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Flexible Numerical Integration for Efficient Representation of Switching in Real Time
1276
IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 19, NO. 3, JULY 2004
Flexible Numerical Integration for Efficient
Representation of Switching in Real Time
Electromagnetic Transients Simulation
Kai Strunz
AbstractIn real time digital simulation of electromagnetic
transients for equipment testing, solution output is provided
at equidistantly spaced time points in order to synchronize
the data exchange with external equipment. Switching events,
which can occur in between these time points of solution output,
are pinpointed by backward interpolation. To recuperate the
time interval lost through interpolation, the simulation must be
resynchronized with the real time clock. For this purpose, the use
of a flexible time step size is desirable. However, in EM transients
simulators based on nodal analysis techniques, time step size
changes have so far involved changes of the nodal admittance
matrix and, thus, imposed a large computational burden. This is
avoided through the method flexible integration for readjustment
in simulation of transients (FIRST), presented in this paper.
Through this method, the characteristics of numerical integration
are modulated such that time step size changes are achieved in a
computationally very efficient manner and without modifying the
nodal admittance matrix.
Index TermsAlgorithms, discontinuities, electromagnetic tran-
sients, HVDC transmission, modeling, power electronics, power
system simulation, real time systems.
I. I
NTRODUCTION
T
HE RAPID progress in computer hardware performance
and the creation of both efficient and accurate simulation
algorithms have contributed to the development of real time dig-
ital simulators [1][6]. In real time simulation for equipment
testing, the simulation time advances at the speed of the real time
so that results are output in synchronism with the real time clock.
A particular challenge in this context is the real time simulation
of switching events as, for example, caused through power elec-
tronic system operation. In this work, the novel method flexible
integration for readjustment in simulation of transients (FIRST),
which allows to generate accurate simulation results for the real
time simulation of such switching, is described.
In Section II of this paper, the state of the art is reviewed.
This concerns techniques implemented in real time simulators
and in programs which do not perform real time simulation as
the Electromagnetic Transients Program (EMTP) [7] and the
Simulation Program with Integrated Circuit Emphasis (SPICE)
[8]. The discussions are intended to provide the reader with the
relevant background knowledge for the subsequent sections. In
Manuscript received December 31, 2002.
The author is with the University of Washington, Seattle, WA 98195-2500
USA.
Digital Object Identifier 10.1109/TPWRD.2004.824387
Fig. 1.
Current and voltage conventions for inductor.
Fig. 2.
Associated discrete circuit model of inductor.
Section III, the novel concept for the real time simulation of
switching is dealt with. Validation tests are performed in Sec-
tion IV. Conclusions are given in Section V.
II. F
UNDAMENTALS OF
R
EAL
T
IME
EM
T
RANSIENTS
S
IMULATION
A. Associated Discrete Circuit Model
Prior to the start of the simulation, the differential equations
describing individual network branches are approximated by
means of numerical integration. For example, the behavior of
the inductor in Fig. 1 is described through the following differ-
ential equation:
(1)
Using the trapezoidal method with the time step size
to dis-
cretize this differential equation yields the following algebraic
equation:
(2)
By substituting:
(3)
(4)
the inductor is modeled through the associated discrete circuit
model [7], [9] comprising a resistor and a current source de-
picted in Fig. 2 and described through the following equation:
(5)
0885-8977/04$20.00 © 2004 IEEE STRUNZ: FLEXIBLE NUMERICAL INTEGRATION FOR EFFICIENT REPRESENTATION OF SWITCHING
1277
Fig. 3.
Satisfied real time constraint.
The application of the backward-Euler method to discretize
(1) yields:
(6)
Thus, the following associated discrete circuit model is obtained
with the backward-Euler method:
(7)
(8)
The recommended choice of the integration method for the
creation of the associated discrete circuit models depends on
the waveform types to be simulated [10], [11]. The trapezoidal
method is better suited for the representation of sinusoidal
waveforms, while the backward-Euler method has advantages
concerning the representation of piecewise linear waveforms as
they can appear in power electronic systems.
B. Network Model
The network model is obtained by connecting the associated
discrete circuit models according to the real world network. This
leads to a network model consisting or resistors and sources and
for which a nodal equation system is established:
(9)
where matrix
is the nodal admittance matrix,
is the
vector of the unknown nodal voltages,
is the vector of the
source-dependent nodal current injections. At each time step ,
(9) is solved for the unknown nodal voltages.
C. Real Time Constraint
In real time digital simulation for fully interactive equipment
testing, the real time constraint must be satisfied. All computa-
tions to be performed in any single time step must be completed
within an interval which does not exceed the time step size .
A situation in which the real time constraint is satisfied during
time step
is shown in Fig. 3. Solution output is provided at the
real time points
and
, which are separated by
:
.
D. Tracking of Switching Events
The simulation of switching events in early releases of the
EMTP is explained by means of Fig. 4. The simulation time after
the th time step is denoted by
. The constant time step
size
is employed. In the situation shown, the current of a diode
switch is turned off leading to a discontinuity of the conductance
value of the associated discrete circuit model representing the
diode [12]. The switch is assumed to be connected in series with
an inductor. Since the current through the inductor constitutes
Fig. 4.
Switching-off event simulated with basic method.
a state variable, a correct simulation of the switching event is
particularly important.
At simulation time point
, the current is positive
and the switch conducts. In time step
, from
to
, the network solution yields a positive value for
the current again. At
, a negative value is obtained. This
negative current does not constitute a plausible solution because
the switch is considered to be a diode. Nonetheless, it is output
as a solution point.
Due to the change of the sign of the diode switch current with
respect to the solution obtained at
, the diode model
requests a change of status in time step . As a consequence,
the nodal admittance matrix in (9) is modified to consider the
opening of the diode switch. This change is only accounted for
in the solution of time step
. Consequently, in time step
the solution is not correct for the fraction of the time step size
following the zero crossing of the diode switch current.
The solution process is non-iterative. If several switching de-
vice models request changes of status in one time step, then the
network model is modified in accordance with all requests. The
latter are only accounted for in the solution of the following time
step. The solution is therefore erroneous for the fraction of the
time step size following the occurrence of the first request.
This error can result in the occurrence of switching mistakes,
incorrect power flows, and noncharacteristic harmonics. Mea-
sures which are taken to reduce the involved error commonly
concern modifications of the time step size and the use of inter-
polation methods.
1) Reduced Time Step Size: The smaller the selected time
step size, the smaller is also the maximal error which occurs
due to the fact that switching events generally occur in between
the discrete time points for which solutions are calculated. In
real time simulators, however, the real time constraint requires
a minimal value of the time step size. Depending on the network
to be simulated, this value might not be small enough to simulate
switching events with the desired accuracy.
2) Controlled Time Step Size: The use of a time step control
algorithm, which dynamically adjusts the time step size during
the simulation, can be helpful in improving the accuracy con-
cerning the simulation of switching events. The changing time
step size must not go below a minimal value in order to ensure a
strict adherence to the real time constraint. However, this value
might not be small enough to achieve the desired accuracy.
3) Interpolation: The implementation