Today’s popular commercial power system dynamics simulation (PSDS) software
use explicit integration to handle the time-increment response of power
system controls. The basis of this goes back to as early as 1960, when an
integration technique known as the Fortran Analog Computer Equivalent, or
FACE, was proposed. Explicit integration allows for simpler
programming by assuming the response to stimulus can be represented in the
next time step, thus avoiding iteration. However, explicit integration
leads to a slight numerical error that is cumulative. This error leads
to phenomena such as the “lie” and “drift” (illustrated in a case study
below) encountered during dynamic simulation.
While the error was tolerable when studying transient stability
problems that could be conducted within simulation times of 2-5 seconds,
today’s brand of simulation problems present a concern. As
transmission systems are stressed, there is an increasing need to study
voltage stability. The transients in voltage stability, which may
include self-adjusting load, may take anywhere from seconds to minutes to
resolve, extending the simulation time and number of time steps required to
conduct the analysis. In addition, the increase in induction
motor and generator equipment, combined with other existing voltage-varying
or fast-acting controls such as FACTS and HVDC, lead to an increased
requirement for even smaller explicit time steps to assure network
convergence in the simulation.
To improve PSDS, the best alternative is to use an implicit integration algorithm.
the implicit method offers a “stable” numerical
solution. “Stable” in this context means the simulation progresses
smoothly with each time step converging within an acceptable tolerance.
An implicit solution further allows adaptive stepping, wherein the size of
the time step is varied in accordance with the simulated dynamic response.
As the simulation approaches steady-state, the time step approaches
infinity. A trigger can be set such as to automatically stop the
simulation when the response has sufficiently damped. This latter is a
distinct advantage that would improve dynamic simulation performance in
online and batch-mode applications. Furthermore, implicit integration
allows multirate simulation wherein multiples of the time step can be
applied to the set of state variables in the simulation as a function
of the rate of change.
In the implementation of the implicit algorithm just described, it is
important to introduce a frequency filter as the time step size increases to
avoid inadvertent sampling of higher frequency control response. The
filter has a net effect of introducing damping to the simulation.
Effective implementation of the filter would minimize damping. In
future implementations, we believe it is possible to conduct a dynamic
simulation focusing only on control response of tap-changers, switched
shunts, AGC, time-varying load models and special protection schemes such as
to replace the steady-state contingency analysis now presently performed
with power flows.
In addition to the improved simulation accuracy, and being able to avoid the
lie and drift phenomena, the implicit algorithm with adaptive and multirate stepping
(IIAMS) requires less computation time. This is perhaps a lesser consideration
with today’s breed of fast and powerful processors, but could be an
important characteristic that can bring to reality real-time dynamic
The following case study illustrates the application of the IIAMS to power
system dynamic simulation. The case study comes out
of a larger work to examine the dynamic behavior of power systems operating
at the voltage ledge (see “The In-Between Voltage State,”
Techblog of December 2005).
The study system is representative of today’s modern systems, that is, one with
amounts of induction generators in the form of wind farms and other
renewables. The combination of initial conditions and contingency
define an operating state that would be on the ledge of the P-V curve; i.e.,
a condition that is close to the voltage stability limit for transfers on
the study system.
Simulating the contingency with PSDS using explicit integration leads us to
a “lie” (see
(click on the figure at right to enlarge) for a plot of terminal voltage of an induction
The “lie” is a term, perhaps not too commonly used, to describe a time plot of stability
simulations which has become numerically unstable.
It’s called the “lie” because the sudden and sharp swings in the time plot
resemble the plot of nervous response of someone who has become untruthful
during a lie detector test. The “lie” is typically a consequence of
the network solution failing to converge at some point during a dynamic
simulation. After the first instance of non-convergence, the next time
steps are suspect and likely to fail convergence as well. There is
recovery other than to solve the cause of non-convergence. The “lie” is
numerical rather than
representative of any actual physical event on the power system. Options to resolve
- decelerate the iterative network solution
- reduce the convergence tolerance
- reduce the size of the time step from the standard half cycle
(about 8.3 msec for a 60 Hz system)
In the study case, the source of the lie is the fairly large
components of induction
time constants are the same
order of magnitude as the explicit time step. In addition, the
combination of induction generators, taking into account the impedance
of their network connections, could lead to even lower equivalent time
constants. Figure 2 shows the induction generator’s terminal
voltage using a quarter cycle time step. The lie has been moved
back a few seconds but is still present. With an eighth of a cycle
the simulation runs far enough to allow an assessment of
voltage stability (see Figure 3) before the lie shows up again.
Simulating the same contingency using IIAMS gives a similar
response for terminal voltage as the explicit integration with a very
small time step. This is shown in Figure 4. The plots are nearly coincident,
and for purposes of the
voltage evaluation, lead to the same assessment of operation on the
However, differences can be seen when viewing plots of rotor angle at a
neighboring generator, as shown in Figure 5. The swing
curve for the IIAMS starts to deviate from explicit
solution with a small time step at around 5 seconds into the simulation.
This is a
of “drift” in the
explicit solution and damping from the high frequency filter of the IIAMS. Drift is
the accumulated error as the number of time steps increase in the
explicit integration method.
Timing comparisons for the various solution methods applied to the same
contingency are as follows:
|Solution Method||Relative Simulation Time|
|Explicit Integration with Half-Cycle Time Step||1.8|
|Explicit Integration with Quarter-Cycle Time Step||4.1|
|Explicit Integration with Eighth-Cycle Time Step||9.0|
|Implicit Integration with Adaptive and
Multirate Time Step
Improvements in dynamic simulation performance and timing are possible using
implicit integration approach with adaptive
and multirate stepping (IIAMS). The
characteristics of IIAMS could help avoid the instances of lie and
drift that is common in power system dynamic simulation based on explicit
integration. In particular, voltage stability studies and studies of
systems with large induction load and generator components are more feasible
with IIAMS. In addition, IIAMS may improve online and batch-mode
applications by providing indicators when near steady-state conditions
are reached. The IIAMS approach
may be extended to simulate what is currently termed steady-state
The relative timing benefits to using IIAMS may make it feasible for use in online dynamic security assessment.
- J. J.
Sanchez-Gasca, et al, “Extended-term Dynamic Simulation Using
Variable Time-Step Integration,” IEEE Computer Applications in
Power, pp. 23-28, Oct. 1993.
Kurita, et al, “Multiple Time Scale Power System Dynamic
Simulation,” IEEE Transactions on Power System, vol. 8, Feb, 1993.
- M. L.
Crow and J. G. Chen, “The Multirate Method for Simulation of Power
System Dynamics,” IEEE Transactions on power Systems, vol 9, Aug.
Jingjia Chen, Mariesa L. Crow, Badrul H. Chowdhury, Levent Acar,
“An Error Analysis of the Multirate Method for Power System
Transient Stability Simulation,” 2004 Power Systems Conference &
Exposition (PSCE), Oct, 2004.
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