Static FC/TCR compensator

Guide for electrical design engineers

Power Quality Grzegorz TKACZEWSKI & Artur KOS AGH-University of Science & Technology

STATIC FC/TCR COMPENSATOR FOR ARC FURNACE COMPENSATION

Power Quality

Power Quality www.leonardo-energy.org

1. INTRODUCTION Industrial facilities are source of major disturbances to power system due to more and more large power loads being installed that, apart of their good functional properties, are characterized by negative impact on the quality of power. Such loads are the cause of the supply voltage distortion, unbalance and fluctuations. Depending on the load type, such disturbances may occur separately or concurrently. These disturbances are propagated through distribution systems to other users' networks, impair operating conditions of equipment and, in extreme cases, prevent operation of electrical equipment sensitive to such disturbances. Among industrial loads, having the most adverse impact on a power system due to the emitted disturbances, are steelworks electric arc furnaces. They cause mainly: • • •

unbalance of currents and voltages, current and voltage distortion, fluctuations of active and reactive power, supply voltage fluctuations and flicker [6].

The adverse impact of non-linear loads on a power system can be mitigated by means of compensation equipment like fixed capacitor (FC), thyristor controlled reactor (TCR) - FC/TCR. The purpose of FC/TCR compensator is compensation of the fundamental component reactive power and filtering selected current harmonics. Such compensator is an example of the indirect compensation method in which, depending on the needs of the voltage restorer or the reactive power compensator function, the value of the sum of two current components is controlled: • fundamental harmonic of the capacitor current iFC, operated mostly as high harmonics filter(s) (the FC section) • fundamental harmonic of the reactor current iTCR controlled by means of a phase-controlled thyristor AC switch (the TCR section) [8].

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Static FC/TCR Compensator for Arc Furnace Compensation www.leonardo-energy.org

2. FC/TCR COMPENSATOR

Fig. 1. Diagram of the static VAR compensator comprising a fixed capacitor bank and reactors whose reactive currents are controlled by means of thyristor AC switches

Fig. 2. Diagram of the three-phase static compensator with a phase-controlled reactor section

The FC/TCR compensator consists of a fixed capacitor bank divided into several three-phase sections incorporating reactors, utilized also as high-order current harmonics filters, and parallel reactors whose fundamental current harmonic is controlled means of thyristor AC switches. The reactors' current can be controlled in a continuous manner from zero, if the switch is turned off, to its maximum value, when the reactor is directly connected to the source. The compensator schematic diagram is shown in figure 1. The capacitor banks generate capacitive reactive current of non-controlled value, whereas the reactor section (TCR) current is controlled within the range from zero to the current of a reactor being connected directly to the source. The reactive current fundamental harmonic of such compensator is: (1)

ik = -iC + iL where: iC – the fundamental harmonic of the capacitor bank current (noncontrolled value) iL – the fundamental harmonic of the inductive unit current (controlled value) ik – the fundamental harmonic of the compensator current.

Under symmetrical control conditions and balanced circuit parameters (the same phase reactances and equal thyristors' control angles) the 3-rd harmonic and its multiples do not occur. 3


The compensator enables continuous control of reactive current over the range from -Ic to -Ic + IL. Maximum voltage across the AC switches' thyristors does not exceed the amplitude of the phase-to-phase voltage of a power system, i.e. it is more than two times smaller than the voltage across the thyristors switching the

3

capacitor bank sections. The phase current of FC-TCR reactor section is times larger than the thyristor switches (reactors') currents [2]. The three-phase compensator circuit is shown in figure 2. The compensator comprises delta connected, fixed inductance reactors (L/2) whose current fundamental harmonic is controlled by the phase control of thyristor AC switches (ST) in each delta branch. AC switches are controlled by the control system in order to control either the supply voltage (voltage restores) or to compensate the load reactive current, depending on the compensation system purpose. Optimum utilization of the applied thyristors is often provided by a step-down transformer (Tr) with the leakage reactance higher than in typical applications, hence the reactors' reactance can be reduced (in the extreme case to zero). The delta connection of reactors is justified in both technical and economic terms. The presented configuration allows reducing the thyristors' current ratings and considerably reduces the supply current harmonic content as compared to a star -connected circuit of the same power. Thus, the compensator voltage-current characteristic encompasses the area of inductive and capacitive loadings within boundaries determined by the capacitor bank and reactors' powers. It should be emphasized that the compensator is a source of odd harmonics and, if the control angles of antiparallel connected thyristors are unequal, even harmonics also occur. Triplen harmonics in the compensator current are cancelled by delta connection of reactor branches. Odd harmonics can be mitigated by means of two 6-pulse circuits supplied from a three-winding Yyd transformer with 30° phase shift between the secondary side voltages, or by the use of suitable filters [10].

3. HIGH-ORDER HARMONICS FILTERS Passive filtering of high harmonic consists in connecting in parallel with the load generating a given harmonic, a series LC circuit whose series resonant frequency is tuned to the filtered harmonic frequency (Fig. 3).

4


Fig. 3. Schematic diagram of the supply network, a non-linear load represented by a current source, and the high-harmonic filter; CF, LF, RF – the filter capacitance, inductance and resistance, respectively The inductive and capacitive reactances of LC series filter are subtracting one from the other. For the series resonance frequency their absolute values are equal and their difference is zero. Thus, at this specific frequency, the filter is practically a short circuit. The remaining equivalent resistance, mainly that of the reactor winding, is very small. The reactance of the filter LC components connected in series is: ⎛ X ⎛ h 2 − ν F2 ⎞ X 1⎞ ⎟⎟ X F (h ) = X L (h ) − X C (h ) = hX L − C = X C ⎜⎜ h L − ⎟⎟ = X C ⎜⎜ 2 h ⎝ ν Fh ⎠ ⎝ XC h ⎠ (2) where: h harmonic order,

νF

-

the filter natural relative frequency ,

Xc XL

-

reactance of the filter capacitor for the fundamental harmonic, reactance of the filter reactor for the fundamental harmonic.

As follows from the relation (2), the filter reactance is near to zero for the harmonic h, whose frequency is close to the filter natural relative frequency nF. In consequence of connecting the filter between the source phases the current with the frequency close to the filter natural frequency, generated by a non-linear load, flows through the filter thereby reducing the harmonic current flow through the source. The filter reactance for h>nF is inductive, whereas for h
3.1. Principles of passive filters design The basic data necessary for the filter design are: • data concerning the source of high harmonics, i.e. amplitude-frequency spectrum of the non-linear load, the fundamental harmonic reactive power required for the compensation purposes, etc. • data concerning the power supply network, i.e. frequency characteristic of the power system impedance at the point of common connection (PCC) or, in the absence of such characteristic, the short-circuit capacity together with the schematic diagram and technical specification of neighbourhood of the considered point of the filter connection, the spectrum of initial voltage distortion at the considered point, permissible voltage harmonic distortion factor THD as per conditions of supply, and harmonic content (p.u.) etc.

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data concerning the filter, i.e. the location of its installation, the selected structure, technical specifications of passive elements to be used, etc. [7].

•

All further considerations are carried out under the following simplifying assumptions: • the source of high harmonics is an ideal current source, • the filter resistance RF, inductance LF and capacitance CF, are lumped elements and their values are constant over the considered frequency interval, • the filter is exclusively loaded with the fundamental harmonic and the harmonic to which it is tuned. The filter-network-load equivalent circuit diagram is shown in figure 3. The power supply network is represented by the ideal AC voltage source and the equivalent resistance and inductance: X

S

= 1 .1 ×

U 2 S SC

⇒

Ls

= 1 .1 ×

U 2 1 × S SC 100 π

R

S

= 0 .1 X

S

(3)

where: SSC is the supply network short-circuit capacity, U is phase-to-phase voltage at the point of connection. The load is represented by the source high-harmonic currents and the load impedance Zo. The purpose of the filter is: • compensation of the load fundamental harmonic reactive power, • mitigation of high-harmonics emitted by the load to the power system [4].

3.2. Single branch filters The single branch filter is a simple structure which can easily be analyzed. In this section a matrix method for designing a group of single-frequency filters is used. The diagram of the single-frequency filter and its frequency-impedance characteristic are shown in figure 4 [4]. Assuming RF = 0, the impedance at the filtered frequency is 0 (this assumption will hold true in further considerations):

Z F (ω r ) = jω r LF − j

1 = 0 ω r = nr * ω1 = ωr CF

6

1 LF C F


LF =

1 1 = 2 2 ω C F n r ω1 C F 2 r

where: Nr ω r = nr * ω1

ω1

(4)

- the resonant frequency order - angular frequency of the filter series branch - the fundamental harmonic angular frequency ().

For the harmonic with angular frequency ω:

ω L F C F −1 1 = j = j Z F (ω r ) = jωLF − j ωC F ωC F 2

ω2

1 n r2ω12 − ω 2 Z F (ω r ) = − j C F n r2ω r2ω

1 n ω12 C F ωC F 2 r

C F −1 = j

ω 2 − n r2ω r2 n r2ω r2 C F ω

(5)

(6)

Fig. 4. The single-frequency filter diagram and its frequency-impedance characteristics for RF ≠ 0, Xc – the filter capacitive reactance, XL- the filter inductive reactance, ZF – the filter impedance, IF – the filter current, U – the filter operating voltage

7


ω = k * ω1

, k=1 – filter tuned to the fundamental harmonic.

Z F (ω r ) = j

ω12 − nr2ω12 1 − nr2 = j nr2ω12 C F ω1 nr2 C F ω1

,

(7)

At the fundamental angular frequency the imaginary part of the filter impedance is directly proportional to the filter operating voltage and inversely proportional to reactive power to be compensated:

U N2 Im(Z F ( n ) ) = − j Q −j −j

U 1− n 1 = j 2 * Q C F ω1 nr 2 N

U n −1 1 =−j 2 * Q C F ω1 nr 2 N

,

(8)

2 r

2 r

,

(9) CF =

,

n r2 − 1 Q * 2 nr ω 1U

2 N

(10)

Q- the load's reactive power to be compensated. The capacitance and inductance values in given branches are computed from relations:

⎧ n r2 − 1 1 Q F ⋅ ⋅ ⎪C i = ω1 U N 2 n r2 ⎪ ⎨ 1 ⎪L = 1 = i 2 2 ⎪⎩ ω r C i n r ω12 C i

(11)

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4. AN EXAMPLE OF FC/TCR COMPENSATOR DESIGN

Fig. 5. Diagram of the network supplying a non-linear load – an arc furnace, to be compensated using the FC/TCR compensator

4.1. Determining the compensation power value In order to determine the compensator power shall be known the economically justified (specified in a contract with utility company) power factor at the given point of a power system as specified by the utility company. In the general case the capacitor bank power is determined from the formula:

Qk = P tgϕ1 − tgϕ2 ,

(12)

where: Qk –reactive power to be installed, P – the load(s) active power,

tgϕ1

– power factor prior to compensation,

tgϕ2

– power factor after compensation.

The capacitor bank design and sizing should take into account problems that may occur during the system operation due to the presence of high order harmonics and resonance effects. Detuning from resonance is achieved using an antiresonance reactor connected in series with capacitor bank. The antiresonance reactor is used when the voltage distortion level is within 9


acceptable limits, the capacitor bank shall also be protected against overloading with high harmonic currents. An inadequate selection of the capacitor bank parameters, as well as the cooperating reactors, may lead to capacitor damages due to overvoltages and overcurrents and, consequently, preclude operation of the compensation system [1,9]. Short-circuit capacity of the 110kV network is SSC110kV = 500 [MVA] Short-circuit capacity of the 6kV network is SSC6kV = 16.9 [MVA] Transformer rated power STr = 20 [MVA] Transformer short-circuit voltage uz% = 13.5 Maximum averaged consumed active power: 10.13 [MW] Maximum averaged consumed reactive power: 13.54 [MVAr] Apparent power: SP =

PP2 + Q P2 = 16 . 91[ MVA ]

(13)

The arc furnace load current:

I obc =

SP 16,91 * 106 = = 1626.9[ A] 3 *U N 3 * 6000

(14)

Hence the power factor:

tg ϕ

=

Q P

P

=

1.34

(15)

P

Required power factor:

tgϕ dyr =

0.3

(16)

Required compensation power:

Qkomp = P (tgϕ − tgϕ dyr ) = 10.5 [MVAr]

(17)

It is assumed that reactive power that should be compensated will increase up to 11 [MVAr]. The transformer reactance:

10


X Tr

u z % * U N2 = 0.24[Ω] = 100 * S Trt

(18)

The network short-circuit reactance is: 0.35 [Ω]

4.2. Determining the voltage distortion factor prior to compensation a) At the 6kV side Harmonic currents In measured at the 6 kV side during the arc furnace operation as the only load in the power network, are listed in table 1. The voltage distortion factor THD at the 6kV busbars without the compensation circuit is calculated using the formula:

THDU =

∑ (U ) 13 2

2

N%

(19)

where: UN% - the percentage voltage harmonic content of the given harmonic.

U n % = 3 * I n * n * X zw( 6 kV ) *

100 UN

(20)

where: In- In- harmonic current, n- harmonic order, XZW(6kV)- the network reactance at the 6kV side, UN- voltage at busbars (6kV)

X zw ( 6 kV ) = X S + X Tr

=0.59 [Ω]

(21)

Table 1. Harmonic current values and corresponding harmonic voltage content

Harmonic order

Harmonic current In [A]

Voltage harmonic content [%]

1 2 3 4 5 6 7 8 9 10 11 12 13

1626.9 21.93 32.55 9.30 38.73 8.83 7.85 5.95 4.60 2.88 4.03 1.15 2.30

1.49 3.32 1.27 6.59 1.80 1.87 1.62 1.41 0.98 1.51 0.47 1.02

THDU6kV= 8,61%, THDdop6kV = 5% [11] 11


(22)

THDU6kV > THDdop6kV The above inequality shows the necessity of high harmonics compensation.

b) At the 110kV side The voltage distortion factor is determined from relation (19), but the percentage voltage harmonic content for the given harmonic UN% is calculated form another formula:

U N % = 1.1 * bN % * n *

S10 max S SC

(23)

where: bn% - percentage content of the harmonic of order n in a nonlinear load current; it depends on the load type, the factor values are listed in table 2., n- harmonic order, SSC - short-circuit capacity (500 MVA), S10max - maximum 10-minute apparent power of a nonlinear load, equal to the arc furnace maximum power calculated from the formula (13). Table 2. The percentage content b% of the nth harmonic in the arc furnace current versus the transformer nominal power

Furnace transformer rated power i

MVA 2.5 5 10 16 50

Harmonic order 2 % 36 26 26 16 7

3 % 25 20 13 18 10

4 % 8 5 4 6 4

5 % 10 7 5 8 5

6 % 4 2 1 3 1

7 % 3

3 2 3 2

9 % 2 2 1 2 2

11 % 1 1 1 1 1

13-25 % 0 0 0 0 0

Table 3. Harmonic current values and corresponding harmonic voltage content Harmonic order

bn [%]

Voltage harmonic content [%]

2 3 4 5 6 7 9 10 13

16 18 6 8 3 3 2 1 0

1.19 2.01 0.89 1.49 0.67 0.78 0.67 0.41 0.00

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THDU110kV= 3.18%, THDdop110kV = 1.5% [11] THDU110kV > THDdop110kV

(24)

The above inequality also shows the necessity for high harmonics compensation.

4.3. Guidelines for sizing high harmonics filters In the analysed example the static compensator and passive filters section shall operate under the following conditions: • the arc furnace is supplied from a 110/6 kV transformer • voltage distortion factor THDU at the point of common connection at 110kV

busbars should not exceed 1.5%

• reactive power to be compensated is 11 MVAr • dominant harmonic currents at 6kV busbars are the 3-rd and 5-th harmonic

currents (32.55A for the 3-rd harmonic and 38.73A for the 5-th harmonic, respectively) • the FC/TCR compensator will be connected at 6kV.

4.4 Selecting high harmonics filters parameters Initial design data are: • the power network nominal voltage Us • maximum reactive power the filter can inject into the power system.

The capacitor bank rated power depends on: • • •

maximum magnitude of the filtered current harmonic capacitor overload current (as specified by the manufacturer) capacitor voltage utilization factor, expressed by the formula (25) [5]:

ku =

US 3U Nbat

(25)

where: US is the supply network voltage, UNbat is the capacitor bank rated voltage. In order to reduce THD factor to the required level, the harmonic of the largest magnitude shall be filtered out while reactive power consumed by the arc furnace shall be compensated. 4.4.1. Determining the required power of the 5-th harmonic filter a) The required compensation power has been allocated between the fitter braches proportionally to the values of eliminated harmonic currents: 13


Qkomp = 11 [Mvar] the 5-th harmonic current is: I5 = 38.8 [A] b) Due to series connection of the resonance reactor and capacitor bank the voltage across capacitors will be knf times higher than the busbars voltage:

k nf

n F2 = 2 nF − 1

(26)

where: nF- harmonic order. The voltage rise is (table 4): For the 5-th harmonic

•

knf =

52 = 1.042 52 − 1

(27)

Table 4. The knf factor value of the nth harmonic order. nF

2

3

4

5

7

11

knf

1.333

1.125

1.067

1.042

1.021

1.008

c) The nominal voltage of the filter capacitor bank shall satisfy the relation: UNbat.F5 ≥ 1.042* 6 *1.1= 6.88 kV

(28)

The value1.1 results from power network voltage variations +/-10% UN (11) d) Considering the above requirements the capacitor bank of "Y" company make, with the following parameters has been selected: Rated reactive power…………………………………………... Rated current……………………………………………………. Rated voltage……………………………………………………. Capacitance……………………………………………………... Capacitance tolerance………………………………………….. Current overload…………………………………………………. Voltage overload (Voltage overload factor)…………………….

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19 MVAr 1000 A 7600V 21.3 uF -5/+10 % [12] 1.5 In [12] 1.1 Un [12]


e) Since the reactive power delivered by the 5-th harmonic filter is expressed by formula (29) [2]:

QUz 5

⎛U = Q NbatF 5 ⋅ ⎜⎜ S ⎝ U CN

2

⎞ ⎟⎟ = Q NbatF 5 ⋅ (k u )2 ⎠

(29)

where: QUz5- the reactive power injected by capacitor to power system, QNbatF5the capacitor rated power, US- power network voltage, UCN- capacitor bank voltage, kU- capacitor voltage utilization factor. Thus, the rated power of the 5-th harmonic filter capacitor bank is:

QNbatF 5

⎛U = QUz 5 ⋅ ⎜⎜ CN ⎝ US

2

⎞ 7600 ⎞ ⎟⎟ = 11 * 106 ⋅ ⎛⎜ ⎟ = 17.65[ M var] ⎝ 6000 ⎠ ⎠ 2

(30)

4.4.2. The reactor sizing [5] From the condition for series resonance:

X DF 3 =

X bat n 2 SR

(31)

where: nSR– the series resonance frequency. a) The required reactance of reactor is determined:

X NbatF 5 = X DF 3 = LDF 3 =

2 U NbatF 7600 5 = = 3.27[Ω] QNbatF 5 17.65 * 106

X bat 3.04 = = 0.13[Ω] n 2 SR 25 X DF 3

ϖ1

=

(32)

(33)

0.12 = 0.41[ mH ] 314

(34)

b) For the filter detuned from resonance below nrsz the inductance values will be 15


larger. For the series resonant frequency order equal 4.5 the inductance will be:

LDF 4,5 = LDF 5

52 = 0.52[mH ] 4.52

(35)

Whereas for the series resonant frequency order equal 4.7 the inductance will be:

LDF 4, 7

52 = LDF 5 = 0.47[mH ] 4.7 2

(36)

For both filters have been selected special design reactors of "Z" company make, with inductance values calculated as above and 2% inductance toleration. The reactors are provided with taps that allow matching their inductance to 4.5 or 4.7 detuning. 4.4.3. Filter configuration for single current harmonic Capacitor banks are connected in a double star configuration with star points connected. In such configuration a malfunction (short circuit) of a single does not cause significant increase in phase currents. The designed topology allows for a simple and cheap protection against the battery internal failures, or capacity changes due to internal short circuits, by measuring the current in the conductor connecting star points using a current transformer.

Fig. 6. Diagram of the single current harmonic filter comprising reactors, capacitor bank divided into two sections and symmetry control by means of measuring the equalizing current I between neutral points of both sections

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4.4.4. Simulation tests of the designed filter The frequency-impedance characteristics for the designed filter were determined using the Matlab software package. a)

For series resonant frequency order 4.7 ‑ charts 1, 2 and 3

10 9 8

Z(w) [Ohm]

7 6 5 4 3 2 1 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 1. Frequency-impedance characteristics of the supply network (blue) and the 5-th harmonic filtering branch (red); each characteristic determined individually 5 4.5 4

Z(w) [Ohm]

3.5 3 2.5 2 1.5 1 0.5 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 2. Frequency-impedance characteristics of the supply network (blue) and the 5-th harmonic filtering branch (green) and the equivalent impedance (red) 17


10 9 8

Z(n)/Z(wo)

7 6 5 4 3 2 1 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 3. Frequency characteristic of the supply network and the filter equivalent impedance related to the equivalent impedance at f=50Hz b) For series resonant frequency order 4.5 – charts 4, 5 and 6 10 9 8

Z(w) [Ohm]

7 6 5 4 3 2 1 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 4. Frequency-impedance characteristics of the supply network (blue) and the 5-th harmonic filtering branch (red); each characteristic determined individually

18


5 4.5 4

Z(w) [Ohm]

3.5 3 2.5 2 1.5 1 0.5 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 5. Frequency-impedance characteristics of the supply network (blue) and the 5-th harmonic filtering branch (green) and the equivalent impedance (red)

10 9 8

Z(n)/Z(wo)

7 6 5 4 3 2 1 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 6. Frequency characteristic of the supply network and the filter equivalent impedance related to the equivalent impedance at f=50Hz 19


4.4.5. Parallel resonance calculations The equivalent impedance of the designed n-th harmonic filter with equivalent circuit as in figure 7:

Fig. 7. The equivalent circuit of the source and the parallel filter for a single harmonic The equivalent impedance of the above circuit is expressed by the formula:

Z z ( n) =

(

)

nX tr n 2 X d − X c n 2 ( X tr + X d ) − X c

Parallel resonance occurs when Z

∞; it is expressed by the relation:

n 2 ( X Tr − X d ) − X c = 0

nr =

(37)

(38)

Xc X tr + X d

(39)

The 5-th harmonic filter parallel resonance relative frequency is: • •

for detuning degree equal 4.5: for detuning degree equal 4.7:

2.82 2.11

4.4.6. Verification of the 5-th harmonic capacitor bank for overload current The filter power (the capacitor bank and reactor connected in series) is:

' NF 5

Q

2 2 ( ( UN ) 6000) = =

X Z (1)

2.92

= 11.46MVar (40) 20


and:

X Z (1) = X DF 5(1) + X NbatF 5(1) = 3.14[Ω]

(41)

where: XdF5(1)- the reactor reactance at the frequency of 50 Hz, XbatF5(1)- the capacitor reactance at the frequency of 50 Hz The rated current of the 5-th harmonic capacitor battery [5] is:

I Nbat , F 5 =

' QNF 12,34 * 106 5 = = 870.46[ A] 3 * U Nbat 3 * 7600

(42)

In order to verify the 5-th harmonic capacitor battery for the overload current condition, it has been assumed that both the fundamental harmonic and the filtered harmonic currents flow in the filter. The current in the 5-th harmonic filtering branch is calculated from the formula:

I batF = I 12 + I n2

(43)

The following condition shall be verified to prevent the capacitors current overload:

I n2 = ki I n p 2 − b 2 k u2

I Nbat . F ≥

(44)

where: b- capacitors' maximum voltage overload factor, p- capacitors' maximum current overload factor (≤ 1.1) [12] ku – capacitors' voltage utilization factor, formula (25) In - the filtered n-th harmonic current. The capacitors' current utilization factor ki is:

ki =

1

p − ku2 2

(45)

The overload factor p determines maximum current overload of a capacitor (for capacitors used in filters its value is 1.5 [12]). It is assumed that filter branch for the given harmonic is loaded with this harmonic current and, additionally, with small currents of other non-filtered harmonics. Their influence is taken into account by reducing the p2 value in a manner indicated in table 5:

21


Table 5. Overload factor High-order harmonics taken into account

p=1.3 1.6

(1.5) 2.16

High-order harmonics not taken into account

1.69

2.25

In the case of considered 5-th harmonic filter the current will be:

I1− 5 = ku I Nbat .F 5 = 0.789 * 870.46 = 687.20[ A]

ku = where:

6000 US = = 0.789 3U Nbat 7600

I batF 5 = I12− 5 + I n2 = 687.202 + 38.82 = 688.29[A] ki =

1

p 2 − ku2

(46)

(47) (48)

= 0.784 (49)

ki I batF 3 = 0.784 * 688.29 = 539.65 [A]

(45)

Thus, the relation:

870.46 [ A] ≥

I Nbat . F ≥

I n2 = 539.65[A] p 2 − b 2 k u2

I n2 = k i I batF 5 p 2 − b 2 k u2

(51)

(52)

is satisfied.

4.4.7. Verification of THDU value Total harmonic voltage distortion factor THDU is expressed by the formula:

THDU =

∑U

2 (n)

Uf

(53)

22


where:

Un Uf = 3

(54)

U (n) = I (n) Z (n)

(55)

UN= 6kV, Z(n)- from the formula (37). Table 6. Harmonic content and total harmonic voltage distortion factor THDU

Current harmonic order 2 3 4 5 6 7 8 9 10 11 12 13 THDU

Un for detuning 4.7 209.35 36.92 2.56 3.58 3.04 4.33 4.36 4.15 3.05 4.89 1.57 3.48 6.15

Un for detuning 4.5 241.31 33.38 1.90 6.23 3.68 4.94 4.87 4.58 3.34 5.33 1.71 3.77 7.04

As seen from table 6, the voltage distortion factor was reduced but its value still exceeds the required level 6.15%. THDUF5= 6.15%, THDdop6kV = 5% [11], THDU6kV > THDdop6kV. It is thus necessary to design from the beginning a compensator that eliminates more than one harmonic, comprising a larger number of branches.

4.5. Selection of components for the 3-rd and 5-th harmonic filters

4.5.1 Determining the required filter power a) The required compensation power has been allocated between the fitter braches proportionally to the values of eliminated harmonic currents:

23


Qkomp = QuzF3 + QuzF5 = 11 Mvar I3 = 32.5 A I5 = 38.8 A I3 + I5 =131.4 A

Quz 3 =

(56)

I3 ⋅ Qkomp = 4.8[ MVar ] I5 + I3

QuzF 3 =

(57)

I5 ⋅ Qkomp = 5.71[ MVar ] I5 + I3

(58)

b) The voltage rise is: •

•

for the 3-rd harmonic

ku =

32 = 1.125 32 − 1

(59)

for the 5-th harmonic

ku =

52 = 1.042 52 − 1

(60)

c) The nominal voltages of filters capacitor banks will satisfy the relation: •

for the 3-rd harmonic UNbat.F3 ≥ 1.125*6*1.1 = 7.43 [kV]

(61)

•

for the 5-th harmonic UNbat.F3 ≥ 1.042* 6 *1.1= 6.88 [kV]

(62)

The value1.1 results from the assumed power network voltage possible rise by 10% [11]. d) Considering the above requirements the capacitor bank of "Y" company make with following parameters has been selected: Rated reactive power ......................................................................... 9 MVar Rated current ...................................................................................... 1000 A Rated voltage….….……………………………………………………….7800 V Capacitance ...................................................................................... 21.3 uF Capacitance tolerance .............................................................. -5/+10 % [12] Current overload ............................................................................. 1.5 In [12] Voltage overload (Voltage overload factor) ....................................1.1 Un [12]

e) The required powers of the 3-rd and 5-th harmonic filters are expressed by formulas [2]:

QUz 3

⎛U = Q NbatF 3 ⋅ ⎜⎜ S ⎝ U CN

2

⎞ ⎟⎟ = Q NbatF 3 ⋅ (k u )2 ⎠ 24

(63)


QUz 5

⎛U = Q NbatF 5 ⋅ ⎜⎜ S ⎝ U CN

2

⎞ ⎟⎟ = Q NbatF 5 ⋅ (k u )2 ⎠

(64)

Hence rated powers of capacitor banks are:

QNbatF 3

QNbatF 5

2

⎛U = QUz 3 ⋅ ⎜⎜ CN ⎝ US

⎞ 7600 ⎞ ⎟⎟ = 4.80 * 106 ⋅ ⎛⎜ ⎟ = 7.70[ M var] ⎝ 6000 ⎠ ⎠

⎛U = QUz 5 ⋅ ⎜⎜ CN ⎝ US

⎞ 7600 ⎞ ⎟⎟ = 5.71 * 106 ⋅ ⎛⎜ ⎟ = 9.16[ M var] ⎝ 6000 ⎠ ⎠

2

2

(65)

2

(66)

4.5.2 Sizing the reactors [5] From the condition for series resonance:

X DF 3 =

X bat n 2 SR

(67)

where: nSR – the series resonance frequency, a) The required reactance of the reactor has been determined: •

for the 3-rd harmonic filter

X NbatF 3 X DF 3 = LDF 3 =

2 U NbatF 3 = = 7.51[Ω] QNbatF 3

X bat = 0.83[Ω] n 2 SR X DF 3

ϖ1

(55) (56)

= 2.66[mH ] (57)

for the 5-th harmonic filter

X NbatF 4 = X DF 5 =

2 U NbatF 5 = 6.31[Ω] QNbatF 5

(58)

X bat = 0.25[Ω] n 2 SR

(59) 25


LDF 5 =

X DF 5

ϖ1

= 0.80[mH ] (60)

b) For the filter detuned from resonance below nrsz the inductance values will be larger. For the degree of detuning from resonance equal 2.5 and 4.5 for the 3-rd and 5th harmonic, respectively, the inductances will be:

LDF 3− 4,5 = LDF 3 LDF 5 − 4, 7 = LDF 5

32 = 3.82[mH ] 2.52

(61)

2

5 = 0.99[mH ] 4.52

(61)

Whereas for degree of detuning from resonance equal 2.7 and 4.7 for the 3-rd and 5-th harmonic, respectively, the inductances will be:

LDF 3− 4, 7 = LDF 3 LDF 5 − 4, 7 = LDF 5

32 = 3.28[mH ] 4.7 2

(61)

2

5 = 0.91[mH ] 4.7 2

(61)

For both filters have been selected special design reactors of "Z" company make, with inductance values calculated as above, and 2% inductance toleration. The reactors are provided with taps that allow matching their inductance to the degree of detuning: 2.5 and 4.5, 2.7 and 4.7, 2.8 and 4.8. 4.5.3. Simulation tests of the designed filters The designed filter frequency-impedance characteristics were determined for selected parameters using the Matlab software package: a) For series resonant frequency order equal 2.7 and 4.7 ‑ charts 7,8 and 9

26


10 9 8 7 Z(w) [Ohm]

6 5 4 3 2 1 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 7. Frequency-impedance characteristics for the degree of detuning 2.7 and 4.7 of the supply network (blue), the 3-rd harmonic filtering branch (green) and the 5-th harmonic filtering branch (red); each characteristic determined individually

5 4.5 4

Z(w) [Ohm]

3.5 3 2.5 2 1.5 1 0.5 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 8. Frequency-impedance characteristics of the supply network (blue), the 3-rd harmonic filtering branch (blue), the 5-th harmonic filtering branch (green) and the equivalent impedance (red) 27


10 9 8 7

Z(n)/Z(wo)

6 5 4 3 2 1 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 9. Frequency characteristic of the supply network and the filter equivalent impedance related to the equivalent impedance at f=50Hz For series resonant frequency order equal 2.5 and 4.5 ‑ charts 10,11 and 12

10 9 8

Z(w) [Ohm]

7 6 5 4 3 2 1 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 10. Frequency-impedance characteristics of the supply network (blue), the 3-rd harmonic filtering branch (green) and the 5-th harmonic filtering branch (red); each characteristic determined individually (red)

28


5 4.5 4

Z(w) [Ohm]

3.5 3 2.5 2 1.5 1 0.5 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 11. Frequency-impedance characteristics of the supply network (blue), the 3-rd harmonic filtering branch (blue), the 5-th harmonic filtering branch (green) and the equivalent impedance

10 9 8

Z(n)/Z(wo)

7 6 5 4 3 2 1 0

0

100

200

300

400 f [Hz]

500

600

700

Chart 12. Frequency characteristic of the supply network and the filter equivalent impedance for the degree of detuning 2.5 and 4.5 related to the equivalent impedance at f=50Hz

29


4.5.4. Parallel resonance calculations: The schematic diagram and formulas as in subsection 4.4.5. The parallel resonance relative frequency for the detuning degree 2.5 and 4.5 is: • for the 3-rd harmonic filter: 2.64 • for the 5-th harmonic filter: 3.58 and for the detuning degree 2.7 and 4.7 it is: • •

for the 3-rd harmonic filter: for the 5-th harmonic filter:

1.46 2.34

4.5.5. Verification of the capacitor bank for overload current: a) For the 3-rd harmonic filter: The filter power is: ' NF 5

Q

2 2 ( ( UN ) 6000) = =

X Z (1)

6.67

= 5.40[ MVar ] (62)

and:

X Z 3(1) = X DF 3(1) + X NbatF 3(1) = 6.67[Ω]

(63)

The rated current of the 3-rd harmonic capacitor bank is:

I Nbat , F 3 =

' QNF 5,4 * 106 3 = = 409.91[ A] 3 * U Nbat 3 * 7600

(64)

The condition for preventing capacitors current overload is:

I1− 3 = ku I Nbat .F 3 = 0,78 * 391.25 = 323.61[ A]

I batF 3 = I12 + I n2 = 323.612 + 32.52 = 325.24[A]

(65)

(66)

Thus, the relation:

409.91[ A] ≥

I n2 = 255.01[A] p 2 − b 2 ku2

30

(67)


I n2 = k i I batF 3 p 2 − b 2 k u2

I NbatF ≥

(68)

is satisfied. b) For the 3-rd harmonic filter The filter power is:

(U N )2

' Q NbatF 5 =

X Z (1)

= 5.94[ MVar ] (69)

and:

X Z 5(1) = X DF 5(1) + X NbatF 5(1) = 6.06[Ω]

(70)

The rated current of the 3-rd harmonic capacitor bank is:

I Nbat , F 5 =

' QNbatF 5 = 451.55[ A] 3 * U Nbat

(71)

The condition for preventing capacitors current overload is:

I1− 5 = ku I NbatF = 356.48[A]

I batF 5 = I12 + I n2 = 358.58[A]

(72) (73)

Thus, the relation:

451.55[A] ≥

I n2 = 281.14[A] p 2 − b 2 ku2

I NbatF ≥

I n2 = k i I batF 5 p 2 − k u2

(74)

(75)

is satisfied. 4.5.6. Verification of THDU value Calculations of total harmonic voltage distortion factor THDU have been carried out as in subsection 4.4.7. 31


Table 7. Voltage harmonic percentage content and total harmonic voltage distortion factor THDU for the 3-rd and 5-th harmonic filter

Current harmonic Un for detuning degree Un for detuning degree order 2.7 and 4.7 2.5 and 4.5 8.73 9.32 2 29.63 31.62 3 18.93 8.79 4 6.90 10.81 5 4.43 5.00 6 5.55 6.00 7 5.23 5.56 8 4.76 5.02 9 3.40 3.57 10 5.34 5.60 11 1.69 1.76 12 3.70 3.86 13 1.12 1.1 THDU THDUF5= 1,1%, THDdop6kV = 5% [11], THDU6kV
5. Calculation of currents distribution after installation of FC compensator The equivalent circuit for calculation of currents distribution is shown in figure 8. The actual distribution of currents is tabulated in table 8.

Fig. 8. Schematic diagram of power network supplying the arc furnace and the designed capacitor banks 32


Total powers of the formerly sized filters are:

Q 'uzbatF 3 = 5.40[MVar]

(76)

Q uzbatF 5 = 5.94[MVar] '

(77)

The filters' compensating power is:

QFC = 1.07 ⋅ (QuzbatF 3 + QuzbatF 5 ) = 12.13[ MVar ]

(78)

The value 1.07 results from the manufacturing tolerances +/-2% for reactors and +/- 5% for capacitors. The output power to be compensated was 11 Mvar. The

Q sk = Q p − Q FC = 1 3 ,52 − 12,13 = 1,41[ MVar ]

output power after compensation is: (79)

Apparent power:

S p' = PP2 + Qsk2 = 10,22[ MVA]

(80)

The load current after compensation: ' I obc =

S p' 3 *U N

= 983,63[ A] (80)

Power factor after compensation:

tϕ =

QSK = 0,14 PP

(81)

Currents for individual branches are calculated from relation:

i Fk =

z 0 ( n ) * i 0 ( n) z k ( n)

(82)

where: iFk z0(n)

– the k-th harmonic filter current; – the equivalent impedance for the n-th harmonic of the power k, transformer and filters, as seen from the point of the arc furnace connection; 33


zk(n) iO(n)

– the impedance for the n-th harmonic of the k-th harmonic filter; – the arc furnace n-th harmonic current.

Table 8. The distribution of currents among the filter branches and 6kV network for the assumed degree of detuning 2.7 and 4.7

Harmonic order

The arc furnace current [A]

F3 [A]

F5 [A]

Tr [A]

1 2 3 4 5 6 7 8 9 10 11 12 13

759.62 21.93 32.55 9.30 38.73 8.83 7.85 5.95 4.60 2.88 4.03 1.15 2.30

109.21 8.47 12.84 0.84 2.19 0.89 0.93 0.75 0.60 0.39 0.55 0.16 0.32

114.81 4.52 12.10 7.45 33.28 6.47 5.30 3.83 2.87 1.75 2.41 0.68 1.35

983.63 8.94 7.60 1.01 3.26 1.47 1.62 1.37 1.13 0.74 1.06 0.31 0.63

6. Sizing the reactor section of FC/TCR compensator Reactors in the delta arms are divided into two equal parts as in figure 9. Advantages of such configuration are: better voltage distribution, easy manufacturing and limiting possible short-circuit currents.

Fig. 9. Schematic diagram of TCR reactor section

34


a) The nominal current in one arm of the delta connection is:

I DTCR = QFC / 3U s = 683[ A]

(83)

b) Hence, the reactance in one delta arm has been calculated:

X DTCR = U 2 s / 3QFCR = 2.97Ω

(84)

c) Then the required inductance:

LDTCR = QFC / ω = 9.45mH

(85)

The designed reactor consists of two parts with inductances of 0123 mH each. the selected reactors are special design reactors of "Z" company make, with inductance values calculated as above and 2% inductance toleration.

7. Conclusions The calculation results allow us to conclude that: •

• •

The voltage distortion factor at 6kV during the arc furnace operation without the compensation circuit is THD = 8.6%, whereas with the compensation circuit connected THD = 1.1% (the 3-rd and 5-th harmonics are reduced considerably) confirms proper operation of filters F3 and F5. The current loading of F3 filter is 255A and does not exceed its nominal current of 409A. The current loading of F3 filter is 281A and does not exceed its nominal current of 409A.

It follows from the calculations that the designed compensation system will not be overloaded with high-harmonic currents and will operate properly. The voltage distortion is within acceptable limits.

8. References 1. Strojny J., Strzałka J.: Projektowanie urządzeń elektroenergetycznych . Kraków , Uczelniane Wydawnictwa Naukowo-Dydaktyczne, 2008 [Electrical Power Equipment Design. AGH-UST Publishers, Krakow, 2008.] 2. Piróg S.: Energoelektronika. Układy o komutacji sieciowej i o komutacji twardej. Kraków, AGH Uczelniane Wydawnictwa Naukowo-Dydaktyczne 2006, [Power Electronics. Line Commutated and Hard Commutated Systems. AGH-UST Publishers, Krakow, 2006] 3. Strzelecki R., Supronowicz H. Współczynnik mocy w systemach zasilania prądu przemiennego i metody jego poprawy. Warszawa : Oficyna Wydaw. 35


PW, 2000 [Power Factor in AC Power Supply Systems and Methods for its Improvement. Warsaw University of Technology Publ., Warsaw, 2000.] 4. Klempka R., Stankiewicz A. Modelowanie i symulacja układów dynamicznych : wybrane zagadnienia z przykładami w Matlabie. Kraków, Uczelniane Wydawnictwa Naukowo-Dydaktyczne AGH, 2007 [Modelling and Simulation of Dynamic Systems: Selected Problems and Examples in Matlab Environment. AGH-UST Publishers, Krakow, 2007] 5. Praca zbiorowa: Poradnik inżyniera elektryka. T.2. Warszawa : Wydaw. Nauk. -Techniczne, 1997 [Joint publication: Electrical Engineer Guide, vol.2. Wydawnictwa Naukowo-Techniczne, Warsaw, 1997.] 6. Hanzelka Z. Skuteczność statycznej kompensacji oddziaływania odbiorników niespokojnych na sieć zasilającą. Rozprawy-Monografie. Kraków , Wydaw. AGH, 1994 Effectiveness of Static Compensation of Fluctuating Loads Impact on Power System. AGH-UST Monographs, Krakow, 1994. 7. Hanzelka Z., Klempka R.: Pasywne filtry wyższych harmonicznych. „elektro.info" 6/2003 [Passive High Harmonics Filters. „elektro.info" 6/2003.] 8. Hanzelka Z. JAKOSC ENERGII ELEKTRYCZNEJ CZĘŚĆ 3 - Wahania napięcia, (http://www.twelvee.com.pl) [Electric Power Quality, Part 3 – Voltage fluctuations (http://www.twelvee.com.pl] 9. Hanzelka Z. JAKOSC ENERGII ELEKTRYCZNEJ CZĘŚĆ 4 - Wyższe harmoniczne napięć i prądów (http://www.twelvee.com.pl) [Electric Power Quality, Part 4 – Voltage and Current High Harmonics (http:// www.twelvee.com.pl] 10. Hanzelka Z. Kompensator statyczny ze sterownikiem prądu indukcyjnego. Rozprawy elektrotechniczne 34, 1988 [A Static Compensator with Inductive Current Controller. Electrical Engineering Transactions No.34, 1988] 11. ROZPORZĄDZENIE MINISTRA GOSPODARKI z dnia 4 maja 2007 r. w sprawie szczegółowych warunków funkcjonowania systemu elektroenergetycznego [The ordinance of the Minister of Economy of May 4 2007 in the matter of detailed terms and conditions of power system operation] 12. Norma PN-EN-60871-1:2006 Tytuł: Kondensatory do równoległej kompensacji mocy biernej w sieciach elektroenergetycznych prądu przemiennego o napięciu znamionowym powyżej 1 kV, Wymagania ogólne [Standard EN 60871-1:2005 Shunt capacitors for a.c. power systems having a rated voltage above 1000 V. General] 13. Geppart A., Polaczek A.: Wskazówki projektowania dotyczące ograniczania odkształceń i wahań napięcia w sieciach ŚN i nn energetyki zawodowej. Instytut Energetyki. Warszawa - Katowice 1987 [Design Guidelines Concerning Voltage Distortion and Fluctuation Mitigation in MV and LV Power Systems. Institute of Power Engineering, Warsaw‑ Katowice, 1987.]

36

Static FC/TCR compensator

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