1
High-Power High-Frequency Converter Modelling
Using Dommel’s and Runge-Kutta Methods in ABC
and DQ frame
Wei-Xing Lin, Member IEEE and Dragan Jovcic, Senior Member, IEEE
weixinglin@abdn.ac.uk, d.jovcic@abdn.ac.uk
School of Engineering, University of Aberdeen, Aberdeen, UK
Abstract—As the future DC grids will involve numerous
converter systems, the accuracy and speed of their modeling
becomes of high importance. This paper develops and compares
four average models for high frequency converters like DC/DC
or DC hubs: the average model in the ABC frame using
Dommel’s method (Model 1), the average model in the dq frame
using Dommel’s method (Model 2), the average model in the ABC
frame using Runge-Kutta method (Model 3) and the average
model in the dq frame using Runge-Kutta method (Model 4).
Schematics and principles of the numerical algorithms for the
four models are presented. Detailed switching model of a
500MW, 2kHz, ±200kV /±400kV DC/DC converter is taken as
benchmark for comparing the four average models. It is found
that Model2 is numerically unstable. Model 1 and Model3 need
the same small solution time step as the detailed switching model
to give accurate results. Model 4 is the recommended average
model for high frequency (1-3kHz) converter. It gives accurate
result as the detailed switching model at the benefit of 100 times
faster simulation speed.
Index
Terms—Converter
modeling,
High
Frequency
Components, DC Power Systems, DC power transmission,
DC-DC power conversion, Numerical simulation methods
I. INTRODUCTION
DC grids and micro grids have gained significant interest
both in the industry and academic in recent years [1]-[2]. In
each of these new grids, large number of power electronic
converters will be used. VSC AC/DC converters will provide
interface with local AC networks while AC/DC/AC converter
systems provide connection to various distributed resources.
As the voltage levels inside a DC grid or inside a micro grid
cannot be the same, DC/DC converter will be used. In most
DC/DC converter concepts there will be a two stage
DC/AC/DC conversion system [3]-[4], and high frequency
operation will be used in the inner AC circuit in order to
reduce the size and cost. In [3] a high frequency
Inductor-Capacitor-Inductor (LCL) circuit is employed while
in [4] high frequency solid-state transformer is used.
50Hz/60Hz grid is 50μs which implies around 400
calculations per cycle [5]. The AC/DC converter systems
operating at around 1kHz switching frequency require 2-10μs
step implying 100-400 calculations per switching event.
To improve the simulation speed, fast average value model
(AVM) of power converters are required. The average models
employed with EMTP [6],[7] are mainly focused on 50/60HZ
SPWM AC/DC converters or buck/boost converters.
If high frequency DC/DC converters are employed the
situation becomes more complex. Here we have 1-3kHz
fundamental frequency in the inner circuit and depending on
the topology the converter switching frequency will be
2-5kHz. The required simulation step will be around 1μs.
Simulation step for the whole DC grid must be same as for
slowest converters and this becomes unacceptably slow when
complex DC grids are studied.
We will illustrate in this study that the average value
modeling in ABC or DQ frames is not suitable for high
fundamental frequency on EMTP platforms. We will examine
if the default solvers used with EMTP, based on Dommel's
methods [8] can be explored for high frequency component
modeling. In order to provide accurate and fast models we will
also employ other numerical algorithms like Runge-Kutta
which are used in other simulation platforms [9][10],[11].
We further study both ABC frame and DQ frame modeling
since with high frequency components these frames give much
different simulation speeds and accuracy.
II. DC-DC CONVERTER TESTS SYSTEM
This paper takes the high power LCL DC/DC converter [3]
as an example of high frequency component. Topology of the
4-phase LCL DC/DC converter is shown in Fig. 1. Rating of
the DC/DC converter is ±200kV/±400kV, 500MW and its key
parameters are shown in Table 2 in the Appendix. The
parameters in Table 2 are designed at 550MW at a rated
modulation index of 1 to provide 10% control margin.
Typical simulation tool for such power electronic based
grids is PSCAD/EMTDC [5] or other platform from the
EMTP (Electromagnetic transients program) family.
However, PSCAD/EMTDC was designed to primarily
simulate 50Hz/60Hz AC grid and few AC/DC converters
coupled to the grid (including HVDC systems). Typical
simulation step for the electro-magnetic transients of
This project is funded by European Research Council under the Ideas
program in FP7; grant no 259328, 2010.
Fig. 1. Circuit diagram of 4-phase high frequency LCL DC/DC converter
2
III. DOMMEL'S METHOD MODELING
A. Background of Dommel’s method
The Dommel’s method [8] is widely used in
electromagnetic transient programs. It is based on trapezoidal
rule to convert the network differential equations to algebraic
equations [10]. The trapezoidal method is an implicit solver
which is A-stable (stability guaranteed for any time step) [11].
However the solution is not guaranteed for the iterative
process, which is involved with non-linear systems, and this
indirectly affects stability. Also the accuracy of trapezoidal
method is among lowest of all solvers.
Fig. 2 shows the representation of inductor and capacitor in
Dommel’s method. Both the inductor and capacitor are
represented by an artificial resistor in parallel with controlled
current source. A set of algebraic equations can then be
formulated using the nodal analysis method to compute the
unknown node voltages at each time t, as explained in [8].
C. AVM using Dommel’s Method in dq Frame (Model 2)
AVM modeling can significantly improve simulation speed
for a 50Hz/60Hz SPWM modulated VSC converter. However
with ABC AVM model of the test DC/DC converter the
dominant frequency of the inner circuit is 2kHz and the
solution time step still needs to be small (around 1-2μs) in
order to provide around 200-400 calculations per cycle.
To increase the solution time step of LCL DC/DC converter
we propose to represent and solve the dynamics of the high
frequency inner LCL circuit in the dq rotating frame. In such
case, all the electrical variables become DC variables. The dq
type of modeling is used frequently in order to develop
converter controllers. On the downside dq modeling will not
properly represent unbalanced AC system events.
In a dq frame, the dynamics of inductor and capacitor are:
dIid 1
= (− Ri I id + ωo Li Iiq + Vid − Vcd )
dt
Li
dIiq
Fig. 2. Representation of inductor and capacitor in Dommel’s method
B. AVM using Dommel’s Method in ABC Frame (Model 1)
The common way of AVM modeling for VSC converters is
to neglect the switching dynamics and use controlled AC
voltage source at fundamental frequency [6], and this method
will be used for modeling complex DC grids [7].
Fig. 3 shows the DC/DC average model in the ABC frame.
The two VSC converter bridges in Fig. 1 are replaced by two
4-phase controlled AC voltage sources (v1A-v1D and v2A-v2D).
The controlled AC voltage sources are given as:
v1 A = m1AV1dc , v1B = m1BV1dc ,
v1C = m1CV1dc , v1D = m1DV1dc
(1)
Where m1A-m1D are modulation indices in the ABC frame.
The inner LCL circuit of Fig. 3 will be solved using graphical
interface in PSCAD /EMTDC, since PSCAD /EMTDC uses
only the Dommel’s method [6].
=
dt
dVcd 1
= ( I1d + I 2d + ωo CVcq )
dt
C
dVcq 1
= ( I1q + I1q − ωo CVcd )
dt
C
(3)
Using constant amplitude dq transformation, active power
of a Mp-phase VSC converter at the AC side is
Paci =
Mp
2
(Vid I id + Viq I iq ) =
Mp
2
kmVidc ( M id I id + M iq I iq )
(4)
= Pdci = 2Vidc Iidc
Therefore, the AC/DC interface equation is
Iidc =
Mp
4
km (M id I id + M iq I iq )
(5)
where km is a coefficient representing the modulation method.
km approximates to 4/π for the modulation method adopted in
the simulations in this paper and used in [3].
Fig. 4 shows the schematic of DC/DC average model in the
dq frame, and Dommel’s solving method is used since model
is implemented in PSCAD/EMTDC. An artificial d-axis
circuit and q-axis circuit are built in PSCAD/EMTDC to
represent the state-space equations of (2)-(3). Relationships
between Vid, Viq and the modulation indexes Mid and Miq are:
Vid = M id Vidc
Viq = M iqVidc
Fig. 3. Average Model using Dommel’s method in ABC frame (Model1)
(2)
1
(− Ri I iq − ωo Li Iid + Viq − Vcq )
Li
(6)
The power balance between AC and DC systems is
represented by controlled current sources, using (5) as shown
in Fig. 4.
3
Fig. 4. Average model using Dommel’s method in dq frame(Model 2)
IV. RUNGE KUTTA SIMULATION
A. Background of Fourh Order Runge-Kutta Method
The Dommel’s method is based on trapezoidal solver,
where the order of its global truncation error is proportional to
square of simulation step (O(h2)). The Runge-Kutta method is
a high precision method for solving differential equations.
Order of its global truncation error is proportional to fourth
order of simulation step (O(h4)), and therefore accuracy is
better. On the downside Runge-Kutta approach belongs to
explicit methods which are not A-stable and therefore
instability will occur for a sufficiently large time step [11]. If
time step is sensibly selected, the Runge-Kutta method will
give both: good stability and accuracy and it is widely
employed for simulation of engineering systems [9]-[11].
A general nonlinear state space model dx/dt=f(x,u) can be
solved using the Runge-Kutta’s method in the following steps:
1) calculate k1 = Δtf ( x (t ), u(t ))
2) calculate x1 = x (t ) + 0.5k1
3) calculate k2 = Δtf ( x1 , u(t ))
4) calculate x2 = x (t ) + 0.5k2
5) calculate k3 = Δtf ( x2 , u(t ))
6) calculate x3 = x (t ) + k3
7) calculate k4 = Δtf ( x3 , u(t ))
8) x (t + Δt) = x (t) +1/ 6(k1 + 2k2 + 2k3 + k4 )
Fig. 5 shows the schematic of DC/DC average model in the
ABC frame using Runge-Kutta solving method. As the
PSCAD/ETMDC has only one solving method, a user defined
solving algorithm needs to be developed in order to implement
the Runge-Kutta method. The user model takes L, R and C as
the input parameters and the modulation indexes m1A-m1D and
m2A-m2D, E1 and E2 as the input variables. For every simulation
step, the model will calculate the state variables following the
procedures of IV.A. The outputs of the user model are the
inductor currents and capacitor voltages in the ABC frame and
the DC currents interfacing the DC system. These variables
are then used as inputs for the controllers in the next
simulation step.
Fig. 5. AVM of Runge Kutta method in ABC frame (Model 3)
C. AVM using Runge-Kutta Method in dq Frame (model 4)
Fig. 6 shows the schematic of DC/DC average model in the
dq frame using Runge-Kutta solving method. It is quite similar
to Fig. 5 except that the state-space equations of (2)-(3) instead
of (7) will be used. Assuming symmetrical and balanced
system, a number of phases can be converted to DQ frame
with only two variables per dynamic element. The advantage
is that there are 12 states for a 4-phase DC/DC converter
model in the ABC frame while only 6 states in the dq frame.
B. AVM using Runge-Kutta Method in ABC Frame(Model 3)
While first order differential equations are employed in the
Dommel’s method, the Runge-Kutta’s method is based on
state-space models. The state-space model of one phase of the
inner LCL circuit is:
Fig. 6. AVM of Runge Kutta method in ABC frame(Model 4)
0
⎡i1A ⎤ ⎡− R1 / L1
d ⎢ ⎥ ⎢
i
=
−
R
0
2A
2 / L2
dt ⎢ ⎥ ⎢
⎢⎣vcA ⎥⎦ ⎣⎢ 1/ C
1/ C
−1/ L1 ⎤ ⎡i1 A ⎤ ⎡1/ L1
0 ⎤
⎡ m E / 2 ⎤ (7)
−1/ L2 ⎥⎥ ⎢⎢i2 A ⎥⎥ + ⎢⎢ 0 1/ L2 ⎥⎥ ⎢ 1 A 1 ⎥
m E /2
0 ⎦⎥ ⎢⎣vcA ⎥⎦ ⎣⎢ 0
0 ⎦⎥ ⎣ 2 A 2 ⎦
Similar as the derivation of (5), the DC current is calculated
by:
i1dc =
2
π
(M1Ai1 A + M1B i1B + M1C i1C + M1D i1D )
(8)
V. MODEL TESTING
A. Numericals stability
The detailed switching model of the 2kHz LCL DC/DC
converter with solution time step of 1us is used as benchmark
for comparing the performances of the four average models.
It is found that Model2 is numerically unstable. Large
artificial resistors in series with the inductors (R1 and R2 in Fig
4) are required to make Model2 numerically stable.
4
Fig. 7 compares the output of Model2 with the outputs from
detailed switching model. Power order is stepped from 0 to
1pu at 1.0s, which is then reversed to -1.0pu at 2.0s, and a
permanent DC fault is applied at 3.0s.
We can see from Fig. 7 that Model2 requires artificial
resistor to make it stable. A 30Ω resistor will make Model2
numerically stable but this resistance is unrealistic, and there is
significant steady state error between Model2 and the detailed
switching model.
Although this modeling approach is adequate with 50Hz
AC/DC converters, it is found to be inadequate for any higher
frequency converters.
600
Pdc1(MW)
400
Model2-30Ω
200
0
Model2-20Ω
-200
-400
Detail
-600
0.5
1
1.5
2
Time(s)
2.5
3
3.5
Fig. 7. Numerical instability of Model2
idc2(kA)
B. Accuracy
Fig. 8 compares the accuracy of the Model1, Model3 and
Model4 with the detailed switching model. The applied system
disturbance is same as in Fig. 7 but DC current is monitored.
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
Detail
Doml-ABC-1us
Doml-ABC-10us
0.5
1
1.5
2
2.5
Time(s)
3
3.5
4
idc2(kA)
(a) Dommel’s Method in ABC frame(Model1)
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
Detail
Rung-ABC-1us
Rung-ABC-10us
0.5
1
1.5
2
2.5
Time(s)
3
3.5
4
idc2(kA)
(c) Runge-Kutta Method in ABC frame(Model3)
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
1
1.5
2
2.5
Time(s)
3
3.5
Fig. 8(b) shows that behaviour of Model3 is almost the same
as the behaviour of Model1. Model3 is able to generate similar
result as the detailed switching model only with solution step
of 1μs.
Fig. 8(c) shows that the output from Model4 is well
matching the detailed switching model. With this model large
solving time step can be adopted, and even 50μs gives
excellent accuracy.
C. Model final comparisons
Table 1 summarizes the overall comparisons of the 4
average models. Solution time step for Model1-Model3 is 1μs
while it is 50μs for Model4. The maximum allowable
simulation step for Model1 and Model3 to give similar result
as the detailed switching model is 1μs. The maximum time
step is 50μs for Model4. Table 1 shows that Model4 can give
accurate results at the benefits of increasing the simulation
speed by nearly 100 times.
Table 1 Overall comparison of the 4 average models for 2kHz
DC/DC Converter
Simulation Accuracy is
Solution
Numerical
Models
time for 4s
good for
Method
stability
real time
time step
Detailed Model
stable
216.3s
≤1μ s
Model 1 (ABC,
Domel
201.3
stable
≤1μ s
1us)
Method
Model 2 (DQ,
\
\
unstable
1us)
Model 3
65.8
stable
Runge≤1μ s
(ABC,1us)
Kutta
Model 4
Method
stable
2.1
≤50μ s
(DQ,50us)
D. Influence of operating frequency
The above four models are further tested on a reduced
frequency (200Hz) LCL DC/DC converter. Voltage and power
ratings are the same with the 2kHz LCL DC/DC converter.
Therefore, the AC inductors and capacitors of the inner LCL
circuit will be increased by 10 times. Model2 is still
numerically unstable with 200Hz converter and will not be
further discussed.
Power order at terminal1 is stepped from 0pu to 1.0pu at
1.0s and again stepped down from 1.0pu to -1.0pu at 3.0s. A
permanent DC fault is applied at terminal 2 at 3.5 s. Fig. 9
shows outputs of the 3 models using different simulation
steps. As the operating frequency is reduced from 2kHz to
200Hz, the simulation step for the detailed model is increased
from 1us to 10μ s.
Detail
Runge-dq-10us
Runge-dq-50us
0.5
Fig. 8(a) shows that Model1 is able to generate similar
results as the detailed switching model, but only with the small
solution time step which is similar as the step in the detailed
switching model. When the solution step is increased up to
10μs, accuracy of Model1 deteriorates significantly.
4
(c) Runge-Kutta Method in dq frame(Model4)
Fig. 8. Accuracy of Model1, Model3 and Model4 for 2kHz DC/DC Converter
Conclusions for the 3 models are the same as in the 2kHz
case. Model 1 and Model3 can generate similar results as the
detailed switching model in both normal operation condition
and during DC faults if the solution step is the same as the
5
detailed model. Model4 is able to generate almost the same
result as the detailed switching model at significantly
increased time step and improved simulation speed.
Fig. 9 (d) gives a very interesting result. Model4 is still able
to generate almost the same result as the detailed switching
model even when the solution time step is as high as 500μs,
which testifies robustness of the proposed modelling
approach. We can envisage the solution step of a DC grid
could be larger than 50μ s if all the related AC systems are
modelled in the dq frame.
1000
E1,E2(kV)
800
600
400
E1
E2
200
0
-200
2.9
3
3.1
3.2
3.3
Time(s)
3.4
3.5
3.6
(a) DC voltages
1
Detail-10us
Doml-ABC-10us
Doml-ABC-50us
idc2(kA)
0.5
0
VI. CONCLUSION
Four average models for high frequency converters are
developed and tested. They are the Dommel’s method in the
ABC frame (Model1), the Dommel’s method in dq frame
(Model2), the Runge-Kutta Method in the ABC frame
(Model3) and the Runge-Kutta method in the dq frame
(Model4). It is concluded that Model2 is numerically unstable,
Model1, Model3 and Model4 can generate similar results as
detailed switching model in both normal operation and during
large transients such as DC faults. However simulation speed
of Model1 and Model3 is similar as the detailed switching
model. Simulation speed of Model4 is around 100 times faster
than the detailed switching model.
Model4 is recommended for modeling DC grids and micro
grids in which high frequency components such as DC/DC
converter or DC hub will be included. Model4 is accurate
enough to replace detailed switching model in electromagnetic
study of DC grids, while the simulation speed is hundreds of
times faster than the detailed switching model. The DQ axis
Runge-Kuta modeling is expected to become more complex
for large DC grids, which will become topic for our future
research.
-0.5
VII. APPENDIX
-1
Table 2 Parameter of a 2kHz LCL DC-DC converter
-1.5
2.9
3
3.1
3.2
3.3
Time(s)
3.4
3.5
3.6
V1dc(kV)
V2dc(kV)
L1(H)
L2(H)
C(uF)
Prate(MW)
f(Hz)
±200
±320
0.0409
0.0498
0.1667
500
2000
(b) Dommel’s Method in ABC frame(Model1)
1
0
idc2(kA)
Table 3 Parameter of a 200Hz LCL DC-DC converter
Detail-10us
Rung-ABC-10us
Rung-ABC-50us
0.5
-0.5
L1(H)
L2(H)
C(uF)
Prate(MW)
f(Hz)
±200
±320
0.409
0.498
1.667
500
2000
D Jovcic, K.Linden, D. Van Hartem, J.P. Taisne “Feasibility of DC
transmission Networks” ISGT Europe, Panel session proceedings,
Manchester, December 2011.
[2] H. Tao, A. Kotsopoulos, J. L. Duarte and M. A. M. Hendrix, “Family of
multiport bidirectional DC-DC converters,” IEE Proc.-Electr. Power
Appl., vol. 153, no. 3, pp. 451-458, May. 2006.
[3] D Jovcic, and L Zhang, “LCL DC/DC converter for DC grids” IEEE
Trans. Power Del., vol 28, no. 4, 2013, pp 2071-2079.
[4] S. Falcones, R. Ayyanar and X. Mao, “A DC-DC multiport converter
based solid state transformer integrating distributed generation and
storage,” IEEE Trans. Pow. Elec. vol. 28, no. 5, pp. 2192-02, May 2013.
[5] Manitoba-HVDC Research Centre, User’s guide on PSCAD, 2005.
[6] S. Chiniforoosh, J. Jatskevich, A. Yazdani, V. Sood, V. Dinavahi, J. A.
Martinez and A. Ramirez, ‘‘Definitions and applications of dynamic
average models for analysis of power systems,’’ IEEE Trans. Power
Del., vol 25, no. 4, pp. 2655-2669, Oct. 2010.
[7] T K Vrana, Y Yang, D Jovcic, S Dennetière, J Jardini, H Saad, ‘The
CIGRE B4 DC Grid Test System’’, ELECTRA vol. 270, Oct. 2013, pp
10-19.
[8] H. W. Dommel, ‘‘Digital computer solution of electromagnetic
transients in single- and multiphase networks,’’ IEEE Trans. Power
App. Syst., vol. PAS-88, no.4, pp. 388-399, Apr. 1969.
[9] Dessaint, L.-A.; Al-Haddad, K. ; Le-Huy, H. ; Sybille, G. " A power
system simulation tool based on Simulink" IEEE Trans. Ind. Electron.,
vol. 46, no. 6, pp. 1252-54, Dec 1999.
[10] L. V. D. Sluis, “Transients in power systems,” John Wiley&Sons Ltd, pp.
137-139, 2001.
[11] T. Hartley, G. Bealey, S. Chicatelli “Digital Simulation of Dynamic
Systems : A Control Theory Approach” PTR Prentice Hall, 1994
[1]
-1.5
2.9
3
3.1
3.2
3.3
Time(s)
3.4
3.5
3.6
(c) Runge-Kutta Method in ABC frame(Model3)
1
Detail-10us
0.5
idc2(kA)
V2dc(kV)
REFERENCES
-1
Rung-dq-50us
0
Rung-dq-500us
-0.5
-1
-1.5
2.9
3
3.1
3.2
3.3
Time(s)
3.4
3.5
3.6
(d) Runge-Kutta Method in dq frame(Model4)
0
idc2(kA)
V1dc(kV)
-0.5
-1
-1.5
3.49
Detail-10us
3.51 3.53
Rung-dq-50us
3.55 3.57 3.59 3.61
Time(s)
3.63 3.65
(e) High precision version of Fig. 9(d) from 3.49s to 3.65s
Fig. 9. Accuracy of Model1, Model3 and Model4 for 0.2kHz Converter