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Short-circuits – Lecture 14 Short-circuits calculations according to standard IEC 60909

Prof. Désiré Rasolomampionona, Prof. dr hab. Jan Machowski

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Outline of the lecture •Definitions •Schematic diagram of short-circuit current •Low Voltage factors •short-circuits fed from non-meshed networks •Short-circuit currents inside a power station unit with on-load tap-changer •Short-circuit currents inside a power station unit without on-load tap-changer •short-circuits in meshed networks •Symmetrical short-circuit breaking current •steady-state short-circuit current •Joule integral and thermal equivalent short-circuit current

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Short-circuits Standards Definitions Definition

Symbol according to IEC 60909

Peak short-circuit current

" IK ip

Symmetrical short-circuit breaking current

Ib

Initial symmetrical short-circuit current

µ

Factor for the calculation of the symmetrical short-circuit breaking current Decaying aperiodic component of short-circuit current

idc

Thermal equivalent short-circuit current

I th

Assymmetrical short-circuit breaking current Initial symmetrical short-circuit power Duration of the short-circuit current

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I b asym " SK TK

Short-circuits Standards

a)

Current

1 - top enveloppe

2 2 IK

(a)

A

ip

2 2 IK

idc - DC component idc of the short-circuit current

Fig. 1. Schematic diagram of short-circuit current

Time

(b) 2 - bottom enveloppe b) Current 1 - top enveloppe

short-circuit current of far from generator shortcircuit with decaying AC component short-circuit current of near to generator shortcircuit with constant AC component.

idc - DC component idc of the short-circuit current

A

2 2 IK = 2 2 IK

ip

2 2 IK

dc

2 - bottom enveloppe Teaching materials distributed for free.

Time

All the other symbols (A, IK, ip, Idc) were given in the definitions, 1. Top enveloppe, 2. bottom enveloppe

Short-circuits Standards •Table 1. Selection of Voltage factor c (equivalent voltage source) Voltage factor c to be calculated Nominal voltage Un

of maximum voltage short-circuit

of minimum voltage shortcircuit

Low voltage (100 – 1000V) a) 230 – 240V b) Other voltages

1.00 1.05

0.95 1.00

Medium voltage (1 – 35kV)

1.10

1.00

High voltage > 35kV

1.10

1.00

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Short-circuits Standards Selection of Voltage factor c (equivalent voltage source) In all hitherto used formulas the initial short-circuit and short-circuit currents were calculated using the U0 voltage (equivalent voltage source) derived from the Thevenin’s theorem. by corresponding to the prefault voltage at short-circuit location node. The IEC standard defines this voltage from the network nominal voltage mutipiled by the c factor given in the previous slide table as follows : U o = c ⋅U n

After defining such a voltage the initial three phase short-circuit current may be calculated using the following expression : " IK =

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cU n 2 2 3 ⋅ RK + XK

=

cU n 3Z K

Short-circuits Standards Short-circuits fed from non-meshed networks For a far-from-generator short-circuit fed from a single source (see Figure 1), the shortcircuit current is calculated using the equation presented at the bottom of the previous slide. G When there is more than one source 3 contributing to the short-circuit current, and the Q M sources are unmeshed, as shown for instance 3 in the figure at right, The initial symmetrical short-circuit current at I KPSU I KT I KM IK i pPSU i pT i pM the short-circuit location F is the sum of the ip I bPSU I bM Ib individual branch short-circuit currents. K3 IK Each branch short-circuit current can be F calculated as an independent single-source three-phase short-circuit current in Fig. 2. Example of a non-meshed network accordance with equation " IK =

cU n 2 2 3 ⋅ RK + XK

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=

cU n 3Z K

Short-circuits Standards Short-circuits fed from non-meshed networks The initial symmetrical short-circuit current is calculated with the corrected impedances of the generator and the power station unit in series with a line impedance. The shortcircuit impedances for the different cases are given by the following equations: •short-circuit fed from one power station unit (generator and unit transformer with or without on-load tap-changer)

(

)

2 Z K = Z PSU + Z L = K PSU t r1 Z G + Z THV + Z L

•Equivalent circuit with unit transformer and transmission line 2 Z K = Z Q t r2 + K T Z THV + Z L

•Set of HV motors and transmission line ZK = ZM +ZL Teaching materials distributed for free.

Short-circuits Standards Short-circuits fed from non-meshed networks The initial short-circuit current at the short-circuit location F is the phasor sum of the individual partial short-circuit currents :

I "K = ∑ I "Ki i

Within the accuracy of this standard, it is often sufficient to determine the short-circuit current at the short-circuit location F as being the sum of the absolute values of the individual partial short-circuit currents.

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Short-circuits Standards Short-circuit currents inside a power station unit with on-load tap-changer I "KG

For calculating the partial short-circuit currents and I "KT with a short-circuit at F1 (Fig. 3), in the case of a power station unit with on-load tapchanger, the partial initial symmetrical short-circuit currents are given by: I "KG

I "KT =

cU rG

=

G G 3

I KG

K GS =

c max 1 + X d" sin ϕ rG

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1:t

IKF2 K3

Q UnQ IKQmax UQ

F2 t rAT 1

AT

IKAT

cU rG

Where KGS is a correcting factor given by the following expression

T

F1

3K GS Z G

3 Z TLV + 12 Z Qmin tr

K3 IKT

K3 F3

A

Fig. 3. Short-circuit currents and partial short-circuit currents for three-phase short-circuits between generator and unit transformer with or without on-load tap-changer, or at the connection to the auxiliary transformer of a power station unit and at the auxiliary busbar A

Short-circuits Standards Short-circuit currents inside a power station unit with on-load tap-changer And the rest of the symbols are defined as follows: ZG is the subtransient impedance of the generator X d" is the subtransient reactance referred to the rated impedance:

ZTLV is the transformer short-circuit impedance referred to the low-voltage side tr is the rated transformation ratio; ZQmin is the minimum value of the impedance of the network feeder, corresponding to the maximum short-circuit power

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Short-circuits Standards Short-circuit currents inside a power station unit with on-load tap-changer For the calculation of the partial short-circuit current feeding into the short-circuit location F2, for example at the connection to the high-voltage side of the auxiliary transformer AT in the figure, it is sufficient to take: cU rG 1 1 " I KF2 = + 3 K GS Z G K TS Z TLV + 12 Z Qmin tr

With KTS given by the following expression K TS =

And KGS as it was given before K GS =

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cmax 1 − X Tp.u. sin ϕ rG c max

1 + X d" sin ϕ rG

Short-circuits Standards Short-circuit currents inside a power station unit without on-load tap-changer For a power station unit without on-load tap-changer of the unit transformer, the partial initial symmetrical short-circuit currents (the same figure) are defined by the same formulas as for the case with on-load tap-changer, with small modifications of the correcting factors KGS and KTS, which become KGS0 and KTS0 I

" KG

I "KT =

cU rG = 3K GS0 Z G

I

" KF2

cU rG = 3

cU rG 3 Z TLV + 12 Z Qmin tr

1 1 + K GS0 Z G K TS0 Z TLV + t1r2 Z Qmin

(

with K GSO =

c max 1 ⋅ 1 + p G 1 + X d" sin ϕ rG

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K TSO =

c max 1 ⋅ 1 + p G 1 − X T p.u. sin ϕ rG

)

Short-circuits Standards Short-circuit currents inside a power station unit without on-load tap-changer If the unit transformer has an on-load tap-changer on the high-voltage side, it is assumed that the operating voltage at the terminals of the generator is equal to UrG. If, even in this case, the voltage region of the generator UG = UrG(1±pG) is used permanently, take equations for the case without on-load tap changer than those for the case with on-load tap changer. The total short-circuit current in F1 or F2 is found by adding the partial short-circuit current, caused by the medium- and low-voltage auxiliary motors of the power station unit.

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Short-circuits Standards Short-circuit currents in meshed networks In meshed networks, such as those shown in figure at right, it is generally necessary to determine the short-circuit impedance Zk = Z(1) by network reduction (series connection, parallel connection, and delta-star transformation, for example) using the positive-sequence short-circuit impedances of electrical equipment. The impedances in systems connected through transformers to the system, in which the shortcircuit occurs, have to be transferred by the square of the rated transformation ratio. If there are several transformers with slightly differing rated transformation ratios (trT1 trT2... trTn), in between two systems, the arithmetic mean value can be used. Teaching materials distributed for free.

Q

G 3 F

M 3

M 3

M 3

Fig. 4. System diagram

Short-circuits Standards Short-circuit currents in meshed networks The initial symmetrical short-circuit current shall be calculated with the equivalent voltage source cU n 3 at the short-circuit location using equation

" IK =

cU n 3⋅

2 2 RK + XK

=

Q

G 3 F

cU n 3Z K M 3

M 3

M 3

Fig. 4. System diagram

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Short-circuits Standards Short-circuit currents in meshed networks For three-phase short-circuits fed from non-meshed networks as in figures 2 and 3, the contribution to the peak short-circuit current from each branch can be expressed by: " ip = χ 2 I K

The factor χ for the R/X ratio shall be obtained from Figure 5 or calculated by the following expression:

χ = 1,02 + 0,98e −3R/X

2,0 1,8 1,6 x 1,4 1,2 1,0

0

0,2 0,4 0,6 0,8 1,0 1,2 R /X

Fig. 5. Factor χ for series circuit as a function of ratio R/X

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Short-circuits Standards Short-circuit currents in meshed networks Equations of ip and χ of the previous slide presume that the short-circuit starts at zero voltage, and that ip is reached approximately after one half-cycle. For a synchronous generator use RGf.. The peak short-circuit current ip at a short-circuit location F, fed from sources which are not meshed with one another, in accordance with Figure 2, is the sum of the partial shortcircuit currents:

And from Fig. 2

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Short-circuits Standards Short-circuit currents in meshed networks When calculating the peak short-circuit current ip in meshed networks, the equation of ip shall be used with χ determined using one of the following methods a), b), or c). a) Uniform ratio R/X or X/R For this method the factor χ is determined from figure 5 taking the smallest ratio of R/X or the largest ratio of X/R of all branches of the network. It is only necessary to choose the branches which carry partial short-circuit currents at the nominal voltage corresponding to the short-circuit location and branches with transformers adjacent to the short-circuit location. Any branch may be a series combination of several impedances. In practise, considering branches, through which the flowing current is about 80% of the shortcircuit current is sufficient.

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Short-circuits Standards Short-circuit currents in meshed networks b) Ratio R/X or X/R at the short-circuit location For this method the factor χ is multiplied by a factor 1,15 to cover inaccuracies caused by using the ratio Rk / Xk from a network reduction with complex impedances.

As long as R/X remains smaller than 0,3 in all branches, it is not necessary to use the factor 1,15. It is not necessary for the product 1,15 . χ(b) to exceed 1,8 in low-voltage networks or to exceed 2,0 in medium- and high-voltage networks. The factor χ(b) is found from figure5 for the ratio Rk / Xk given by the short-circuit impedance Zk = Rk + jXk at the short-circuit location F, calculated for frequency f = 50 Hz

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Short-circuits Standards Short-circuit currents in meshed networks c) Equivalent frequency fc An equivalent impedance Zc of the system as seen from the short-circuit location is calculated assuming a frequency fc = 20 Hz The R/X or X/R ratio is then determined according to the following equation :

where

R Rc f c = ⋅ X Xc f

Zc = Rc + jXc is the equivalent impedance of the system as seen from the short-circuit location for the assumed frequency fc; Rc is the real part of Zc (Rc is generally not equal to the R at nominal frequency) Xc is the imaginary part of Zc (Xc is generally not equal to the X at nominal frequency).

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Short-circuits Standards Short-circuit currents in meshed networks c) Equivalent frequency fc The factor χ is found from figure 5 using the R/X or X/R ratio from equation (*), or with equation (**). Method c) is recommended in meshed networks (see IEC 60909-1). R Rc f c = ⋅ X Xc f

(*)

χ = 1,02 + 0,98e −3R/X

(**)

When using this method in meshed networks with transformers, generators and power station units, the impedance correction factors KT, KG and KS, respectively KSO, shall be introduced with the same values as for the 50 Hz or 60 Hz calculations.

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Short-circuits Standards Symmetrical short-circuit breaking current: Single-fed three- phase short-circuit For a near-to-generator short-circuit, in the case of a single fed short-circuit or from nonmeshed networks (fig. 2), the decay to the symmetrical short-circuit breaking current Ib (*) is taken into account by the factor m according to equations (**). " I b = µ I K" (*) - for tmin = 0,02 s µ = 0,84 + 0,26 e- 0 ,26 I KG / I rG The factor m depends on the minimum time delay tmin and the ratio I K" / I rG where IrG is the rated generator current.

"

- for tmin = 0,05 s

µ = 0,71 + 0,51 e- 0 ,30 I KG / I rG

- for tmin = 0,10 s

" /I - 0 ,32 I KG rG

µ = 0,62 + 0,72 e

"

- for tmin ≥ 0,25 s µ = 0,56 + 0,94 e- 0 ,38 I KG / I rG

The values of m in equation (**) apply if synchronous machines are excited by rotating exciters or by static converter exciters (provided, for static exciters, the minimum time delay tmin is less than 0,25 s and the maximum excitation voltage is less than 1,6 times rated load excitation-voltage). For all other cases take µ = 1 if the exact value is unknown.

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(**)

Short-circuits Standards Symmetrical short-circuit breaking current: Single-fed three- phase short-circuit If I K" / I rG is not greater than 2, apply µ = 1 for all values of the minimum time delay tmin. 1,0 The factor µ may also be obtained from figure 6. For other values of minimum time delay, linear interpolation between curves is acceptable.

µ

Minimum time delay tmin

0,9

0,02 s

0,8

0,05 s

0,7

0,1 s 0,25 s

0,6

Figure 6 can be used also for compound excited low-voltage generators with a minimum time delay tmin not greater than 0,1 s.

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0,5 0 1 2 3 4 5 6 7 8 Three-phase short circuit IkG /I rG or I kM /IrM

Fig. 6. Factor µ

9

Short-circuits Standards Symmetrical short-circuit breaking current: Single-fed three- phase short-circuit For three-phase short-circuits in non-meshed networks as in figure 2, the symmetrical breaking current at the short-circuit location can be calculated by the summation of the individual breaking current contributions: " " I b = I b PSU + I KT + I b M where I bM = qµI KM m is taken from equation (** - slide 24) or figure 6 for synchronous generators and asynchronous motors. The factor q for the calculation of the symmetrical short-circuit breaking current for asynchronous motors may be determined as a function of the minimum time delay tmin (fig. 7). Where m is the ratio between is the rated − for tmin = 0,02 s q = 1,03 + 0,12 ln m active power in MW and the number of − for tmin = 0,05 s q = 0,79 + 0,12 ln m (*) pairs of poles of the motor. − for tmin = 0,10 s q = 0,57 + 0,12 ln m m = PrM / p − for tmin ≥ 0,25 s q = 0,26 + 0,10 ln m Teaching materials distributed for free.

Short-circuits Standards Symmetrical short-circuit breaking current Three-phase short-circuit in meshed networks 1,0

The short-circuit breaking current Ib in meshed networks shall be calculated by:

0,9 0,8

Minimum time delay tmin 0,02 s

0,7

I b = I K"

0,6

0,05 s

0,5

At first the current at the short-circuit location is calculated for the time of breaking, and then the partial currents in the branches where the circuit breakers are located.

q

0,4

0,1 s

0,3 0,2 0,25 s 0,1 0 0,01 0,02 0,04 0,1 0,2 0,4 1 2 4 MW 10 Active power of the motor per pair of poles m

∆U Mj ∆U G" i Fig. 7. Factor q for the calculation of the " " ( ( Ib = I − ∑ 1 − µi )I KG − 1 − µ j q j )I KM ∑ i j i cU n j cU n symmetrical short-circuit breaking current 3 3 of asynchronous motors "

" K

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Short-circuits Standards Symmetrical short-circuit breaking current Three-phase short-circuit in meshed networks ∆U M" j ∆U G" i " " " ( ( Ib = IK − ∑ 1 − µi )I KGi − ∑ 1 − µ j q j )I KM j cU cU i

n

j

n

3 3 where µi, µj are the values given in equation (** - slide 24) for both synchronous (i) and asynchronous (j) machines; qj are the values given in equation (* - slide 26) for asynchronous (j) motors;

I b , I K" are respectively the initial symmetrical short-circuit current and the symmetrical shortcircuit breaking current with influence of all network feeders, synchronous machines and asynchronous motors; ∆U G" i , ∆U M" j are the initial voltage drops at the terminals of the synchronous machines (i) and the asynchronous motors (j); " " I KG i , I KMj are the contributions to the initial symmetrical short-circuit current from the synchronous machines (i) and the asynchronous motors (j) as measured at the terminals of the machines.

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Short-circuits Standards Symmetrical short-circuit breaking current The symmetrical short-circuit breaking current may be considered as : the difference between the symmetrical short-circuit breaking current with influence of all network feeders, synchronous machines and asynchronous motors and

IK2 IK3

IK,Ib

IK4

IK1 IK6 IK5

Fig. 8. Calculation of short-circuit the contributions to the initial symmetrical breaking current Ib in meshed networks short-circuit current from the synchronous machines (i) and the asynchronous motors (j) I b = I K" − ∑ I zGi − ∑ I zMj (*) as measured at the terminals of the i j machines. Both last current sum can be considered as decaying components and can be determined using the following equations, providing that the short-circuit occurs at the terminals of the ith generator (jth motor ) : " " I zGi = (1 − µ i )I KG I zMj = (1 − q j µ j )I KM i , j Teaching materials distributed for free.

(**)

Short-circuits Standards Symmetrical short-circuit breaking current The short-circuit is a far-from-generator short-circuit, hence the decaying process of the short-circuit current sinusoidal component is dynamic (Fig. 1). We can assume that the decaying factor is equal to a coefficient αi, which is a real number, the absolute value of which less than 1 (αj for motors), multiplied by the current defined for near-togenerator short-circuit. " I zGi = α i (1 − µ i )I KG i,

(

)

" I zMj = α j 1 − q j µ j I KM j

(*)

The coefficient αi defines the distance between the short-circuit location and the generator terminals (for near-to-generator short-circuits, its value is near 1, for very farfrom-generator short-circuits almost equal to 0). In case of non-meshed networks this coefficient can be determined in correlation with the impedance of the portion of line from the short-circuit location and the generator.

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Short-circuits Standards Symmetrical short-circuit breaking current In case of meshed networks it is difficult to define such a line, then the distance between the short-circuit location and the generator is assumed as equal to the ration of voltage drop in the corrected generator (motor) reactance and the network phase voltage, hence.

αi =

∆U G" i cU n 3

=

" " I KG i X dK cU n 3

(*)

αj =

" ∆U Mj cU n 3

=

I kMj X M cU n 3

(**)

Considering the equations (* - previous slide), (*) and (**) leads to a new form of the equation (* - slide nr 28). In case of greater number of voltage sources, it is much more convenient to use special computer methods for short-circuit calculations.

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Short-circuits Standards Maximum steady-state short-circuit current For near-to-generator three-phase short-circuits fed directly from one synchronous generator or one power station unit only, according to figure 11b or 11c, the steady-state short-circuit current Ik depends on the excitation system, the voltage regulator action, and saturation influences. Synchronous machines (generators, motors, or compensators) with terminal-fed static exciters do not contribute to Ik in the case of a short-circuit at the terminals of the machine, but they contribute to Ik if there is an impedance between the terminals and the short-circuit location. A contribution is also given if, in case of a power station unit, the short-circuit occurs on the highvoltage side of the unit transformer For the calculation of the maximum steady-state short-circuit current, the synchronous generator may be set at the maximum excitation. I K max = λmax I rG For static excitation systems fed from the generator terminals and a short-circuit at the terminals, the field voltage collapses as the terminal voltage collapses, therefore take λmax = λmin = 0 in this case. Teaching materials distributed for free.

Short-circuits Standards Maximum steady-state short-circuit current a) seria pierwsza

λmax may be obtained from figures 9 or 10 for cylindrical rotor generators or salient-pole generators. The saturated reactance xdsat is the reciprocal of the saturated no-load short-circuit ratio.

λmax curves of series 1 are based on the highest possible excitation voltage according to either 1,3 times the rated excitation at rated apparent power and power factor for cylindrical rotor generators (figure 9a) or 1,6 times the rated excitation voltage for salient-pole generators (figure 10a). Teaching materials distributed for free.

2,8 2,6 2,4 2,2 2,0 1,8 1,6 1,4 λ 1,2 1,0 0,8 0,6 0,4 0,2 0

λ max

X d sat 1,2 1,4 1,6 1,8 2,0 2,2

λ min

0 1 2 3 4 5 6 7 8 Zwarcie trójfazowe IKG /IrG

b) seria druga 2,8 X d sat 1,2 2,6 1,4 2,4 1,6 λ max 1,8 2,2 2,0 2,0 2,2 1,8 1,6 1,4 λ 1,2 1,0 0,8 λ min 0,6 0,4 0,2 0 0 1 2 3 4 5 6 7 8 Zwarcie trójfazowe I KG /IrG

Fig. 9. λmin and λmax factors of series 1 (left) λmin and λmax factors of series 2 (right)

Short-circuits Standards Maximum steady-state short-circuit current λmax -curves of series 2 are based on the highest possible excitation-voltage according to either 1,6 times the rated excitation at rated apparent power and power factor for cylindrical rotor generators (figure 9b), or 2,0 times the rated excitation voltage for salient-pole generators (figure 10b).

λmax -curves of series 1 or 2 may also be

a) seria pierwsza 5,5 5,0 4,5 4,0 3,5 3,0 λ 2,5 2,0 1,5 1,0 0,5 0

applied in the case of terminal-fed static exciters, if the short-circuit is at the highvoltage side of the unit transformer of a power station unit or in the system, and if the maximum excitation voltage is chosen with respect to the partial breakdown of the terminal voltage of the generator during the short-circuit. Teaching materials distributed for free.

λ max

b) seria druga X d sat 0,6 0,8 1,0 1,2 1,7 2,0

λ min

0 1 2 3 4 5 6 7 8 Zwarcie trójfazowe I KG /IrG

5,5 5,0 4,5 4,0 3,5 3,0 λ 2,5 2,0 1,5 1,0 0,5 0

λ max

X d sat

0,6 0,8 1,0 1,2 1,7 2,0

λ min

0 1 2 3 4 5 6 7 8 Zwarcie trójfazowe I KG /IrG

Fig. 10. λmin and λmax factors of series 1 (left) λmin and λmax factors of series 2 (right)

Short-circuits Standards Minimum steady-state short-circuit current For the minimum steady-state short-circuit current in the case of a single-fed shortcircuit from one generator or one power station unit, constant no-load excitation (voltage regulator not being effective) of the synchronous machine is assumed:

I K min = λmin I rG λmin may be obtained from figures 9 and 10. In the case of minimum steady-state short-circuit introduce c = cmin, according to table 1 (slide 5).

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Short-circuits Standards Minimum steady-state short-circuit current The calculation of the minimum steady-state short-circuit current in the case of a neartogenerator short-circuit, fed by one or several similar and parallel working generators with compound excitation, is made as follows:

For the effective reactance of the generators, introduce:

IkP is the steady-state short-circuit current of a generator at a three-phase terminal short-circuit. The value should be obtained from the manufacturer.

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Short-circuits Standards DC component of the short-circuit current The maximum d.c. component id.c. of the short-circuit current as shown in figures 1 and 2 may be calculated with sufficient accuracy by the following equation : " −2πft K R / X idc = 2 I K e

where I K" is the initial symmetrical short-circuit current;

f

is the nominal frequency;

t

is the time;

R/X s the ratio according to Fig. 5 or the ratios according to the methods a) and c) slides 19 - 22.

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Short-circuits Standards Joule integral and thermal equivalent short-circuit current The joule integral is a measure of the energy generated in the resistive element of the system by the short-circuit current. In this standard it is calculated using a factor m for the time-dependent heat effect of the d.c. component of the short-circuit current and a factor n for the time-dependent heat effect of the a.c. component of the short-circuit current (see figures 11 and 12) The thermal equivalent short-circuit current is: " I th = I K m+n

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Short-circuits Standards Joule integral and thermal equivalent short-circuit current For a series of i ( i = 1, 2,....,r) three-phase successive individual short-circuit currents, the following equation shall be used for the calculation of the Joule integral or the thermal equivalent short-circuit current. where I " Ki (*)

is the initial symmetrical three-phase short-circuit current for each short-circuit

I th

is the thermal equivalent short-circuit current

mi

is the factor for the heat effect of the d.c. component for each short-circuit current

ni

is the factor for the heat effect of the a.c. component for each short-circuit current

(**) (***)

The Joule integral and the thermal equivalent shortcircuit current should always be given with the short-circuit duration with which they are associated.

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TKi

is the duration of the short-circuit current for each short-circuit

TK is the sum of the durations for each short-circuit current

Short-circuits Standards Joule integral and thermal equivalent short-circuit current The factors mi are obtained from Figure 11 using f · Tki and the factor k derived in [2]. The " factors ni are obtained from Figure 12 using Tki and the quotient I K I K , where Iki is the steadystate short-circuit current for each short-circuit.

a)

b)

2,0

1,6

1,6

9 1, 8 χ= , 1 1, 7

2,0 1,2

1, 1,

1,

6

5

m 0,8

1,

0,4 1,3 4 2 1 ,1 0 10 -2 2 4 6 10-12 4 6 s1 tK

Fig. 11. Factor m for the heat effect of the d.c. component of the short-circuit current

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IK/IK =10

1,2 3

1,25 1 ,5 2 ,0

,0 n 0,8 5 , 0 4, 0 0,4 6 ,0 0 10 -2 2 4 6 10-12 4 6 1 2 4 6 s 10 tK Fig. 12. Factor n for the heat effect of the a.c. component of the short-circuit current

Short-circuits Standards Joule integral and thermal equivalent short-circuit current When a number of short-circuits occur with a short time interval in between them, the resulting Joule integral is the sum of the Joule integrals of the individual short-circuit currents, as given in equation (* slide 38) For distribution networks (far-from-generator short-circuits) usually n=1 can be used. For far-from-generator short-circuit with the rated short-circuit duration of 0,5 s or more, it is permissible to take m + n = 1. If the Joule integral or the thermal equivalent short-circuit current shall be calculated for unbalanced short-circuits, replace I Ki" with the appropriate unbalanced shortcircuit currents.

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Bibliography 1. Machowski J, Kacejko P. Zwarcia w systemach elektroenergetycznych (Power System Short-Circuits – in Polish), Wydawnictwo Naukowo-Techniczne, Warszawa 2002 2. INTERNATIONAL STANDARD IEC 60909-0 „Short-circuit currents in three-phase a.c. systems” Calculation of currents

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Prof. Désiré Rasolomampionona, Prof. dr hab. Jan Machowski

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Outline of the lecture •Definitions •Schematic diagram of short-circuit current •Low Voltage factors •short-circuits fed from non-meshed networks •Short-circuit currents inside a power station unit with on-load tap-changer •Short-circuit currents inside a power station unit without on-load tap-changer •short-circuits in meshed networks •Symmetrical short-circuit breaking current •steady-state short-circuit current •Joule integral and thermal equivalent short-circuit current

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Short-circuits Standards Definitions Definition

Symbol according to IEC 60909

Peak short-circuit current

" IK ip

Symmetrical short-circuit breaking current

Ib

Initial symmetrical short-circuit current

µ

Factor for the calculation of the symmetrical short-circuit breaking current Decaying aperiodic component of short-circuit current

idc

Thermal equivalent short-circuit current

I th

Assymmetrical short-circuit breaking current Initial symmetrical short-circuit power Duration of the short-circuit current

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I b asym " SK TK

Short-circuits Standards

a)

Current

1 - top enveloppe

2 2 IK

(a)

A

ip

2 2 IK

idc - DC component idc of the short-circuit current

Fig. 1. Schematic diagram of short-circuit current

Time

(b) 2 - bottom enveloppe b) Current 1 - top enveloppe

short-circuit current of far from generator shortcircuit with decaying AC component short-circuit current of near to generator shortcircuit with constant AC component.

idc - DC component idc of the short-circuit current

A

2 2 IK = 2 2 IK

ip

2 2 IK

dc

2 - bottom enveloppe Teaching materials distributed for free.

Time

All the other symbols (A, IK, ip, Idc) were given in the definitions, 1. Top enveloppe, 2. bottom enveloppe

Short-circuits Standards •Table 1. Selection of Voltage factor c (equivalent voltage source) Voltage factor c to be calculated Nominal voltage Un

of maximum voltage short-circuit

of minimum voltage shortcircuit

Low voltage (100 – 1000V) a) 230 – 240V b) Other voltages

1.00 1.05

0.95 1.00

Medium voltage (1 – 35kV)

1.10

1.00

High voltage > 35kV

1.10

1.00

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Short-circuits Standards Selection of Voltage factor c (equivalent voltage source) In all hitherto used formulas the initial short-circuit and short-circuit currents were calculated using the U0 voltage (equivalent voltage source) derived from the Thevenin’s theorem. by corresponding to the prefault voltage at short-circuit location node. The IEC standard defines this voltage from the network nominal voltage mutipiled by the c factor given in the previous slide table as follows : U o = c ⋅U n

After defining such a voltage the initial three phase short-circuit current may be calculated using the following expression : " IK =

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cU n 2 2 3 ⋅ RK + XK

=

cU n 3Z K

Short-circuits Standards Short-circuits fed from non-meshed networks For a far-from-generator short-circuit fed from a single source (see Figure 1), the shortcircuit current is calculated using the equation presented at the bottom of the previous slide. G When there is more than one source 3 contributing to the short-circuit current, and the Q M sources are unmeshed, as shown for instance 3 in the figure at right, The initial symmetrical short-circuit current at I KPSU I KT I KM IK i pPSU i pT i pM the short-circuit location F is the sum of the ip I bPSU I bM Ib individual branch short-circuit currents. K3 IK Each branch short-circuit current can be F calculated as an independent single-source three-phase short-circuit current in Fig. 2. Example of a non-meshed network accordance with equation " IK =

cU n 2 2 3 ⋅ RK + XK

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=

cU n 3Z K

Short-circuits Standards Short-circuits fed from non-meshed networks The initial symmetrical short-circuit current is calculated with the corrected impedances of the generator and the power station unit in series with a line impedance. The shortcircuit impedances for the different cases are given by the following equations: •short-circuit fed from one power station unit (generator and unit transformer with or without on-load tap-changer)

(

)

2 Z K = Z PSU + Z L = K PSU t r1 Z G + Z THV + Z L

•Equivalent circuit with unit transformer and transmission line 2 Z K = Z Q t r2 + K T Z THV + Z L

•Set of HV motors and transmission line ZK = ZM +ZL Teaching materials distributed for free.

Short-circuits Standards Short-circuits fed from non-meshed networks The initial short-circuit current at the short-circuit location F is the phasor sum of the individual partial short-circuit currents :

I "K = ∑ I "Ki i

Within the accuracy of this standard, it is often sufficient to determine the short-circuit current at the short-circuit location F as being the sum of the absolute values of the individual partial short-circuit currents.

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Short-circuits Standards Short-circuit currents inside a power station unit with on-load tap-changer I "KG

For calculating the partial short-circuit currents and I "KT with a short-circuit at F1 (Fig. 3), in the case of a power station unit with on-load tapchanger, the partial initial symmetrical short-circuit currents are given by: I "KG

I "KT =

cU rG

=

G G 3

I KG

K GS =

c max 1 + X d" sin ϕ rG

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1:t

IKF2 K3

Q UnQ IKQmax UQ

F2 t rAT 1

AT

IKAT

cU rG

Where KGS is a correcting factor given by the following expression

T

F1

3K GS Z G

3 Z TLV + 12 Z Qmin tr

K3 IKT

K3 F3

A

Fig. 3. Short-circuit currents and partial short-circuit currents for three-phase short-circuits between generator and unit transformer with or without on-load tap-changer, or at the connection to the auxiliary transformer of a power station unit and at the auxiliary busbar A

Short-circuits Standards Short-circuit currents inside a power station unit with on-load tap-changer And the rest of the symbols are defined as follows: ZG is the subtransient impedance of the generator X d" is the subtransient reactance referred to the rated impedance:

ZTLV is the transformer short-circuit impedance referred to the low-voltage side tr is the rated transformation ratio; ZQmin is the minimum value of the impedance of the network feeder, corresponding to the maximum short-circuit power

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Short-circuits Standards Short-circuit currents inside a power station unit with on-load tap-changer For the calculation of the partial short-circuit current feeding into the short-circuit location F2, for example at the connection to the high-voltage side of the auxiliary transformer AT in the figure, it is sufficient to take: cU rG 1 1 " I KF2 = + 3 K GS Z G K TS Z TLV + 12 Z Qmin tr

With KTS given by the following expression K TS =

And KGS as it was given before K GS =

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cmax 1 − X Tp.u. sin ϕ rG c max

1 + X d" sin ϕ rG

Short-circuits Standards Short-circuit currents inside a power station unit without on-load tap-changer For a power station unit without on-load tap-changer of the unit transformer, the partial initial symmetrical short-circuit currents (the same figure) are defined by the same formulas as for the case with on-load tap-changer, with small modifications of the correcting factors KGS and KTS, which become KGS0 and KTS0 I

" KG

I "KT =

cU rG = 3K GS0 Z G

I

" KF2

cU rG = 3

cU rG 3 Z TLV + 12 Z Qmin tr

1 1 + K GS0 Z G K TS0 Z TLV + t1r2 Z Qmin

(

with K GSO =

c max 1 ⋅ 1 + p G 1 + X d" sin ϕ rG

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K TSO =

c max 1 ⋅ 1 + p G 1 − X T p.u. sin ϕ rG

)

Short-circuits Standards Short-circuit currents inside a power station unit without on-load tap-changer If the unit transformer has an on-load tap-changer on the high-voltage side, it is assumed that the operating voltage at the terminals of the generator is equal to UrG. If, even in this case, the voltage region of the generator UG = UrG(1±pG) is used permanently, take equations for the case without on-load tap changer than those for the case with on-load tap changer. The total short-circuit current in F1 or F2 is found by adding the partial short-circuit current, caused by the medium- and low-voltage auxiliary motors of the power station unit.

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Short-circuits Standards Short-circuit currents in meshed networks In meshed networks, such as those shown in figure at right, it is generally necessary to determine the short-circuit impedance Zk = Z(1) by network reduction (series connection, parallel connection, and delta-star transformation, for example) using the positive-sequence short-circuit impedances of electrical equipment. The impedances in systems connected through transformers to the system, in which the shortcircuit occurs, have to be transferred by the square of the rated transformation ratio. If there are several transformers with slightly differing rated transformation ratios (trT1 trT2... trTn), in between two systems, the arithmetic mean value can be used. Teaching materials distributed for free.

Q

G 3 F

M 3

M 3

M 3

Fig. 4. System diagram

Short-circuits Standards Short-circuit currents in meshed networks The initial symmetrical short-circuit current shall be calculated with the equivalent voltage source cU n 3 at the short-circuit location using equation

" IK =

cU n 3⋅

2 2 RK + XK

=

Q

G 3 F

cU n 3Z K M 3

M 3

M 3

Fig. 4. System diagram

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Short-circuits Standards Short-circuit currents in meshed networks For three-phase short-circuits fed from non-meshed networks as in figures 2 and 3, the contribution to the peak short-circuit current from each branch can be expressed by: " ip = χ 2 I K

The factor χ for the R/X ratio shall be obtained from Figure 5 or calculated by the following expression:

χ = 1,02 + 0,98e −3R/X

2,0 1,8 1,6 x 1,4 1,2 1,0

0

0,2 0,4 0,6 0,8 1,0 1,2 R /X

Fig. 5. Factor χ for series circuit as a function of ratio R/X

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Short-circuits Standards Short-circuit currents in meshed networks Equations of ip and χ of the previous slide presume that the short-circuit starts at zero voltage, and that ip is reached approximately after one half-cycle. For a synchronous generator use RGf.. The peak short-circuit current ip at a short-circuit location F, fed from sources which are not meshed with one another, in accordance with Figure 2, is the sum of the partial shortcircuit currents:

And from Fig. 2

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Short-circuits Standards Short-circuit currents in meshed networks When calculating the peak short-circuit current ip in meshed networks, the equation of ip shall be used with χ determined using one of the following methods a), b), or c). a) Uniform ratio R/X or X/R For this method the factor χ is determined from figure 5 taking the smallest ratio of R/X or the largest ratio of X/R of all branches of the network. It is only necessary to choose the branches which carry partial short-circuit currents at the nominal voltage corresponding to the short-circuit location and branches with transformers adjacent to the short-circuit location. Any branch may be a series combination of several impedances. In practise, considering branches, through which the flowing current is about 80% of the shortcircuit current is sufficient.

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Short-circuits Standards Short-circuit currents in meshed networks b) Ratio R/X or X/R at the short-circuit location For this method the factor χ is multiplied by a factor 1,15 to cover inaccuracies caused by using the ratio Rk / Xk from a network reduction with complex impedances.

As long as R/X remains smaller than 0,3 in all branches, it is not necessary to use the factor 1,15. It is not necessary for the product 1,15 . χ(b) to exceed 1,8 in low-voltage networks or to exceed 2,0 in medium- and high-voltage networks. The factor χ(b) is found from figure5 for the ratio Rk / Xk given by the short-circuit impedance Zk = Rk + jXk at the short-circuit location F, calculated for frequency f = 50 Hz

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Short-circuits Standards Short-circuit currents in meshed networks c) Equivalent frequency fc An equivalent impedance Zc of the system as seen from the short-circuit location is calculated assuming a frequency fc = 20 Hz The R/X or X/R ratio is then determined according to the following equation :

where

R Rc f c = ⋅ X Xc f

Zc = Rc + jXc is the equivalent impedance of the system as seen from the short-circuit location for the assumed frequency fc; Rc is the real part of Zc (Rc is generally not equal to the R at nominal frequency) Xc is the imaginary part of Zc (Xc is generally not equal to the X at nominal frequency).

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Short-circuits Standards Short-circuit currents in meshed networks c) Equivalent frequency fc The factor χ is found from figure 5 using the R/X or X/R ratio from equation (*), or with equation (**). Method c) is recommended in meshed networks (see IEC 60909-1). R Rc f c = ⋅ X Xc f

(*)

χ = 1,02 + 0,98e −3R/X

(**)

When using this method in meshed networks with transformers, generators and power station units, the impedance correction factors KT, KG and KS, respectively KSO, shall be introduced with the same values as for the 50 Hz or 60 Hz calculations.

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Short-circuits Standards Symmetrical short-circuit breaking current: Single-fed three- phase short-circuit For a near-to-generator short-circuit, in the case of a single fed short-circuit or from nonmeshed networks (fig. 2), the decay to the symmetrical short-circuit breaking current Ib (*) is taken into account by the factor m according to equations (**). " I b = µ I K" (*) - for tmin = 0,02 s µ = 0,84 + 0,26 e- 0 ,26 I KG / I rG The factor m depends on the minimum time delay tmin and the ratio I K" / I rG where IrG is the rated generator current.

"

- for tmin = 0,05 s

µ = 0,71 + 0,51 e- 0 ,30 I KG / I rG

- for tmin = 0,10 s

" /I - 0 ,32 I KG rG

µ = 0,62 + 0,72 e

"

- for tmin ≥ 0,25 s µ = 0,56 + 0,94 e- 0 ,38 I KG / I rG

The values of m in equation (**) apply if synchronous machines are excited by rotating exciters or by static converter exciters (provided, for static exciters, the minimum time delay tmin is less than 0,25 s and the maximum excitation voltage is less than 1,6 times rated load excitation-voltage). For all other cases take µ = 1 if the exact value is unknown.

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(**)

Short-circuits Standards Symmetrical short-circuit breaking current: Single-fed three- phase short-circuit If I K" / I rG is not greater than 2, apply µ = 1 for all values of the minimum time delay tmin. 1,0 The factor µ may also be obtained from figure 6. For other values of minimum time delay, linear interpolation between curves is acceptable.

µ

Minimum time delay tmin

0,9

0,02 s

0,8

0,05 s

0,7

0,1 s 0,25 s

0,6

Figure 6 can be used also for compound excited low-voltage generators with a minimum time delay tmin not greater than 0,1 s.

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0,5 0 1 2 3 4 5 6 7 8 Three-phase short circuit IkG /I rG or I kM /IrM

Fig. 6. Factor µ

9

Short-circuits Standards Symmetrical short-circuit breaking current: Single-fed three- phase short-circuit For three-phase short-circuits in non-meshed networks as in figure 2, the symmetrical breaking current at the short-circuit location can be calculated by the summation of the individual breaking current contributions: " " I b = I b PSU + I KT + I b M where I bM = qµI KM m is taken from equation (** - slide 24) or figure 6 for synchronous generators and asynchronous motors. The factor q for the calculation of the symmetrical short-circuit breaking current for asynchronous motors may be determined as a function of the minimum time delay tmin (fig. 7). Where m is the ratio between is the rated − for tmin = 0,02 s q = 1,03 + 0,12 ln m active power in MW and the number of − for tmin = 0,05 s q = 0,79 + 0,12 ln m (*) pairs of poles of the motor. − for tmin = 0,10 s q = 0,57 + 0,12 ln m m = PrM / p − for tmin ≥ 0,25 s q = 0,26 + 0,10 ln m Teaching materials distributed for free.

Short-circuits Standards Symmetrical short-circuit breaking current Three-phase short-circuit in meshed networks 1,0

The short-circuit breaking current Ib in meshed networks shall be calculated by:

0,9 0,8

Minimum time delay tmin 0,02 s

0,7

I b = I K"

0,6

0,05 s

0,5

At first the current at the short-circuit location is calculated for the time of breaking, and then the partial currents in the branches where the circuit breakers are located.

q

0,4

0,1 s

0,3 0,2 0,25 s 0,1 0 0,01 0,02 0,04 0,1 0,2 0,4 1 2 4 MW 10 Active power of the motor per pair of poles m

∆U Mj ∆U G" i Fig. 7. Factor q for the calculation of the " " ( ( Ib = I − ∑ 1 − µi )I KG − 1 − µ j q j )I KM ∑ i j i cU n j cU n symmetrical short-circuit breaking current 3 3 of asynchronous motors "

" K

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Short-circuits Standards Symmetrical short-circuit breaking current Three-phase short-circuit in meshed networks ∆U M" j ∆U G" i " " " ( ( Ib = IK − ∑ 1 − µi )I KGi − ∑ 1 − µ j q j )I KM j cU cU i

n

j

n

3 3 where µi, µj are the values given in equation (** - slide 24) for both synchronous (i) and asynchronous (j) machines; qj are the values given in equation (* - slide 26) for asynchronous (j) motors;

I b , I K" are respectively the initial symmetrical short-circuit current and the symmetrical shortcircuit breaking current with influence of all network feeders, synchronous machines and asynchronous motors; ∆U G" i , ∆U M" j are the initial voltage drops at the terminals of the synchronous machines (i) and the asynchronous motors (j); " " I KG i , I KMj are the contributions to the initial symmetrical short-circuit current from the synchronous machines (i) and the asynchronous motors (j) as measured at the terminals of the machines.

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Short-circuits Standards Symmetrical short-circuit breaking current The symmetrical short-circuit breaking current may be considered as : the difference between the symmetrical short-circuit breaking current with influence of all network feeders, synchronous machines and asynchronous motors and

IK2 IK3

IK,Ib

IK4

IK1 IK6 IK5

Fig. 8. Calculation of short-circuit the contributions to the initial symmetrical breaking current Ib in meshed networks short-circuit current from the synchronous machines (i) and the asynchronous motors (j) I b = I K" − ∑ I zGi − ∑ I zMj (*) as measured at the terminals of the i j machines. Both last current sum can be considered as decaying components and can be determined using the following equations, providing that the short-circuit occurs at the terminals of the ith generator (jth motor ) : " " I zGi = (1 − µ i )I KG I zMj = (1 − q j µ j )I KM i , j Teaching materials distributed for free.

(**)

Short-circuits Standards Symmetrical short-circuit breaking current The short-circuit is a far-from-generator short-circuit, hence the decaying process of the short-circuit current sinusoidal component is dynamic (Fig. 1). We can assume that the decaying factor is equal to a coefficient αi, which is a real number, the absolute value of which less than 1 (αj for motors), multiplied by the current defined for near-togenerator short-circuit. " I zGi = α i (1 − µ i )I KG i,

(

)

" I zMj = α j 1 − q j µ j I KM j

(*)

The coefficient αi defines the distance between the short-circuit location and the generator terminals (for near-to-generator short-circuits, its value is near 1, for very farfrom-generator short-circuits almost equal to 0). In case of non-meshed networks this coefficient can be determined in correlation with the impedance of the portion of line from the short-circuit location and the generator.

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Short-circuits Standards Symmetrical short-circuit breaking current In case of meshed networks it is difficult to define such a line, then the distance between the short-circuit location and the generator is assumed as equal to the ration of voltage drop in the corrected generator (motor) reactance and the network phase voltage, hence.

αi =

∆U G" i cU n 3

=

" " I KG i X dK cU n 3

(*)

αj =

" ∆U Mj cU n 3

=

I kMj X M cU n 3

(**)

Considering the equations (* - previous slide), (*) and (**) leads to a new form of the equation (* - slide nr 28). In case of greater number of voltage sources, it is much more convenient to use special computer methods for short-circuit calculations.

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Short-circuits Standards Maximum steady-state short-circuit current For near-to-generator three-phase short-circuits fed directly from one synchronous generator or one power station unit only, according to figure 11b or 11c, the steady-state short-circuit current Ik depends on the excitation system, the voltage regulator action, and saturation influences. Synchronous machines (generators, motors, or compensators) with terminal-fed static exciters do not contribute to Ik in the case of a short-circuit at the terminals of the machine, but they contribute to Ik if there is an impedance between the terminals and the short-circuit location. A contribution is also given if, in case of a power station unit, the short-circuit occurs on the highvoltage side of the unit transformer For the calculation of the maximum steady-state short-circuit current, the synchronous generator may be set at the maximum excitation. I K max = λmax I rG For static excitation systems fed from the generator terminals and a short-circuit at the terminals, the field voltage collapses as the terminal voltage collapses, therefore take λmax = λmin = 0 in this case. Teaching materials distributed for free.

Short-circuits Standards Maximum steady-state short-circuit current a) seria pierwsza

λmax may be obtained from figures 9 or 10 for cylindrical rotor generators or salient-pole generators. The saturated reactance xdsat is the reciprocal of the saturated no-load short-circuit ratio.

λmax curves of series 1 are based on the highest possible excitation voltage according to either 1,3 times the rated excitation at rated apparent power and power factor for cylindrical rotor generators (figure 9a) or 1,6 times the rated excitation voltage for salient-pole generators (figure 10a). Teaching materials distributed for free.

2,8 2,6 2,4 2,2 2,0 1,8 1,6 1,4 λ 1,2 1,0 0,8 0,6 0,4 0,2 0

λ max

X d sat 1,2 1,4 1,6 1,8 2,0 2,2

λ min

0 1 2 3 4 5 6 7 8 Zwarcie trójfazowe IKG /IrG

b) seria druga 2,8 X d sat 1,2 2,6 1,4 2,4 1,6 λ max 1,8 2,2 2,0 2,0 2,2 1,8 1,6 1,4 λ 1,2 1,0 0,8 λ min 0,6 0,4 0,2 0 0 1 2 3 4 5 6 7 8 Zwarcie trójfazowe I KG /IrG

Fig. 9. λmin and λmax factors of series 1 (left) λmin and λmax factors of series 2 (right)

Short-circuits Standards Maximum steady-state short-circuit current λmax -curves of series 2 are based on the highest possible excitation-voltage according to either 1,6 times the rated excitation at rated apparent power and power factor for cylindrical rotor generators (figure 9b), or 2,0 times the rated excitation voltage for salient-pole generators (figure 10b).

λmax -curves of series 1 or 2 may also be

a) seria pierwsza 5,5 5,0 4,5 4,0 3,5 3,0 λ 2,5 2,0 1,5 1,0 0,5 0

applied in the case of terminal-fed static exciters, if the short-circuit is at the highvoltage side of the unit transformer of a power station unit or in the system, and if the maximum excitation voltage is chosen with respect to the partial breakdown of the terminal voltage of the generator during the short-circuit. Teaching materials distributed for free.

λ max

b) seria druga X d sat 0,6 0,8 1,0 1,2 1,7 2,0

λ min

0 1 2 3 4 5 6 7 8 Zwarcie trójfazowe I KG /IrG

5,5 5,0 4,5 4,0 3,5 3,0 λ 2,5 2,0 1,5 1,0 0,5 0

λ max

X d sat

0,6 0,8 1,0 1,2 1,7 2,0

λ min

0 1 2 3 4 5 6 7 8 Zwarcie trójfazowe I KG /IrG

Fig. 10. λmin and λmax factors of series 1 (left) λmin and λmax factors of series 2 (right)

Short-circuits Standards Minimum steady-state short-circuit current For the minimum steady-state short-circuit current in the case of a single-fed shortcircuit from one generator or one power station unit, constant no-load excitation (voltage regulator not being effective) of the synchronous machine is assumed:

I K min = λmin I rG λmin may be obtained from figures 9 and 10. In the case of minimum steady-state short-circuit introduce c = cmin, according to table 1 (slide 5).

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Short-circuits Standards Minimum steady-state short-circuit current The calculation of the minimum steady-state short-circuit current in the case of a neartogenerator short-circuit, fed by one or several similar and parallel working generators with compound excitation, is made as follows:

For the effective reactance of the generators, introduce:

IkP is the steady-state short-circuit current of a generator at a three-phase terminal short-circuit. The value should be obtained from the manufacturer.

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Short-circuits Standards DC component of the short-circuit current The maximum d.c. component id.c. of the short-circuit current as shown in figures 1 and 2 may be calculated with sufficient accuracy by the following equation : " −2πft K R / X idc = 2 I K e

where I K" is the initial symmetrical short-circuit current;

f

is the nominal frequency;

t

is the time;

R/X s the ratio according to Fig. 5 or the ratios according to the methods a) and c) slides 19 - 22.

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Short-circuits Standards Joule integral and thermal equivalent short-circuit current The joule integral is a measure of the energy generated in the resistive element of the system by the short-circuit current. In this standard it is calculated using a factor m for the time-dependent heat effect of the d.c. component of the short-circuit current and a factor n for the time-dependent heat effect of the a.c. component of the short-circuit current (see figures 11 and 12) The thermal equivalent short-circuit current is: " I th = I K m+n

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Short-circuits Standards Joule integral and thermal equivalent short-circuit current For a series of i ( i = 1, 2,....,r) three-phase successive individual short-circuit currents, the following equation shall be used for the calculation of the Joule integral or the thermal equivalent short-circuit current. where I " Ki (*)

is the initial symmetrical three-phase short-circuit current for each short-circuit

I th

is the thermal equivalent short-circuit current

mi

is the factor for the heat effect of the d.c. component for each short-circuit current

ni

is the factor for the heat effect of the a.c. component for each short-circuit current

(**) (***)

The Joule integral and the thermal equivalent shortcircuit current should always be given with the short-circuit duration with which they are associated.

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TKi

is the duration of the short-circuit current for each short-circuit

TK is the sum of the durations for each short-circuit current

Short-circuits Standards Joule integral and thermal equivalent short-circuit current The factors mi are obtained from Figure 11 using f · Tki and the factor k derived in [2]. The " factors ni are obtained from Figure 12 using Tki and the quotient I K I K , where Iki is the steadystate short-circuit current for each short-circuit.

a)

b)

2,0

1,6

1,6

9 1, 8 χ= , 1 1, 7

2,0 1,2

1, 1,

1,

6

5

m 0,8

1,

0,4 1,3 4 2 1 ,1 0 10 -2 2 4 6 10-12 4 6 s1 tK

Fig. 11. Factor m for the heat effect of the d.c. component of the short-circuit current

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IK/IK =10

1,2 3

1,25 1 ,5 2 ,0

,0 n 0,8 5 , 0 4, 0 0,4 6 ,0 0 10 -2 2 4 6 10-12 4 6 1 2 4 6 s 10 tK Fig. 12. Factor n for the heat effect of the a.c. component of the short-circuit current

Short-circuits Standards Joule integral and thermal equivalent short-circuit current When a number of short-circuits occur with a short time interval in between them, the resulting Joule integral is the sum of the Joule integrals of the individual short-circuit currents, as given in equation (* slide 38) For distribution networks (far-from-generator short-circuits) usually n=1 can be used. For far-from-generator short-circuit with the rated short-circuit duration of 0,5 s or more, it is permissible to take m + n = 1. If the Joule integral or the thermal equivalent short-circuit current shall be calculated for unbalanced short-circuits, replace I Ki" with the appropriate unbalanced shortcircuit currents.

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Bibliography 1. Machowski J, Kacejko P. Zwarcia w systemach elektroenergetycznych (Power System Short-Circuits – in Polish), Wydawnictwo Naukowo-Techniczne, Warszawa 2002 2. INTERNATIONAL STANDARD IEC 60909-0 „Short-circuit currents in three-phase a.c. systems” Calculation of currents

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