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LOAD REPRESENTATION IMPACT ON THE DAMPING OF
ELECTROMECHANICAL OSCILLATIONS IN ELECTRICAL POWER
SYSTEMS
JORGE GUILLERMO CALDERON-GUIZAR
Gerencia de Análisis de Redes
Instituto de Investigaciones Eléctricas
Reforma 113, Col. Palmira, Cuernavaca, Morelos, 62490
MEXICO
jgcg@iie.org.mx
considered important in those studies, even though the
impact of load characteristics on system performance was
recognized [6,7,8].
The constant increase in demand for electric energy
together with the economical and severe environmental
constraints for building new transmission lines have
forced the electric utilities to operate their transmission
systems closer to their security limits [9]. As a
consequence of this, modern power systems are designed
and operated with small stability margins [1]. This means
that, nowadays power systems have to operate under
stressed conditions most of the time [9]. Since operating
under stressed conditions makes power systems prone to
instability problems, the use of improved representations
of the power system components for analyzing the overall
system performance following the occurrence of a
disturbance is necessary. This fact has been highlighted in
the analysis of the Swedish blackout reported in reference
[10].
The objective of this paper is not to develop a new model
for representing the load characteristics, but rather to use
already existing models and evaluate the influence of
different combinations of them on the damping of the
electromechanical oscillations in electrical power systems.
ABSTRACT
This paper reports on the influence of load representation
on the small-signal stability of power systems. With this
aim, simulations considering different representations for
load modeling such as; constant impedance, constant
current, constant power, induction motor and a
combination of such representations were performed and
the influence of load representation on the damping of the
electromechanical oscillations and thus on the overall
system stability was analyzed. The results obtained from
the simulations indicate that using the constant impedance
representation yields the lowest damping of the local
modes while the constant power representation leads to
the lowest damping of the inter-area modes.
KEY WORDS
Load modelling, small-signal stability, electromechanical
oscillations.
1. Introduction
Decisions on system reinforcements and/or system
performance are always supported by the analysis of both
load flow and stability studies. Since the reliability of
results from the aforementioned studies it is dependent on
the representation of the power system components, the
use of adequate models for simulating the components
characteristics is of great importance [1,2,3]. In this
context, modeling of synchronous generators and their
controllers, transmission and distribution equipment has
received a great deal of attention. As a consequence of
this, reliable mathematical models for representing the
behavior of such components, in stability studies, are now
available [2,3,4,5]. On the other hand, less effort has been
dedicated to the accurate representation of the load
characteristics [2,6]. This is mainly due to the fact that in
the past, most stability studies focused their attention on
keeping generators in synchronism and the accurate
representation of the load characteristics was not
567-027
2. Electrical Load Modeling
Load modeling characteristics is an important issue in
power system analysis. It is well known that loads located
at bulk power delivery points represent an aggregation of
a number of small individual electrical loads with
different operating characteristics. This fact, turns the
actual load performance difficult to predict at those points
of the system. As a result of this uncertainty, utilities have
tried to use characteristics that would lead to conservative
design for all system configurations [1].
This section presents some of the mathematical models
commonly used to represent aggregated or “lumped”
loads at bulk power delivery points in power system
stability studies.
55
id =
2.1 Static Load Models
[R (V − E ) − X (V − E )]
s
'
d
d
iq =
'
q
q
Rs2 + X 2
[− X (V − E ) + R (V − E )]
'
This type of models, assume that load characteristics may
be represented as algebraic functions of the instantaneous
values of voltage magnitude at load buses.
Mathematically, these assumptions may be represented as
shown below;
'
'
d
d
s
q
'
q
Rs2 + X 2
Te = Ed' id + Eq' iq
;
(3)
Tm = Tnomωk
Where;
2.1.1 Polynomial or ZIP model,
⎡ ⎛V
P = P0 ⎢ Ap ⎜⎜
⎢ ⎝ V0
⎣
2
⎤
⎞
⎟ + Cp⎥
⎟
⎥
⎠
⎦
2
⎤
⎛V ⎞
+ Bq ⎜⎜ ⎟⎟ + C q ⎥
⎥
⎝ V0 ⎠
⎦
⎞
⎛V
⎟ + Bp ⎜
⎟
⎜V
⎠
⎝ 0
E’q , E’d , Vd , Vq, id and iq are the transient and the
terminal voltages and currents on the “dq” axis
synchronous reference frame.
s
is the motor slip.
ω
is the mechanical rotor speed.
Te
is the electrical torque developed by the motor.
Tm
is the mechanical torque at any instant.
Tnom is the mechanical torque at nominal speed.
T ’0
is the transient open circuit time constant.
X’
is the blocked rotor short circuit reactance
X
is the rotor open circuit reactance
k
is the exponent for representing different types of
mechanical loads.
(1)
⎡ ⎛V
Q = Q0 ⎢ Aq ⎜⎜
⎢ ⎝ V0
⎣
⎞
⎟
⎟
⎠
Ap+Bp+Cp=1.0
Where;
;
Aq+Bq+Cq=1.0
P and Q are the active and reactive load powers as a
function of voltage magnitude (V).
P0 and Q0 are the active and reactive load powers at a
voltage magnitude of V0.
Ap and Aq are the portion of active and reactive load
power represented as constant impedance (Z).
Bp and Bq are the portion of active and reactive load
power represented as constant current (I).
Cp and Cq are the portion of active and reactive load
power represented as constant power (P).
B
B
Fig. 1 Induction Motor transient-state equivalent circuit
2.2 Dynamic Loads Models
3. Study Systems
Surveys on electricity consumption in the US have
pointed out that more than 50% of this is due to induction
motors loads [1]. In this paper, the dynamics of
aggregated induction motor loads are represented by a
third-order dynamical model, as shown below;
The study systems considered in this paper are the WSCC
3-generator 9-bus system [5], its single line diagram of the
system is shown in Figure 2, and the four-machine twoarea system [3] shown in Figure 3.
In the 3-machine system, generators are represented using
standard two-axis third-order model for Gen-1 and fourthorder model for Gen-2 and Gen3. Excitation system
action of fast exciters on Gen-2 and Gen-3 is represented
by first-order models. No excitation system action is
considered on Gen-1. Transmission system parameters,
operating condition and generator parameters are given in
[5]. Excitation systems data are shown in appendix I of
the paper.
In the 4-machine two-area system, generator
representation accounts for the rotor transient and
subtransient effects along d and q axes in each machine,
saturation effects are not considered. Excitation system
effects are represented by a thyristor exciter with a
transient gain reduction (TGR) [3]. For this test system,
the loading condition is characterized by a 400 MW
d '
1 ' X − X'
'
Eq = −ω0 s Ed − ' Eq +
id
dt
T0
T0'
1 ' X − X'
d '
'
iq
Ed = ω0 s Eq − ' Ed +
T0'
T0
dt
d
1
(Te − Tload)
ω=
dt
2Hm
(2)
56
power transfer from area 1 to area 2. For operating
condition, transmission system parameters and generator
data see reference [3]. excitation system data used in the
paper are shown in appendix I.
modes of oscillations. According to the frequency of these
modes (1.36 and 2.08 Hz approximately) they may be
classified as local modes [3,11]. The lowest frequency
represents the oscillation of Gen-2 and Gen-3 against
Gen-1. While the highest one basically represents the
oscillation of Gen-3 against Gen-2 as indicated by the
speed eigenvectors shown below;
Output: WW
1.
Eigen: -0.22945
+J8.5561
f = 1.36 Hz
G2
G3
G1
-0.35
Fig. 5 Speed eigenvector associated with the 1.36 Hz
electromechanical mode
Output: WW
1.
Fig. 2
Eigen: -0.96856
3-machine test system 1 [11]
+J13.116
f = 2.08 Hz
G3
G2
A1
A2
G1
-0.33
Fig. 6 Speed eigenvector associated with the 2.08 Hz
electromechanical mode
Fig.3
Participation vectors associated with the 1.36 Hz
electromechanical mode, indicate that the rotor variables
of Gen-2 and Gen-3 are the ones with the highest
participation. While rotor variables of Gen-3 have the
highest participation on the 2.08 Hz electromechanical
mode. Tables 1 and 2 only show those variables whose
participation factors have modules greater than 0.2
4-machine two-area system 2 [3]
4. Analysis Method
The influence of load representation on the damping of
electromechanical modes is achieved using modal
analysis techniques, which are more suitable than time
domain techniques in identifying the most influencing
factors on power system oscillations [11]. For a
comprehensive review on the application of modal
analysis techniques to power system analysis the reader is
referred to references [3, 9, 11].
The linearized models of the study systems as well as
the results used for the eigenanalysis reported in this
paper were obtained using the CEPEL PacDyn software
package [12].
5
Module
Phase
Bus Number
Variable
1.0000
0.0000
2
Delta
0.7396
-0.3739
2
Speed
0.3366
0.3962
1
Speed
0.2073
-1.5312
3
Delta
Table 1 Participation vector of the 1.36 Hz mode
Module
Phase
Bus Number
Variable
1.0000
0.0000
3
Delta
0.9669
-0.2054
3
Speed
0.2275
-2.0780
2
Speed
0.2041
-3.0941
2
Delta
Table 2 Participation vector of the 2.08 Hz mode
Simulation Results
Results from simulations performed on the 3-machine test
system indicate the existence of two electromechanical
57
The influence of different load representations on the
damping and frequency of the electromechanical modes
of test system 1 is shown in the following table, where the
terms LIM and SIM indicate Large and Small Industrial
Motors, respectively.
Load Representation
100% Z
100% I
100% P
50% Z and 50% I
Damping
(%)
Frequency
(Hz)
2.6807
7.3643
2.7349
7.3751
2.8247
7.3816
2.7044
7.3701
2.7429
7.3635
1.3618
2.0875
1.3675
2.0862
1.3746
2.0846
1.3645
2.0869
1.3639
2.0869
small influence on the frequency of oscillation associated
with the system electromechanical modes. Results from
simulations performed on the 2-area test system indicate
that the system electromechanical modes may be
classified as one inter-area mode (0.5401 Hz) and two
local modes (1.0855 Hz and 1.1211 Hz). The inter-area
mode is associated with the oscillations of both generators
of area 1 against those of area 2 as shown in Fig. 7.
1.
Output: WW
Eigen: -0.0035143 + J3.3936
G3
G4
f = 0.5401 Hz
G1
50% Z and 50 % I for
active power and 100%
Z for reactive power
20% of active power as
LIM and the remainder
2.6819
1.3637
active and reactive powers
7.3740
2.0871
as constant impedance Z
50% of active power as
LIM and the remainder
2.6683
1.3669
active and reactive powers
7.3909
2.0864
as constant impedance Z
50% of active power as
SIM and the remainder
2.6727
1.3670
active and reactive powers
7.3899
2.0864
as constant impedance Z
Table 3 Damping of electromechanical modes for
different representations of system load.
G2
-0.14
Fig. 7 Speed eigenvector associated with the 0.54 Hz
inter-area mode
The local or inter-plant modes are associated with the
oscillations of G1 against G2 (1.0855 Hz) and the
oscillations of G3 against G4 (1.1211 Hz) as shown in
Figs 8 and 9 respectively.
1.
Output: WW
Eigen: -0.56009 + J6.8205
G2
f = 1.0855 Hz
G4
G3
G1
-0.83
Results shown in Table 3, indicate that using only static
loads models ( ZIP ) for representing the whole system
load, the lowest value associated with the damping of both
electromechanical modes is obtained when the constant
impedance (100% Z) model is used. While the highest
value associated with the damping of those modes is
obtained when the constant power (100% P) model is
used. It can also be inferred that if a relatively small
portion, in this case 20%, of the whole system load is
represented as equivalent induction motors and the
remainder load as constant impedance the results are
practically the same as the case when the whole system
load is represented as constant impedance (100% Z).
However, for this test system, when representing a greater
portion of the system load as induction motors, in this
case 50% of the whole system load, the damping
associated with one of these modes is less than the value
obtained for the constant impedance case (100% Z). On
the other hand, the other mode is more damped than the
case where the system load is represented as constant
power (100% P). From those results is easily inferred that
in this test system, the load representation has a very
Fig. 8 Speed eigenvector associated with the 1.08 Hz
local mode
1.
Output: WW
Eigen: -0.5641 + J7.0439
G4
f = 1.1211 Hz
G1
G3
G2
-
0.7
Fig. 9 Speed eigenvector associated with the 1.12 Hz
local mode
Participation factors associated with the above
electromechanical modes indicate that the rotor variables
with the highest participation on the inter-area mode (0.54
Hz) are those of G3 and G4. These factors indicate that
58
the rotor variables of the aforementioned generators also
have the highest participation on the local mode which
frequency is 1.12 Hz. While the rotor variables of
generators G1 and G2 have the highest participation on
the local mode which frequency is 1.08 Hz.
Results shown in Table 7, indicate that the influence of
the load representation on the local electromechanical
modes of test system 2 is similar to that identified on the
local modes of test system 1, where only local modes are
present. Regarding the inter-area mode, the results point
out that the highest value for the damping of this mode is
obtained when the whole system load is represented as
constant current (100% I) and the lowest damping value
of this mode is obtained when the whole system load is
represented as constant power (100% P). In fact, for the
operating condition considered in this paper the inter-area
mode turns unstable when the whole system load is
represented as constant power. Furthermore, the results of
these simulations also suggest that when a portion of the
system load is represented as aggregated induction motors
the damping of the inter-area mode increases as the
portion of the system load represented as induction motor
increases.
Finally, the results from the different simulations
performed on the test systems considered in the paper,
suggest that the influence of load representation on the
frequency of oscillation associated with the “local”
electromechanical modes is rather negligible.
Module
Phase
Bus Number
Variable
1.0000
0.0000
3
Delta
0.8806
-1.7344
3
Speed
0.6617
0.3173
4
Delta
0.5761
-1.6435
4
Speed
Table 4 Participation vector of the 0.54 Hz mode
Module
Phase
Bus Number
Variable
1.0000
0.0000
4
Speed
0.9629
0.9421
4
Delta
0.7002
3.2664
3
Delta
0.6666
3.5738
3
Omega
Table 5 Participation vector of the 1.12 Hz mode
Module
Phase
Bus Number
Variable
1.0000
0.0000
2
Delta
0.5463
-1.2538
2
Speed
0.4305
1.9926
1
Speed
Table 6 Participation vector of the 1.08 Hz mode
Load Representation
100% Z
100% I
100% P
50% Z and 50% I
6
Conclusion
Results from the simulations carried out on the two test
systems considered in this paper, lead to the following
conclusions on the influence of load representation on the
damping of the electromechanical modes of a power
system;
The lowest damping of a local mode is obtained when the
whole system load is modeled as constant impedance.
The higher the amount of the system load modeled as
constant impedance the lower the damping of the local
modes.
The highest damping of a local mode is obtained when the
whole system load is represented as constant power.
The highest damping of the inter-area mode is obtained
when the whole system load is represented as constant
current. While the lowest value is obtained when the
whole system load is represented as constant power.
Representing a portion of the system load as aggregated
induction motors yields a better damping of the inter-area
mode than the case where the whole system load is
represented as constant impedance. The higher the portion
of the system load modeled as aggregated induction
motors, the higher the damping associated with the
electromechanical modes.
From a practical point of view, the influence of load
representation the frequency of oscillation of the local
modes may be neglected. This finding may not apply to
the frequency of oscillation of inter-area modes,
particularly when the system load be represented as
constant power.
Finally, the results shown in the paper have pointed out
the influence of the load representation when assessing
Damping Frequency
(%)
(Hz)
0.1033
0.5401
8.1843
1.0855
7.9828
1.1211
0.7569
0.5377
8.2643
1.0828
8.1699
1.1132
-12.774
0.3976
9.1348
1.0696
8.5791
1.0918
0.3303
0.5411
8.2160
1.0844
8.0528
1.1179
0.3574
0.5422
8.2146
1.0843
8.0500
1.1178
0.1238
0.5407
8.1962
1.0853
8.0104
1.1205
50% Z and 50 % I for
active power and 100%
Z for reactive power
7 % of active power as
LIM and the remainder
active and reactive powers
as constant impedance Z
50% of active power as
0.1597
0.5413
LIM and the remainder
8.2120
1.0850
active and reactive powers
8.0497
1.1197
as constant impedance Z
Table 7 Damping of electromechanical modes for
different load representations.
59
the damping associated with the electromechanical modes
of a power system.
Vt -
References
Appendix I
Block diagram and data parameters of the excitation
systems used in the simulations reported in this paper.
Vt -
75
1 + 10.0s
1 + 0.01s
EFD
+
Vref
Fig. 11 Excitation system model used in test system 2
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[12] CEPEL, PacDyn User’s Manual, 2002
200
1 + 0.01s
EFD
1 + 0.01s
+
V ref
Fig. 10 Excitation system model used in test system 1
60
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