1
Abstract—This paper addresses the problem posed by complex,
nonlinear controllers for power system load flows employing
multi-terminal voltage source converter (VSC) HVDC systems.
More realistic dc grid control strategies can thus be carefully
considered in power flow analysis of ac/dc grids. Power flow
methods for multi-terminal VSC-HVDC (MTDC) systems are
analysed for different types of dc voltage control techniques and
the weaknesses of present methods are addressed. As distributed
voltage control is likely to be adopted by practical dc grids, a new
generalized algorithm is proposed to solve the power flow
problems with various non-linear voltage droops, and the method
to incorporate this algorithm with ac power flow models is also
developed. With five sets of voltage characteristics implemented,
the proposed scheme is applied to a five-terminal test system and
shows satisfactory performance. For a range of wind power
variations and converter outages, post-contingency behaviours of
the system under the five control scenarios are examined. The
impact of these controls on the power flow solutions is assessed.
Index Terms—Multi-terminal, Voltage source converter (VSC)
HVDC, MTDC, power flow, droop control, dc grid
I. INTRODUCTION
S voltage source converter (VSC) HVDC technology
rapidly evolves, both the amount and the scale of VSC
HVDC projects continue to rise globally [1-4]. VSC HVDC has
become the most feasible solution to the integration of remotely
located large wind farms, mainly due to its small footprint and
its ability to support very weak ac systems. Advantages offered
by a multi-terminal topology include enhanced reliability of the
dc system, flexibility of power dispatch control, reduction of
intermittence of renewable energy, and efficient utilization of
converters and cables. Multiple plans for dc grids have been
approved or are under consideration [1, 3].
DC voltage is the essential factor that indicates the power
balance and the stability of an MTDC system. Compared with a
conventional ac grid of large inertia, the transient response of a
capacitance-based dc grid is extremely fast. Therefore it is
unrealistic to rely on telecommunications for MTDC control.
Local control per terminal, at most modified by a slow central
coordinating control, seems to be the favoured control option.
Such dc voltage control techniques can be categorised into
Manuscript submitted for review June 19, 2013. This work was supported
by National Grid and The University of Manchester.
W. Wang and M. Barnes are with the School of Electrical and Electronic
Engineering, The University of Manchester, Manchester, M13 9PL, U.K.
(email: wenyuan.wang@student.manchester.ac.uk; mike.barnes@manchester.
ac.uk).
Fig. 1. Basic voltage characteristics for MTDC control.
centralized dc slack bus control, voltage margin control and
distributed voltage droop control.
One terminal may be selected as the dc slack bus to
compensate the power imbalance of the overall dc grid.
However, the slack terminal may have to be over-rated and
connected to a very strong ac network, and the loss of this
converter may lead to the instability of the whole dc grid.
Voltage margin control [5] can be considered as an improved
constant dc voltage control with multiple stages of alternative
slack buses. The role of voltage regulation can be taken over by
another terminal once a slack bus is offline or reaches its limit.
However, several disadvantages remain for voltage margin
control especially in a large MTDC system.
The reliability of the dc grid can be significantly enhanced by
droop control since multiple converters can simultaneously
contribute to the dc voltage stability. Various types of dc
voltage droop characteristics, including voltage-power (V-P)
droop [6, 7] , voltage-current (V-I) droop [8] and voltage droop
with different dead-bands and limits [9], have been proposed
for MTDC control. However, prior art has been focusing on the
dynamic behaviour of voltage droop, while the steady-state
aspect of various droop lines has not been addressed in detail.
Droop control is likely to be employed in the interests of
MTDC dynamics and stability, however the use of such
quasi-steady-state droop characteristics could significantly
increase the complexity of power flows. Under such situations,
the operating point and power sharing of the system vary with
the power disturbances such as those imposed by wind farm
power fluctuations and converter outages. As a grid supervisory
control relying on telecommunications may not be available or
in any case is likely to update the converter set-points only
periodically in a very slow manner, the impact of the
quasi-steady-state droop lines on the dc power flow needs to be
well understood, especially when the standardization of the
supervisory control remains unclear. Moreover, determination
0
(c) Voltage droop
0
(a) Slack bus
0
(b) Voltage margin
Inverter Rectifier Inverter Rectifier Inverter Rectifier
P
max
P
min
P
P
P, I
dc
P
*
, I
*
P
*
V
dc
V
dc
V
dc
V
*
V
*
V
1
*
V
2
*
Power Flow Algorithms for Multi-terminal
VSC-HVDC with Droop Control
Wenyuan Wang, Student Member, IEEE, and Mike Barnes, Senior Member, IEEE
A
2
dc
V
PLL
abc dq
abc
i
abc
v
θ
dq
v
dq
i
abc
e
Converter
Voltage
Control
*
abc
e
AC Grid
HVDC
Network
,
dc dc
PI
,
ac
PQ
,
dq
ii
*
d
i
*
q
i
P Controller
dq Current
Controller
Q Controller
**
,
ac
VQ
**
,
dc ac
PP
**
,
dc dc
VI
Voltage Droop
High-level
Control
PCC
Fig. 2. Voltage droop and converter power control.
of the optimised set-points by the supervisory control could
also require such power flow techniques incorporating droop
characteristics. It is of great interest for grid operators to fully
understand the power flow of the systems using the distributed
droop control as the outermost local converter control.
Previous research has focused on solving MTDC power flow
without taking detailed dc voltage control into consideration. A
unified power flow method for integrated AC/MTDC systems
is proposed in [10] by solving the ac and dc power flow
equations simultaneously, assuming dc slack bus control. In a
the ac/dc power flow algorithm discussed in [11], the ac and dc
network equations are solved sequentially in each iteration.
Research on power flow of dc network is provided in [7,
12-14]. Basic dc power flow analysis has been applied in [13]
and [14] to represent the dc transmission losses and to optimise
the voltage references. DC slack bus voltage control has been
used by most papers examining power flow analysis. The
sequential method is updated by including the V-P droop in
[15]. V-P droop is assessed in detail using linear analysis in [7],
however the non-linear power flow is not performed.
This paper aims to provide a range of power flow solutions
for MTDC systems with different voltage control techniques.
II. C
ONTROL OF MTDC SYSTEMS
From the innermost to the outermost control, a cascaded
MTDC control system can be structured as: converter voltage
control, dq current control, real and reactive power control,
voltage droop control, secondary control (optional) and
supervisory control.
A simplified diagram for a VSC control system is shown in
Fig. 2. The control system is a cascaded system based upon a
fast and decoupled dq current control. The voltage reference for
the VSC is produced by the current controller. The set-point of
the current control is supplied by the real and reactive power
controller, whose input may vary depending on different
applications. The reference for the active power control can be
generated by the voltage droop control or directly scheduled by
the supervisory control via a telecommunication link.
A basic droop characteristic and the higher level grid-stage
control are shown in Fig. 3, where the dashed lines indicate that
communication is required. The power or current reference of
the local VSC control is scheduled according to the pre-defined
droop lines and measured dc voltage. The secondary control,
0
dc
V
,
dc dc
PI
**
,PI
*
V
*
V
or
*
dc
P
* **
,,VPI
**
,PI
ref
P
Secondary
Controller
1/K
ρ
=
1
Supervisory
Control
*
dc
I
dc
V
RectifierInverter
Fig. 3. Voltage droop control and outer supervisory control.
with its reference given by the supervisory control, is mainly
used to adjust the droop references after contingencies in order
to maintain scheduled power flow. The supervisory control is
able to access and modify most control references in order to
achieve an optimal operation of the dc grid. Optimal power
flow (OPF) might be required for this high-level application.
Either the supervisory or the secondary control is not a
necessity for the operation of the dc grid and it is likely that
they may only be active periodically. Analysing the power flow
resulting from the droop control is essential for understanding
the quasi-steady-state behaviour of MTDC system.
III. I
NTEGRATION OF AC/DC POWER FLOW
The main novelty of this paper is the development of the dc
power flow approaches which can be applied with various dc
grid control designs. Since MTDC systems essentially serve ac
networks, the methodology for the integration of the ac and dc
power flow models is presented in this section, with the
associated flow chart illustrated in Fig. 4. The ac and dc power
flow are computed separately, as the power flow of the dc
system is intrinsically determined by the converter dc
voltage/power control setting. This approach starts by solving
the dc power flow, and the computed dc side powers will then
be utilized for the initialization of ac power flow. The power
loss is calculated iteratively according to the results obtained
from the up-to-date ac power flow, with the convergence
indicated by active power at the point of common coupling
(PCC) bus.
Unlike the unified ac/dc power flow algorithms, this method
allows existing ac power flow models to incorporate the MTDC
models with no significant modification. The key step of this
algorithm is the dc system power flow with droop lines, which
will be analysed in detail in Section IV-VI.
From the dc power flow point of view, the converter dc bus
can be represented by a fixed or variable V
dc
bus or P
dc
bus, or
more generally, the droop dc voltage characteristics. To
facilitate ac power flow calculation, the PCC bus of the
converter station can be described as a PV or PQ bus,
depending on the reactive power control design.
A simplified power loss model is provided here to link the
converter dc side power with the power injection into the ac
grid at the PCC bus. According to the detailed power loss
analysis for the state-of-the-art modular multi-level converter
(MMC) performed in [16], the conduction loss of upper arm
(UA) and lower arm (LA) can be represented by (1)-(2), where
and
represent the number of sub-modules in series and in
parallel in each arm respectively, V
o
and R
o
denote the on-state
slope voltage and resistance of IGBT/diode.
3
DC grid power flow
(with droop lines)
AC grid power flow
Calculation of converter
power loss P
loss
Calculate the power P
g
at
PCC based on P
dc
and P
loss
Converged?
PQ bus
PV bus
PCC 1
PCC i
PCC n
AC
Grid
DC
Grid
,
dc dc
VP
,
gg
VP
Fig. 4. Integration of MTDC power flow with ac power flow.
Assumption that the IGBT and diode have similar on-state
characteristics is made to greatly simplify the power loss
calculation with only a slight degradation on the accuracy with
appropriate parameter choice. The total conduction loss for the
converter valve can then be computed by (3).
( ) ( )
2
23 23
ac ac
UA
dc o s dc
con o s
p
It It
I RN I
P VN
N
ωω
= ⋅ ++ ⋅ +
(1)
( )
( )
2
23 23
ac ac
LA
dc o s dc
con o s
p
It It
I RN I
P VN
N
ωω
= ⋅ −+ ⋅ −
(2)
( )
( )
2
0
3
UA LA
con con con
P P Pdt
π
ω
=⋅+
∫
(3)
Switching loss is derived based on the general form [17]:
( ) ( ) (
)
( )
6.
switch s p on off rec
cycle
P NNf EIEIEI= ⋅⋅ + +
∑
(4)
The result in [16] suggests that the switching loss only
corresponds to a fraction of the total MMC valve loss. Accurate
switching loss calculation requires complex non-linear
modelling however this level of modelling fidelity may not be
necessary for power flow calculations. The average phase
current is used in (4) for simplification, as suggested in [18].
By combing (3) and (4), the total loss of the converter station
can be represented as:
( )
22
12 3 4loss ac dc T ac dc
P KI K I K R I KI=⋅ +⋅ + + ⋅+⋅
(5)
where R
T
is the aggregated resistance of converter transformer
and the equivalent converter reactor, and K
1-4
are constants.
IV. P
OWER FLOW OF MTDC WITH A SLACK BUS
Power flow analysis of an MTDC system aims to obtain the
operating point of every terminal in the grid, with provided
generation, loading and control conditions. The problems are
usually represented by a series of non-linear relationships
between voltages and currents. The power flow algorithms
Fig. 5. Flow chart of the NR method for MTDC power flow.
proposed in this paper are based on the well-known
Newton-Raphson (NR) method [19, 20].
A simplified flow chart of the numerical iterative procedure
for solving a generic MTDC power flow is shown in Fig. 5.
Firstly, with the system represented by the power flow and
control equations, the unknown variables V and the specified
parameters P
sp
need to be selected. In MTDC studies, normally
dc voltages are selected as the variables and the power
quantities are chosen to be the specified parameters, which can
be represented using the non-linear parametric functions f (V).
Based upon the existing estimates of the unknown voltages,
the Jacobian matrix J is composed of partial derivatives of the
functions f (V):
()
.
sp
∂
∂
= =
∂∂
P
fV
J
VV
(6)
Based on the inverse of the Jacobian, the new set of the
voltage estimates are calculated by
( )
( 1) () 1 ()
ii i
sp
+−
= +⋅ −
V V J P fV
(7)
where
V
(i)
and V
(i+1)
are the ith and (i+1)th
estimate respectively.
The estimated voltages are updated iteratively until an
acceptable tolerance has been achieved for the mismatch
between the specified parameters and those computed using the
estimated variables.
For power flow studies of various dc networks and control
methodologies, the key difference is the description of the
power flow equations and the resulting Jacobian matrix.
In a dc system with terminals, the steady-state relationship
between the dc voltages and currents can be written as
dc dc
= ⋅I YV
(8)
where I
dc
is the nodal current injection vector, V
dc
is the dc
voltage vector and Y is the admittance matrix of the network.
For a symmetrical monopole HVDC system, if V
dc
is
comprised of pole-to-pole dc voltages, the admittance matrix
needs to be calculated based on the series resistance of the
Identify the unknown
variables V and the specified
parameters P
sp
=f(V)
Initial estimation of the
unknown variables
Calculate the parametric
error ∆P
Update the values of the
variables V
(i+1)
= V
(i)
+∆V
max
ε
∆≤P
Solve for ∆V using the
Jacobian inverse
∆V = J
-1
*[P
sp
– f (V
(i)
)]
Calculate flows of
lines and buses, etc.
Yes
No
Output results
Power Flow and
Control Equations
4
positive-pole and negative-pole cables. Per-unit values are used
for the power flow studies here as the differences between the
bipolar and monopolar configurations can thus be disregarded.
As analysed by a majority of previous literature, in a basic
power flow problem for a dc grid of n buses, dc voltage of the
slack bus is provided. The vector of the (n-1) specified
parameters are comprised of given nodal power injections or
line branch power flows. Wind farm terminals are assumed to
be power buses as normally they are not equipped with proper
dc voltage control capability.
The power injected to the MTDC system by terminal i can be
represented as
1
.
n
i i ij j
j
P V YV
=
= ⋅
∑
(9)
The branch power flow from terminal i to terminal j is
( )
.
ij i ij j i
P VYV V=⋅−
(10)
Accordingly, the Jacobian matrix elements associated with
(9) and (10) can be derived as (11) and (12), respectively. With
the (n-1) non-linear equations tackled by the NR method, this
basic power flow with the slack bus is solved.
1,
2,
n
ii
ii i ij j i ij
j ji
ij
ji
PP
YV YV VY
VV
= ≠
≠
∂∂
=+=
∂∂
∑
(11)
,
2, , 0
ij ij
i
ij j j ij i
i jk
k ik j
PP
P
YV V YV
V VV
≠≠
∂∂
∂
=−= =
∂ ∂∂
(12)
In an MTDC system with large power flows and long
transmission distances, the voltages might be significantly
different from each other. From the system planning point of
view, it might be more reasonable to specify the mean voltage
of all the buses instead of that of the slack bus. A new problem
can be presented as how to solve the power flow with the given
powers and the mean voltage. An algorithm has been developed
from the former power flow method to solve this problem.
Under this circumstance, all the n voltages are configured to
be the variables and an additional equation is built to represent
the average voltage. The supplementary parametric equations
and the associated Jacobian element are
11
1 11
() ,
nn
av
av i i
ii
ii
V
VV V
n V Vn n
= =
∂
∂
= = =
∂∂
∑∑
V
(13)
where
is the specified average voltage of the grid. The other
Jacobian elements can be computed according to (11) and (12),
as the power flow equations remain as (9) and (10).
If the power injections are specified for all terminals except
for the terminal m, this bus can be seen as a ‘floating slack bus’
aiming to achieve the given mean voltage. This power flow
does not stand for any dc grid control strategy but could be very
useful for system planning and converter reference setting.
From the control perspective, the two methods discussed in
this section can be employed to configure the nominal voltage
and power/current references for the droop line set.
V. P
OWER FLOW OF MTDC WITH DROOP CONTROL
Voltage-power (V-P) and voltage-current (V-I) characteris-
tics are the two most widely proposed droop control
approaches. From a local converter control point of view, their
behaviours are very similar to each other especially when the dc
voltage is controlled close to its rated value. To compare the
steady-state performance of these two types of droop, accurate
power flow calculation methods are required.
A. Voltage Power (V-P) Droop
Frequency-power droop is used by synchronous generators
in conventional ac systems to achieve a distributed frequency
control. Based on the experience of ac grids, voltage-power
droop is a strong candidate to be adopted in future MTDC grids.
However, there is no equivalent ‘AGC’ type of control in
MTDC systems to restore the dc voltage to its nominal value
thus the V-P droop needs to be incorporated in power flows.
Generally, if V-P droop is employed by a dc grid of n buses,
the power flow problem can be described as how to solve the
operating point of the system with a series of m specified V-P
characteristics and (n-m) given nodal or branch powers.
If the V-P droop is used for terminal , the converter
rectifying power would be controlled according to:
**
()
i ii i i
P KV V P= −+
(14)
where the voltage and power references of the droop line are
denoted by
and
. The droop control gain, which indicates
the sensitivity of the converter power to the local dc voltage, is
represented by K
i
. Note that rectifier orientation is used through
this paper for the direction of dc side power and current.
By setting K to zero, a VSC terminal in power control mode
or with known power generation can also be represented by
equation (14). This feature of V-P droop makes it easier to
analyse the power flow of the grid in a more generic way.
Voltages of all dc buses are selected as the variables to be
solved. The vector of the specified parameters is comprised of
the power references of the converters in droop control and the
given power profile related to other converters:
** *
12
[ ].
T
sp n
PP P=P
(15)
The non-linear functions related to the given nodal and
branch powers are shown in (9) and (10). By combining (9) and
the V-P characteristic equation (14), the parametric functions
related to the droop are computed as:
2 **
1,
() .
n
i i ii i ij j i i i i i
j ji
f V Y V YV KV KV P
= ≠
= + − +=
∑
V
(16)
The corresponding Jacobian elements can thus be obtained:
1,
2 ,.
n
ii
ii i ij j i i ij
j ji
ij
ji
PP
YV YV K VY
VV
= ≠
≠
∂∂
=++ =
∂∂
∑
(17)
The offline converters can be considered as a specific V-P
droop with both the gain K and the power reference P
*
equal
zero. If there are no branch power flows specified, all the
non-linear equations and the Jacobian elements can be written
in the form of (16) and (17), respectively.
5
B. Voltage Current (V-I) Droop
The rectifying power of terminal equipped with a typical
V-I droop can be represented as:
**
()
i i ii i i
P V KV V I
=⋅ −+
(18)
where the current reference is represented by
. Considering
that the power
can also be derived as (9) based on (8), the
following objective function is obtained for terminals in V-I
droop mode as
( )
( )
2 **
1,
( ) 0.
n
i ii i i i ij j i i i
j ji
f Y K V V YV I KV
= ≠
=+ + −+ =
∑
V
(19)
Accordingly, the specified parameters are obtained as:
12
[0 0 0 ]
T
sp m m n
V I droop control
P control
PP P
++
−
=P
(20)
where zero is chosen to indicate the effectiveness of the droop
control while the other (n-m) parameters are comprised of the
specified nodal or line powers.
The Jacobian elements associated with (19) are derived as
( )
**
1,
2( )
n
i
ii i i ij j i i i
j ji
i
f
Y K V YV I KV
V
= ≠
∂
= + + −+
∂
∑
(21)
.
i
i ij
j
ji
f
VY
V
≠
∂
=
∂
(22)
So far, the steady-state equations for converters in slack bus
control, constant power control, basic V-P and V-I droop
control, and off-line operations have been addressed.
Therefore, generalized MTDC power flow can be solved for dc
grids with a mixture of these control strategies by appropriately
integrating these equations using the NR method.
VI. P
OWER FLOW WITH GENERALIZED V-I/V-P
CHARACTERISTICS
In the algorithms discussed in Section IV and V, the
operating mode of a converter is fixed and the converter limits
have not been considered in detail. Furthermore, the droop
control used in Section V has not been sufficiently generic.
Realistically, the droop characteristics could be a combination
of multiple linear or non-linear functions of dc voltage, such as
the two types of droop lines illustrated in Fig. 6.
In a practical dc grid, a degree of voltage control capability
could be required for all the VSCs. When the voltage droop
with a power dead-band shown in Fig. 6(a) is implemented, the
scheduled power will be produced by the converter as the dc
voltage is maintained close to its nominal value. Once the
voltage exceeds the dead-band zone, the converter power will
adjust to contribute to the stabilization of the dc grid.
Shown in Fig. 6(b), voltage limits can be enabled by
introducing two more stages of droop. As the overvoltage or
undervoltage threshold is violated, the droop control gain K
will be increased in order to maintain the voltage within an
acceptable range. The increased gain K
max
should be kept within
the dynamic requirement of the stability margin.
0
dc
V
dc
P
*
dc
P
Rectifier
Inverter
min
P
max
P
*
low
V
*
high
V
max
V
min
V
max
K
max
K
K
0
dc
V
dc
P
*
low
V
*
dc
P
Rectifier
Inverter
*
high
V
1
K
min
P
max
P
2
K
(a)
(b)
Fig. 6. (a) V-P droop with power dead-band; (b) V – P droop with
dead-band and voltage limits.
An algorithm is proposed here to solve the MTDC power
flow involving more complex droop characteristics with
multiple control modes. This generalized approach can be
applied to most types of VSC power and dc voltage control.
Converter limit checking is included by enabling an additional
outer iteration loop. The key procedure of this method is to
iteratively update the parameters of V-P or V-I functions
according to the newly estimated voltages.
Each voltage droop line is essentially a function between the
voltage and desired current/power output. For example, the
voltage droop with a dead-band and voltage limits is comprised
of multiple linear functions. Differentiated by the voltage level,
these linear functions of the converter power
(
) can all be
represented in a form of the typical droop lines:
( )
( )
( )
( )
( )
( )
( )
**
max max max max
** *
max
* * **
** *
min
**
max min min min
,
,
0,
,
,.
dc high dc dc
high dc dc high dc
high dc dc low dc high
low dc dc dc low
dc low dc dc
K V V K V V P for V V
K V V P for V V V
V V P for V V V
K V V P for V V V
K V V K V V P for V V
−+ − + ≥
− + <<
⋅ − + ≤≤
− + <<
−+ − + ≤
(23)
In fact, like the constant power control of an effective K of
zero, the slack bus control can be modelled as another extreme
case of droop:
( )
*
() , .
dc dc
PV K V V K= − →∞
(24)
With the gain of the slack bus approximated by a sufficiently
large number, a good accuracy can be achieved. Based on this,
the voltage margin characteristic can be considered as a specific
case of the voltage droop with a power dead-band. Basically, all
the linear stages of different voltage control methods can be
represented in the form of the droop function. This will
significantly facilitate the power flow programming.
More generally, if the steady-state V-I function
(
) or V-P
function
(
) is non-linear at some stages, the power flow
equation representing the related terminal i can be written as:
1
() 0
n
i i i i i ij j
j
f V I V V YV
=
=⋅ −⋅ =
∑
(25)
1
() 0
n
i i i i ij j
j
f PV V YV
=
= −⋅ =
∑
(26)
