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Current-limiting characteristics of saturated iron-core fault

current limiters in VSC-HVDC systems based on electromagnetic

energy conversion mechanism

Botong LI

1

, Hanqing CUI

1

, Fangjie JING

1

, Bin LI

1

, Yichao LIU

1

Abstract A common method to examine the current-lim-

iting performance of saturated iron-core fault current lim-

iter (SI-FCL) in high-voltage direct-current transmission

based on voltage source converter (VSC-HVDC) systems

is to solve differential equations based on the system fault

transient characteristics and the equivalent inductance

calculation equation. This method analyzes the fault cur-

rent of the VSC-HVDC system in the time domain. How-

ever, it is computationally complex and cannot directly

reﬂect the relationship between parameters and the current-

limiting effect of the SI-FCL. In this paper, the relationship

between the magnetic ﬂux density and magnetic ﬁeld

energy of the SI-FCL is analyzed. The energy exchange

between the DC capacitor and the SI-FCL in the DC short

circuit fault process is analyzed. From the perspective of

electromagnetic energy conversion, the criterion for

determining the current-limiting ability of the SI-FCL in

the transient process is given based on the parameters of

the SI-FCL and VSC-HVDC system. On this basis, the

characteristics of the DC side fault current and the capac-

itor voltage when the SI-FCL has current-limiting ability

are examined. Based on the parameters of the SI-FCL and

VSC-HVDC system, a method for calculating the fault

current peak value and capacitor voltage drop time is

given. Finally, the accuracy of the analysis of the SI-FCL

in the VSC-HVDC system based on the electromagnetic

energy conversion mechanism is demonstrated through a

case study and simulation results of the VSC-HVDC sys-

tem with different SI-FCLs.

Keywords Electromagnetic energy, Saturated iron-core

fault current limiter (SI-FCL), High-voltage direct-current

transmission based on voltage source converter (VSC-

HVDC) system, Fault analysis

1 Introduction

High-voltage direct-current transmission based on volt-

age source converter (VSC-HVDC) systems are a new type

of DC transmission technology with fully controllable

power electronic devices, which is one key technology in

the research and development of the smart grid [1]. The

characteristics of the converter structure and the control

strategy of the VSC-HVDC system result in high peak

value and fast rising speed of the DC side fault current. A

large fault current can easily damage converter components

and protective devices [2,3], yet is difﬁcult to eliminate

immediately. Therefore, current-limiting technology is

introduced to suppress the DC side fault current, and slow

down the capacitor voltage drops. Meanwhile, this tech-

nology can effectively protect equipment [4], relax the

requirements on the DC breaker in terms of the speed and

CrossCheck date: 13 July 2018

Received: 10 November 2017 / Accepted: 13 July 2018

The Author(s) 2018

&Botong LI

libotong@tju.edu.cn

Hanqing CUI

cuihanqing12@163.com

Fangjie JING

jing_fangjie@126.com

Bin LI

binli@tju.edu.cn

Yichao LIU

liuyichao@tju.edu.cn

1

Key Laboratory of Smart Grid of Ministry of Education,

Tianjin University, Tianjin 300072, China

123

J. Mod. Power Syst. Clean Energy

https://doi.org/10.1007/s40565-018-0459-4

capacity, and reduce the difﬁculty of designing the pro-

tection scheme.

As current-limiting technology is critical to the VSC-

HVDC system, much related research has been performed

and results achieved. Reference [5] analyzed the inﬂuence

of the location of resistance-type superconductive fault

current limiters (SFCLs) on the DC side fault. They pointed

out that installing the resistance-type SFCL on the DC side

could successfully limit the DC fault current in the case of

a DC line fault. Reference [6] studied the current-limiting

effect of the resistance-type SFCL in the single-pole

ground fault and inter-pole fault at the DC side, and the

simulation results revealed that the resistance-type SFCL

has a signiﬁcant effect on the DC side fault. Reference [7]

examined application cases of the resistance-type SFCL in

a three-terminal VSC-HVDC system, and proved the lim-

iting effect of this STCL on various types of DC side fault

current. Reference [8] analyzed the performance of the

resistance-type SFCL in a multi-terminal VSC-HVDC

system connected with an offshore wind farm, and their

transient analysis for different perturbations proved that the

resistance-type SFCL can reduce the fault current for both

DC and AC faults. Reference [9] analyzed the coordination

of fault current limitation in an electric power grid with

multiple SFCLs to further limit fault current rise. Refer-

ence [10] proposed a new resistance-type SFCL that can

reduce the fault current to 50% of the original value within

2 ms. To limit the fault current, the resistance of the fault

circuit is increased by installing a resistance-type SFCL in

the VSC-HVDC system.

Some literatures have also proposed limiting the fault

current by increasing the circuit inductance [11]. Reference

[12] designed an inductive SFCL and the simulation results

revealed that this SFCL has a signiﬁcant effect on the DC

side fault. Reference [13] analyzed the impact of different

inductance values in the VSC-HVDC system on the DC

side fault current. Reference [14] offered an optimal design

scheme of installing the current-limiting inductor in a

multi-terminal VSC-HVDC system, and proved that this

scheme can effectively deaccelerate the rise in fault cur-

rent. Reference [15] designed a protective inductor for the

VSC-HVDC system of an offshore wind farm to limit the

current. The coordination relationships among converter,

DC circuit breaker and DC SFCL were studied and some

technical indicators were proposed to evaluate the perfor-

mance of the SFCL in [16].

Though inductors at the DC side can serve as current

limiters, large reactance will reduce the dynamic response

speed and can easily cause parallel resonance. In normal

operation, the equivalent inductance of saturated iron-core

fault current limiters (SI-FCL) is so small that it has no

effect on the VSC-HVDC system. When a DC side fault

occurs, the equivalent inductance increases rapidly to limit

the fault current. Thus, this approach is superior to instal-

ling inductors.

Some researches have also investigated the current-

limiting effect of the SI-FCL in the VSC-HVDC system.

Reference [11] proved the current-limiting effect of the SI-

FCL in the VSC-HVDC system. Reference [17] analyzed

the transient process of the DC side fault in the VSC-

HVDC system with the SI-FCL, and gave the relationship

between the peak value of the fault current and the SI-FCL

parameters. Reference [18] analyzed the leakage magnetic

ﬂux of the SI-SFCL and proposed the equivalent magnetic

circuit model considering the effect of leakage magnetic

ﬂux. Reference [19] introduced the application of the

active SI-SFCL in the VSC-HVDC system.

The analysis method in these papers solves the DC side

circuit differential equation with the equivalent inductance

of the SI-FCL in the DC side fault condition. However,

owing to the nonlinearity of the equivalent inductance, the

calculation involved is enormous and error prone. Other

researches mainly focused on the application of the SI-FCL

to the AC system, which is not applicable to the VSC-

HVDC system [20–22].

To analyze the current-limiting effect of the SI-FCL in

the VSC-HVDC system, this paper analyzes the energy

exchange between the SI-FCL and the DC capacitor after

the DC side fault occurs. The relationship between the DC

side fault current and the parameters of the SI-FCL is

studied, and a rapid and effective analytical method of the

current-limiting performance of the SI-FCL is proposed in

this paper.

2 Analysis of energy variation process of SI-FCL

with changing magnetic ﬂux density

The basic structure of the SI-FCL is shown in Fig. 1,

which includes one iron-core, one coil connected with the

DC side of the VSC-HVDC system, and one excitation coil

with a DC-biased source circuit. To reduce the run-time

Coil 2

Coil 1

DC source

N2N1

u2

i2

i1

Fig. 1 Structure of SI-FCL for VSC-HVDC system

Botong LI et al.

123

loss, the excitation coil can be made of superconductive

materials.

In Fig. 1, coil 1 is the excitation coil and coil 2 is the

coil connected with the DC side of the VSC-HVDC sys-

tem; N

1

is the turn number of coil 1 and N

2

is the turn

number of coil 2; i

1

is the current of coil 1 generally

excited by a constant current source; thus, i

1

is considered

as constant; i

2

is the current of the DC side of the VSC-

HVDC system.

The topology of the VSC-HVDC system with SI-FCLs

is shown in Fig. 2. To limit the DC side fault current, SI-

FCLs are installed in the converter station export. The

locations are also shown in Fig. 2.

As shown in Fig. 3, the magnetic ﬂux density–magnetic

ﬁeld intensity B–H curve of the iron core used in the SI-

SFCL is divided into the saturated region and the unsatu-

rated region signiﬁed by points K

1

and K

2

. When the VSC-

HVDC system works normally, the magnetic ﬂux density

of the iron core is in the positive saturated region, the

permeability of the iron core is small, and the equivalent

inductance of the SI-FCL is extremely small; thus, its

inﬂuence on the VSC-HVDC system is slight. When a DC

side short-circuit fault occurs, the greatly increased fault

current forces the magnetic ﬂux density of the core into the

non-saturated region. Subsequently, the equivalent induc-

tance of the SI-FCL rises rapidly with the increase in

permeability, and the fault current is limited by the SI-

FCL.

From the perspective of electromagnetic energy con-

servation, the electric energy is absorbed and converted

into magnetic ﬁeld energy by the SI-FCL after a DC side

fault. Thereafter, the magnetic ﬁeld energy draws the

magnetic ﬂux density from the positive saturated region

into the unsaturated region. During this process, the mag-

netic permeability of the iron core increases, and so does

the equivalent inductance of the SI-FCL. The energy

conversion of the SI-FCL is analyzed at the stage when the

magnetic ﬂux density moves from the positive saturated

region into the unsaturated region.

2.1 Theoretical analysis of energy change of SI-FCL

with varying magnetic ﬂux density

For simplicity, it is assumed that the cross-section areas

of the iron core denoted by Sare identical. The magnetic

ﬂux density is B. The relationship between the magnetic

ﬂux density Band the current of the DC side of the VSC-

HVDC system i

2

can be expressed by the following

equation according to the relevant knowledge of magnetic

circuit analysis [23]:

i2¼N1i1Hl

N2

ð1Þ

where lis the overall length of the core; His the magnetic

ﬁeld intensity. From the energy calculation formula, we

can see that the energy variation of the SI-FCL in the

period of t

1

–t

2

is:

DWL¼Zt2

t1

u2i2dt¼Sl ZB2

B1

B

l

dBSN1i1B2þSN1i1B1

ð2Þ

where B

1

is the magnetic ﬂux density at time t

1

;B

2

is the

magnetic ﬂux density at time t

2

;Sis the cross-section of the

iron core.

According to the B–Hcurve in Fig. 3, the energy vari-

ation of the SI-FCL after the fault is:

DWL¼Sl ZBk

Bm

B

l

dBSN1i1BkþSN1i1Bm

þSl ZBt

Bk

B

l

dBSN1i1BtþSN1i1Bk

ð3Þ

where B

m

is the magnetic ﬂux density before the fault and

in the saturated region; B

t

is the magnetic ﬂux density at a

time point after the fault and in the non-saturated region; B

k

is the magnetic ﬂux density of point K

1

.

Because the variation of the magnetic ﬂux density in the

saturated region is negligible, B

m

&B

k

. Hence, (3) can be

simpliﬁed as follow:

idc

ic

usa

usb

usc

Ls

Rs

ua

ub

uc

D1

iabc

/2R/2L

D3 D5

D6D4 D2

Udc

SI-FCL

SI-FCL

Fig. 2 Topology of VSC-HVDC system with SI-FCLs

-3000 -2000 -1000 0 1000 2000 3000

Magnetic field intensity H (m/A)

-3

-2

-1

0

1

2

3

Magnetic flux density B (T)

K1

K2 Positive saturated

region

Negative saturated

region

Unsaturated region

Fig. 3 B–Hcurve of iron core

Current-limiting characteristics of saturated iron-core fault current limiters in VSC-HVDC…

123

DWL¼Sl ZBt

Bk

B

l

dBSN1i1BtþSN1i1Bkð4Þ

According to the B–Hcurve of the iron core in Fig. 3,

we obtain the permeability–magnetic ﬂux density l

1

–

Bcurve shown in Fig. 4that depicts the l–Brelationship of

the core. It can be seen from the Fig. 4, that the l–Bcurve

is linear in the unsaturated region.

Therefore, the ﬁrst-order polynomial can be used to ﬁt

the relationship between the permeability land the mag-

netic ﬂux density Bwith high accuracy. The ﬁtting equa-

tion is:

l¼mB þn0BBk

mB þnBkB\0

ð5Þ

where the values of mand ndepend on the material of the

iron core.

The l

2

–Bcurve in Fig. 4is obtained from (5). Com-

pared with the l

1

–Bcurve, it can be seen that (5) can well

ﬁt the non-saturated region of the l–B curve, but is not

consistent with the saturated region.

According to (3) and (5), when the magnetic ﬂux density

Bchanges from the positive saturation region to the

unsaturated region, the relationship between the energy

variation DW

L

and the magnetic ﬂux density B

t

is:

DWL¼

Slm

k2ln lBk

klBtþm

Sl

kBkBt

ðÞ

þSN1i1BkBt

ðÞ0BtBk

Slm

k2ln lBk

klBtþm

Sl

kBkþBt

ðÞ

þSN1i1BkBt

ðÞBkBt\0

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

ð6Þ

where l

Bk

is the permeability when the magnetic ﬂux

density is B

k

.

When the magnetic ﬂux density is in the unsaturated

region, the relationship between the energy variation and

the magnetic ﬂux density can be reﬂected by (6). Thus, the

variation of magnetic ﬁeld energy can be calculated by (6)

only when the magnetic ﬂux density varies from the knee

point to the unsaturated region.

2.2 Simulation and veriﬁcation of SI-FCL energy

change with varying magnetic ﬂux density

To verify the correctness of (6), a simulation model of

the SI-FCL is established in PSpice. A single-phase

transformer model with saturated characteristics was built

to simulate the SI-FCL, and the parameters of the model

are listed in Table 1.

The excitation coil is connected to a ramp-type current

source that generates current to drive the magnetic ﬂux

density of the SI-FCL from the positive saturated region

across the unsaturated region into the negative saturated

region. Current i

2

, voltage u

2

, and the magnetic ﬂux density

are recorded constantly in the simulation. According to (2),

the energy variation of the SI-FCL from the simulation

start time to a certain time can be calculated, and then the

relationship between the energy variation and magnetic

ﬂux density can be obtained, which is shown as the DW

L1

–

Bcurve in Fig. 5with cross marks.

Based on (6), the relationship between the energy vari-

ation and the magnetic ﬂux density is calculated and shown

as the DW

L2

–B curve in Fig. 5with circle marks.

In Fig. 5, the two curves coincide well when the mag-

netic ﬂux density varies from the positive saturated region

near the point K

1

to the unsaturated region. Equation (6)

proves to be accurate in calculating the energy variation of

the SI-FCL with the magnetic ﬂux density ranging from the

positive saturated region to the unsaturated region.

3 Current-limiting performance analysis of SI-

FCL in VSC-HVDC system

The structure of the two-level VSC-HVDC system is

shown in Fig. 2. The DC side faults in the system can be

categorized as two-pole short-circuit fault, single-pole

short-circuit fault and disconnection fault, among which

-3 -2 -1 0 1 2 3

-0.005

0

0.005

0.010

0.015

0.020

μ

1−B

μ

2−B

Magnetic flux density B (T)

Permeability

Unsaturated

region

Positive

saturated region

Negative

saturated region

Fig. 4 l–Bcurve of iron core

Table 1 Parameters of SI-FCL simulation model

Parameter Value

Length of the core (l)6m

Cross-section areas (S) 1.1 m

2

Winding number of coil 1 (N

1

) 1500

Winding number of coil 2 (N

2

) 150

Current of the excitation coil (i

1

)30A

Magnetic ﬂux density of point K(B

k

) 2.3 T

Botong LI et al.

123

the two-pole short-circuit fault is the most harmful.

Therefore, this paper analyzes the energy exchange process

of the DC side in the case of a two-pole short-circuit fault.

On this basis, the current-limiting performance of the SI-

FCL is studied.

When a DC side fault occurs, the insulated-gate bipolar

transistors of the inverter will be blocked under self-pro-

tection. At this time, the capacitor voltage U

dc

is greater

than the peak value of the inter-phase voltage of the AC

side. Additionally, the AC side system cannot supply

power to the DC side system through the free-wheel

diodes. The DC side system can be regarded as a circuit

structure in which the capacitor discharges to the short-

circuit point. After the capacitor voltage drops below the

peak value of the inter-phase voltage of the AC side, the

free-wheel diodes will turn on, and will most likely be

damaged by the fault current [3]. Therefore, the SI-FCL

that provides ample time for the breaker to clear the fault

should limit the fault current and delay the voltage drop

time of the capacitor before the free-wheel diodes turns

on.

In the most serious scenario, in which the resistance of

the SI-FCL ignored, and a short-circuit fault occurs at the

junction of the DC rail and the power cable, the DC side

system is equivalent to the simpliﬁed circuit as shown in

Fig. 6until the free-wheel diodes turn on.

For the circuit shown in Fig. 6, the capacitor voltage at

initial time t

1

is denoted as U

dc1

, a time point after the fault

is denoted as t

2

, and the voltage at time t

2

is denoted as

U

dc2

. The energy variation of the capacitor in the period t

1

–

t

2

is:

DWc¼1

2CU2

dc1 1

2CU2

dc2 ð7Þ

As observed from the equivalent circuit in Fig. 6, the

electric ﬁeld energy variation of the capacitor is equal to

the magnetic ﬁeld energy variation of the SI-FCL, DW

c

=

DW

L

. According to the relationship between DW

L

and the

magnetic ﬂux density Bshown in Fig. 5, it can be seen that

the magnetic ﬂux density Bwill gradually alter from the

positive saturated region to the negative saturated region as

the energy variation increases.

As can be seen from the B–H curve shown in Fig. 3, the

magnetic ﬁeld intensity His reduced rapidly when the

magnetic ﬂux density Bchanges from the positive saturated

region to the unsaturated region. Meanwhile, the current i

2

increases rapidly, which can be seen from (1). When the

magnetic ﬂux density Bis in the unsaturated region, the

magnetic ﬁeld intensity Hconverts slowly, and the current

i

2

is basically stable at a ﬁxed value. When the magnetic

ﬂux density Benters the negative saturated region, the

magnetic ﬁeld intensity Hwill decrease rapidly again, the

current i

2

will dramatically increase, and the SI-FCL will

no longer limit the fault current of the DC side. Therefore,

to achieve the desired current-limiting effect, the energy

released by the capacitor should remain below what can

drive the magnetic ﬂux density Bfrom the positive satu-

rated region across the unsaturated region into the negative

saturated region.

Combining (6) and (7), when the magnetic ﬂux density

Bvaries from the positive saturated region into the unsat-

urated region, the relationship between the energy variation

of the SI-FCL and the energy variation of the capacitor is:

1

2CU2

dc1 1

2CU2

dc2

¼

Slm

k2ln lBk

klBtþm

Sl

kBkBt

ðÞ

þSN1i1BkBt

ðÞ0BtBk

Slm

k2ln lBk

klBtþm

Sl

kBkþBt

ðÞ

þSN1i1BkBt

ðÞBkBt\0

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

ð8Þ

It is noted that in (8) the magnetic ﬂux density B

t

is in

the unsaturated region. Once the magnetic ﬂux density

moves from the non-saturated region to the negative

saturated region, the l–Bcurve is no longer consistent with

(5), and (8) fails to reﬂect the relationship between the

energy variation of the SI-FCL and that of the capacitor.

The capacitor voltage that turns on the free-wheel diodes

is U

dc,min

. In the extreme case that the capacitor voltage

drops to U

dc,min

, the energy absorbed by the SI-FCL is not

-2 -1 0 1 2

Magnetic flux density B (T)

0

50

100

150

200

250

ΔWL2

Energy variation WL(kJ)

3-3

ΔWL1

Fig. 5 Relationship between energy variation and magnetic ﬂux

density

SI-FCL

Udc

Coil 2

Coil 1

Fig. 6 Equivalent circuit of DC side system after a two-pole short-

circuit fault

Current-limiting characteristics of saturated iron-core fault current limiters in VSC-HVDC…

123

sufﬁcient to draw the magnetic ﬂux density Binto the

negative saturated region, signiﬁed by point K

2

.

According to (2), for Bnot to reach the negative satu-

rated region, the condition is:

2SN1i1Bk1

2CU2

dc1 1

2CU2

dc;min ð9Þ

Let DWL;max ¼2SN1i1Bk, then DW

L,max

denotes the

maximum energy absorbed by the SI-FCL when the

magnetic ﬂux density Bstays out of the negative

saturated region. Let DWc;max ¼1=2CU2

dc1 1=2CU2

dc;min,

then DW

c,max

represents the energy variation of the

capacitor when the voltage drops to U

dc,min

, and (9) can

be expressed as:

DWL;max DWc;max ð10Þ

Equation (10) is a criterion used to determine whether

the SI-FCL has current-limiting ability all the time before

the capacitor voltage drops to U

dc,min

.

When the capacitance and voltage level of the VSC-

HVDC system are given, if the criterion is met, the SI-FCL

will have current-limiting ability all the time before the

capacitor voltage drops to U

dc,min

and the fault current i

2

will not surge. Otherwise, the SI-FCL will not always have

current-limiting ability and thus cannot achieve the desired

performance.

The SI-FCL is designed to limit the fault current to an

acceptable level. In addition, the SI-FCL applied in the

VSC-HVDC system is also used to extend the time before

the voltage drops to U

dc,min

. Therefore, in addition to

analyzing whether the SI-FCL has current-limiting ability

in the process when the capacitor voltage drops from the

initial state U

dc1

to U

dc,min

, the maximum fault current

(i

2,max

) and the duration of this process (T) also needs to be

obtained.

When the magnetic ﬂux density Bis in the unsaturated

region, the current i

2

can be calculated by (11) derived

from (1) and (5).

i2¼

N1i1Bl

kB þm

N2

0BBk

N1i1Bl

kB þm

N2

BkB\0

8

>

>

>

>

>

<

>

>

>

>

>

:

ð11Þ

Depending on whether (10) is met before the capacitor

voltage drops to U

dc,min

, the calculation of i

2,max

can be

divided into two cases:

Case 1: the SI-FCL has current-limiting ability con-

stantly during this voltage drop process. In this case, (8)is

used to calculate the magnetic ﬂux density B

min

when the

capacitor voltage is U

dc,min

. With B

min

substituted into (11),

the maximum fault current i

2,max,I

is known.

Case 2: the SI-FCL does not always have current-lim-

iting ability during this process. In this case, the magnetic

ﬂux density of the SI-FCL will enter the negative saturated

region, after which the fault current will surge before the

capacitor voltage drops to U

dc,min

. By substituting the

magnetic ﬂux density at point K

2

into (11), the maximum

fault current i

2,max,II

can be obtained.

In normal operation, the iron core of the SI-FCL works

in the saturated state so that the equivalent inductance is

small. To ensure that the core is saturated, N

1

i

1

must be

very large. When a DC side fault occurs, the magnetic ﬂux

density enters the unsaturated region. As shown in Fig. 3,

when the core is in the unsaturated state, the magnetic ﬁeld

intensity is small and changes little. Meanwhile, the vari-

ation range of the current i

2

is small, and it can be assumed

that i

2

is approximately constant.

According to the capacitor discharge formula, it is

known that:

i2¼CdUdc

dtð12Þ

Setting Tis the time for the capacitor voltage to fall to

U

dc,min

. Depending on whether the condition (10) can be

satisﬁed during the process before the capacitor voltage

drops to U

dc,min

, the calculation of Tis divided into two

cases:

Case 1: the SI-FCL has current-limiting ability all the

time during this process. In this case, the maximum fault

current is i

2,max,I

. If the fault current is set to i

2,max,I

, the

shortest time T

min,I

for capacitor voltage to fall from U

dc1

to U

dc,min

is:

Tmin;I¼CU

dc1 Udc;min

i2;max;I

ð13Þ

Case 2: the SI-FCL does not always have current-

limiting ability during this process. In this case, by

substituting the magnetic ﬂux density at point into (8),

the voltage U

dc,s

when the fault current begins to surge can

be obtained as:

Udcs ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

U2

dc1 4SN1i1Bk=C

qð14Þ

As the variation range of the current i

2

is narrow, it is

assumed that the current i

2

is i

2,max,II

. In addition, the

shortest time for capacitor voltage to fall from U

dc1

to U

dc,s

is:

Tmin;s¼CU

dc1 Udc;s

i2;max;II

ð15Þ

Assuming that the fault current remains i

2,max,II

, the

shortest time for capacitor voltage to fall from U

dc1

to

U

dc,min

is:

Botong LI et al.

123

T0

min;s¼CU

dc1 Udc;min

i2;max;II

ð16Þ

The time for capacitor voltage to fall from U

dc1

to

U

dc,min

is T

min,II

, and T

min,II

is greater than T

min,s

. After

the capacitor voltage drops to U

dc,s

, the DC side fault

current surges, and T

min,II

must be less than T0

min;s. Thus,

T

min,II

satisﬁes the condition as follow:

Tmin;s\Tmin;II\T0

min;sð17Þ

Through the analysis of the SI-FCL in the VSC-HVDC

system, the criterion is found for determining whether the

SI-FCL has current-limiting ability all the time before the

capacitor voltage drops to U

dc,min

. By further analyzing the

process in which the capacitor voltage drops from the

initial state U

dc1

to U

dc,min

, the current level to which the

fault current is limited, and the time for the capacitor

voltage to drop from U

dc1

to U

dc,min

are obtained.

4 Case study and simulation veriﬁcation

To verify the correctness of the theoretical analysis

above, a two-level VSC-HVDC system model of ±35 kV

(U

dc

= 70 kV) is built in PSpice. The model structure is

shown in Fig. 2, and the parameters of the model are listed

in Table 2.

The transmission capacity is 20 MW, and the rated

current i

2rated

is calculated as 0.3 kA. For the two-level

VSC-HVDC system, the free-wheel diodes turn on once

the capacitor voltage drops to the peak value of the inter-

phase voltage at the AC side, so U

dc,min

is calculated as

49.497 kV.

The capacitor voltage and fault current are shown in

Figs. 7and 8, when a two-pole short-circuit fault occurs 5

km away from the rectiﬁer at 0.1 s. It can be seen that the

capacitor voltage drops rapidly to zero without the SI-FCL

while the fault current rises rapidly and reaches 17.69 kA,

approximately 60 times higher than the rated current.

To verify the correctness of the theoretical analysis of

the SI-FCL, the SI-FCLs A and B are separately connected

in the VSC-HVDC system. The parameters of SI-FCL A

are listed in Table 3.

The current-limiting performance of SI-FCL A and B

can be analyzed by the theory presented in Section 3.

Assuming that the two-pole short-circuit fault occurs at

the junction of the DC rail and the power cable, the energy

variation of the capacitor DW

c,max

is 2.45910

6

J, and the

energy variation of the SI-FCL A DW

L,max

is 2.48910

6

J. It

can be seen from (10) that the SI-FCL has current-limiting

Table 2 Parameters of the two-level VSC-HVDC system model

Parameter Value

DC side voltage ±35 kV

RMS inter-phase voltage of the AC side 35 kV

Length of the DC side cable 10 km

Transmission capacity 20 MW

Resistance of the DC cable 0.15 X/km

Inductance of the DC cable 2.5 mH/km

Inductance of the AC side 0.01 H

DC capacitor 2.0 mF

0.08 0.10 0.12 0.14 0.16

-20

0

20

40

60

80

Time (s)

Capacitor voltage U

dc

(kV)

Fig. 7 Capacitor voltage of the VSC-HVDC system without an SI-

FCL

0.08 0.10 0.12 0.14 0.16

0

5

10

15

20

Time (s)

Fault current i

2

(kA)

Fig. 8 DC side fault current of VSC-HVDC system without an SI-

FCL

Table 3 Parameters of SI-FCL A

Parameter Value

Length of the core (l)4m

Cross-section areas (S) 0.6 m

2

Winding number of coil 1 (N

1

) 6000

Winding number of coil 2 (N

2

) 900

Current of the excitation coil (i

1

) 150 A

Magnetic ﬂux density of point K(B

k

) 2.3 T

Current-limiting characteristics of saturated iron-core fault current limiters in VSC-HVDC…

123

ability before the capacitor voltage drops to U

dc,min

. The

SI-FCL A meets the condition of case 1. Therefore, the

magnetic ﬂux density B

min

is about 2.238 T, the maximum

fault current i

2,max,I

is 1.017 kA, and T

min,I

is about 40.2

ms. The above theoretical calculations were made for the

most serious fault conditions (resistance of the DC side is

0). When a DC side fault occurs at other locations, the line

resistance is greater than 0, so the current-limiting effect of

the SI-FCL can be better.

When a two-pole short-circuit fault occurs at 5 km away

from the rectiﬁer at 0.1 s, the capacitor voltage and fault

current of the VSC-HVDC system with SI-FCL A are

shown in Figs. 9and 10. It can be seen that the time Tfor

capacitor voltage to fall from U

dc1

to U

dc,min

is extended to

40.8 ms by the SI-FCL A. In addition, the maximum fault

current is about 1.04 kA. The results obtained by the

simulation and theoretical analysis of SI-FCL A are basi-

cally the same.

The parameters of SI-FCL B are listed in Table 4.

Assuming that the two-pole short-circuit fault occurs at

the junction of the DC rail and the power cable in the VSC-

HVDC system with SI-FCL B. The energy variation of the

capacitor DW

c,max

is 2.45910

6

J, and the energy variation

of the SI-FCL B is 1.49910

6

J. It can be seen from (10)

that the SI-FCL will lose current-limiting ability before the

capacitor voltage drops to U

dc,min

. The SI-FCL B meets the

condition of case 2, and the current surge occurs before the

capacitor voltage drops to U

dc,min

. The magnetic ﬂux

density B

k

substituted into (11) and (14), the maximum

fault current before the current surges is about 1.389 kA,

and the voltage U

dc,s

is 57.271 kV. The shortest time T

min,s

for the capacitor voltage to fall from U

dc1

to U

dc,s

is 18.3

ms. Assuming that the fault current remains 1.389 kA, the

shortest time T0

min;sfor capacitor voltage to fall from U

dc1

to U

dc,min

is 29.5 ms. Therefore, the time for the capacitor

voltage to drop to U

dc,min

is longer than 18.3 ms and less

than 29.5 ms.

When a two-pole short-circuit fault occurs 5 km away

from the rectiﬁer at 0.1 s, the capacitor voltage and fault

0.12

0.08 0.10 0.14 0.16

30

40

50

60

70

80

Capacitor voltage U

dc

(kV)

Time (s)

T

Fig. 9 Capacitor voltage of VSC-HVDC system with SI-FCL A

Fault current i

2

(kA)

0.08 0.10 0.12 0.14 0.16

Time (s)

0

2

4

6

T

Fig. 10 DC side fault current of VSC-HVDC system with SI-FCL A

Table 4 Parameters of SI-FCL B

Parameter Value

Length of the core (l)4m

Cross-section areas (S) 0.6 m

2

Winding number of coil 1 (N

1

) 4500

Winding number of coil 2 (N

2

) 400

Current of the excitation coil (i

1

) 120 A

Magnetic ﬂux density of point K(B

k

) 2.3 T

0.08 0.10 0.12 0.14 0.16

-20

0

20

40

60

80

T

T

Capacitor voltage Udc (kV)

Time (s)

Fig. 11 Capacitor voltage of VSC-HVDC system with SI-FCL B

Fault current i2 (kA)

Time (s)

0.08 0.10 0.12 0.14 0.16

0

2

4

6

8

10

T

T

Fig. 12 DC side fault current of VSC-HVDC system with SI-FCL B

Botong LI et al.

123

current of the VSC-HVDC system with SI-FCL B are

shown in Figs. 11 and 12. It can be seen that the time Tfor

capacitor voltage to fall from U

dc1

to U

dc,min

is 23.8 ms,

and the maximum fault current is approximately 1.409 kA

before the current surges. When the fault current surges,

the capacitor voltage is 57.653 kV, and the time T0for

capacitor voltage to fall from U

dc1

to U

dc,s

is 18.7 ms. The

results obtained by the simulation and theoretical analysis

of SI-FCL B are basically the same.

The simulation of the VSC-HVDC system with different

SI-FCLs shows that the method analyzing the performance

of the SI-FCL based on the electromagnetic energy con-

version mechanism is correct. The method gives the cri-

terion for determining whether the SI-FCL has current-

limiting ability constantly before the capacitor voltage

drops to U

dc,min

. In addition, the maximum fault current

and the time for capacitor voltage to fall from U

dc1

to

U

dc,min

can also be calculated accurately by this method.

5 Conclusion

This paper has examined the performance of the SI-FCL

in a VSC-HVDC system. From the perspective of elec-

tromagnetic energy conversion, the relationship between

the electromagnetic energy and the magnetic ﬂux density

of the SI-FCL has been analyzed. Based on the study of the

electromagnetic energy conversion process after the DC

side fault, the relationship between the fault current and the

magnetic ﬂux density of the SI-FCL and that between the

time for the capacitor voltage dropping to U

dc,min

and the

DC side fault current have been given.

The method identifying the current-limiting character-

istic of the SI-FCL in the VSC-HVDC system is based on

the analysis of the energy conversion of the DC side fault.

On this basis, the time for capacitor voltage to drop to

U

dc,min

and the maximum fault current in the case of a

constantly current-limiting-capable SI-FCL can be

obtained if the parameters of the SI-FCL and the VSC-

HVDC system are given. This method avoids the complex

differential calculation of solving the DC side fault tran-

sient process in the time domain, and its accuracy has been

demonstrated through a case study and simulation.

Acknowledgements This research work was supported in part by the

National Key R&D Program of China (No. 2018YFB0904600), and

in part by the National Nature Science Foundation of China (NSFC)

under Grant 51677125.

Open Access This article is distributed under the terms of the

Creative Commons Attribution 4.0 International License (http://

creativecommons.org/licenses/by/4.0/), which permits unrestricted

use, distribution, and reproduction in any medium, provided you give

appropriate credit to the original author(s) and the source, provide a

link to the Creative Commons license, and indicate if changes were

made.

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Botong LI received the Ph.D. degree in electrical engineering and

automation from Tianjin University, Tianjin, China in 2010. He is an

associate professor in the School of Electrical and Information

Engineering, Tianjin University, Tianjin, China. His research interests

include power system protection and control of VSC-HVDC trans-

mission system.

Hanqing CUI is currently pursuing the M.S. degree in electrical

engineering and automation at Tianjin University, Tianjin, China. His

research interests include protection of VSC-HVDC.

Fangjie JING is currently pursuing the M.S. degree in electrical

engineering and automation at Tianjin University, Tianjin, China. His

research interests include protection of VSC-HVDC.

Bin LI received the Ph.D. degree in electrical engineering and

automation from Tianjin University, Tianjin, China in 2005. He is

currently a professor in the School of Electrical and Information

Engineering, Tianjin University, Tianjin, China. His research interests

include power system protection and control.

Yichao LIU is currently pursuing the M.S. degree in electrical

engineering and automation at Tianjin University, Tianjin, China. His

research interests include protection of VSC-HVDC.

Botong LI et al.

123