Short-circuits in ES (2nd part)

Unbalanced short-circuits equivalence with three-phase short-circuit

Positive-sequence current component calculation by means of an additional impedance (in the short-circuit place according to short-circuit type)

Generalized relation

ZZEI

11

Short-cir. type 3ph 1ph 2ph 2ph to ground

Z 0 02 ZZ 2Z 02

02

ZZZZ

Current

components 1I 021 III 12 II 021 III

m ( 1k ImI ) 1 3 3 202

02

XXXX13

Substitution diagram

Individual short-circuit types comparison

For: t = 0 (I’’), R = 0, X1 = X2

ratio X0/X1 can be changed from 0 to ∞ reference 3ph short-circuit

Three-phase short-circuit

11

)3(k X

EII

Single-phase-to-ground short-circuit )3(

k

1

0

)3(k

01

1

0211

)1(k I

XX2

3IXX2

X3XXX

E3I3I

)3(k

)1(k I5,10I

Phase-to-phase short-circuit )3(

k)3(

k)3(

k1

1

211

)2(k I866,0I

23I

X2X3

XXE3I3I

Double-phase-to-ground short-circuit

1

0

1

0

)3(k

2

1

0

1

0

02

021

202

02)z2(k

XX1

XX

1

I

XX1

XX

13

XXXXX

EXXXX13I

)3(k

)z2(k I3

23I

5 10 15 20x0x1 -

0.5

1.0

1.5

ik -

2fz

2f

1f

3f

Arc influence during short-circuit

Single-phase-to-ground s.-c. Three-phase s.-c.

R3ZZZEI

0211 RZ

EI1

1

Phase-to-phase s.-c.

RZZEI

211

Note: Double-phase-to-ground s.-c. doesn’t have a symmetrical resistance segment.

Phase interruption

Single-phase interruption (analogy with double-phase-to-ground s.-c.)

0I;0UU ACB

Components

A021 U31UUU

ZZ

E

ZZZZZ

EI1

20

201

1

Double-phase interruption (analogy with single-phase-to-ground s.-c.)

0II;0U CBA

Components

A021

021 I31

ZZZEIII

Analogy between interruptions and short-curcuits

Sequence impedances short-circuits – between s.-c. place and the ground interruption – between places on both sides of the interruption

Similarly for the additional impedance. Sources always by means of impedances to the ground.

Multiple unbalances in ES

Single-phase-to-ground short-circuit in phase B, reference phase A

B

B2

B2

2

ABC1

120

IIaIa

31

0I0

111aa1aa1

31ITI B

B002

201 I31I;IaI;IaI

aa1

IIp

aa1

IIp

1p

22

02

2

1

01

0

Note: TRF ratio (1ph)

pis

pus

ipi

upu

power invariance *pp

*p

*ipu

*ss iuipupiu

*i

u p1p

If apu then u2*i paa1

a1p .

It is valid for reference phase B 0B02B21B1 UU,UU,UU

0212

C

021B

022

1A

UUaUaU

UUUU

UUaUaU

1aa1111aa

T2

2

111a1aa1a

31T 2

2

1

B

B

B

B2

2

ABC1

120

III

31

0I0

111a1aa1a

31ITI

Two simultaneous single-phase-to-ground short-circuits in different ES places

More than 2 faults → sequence diagrams interconnection is complicated → rather calculation in phases.

Short-circuit impedance matrix

short-circuit is replaces by two sources with U[k] voltage and the opposite orientation

U[k] voltage size equals to voltage value in the node k just before the fault

superposition principle

AES (active ES) => ES steady-state just before the short-circuit,

sources modelled by an ideal voltage source and generator reactance PES (passive ES) => without sources: fault state, generators replaced

only by sub-transient reactance against the ground

UYI

iMm

km)k,k( YY km)m,k( YY ES simplified diagram a) corresponding with node admittance matrix b) corresponding with short-circuit admittance matrix

All node currents are zero except the short-circuit place, here an ideal voltage source → short-circuit admittance matrix.

UYI k

n

k

1

kk

U

U

U

Y

0

I

0

Conversion to short-circuit impedance matrix IZIYU k

1k

0

I

0

ZZ

Z

ZZ

U

U

U

k

)n,n()1,n(

)k,k(

)n,1()1,1(

n

k

1

Short-circuit current

)k,k(

kk Z

UI

)k,k(Z short-circuit impedance in the node k

Voltage in any node k)k,j(j IZU

Current in a branch

kij

)k,i()k,j(

ij

jiij I

ZZZ

ZUU

I

ijZ impedance of the branch between the nodes i and j

Real node voltages and real branch currents during short-circuits are: fault = AES + PES Iij

k = I(ij) + Iij Ujk = U(j) + Uj

where: current in the branch between the nodes i and j, node j voltage Iij

k, Ujk during short-circuit

I(ij), U(j) just before the short-circuit origin Iij, Uj self fault state

Short-circuit currents impacts

Mechanical impacts Influence mainly at tightly placed stiff conductors, supporting insulators, disconnectors, construction elements,… Forces frequency 2f at AC → dynamic strain.

Force on the conductor in magnetic field )N(sinlIBF )T(HB )m/H(104 7

0

α – angle between mag. induction vector and the conductor axis (current direction)

Magnetic field intensity in the distance a from the conductor

)m/A(a2

IH

2 parallel conductors → force perpendicular to the conductor axis (sin α = 1) → it is the biggest

)N(laI102Il

a2I104F

277

The highest force corresponds to the highest immediate current value → peak short-circuit current Ikm (1st magnitude after s.-c. origin)

)A(I2e1I2I 0kT/01,0

0kkmk

κ – peak coefficient according to grid type (κLV = 1,8; κHV = 1,7) theoretical range κ = 1 ÷ 2 Tk – time constant of equivalent short-circuit loop (Le/Re) i.e. for DC component of short-circuit current 0kI - initial sub-transient short-circuit current

Real value differs according to the short-circuit origin moment. AC component decreasing slower than for DC therefore neglected.

Max. instantaneous force on the conductor length unit

)m/N(a

I10kk2f2

km721

k1 – conductor shape coefficient

k2 – conductors configuration and currents phase shift coefficient

a – conductors distance

Heat impacts Key for dimensioning mainly at freely placed conductors. They are given by heat accumulation influenced by time-changing current during short-circuit time tk (adiabatic phenomenon).

Heat produced in conductors

)J(dt)t(i)(RQkt

0

2k

Thermal equivalent current – current RMS value which has the same heating effect in the short-circuit duration time as the real short-circuit current

kt

0

2kk

2ke dt)t(itI

)A(dt)t(it1I

kt

0

2k

kke

Calculation according to ke coefficient as kI multiple keke IkI

Coefficient ke

S.-c. in ES

Short-circuit duration time

tk (s)

S.-c. on generator terminals

HV, MV LV

pod 0,05 1,70 1,60 1,50 0,05 – 0,1 1,60 1,50 1,20 0,1 – 0,2 1,55 1,40 1,10 0,2 – 1,0 1,50 1,30 1,05 1,0 – 3,0 1,30 1,10 1,00 nad 3,0 1,15 1,00 1,00