The stability of a multiterminal direct current (MTDC) system is often influenced by its control strategy. To improve the stability of the MTDC system, the control strategy of the MTDC system must be appropriately adopted. This paper deals with estimating stability of a MTDC system based on the linecommutated converter based high voltage direct current (LCC HVDC) system with an inverter with constant extinction angle (CEA) control or a rectifier with constant ignition angle (CIA) control. In order to evaluate effects of two control strategies on stability, a MTDC system is tested on two conditions: initialization and changing DC power transfer. In order to compare the stability effects of the MTDC system according to each control strategy, a mathematical MTDC model is analyzed in frequency domain and time domain. In addition, Bode stability criterion and transient response are carried out to estimate its stability.
1. Introduction
A multiterminal direct current (MTDC) system had been researched since early 1963
[1]
. With the benefits of the MTDC system such as flexibility in energy exchange and economics, extending HVDC systems into the MTDC system became an technical issue
[2
,
3]
. A MTDC system can be classified into three categories according to composed threephase converters: a MTDC system based on the voltage source converter (VSCMTDC), a MTDC system based on the linecommutated converter (LCCMTDC), and a hybrid MTDC system based on both VSC and LCC converters. The MTDC system based on VSC HVDC technology is the most investigated owing to its advantages such as operation in AC grids with low shortcircuit capacity, fast transient response, and black start capability. However, the disadvantages of this configuration are the high investment cost, high power losses, and low power capability. Nowadays, the development of modular multilevel converters (MMC) is improving the power capability of VSC HVDC with reduced power losses. Thus, a MTDC system based on MMC HVDC will be more attractive with respect to reliability and fault handing. By comparison, a MTDC system based LCC HVDC technology still remains the dominant technology for bulk power transmission over long distances owing to low investment cost and much high maximum power transfer capacities. A hybrid MTDC system has combined advantages of both VSC and LCC HVDC technologies. It is estimated that each type of the MTDC system plays an important role in the HVDC network connecting offshore and onshore grids. That is, the VSCMTDC system can be a potential solution for integration offshore wind farms as well as connection offshore wind farms to the onshore grids
[4

9]
, the hybrid MTDC system with the unidirectional power transfer is suitable for connection offshore to onshore grids
[10
,
11]
, and the LCCMTDC system is often placed on onshore grid
[12

14]
. Especially, the LCCMTDC system has been developed in few projects, such as the QuebecNew England HVDC, the NorthEast Agra link, and the Northwest China
[13

15]
.
Meanwhile, control strategy plays an important role in the LCCMTDC system due to interaction between AC and DC systems. In 1980, several types of control strategies of LCCMTDC systems were demonstrated and compared with different operating conditions but they didn’t deal with clearly the preferable control strategy for the LCCMTDC system
[16
,
17]
. Therefore, the main purpose of this study is to propose the suitable control strategy for the LCCMTDC system.
Power system stability is the property of an electric power system that enables it to remain in a state of operating equilibrium under normal operating conditions and to regain an acceptable state of equilibrium after being subjected to a disturbance
[18]
. As a result, in this study, initialization and power change conditions are applied to estimate stability of the LCCMTDC system according to control strategies. LCCMTDC system stability is estimated by using time domain analysis or frequency domain analysis of its mathematical model. Several methods were used to estimate stability of the LCCMTDC system. Stability of the LCCMTDC system was estimated by analysis its mathematical model in frequency domain
[19
,
20]
. In 1980, the LCCMTDC system with 3terminals and 7terminals were modeled in a transient stability program, but only one type of control strategy was applied to the LCCMTDC system
[21]
. The stability of the LCCMTDC system was also estimated by digital simulation
[22]
or eigenvalues analysis
[23]
. This paper analyzes stability of the LCCMTDC system with its mathematical model.
In this paper, firstly, a MTDC system based on LCCHVDC with 3terminals shown in
Fig. 1
is modeled in PSCAD/EMTDC under two types of control strategies: CEAtype control based on constant extinction angle (CEA) control on inverter 1 as a receiving converter and CIAtype control based on constant ignition angle (CIA) control on a rectifier as a sending converter. Secondly, a mathematical MTDC model is developed based on the impulse model. MATLAB/SIMULINK program is used for modeling mathematical MTDC system with different types of control strategies. Bode stability criterion is used to estimate the stability of the MTDC system in frequency domain. The bandwidth of the closeloop system is also carried out to evaluate the transient response of the system in time domain. Finally, the MTDC system is evaluated on two operation conditions such as initialization and changing DC power transfer.
MTDC System
2. MTDC Modeling
This paper considers a MTDC system with 3terminals connected in parallel as shown in
Fig. 1
as mentioned before. The AC sides of the MTDC system include AC supply networks, AC filters, and transformers on each AC side of converters. AC supply networks are modeled by voltage sources connected in series with a thevenin equivalent impedance of the AC system
[24]
. The strength of the AC system illustrated by the short circuit ratio (SCR) has a significant impact on AC/DC system interactions. AC filters such as single tuned and high pass filters are used to eliminate harmonic voltage and current generated by converters. Additionally, they also provide the reactive power to converters
[25]
. Transformers are added on both a rectifier and inverter sides for the purpose of providing the appropriate voltage level to valve bridges.
The DC side of the MTDC system is composed of DC smoothing reactors and DC cables with the length of 100 km for each line. The cables are based on the Frequency Dependent (Phase) model in PSCAD/EMTDC to study the transient behavior of the cable
[26]
.
Converters are modeled by the 6pulse Graetz converter bridge in PSCAD / EMTDC. Thyristor valves are used as an ideal switch parallel with a snubber circuit for each thyristor.
3. Control Strategy of MTDC System
Firing angle (α), extinction angle (γ), and direct current at each converter are used as input signals to control systems such as constant current (CC), CEA or CIA. Nevertheless, each converter has different type of the control mode depending on control strategy.
Fig. 2
shows the VI characteristics of each control strategy. In CEAtype control, the rectifier and inverter 2 have CC control. Besides, inverter 1 has CEA and CC control. Under normal condition, the rectifier and inverter 2 operate at the current control mode and inverter 1 operates at the voltage control mode. In this type, inverter 1 is the voltage setting terminal. By comparison, in CIAtype control, the rectifier has CIA and CC control while inverters 1 and 2 have CC control. At the normal condition, the rectifier is on the CIA control mode to determine system voltage.
VI characteristics with different control strategy
In addition, significantly decreasing voltage at the inverter terminal may lead to commutation failures and cause the collapse of DC voltage. Therefore, voltage dependent current order limit (VDCOL) is used to reduce the allowable maximum direct current when voltage drops below the predetermined value
[18]
. VDCOL is not shown in
Fig. 2
for the sake of simplicity.
4. Mathematical MTDC Model of Control System
In this section, a MTDC system considering two types of control strategies is modeled mathematically to estimate its stability. The block diagrams of the closeloop control system in CEAtype control and CIAtype control are shown in
Figs. 3
and
4
, respectively. The difference between CEAtype control and CIAtype control is the feedback of the closeloop control system. That is, CEAtype control has rectifier current feedback to remain constant current under the normal condition and CIAtype control has rectifier voltage feedback to remain constant voltage. Both types of control strategies are implemented by proportional  integral (PI) control which is represented by transfer function
G1
in
Figs. 3
and
4
.
Block diagram of MTDC system in CEAtype control
Block diagram of MTDC system in CIAtype control
 4.1 Rectifier model
Output voltage of the rectifier is influenced by change in firing angle and voltage drop due to commutation inductance of AC source. In
[27]
, the rectifier was considered as an ideal sampler. If a small variation Δ
α
of firing angle is considered with constant output current, the increment in output voltage will be consisted of two pulses in each firing period.
Fig. 5 (a)
shows the positive and negative Δ
α
.
Incremental output voltage
The change of output voltage with Δ
α
is determined by area Δ
A
_{1}
and Δ
A
_{2}
as shown in
Fig. 5 (a)
. Area Δ
A
_{1}
equals to area Δ
A
_{2}
because of constant output current. These areas are given by:
where
α
^{0}
is the normal firing angle,
T
_{s}
is the sampling period, and
V
_{d0}
is the ideal noload voltage.
If a small variation Δ
I_{d}
is considered with constant firing angle, the change of output voltage is represented by area Δ
A
in
Fig. 5 (b)
.
where
L_{C}
is commutating inductance per phase.
The block diagram of the rectifier model is shown in
Fig. 3.
The increment voltage Δ
V_{d}
is given by:
The transfer function
G
2(
s
) given by Eq. (5) shows the output voltage of the rectifier with Δα. It means that the incremental rectifier voltage is made of two identical impulses which shown in
Fig. 5(a)
. The first impulse occurred at instant firing angle, and the second impulse is delayed by commutation time,
. The transfer function
G
3(
s
) represents the incremental output voltage with Δ
I_{d}
. This period can be considered as a single pulse delayed by commutation time,
T_{μ}
with respect to the instant firing angle.
 4.2 Inverter model
As shown in
Fig. 3
, the block diagram of inverters 1 and 2 is different owing to the control mode under the steadystate condition. That is, inverter 1 is operated at the CEA mode and inverter 2 is operated at the CC mode. To deal with the CEA control mode of the inverter, this paper uses its lumped block diagram proposed in
[27]
. Beside, inverter 2 has the CC control mode, and its block diagram is similar to the rectifier model with the transfer function
G'
2 and
G'
3 represented by following equations:
where
β = π − α
^{0}
is ignition advance angle.
 4.3 DC line
The DC cable line can be replaced by the T or Π model, which includes series and shunt resistance, inductance, and capacitance of the line. However, for a straightforward 3 terminals MTDC system, the DC cable line is represented by the series element only
[19]
. According to
[21]
, the shunt capacitance of the line can be neglected due to its insignificant effect on transient stability. As results, in this paper, the DC line is replaced by a lumpedparameter RL element which shown in
Fig. 6.
Smoothing inductance is also included in the line parameter. By using Kirchhoff’s circuit laws to solve the circuit shown in
Fig. 6
, the circuit equation matrix of the DC cable line can be obtained by:
DC line Tsection
Therefore, the admittance matrix in term of the Laplace transfer function shown in
Figs. 3
and
4
is given by the following:
where
 4.4 The transfer function of control system
Block diagrams of the control system of the MTDC system shown in
Figs. 3
and
4
are built in MATLAB / SIMULINK. The steadyspace models of
G2
and
G'2
include the internal time delays that cannot be converted to transfer function form in Simulink models. As a consequence, the “
pade
” function that represents a Padé approximant is applied to approximate these transfer function. The openloop transfer function of the MTDC system is carried out by “
linmod
” function.
5. Simulation Results
 5.1 Calculated frequency response for MTDC system
Fig. 7
shows the frequency response of the MTDC system in term of the Bode diagram. Gain margin and phase margin of the openloop control system also are shown in the figure. In order to maintain the stable close loop response, PI parameter was chosen to satisfy as in (11) and (12)
[28]
.
Bode diagram for openloop control system
As shown in
Fig. 7
, the gain margins of the openloop MTDC system in CEAtype control and CIAtype control are 28.5 dB and infinite, respectively. The phase margin is 45° for both types of control. Obviously, the system with large gain margin is more stable than one with smaller gain margin by same phase margin
[29]
. As a result, the MTDC system under CIAtype control is more stable than that under CEAtype control. Additionally, the bandwidth of the control system gives indication on the transientresponse properties in the time domain with larger bandwidths corresponds to faster rise time. In this paper, the bandwidths of the MTDC system under CEAtype control and CIAtype control are 16.509 Hz and 573 Hz, respectively. Thus, the MTDC system under CIAtype control has a faster rise time than CEAtype control.
 5.2 Operation characteristic of MTDC system
In order to estimate the stability of the MTDC system, the simulation is carried out on initialization and changing DC power transfer conditions.
Fig. 8
shows the system characteristics under the initialization condition with two types of control strategies. In this figure, direct current, direct voltage, and firing angle are illustrated. The current orders at inverters 1 and 2 are 0.7 and 0.3 pu, respectively. The transient response of direct current at the initialization is shown in
Figs. 8 (a)
and
(b)
, where it can be seen that the rise time of direct current in CEAtype control with 0.48 s is about fourth times higher than that in CIAtype control. This trend is similar to the direct voltage and firing angle at the converter under two types of control strategies. Furthermore, the fluctuations of direct voltage and current under initialization process in CEAtype control are larger than CIAtype control. In other words, the MTDC system under CIAtype control is more stable than CEAtype control.
Initialization of MTDC system
The characteristics of the MTDC system under changing DC power transfer are shown in
Fig. 9
. DC power transfer from rectifier to inverters 1 and 2 is changed by adjusting current order to each converter.
Fig. 9
shows that transient response of the MTDC system under CIAtype control is faster than CEAtype control, and the MTDC system under CIAtype control is more stable than CEAtype control. Additionally, when the current order to inverters 1 and 2 is adjusted at 2 s, the transient response of current in CIAtype control is nearly immediate. By comparison, the transient response of direct current in CEAtype control is slower than CIAtype control. Besides, after changing DC power transfer, fluctuation of direct voltage in CEAtype control is bigger than CIAtype control. In CEAtype control, direct voltage reached to 1.1 pu at the new stable operating point while direct voltage at converter was remained unchanged at 1 pu.
Changing DC power transfer
6. Conclusion
The VSCMTDC system and the hybrid MTDC system become more attractive. However, with their lower power capability and higher investment cost compared to the LCCMTDC system, it is anticipated that the LCCMTDC still plays an important position in the future grids, such as integration onshore grids over long distances and bulk power transmission. This paper dealt with the proper control strategy for the LCCMTDC system with high stability and fast response.
Two types of control strategies for the LCCMTDC system were analyzed using a LCCMTDC system modeled in PSCAD / EMTDC program as well as MATLAB / SIMULINK under different operation conditions. The simulations results showed that the LCCMTDC system based on CIAtype control is more stable and has faster response than CEAtype control. It can be explained as follows;

a) Since inverter 1 plays a role of voltage setting terminal in CEAtype control, it cannot control the current at its terminal in case of system disturbance or load change. However, in CIAtype control, all inverters can control current at its terminals.

b) In addition, system voltage can be controlled by a rectifier with largest power at its terminal. Thus, the LCCMTDC system based on CIAtype control is more stable and faster response than other ones.
In case of a LCCMTDC system with multiple rectifiers, additionally, we suggest that a rectifier terminal with the largest power should be the voltage setting terminal to operate stably the LCCMTDC system.
BIO
ThaiThanh Nguyen received his B.S degree in Electrical Engineering from Hanoi University of Science and Technology, Vietnam, in 2013. Currently, he is a combined Master and Ph. D. student in the Department of Electrical Engineering, Incheon National University, Korea. His research interests include power system analysis & modeling, FACTS, and HVDC.
HoIk Son received his B.S degree in Electrical Engineering from Incheon National University, Korea, in 2012. Currently, he is a combined Master and Ph. D. student in the Department of Electrical Engineering, Incheon National University, Korea. His research interests include power system analysis & modeling, FACTS, and HVDC.
HakMan Kim received his first Ph. D. degree in Electrical Engineering from Sungkyunkwan University, Korea in 1998 and received his second Ph. D. degree in Information Sciences from Tohoku University, Japan, in 2011, respectively. He worked for Korea Electrotechnology Research Institute (KERI), Korea from Oct. 1996 to Feb. 2008. Currently, he is a professor in the Department of Electrical Engineering and also serves as the Vice Dean of College of Engineering, Incheon National University, Korea. His research interests include power system analysis & modeling, HVDC, FACTS, Microgrid, and LVDC.
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