Symmetrical system components - PowerWiki

Symmetrical system components

Decomposition of unsymmetrical (unbalanced) voltage:

0C2C1CC

0B2B1BB

0A2A1AA

UUUU

UUUU

UUUU

Positive sequence (1), negative (2) and zero (0) sequence.

Hence (reference phase A)

022

1C

0212

B

021A

UUaUaU

UUaUaU

UUUU

02

21C

0212

B

021A

IIaIaI

IIaIaI

IIII

where 3

2je

23j

21a

3

4j2 e23j

21a

Matrix

120

0

2

1

2

2

C

B

A

ABC UTUUU

1aa1aa111

UUU

U

Inversely

ABC1

C

B

A2

2

0

2

1

120 UTUUU

111aa1aa1

31

UUU

U

3ph power

*ABCT

ABC

*

C

B

A

CBA*CC

*BB

*AA IU

III

UUUIUIUIUS

*120*TT

120*

120T

120 ITTUITUTS

E3300030003

1aa1aa111

111aa1aa1

TT2

22

2

*T

*120T

120 IU3S

*00

*22

*11 IUIUIU3S

Series symmetrical segments in ES

C

B

A

C

B

A

III

ZZZZZZZZZ

UUU

ABCABCABC IZU 120ABC120 ITZUT 120120120ABC

1120 IZITZTU

TZTZ ABC1

120

Z2Z00

0ZZ000ZZ

Z000Z000Z

Z

0

2

1

120

Multiple lines

ABC

2V

1V

ABC

ABC

2V

1V

...II

Z...UU

120

2V

1V

ABC

120

2V

1V

...II

...000T000T

Z...UU

...000T000T

120

2V

1V

ABC

1

120

2V

1V

...II

...000T000T

Z...000T000T

...UU

120

2V

1V

ABC1

1

120

2V

1V

...II

...000T000T

Z...000T000T

...UU

Shunt symmetrical segments in ES

ABCNABCABCABC IZIZU

NNN

NNN

NNN

N

ZZZZZZZZZ

Z

120N1

120ABC1

120 ITZTITZTU

TZZTZ NABC1

120

N

120

Z3Z2Z000ZZ000ZZ

Z

Symmetrical components voltages in the symmetrical segments depend only on the corresponding component current and component impedance.

Inductors and capacitors in ES

a) Series inductors

- reactors are used to limit short-circuit currents - used in grids up to 35 kV, single-phase (In > 200A) or three-phase

(In < 200A), usually air-cooled (small L) - the same design in LC filters for harmonics suppression

Input: Xtl%, Stl, Un, In

Calculation: nntl .I.U3S

tl

2nt%

n

nt%tl S100

UXI3100

UXX

IZI)Xj(RUUU tttf2f1f EZZZ tt012tabc - 3ph inductor → self-impedance tZ , mutual impedances 0

In fault-free state the inductor can be bypassed by a fuse to reduce a voltage drop.

Rtl << Xtl

b) Shunt (parallel) inductors

- in the systems UN > 220 kV, oil cooling, Fe core - used to compensate capacitive (charging) currents of

overhead lines for no-load or small loads → U control:

n tl

2n tl

n tl

n tltl Q

UI3

UX

0tlt2tl1tl Z,ZZZ Connection in the system:

a) galvanic connection to the line - Y winding, neutral point connected to the ground

through V only during auto-reclosing (disturbances)

b) inductor connection to transformer tertiary winding - lower voltage Un ≈ 10 ÷ 35 kV - problem with switch-off (purely inductive load)

Kočín 400 kV

c) Neutral point inductors

- used in networks with indirectly earthed neutral point to compensate currents during ground fault

- value of the fault current does not depend on the ground fault position and the current is capacitive

- inductor reactance Xtl should assure that the value of the inductive current is equal to the value of capacitive current → arc extinction

- for voltage 6 to 35 kV (rated at Ufn), reactor is single-phased!, oil cooling - capacitive current change (network reconfiguration) → change in

inductance (air gap correction in the magnetic circuit) = arc-suppression coil (Peterson coil)

- it doesn’t occur in positive and negative sequence component, tl0 3XX

6 MVAr, 13 kV, Sokolnice

d) Series capacitors

- capacitors in ES = capacitor banks = series and parallel connection - to improve voltage conditions (MV lines) or adjusting parameters (long

HV lines) - voltage and power of the capacitor varies with the load - during short-circuits and overcurrents there appears overvoltage on the

capacitor (very fast protections are used)

IC

1jUC

- C must be insulated against the ground (insulated platforms) – C under voltage

- drawback – allows harmonic currents flow - current distribution among parallel transmission lines could be achieved

Canada 750 kV

e) Shunt capacitors

- used in industrial networks up to 1 kV - connection: a) wye - Y

b) delta - Δ (D)

Y2fCff CUIUQ CUIUQ 2

Cf Y

2Y

2f CUCU3Q CU3Q 2

- with the same reactive power

CU3CU 2Y

2 → C3CY → rather delta

- reactive power compensation a) QC < Q under-compensated b) QC = Q exact compensation c) QC > Q over-compensated

→ power factor improvement, lower power losses, voltage drops - individual or group compensation could be used

Transformer parameters

350 MVA, 400/110 kV YNauto - d1, Sokolnice

a) Two-winding transformers

- winding connection Y, Yn, D, Z, Zn, V Yzn – distribution TRF MV/LV up to 250 kVA, for unbalanced load Dyn – distribution TRF MV/LV from 400 kVA Yd – block TRF in power plants, the 3rd harmonic suppression Yna-d, YNynd – power grid transformer (400, 220, 110 kV) YNyd – power grid transformer (e.g. 110/23/6,3 kV)

- equivalent circuit: T – network σppσp jXRZ σssσs jXRZ qqq jBGY

- each phase can be considered separately (unbalance is neglected), i.e. operational impedance (positive sequence) is used

- values of the parameters are calculated, then verified by no-load and short-circuit tests

o no-load test – secondary winding open, primary w. supplied by rated voltage, no-load current flows (lower than rated one)

o short-circuit test – secondary winding short-circuit, primary w. supplied by short-circuit voltage (lower than rated one) so that rated current flows

P0 (W), i0 (%), Pk (W), zk = uk (%), Sn (VA), Un (V) uk ≈ 4 ÷ 14 % (increases with TRF power) pk ≈ 0,1 ÷ 1 % (decreases with TRF power) p0 ≈ 0,01 ÷ 0,1 % (decreases with TRF power)

- shunt branch:

n

0q S

ΔPg 100iy 0%

q 2q

2qq gyb

qq

2

n

02

0%

n

0q bjg

SΔP

100ij

SΔPy

qq

2

n

02

0%

n

02n

n2n

nqq BjG

SΔP

100ij

SΔP

US

USyY

- series branch:

n

kk S

ΔPr 100

uz k%k

2k

2kk rzx

kk

2

n

k2

k%

n

kk xjr

SΔP

100uj

SΔPz

kk

2

n

k2

k%

n

k

n

2n

n

2n

kk XjRSΔP

100uj

SΔP

SU

SUzZ

σsσpspkσps XXjRRZZ

- we choose σsσpsσp ZZ5,0Z

- this division is not physically correct (different leakage flows, different resistances)

- usage of T-network to calculate meshed systems is not appropriate sometimes (it adds another node)

- therefore calculation using -network, -network

b) Three-winding transformers

- parameters are calculated, then verified by no-load and short-circuit measurements (3 short-circuit tests: 1 winding no-load, 1 short-circuit and 1 supplied): P0 (W), i0 (%), Pk (W), zK = uK (%), Sn (VA), Un (V)

- powers needn’t be the same: e.g. SSn = STn = 0,5·SPn

- equivalent circuit:

- no-load measurement: related to the primary rated power and rated voltage SPn a UPn (supplied)

2

Pn

02

0%

Pn

0qqq S

ΔP100ij

SΔPbjgy

denominated value (S) – related to UPn

2

Pn

02

0%

Pn

02Pn

Pnqq2

Pn

Pnqq S

ΔP100ij

SΔP

USBjG

USyY

- short-circuit measurement: (3x, supply – short-circuit – no-load)

provided: SPn ≠ SSn ≠ STn

measurement between P - S P - T S - T short-circuit losses (W) PkPS PkPT PkST short-circuit voltage (%) ukPS ukPT ukST measurement corresponds to power (VA) SSn STn STn

short-circuit tests S – T: parameter to be found:

σTσSST ZZZ σSSσS XjRZ - recalculated to UPn σTσSST zzz - recalculated to UPn, SPn

Pk for ITn → PkST = 3·R+ST·I2Tn , Tn

TnTn U3

SI

R+ST….resistance of secondary and tertiary windings (related to UTn) 2Tn2

Tn

kSTST U

SPR

2Tn

2Pn

STST UURR

→ 2Pn2

Tn

kSTTSST U

SPRRR

RS (RT)…resistance of sec. and ter. windings recalculated to primary

Pn2Tn

kST2Pn

PnSTST S

SP

USRr

- impedance:

Tn

PnkST%ST S

S100

uz , Tn

2PnkST%

Pn

2Pn

STST SU

100u

SUzZ

STSTST xjrz , 2

ST2STST rzx , σTσSST xxx

- based on the derived relations we can write: P - S:

2

Pn2Sn

kPS

2

Sn

PnkPS%Pn2

Sn

kPSPSPS S

SΔP

SS

100ujS

SΔPxjrz

PS

22Pn2

Sn

kPS

2

Sn

2PnkPS%2

Pn2Sn

kPSPSPS U

SΔP

SU

100ujU

SΔPXjRZ

PS

- analogous for P – T and S – T

- leakage reactances for P, S, T: )ZZZ(5,0XjRZ STPTPSPPσP )ZZZ(5,0XjRZ PTSTPSSSσS )ZZZ(5,0XjRZ PSSTPTTTT

- knowledge of the series impedances and shunt admittances allows to study voltage and power conditions of 3-winding transformers

- mentioned impedances are valid for positive and negative sequences

Transformers zero sequence impedances

Series parameters are the same as for the positive sequence, the shunt always need to be determined. Assumptions:

- Zero sequence voltage supplies the primary winding. - The relative values are related to UPn and SPn. - We distinguish free and tied magnetic flows (shell x core TRF).

Z0 depends on the winding connection.

note: reluctance, inductance

magnetic resistance (reluctance)

Sl1R m

analogy Sl

Sl1R e

magnetic flux (Hopkinson’s law)

mRIN

analogy (Ohm’s law) eR

UI

m

2c

RN

IN

IL

LLRR Fe0mmFe0Fe

TRF magnetic circuit

LL

NI

NIL

NIL

RIN

RIN

FeFe

0mmFeh

a) Y / any connection 0i3 0

0

00 i

uz 0p0z ps0z

b) D / any connection Zero sequence voltage is attached to D → voltage at each phase

0uu 00 → 0iii cba → 0i0

0

00 i

uz 0p0z ps0z

Primary winding Secondary winding

Any connection


Any connection

c) YN / D Currents in the primary winding i0 induce currents i0’ in the secondary winding to achieve magnetic balance. Currents i0’ in the secondary winding are short-closed and do not flow further into the grid.

0qp0p0 zzz

0qs

0qsp

0

00 zz

zzz

iuz

shell s

1q0q zyz → k1ps0 zzz

3-core

1q0q yz → ps0 z9,07,0z


d) YN / Y Zero sequence current can’t flow through the secondary winding. Current i0 corresponds to the magnetization current.

ps0z 0qp0p00 zzzz

shell

1q0q yz → 0z

3-core

1q0q yz → 13,0z0


e) YN / YN If element with YN or ZN behind TRF → points a-b are connected → as the positive sequence. If element with Y, Z or D behind TRF → a-b are disconnected → as YN / Y.


f) ZN / any connection Currents i0 induce mag. balance on the core themselves → only leakages between the halves of the windings.

ps0z ps0p00 z3,01,0zz

p0 rr


g) impedance in the neutral point Current flowing through the neutral point is 3i0. Voltage drop: 0u0uuz iz3i3zu → in the model uz3 in series with the leakage reactance

h) three-winding TRF

Tertiary winding

Primary winding

Secondary winding

System equivalent

Impedance (positive sequence) is given by the nominal voltage and short-circuit current (power). Three-phase (symmetrical) short-circuit: )kA(I),MVA(S kk

knk IU3S

k

n

k

2n

s I3U

SUZ

CR: 400 kV MVA300006000Sk kA459Ik 220 kV MVA120002000Sk kA302Ik

110 kV MVA3000x100Sk kA15xIk

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Symmetrical system components - PowerWiki