INCLUSION OF SMALL SIGNAL STABILITY ASSESSMENT TO
ELECTROMAGNETIC TRANSIENT PROGRAMS
J. R. Lucas, U. D. Annakkage*, C. Karawita*, D. Muthumuni#, R. P. Jayasinghe#
University of Moratuwa Sri Lanka, *University of Manitoba, #Manitoba HVDC Research Centre
lucas@elect.mrt.ac.lk, annakkag@ee.umanitoba.ca, chandana@ee.umanitoba.ca, dharshana@hvdc.ca, jayas@hvdc.ca
ABSTRACT
This paper presents the inclusion of a linearized statespace small-signal stability (SSS) module to the transient
simulation program PSCAD/EMTDC. It combines smallsignal stability assessment, with the proven user interface
capabilities of PSCAD. The motivation for developing
the package is to provide small-signal stability assessment
of case studies on PSCAD, and an adaptable simulation
background for users to carry out conventional controller
design without the need for an outside package. The SSS
module, once fully developed, will be capable of
incorporating all the system models available on PSCAD.
The parameters necessary for the SSS analysis are self
generated using a Software program. SSS is capable of
carrying out Eigenvalue analysis. In addition it can plot
step and pulse responses on the PSCAD output windows
together with the EMT results. The capabilities of SSS are
illustrated using the benchmark 12-bus system.
Validation of the development has been checked by
comparison of the Eigen variables with those of SSAT,
and pulse responses with PSCAD. The system is further
analysed to show how Eigenvalues obtained from SSS
analysis of an unstable system may be made use of to
stabilize the system before EMT simulation.
KEY WORDS
Small-signal-stability, state-space analysis, power system
oscillations, EMT simulation, modes, participation factors
1. Introduction
“Power system stability is the ability of an electric power
system, for a given initial operating condition, to regain a
state of operating equilibrium after being subjected to a
physical disturbance, with most system variables bounded
so that practically the entire system remains intact”[1].
This physical disturbance is classified into (a) large
disturbances related to: high frequency transients over
short periods of time (electromagnetic phenomena), low
frequency transients over much longer periods (electromechanical phenomena) and (b) small disturbances,
related to the small signal stability.
PSCAD/EMTDC[2], EMTP, ATP are widely used
computer simulation tools for the analysis of high
frequency transients.
Such tools do not give an insight into the parameters, such
as Eigenvalues, determining the small signal stability of
the system. Also they are time consuming to obtain the
system response over long durations. Separate analytical
tools [3-7], such as SSAT of DSAPowerTools, PSS/E and
DIgSILENT PowerFactory, are generally required for
studying these small signal stability issues and developing
controllers for satisfactory operation of power systems to
sufficiently damp out the system oscillations, without the
need to do them heuristically.
A principle objective of the development is to enhance the
usefulness of PSCAD by introducing a small signal
stability module within the package itself. The necessary
analytical tools were developed within the package, to
evaluate the small signal stability of a power system at a
chosen operating point.
Having the ability to perform small-signal stability
assessment within the EMT package, facilitates this task
seamlessly. If the same database is used, the possibility of
inadvertently using different data in the two programs is
completely eliminated and saves the time of the user. A
Software program has been developed to extract the
required data, from the EMTDC database and PSS/E type
power flow results, for the small signal stability
assessment.
It is important to validate the linearized models
developed, which give the necessary parameters.
Validation is carried out using the benchmark 12-bus
system [8]. The Eigen variables are compared with those
of SSAT, and small perturbation simulations with those of
the more accurate EMT type simulations (EMTDC).
2. Modelling Philosophy
The power system to be simulated is set up as a EMT case
with each dynamic component on a separate page module.
The formulation commences with the identification of the
dynamic devices (generators and auxiliary components,
other dynamic devices etc.,) and other power system
components (transformers, transmission lines, loads etc).
For small disturbances, the stability properties of the
oscillatory modes can be considered independent of the
size of the system disturbance. Thus they can be analyzed
by linearizing the system about the steady state operating
point and using analysis methods applicable to linear
systems [10].
2.1 Modular computation of State-space Equations
Each dynamic device i of the power system is described
by a set of differential-and-algebraic equations (DAE),
linearized about a chosen steady state operating point, of
the form:
•
∆ X i = [ADi] ∆Xi + [BDi] ∆Ui + [EDi] ∆Vi
∆Ii = [CDi] ∆X + [DDi] ∆Ui – [YDi] ∆Vi
The DAE are then assembled in a modularized form. For
example, the DAE for each generator is compiled to give
the state-space matrices corresponding to the DAE. The
program determines the auxiliary devices associated with
the particular generator and combines their state-space
equations sequentially, to give the overall state-space
matrices for the generator with auxiliary devices
combined, also of the above form.
The state-space matrices for each generator with
auxiliaries is then combined together.
The algebraic equations are used
steady state operating condition
program. The network model is
steady state operating condition
model.
to obtain the related
using a power flow
linearized around the
yielding a linearized
The algebraic equations for the Network are given as
∆I = [Ybus] ∆V
which on combination with the state-space equations of
the devices give the general state-space equations
•
where
∆ X = [A] ∆X + [B] ∆U
[A] = [AD] + [ED][Ybus+YD]-1[CD],
[B] = [BD] + [ED][Ybus+YD]-1[DD]
2.2 Small Perturbation Analysis and Simulation
The Eigenvalues [Λ] and Eigenvectors [ψ] of the statespace matrix [A] are obtained using
[Λ] = [φ][A][ψ]
where [ψ] = [φ]–1
The state space equations are then transformed to
decoupled modal form as
•
Z = [Λ] Z + [β] U, Y = [γ] Z
where
Z = [ψ] X, [β] = [ψ][B], [γ] = [C] [φ]
The small-signal stability of the system can be computed
from the Eigenvalues of the state matrix around the steady
state operating point.
It is usually necessary to determine the participation of
each of the state variables in each mode to take any
necessary corrective action regarding stability. The
participation of the kth state variable in the ith mode are
determined as
pki = ψki φik
Considering a small disturbance to one input, U has one
element, and [Λ] is diagonal, so that the transformed state
variables become independent modes, giving the set of
decoupled equations
•
zi = [λi] zi + [βi] u
which transforms, in the Laplace domain, to
s Zi(s) = λi Zi(s) + [βi] U(s)
This has an analytical solution of the form
zi(t) = zi(0). eλit
The resulting modes Z are then re-transformed to obtain
the original state variables X = [φ] Z
2.3 Interfacing with Simulation Database
While EMT simulation programs normally study fast
transients, they require other programs to carry out smallsignal state-space analysis to get an insight into the power
system.
Thus for large system studies, where electromagnetic and
electromechanical phenomena are investigated, in
addition to the simulation tool, a small signal assessment
has to be often carried out using a different package. It
would thus be convenient if all the analysis could be
simulated using a common tool and a single database.
The SSS module requires both a dynamic file, as well as a
network file, to carry out the SSS assessment. The
network data file basically contains the information
obtained from the Power flow in a form suitable for the
SSS analysis, while the dynamic data file contains the
parameters required for the DAE of the dynamic devices.
These files may be hand compiled (in the customized
format – in this case the EMTDC case does not have to be
set up). To eliminate human error in transforming
variables from the power flow solution and the EMT case,
Software programs have been developed to extract the
required information in the required format directly from
the relevant PSS/E file, and the PSCAD case file.
SSS has been developed to give a computationally
efficient algorithm with an increased level of flexibility
for research and development using PSCAD. SSS has
been developed in a fully modular basis, allowing for the
easy addition of new components by the user.
2.4 Comparison of Dynamic Simulations
Provision is available for the selection of the disturbance
as either a step input or a short duration pulse input and of
defined magnitude. The state variables may be compared
on the EMT simulation plot window from both the SSS
module and the EMTDC simulation for the same case.
2.5 Graphical User Interface (GUI)
A user friendly GUI has been provided to allow the user
to make choices and to integrate and communicate within
PSCAD through high resolution graphics. The GUI
permits the selection of Eigenvalue analysis and timedomain small perturbation simulation as appropriate.
The governor models and the turbine models being
different from those in SSAT, a comparison of
Eigenvalues was not attempted with SSAT with these
added.
Table 2 - Comparison of some Non-oscillatory modes
Eigen values with SSS
Eigen values with SSAT
–42.60
–34.53
–26.91
–16.35
–13.81
–3.30
–2.10
–42.58
–34.74
–26.94
–16.31
–13.83
–3.35
–2.14
3. Case Study
The PSCAD/EMTDC simulation software, and SSAT
software are used for the module verification. The case
study used is the benchmark 12 bus system[8] shown in
figure 1.
Exc
230 kV
Bus 5
22 kV
Bus 10
230 kV
Bus 4
A further validation was made, by including the
dynamics [11] of the governors (Mechanical-hydraulic
controls GOV1) and the turbines (Non-elastic water
column without surge tank TUR1), by comparing
simulation results of SSS with PSCAD.
state variables,Gen9
G10
230 kV
Bus 2
Infinite Bus
22 kV
Bus 12
G9
Tur Gov
345 kV
Bus 7
22 kV
Bus 11 Exc
345 kV
Bus 8
G11
230 kV
Bus 3
Gov Tur
Figure 1 - 12 bus benchmark system
The linearized small signal stability modelling of the
network was validated against the SSAT software. In this
analysis the generators[9] (generators 9 and 10 – 5th order
GENSAL, generator 11 – 6th order GENROU) and their
exciters (alternator supplied rectifier excitation system
AC4A) are simulated [10].
The generator 12 is
represented as a source without dynamics.
Table 1 - Comparison of Oscillatory modes
Eigen values with SSS
Eigen values with SSAT
– 23.26 ± j 2.24
–23.36 ± 2.25
– 7.66 ± j 2.38
– 0.58 ± j 8.17
–7.67 ± 2.25
–0.56 ± j8.12
– 0.48 ± j 6.20
– 0.27 ± j 5.06
–0.48 ± j6.19
–0.24 ± j5.05
Table 1 and Table 2 show a comparison of the
Eigenvalues obtained from SSS compared with those
obtained from SSAT, for oscillatory and non-oscillatory
modes respectively. It is seen that there is a very close
match, validating the model developed.
speed (rad/s)
Exc
22 kV
Bus 9
230 kV
Bus 6
Field voltage (pu)
230 kV
Bus 1
Internal angle (rad)
Tur Gov
0.125
0.100
0.075
0.050
0.025
0.000
-0.025
-0.050
-0.075
-0.100
0.8m
0.6m
0.4m
0.2m
0.0
-0.2m
-0.4m
-0.6m
-0.8m
0.40
0.30
0.20
0.10
0.00
-0.10
-0.20
-0.30
-0.40
x11-SSS
iAng1-PSCAD
x21-SSS
w1-PSCAD
x61-SSS
Efd1-PSCAD
20.0
22.0
24.0
26.0
28.0
30.0
Figure 2 - Comparison of state variables on Plot window
A 2% pulse of duration 2s is applied to the field voltage,
at time t=20s, allowing PSCAD to stabilize before the
application of the pulse.
Figure 2 compares the simulation responses (thin line)
and the SSS model responses (thick line) for 3 of the
system states – rotor angle, speed, and field voltage.
The output shown is directly reproduced from the
simulation window of PSCAD. A good match is observed
for each of the 3 state variables. System data was so
selected as to produce an unstable situation.
Figure 3 shows a plot of the eigenvalues of the 12 bus
system, as shown on the Eigen plot window. The plot
window enables the user to obtain an overall view of the
location of the Eigenvalues and whether there are unstable
modes. Facility is available for expanding a given region.
Eigenvalue plot
10.0
+y
Table 4 shows the participation factors of some of the
selected modes, grouped as per rotor, machine, exciter,
governor and turbine, and listed for generators 9 and 10.
The participation factors can be used to determine which
of the parameters are responsible for the unstable modes.
Table 4 – Participation (%) of selected modes
Mode Rot9 Mac9 Exc9 Gov9 Tur9 Rot10 Mac10 Exc10 Gov10 Tur10
4
5.0
50.5
9.9
5.2
10
22.3
37.7
10.8
22.6
49.5
50.0
49.7
4.4
1.4
1.2
25.7
21,39 46.5
4.5
29
30
13.3
0.9
51.4
3.3
27
+x
9.4
7.5
12,16
23
-x
22.8
27.6
49.1
1.6
5.1
0.1
0.0
1.2
29.6
2.9
15.6
0.7
0.6
50.3
2.4
43.8
41.0
45.7
2.2
1.8
30.5
3.0
96.5
1.7
1.2
96.2
Table 4 shows the participation of 2 generators 9 (which
has been excited) and 10 (one which has not been excited)
in the oscillatory modes. It is seen that the unstable modes
25,40 have their main contributions from the rotor
dynamics (state variables – rotor angle and speed) of
generators 9 and 10. It was also observed that generator
11 (for which the participation factors are not displayed)
had little contribution to these modes.
-2.5
-5.0
-7.5
-y
-50
-40
-30
-20
-10
0
Figure 3 - Eigenvalue plot window
Table 3 - Eigenvalues of oscillatory modes
Mode No.
6, 7
8, 9
11,15
12,16
13,17
18,38
21,39
25,40
Eigenvalue
–26.680 ± 0.172
–24.257 ± 2.198
–10.069 ± 10.306
–10.066 ± 10.302
–10.045 ± 10.303
– 0.643 ± 8.137
– 0.517 ± 6.140
0.161 ± 2.805
Frequency (Hz)
0.027
0.350
1.640
1.640
1.640
1.295
0.977
0.447
state variables,Gen9
10
Damping (%)
1.000
0.996
0.699
0.699
0.698
0.079
0.084
– 0.057
Internal angle (rad)
-10.0
The Eigenvalues of the Real modes are all stable modes
and are not listed in this paper.
x11-SSS
iAng1-PSCAD
x21-SSS
w1-PSCAD
x61-SSS
Efd1-PSCAD
0.10m
0.05m
0.00
-0.05m
-0.10m
-0.15m
Field voltage (pu)
Table 3 shows the Eigenvalues of the oscillatory modes in
tabular form with their frequencies of oscillation and their
damping.
It is seen that the Eigenvalue pair (modes 25 and 40) has a
positive real part giving an unstable mode. This is the
cause of the instability demonstrated in the simulation of
figure 2.
0.020
0.010
0.000
-0.010
-0.020
-0.030
-0.040
-0.050
-0.060
0.15m
speed (rad/s)
0.0
46.6
11,15
25,40 39.0
2.5
0.8
30.9
28.5
18,38
19
7.5
1.1
6, 7
8, 9
0.100
0.080
0.060
0.040
0.020
0.000
-0.020
-0.040
-0.060
20.0
22.0
24.0
26.0
28.0
Figure 4 - Comparison after stabilization
30.0
Based on the above, the exciter gains of generators 9 and
10 were changed so as to yield all negative eigenvalues,
as shown in table 5. The new simulation results, of
comparison of SSS with PSCAD (figure 4) shows that the
simulation again agrees closely, but that the system is
now stable.
Table 5 - Eigenvalues of oscillatory modes (new)
Mode No.
8, 9
11,15
12,16
13,17
18,38
21,39
25,40
Eigenvalue
–22.752 ± 1.929
–10.069 ± 10.306
–10.066 ± 10.302
–10.045 ± 10.303
– 0.578 ± 8.098
– 0.490 ± 6.157
– 0.095 ± 2.590
Frequency (Hz)
0.307
1.640
1.640
1.640
1.289
0.980
0.412
Damping (%)
0.996
0.699
0.699
0.698
0.071
0.079
0.037
5. Conclusions
A small-signal stability (SSS) module has been
formulated for use with the transient simulation tool
PSCAD. The developed SSS module is the outcome of
the ongoing research conducted at the University of
Manitoba on linearization of dynamic devices and its
implementation into small-signal stability analysis. SSS
integrates the abilities of the EMT package and its GUI,
with the SSS analysis module. The common database,
from which data for SSS is extracted, aids the user by
having a single tool for design studies. Simulation results
for the benchmark 12 bus system show that it is a
powerful and promising tool for carrying out small-signal
analysis together with transient analysis, especially for
power system controller design to prevent instabilities.
Validation has been carried out by comparison of
Eigenvalues with SSAT and time domain responses with
PSCAD/EMTDC.
SSS could be effectively used to study the stability of a
proposed power system before detailed simulation on an
EMT type simulation is carried out.
SSS could also enhance power engineering courses by use
in the classroom for instructional purposes for the
students to get an insight into controller design.
The SSS module has been developed on a modular basis,
and will in the future permit incorporation of user defined
devices by inputting their state matrices.
Acknowledgements
The financial assistance of Manitoba HVDC Research
Centre for the project through the University of Manitoba
is gratefully acknowledged.
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