negative sequence component

( ) ( ) ( ) ( )

−=

−==3

2,

3,

Ttftf

Ttftftftf pp

cpp

bpp

a

( ) ( )

( ) ( ) ( ) ( )

+=

+==

−+

−+=

3

2,

3,

3

2

33

1

Ttftf

Ttftftftf

Ttf

Ttftftf

nnc

nnb

nna

hc

hb

ha

n


Residual components

( )

( )

( ) 23

32

3

−+

−+

TT

Ttf

Ttftf

hhh

ha

ha

ha

r

71

( )( )( )

( )

( )3

32

3

3

32

3

−+

−+

−+

−+==

Ttf

Ttftf

Ttf

Ttftf

tf

tf

tf

f

hc

hc

hc

hb

hb

hb

rc

rb

ra

r

Note: these components are computed independently for each phase and vanish in sinusoidal operation


Resulting decomposition

( )( ) ( ) ( ) ( )tftftftf

tf

ra

npz

a

+++

72

( )( )( )

( ) ( )

( ) ( )tfT

tfT

tftf

tfT

tfT

tftf

tf

tf

tf

f

rc

npz

rb

npz

c

b

a

+

++

−+

+

++

−+==

3

2

3

233

4. Analysis in the frequency domain4. Analysis in the frequency domain

Expressing variables fa(t), fb(t), fc(t) in Fourier series:

( ) ( ) ( )

( ) ( ) ( )∑∑

∑∑∞∞

∞

=

∞

=

+==

+==11

sin2

sin2

bkbkbkb

kakak

kaka

tkFtftf

tkFtftf

αω

αω

73

we can determine, for each harmonic, the zero, posit ive, and negative components f zk(t), f pk(t), f nk(t). Instead, residual harmonic components are zero because harmonic quantities are sinusoidal.

( ) ( ) ( )

( ) ( ) ( )∑∑

∑∑∞

=

∞

=

==

+==

+==

11

11

sin2

sin2

kckck

kckc

kbkbk

kbkb

tkFtftf

tkFtftf

αω

αω

Contribution of harmonic sequence components to generalized sequence components

[ ]zz

knn

kpp

k

mmkorderHarmonic

ffffff ⇒⇒⇒

∞∈∀+=

,,

,013:

4. Analysis in the frequency domain4. Analysis in the frequency domain

74

[ ]

[ ]zz

krn

krp

k

zz

kpn

knp

k

mmkorderHarmonic

mmkorderHarmonic

ffffff

ffffff

⇒⇒⇒

∞∈∀=

⇒⇒⇒

∞∈∀+=

,,

,03:

,,

,023:

5. Orthogonality of components5. Orthogonality of components

Given two sets of three-phase quantities f and g, their sequence components obey the following general rules:

Scalar product0==== rznzpzhz gfgfgfgf oooo

75

Internal product

Norm

0==== gfgfgfgf oooo

2222222 rnpzhz fffffff +++=+=

rrnnppzzhhzz

rnrpnp

gfgfgfgfgfgfgf

gfgfgf

,,,,,,,

0,,,

+++=+=

===

;)tsin()tsin()tsin()tsin()t(v



c

b

a

10

1

10

1

3

25

5

1

3

23

3

1

3

210

1

10

1

3

25

5

1

3

23

3

1

3

210

1

10

15

5

13

3

1

++++−++=

++−+++−=

++++=

ωπωπωπω

ωπωπωπω

ωωωω

6. Application Example6. Application Example

Line to Neutral Voltages

760 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5Line to Neutral Voltages

ms

V

Generalized Zero Sequence Voltages




c

b

a

10

1

10

1

3

25

5

1

3

23

3

1

3

210

1

10

1

3

25

5

1

3

23

3

1

3

210

1

10

15

5

13

3

1

++++−++=

++−+++−=

++++=

ωπωπωπω

ωπωπωπω

ωωωω


770 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5Generalized Zero Sequence Voltages

ms

V

Generalized Direct Sequence Voltages




c

b

a

10

1

10

1

3

25

5

1

3

23

3

1

3

210

1

10

1

3

25

5

1

3

23

3

1

3

210

1

10

15

5

13

3

1

++++−++=

++−+++−=

++++=

ωπωπωπω

ωπωπωπω

ωωωω


780 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5Generalized Direct Sequence Voltages

ms

V

Generalized Inverse Sequence Voltages




c

b

a

10

1

10

1

3

25

5

1

3

23

3

1

3

210

1

10

1

3

25

5

1

3

23

3

1

3

210

1

10

15

5

13

3

1

++++−++=

++−+++−=

++++=

ωπωπωπω

ωπωπωπω

ωωωω


790 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5Generalized Inverse Sequence Voltages

ms

V

Residual Voltages




c

b

a

10

1

10

1

3

25

5

1

3

23

3

1

3

210

1

10

1

3

25

5

1

3

23

3

1

3

210

1

10

15

5

13

3

1

++++−++=

++−+++−=

++++=

ωπωπωπω

ωπωπωπω

ωωωω


800 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5Residual Voltages

ms

V

9. 9. Summary Summary -- 11

An extension of the sequence components in case of non-sinusoidal periodic operation has been proposed .

It has been shown that three-phase currents (or voltages) cannot always be derived from generalized positive sequence, generalized negative sequence and generalized zero-sequence components. A residual component may be required.

81

component may be required. To compute the generalized positive and negative

sequence components, the zero-sequence components should first be subtracted from the phase quantitie s, in contrast with the sinusoidal case where this is not necessary.

In the sinusoidal case the residual component is ab sent and the other components reduce to the classical symmetrical components.

The generalized positive sequence, negative sequence , and zero-sequence components have complete phase symmetry. This implies that the three-phase analysi s can be reduced to a single-phase analysis.

The residual components do not have the same symmetry, and the corresponding three -phase analysis

9. Summary 9. Summary -- 22

82

symmetry, and the corresponding three -phase analysis cannot be reduced to single-phase analysis. It corresponds to a periodic time function in each of the three phases with a period which is 1/3 of the line period; this simplifies the analysis because only 1/3 of th e period must be studied.

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

Derivation of homopolar component 0

0.5

1

1.5

Voltage phase a

Voltage phase b

Homopolar V

ms

83

component

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

Voltage phase c

Homopolar component

V

ms

ms

V

.

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

0.5

1

1.5

Derivation of heteropolar component

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

0.5

1

1.5

Opposite of the homopolar component

Voltage phase a

V

V

ms ms


84

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

component

.

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

.

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

Voltage phase b

Voltage phase c

VV V

V

ms

ms ms

ms


0 10 20 30 40 50 60-1.5

-1

-0.5

0

0.5

1

1.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

0

0.5

1

1.5

Derivation of positive sequence component

(phase a)

Voltage phase a

V V

msms

85

0 10 20 30 40 50 60-1.5

-1

-0.5

0 10 20 30 40 50 60-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

(phase a)

Voltage phase b

Voltage phase c

Positivesequence component

VV V

V

ms

msms

ms

0 10 20 30 40 50 60-1.5

-1

-0.5

0

0.5

1

1.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

Derivation of negative sequence component

(phase a) 0

0.5

1

1.5

Voltage phase a

V V

ms ms

86

0 10 20 30 40 50 60-1.5

-1

-0.5

0 10 20 30 40 50 60-1.5

-1

-0.5

0

0.5

1

1.5

(phase a)

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

Voltage phase b

Voltage phase c

Negativesequence component

VV

VV

ms

ms ms

ms

Derivation of residual component (phase a)

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

0

0.5

1

1.5

0

0.5

1

1.5

Opposite of the homopolar comp.

Opposite of the positive sequence comp.

V V

ms ms

87

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

Opposite of the negative sequence comp.

Residual component

VV

VV

ms ms

msms


2. Mathematical and physical foundations of the theor y

3. Instantaneous and average power & energy terms in poly-phase networks

4. Definition of current and power terms in single-pha se networks under non -sinusoidal conditions

88

networks under non -sinusoidal conditions



7. Measurement & accountability issues (basic approach)

Measurement & accountability issuesMeasurement & accountability issues

Active and reactive current (and power) terms are affected by the presence of negative-sequence, zero-sequence and harmonic voltages

A proper accountability approach must be adopted to depurate the power and current terms from the effects of voltage non-idealities, which are not under load responsibility

89

responsibility

ωω

pf

pf

n

pf

pfn

Un

UU

UnUU

331,

331,

=⇒÷==

=⇒÷==

U

U

))

If we assume that the supply voltages are sinusoidal and symmetrical with positive sequence , we have:

Assuming that the equivalent phase resistanceremains the same irrespective of supply conditions , wecan express the active current and power accountableto the load in each phase as:

Phase current and power termsPhase current and power terms

2

2

,pf

nnapfnn

pfnnna U

UPiuPuGi ==⇒=

lll∑

=

=3

21npa P

U ll

I

90

Similarly, if the equivalent phase reactivity remains thesame irrespective of supply conditions , we canexpress the reactive current and power accountable tothe load in each phase are:

2

2

,n

pf

nnrpfnn

pfnnnr U

UWiuWuBi )

)

))

lll

==⇒=

2n

nnafnnfnnna Ulll∑

=1npfU

∑=

=3

1

21

nnp

fr Q

U ll

I

The total power terms accountable to the load are:

Balanced current and power termsBalanced current and power terms

2

3

1 3 pf

b

nn

U

PGPP l

lll

=⇒=∑=

22

3

1 33 pf

pf

b

nn

U

Q

U

WBWW ll

lll)

ω==⇒=∑=

91

The balanced current terms accountable to the loadare:

pf

pf

br

pfn

bbr U

Q

U

WuBi ll

lll

)

)

3

1

3

1 ==⇒= Ipf

ba

pf

bba U

PuGi l

lll

3

1=⇒= I

The unbalanced active current and power accountable to the load are :

Unbalanced current and power termsUnbalanced current and power terms

( ) pnf

bn

bnana

una uGGiii

llll

−=−=

( )3

1 23

1

223

1

2l

lll

PP

UUGG

nnp

f

pf

n

bn

ua ∑∑

==

−=−=I

92

The unbalanced reactive current and power accountable to the load are :

( ) pfn

bn

bnrnr

unr uBBiii

)

llll

−=−=

311 U nfn ==

( )3

1 23

1

223

1

2l

lll

) QQ

UUBB

nnp

f

pf

n

bn

ur −=−= ∑∑

==

I

Void current and powerVoid current and power

The void currents satisfy the condition:

0,,0,

0,,0,

=+++⇒=

=+++⇒=

vhzf

nfv

pfv

vhzf

nfv

pfv

iuuuiuiu

iuuuiuiu

)))))

The void current terms which can be accounted

93

The void current terms which can be accountedto the load are therefore given by:

pfp

f

vpfp

fpf

vpf

vv uU

iuu

U

iuii

)

)

)

l 223

,

3

,−−=

In fact, the fundamental component of the voidcurrent has been already accounted for in theactive and reactive current terms.

Currents terms accountable to the loadCurrents terms accountable to the load

lllllllll vur

ua

br

bavra iiiiiiiii ++++=++=

All current terms are orthogonal, thus:All current terms are orthogonal, thus:

Summary of current decomposition Summary of current decomposition

94

22222llllll v

ur

ua

br

ba IIIIII ++++=

Apparent power accountable to the loadApparent power accountable to the load22222

2

lllllll

43421

l

VNNQPA

N

rapf ++++== IU

Case I: Symmetrical sinusoidal voltagesCase I: Symmetrical sinusoidal voltagesCase II: Asymmetrical sinusoidal voltagesCase II: Asymmetrical sinusoidal voltages

Case III: Symmetrical nonCase III: Symmetrical non--sinusoidal voltagessinusoidal voltagesCase IV: Asymmetrical nonCase IV: Asymmetrical non--sinusoidal voltagessinusoidal voltages

Application Examples: 3Application Examples: 3 --phase 3phase 3 --wirewire

Case I Case II U1 = 127∠∠∠∠0º Vrms U1 = 127∠∠∠∠0º Vrms

U2 = 127∠∠∠∠-120º Vrms U2 = 113∠∠∠∠-104,4º VrmsU3 = 127∠∠∠∠120º Vrms U3 = 147,49∠∠∠∠144º Vrms

95

BalancedBalanced LoadLoad

L3

L2

L1

i3

i1

u21

u32

3

2

1

uSn

LLnRLn

LOADPCCLINESOURCE

n 1,2,3

R C

Distorting LoadDistorting Load

U3 = 127∠∠∠∠120º Vrms U3 = 147,49∠∠∠∠144º Vrms

cases III and IV are the same of cases I and II with the addition of 10% of 5 th and 7 th harmonics

The line parameters are : RL1= RL2= RL3= 1mΩΩΩΩ and LL0 = LL1= LL2= LL3= 10 µµµµH.

CASE ICASE I CASE IICASE II CASE IIICASE III CASE IVCASE IV

PCCPCC LOADLOAD PCCPCC LOADLOAD PCCPCC LOADLOAD PCCPCC LOADLOAD

AA 1,00001,0000 1,00001,0000 1,0000 0,96340,9634 1,0000 0,98400,9840 1,0000 0,95380,9538

PP 0,79850,7985 0,79850,7985 0,7985 0,76930,7693 0,7913 0,77580,7758 0,7945 0,75560,7556

QQ 0,60200,6020 0,60200,6020 0,6020 0,58000,5800 0,6023 0,59620,5962 0,6022 0,57700,5770


Example Example ######## 1: 1: BalancedBalanced LoadLoad

96

QQ 0,60200,6020 0,60200,6020 0,6020 0,58000,5800 0,6023 0,59620,5962 0,6022 0,57700,5770

NN 0,0000 0,0000 0,0000 0,0000 0,0002 0,00020,0002 0,0056 0,00610,0061

VV 0,0000 0,0000 0,0000 0,0000 0,1054 0,10380,1038 0,0782 0,07600,0760

λλλλλλλλ 0,79850,7985 0,79850,7985 0,79850,7985 0,79850,7985 0,7913 0,7885 0,7945 0,7922

The load is accounted for less active, reactive and voidpower than the PCC

CASE ICASE I CASE IICASE II CASE IIICASE III CASE IVCASE IV

PCCPCC LOADLOAD PCCPCC LOADLOAD PCCPCC LOADLOAD PCCPCC LOADLOAD

AA 1,0000 1,0009 1,0000 0,9050 1,0000 0,98320,9832 1,0000 0,8973

PP 0,8275 0,8280 0,8841 0,7668 0,8254 0,80980,8098 0,8808 0,7570

QQ 0,2432 0,2451 0,2135 0,22450,2245 0,2422 0,24170,2417 0,2135 0,22320,2232

Example Example ######## 2: 2: Distorting lDistorting loadoad


97

QQ 0,2432 0,2451 0,2135 0,22450,2245 0,2422 0,24170,2417 0,2135 0,22320,2232

NN 0,5060 0,5060 0,4156 0,42500,4250 0,5055 0,49800,4980 0,4185 0,42310,4231

VV 0,0000 0,0000 0,0000 0,0000 0,0674 0,06660,0666 0,0581 0,0566

λλλλλλλλ 0,8275 0,8273 0,8841 0,8473 0,8254 0,8237 0,8808 0,8437

Again the apparent and active power accounted to theload are always lower than those computed at PCC dueto the depuration of the effects of voltage asymmetryand distortion

Defects of proposed accountability Defects of proposed accountability approachapproach

The equivalent phase conductance and reactivity are The equivalent phase conductance and reactivity are computed by computed by considering the effect of fundamental and considering the effect of fundamental and harmonic supply voltages as a wholeharmonic supply voltages as a whole . In practice, the . In practice, the load can have different response to fundamental and load can have different response to fundamental and harmonic voltages.harmonic voltages.

98

harmonic voltages.harmonic voltages.

The The void currents are not orthogonal void currents are not orthogonal to supply voltages to supply voltages if these latter become sinusoidal.if these latter become sinusoidal.

Only the load impedance is modeled, while Only the load impedance is modeled, while supply supply impedance is not estimatedimpedance is not estimated and enters the computation and enters the computation process indirectly, in a way that does not allow to fully process indirectly, in a way that does not allow to fully analyze its effect.analyze its effect.


2. Mathematical and physical foundations of the theor y

3. Instantaneous and average power & energy terms in poly-phase networks

4. Definition of current and power terms in single-pha se networks under non -sinusoidal conditions

99

networks under non -sinusoidal conditions



7. Measurement & accountability issues (extended approach)

Extended approach to accountabilityExtended approach to accountabilityBoth load and supply are modeled based on

measurements made at PCCLoad modeling is done under sinusoidal conditions

o This makes the load model more reliable, since harm onic effects are depurated

o Moreover, the harmonic currents generated by the lo ad are represented separately and their effect can directl y be accounted for accountability purposes

100

for accountability purposes

Supply modeling is made for three-phase symmetrical systems and allows estimation of no-load supply voltages and line impedances

The extended accountability approach is more reliab le than the basic one, and possibly avoids under- and o ver-penalization of the loads.

Of course, better results can be achieved if a more accurate modeling of the load is available.

The passive parameters of the equivalent circuit ar e computed to suit the circuit performance at fundamental frequency, i .e.:

Load modeling (3Load modeling (3 --phase 4phase 4 --wire) wire) -- 11

Single-phase equivalent circuit of 3-phase load seen from PCC

101

the circuit performance at fundamental frequency, i .e.:

=⇒==

=⇒==⇒+=

fm

fm

mm

fmf

mfm

fm

fm

fm

mm

fmf

mfm

fm

m

fm

m

fmf

m

W

UL

L

Ui,uW

P

UR

R

Ui,uP

L

u

R

ui

22

22

With this assumption current jm is purely harmonic. In fact:

fm

hm

fm

hmh

mfm

hm

fm

fm

hm

fmh

mf

mfm

mf

m

mmm

L

u

R

ui

L

uu

R

uuii

L

u

R

uij

))))

−−=+

−+

−+=−−=

For the validity of the model we must assume that the equivalent circuit parameters remain the same withi n

Load modeling (3Load modeling (3 --phase 4phase 4 --wire) wire) -- 22


102

equivalent circuit parameters remain the same withi n reasonable variations of the voltage supply , both in terms of asymmetry and distortion.

This is only approximately true in real networks, b ut it makes possible an accountability approach based on measurement at the load terminals , without requiring a precise knowledge of the load itself.

The passive parameters of the equivalent circuit ar e the same for all phases, due to supply lines symmetry. The circuit e quations are:

Supply modeling (3Supply modeling (3 --phase 4phase 4 --wire) wire) -- 11

Single-phase equivalent circuit of 3-phase supply seen from PCC

103

phases, due to supply lines symmetry. The circuit e quations are:

dt

dLR SS

iiue ++=

Due to its linearity, this equation can be applied separately to the fundamental positive-sequence voltage and current t erms (index p)and the remaining terms (index n), which represent the unwanted current and voltage components.

++=

++=⇒

dt

dLR

dt

dLR

n

Sn

Snn

p

Sp

Spp

iiue

iiue

The passive parameters of the equivalent circuit ar e selected so as to



104

The passive parameters of the equivalent circuit ar e selected so as to minimize the unwanted components of the supply volt ages en, i.e., the function:

dt

dL,L,R

dt

dLR

e,e

n

Sn

Snn

S

n

Sn

Sn

M

m

nm

nm

n

iuiu

iiu

e

22

2

2222

1

2

++++=

===ϕ ∑=

The result is expressed as a function of the quanti ties measured at PCC in the form:



105

PCC in the form:

2

2

0

0

dt

d

dt

d,L

L

,RR

nnn

SS

nnnS

S

iiu

iiu

−=⇒=∂

ϕ∂

−=⇒=∂

ϕ∂

Obviously, only positive solutions are acceptable f or RS and LS. In case of negative solution, the corresponding parame ter is set to zero.

Given RS and LS, we may compute the positive -sequence supply



106

Given RS and LS, we may compute the positive -sequence supply voltages to be included in the equivalent circuit :

dt

dLR

p

Sp

Spp i

iue ++=

Note that ep, up, ip are the fundamental positive sequence components of the related voltages and currents, wh ile en, un, in are calculated by difference from the original voltages and currents.

1. From the voltages and currents measured at PCC we estimate the phase parameters R and L and the current source j of the

Accountability Accountability – – Procedure (1)Procedure (1)


107

phase parameters Rm and Lm and the current source jm of the equivalent circuit.

fm

fm

mfm

fm

mW

UL

P

UR

22

==

fm

hm

fm

hmh

mmL

u

R

uij

)

−−=



2. From the voltages and currents measured at PCC we estimate the supply line parameters R and L and the fundamental positive -

108

supply line parameters RS and LS and the fundamental positive -sequence supply voltages ep

22

dt

d

dt

d,L,R

nnn

Snnn

Sii

uiiu −=−=

dt

dLR

p

Sp

Spp i

iue ++=


Equivalent circuit for the computation of fundamental voltages at PCC

4. Applying now the positive-sequence supply voltage s at the input terminals of the equivalent circuit, we may determi ne the

109

terminals of the equivalent circuit, we may determi ne the fundamental phase currents absorbed by the loa d under these supply conditions and the corresponding fundamental phase voltages appearing at the PCC terminals.

Note that the currents and voltages at PCC may re sult asymmetrical due to load unbalance. This non-ideali ty must obviously be ascribed to the load, since the voltag e supply and distribution lines are symmetrical

fl

i

fl

u


5. Finally, the load voltages and currents at PCC, w hich are accountable to the load are given by:

We can now compute all power terms accountable to t he load and

hf

hf

lll

lll

iii

uuu

+=

+=

110

We can now compute all power terms accountable to t he load and the corresponding performance factors.

v

ur

ua

br

ba

bbramm

ra

M

mm

M

mm

mmmmmm

,

,,B,G

,,B,G

V

NN

WWPP

i,uWi,uP

l

ll

llll

llll

l

ll

llll

llllll

KK

)

I

II

II

II

⇒

==

==

==

∑∑== 11

Distortion factor:

Performance factorsPerformance factors

2

222

2

2

1A

NQP

A

VD

++=−=λ

Unbalance factor: 222

22

222

2

1NQP

QP

NQP

NN ++

+=++

−=λ

111

Reactivity factor: 22

2

1QP

QQ +

−=λ

Power factor: DNQ

ra DNNQP

P

A

P λλλ=++++

==λ22222

Load circuitsLoad circuits


A. Unbalanced linear load

112Line parameters: R L1= RL2= RL3= 10.9 mΩΩΩΩ - LL1= LL2= LL3= 38.5 µµµµH

B. Unbalanced nonlinear load

Supply conditionsSupply conditionsCase 1: Symmetrical sinusoidal voltagesCase 1: Symmetrical sinusoidal voltages

Case 2: Symmetrical nonCase 2: Symmetrical non--sinusoidal voltagessinusoidal voltages


Case 1 Case 2 1 = 127∠0 Vrms 1 = + ∑ 1 Vrms

113

1 = 127∠0 Vrms 1 = + ∑ 1 Vrms 2 = 127∠ − 120 Vrms 2 = + ∑ 2 Vrms

3 = 127∠120 Vrms 3 = + ∑ 3 Vrms

In case 2 the terms called ΣΣΣΣH represent the harmonic contents of the phase voltages.

Each phase voltage includes 2% of 3 rd harmonic, 2% of 5 th harmonic.

The phase angle of each harmonic term is the phase angle of the fundamental voltage (as in Case 1) multiplied by th e harmonic order.


Case ACase A: : UnbUnbalanced linear loadalanced linear load

Case A.1 Case A.2 PCC Load PCC Load

A [KVA] 95.522 98.722 95.263 96.571 P [KW] 65.224 67.862 65.231 65.089

Q [KVA] 67.698 69.813 67.729 69.597 N [KVA] 15.158 16.334 15.163 15.582 D [KVA] 0.017 0.074 1.627 1.649

114

The load is penalized for its unbalance, especially in case A.1 (sinusoidal and symmetrical supply voltages)

D [KVA] 0.017 0.074 1.627 1.649 λλλλ 0.6828 0.6874 0.6847 0.6740 λλλλQ 0.6938 0.6970 0.6937 0.6831 λλλλN 0.9872 0.9862 0.9872 0.9869 λλλλD 1.0000 0.9999 0.9999 0.9999


Case B.1 Case B.2 PCC Load PCC Load

A [KVA] 93.267 94.007 89.494 91.207 P [KW] 63.909 63.334 62.738 62.763

Q [KVA] 33.274 35.376 33.599 36.159 N [KVA] 20.873 21.143 20.619 21.296 D [KVA] 55.421 55.916 50.190 51.171

Case BCase B: : UnbUnbalanced nonlinear loadalanced nonlinear load

115

The apparent, reactive and unbalance power accounted to the load are higher than those compute d

at PCC

D [KVA] 55.421 55.916 50.190 51.171

λλλλ 0.6852 0.6737 0.7010 0.6881 λλλλQ 0.8870 0.8730 0.8815 0.8665 λλλλN 0.9605 0.9601 0.9605 0.9594 λλλλD 0.8043 0.8039 0.8279 0.8278

conservative power theory cpt

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