# Transient Behaviour of Superconducting Magnet Systems of Fusion Reactor ITER during Safety Discharge

Authors: A. M. Miri; S. Fink; W. H. Fietz
Journal: Modelling and Simulation in Engineering
 (2008)

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## Abstract

To investigate the transient behaviour of the toroidal and poloidal field coils magnet systems of the International Thermonuclear Experimental Reactor during safety discharge, network models with lumped elements are established. Frequency-dependant values of the network elements, that is, inductances and resistances are calculated with the finite element method. That way, overvoltages can be determined. According to these overvoltages, the insulation coordination of coils has to be selected.


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## Body

## 1. Introduction

The  International
Thermonuclear  Experimental  Reactor (ITER) is the step between today’s
studies of plasma physics and tomorrow's production of electricity by fusion
power plants [1]. The plasma is confined by the magnetic fields of the toroidal
field (TF) coils, the poloidal
field (PF) coils, and the central solenoid (see Figure 1).Figure 1
Sectional view of ITER tokamak-reactor.Due to the required huge
magnetic fields, the coils need to be superconducting. In the event of a
quench, when the conductor changes from superconducting to normal conducting
state, the stored magnetic energy must be discharged from the coil by a fast
discharge circuit. Thus damage through overheating is avoided. To initiate the
fast discharge, counteracting current switches are used which can cause high-voltage
transients across the winding. These transients can lead to internal
oscillations with overvoltages.In this
investigation, the transient behaviours of the ITER toroidal field (TF) and
poloidal field (PF) coils are simulated during safety discharge.The TF coils are equal
and in series. The TF coil system consists of 18 TF coils and 9 fast discharge
units (FDUs). The structure of the system is shown in Figure2. After the determination of the transient behaviour
of the system in Figure
7, the resulted voltage across one TF coil terminal
will be applied for further investigations of the detailed model of one coil.Figure 2
Simplified analogue circuit of the TF coil system during a fast
discharge (power supply already disconnected).The PF coils have
different diameters, winding numbers, and structures. The PF coils 2, 3, 4, and
5 are connected in parallel and are excited according to Figure3. PF coils 1 and 6 are excited separately. Because of
their geometry, for PF3 and PF6 the lowest resonance frequency is assumed. So
the models of these two coils are established to determine the insulation
stress inside the windings of the PF coils during safety discharge.Figure 3
Outer PF coil electrical circuit PF2, PF3, PF4, and PF5.For PF coil 3, the lowest
resonance frequency was determined. So in safety discharge, only the electrical
circuit of PF2 to PF5 is considered. All coils and converters are in parallel.
The main purpose of the vertical  converter is to control the vertical plasma displacements. To verify
the calculation methods, some measurements were performed with a single TF
model coil, which has a reduced size compared to the future TF coils [2–6].


## 2. The ITER TF and PF Coils and Their Network Models

The TF coils are “D”
shaped and consist of 192 windings. The winding pack is enclosed in a steel
case which consists of 7 radial plates (see Figure4). The conductors are surrounded by an insulation made
of epoxy resin impregnated kapton and glass [7]. The coil terminals project out
of the case on the lower curved part of the case. All other double pancake
joints (i.e., joints linking radial plates) are contained within a local
extension of the case.Figure 4
Cross-section of
one TF coil.The PF coils are in ring form and have
different number of turns. The structure is given in Figure5.Figure 5
Part of the cross-section of
an ITER PF coil.Figure 6
Part of the cross-section of an ITER TF coil showing the three
different types of the electrical insulation.Figure 7
Network model for one TF coil.The 6 PF coils are placed circular around the outboard legs of the TF coils.
The PF coils are self-supporting as regards the radial magnetic loads and the vertical loads transmitted to the TF coil
cases by supports designed to allow a free radial expansion of the coils.
The lead of the PF coil is made up of square steel conduit, which contains the
circular superconductor. The superconducting material is the alloy
niobium-titanium. It is included as multiple filaments strands in the circular
conductor. The square conduit is surrounded by insulation made of epoxy resin-impregnated
polyimide film and dry glass.


### 2.1. TF Network Model

Each
TF coil consists of 134 turns embedded in radial plates (see Figure4). Each turn has been modelled by a resistance, a
capacitance,  and an inductance. The 7 double pancakes are connected in series.
The capacitance represents insulation between the turn and the radial plate.
The radial plates have capacitive coupling between each other and to the grounded case. From [8], it is proved that the radial
plates connected with the centre of double pancake (location of helium inlet)
through a resistance is the best linking method. For ITER TF, the value of the
resistance is given as 2*50kΩ=100kΩ. Additionally, the influences of instrumentation
cables have been considered here. The instrumentation cables are represented
with additional capacitances between helium inlet and ground. Figure 7
shows the built network model for one TF coil. The
frequency-dependant damping of the radial plates is treated by an FEM model.


#### 2.1.1. Frequency Sensitivity of Resistances

Measurements have shown that the
superconducting filaments do not participate in high-frequency oscillations for
fast transient processes in cable-in-conduit conductors (CICCs) of the type
similar to the ones used in the ITER TF [3, 4]. A possible reason is that at
4 K, the skin depth in the copper matrix surrounding the filaments is very
small already for frequencies in the range of kHz. Furthermore, the flux
penetration of the superconductor is also related to a time constant. In other
words, the transient wave propagates in the surrounding copper matrix. Hence,
for the network model, the superconducting strands were regarded conducting in
a normal way.For
the total DC resistance of the coil, a value of 1.0 mΩ was found,
assuming a residual resistance ratio (RRR) [9] of the stabilizing copper of
120. It can be proved that there is no effective skin or proximity effect for
the conductor as a whole. A frequency dependence of the resistance is only
effective due to a local current displacement inside the strands by a local
skin effect and due to a local proximity effect among the strands. To calculate
the frequency-dependent resistance R_skin(f), an analytic formula or a finite
element method (FEM) can be used. The FEM (it refers to the programme Maxwell
[9] in this project) has been chosen to do the calculation of the ratio of
resistance by certain frequencies to DC values. Some results are shown in Table 1.Table 1
Resistance ratio.
Resistance ratioFrequency (Hz)25100371701084705332000086050000


#### 2.1.2. Turn-to-Radial Plate Capacitance

The  capacitance between turns and radial plates can be
regarded as cylindrical capacitors [2]. The capacitance between one turn of
layer k and the radial plate is given by (1)Ck,TtoR=2πε0εrlkln(Ra/Ri), where Ra and Ri are  the outer and inner radii of
the winding. lk is the length of the winding of
layer k(k=1,…,11).


#### 2.1.3. Radial Plate to Radial Plate Capacitance

The  capacitance between
neighbouring radial plates can be regarded as plate capacitors. The capacitor between
two radial plates is given by(2)CRtoR=ε0εrL*Wd, where
L=34.1m(8.) is the length of the coil centre line. W is the width of the
radial plate in the direction from layer 1 to layer 11 and d is the insulation
thickness between the radial plates. It can be proved that this formula has a
good approximation compared with the formula with the application of different radii.


#### 2.1.4. Ground Capacitance

There are considerable ground capacitance at the innermost and the
outermost layer. For the calculation of the capacitances, (2) applies.


#### 2.1.5. Capacitance of Instrumentation Cables

The  instrumentation cables of
60 m length cause considerable ground capacitances. It is assumed that the capacitance
per unit length is 0.4 nF/m.


#### 2.1.6. Self- and Mutual Inductances

The self- and mutual inductances of DC value are to be
calculated using analytical formulae for circular cylindrical conductors [10]. The
self-inductance of one turn of layer k is given by (3)Lk=μ02π*lk*(ln(4lkd)−0.75), where lk is
the length of layer k and d is the diameter of the winding.The mutual inductance betweenkth winding and jth winding is calculated by
using the Neumann equation (4)M=μ04π∗∮l1∮l2dl1dl2r12, where l1 is  the length of the kth
winding and l2 is the length of the jth winding. dl1 is
the length element of the
kth winding, dl2 is the length element of the jth
winding, and r12 is the distance between the two windings.The  total inductance of one TF coil was
calculated in program MATLAB with the value of
0.3498H, which agrees well with the value of 0.349H found in literature [5].
The calculations for the DC values of self- and mutual inductance are discussed
above.During the transient process, there are many different
frequency portions, which are included in a voltage signal or a current signal.
To get the right frequency response, it is necessary to set the specific
inductance values in the model for the corresponding frequencies. That means
the frequency-dependent inductances need to be calculated. This has been done
with the use of the finite element method (FEM), because the eddy current effect
cannot be described through analytical formula.The FEM
discretises 2D or 3D areas. Areas with complex structures are divided into
small triangles, so that the interior of a triangle can approximately be
handled as homogeneous. A continuum is idealised with a finite number of
elements that are connected with nodes. This solution converges towards the
exact solution. More detailed information can be read in [11].


#### 2.1.7. Confirmation of the First Dominant Resonant Frequency

To make a better statement about the frequency response to the
stimulating signals, which contain different frequencies, it is necessary to
know the first dominant resonant frequency or the so-called resonance frequency
of the single coil network model. Due to the skin effect, at
high frequencies, the current is displaced from the inner area of the
conductor. So there is a frequency dependence: the AC resistance of the
conductor increases and the inductance decreases.So the real resonance frequency of the coil cannot be calculated directly.
Application of iterative method is necessary: with the FEM-using programme
Maxwell, the DC values for the resistances and inductances are calculated. Then
the values for the lumped elements are inserted in the transient program ATP
[12] and the resonance frequency is determined. With this new resonance
frequency, the FEM is applied again. This procedure is carried out iteratively
until the resonance frequency converges to the real resonance frequency of the
coil. At last 50 kHz resulted in this calculation for a single ITER TF coil.


#### 2.1.8. Simplified Network Model of ITER TF Coil System without and with Earth Capacitances

For
the magnets, one of the important parameters is overvoltage on the coil
terminals that can cause an insulation breakdown and can trigger a chain of other fault events.
To get the voltage across the coil terminals and for the transient analysis in
the fast discharge unit (FDU), the circuit for
TF coil system analysis [16] is used (see Figure8). The TF coil here has been simplified as a pure inductance.
This is true when the event of discharge is only analysed without any further
fault conditions. But if there is a short circuit on the coil terminals to
ground, the capacitances between the coil terminals and ground must be
considered for the analysis of the voltage across the coil terminals and other
parameters in the FDUs [8].Figure 8
Circuit for TF coil system analysis.A replacement of the 18
pure inductances in Figure8 by 18 coupled single coil models like in Figure 7
would cause huge calculation time. Therefore,
capacitances are summarised and distributed within the circuit. The network
model of the complete system with the simplified coils is shown in Figure 9. In both cases, the coupling factors
among all the 18 TF
coils were considered. The values for the coupling factors were taken from [5].Figure 9
Network model of the ITER TF coil system, the 9 FDUs are shown as blocks.In Figure9, the earth capacitances are divided into two parts.
One part represents the resulted insulation between the radial plates and
ground; additionally the resistance with the value of 1Ω formed by the case has
been taken into account as well. Another part represents the capacitances
formed by the instrumentation cables, hence the inductance of μH from the
instrumentation cable has also been considered.


#### 2.1.9. The Admissibility of the Network Model

The
admissibility  of the network model can be evaluated by calculating the frequency
range. The propagation velocity is given by(5)c=c0μrεr=1.5*108ms.


### 2.2. PF Network Model

The maximum
permitted frequency for the coil is(6)fmax=c8l=1.1MHz(withl=34.1m).As the
resonance frequency is much lower, the network model is admissible.For the establishment of the models for PF coil 2 and coil 3, the procedure
is the same as for the TF coils. In Figure10, the network of PF3 is shown.Figure 10
Network model of one double pancake of PF coil 3.


## 2.1. TF Network Model

Each
TF coil consists of 134 turns embedded in radial plates (see Figure4). Each turn has been modelled by a resistance, a
capacitance,  and an inductance. The 7 double pancakes are connected in series.
The capacitance represents insulation between the turn and the radial plate.
The radial plates have capacitive coupling between each other and to the grounded case. From [8], it is proved that the radial
plates connected with the centre of double pancake (location of helium inlet)
through a resistance is the best linking method. For ITER TF, the value of the
resistance is given as 2*50kΩ=100kΩ. Additionally, the influences of instrumentation
cables have been considered here. The instrumentation cables are represented
with additional capacitances between helium inlet and ground. Figure 7
shows the built network model for one TF coil. The
frequency-dependant damping of the radial plates is treated by an FEM model.


### 2.1.1. Frequency Sensitivity of Resistances

Measurements have shown that the
superconducting filaments do not participate in high-frequency oscillations for
fast transient processes in cable-in-conduit conductors (CICCs) of the type
similar to the ones used in the ITER TF [3, 4]. A possible reason is that at
4 K, the skin depth in the copper matrix surrounding the filaments is very
small already for frequencies in the range of kHz. Furthermore, the flux
penetration of the superconductor is also related to a time constant. In other
words, the transient wave propagates in the surrounding copper matrix. Hence,
for the network model, the superconducting strands were regarded conducting in
a normal way.For
the total DC resistance of the coil, a value of 1.0 mΩ was found,
assuming a residual resistance ratio (RRR) [9] of the stabilizing copper of
120. It can be proved that there is no effective skin or proximity effect for
the conductor as a whole. A frequency dependence of the resistance is only
effective due to a local current displacement inside the strands by a local
skin effect and due to a local proximity effect among the strands. To calculate
the frequency-dependent resistance R_skin(f), an analytic formula or a finite
element method (FEM) can be used. The FEM (it refers to the programme Maxwell
[9] in this project) has been chosen to do the calculation of the ratio of
resistance by certain frequencies to DC values. Some results are shown in Table 1.Table 1
Resistance ratio.
Resistance ratioFrequency (Hz)25100371701084705332000086050000


### 2.1.2. Turn-to-Radial Plate Capacitance

The  capacitance between turns and radial plates can be
regarded as cylindrical capacitors [2]. The capacitance between one turn of
layer k and the radial plate is given by (1)Ck,TtoR=2πε0εrlkln(Ra/Ri), where Ra and Ri are  the outer and inner radii of
the winding. lk is the length of the winding of
layer k(k=1,…,11).


### 2.1.3. Radial Plate to Radial Plate Capacitance

The  capacitance between
neighbouring radial plates can be regarded as plate capacitors. The capacitor between
two radial plates is given by(2)CRtoR=ε0εrL*Wd, where
L=34.1m(8.) is the length of the coil centre line. W is the width of the
radial plate in the direction from layer 1 to layer 11 and d is the insulation
thickness between the radial plates. It can be proved that this formula has a
good approximation compared with the formula with the application of different radii.


### 2.1.4. Ground Capacitance

There are considerable ground capacitance at the innermost and the
outermost layer. For the calculation of the capacitances, (2) applies.


### 2.1.5. Capacitance of Instrumentation Cables

The  instrumentation cables of
60 m length cause considerable ground capacitances. It is assumed that the capacitance
per unit length is 0.4 nF/m.


### 2.1.6. Self- and Mutual Inductances

The self- and mutual inductances of DC value are to be
calculated using analytical formulae for circular cylindrical conductors [10]. The
self-inductance of one turn of layer k is given by (3)Lk=μ02π*lk*(ln(4lkd)−0.75), where lk is
the length of layer k and d is the diameter of the winding.The mutual inductance betweenkth winding and jth winding is calculated by
using the Neumann equation (4)M=μ04π∗∮l1∮l2dl1dl2r12, where l1 is  the length of the kth
winding and l2 is the length of the jth winding. dl1 is
the length element of the
kth winding, dl2 is the length element of the jth
winding, and r12 is the distance between the two windings.The  total inductance of one TF coil was
calculated in program MATLAB with the value of
0.3498H, which agrees well with the value of 0.349H found in literature [5].
The calculations for the DC values of self- and mutual inductance are discussed
above.During the transient process, there are many different
frequency portions, which are included in a voltage signal or a current signal.
To get the right frequency response, it is necessary to set the specific
inductance values in the model for the corresponding frequencies. That means
the frequency-dependent inductances need to be calculated. This has been done
with the use of the finite element method (FEM), because the eddy current effect
cannot be described through analytical formula.The FEM
discretises 2D or 3D areas. Areas with complex structures are divided into
small triangles, so that the interior of a triangle can approximately be
handled as homogeneous. A continuum is idealised with a finite number of
elements that are connected with nodes. This solution converges towards the
exact solution. More detailed information can be read in [11].


### 2.1.7. Confirmation of the First Dominant Resonant Frequency

To make a better statement about the frequency response to the
stimulating signals, which contain different frequencies, it is necessary to
know the first dominant resonant frequency or the so-called resonance frequency
of the single coil network model. Due to the skin effect, at
high frequencies, the current is displaced from the inner area of the
conductor. So there is a frequency dependence: the AC resistance of the
conductor increases and the inductance decreases.So the real resonance frequency of the coil cannot be calculated directly.
Application of iterative method is necessary: with the FEM-using programme
Maxwell, the DC values for the resistances and inductances are calculated. Then
the values for the lumped elements are inserted in the transient program ATP
[12] and the resonance frequency is determined. With this new resonance
frequency, the FEM is applied again. This procedure is carried out iteratively
until the resonance frequency converges to the real resonance frequency of the
coil. At last 50 kHz resulted in this calculation for a single ITER TF coil.


### 2.1.8. Simplified Network Model of ITER TF Coil System without and with Earth Capacitances

For
the magnets, one of the important parameters is overvoltage on the coil
terminals that can cause an insulation breakdown and can trigger a chain of other fault events.
To get the voltage across the coil terminals and for the transient analysis in
the fast discharge unit (FDU), the circuit for
TF coil system analysis [16] is used (see Figure8). The TF coil here has been simplified as a pure inductance.
This is true when the event of discharge is only analysed without any further
fault conditions. But if there is a short circuit on the coil terminals to
ground, the capacitances between the coil terminals and ground must be
considered for the analysis of the voltage across the coil terminals and other
parameters in the FDUs [8].Figure 8
Circuit for TF coil system analysis.A replacement of the 18
pure inductances in Figure8 by 18 coupled single coil models like in Figure 7
would cause huge calculation time. Therefore,
capacitances are summarised and distributed within the circuit. The network
model of the complete system with the simplified coils is shown in Figure 9. In both cases, the coupling factors
among all the 18 TF
coils were considered. The values for the coupling factors were taken from [5].Figure 9
Network model of the ITER TF coil system, the 9 FDUs are shown as blocks.In Figure9, the earth capacitances are divided into two parts.
One part represents the resulted insulation between the radial plates and
ground; additionally the resistance with the value of 1Ω formed by the case has
been taken into account as well. Another part represents the capacitances
formed by the instrumentation cables, hence the inductance of μH from the
instrumentation cable has also been considered.


### 2.1.9. The Admissibility of the Network Model

The
admissibility  of the network model can be evaluated by calculating the frequency
range. The propagation velocity is given by(5)c=c0μrεr=1.5*108ms.


## 2.1.1. Frequency Sensitivity of Resistances

Measurements have shown that the
superconducting filaments do not participate in high-frequency oscillations for
fast transient processes in cable-in-conduit conductors (CICCs) of the type
similar to the ones used in the ITER TF [3, 4]. A possible reason is that at
4 K, the skin depth in the copper matrix surrounding the filaments is very
small already for frequencies in the range of kHz. Furthermore, the flux
penetration of the superconductor is also related to a time constant. In other
words, the transient wave propagates in the surrounding copper matrix. Hence,
for the network model, the superconducting strands were regarded conducting in
a normal way.For
the total DC resistance of the coil, a value of 1.0 mΩ was found,
assuming a residual resistance ratio (RRR) [9] of the stabilizing copper of
120. It can be proved that there is no effective skin or proximity effect for
the conductor as a whole. A frequency dependence of the resistance is only
effective due to a local current displacement inside the strands by a local
skin effect and due to a local proximity effect among the strands. To calculate
the frequency-dependent resistance R_skin(f), an analytic formula or a finite
element method (FEM) can be used. The FEM (it refers to the programme Maxwell
[9] in this project) has been chosen to do the calculation of the ratio of
resistance by certain frequencies to DC values. Some results are shown in Table 1.Table 1
Resistance ratio.
Resistance ratioFrequency (Hz)25100371701084705332000086050000


## 2.1.2. Turn-to-Radial Plate Capacitance

The  capacitance between turns and radial plates can be
regarded as cylindrical capacitors [2]. The capacitance between one turn of
layer k and the radial plate is given by (1)Ck,TtoR=2πε0εrlkln(Ra/Ri), where Ra and Ri are  the outer and inner radii of
the winding. lk is the length of the winding of
layer k(k=1,…,11).


## 2.1.3. Radial Plate to Radial Plate Capacitance

The  capacitance between
neighbouring radial plates can be regarded as plate capacitors. The capacitor between
two radial plates is given by(2)CRtoR=ε0εrL*Wd, where
L=34.1m(8.) is the length of the coil centre line. W is the width of the
radial plate in the direction from layer 1 to layer 11 and d is the insulation
thickness between the radial plates. It can be proved that this formula has a
good approximation compared with the formula with the application of different radii.


## 2.1.4. Ground Capacitance

There are considerable ground capacitance at the innermost and the
outermost layer. For the calculation of the capacitances, (2) applies.


## 2.1.5. Capacitance of Instrumentation Cables

The  instrumentation cables of
60 m length cause considerable ground capacitances. It is assumed that the capacitance
per unit length is 0.4 nF/m.


## 2.1.6. Self- and Mutual Inductances

The self- and mutual inductances of DC value are to be
calculated using analytical formulae for circular cylindrical conductors [10]. The
self-inductance of one turn of layer k is given by (3)Lk=μ02π*lk*(ln(4lkd)−0.75), where lk is
the length of layer k and d is the diameter of the winding.The mutual inductance betweenkth winding and jth winding is calculated by
using the Neumann equation (4)M=μ04π∗∮l1∮l2dl1dl2r12, where l1 is  the length of the kth
winding and l2 is the length of the jth winding. dl1 is
the length element of the
kth winding, dl2 is the length element of the jth
winding, and r12 is the distance between the two windings.The  total inductance of one TF coil was
calculated in program MATLAB with the value of
0.3498H, which agrees well with the value of 0.349H found in literature [5].
The calculations for the DC values of self- and mutual inductance are discussed
above.During the transient process, there are many different
frequency portions, which are included in a voltage signal or a current signal.
To get the right frequency response, it is necessary to set the specific
inductance values in the model for the corresponding frequencies. That means
the frequency-dependent inductances need to be calculated. This has been done
with the use of the finite element method (FEM), because the eddy current effect
cannot be described through analytical formula.The FEM
discretises 2D or 3D areas. Areas with complex structures are divided into
small triangles, so that the interior of a triangle can approximately be
handled as homogeneous. A continuum is idealised with a finite number of
elements that are connected with nodes. This solution converges towards the
exact solution. More detailed information can be read in [11].


## 2.1.7. Confirmation of the First Dominant Resonant Frequency

To make a better statement about the frequency response to the
stimulating signals, which contain different frequencies, it is necessary to
know the first dominant resonant frequency or the so-called resonance frequency
of the single coil network model. Due to the skin effect, at
high frequencies, the current is displaced from the inner area of the
conductor. So there is a frequency dependence: the AC resistance of the
conductor increases and the inductance decreases.So the real resonance frequency of the coil cannot be calculated directly.
Application of iterative method is necessary: with the FEM-using programme
Maxwell, the DC values for the resistances and inductances are calculated. Then
the values for the lumped elements are inserted in the transient program ATP
[12] and the resonance frequency is determined. With this new resonance
frequency, the FEM is applied again. This procedure is carried out iteratively
until the resonance frequency converges to the real resonance frequency of the
coil. At last 50 kHz resulted in this calculation for a single ITER TF coil.


## 2.1.8. Simplified Network Model of ITER TF Coil System without and with Earth Capacitances

For
the magnets, one of the important parameters is overvoltage on the coil
terminals that can cause an insulation breakdown and can trigger a chain of other fault events.
To get the voltage across the coil terminals and for the transient analysis in
the fast discharge unit (FDU), the circuit for
TF coil system analysis [16] is used (see Figure8). The TF coil here has been simplified as a pure inductance.
This is true when the event of discharge is only analysed without any further
fault conditions. But if there is a short circuit on the coil terminals to
ground, the capacitances between the coil terminals and ground must be
considered for the analysis of the voltage across the coil terminals and other
parameters in the FDUs [8].Figure 8
Circuit for TF coil system analysis.A replacement of the 18
pure inductances in Figure8 by 18 coupled single coil models like in Figure 7
would cause huge calculation time. Therefore,
capacitances are summarised and distributed within the circuit. The network
model of the complete system with the simplified coils is shown in Figure 9. In both cases, the coupling factors
among all the 18 TF
coils were considered. The values for the coupling factors were taken from [5].Figure 9
Network model of the ITER TF coil system, the 9 FDUs are shown as blocks.In Figure9, the earth capacitances are divided into two parts.
One part represents the resulted insulation between the radial plates and
ground; additionally the resistance with the value of 1Ω formed by the case has
been taken into account as well. Another part represents the capacitances
formed by the instrumentation cables, hence the inductance of μH from the
instrumentation cable has also been considered.


## 2.1.9. The Admissibility of the Network Model

The
admissibility  of the network model can be evaluated by calculating the frequency
range. The propagation velocity is given by(5)c=c0μrεr=1.5*108ms.


## 2.2. PF Network Model

The maximum
permitted frequency for the coil is(6)fmax=c8l=1.1MHz(withl=34.1m).As the
resonance frequency is much lower, the network model is admissible.For the establishment of the models for PF coil 2 and coil 3, the procedure
is the same as for the TF coils. In Figure10, the network of PF3 is shown.Figure 10
Network model of one double pancake of PF coil 3.


## 3. Safety Discharge by TF and PF Coils in Normal and in Fault Cases

If a quench (i.e., the conductor is no longer superconducting) is
detected, the fast discharge is initiated to extract the stored energy of the
coils and to avoid
damage.The current is commutated from the continuous bypaths to the vacuum
circuit breaker paths (see Figure11) and the power supply is short circuited
and switched off. The switches connect the charged counter pulse capacitors.
The counter current runs through the vacuum circuit breaker (VCB) in the
opposite direction to the coil current. The resulting current decreases.Figure 11
Model of the fast discharge
circuit of the TF coil.After this current reaches a zero-crossing and the arc in the circuit
breaker extinguishes, the circuit breaker opens. The coil current commutates
from the short current path to the damp resistor path where the energy is
abolished in the damping resistor (Rd).The  maximum voltage across the damp resistor of the TF system is in the
case of 60 KA about 6 kV (see Figure12). As at the moment of commutation, the counter pulse
capacitor is not completely discharged, the discharge continues over the
snubber circuit in the snubber resistance (Rsn), resulting in a 2 kV pulse at
the FDU terminal with a rise time of about 1 microsecond. The currents are
shown in Figure
13.Figure 12
Results of TF fast discharge transient analysis;UAB: voltage across
the FDU; Uc: voltage on the capacitor; the above results have a good agreement
compared with [10].Figure 13
Results of TF fast discharge transient analysis;IVCB: current through the VCB;
IRd: current in the discharge resistor; ISN: current in the snubber circuit.The transient behaviour of
the coils is determined for two different cases.


### 3.1. Normal Case

A quench is detected and the energy stored in the coil must be extracted.
The TF coil system is excited with a current of 68 kA. When the current is
switched off, the
safety discharge is carried out.The maximum voltage across the TF coil terminals is investigated with the
model in Figure2. The detailed model like in Figure 9
of a single TF coil is excited with this voltage and
the maximum voltage in this coil is determined [6].For the PF coils, as
excitation, a step function with a rise time of 1 millisecond from 1 kV
to 9 kV to the network (see Figure10) is applied.


### 3.2. Fault Case

The single TF coil models
(1 kHz, 50 kHz) are examined for fast discharge (see Figure14). Maximum voltages are determined for normal case and
two fault cases on every type of insulation. The results are summarised in Table 2. For all cases, the first 5 seconds are used for charging
and running a DC current of 68 kA through the coils.Table 2
Overview of the calculated
maximum voltage at ground, radial plate, and conductor insulation for the 1 kHz
and 50 kHz model (the elements of network are calculated for 1 kHz and 50 kHz,
resp.) of TF coil.
f(kHz)Maximum calculated voltage (kV)Insulation typeFast dischargeFault case 1Fault case 2Ground13.58.116.4503.58.116.4Radial plate10.70.74.8500.60.74.8Conductors10.60.84.3500.50.83.7Figure 14
Fast discharge without fault. Terminal voltage to ground (e.g.,L8:2), 3467 V (t=5.0085 seconds), rise
time from zero point to maximum: 1.6 milliseconds, capacitor discharge: −696 V,
rise time: 1.1 microsecond (t=5.00100 seconds), terminal to terminal voltage (e.g., L8): 3467 V, rise time: 1.6 milliseconds (t=5.0085 seconds).The malfunction of the two FDUs was examined in fault
case 1, that is, the commutation in the resistor path is only successful for
seven FDUs while the current remains in the bypass path for two FDUs. The
voltages from different terminals to ground can be seen in Figure15.Figure 15
Voltages to ground on coil 3, 8,
14, and 15 during fast discharge with failure of FDU2 and FDU3.For the second fault case, it was assumed that the voltage imbalance in
the system, caused by fault 1, with the increase of terminal to earth voltages,
leads to an earth fault at the time and location of the maximum voltage to
earth. For this second fault case, the rise times are in the range of few microseconds
for the maximum radial plate and conductor insulation voltages, and in the
range of some milliseconds for the maximum ground insulation voltage. The
reason for such a difference among
the rise times is that the maximum terminal voltage to ground and the maximum
voltage between the terminals of a coil were not found at the same coil in Table3 [14]. The maximum voltages are shown in Figure 16.Table 3
DC and AC (peak value) test voltages of TF coils for
ground, radial plate, and conductor insulation based on the calculated voltages
for the two different fault cases.
Test voltages (kV)InsulationBased on fault case 1Based on fault case 2Ground1834Radial plate511Conductor511Figure 16
Fast discharge with failure of 2 FDUs and earth fault. Maximum
voltage at terminal:L8:2, U = 16.35 kV with oscillations t = 5.087 seconds,
rise time from 8.06 kV to maximum voltage: 3, 5 milliseconds, maximum terminal
to terminal voltage: L2, U=13.49kV at t = 5.0819  seconds, rise time from 2.69 kV to maximum kV:
2.4 microseconds.The TF coil system was analysed with PSpice programme
package where the coupling matrix could be included. Calculations lead to the
result, that it can be disregarded for increasing frequency, especially
resonance frequency.To determine the PF coil system, similar analyses are performed.
During fast discharge in a circuit as shown in Figure3, all
possible combinations of malfunction of one and two FDUs are simulated. The
maximum potential to ground of 4.6 kV occurs at the upper terminal of PF coil 4
with FDU5 not opening.So for the PF coils, fault case 1 means the
malfunction of only one FDU, fault case 2 is fault 1 with an earth fault. This
earth fault occurs at the terminal where the potential maximum appears after
switching off the power supply. The detailed coil model of PF coil 4 is similar
to the model of PF coil 3 shown in Figure10. With
the values of the lumped elements at resonance frequency, it is charged by a
step function with the maximum occurring potential of 4.6 kV in fault case. Then
the power supply is interrupted and the rise time of the potential is determined
to be 8 microseconds.Now a step  function with that rise time is applied to the detailed model. Three cases are considered after a
voltage of 4.6 kV is reached:(i)
voltage keeps constant (step
function);(ii)
the power supply is switched
off (fault case 1);(iii)
the power supply is switched
off and an earth fault occurs (fault case 2). Because of the capacitances in the detailed
model, there are oscillations in the coil which result in overvoltages. Results
can be seen in Table 4.Table 4
Overview of
the calculated maximum voltages for the 21.5 kHz model of PF coil 4.
Maximum calculated voltage (kV)Insulation typeStep functionFault case 1Fault case 2Ground4.83.32.9Radial plate0.90.91.0Conductors1.00.91.2For modelling the PF coils,
the software ATP [12] was used. ATP does not allow including the coupling matrix of the
complete PF system. So the influence of the other PF coils was not considered.


## 3.1. Normal Case

A quench is detected and the energy stored in the coil must be extracted.
The TF coil system is excited with a current of 68 kA. When the current is
switched off, the
safety discharge is carried out.The maximum voltage across the TF coil terminals is investigated with the
model in Figure2. The detailed model like in Figure 9
of a single TF coil is excited with this voltage and
the maximum voltage in this coil is determined [6].For the PF coils, as
excitation, a step function with a rise time of 1 millisecond from 1 kV
to 9 kV to the network (see Figure10) is applied.


## 3.2. Fault Case

The single TF coil models
(1 kHz, 50 kHz) are examined for fast discharge (see Figure14). Maximum voltages are determined for normal case and
two fault cases on every type of insulation. The results are summarised in Table 2. For all cases, the first 5 seconds are used for charging
and running a DC current of 68 kA through the coils.Table 2
Overview of the calculated
maximum voltage at ground, radial plate, and conductor insulation for the 1 kHz
and 50 kHz model (the elements of network are calculated for 1 kHz and 50 kHz,
resp.) of TF coil.
f(kHz)Maximum calculated voltage (kV)Insulation typeFast dischargeFault case 1Fault case 2Ground13.58.116.4503.58.116.4Radial plate10.70.74.8500.60.74.8Conductors10.60.84.3500.50.83.7Figure 14
Fast discharge without fault. Terminal voltage to ground (e.g.,L8:2), 3467 V (t=5.0085 seconds), rise
time from zero point to maximum: 1.6 milliseconds, capacitor discharge: −696 V,
rise time: 1.1 microsecond (t=5.00100 seconds), terminal to terminal voltage (e.g., L8): 3467 V, rise time: 1.6 milliseconds (t=5.0085 seconds).The malfunction of the two FDUs was examined in fault
case 1, that is, the commutation in the resistor path is only successful for
seven FDUs while the current remains in the bypass path for two FDUs. The
voltages from different terminals to ground can be seen in Figure15.Figure 15
Voltages to ground on coil 3, 8,
14, and 15 during fast discharge with failure of FDU2 and FDU3.For the second fault case, it was assumed that the voltage imbalance in
the system, caused by fault 1, with the increase of terminal to earth voltages,
leads to an earth fault at the time and location of the maximum voltage to
earth. For this second fault case, the rise times are in the range of few microseconds
for the maximum radial plate and conductor insulation voltages, and in the
range of some milliseconds for the maximum ground insulation voltage. The
reason for such a difference among
the rise times is that the maximum terminal voltage to ground and the maximum
voltage between the terminals of a coil were not found at the same coil in Table3 [14]. The maximum voltages are shown in Figure 16.Table 3
DC and AC (peak value) test voltages of TF coils for
ground, radial plate, and conductor insulation based on the calculated voltages
for the two different fault cases.
Test voltages (kV)InsulationBased on fault case 1Based on fault case 2Ground1834Radial plate511Conductor511Figure 16
Fast discharge with failure of 2 FDUs and earth fault. Maximum
voltage at terminal:L8:2, U = 16.35 kV with oscillations t = 5.087 seconds,
rise time from 8.06 kV to maximum voltage: 3, 5 milliseconds, maximum terminal
to terminal voltage: L2, U=13.49kV at t = 5.0819  seconds, rise time from 2.69 kV to maximum kV:
2.4 microseconds.The TF coil system was analysed with PSpice programme
package where the coupling matrix could be included. Calculations lead to the
result, that it can be disregarded for increasing frequency, especially
resonance frequency.To determine the PF coil system, similar analyses are performed.
During fast discharge in a circuit as shown in Figure3, all
possible combinations of malfunction of one and two FDUs are simulated. The
maximum potential to ground of 4.6 kV occurs at the upper terminal of PF coil 4
with FDU5 not opening.So for the PF coils, fault case 1 means the
malfunction of only one FDU, fault case 2 is fault 1 with an earth fault. This
earth fault occurs at the terminal where the potential maximum appears after
switching off the power supply. The detailed coil model of PF coil 4 is similar
to the model of PF coil 3 shown in Figure10. With
the values of the lumped elements at resonance frequency, it is charged by a
step function with the maximum occurring potential of 4.6 kV in fault case. Then
the power supply is interrupted and the rise time of the potential is determined
to be 8 microseconds.Now a step  function with that rise time is applied to the detailed model. Three cases are considered after a
voltage of 4.6 kV is reached:(i)
voltage keeps constant (step
function);(ii)
the power supply is switched
off (fault case 1);(iii)
the power supply is switched
off and an earth fault occurs (fault case 2). Because of the capacitances in the detailed
model, there are oscillations in the coil which result in overvoltages. Results
can be seen in Table 4.Table 4
Overview of
the calculated maximum voltages for the 21.5 kHz model of PF coil 4.
Maximum calculated voltage (kV)Insulation typeStep functionFault case 1Fault case 2Ground4.83.32.9Radial plate0.90.91.0Conductors1.00.91.2For modelling the PF coils,
the software ATP [12] was used. ATP does not allow including the coupling matrix of the
complete PF system. So the influence of the other PF coils was not considered.


## 4. Conclusion

The numerical investigations of the
transient behaviour of the ITER TF coils during fast discharge in normal case
and in fault case show a significant increase of the radial plate and conductor
insulation stress as a consequence of a malfunction of two adjacent FDUs
followed by an additional earth fault. The design and test voltages must be
chosen depending on the realistic fault scenarios identified by the ITER community.For the poloidal field coils, the PF coil
4 shows the highest terminal potential during fast discharge with the fault of
one FDU. As it displays overvoltages, the insulation coordination has to be
selected.Further activities have
to be done on central solenoid and its magnetic interactions with PF coils and
the plasma under fault conditions during fast discharge.



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*Source: 359210-2008-12-28.xml*