72 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO.

1, JANUARY 2014

Electromechanical Transient Modeling

of Modular Multilevel Converter Based
Multi-Terminal HVDC Systems
Sheng Liu, Zheng Xu, Member, IEEE, Wen Hua, Geng Tang, and Yinglin Xue, Student Member, IEEE

Abstract—This paper studies the techniques for modeling mod- MMC-MTDCs are regarded as grid shock absorbers. Al-
ular multilevel converter (MMC) based multi-terminal HVDC though there is no MMC-MTDC system in practice at present,
(MTDC) systems in the electromechanical transient mode. Firstly, several MMC-MTDC projects are being designed in China.
the mathematical model of the MMC and its corresponding
Therefore, studying the interaction between MMC-MTDC sys-
equivalent circuit are established, which are similar to those of the
two level converters. Then, a power flow calculation method for tems and AC systems has become an important task. However,
AC/DC systems containing MMC-MTDC systems is developed. before this can be done, a valid model of the MMC-MTDC
Two dynamic models for MMC-MTDC systems are developed in system needs to be developed at first.
the paper. One is the detailed model, taking into account of the Research in the field of MMC modeling and control has be-
AC side circuit, the inner controllers, the modulation strategies, come more and more active and some significant modeling ap-
the outer controllers and the MTDC circuit. The other is the proaches have been proposed [5]–[10]. A continuous model to
simplified model, which only reserves the outer controllers and
partial dynamics of the MTDC circuit based on a quantitative describe the operation of the MMC system was derived and
analysis of the detailed model’s dynamic processes, and it can be the principle allowing the control of the total energy and the
used in electromechanical transient simulation with a larger step balance between the upper and lower arm of a phase was pre-
size. Both the detailed and the simplified models are implemented sented in [5]. Furthermore, [6] evaluates four control methods
on PSS/E and compared with the accurate electromagnetic tran- for the MMC together with an experimental comparison. Refer-
sient models on PSCAD in a four terminal MMC-MTDC system; ence [7] develops a comprehensive mathematical model based
the result proves the validity of the developed models. Lastly,
a stability study of a modified New England 39-bus system is on the negative and positive sequence decomposition technique.
executed, and the result shows that the AC fault can be isolated For modeling the switching behavior of the power electronic
well in an MMC-MTDC asynchronously connected AC grid. devices in MMC-MTDC systems, simulation in electromag-
Index Terms—Electromagnetic transient, electromechanical
netic transient mode is a convenient method, which can study
transient, modeling, modular multilevel converter, multi-terminal the dynamic behavior of the MMC accurately [9]–[11]. Based
HVDC (MTDC), simplified model. on the electromagnetic transient simulation, different control
strategies for VSC-MTDCs [12]–[15] and applications of the
VSC-MTDCs in wind farms [16], [17] have been studied. More-
I. INTRODUCTION over, there was a breakthrough on the efficient modeling of
MMC in the electromagnetic transient simulation mode, which
W ITH the rapid development of high-voltage, high-power
and full-controlled power electronic devices,
voltage source converter based high voltage direct current
significantly increases the simulation speed and makes it pos-
sible to simulate an MMC-MTDC system even if the converters
contain hundreds of submodules (SM) [9].
(VSC-HVDC) systems have become an attractive innovation
However, electromagnetic transient simulation is still
for their immunity against commutation failure and relatively
not suitable for studying large-scale systems at present. To
easy extension to multi-terminal (MTDC) configurations. There
study the stability of large-scale AC/DC systems containing
has been a variety of topologies for VSC, and the modular mul-
MMC-MTDC systems, the electromechanical transient simula-
tilevel converter (MMC), due to its low switching-frequency,
tion is preferred. Therefore, the MMC-MTDC model suitable
low losses, small harmonic components and other benefits
for electromechanical transient simulation is necessary and
[1]–[4], is gaining increasing attention.
still needs further studies. Several steady-state VSC-MTDC
models for power flow analysis were proposed [18], [19]. And
Manuscript received September 07, 2012; revised January 23, 2013, April 11, a generalized dynamic VSC-MTDC model for power system
2013, and June 08, 2013; accepted August 06, 2013. Date of publication August
30, 2013; date of current version December 16, 2013. This work was supported
stability studies has been derived and implemented on MatDyn
by the National High Technology Research and Development Program of China [20], [21], which presented a comprehensive analysis for the
under Project 2012AA050205. Paper no. TPWRS-01035-2012. electromechanical transient modeling of the VSC-MTDC.
S. Liu, Z. Xu, G. Tang, and Y. Xue are with the Department of Elec- Furthermore, based on the generalized dynamic model, a dis-
trical Engineering, Zhejiang University, Hangzhou 310027, Zhejiang, China
(e-mail: lszju@zju.edu.cn; xuzheng007@zju.edu.cn; tanggeng7@zju.edu.cn; tributed DC voltage control strategy was proposed and tested
yinglinxue@gmail.com). [22]. The generalized dynamic model was derived based on the
W. Hua is with the Zhejiang Electric Power Corporation Research Institute, 2-level VSC. Though the MMC and the 2-level VSC are sim-
Hangzhou 310014, Zhejiang, China (e-mail: huawenpin@gmail.com).
Color versions of one or more of the figures in this paper are available online
ilar, there are still some differences among their modeling. In
at http://ieeexplore.ieee.org. the field of modeling the MMC for stability studies, a simplified
Digital Object Identifier 10.1109/TPWRS.2013.2278402 two-terminal MMC-HVDC dynamic model was developed in

0885-8950 © 2013 IEEE

LIU et al.: ELECTROMECHANICAL TRANSIENT MODELING OF MODULAR MULTILEVEL CONVERTER BASED MULTI-TERMINAL HVDC SYSTEMS 73

II. BASIC THEORY

The MMC and other VSCs are similar in some aspects.

The important structural differences for modeling between
the MMC and the well-known 2-level converter are that: the
former adopts arm series inductors and SM capacitors, while
the latter adopts phase inductors and two DC capacitors. And
these differences will be discussed in the following text.
This section introduces the basic theory of the MMC and
derives an equivalent circuit on the AC side, which is the basic
model for the following steady-state analysis and dynamic
modeling.
The basic structure of an MMC is shown in Fig. 1, where
is the series arm inductor and is the equivalent resistor of
each arm.
The following equations can be derived:
Fig. 1. Basic structure of an MMC.

(1)

where and are the upper and lower arm currents in

phase- , respectively, is the corresponding line
current, is the DC current injected into the DC side and
is the corresponding circulating current.
Take phase- as an example, the voltages of point and
point are

(2)

(3)

Fig. 2. Equivalent model of the MMC on the AC side.

The sum of (2) and (3) divided by 2 is

PSLF [8] and applied in modeling the Trans Bay Cable Project (4)
[23].
A proper MMC-MTDC electromechanical transient model Now introduce a new variable , which can be regarded as
will contribute to the stability study of large-scale AC/DC sys- a virtual potential [24]
tems. However, there is no standard MMC-MTDC model in
commercial electromechanical transient simulation software so
(5)
far, thus the aim of this paper is to develop a valid model on
PSS/E.
This paper is organized as follows. In Section II, the Equation (4) for three phases can be written as
basic theory about the MMC is described. In Section III, a
power flow calculation method based on PSS/E for systems
with MMC-MTDC is presented. In Section IV, a detailed
MMC-MTDC electromechanical transient model is given, (6)
focusing on the modeling of the SM capacitors and the MTDC
circuit. The limitation of the detailed model is also discussed.
Section V discusses how to increase the simulation step size
by simplifying the very fast dynamic processes in the detailed The equivalent circuit of the MMC on the AC side is shown
model, and thus proposes a simplified model. In Section VI, the in Fig. 2, where is the resistance of the converter transformer,
validity of both the detailed and the simplified models is tested is the leakage inductance of the converter transformer.
by their comparison with the accurate electromagnetic transient The above equivalent model of the MMC transforms the arm
MMC-MTDC models on PSCAD, and then a stability study series inductors into phase inductors theoretically and is the fun-
of a modified New England 39-bus system is implemented. damental for the following steady-state analysis and dynamic
Conclusions are presented in Section VII. modeling.
74 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 1, JANUARY 2014

Step 3): Estimating the variable losses of the constant power

converters . Note
that does not include the fixed loss . In this
paper is estimated by (7):
(7)
where “ ” denotes the element-by-element multi-
Fig. 3. AC side steady-state equivalent circuit of MMC. plication. and is the
coefficient of the variable loss.
Step 4): Calculating the DC powers of the
III. STEADY-STATE constant power nodes
. The
Power flow calculation is the basis of simulation. This section DC side powers of the constant power converters
gives a method to calculate power flow on PSS/E. Since there can be obtained
is no power flow model for MMC-MTDC on PSS/E, each con- by
verter is modeled as a generator in the power flow.
The AC side steady-state equivalent circuit of the MMC is (8)
shown as Fig. 3 where the positive conventions for active and
reactive power are from AC side into the converter. In Fig. 3, The injection powers of the joint buses in DC net-
is the voltage behind the transformer, is the voltage of the work are all zeros:
virtual potential point, is the equivalent phase resistor rep- (9)
resenting the losses of the converter circuit and denotes the
fixed losses of the converter station. Each AC bus of the con- Thus can be obtained:
verter can be regarded as a PQ or PV bus. A typical steady-state
control strategy for the MMC-MTDC system is that one con- (10)
verter regulates the DC voltage (slack converter) and the others
control the AC side active power (constant power converters). Step 5): Calculating the DC voltages of the constant power
Another control strategy is that some converters use DC voltage nodes . The
droop control (slack converters) and the others use active power equation set of the DC power flow can be repre-
control. The two control strategies are essentially the same for sented as
power flow calculation.
The method for power flow calculation takes the following
steps:
Step 1): Ordering the DC nodes. Assume that there are (11)
slack converters and joint nodes in an -ter-
minal MMC-MTDC system. The joint nodes refer Equation (12) can be derived from (11):
to the nodes which are not connected directly to
converters. Let the 1st to the th nodes be slack
converters, the th to the th nodes be con- (12)
stant power converters, and the th to the
th nodes be joint nodes. In fact, since the In (12), only is unknown, and (12) can be
injection powers of the joint nodes are all zero, solved by the Newton-Raphson method, where the
they can also be regarded as constant power nodes. initial values of can be set to the nominal DC
Therefore, the th to the th nodes are voltage. Therefore, can be obtained.
regarded as constant power nodes together. Step 6): Calculating the DC side powers of the slack con-
In the following steps, the subscript “ ” of each verters . From
variable means that the variable corresponds to the (11), can be derived:
th converter. The subscripts “slack”, “set”, “joint”
and “ ” of each vector mean that the vector corre-
sponds to the slack converters, the constant power (13)
converters, the joint nodes and the constant power
nodes, respectively. Step 7): Calculating the AC side active powers of the slack
Step 2): Determining known quantities. The DC volt- converters and their
ages of the slack converters variable losses .
, the AC side ac- The associated equations are
tive powers of the constant power converters (14)
, all converters’
AC side reactive powers (or AC voltage magni- where . Since can
tudes ) and the DC network admittance matrix be estimated, can be derived according to
can be given at first. (14).
LIU et al.: ELECTROMECHANICAL TRANSIENT MODELING OF MODULAR MULTILEVEL CONVERTER BASED MULTI-TERMINAL HVDC SYSTEMS 75

Fig. 4. AC side dynamic equivalent circuit of MMC.

Fig. 5. Structure of an MMC with inner current decoupled controllers and
modulations.
After are derived, can be
obtained:
model, assuming that the phase-locked loop is ideal and the
(15) d-axis is always aligned with the AC system voltage. Because
the phase-locked loop can be safely neglected in the study of
Step 8): Calculating the power flow of the whole AC/DC power system stability [20].
system. Since the AC side active powers of all con-
verters are known, the power flow of the whole A. AC Side Modeling
AC/DC system can be calculated by PSS/E, and all
AC buses’ voltages , active powers and reac- The AC side dynamic model of the MMC can be derived from
tive powers are obtained, as well as the AC side Fig. 2 and is shown as Fig. 4. In this dynamic model, the effects
injection current of each converter: of the transformer, the phase resistor and the phase reactor are
represented as an equivalent resistor and an equivalent reactor
(16) :

where “conj” means the complex conjugation. (20)

Step 9): Calculating the equivalent phase resistor : (21)
(17)
B. Controllers Modeling
Step 10): Calculating and :
Generally, controllers of an MMC-MTDC system consist of
(18) inner controllers, modulations and outer controllers.
(19) The structure of an MMC with inner controllers and modula-
tions is shown in Fig. 5.
During the process, both the DC network power flow and Outer controllers have five typical control modes [25]: 1) DC
the power flow of the whole AC/DC system are calculated only voltage droop control, 2) active power control, 3) DC voltage
once. control, 4) reactive power control and 5) AC voltage control.
It should be noted that the above method to represent the con- For an MMC-MTDC system, there are two typical DC
verter station losses may not be the best way, because it only voltage control strategies [22]. One is the DC voltage margin
takes account of fixed losses and variable losses proportional control strategy which is similar to the mode shift in LCC based
to the square of the converter current, and it may be more pre- HVDC systems. If some faults happen at the slack converter,
cise to represent the converter station losses by a generalized the DC voltage may lose control and fluctuate in a large range,
losses formula with the losses quadratically dependent on the and at this moment the DC voltage control would be taken
converter current [19]. However, the losses of the converter sta- over by another converter which adopts the DC voltage margin
tions are very small compared with their nominal powers, and it control. The other one is the DC voltage droop control strategy,
would cause quite limited impact on the result of the power flow where converters with DC voltage droop controllers will adjust
and the transient stability. Moreover, the method avoids solving their active powers simultaneously to control the DC voltage.
AC and DC equations simultaneously and is easy to realize in Section VI will test these two DC control strategies.
PSS/E. Therefore, it is a trade-off between accuracy and easy
implementation. C. DC Side Modeling
1) Equivalent Method of SM Capacitors: The SM capacitors
IV. MMC-MTDC DYNAMIC MODELING of the MMC are used to store energy and control DC voltage. In
This section introduces the modeling processes of the AC the electromechanical transient simulation, the outer dynamic
side, the controllers and the DC side of the MMC-MTDC behavior of the whole converter rather than that of the SMs is
system, during which the method of modeling the SM capaci- the focus. For this purpose, this paper tries to represent all the
tors and the MTDC circuit is presented. The limitation of the SM capacitors by an equivalent capacitor .
detailed model is also discussed. Assuming that an arm of the MMC contains SMs, then
This section builds a detailed model of the MMC-MTDC in per phase there will always be SMs inserted and the rest
system based on its fundamental frequency mathematical SMs are bypassed. However, no matter which state the
76 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 1, JANUARY 2014

capacitor and . is the current flowing from node

to node .
For joint node :

(27)

Note that if a converter is blocked, the corresponding node

will become a joint node.
For DC line between node and node :

(28)

(29)

(30)
(31)
Fig. 6. Equivalent DC side circuit of a single MMC.

If all the dynamics of the DC lines are considered, the MTDC

circuit will become a very complex RLC circuit, and it will be
SMs are in, voltages of all the 6 SMs in the six arms are
a very difficult task to implement a generalized MTDC model
almost same because of the capacitor voltage balancing control in a commercial software. Moreover, the time constants associ-
[26], [27]. And the above characteristic makes the overall effect, ated with the RLC dynamics of the DC lines are much shorter
produced by SMs in per phase, on the DC voltage similar
than those normally used by PSS/E, and the simulation might
to that of a single DC capacitor. Therefore, the value of be prone to numerical instability when a larger time step size is
can be calculated according to the principle that at any time the used.
energy stored in the equivalent capacitor equals that stored in
To enhance the generality of the model, this paper absorbs
the whole SM capacitors: the PSS/E’s modeling philosophy of pseudo steady-state HVDC
models, such as two-terminal models (CDC4, CDC6) and LCC-
(22) MTDC models (MTDC01, MTDC03). Because the response of
the DC current is so rapid in relation to the time scale of most
(23) electromechanical transient simulations in general, the above
DC transmission models are not concerned with the internal dy-
The left-hand side of (22) represents all the energy stored in namic behavior of DC lines, just as the internal transient be-
the six arms of an MMC, where is the capacitance of an havior of transformers and AC transmission lines are not con-
SM, is the voltage of an SM and is the DC voltage. cerned in the AC network model.
The value of can be derived by substituting (23) in (22): According to the similar idea and also considering that the
capacitances of the cables may contribute significantly to the
(24) overall system capacitance, this paper represents each DC line
as a simplified “ ” type RC circuit, which consists of a lumped
resistor and two lumped capacitors. After the simplification,
Through the above equivalent, the MMC and the 2-level con-
(29), the dynamics of the DC line inductor, is removed from
verter are now almost the same. And the DC side equivalent cir-
the DC side equations and the MTDC circuit becomes an RC
cuit of the MMC can be represented as Fig. 6, which can also be
network as shown in Fig. 7. In order to verify the validity of the
applied to the 2-level converter. In the DC circuit, the DC line
simplified MTDC circuit, Section VI specifically compares the
is represented as the “ ” type RLC circuit.
results of the MTDC circuit represented by the Bergeron model
2) DC Side Equations: DC side equations can be obtained
[28], [29] and the simplified “ ” type RC circuit.
from Fig. 6. For converter node :
To facilitate the representation of the simplified MTDC cir-
cuit, , the node lumped capacitance, is defined at first.
(25) can be regarded as an equivalent capacitance comprising all the
lumped capacitance of the DC lines connected to node , and it
(26) can be calculated as (32):

where , the total current injected from the converter into (32)
its DC side, consists of the current flowing to the equivalent
LIU et al.: ELECTROMECHANICAL TRANSIENT MODELING OF MODULAR MULTILEVEL CONVERTER BASED MULTI-TERMINAL HVDC SYSTEMS 77

system. However, it has the limitation that some of its processes

are very fast, which means it needs to be simulated with very
small step size. If the step size is too small, it will take consider-
able time to complete an electromechanical transient simulation
on a large scale AC/DC power system. Therefore, to increase the
simulation step size and to improve the simulation efficiency,
the model needs to be simplified.

V. SIMPLIFIED MODEL OF MMC-MTDC

The objective of this section is to derive a simplified model
which is suitable for the electromechanical transient step size
(usually the default is 10 ms). The section will discuss how to
complete the simplification by quantitative analyses of the dif-
ferent parts of the detailed model. The main idea for the simpli-
Fig. 7. Simplified MTDC circuit of MMC-MTDC. fication is to neglect the dynamic processes of very small time
constants, which means that the very fast dynamic processes are
Still assuming there are joint nodes in the -terminal MMC- considered to be completed instantly.
MTDC system. The DC side equation sets can be rewritten in To compare the time constants of different equations in a con-
matrix forms: venient way, all the components are per-unitized. The variables
with the superscript “*” are of per-unit values.
(33)
A. Modulation Simplification
(34)
Firstly, if the modulation block is ideal, the time delay can be
(35) eliminated as follows:
(37)
where “ ” denotes the element-by-element multipli-
cation. In (33), (38)

. B. Simplification of MMC With Inner Controllers

In (34),
Secondly, let us consider the dynamic processes of the inner
controllers and the MMC. In Fig. 5, the structures of the current
. In (35), is loops along axes d and q are symmetrical, so take the d axis
of dimension , and current loop for analysis. Then (39) can be derived from Fig. 5:
.
In summary, once the initial values of the state variables (39)
( and ) and the algebraic variables (
and so on) of the MTDC circuit are determined in
Now one should consider the range of and . Usu-
Section III, the differential algebraic equation set (33)–(35) can
ally the per-unit value of the MMC’s equivalent reactance
be solved at every simulation step by the following procedures.
ranges from 0.1 pu to 0.3 pu, and it can be converted to
In (33)–(35), and are state variables which need
through (40):
to be updated at every step. At the beginning of every simulation
step, and are already known, thereby can be (40)
derived from (36) at first:

(36) The equivalent resistor is used to indicate the variable

losses of the converter, which is usually about 1% of the rated
where and “./” denotes element-by- transmission power, so the per-unit value of the equivalent re-
element division. sistor is
Next and can be calculated from (35). After
and have been obtained, and can be (41)
calculated as (33) and (34), and then and for the
next simulation step can be obtained by the Modified Euler in-
The gain coefficient of the inner PI controller is usually
tegration method.
large enough for the system to achieve the ideal response, and
Now the detailed model, which consists of the AC side cir-
the following relationship will be established:
cuit, inner controllers, modulations, outer controllers and the
simplified MTDC circuit, has been built. (42)
D. Limitation of the Detailed Model (43)

In theory the detailed model can reflect the accurate funda- From (39), it can be observed that the time domain expression
mental frequency dynamic characteristic of an MMC-MTDC of transfer function will contain components with time
78 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 1, JANUARY 2014

constants and . These components are usually

too fast to simulate in the electromechanical transient simulation
with the default step size, so in order to adapt the MMC-MTDC
model to a relatively large simulation step, (39) can be simpli-
fied as follows:

(44)

Through the assumptions and simplifications above, the dy-

namics of the inner controller and the MMC are completely ne-
glected, which means that and are the
same.
Note that if (42) or (43) is false, will not track
instantaneously. As a complement to the simplified
model, a first order inertia block can be inserted between Fig. 8. Structure of the MMC-MTDC simplified model.
and , whose time constant can be adjusted
to achieve the desired response characteristics approximately.
s, and it is already a relatively large time constant. So
C. Discussion of DC Side Dynamics
(33) represent relatively slow dynamics. No matter how large
Thirdly, the DC side differential equations (33) will be dis- the time constants of node lumped capacitances are,
cussed. The per-unit value of the equivalent capacitor , (33) should be reserved in the simplified model.
which is also the time constant of the equivalent capacitor, can Lastly, (34), dynamics of the joint buses, should be discussed.
be calculated by (45): Generally the time constant of a DC line is determined by its
type, length, and . If , the time constant of the node
(45) lumped capacitance at a joint node, is larger than the simulation
step, its dynamics should be reserved. Otherwise, its dynamics
To calculate should be known at first. Before this, should be removed from (34) to avoid numerical instability, and
parameters of the SM capacitors are needed and can be derived the joint node will become a floating node which can be con-
from the following equation [27]: tracted to the DC resistive network.

D. MMC-MTDC Simplified Model

(46)
Now the simplified MMC-MTDC model, which removes the
dynamic processes of the inner controllers, the converters, the
modulations and only reserves the outer controllers and the par-
where is the rated active power of the converter, the
tial dynamics of the MTDC circuit, is obtained. The structure
power factor, the voltage modulation index, the number
of the MMC-MTDC simplified model, containing five kinds of
of SMs per arm, the fundamental frequency, the voltage
outer controllers, is shown in Fig. 8. Three and two
ripple of the capacitor, and the nominal DC voltage.
are selected separately according to the control mode before the
After substitution of (46) for in (24), the expression for
transformation block, for example, if the control mode of the
is
converter is the DC voltage droop control, will be .
The transformation is used to transform the variables from
(47) the dq reference frame to the common R-I reference frame of
the AC network. The equation is

When the converter is in the rated operating condition

(51)
(48)
(49) where is the angle from the R-axis to the d-axis.
After and are determined, can be calculated by the
Substitute (47), (48) and (49) in (45) to get transformation.
Let
(50) (52)
(53)
Equation (50) demonstrates that is independent of
and . Instead it is determined by and . Since we have assumed that the phase-locked loop works ide-
Values of these four quantities change in a relatively small range ally, the following equation can be obtained:
in general. Let us take a set of typical values,
rad/s, % and . Then we obtain (54)
LIU et al.: ELECTROMECHANICAL TRANSIENT MODELING OF MODULAR MULTILEVEL CONVERTER BASED MULTI-TERMINAL HVDC SYSTEMS 79

TABLE I
COMPARISON OF POWER FLOW

Fig. 9. Structure of the MMC-MTDC system.

The transformation block can be expressed as

(55)

Then can be obtained from (53) and (55).

The advantage of the simplified model is that a larger step size
can be adopted in the simulation. The basic assumption of the
simplified model is that the inner controllers and modulations
work ideally, that is to say, can track
instantaneously from the viewpoint of electromechanical tran-
sient simulation. However, it should be noted that the above as-
sumption may fail under some cases, such as the DC line faults
and the SM faults, and the simplified model will be unable to 2) Transient Simulation and Comparison: The step size on
accurately reflect the behavior of the converter. PSCAD is 20 s, and on PSS/E is 0.1 ms for the detailed model
and 10 ms for the simplified model. It should be noted that the
VI. MODEL VALIDATION AND SIMULATIONS numerical instability will occur if the step size is larger than 0.4
ms for the detailed model on PSS/E.
This section will verify the validity of the detailed model From Table IV, the time constant of the DC cable lumped
and the simplified model with a four-terminal MMC-MTDC capacitance can be calculated:
system and study the stability of a modified New England
39-bus system containing an MMC-MTDC.

A. Four-Terminal System (56)

A four-terminal MMC-MTDC system (Fig. 9) is simulated
to verify the validity of the detailed model and the simplified Since the four cables are connected to DC node D5, the time
model, and the data are included in the Appendix. The AC1 and constant of the node lumped capacitance at D5 is
AC3 are the sending ends, while AC2 and AC4 are the receiving (57)
ends. The steady-state control parameters are given in Table V.
This section will test the performance of the two control is a relatively large time constant even when com-
strategies. In strategy A, MMC2 controls the DC voltage and pared with the simulation step of the simplified model, so its
other MMCs control their active powers at the normal condi- dynamics should be reserved in the simplified model.
tion, when MMC2 loses its control ability owing to some faults, When the system runs to 2.0 s, a three-phase short-circuit
MMC1 will activate the DC voltage margin control. In strategy fault is applied to bus B2 and is cleared 0.1 s later. The
B, all MMCs use DC voltage droop controllers shown in Fig. 8. same faults under two different control strategies are simulated.
In both strategies, all MMCs control their reactive powers. Figs. 10 and 11 show the responses of the DC voltage, the active
Two accurate electromagnetic transient MMC-MTDC sys- power and the reactive power. The DC voltage is per-unitized
tems (EMTDC MMC-MTDC switching models) are modeled on the base value of 300 kV, and the positive conventions of
and simulated on PSCAD. In one case the Bergeron model is the active power and the reactive power are from AC side into
used to simulate the cable while in the other case the simpli- the converter.
fied “ ” type RC lumped circuit is used. Another two electro- Figs. 10 and 11 demonstrate that the dynamic responses of
mechanical transient MMC-MTDC models, the detailed model the four models coincide well. The results of the PSCAD models
and the simplified model, are implemented on PSS/E as user- contain some very high frequency oscillations, but it is clear that
written models. the effect of the high frequency oscillations on the transient sta-
1) Power Flow Comparison: The power flow results of the bility is limited. The results of the detailed model and the simpli-
MMC-MTDC system are shown in Table I. The power flow re- fied model on PSS/E are almost same. The differences between
sult of the detailed model is the same as that of the simplified one the four cases are mainly caused by the fact that high frequency
on PSS/E. Because the output quantities of PSCAD are always information is included in the electromagnetic transient model
changing over a very small range, only one decimal is reserved. on PSCAD, while the detailed model and the simplified model
In the steady state, there is little difference between the case on PSS/E only consider the fundamental frequency information.
of the Bergeron model and that of the RC lumped circuit; and a) Strategy A, Fault Applied at B2: In strategy A, MMC2
the power flow on PSCAD and PSS/E are almost the same. controls DC voltage and other MMCs control active powers
80 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 1, JANUARY 2014

Fig. 10. Dynamic responses of the MMC-MTDC in strategy A. (a) DC voltage Fig. 11. Dynamic responses of the MMC-MTDC in strategy B. (a) DC voltage
of MMC1. (b) DC voltage of MMC2. (c) Active power of MMC1. (d) Active of MMC1. (b) DC voltage of MMC2. (c) Active power of MMC1. (d) Active
power of MMC2. (e) Active power of MMC3. (f) Active power of MMC4. power of MMC2. (e) Active power of MMC3. (f) Active power of MMC4.
(g) Reactive power of MMC1. (h) Reactive power of MMC2. (i) Reactive power (g) Reactive power of MMC1. (h) Reactive power of MMC2. (i) Reactive power
of MMC3. (j) Reactive power of MMC4. of MMC3. (j) Reactive power of MMC4.

at the normal condition. When MMC2 loses its control ability tive and reactive power injected from MMC2 to AC2 becomes
owing to some faults, the DC voltage increase obviously, and zero, depriving MMC2 of the ability to control the DC voltage.
MMC1 will shift to the DC voltage margin control, which is Nevertheless, MMC1, MMC3 and MMC4 still control their ac-
similar to the mode shift in LCC based HVDC systems. The tive and reactive powers at the reference value. And the excess
DC voltage margin control logic of MMC1 is set as follows: active power in the DC network causes the DC voltage to rise.
1) If MMC1 is under the active power control and its DC At about 2.01 s, the DC voltage of MMC1 reaches 1.06 pu, thus
voltage is higher than 1.06 pu, its control mode will shift it activates the DC voltage margin control and decreases its ac-
to the DC voltage control and maintain at least 0.1 s; tive power quickly, making the DC voltage begin to decrease.
2) If MMC1 is under the DC voltage control and its DC After the fault is clear at 2.1 s, the active power of MMC2 re-
voltage is lower than 1.02 pu, its control mode will shift covers fast and the DC voltage continues to decrease. At about
to the active power control and maintain at least 0.1 s. 2.11 s, the DC voltage of MMC1 is lower than 1.02 pu, and
Fig. 10 shows the dynamic responses of the MMC-MTDC. MMC1 shifts to the active power control. The system becomes
During the fault, since B2 is three-phase short-circuited, the ac- stable quickly.
LIU et al.: ELECTROMECHANICAL TRANSIENT MODELING OF MODULAR MULTILEVEL CONVERTER BASED MULTI-TERMINAL HVDC SYSTEMS 81

TABLE II
MMC DATA

TABLE III
POWER FLOW SOLUTION

Fig. 12. Modified New England 39-bus system.

b) Strategy B, Fault Applied at B2: In strategy B, all con-

verters use DC voltage droop controllers whose structures are
shown in Fig. 8. Note that the coefficients of DC voltage devia-
tions of the droop controllers are relatively large on this
simulation, so as to make the controlling of their DC voltages
rather than the active powers the primary objective. During the
simulation the same fault as above is applied, and Fig. 11 shows
the dynamic responses of the MMC-MTDC. and the fault is cleared. Fig. 13 shows the simulation results. The
During the short-circuited fault, the active power and the re- simplified model of the MMC-MTDC is used in the simulation.
active power injected from MMC2 to AC2 become zero. The When bus 16 is three-phase short-circuited, the active power
excess power in the DC network makes the DC voltage rise of MMC1 becomes zero suddenly. Active powers of both
abruptly. Because other converters use DC voltage droop con- MMC3 and MMC4 also reduce since their AC voltages drop
trollers, MMC1 and MMC3 reduce their active powers injected quickly. The DC voltage starts to rise. With the DC voltage
to the converters, and MMC4 increases its active power ab- droop control, MMC4 decreases its active power further to
sorbed from the converter. They adjust their active powers to control the DC voltage. However, since the adjustment speed of
decrease the DC voltage simultaneously, and the DC voltage MMC4 is not fast enough, the DC voltage continues to rise. At
starts to decrease. After the fault is clear at 2.1 s, the active and 1.01 s, the DC voltage of MMC2 reaches 1.06 pu, and MMC2
reactive powers of MMC2 recover fast and the DC voltage de- activates the DC voltage margin control and decreases its active
creases more quickly. Then the system is adjusted to a stable power quickly, making the DC voltage begin to decrease. After
condition. During the transient process, the DC voltage fluctu- the fault is clear at 1.1 s, The AC voltages and the active and
ates in a relatively small range. reactive powers of all MMCs recover fast. At 1.11 s, the DC
voltage of MMC2 is below 1.02 pu, and MMC2 shifts to the
B. Modified New England 39-Bus System active power control. During the transient process, MMC1 and
To study the interaction between the AC system and the MMC3 can provide the reactive power support.
MMC-MTDC system, a stability study is performed on a In this contingency, all generators can keep the rotor angle
modified New England 39-bus system which consists of two stability. Even though MMC2 adjusts its active power during a
asynchronous AC grids (Fig. 12). In Fig. 12, four MMCs are short time, this adjustment has quite limited effects on its AC
configured at bus 16, 19, 21, and 22, respectively. The dash system. Therefore, the AC fault is isolated well.
lines refer to DC lines. They constitute a four-terminal 400-kV
MMC-MTDC system in the 345-kV AC system. VII. CONCLUSION
The data of the MMCs are shown in Table II. Table III sum- In this paper, two kinds of electromechanical transient MMC-
marizes the power flow solution of the MMC-MTDC system, MTDC models are presented. The main features of the detailed
showing that the converter transformers and the phase reac- model are: 1) the SM capacitors are represented as an equiva-
tors consume reactive powers and make an impact on reactive lent capacitor, which makes the MMC and the 2-level converter
power dispatch and voltage magnitudes in the network. The ac- almost the same; 2) the DC line is represented as the simplified
tive power losses of the converters are relatively small. “ ” type RC circuit, which facilitates the solving of the DC net-
To study the transient interaction between the AC system and work. However, the limitation of the detailed model is that some
the MMC-MTDC system, this paper analyzes a contingency as of its processes are too fast and require quite small step size
an example. When the system runs to 1.0 s, a three-phase short- to simulate. To overcome the disadvantage, a simplified model
circuit fault is applied to bus 16; 0.1 s later, AC line 16–17 trips suitable for the electromechanical transient step size is derived
82 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 29, NO. 1, JANUARY 2014

TABLE IV
PARAMETERS OF MMC-MTDC SYSTEM’S SIMULATION PLATFORM

TABLE V
CONTROL PARAMETERS OF STEADY-STATE CONTROL STRATEGY

Fig. 13. Dynamic responses of the modified New England 39-bus system.
(a) Machine angles (Reference machine is Gen39). (b) Machine angles (Refer-
ence machine is Gen34). (c) AC voltages. (d) DC voltages of MMC-MTDC.
(e) Active powers of MMC-MTDC. (f) Reactive powers of MMC-MTDC.

based on the quantitative analysis of the detailed model, and the

limitation of the simplified model is also discussed.
Both the models are implemented on PSS/E and compared
with the accurate electromagnetic transient MMC-MTDC
models on PSCAD. The consistency of their simulation results
validates the models. During the validation, the DC voltage
margin control strategy and the DC droop voltage control
strategy are tested. The results show that both strategies are
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