J. Beerten, D. Van Hertem, and R. Belmans, “VSC MTDC systems with a distributed DC
voltage control - a power flow approach,” Proc. IEEE PowerTech 2011, Trondheim, Norway,
Jun. 19–23, 2011, 6 pages.
Digital Object Identifier: 10.1109/PTC.2011.6019434
URL:
http://ieeexplore.ieee.org/xpl/articleDetails.jsp?tp=&arnumber=6019434
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1
VSC MTDC Systems with a Distributed DC
Voltage Control – A Power Flow Approach
Jef Beerten, Student Member, IEEE, Dirk Van Hertem, Senior Member, IEEE, and Ronnie Belmans, Fellow, IEEE
Abstract—In this paper, a power flow model is presented to
include a DC voltage droop control or distributed DC slack
bus in a Multi-terminal Voltage Source Converter High Voltage
Direct Current (VSC MTDC) grid. The available VSC MTDC
models are often based on the extension of existing point-to-point
connections and use a single DC slack bus that adapts its active
power injection to control the DC voltage. A distributed DC
voltage control has significant advantages over its concentrated
slack bus counterpart, since a numbers of converters can jointly
control the DC system voltage. After a fault, a voltage droop
controlled DC grid converges to a new working point, which
impacts the power flows in both the DC grid and the underlying
AC grids. Whereas current day research is focussing on the
dynamic behaviour of such a system, this paper introduces
a power flow model to study the steady-state change of the
combined AC/DC system as a result of faults and transients
in the DC grid. The model allows to incorporate DC grids in
a N-1 contingency analysis, thereby including the effects of a
distributed voltage control on the power flows in both the AC
and DC systems.
Index Terms—DC Grids, HVDC transmission, Load flow
analysis, Voltage control.
I. INTRODUCTION
A
T present, the power engineering world is facing enor-
mous challenges. Today’s power systems are operated
more closely to their limits, while system operators are more
and more faced with an increased public opposition to the
construction of new lines. The projected massive integration
of intermittent renewable energy sources also imposes major
technical challenges in terms of a secure grid operation.
The growing need for transmission capacity that accompanies
the recent challenges has lead to an increased interest in High
Voltage Direct Current (HVDC) systems in multi-terminal
(MTDC) configuration as an alternative to grid enforcements
based on AC technology. In Europe, suggestions are even
made to construct a whole new overlaying DC ‘supergrid’,
as DC technology has technical and economical advantages
over traditional AC transmission. The meshed DC supergrid
could thereby interconnect remotely located offshore wind
farms and connect them with various points in the existing
AC infrastructure, to provide a more reliable grid. Special
attention is given to HVDC based on Voltage Source Con-
verter (VSC) technology as it has significant advantages over
Jef Beerten is funded by a research grant from the Research Foundation –
Flanders (FWO).
The authors are with the Department of Electrical Engineering
(ESAT), Division ELECTA, Katholieke Universiteit Leuven,
Kasteelpark Arenberg 10, bus 2445, 3001 Leuven-Heverlee, Belgium
(e-mail: jef.beerten@esat.kuleuven.be, dirk.vanhertem@ieee.org,
ronnie.belmans@esat.kuleuven.be).
both Line-Commutated Converters (LCC) and traditional AC
technology. Contrary to LCC, the VSC technology can support
the AC grid due to a fast and independent control of active
and reactive power. The inherent technical characteristics of
VSCs give the technology much better prospects for a multi-
terminal operation than its LCC counterpart. Furthermore, a
connection to remote wind farms would only be realistic with
VSC technology. As European plans and studies to gradually
construct such an overlaying DC grid are taken more concrete
forms, the technical problems yet still unsolved are becoming
predominantly important in current day research on DC grids.
An outstanding research issues is a distributed control of
the system voltage in such a DC grid. A strong urge has risen
in recent years to thoroughly address this issue and to provide
robust control functions to increase the overall safety of the
DC grid [1]. An extension of the control principles of point-
to-point connections gives rise to one converter controlling
the DC voltage at its terminals while the other converters
control their active power injections in the AC grid. The DC
voltage controlling converter is often referred to as the ‘DC
slack’ converter since the control adapts the output power
automatically to compensate for the losses in the DC system.
Such a centralised DC slack converter has to react fast on DC
grid transients, such as e.g. the loss of converters or DC line.
The converter needs to be oversized and connected to a strong
AC system to cope with severe system transients. Furthermore,
an outage of this converter cannot be covered. The control
function of the slack bus converter can be duplicated to other
converters that would be functioning as a back-up slack to
take over the DC voltage control in case the primary DC
slack converter fails. While a duplication could increase the
overall reliability of the system, it does not disregard the main
disadvantages of a centralised DC slack converter [2]. Beside
the technical problems, the geographic location of such a DC
slack converter might be controversial, as one system operator
would have to cope with all problems on the DC grid.
Alternatively to a centralised DC slack approach, the DC
voltage control can be distributed over a number of converters
using a voltage droop control [3], [4]. In this way, a number
of converters contribute to the control of the DC system
voltage by adopting their active power input when the DC
system voltage alters as a result of changes in the operation
of the grid. Whereas previous research has primarily focused
on the transient behaviour of a VSC MTDC systems with a
distributed voltage control [5], [6], the steady-state behaviour
of distributed voltage control schemes and its integration
in power flow algorithms has mainly remained unaddressed
so far. However, when system studies and N-1 contingency
analyses of systems including DC grids need to be undertaken,
2
Filter
Phase
reactor
Converter
S
s
S
c
P
dc
Fig. 1. VSC HVDC converter station connected to a DC grid.
the steady-state power flows and the effect of this distributed
voltage control are of primary interest [7]. Contrary to the
situation in case of a system with a centralised slack bus, a
converter outage will give rise to a change of the power set-
points of all voltage controlling converters. This paper aims
to provide an answer to this mainly unaddressed research
question by presenting a model to fully integrate a distributed
voltage control in an AC/DC power flow program.
The paper elaborates further on the sequential AC/DC power
flow algorithm presented in [8] and discusses how the algo-
rithm has to be extended to include the effects of a distributed
DC voltage control on the power flows. Section II briefly
discusses the control principles of VSC HVDC converters that
are of interest with regard to power flow algorithms. Special
attention is paid to the proposed voltage droop control strategy.
Section III discusses the representation of the DC grid and
Section IV presents the inclusion of the distributed voltage
control in the AC/DC power flow algorithm. Section V shows
the effects of a converter outage on the power flows by means
of simulation results.
II. VSC CONVERTER CONTROL
As addressed in the previous section, a VSC HVDC con-
verter, shown in Fig. 1, can fully control its apparent power
injection by independently controlling its active and reactive
power, respectively P
s
and Q
s
. To achieve this, a vector-
control scheme with two inner current controllers indepen-
dently controls the converter currents in a rotating dq -reference
frame. Slower outer control loops allow to independently
control the active and reactive power injections by changing
the reference set-points of the inner current control loops.
From a power flow point-of-view, only the steady-state
behaviour of the outer control loops is of importance. A VSC
converter can exhibit two reactive power control functions:
1) Q-control: The reactive power Q
s
injected in the AC
grid is kept constant.
2) U-control: The converter adopts its reactive power in-
jection Q
s
to keep its AC bus voltage magnitude U
s
constant.
As far as active power is concerned, current-day two-
terminal VSC HVDC transmission schemes have two different
control functions for each converter:
1) P -control: The active power P
s
injected in the AC grid
is kept constant.
-
6
k
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
P
dc
U
dc
P
dc,0
U
dc,0
Fig. 2. Standard voltage droop characteristic.
2) U
dc
-control: The converter adapts its active power in-
jection P
dc
to control its DC bus voltage.
This two-terminal concept can be extended to a VSC HVDC
grid by having all-but-one converters fixing their active power
injection. The function of the remaining converter is to control
the voltage at the DC bus to its reference value, thereby
controlling the DC grid voltage profile by clamping the voltage
at one bus. From a power flow point of view, there is no
objection to assign a voltage control function as described
above to more than converter, thereby having multiple slack
buses in one DC grid. One of the problems, however, is that
this control may give rise to unwanted voltage and power
oscillations.
Alternatively, the DC voltage can be controlled by introduc-
ing a voltage droop, as shown in Fig. 2. By doing so, multiple
converters can assist the voltage control by adapting the power
according to their droop characteristic. The lower the value
of the voltage droop k, with k the opposite of the slope in
Fig. 2, the more the converter adapts its output power when
the voltage changes. The limit values of the voltage droop k
are 0 and ∞, at which the converter respectively controls the
DC voltage (DC slack) or the DC power (P -control) to their
reference values.
The DC voltage droop control shows many similarities to
the frequency droop used in AC systems. However, there are
peculiarities which make the DC voltage droop implementa-
tion less straight-forward than the frequency droop counter-
part. Whereas the frequency remains constant in an AC system,
the DC voltage differs from one bus to another in a DC system
as a result of the steady-state power flows in the DC grid. From
this perspective, voltage deviations at different locations in the
DC grid as such do not necessarily reflect transient system
conditions. They might as well be the result of the power
flows in the DC grid or they might be caused by changing
operating conditions. In light of this, [9] suggested to introduce
a Load Reference Set Point for each converter, representing the
target no-load DC voltage for all converters connected to the
grid. Similarly, the implementation put forward in this paper
uses a reference voltage U
dc,0
and power P
dc,0
for the droop-
controlled converters, as shown in Fig. 2. In this paper, these
reference values are set to the system’s working conditions
in steady-state, without a droop control being active. In this
way, the power flows are the same as those in a situation with
a centralised DC slack bus. After a contingency in the DC
3
grid, the system will start operating in a new operation point
determined by the voltage droop characteristics. The droop
control can be extended to include current and voltage limits. If
the dynamic characteristics of the droop control are of interest,
the basic droop characteristic from Fig. 2 has to be extended
to include a dead band around the set-point voltage value,
similar to the frequency droop implementation in governors.
Such a detailed representation is out of the scope of this study.
Instead, the basic droop characteristics from Fig. 2 are used
in the remainder of the paper.
III. DC GRID POWER FLOW
This section discusses the DC grid power flow equations
and the modifications to the algorithm presented in [8] when a
voltage droop is introduced. The power flow equations to solve
a DC grid show many similarities to those of a conventional
AC power flow and the methods used to solve the equations
are also applied here.
In an AC system, the active power through the lines is
mainly linked with the angle difference between different
buses, whereas the magnitude of the voltage at the different
buses is linked to the flow of reactive power. In DC grids, on
the contrary, the power flows are dictated by the differences
in voltage magnitude between the different DC buses. The
current injected at a DC node i can be written as the current
flowing to the other n − 1 nodes in the network:
I
dc
i
=
n
X
j=1
j6=i
Y
dc
ij
· (U
dc
i
− U
dc
j
), (1)
with Y
dc
ij
equal to 1/R
dc
ij
.
Combining all currents injected in an n bus DC network
results in
I
dc
=Y
dc
U
dc
, (2)
with the DC current vector I
dc
given by
I
dc
= [I
dc
1
, I
dc
2
. . . I
dc
k
| {z }
working converters
, 0 . . . 0
| {z }
outage
]
T
, (3)
with n − k zero elements due to converter outages and DC
buses without a power injection. The DC voltage vector is
given by U
dc
= [U
dc
1
, U
dc
2
. . . U
dc
n
]
T
and Y
dc
is the DC bus
matrix. When line outages are taken into account, this involves
altering the DC bus matrix accordingly. For a monopolar,
symmetrically grounded DC grid, the power injections become
P
dc
i
=2U
dc
i
I
dc
i
, ∀i ≤ k. (4)
A similar expression holds for a bipolar configuration when
in steady-state operation.
The current injections I
dc
are unknown prior to the DC
grid power flow. Instead, the active power injections P
dc
of
the converters in P -control are calculated using the results
of the AC power flow. However, the DC slack bus active
power injection P
dc
is not known prior to the DC power flow.
When there are converters with a distributed voltage control,
the active power injection P
dc
at these buses is not known
prior to the DC power flow either.
AC grids
power flow
Data input
per unit
conversion
& internal
numbering
DC slack bus
and/or DC
voltage droop
buses power
estimates
Converter
powers
and losses
DC grids
power flow
with DC
voltage
droop buses
DC slack
bus and/or
DC voltage
droop buses
loss iteration
Converged?Output
per unit
reconversion
& external
numbering
Update
DC slack
bus power
and/or DC
voltage droop
buses power
yes
no
Fig. 3. Flow chart of the sequential VSC AC/DC power flow algorithm with
a distributed voltage control.
Combining (2) and (4) and assuming a monopolar, symmet-
rically grounded DC grid, the known active power injections
can be written as
P
dc
i
=2 U
dc
i
n
X
j=1
j6=i
Y
dc
ij
· (U
dc
i
− U
dc
j
). (5)
The active power injected by the converters in voltage control
depends on the deviation of the bus voltage from its reference
value and is an unknown prior to the power flow. Using Fig.
2, the active power injection can be written as
P
dc
i
=P
dc,0
i
−
1
k
i
(U
dc
i
− U
dc,0
i
), (6)
with k
i
the voltage droop, defined as ∆U
dc
/∆P
dc
.
This system of non-linear equations can be solved with a
Newton-Raphson (NR) method, as shown in the next section.
IV. SEQUENTIAL POWER FLOW WITH DISTRIBUTED DC
VOLTAGE CONTROL
In this section, the DC grid model with a distributed voltage
control is introduced in the sequential AC/DC power flow
algorithm from [8]. Fig. 3 shows the flow chart of the power
flow algorithm when a distributed voltage control scheme
is included. The modifications to the sequential power flow
algorithm from [8] are printed in bold. In [8], one converter
was assigned the task of the DC slack converter, adapting its
power to control the DC system voltage. When a DC voltage
droop is applied by other converters, it is still possible, but not
necessary, to include a slack bus in the DC grid. From a control
point of view, this stems with a converter that controls its DC
4
bus voltage to the reference value using a PI controller, whilst
the converters with a DC voltage droop use a proportional
gain. In this section, the equations are written in their most
general form, with the first converter being a regular DC slack
converter.
The remainder part of the section only briefly discusses the
sequential algorithm as such. Instead, emphasis is put on the
modifications introduced by to the distributed voltage control.
A. AC grid power flow
Similarly to the converters under active power control,
the DC voltage controlling buses are introduced in the AC
power flow algorithm as either PQ or PV buses. When no
generator is present at the AC bus under consideration, a
dummy generator is added to the bus to deliver or absorb
the reactive power needed to keep up the AC system voltage.
When under constant reactive power control, the reactive part
of the load is changed to include the converter’s reactive power
injection.
All active power injections to the AC grid are included for
the AC power flow calculation by adapting the active part of
the load at the selected buses. With the active power injection
defined with respect to the AC system, the powers can be
modified at once. However, when a converter is under DC
voltage control, either a slack bus or droop-based control, the
active power injection in the AC grid is not known beforehand,
since it depends on the active power needed at the DC side to
control the DC voltage. The voltage droop buses can thus be
treated in a way similar to the slack converter with regards to
the AC system.
As a first estimate to initiate the overall iteration, the AC
active power injections of the droop controlled converters are
put equal to the negative of the DC power reference P
dc,0
,
thereby assuming that the DC voltage does not deviate from
the reference value U
dc,0
and neglecting the converter losses
P
loss
. Without lack of generality, we assume a n bus DC
system with the first converter as the DC slack bus and the
subsequent m−1 converters using a DC voltage droop control.
The next k − m buses are under constant active power control.
The remaining n − k buses do not have a connection to the
AC grid or are facing a converter outage. The active power
injection estimate of the m−1 converters under voltage control
is given by
P
(0)
s
i
=−P
dc,0
i
∀i: 2 ≤ i ≤ m. (7)
The active power delivered by the DC slack bus, if present, is
initiated as
P
(0)
s
1
=−
m
X
i=2
P
(0)
s
i
−
n
X
j=m+1
P
s
j
, (8)
with P
(0)
s
i
from (7) and P
s
j
defined prior to the power flow.
After calculating the AC power flow, all converter powers
and losses are calculated to obtain the DC grid’s injected
powers P
dc
for the k DC buses to which converters are
connected, disregarding the ones facing outages.
P
dc
i
= −P
c
i
− P
loss
i
, ∀i < k, (9)
with P
c
the active part of the complex power injected at the
converter side, shown in Fig. 1.
During the iteration, this expression does not hold for the
DC slack converter and the m−1 converters under DC voltage
droop for reasons explained above. However, the intermediate
AC grid states can be used for estimating the losses of these
converters to start the iteration that follows the DC power flow,
shown in Fig. 3.
B. DC grid power flow
With the DC power injections calculated as a result of the
AC power flow, a NR iteration, based on (5)–(6) is used to
calculate the DC grid’s power flow. For the converters under
constant power control, m + 1 to k, the DC power injection
P
dc
as defined by (5), is known as a result of the AC power
flow. For the converters under distributed voltage control, 2 to
m, the DC power injection set-points P
dc,0
are all known. A
modified active power vector P
0
dc
is introduced to group these
variables, hence
P
0
dc
= [P
dc
1
|{z}
slack
, P
dc,0
2
. . . P
dc,0
m
| {z }
voltage droop
, P
dc
m+1
. . . P
dc
k
| {z }
P −control
, 0 . . . 0
| {z }
outage
]
T
.
(10)
Using this modified power vector, the DC bus voltages are
calculated with a NR method:
U
dc
∂P
0
dc
∂U
dc
(j)
·
∆U
dc
U
dc
(j)
= ∆P
0
dc
(j)
. (11)
The equations and terms corresponding to the slack bus are
removed since its voltage is known prior to the DC network
power flow.
In this system of equations, the modified power mismatch
vector ∆P
0
dc
(j)
is given by
∆P
0(j)
dc
i
=
P
dc,0
i
− P
dc,0
i
(U
dc
(j)
) ∀i: 2 ≤ i ≤ m
P
(k)
dc
i
− P
dc
i
(U
dc
(j)
) ∀i: m < i ≤ k
−P
dc
i
(U
dc
(j)
) ∀i: k < i ≤ n
,
(12)
with P
dc,0
i
(U
dc
(j)
) given by
P
(j)
dc,0
i
= P
dc
i
(U
dc
(j)
) +
1
k
i
(U
(j)
dc
i
− U
dc,0
i
), (13)
and superscripts (j) and (k) respectively referring to the inner
NR iteration and the outer AC/DC power flow iteration. The
terms of the Jacobian are given by
U
dc
j
∂P
dc
i
∂U
dc
j
(j)
= −2U
(j)
dc
i
Y
dc
ij
U
(j)
dc
j
, (14)
U
dc
i
∂P
dc
i
∂U
dc
i
(j)
= P
(j)
dc
i
+ 2 U
(j)
dc
i
2
n
X
j=1
j6=i
Y
dc
ij
, (15)
U
dc
j
∂P
dc,0
i
∂U
dc
j
(j)
=
U
dc
j
∂P
dc
i
∂U
dc
j
(j)
, (16)
U
dc
i
∂P
dc,0
i
∂U
dc
i
(j)
=
U
dc
i
∂P
dc
i
∂U
dc
i
(j)
+
1
k
i
U
(j)
dc,i
. (17)
