conservative power theory - dsce.fee.unicamp.br
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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
Power terms in passive networksPower terms in passive networks
20
L
2,,0,, iLiiLiuW
L
uuiuP LL ====== )
)
22
1
2
1 22 LLLL
WiLiL ==Ε=⇒= εεInductorInductor
energyenergy
C
iu
uCdt
duCi
)
(
=
==
Capacitor
Power terms in passive networksPower terms in passive networks
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
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
23
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
Conservative Conservative Power TheoryPower Theory
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,
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)
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
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)
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 −−=
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)
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
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)
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
Orthogonal current decomposition in Orthogonal current decomposition in singlesingle --phase networksphase networks
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,
Orthogonal current decomposition in Orthogonal current decomposition in singlesingle --phase networksphase networks
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
Sinusoidal voltageSinusoidal voltage Non Non –– sinusoidal voltagesinusoidal voltage
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]
Sinusoidal voltageSinusoidal voltage Non Non –– sinusoidal voltagesinusoidal voltage
Reactivecurrent
Voidcurrent
current
i r (t)= 0
i v (t)= 0
Application ExamplesApplication ExamplesExample Example ######## 22
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
Sinusoidal voltageSinusoidal voltage Non Non –– sinusoidal voltagesinusoidal voltage
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 ==
Application ExamplesApplication ExamplesExample Example ######## 22
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 ==
Sinusoidal voltageSinusoidal voltage Non Non –– sinusoidal voltagesinusoidal voltage
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
Application ExamplesApplication ExamplesExample Example ######## 22
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
Sinusoidal voltageSinusoidal voltage Non Non –– sinusoidal voltagesinusoidal voltage
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
Application ExamplesApplication ExamplesExample Example ######## 33
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
Non Non –– sinusoidal voltagesinusoidal voltage
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
Non Non –– sinusoidal voltagesinusoidal voltage
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
Non Non –– sinusoidal voltagesinusoidal voltage
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]
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
45
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
Conservative Conservative Power TheoryPower Theory
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
Orthogonal current decomposition Orthogonal current decomposition Extension to polyExtension to poly --phase: 3phase: 3 --wires / 4wires / 4 --wireswires
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
Orthogonal current decomposition Orthogonal current decomposition Extension to polyExtension to poly --phase: 3phase: 3 --wires / 4wires / 4 --wireswires
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
Orthogonal current decomposition Orthogonal current decomposition Extension to polyExtension to poly --phase: 3phase: 3 --wires / 4wires / 4 --wireswires
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
Orthogonal current decomposition Orthogonal current decomposition Extension to polyExtension to poly --phase: 3phase: 3 --wires / 4wires / 4 --wireswires
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
Orthogonal current decomposition Orthogonal current decomposition Extension to polyExtension to poly --phase: 3phase: 3 --wires / 4wires / 4 --wireswires
43421321321
vra i
gsr
sa
i
ur
br
i
ua
bavra iiiiiiiiiii ++++++=++=
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
bavra iiiiiiiiiii ++++++=++=
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
Application ExamplesApplication 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]
Application ExamplesApplication Examples
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
Application ExamplesApplication Examples
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]
Application ExamplesApplication Examples
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 active currents
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
Application ExamplesApplication Examples
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 active currents
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=
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
64
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
Conservative Conservative Power TheoryPower Theory
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
−−−
==−=
3. Derivation of generalized 3. Derivation of generalized symmetrical componentssymmetrical components
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
3. Derivation of generalized 3. Derivation of generalized symmetrical componentssymmetrical components
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
3. Derivation of generalized 3. Derivation of generalized symmetrical componentssymmetrical components
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
;)tsin()tsin()tsin()tsin()t(v
;)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
;)tsin()tsin()tsin()tsin()t(v
;)tsin()tsin()tsin()tsin()t(v
;)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
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
;)tsin()tsin()tsin()tsin()t(v
;)tsin()tsin()tsin()tsin()t(v
;)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
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
;)tsin()tsin()tsin()tsin()t(v
;)tsin()tsin()tsin()tsin()t(v
;)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
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
;)tsin()tsin()tsin()tsin()t(v
;)tsin()tsin()tsin()tsin()t(v
;)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
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
Opposite of the homopolar component
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
Opposite of the homopolar component
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
Seminar OutlineSeminar Outline1. Motivation of work
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
5. Extension to poly-phase domain: 3-wires / 4-wires
6. Sequence components 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
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
Application Examples: 3Application Examples: 3 --phase 3phase 3 --wirewire
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
Application Examples: 3Application Examples: 3 --phase 3phase 3 --wirewire
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.
Seminar OutlineSeminar Outline1. Motivation of work
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
5. Extension to poly-phase domain: 3-wires / 4-wires
6. Sequence components 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
Single-phase equivalent circuit of 3-phase load seen from PCC
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
Supply modeling (3Supply modeling (3 --phase 4phase 4 --wire) wire) -- 22
Single-phase equivalent circuit of 3-phase supply seen from PCC
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:
Supply modeling (3Supply modeling (3 --phase 4phase 4 --wire) wire) -- 33
Single-phase equivalent circuit of 3-phase supply seen from PCC
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
Supply modeling (3Supply modeling (3 --phase 4phase 4 --wire) wire) -- 44
Single-phase equivalent circuit of 3-phase supply seen from PCC
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)
Single-phase equivalent circuit of 3-phase load seen from PCC
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
)
−−=
Accountability Accountability – – Procedure (2)Procedure (2)
Single-phase equivalent circuit of 3-phase supply seen from PCC
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 ++=
Accountability Accountability – – Procedure (3)Procedure (3)
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
Accountability Accountability – – Procedure (5)Procedure (5)
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
Application Examples: 3Application Examples: 3 --phase 4phase 4 --wirewire
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
Application Examples: 3Application Examples: 3 --phase 4phase 4 --wirewire
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.
Application Examples: 3Application Examples: 3 --phase 3phase 3 --wirewire
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
Application Examples: 3Application Examples: 3 --phase 3phase 3 --wirewire
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