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Chaotic Oscillationofa Three-bus Power System Model
Using ElmanNeural Network

I Made Ginarsa1, Adi Soeprijanto2, Mauridhi Hery Purnomo3

1Dept. of Electrical Engineering, Mataram University, Mataram
2Dept. of Electrical Engineering, Sepuluh Nopember Institute of Technology, Surabaya
3Dept. of Electrical Engineering, Sepuluh Nopember Institute of Technology, Surabaya

e-mail: kadekgin@yahoo.com1, adisup@ee.its.ac.id2, hery@ee.its.ac.id3

Abstrak

Paper ini meneliti dan membahas secara mendalam mengenai osilasi chaotic pada sistem
tenaga listrik.Dengan menggunakan sebuah three-bus pada sistem tenaga listrik, rute mungkin
menyebabkan unjuk kerja chaotic sehingga dievaluasi, digambarkan serta dibahas dalam
penelitian ini. Osilasi chaotic ini dimodelkan menggunakan Elmanneural network karena
bentuknya yang sederhana dan juga melibatkan algoritmabackpropagation dengan adaptive
learning rate dan momentumnya.Unjuk kerja learning rate dan momentumnya lebih baik
dibandingkan jika tanpa momentumnya. Unjuk kerja chaotic dalam sistem tenaga listrik muncul
karena sistem ini dioperasikan dalam mode critical. Unjuk kerja chaotic ini terdeteksi dengan
munculnya sebuahchaotic attractordalam phase-plane trajectory.

Kata kunci:sistem tenaga listrik, Elman neural network, chaotic attractor, phase-plane trajectory

Abstract

Chaotic oscillation of power systems was deeply studied in this paper. By using a three-bus
power system, route may cause chaotic behavior in power systems are evaluated, illustrated
and discussed.Chaotic oscillationof power systems was modeled using Elman neural network
because the Elman neural networkhas a simple form. Backpropagation algorithm with adaptive
learning rate and momentum was proposed in this research. Performance of learning rate with
momentum was better than learning rate without momentum. Chaoticbehaviors in a power
system appeared due to the system operated in critical mode. A Chaotic behavior in power
systems was detected by appearing a strange attractor (a chaotic attractor) in phase-plane
trajectory.

Keywords:power systems, Elman neural network, chaotic attractor, phase-plane trajectory

1. Introduction

In recent years, electric power consuming has grown up rapidly. On the other hand, the power
plants and transmission systems being built are very slow due to environmentaland economical
constraints. This condition will make the power systems operate in critical mode at the boundary
of stability region. Meanwhile, chaotic phenomena is one type of un-deterministic oscillations
exist in deterministic systems such as in power system model.Chiang et al, have builtvoltage
collapse model, both physical explanations and computational considerations of this model are
presented. Static and dynamic models are used to explain the type of voltage collapse, where
the static is used before a saddle-node bifurcation and the dynamic model is employed after the
bifurcation [1]. Lyapunov exponent, measuring how rapidly two nearby trajectories separate
from one another within state space and broad-band spectrum was used to confirm the
observation [2]. Within the range of loading conditions, the sensitive dependence feature of
chaotic behaviors makes the power system unpredictable after a finite time. In addition, within
the range the effectiveness any control scheme was questionable and should bere-evaluated
based on state vector information.Furthermore,nonlinear phenomena including bifurcation,
chaos and voltage collapse occurred in a power system model. The present of the various
nonlinear phenomena was found to be a crucial factor in the inception of voltage collapse in this

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model. The problem of controlled and suppressed of the presence of non-linear phenomena in
power systems were addressed here in this paper. The bifurcation control approach is approach
to modify the bifurcations and to suppress chaos [3,4]. The presence of chaos in a power
system causing seriously unstable problem was studiedby Yu, et al.[5]. The existence chaos in
power systems due to disturbing of energy at rotor speed has been found in Ref.[6]. One
scheme of chaos utility was used on electrical systems for smelting which was based on chaos
control. Lei et al. Demonstratedthat chaotic steel-smelting ovens regulate their heating current
according to chaos control theory [7]. A control system using a neural network controller was
presumed to be able to stabilize the unstable focus points of 2-dimensional chaotic systems;
although, Konishiand Kokame stated that the control system did not require this presumption
[8]. Elman neural network was used to predict short-term load forecasting in power systems [9].
Modeling of chaotic behavior using RNN has been studied in [10]. Various studies on controlling
transient chaos have been carried out, such as those by Dhamala et al., and Dhamala and Lai
attempted to control transient chaos in power systems using a data time series [11,12].
Strategies for controlling chaos in process plants have been tested on the Henon mapdiscrete
chaotic system [13].

In this paper, we focused on the cause of chaotic oscillation in power systems and its model. By
using Elman neural network model is proposed. The reason of using the Elman neural network
because the Elman network is able to traindata both on present input and on past output, and
other reason because an Elman RNN has simple form.

This paper is organized as follows: in advance, power system model used in this research is
given in Section 2. Then, Elman neural network model isexplained in Section 3. Chaotic
behavior due to sensitivityof initialcondition and analysis a chaotic behavior are presented in
Section 4 and 5, respectively. The conclusionis given in the last section.

2. Power System Model

A Synchronous machine was modeled as a voltage (Eq0’) behind a direct reactance (xd’). The
voltage magnitude was assumedas remaining constant at the pre-disturbance value, as shown
in Fig.1(a).De Mello and Concordia as well as Padiyar and Kundur derivedof a machine
connected toan infinite bus [13,14]. Meanwhile, if saturation and the stator resistance were
neglected, the system condition was balanced with a static load. The mechanical mode block
diagram of single-machine connected to infinite bus is shown in Fig.1(b).

Figure 1.Single machine connected to infinite bus.

(a) Circuit equivalent (b)Mechanical mode.

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The machine wasconnected to infinite bus and supplied the load. Then the armature current
flowedfrom the machine to the load. This current causedelectrical torque on the stator winding,
and vice versa. The mechanical torque was produced by flux through the rotor winding.
Meanwhile, whenthe rotor speed wasconstant, the rotor speed followed thesynchronous speed.
When there was imbalanced energy, the rotor speed accelerated or decelerated and caused the
swing equation.The swing equation is represented as follows:

ema TTTDH (1)

Where D, are amping constant and rotor speed deviation, respectively. Eq.1 is a basic
equation for mechanical modeof single machine connected to infinite bus. Furthermore,the the
Eq. 1 can be expressed as follows:

B (2)

DTT
M em
1

(3)

Where Tm, Te, , , D and M are mechanical torque, electrical torque, power angle, speed
rotor, damping constant, inertia constant respectively.The system was developed from Ref.[3]
and shown in Fig.3, which is regarded as one synchronous machine supplying power to a local
dynamic load shunt with a capacitor (Bus 2) and connected by weak tie line to the extern
system (Bus 3). The system equations are:

. (4)

881.1..333.3
087.0sin667.16

d
VLL (5)

333.43333.33
209.0cos667.666

333.93
087.0cos667.166

872.496

Q
V

V
V

(6)

033.7229.5
135.0cos869.104523.14

012.0cos217.26
764.78

Q
V

V
VV

(7)

Table 1. Power system parameters
Y0 Ym 0 m V0 Vm Pm M
20.
0

5.0 5.
0

5.
0

1.0 1.0 1.0 0.3

D T C Kp Kpv Kq Kqv Kqv2
0.0
5

8.5 12.
0

0.4 0.3 0.0
3

2.
8

2.1

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Figure 2.One line diagram power system with 3 buses.

, , d, Qld, L,VL, arethe power angle, rotor speed deviation, damping constant, reactive load,
voltage angle and magnitude at load bus, respectively. Eqs.4,5,6,and7 can be simplified into a
uniform equation in Eq.8.

pn RRxxfx ,,, , (8)

Where x is vector state variables and is vector of parameters. The state variables are x =
[ , , L,VL]

T, superscript T denote transpose of the associate vector.

3. ElmanNeural Network Model

Recurrent Elman network commonly is a two-layer network with feedback from the first-layer
output to the first-layer input. This recurrent connection allows the Elman network to both detect
and generate time-varying patterns. A two-layer Elman network is shown in Fig.3. The Elman
network has tansig neurons in its hidden (recurrent) layer and purelin in its output layer. The
Elman network differs fromconventional two-layer networks in that the first layer has a recurrent
connection. The delay in this connection stores values from the previous time step, which can
be used in the current time step. Thus, even if two Elman networks with the same weight and
bias, are given identical inputs at a given time step, their outputs can be different due to different
feedback states. Because network can store information for future reference, it is able to learn
temporal pattern as well as spatial patterns [15,16,17,18]. The Elman network can be trained to
respond and to generate, both kinds of patterns.

2
1

1,2
2

1
1

1,11,1
1 1tansig

bnaLWpurelinna

bnaLWpIWna
. (9)

The architecture 4:8:8:4 RNN is used in this research. Where p,a1(n), a2(n), IW1,1, LW1,1, LW1,2,
b1 and b2 are the vector input, recurrent-layer output, purelin-layer output, weight first-layer,
weight hidden layer back to first-layer, weight hidden layer to output layer and biases,
respectively.

Figure 3.Elman recurrent neural network block diagram[18]

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The RNN wastrained by using 1000 data points. Tansig and purelin activation function were
used at hidden layer and at output layer, respectively. Data time series were obtained from the
mathematical (exact) model in Eqs.4-7, respectively. The network performance is measured by
mean square error (MSE). Formula of the MSE can be expressed by equation as follow:

i
nn xxk

MSE
1

2ˆ1 (10)

Where k, nx and nx̂ are the size ofdata, input and estimation n

th data.

4. Chaotic Behavior due to Sensitivity of InitialCondition

Chaos definition and its properties have been given by Devaney and Alligood et al.[19,20].
Sensitivity of initial condition is one type of chaos properties. It is described by existing route to
chaotic behavior in power systems caused by sensitivity of initial condition rotor speed ( 0).
Initial rotor speed ( 0) in power systems was presented by disturbing ofenergy (DE). Kinetic
energy disturbance was related to rotor speed deviation only. The large rotor speeddeviation
was implemented as a large DE. When DE was smaller than the value of 1.3824 rad/s
( 0<1.3824 rad/s)a power system converged to a stable equilibrium point. When the DE was
increased, the convergencebecame more difficult. At 0 = 1.3825 rad/s, power systems
produced route to a chaotic behavior in a longer time.When the DE was from1.3825 to 17003
rad/s, the final states were controlled by a chaotic behavior. Furthermore, while the DE excess
than 1.7004 rad/s the system went to divergence or voltage collapse. Based on the simulation
result it is shown that chaotic behavior in power systems due todisturbing of energy at the rotor
speed deviation.

Table 2. System conditionwith different initial rotor speed ( 0)

0(rad/s) Times (s) Final state Time response
0.5 1000 Equilibrium point Fig.4(a)

1.3824 1000 Equilibrium point Fig.4(b)
1.3825 1000 Chaotic Fig.5(a)
1.7003 1000 Chaotic Fig.5(b)
1.7004 10 Divergen -

Figure 4.Simulation results with equilibrium point state

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Figure 5.Simulation results with chaotic state

Figure 6. (a).Chaotic behavior of the rotor speed deviation

(b). Magnified of Fig. 5 fromtime = 0 to time = 50 s

5. Result and Analysis

In this research, RNN initial simulation parameters were taken: learning rate train parameter =
0.17; increment learning rate = 1.2; decrement learning rate = 0.6; and momentum learning rate
= 0.75. The training performance of RNN using adaptive learning rate and adaptive learning rate
with momentum are listed in Table 3.The training process is organized as follows: performances
(MSE) are obtained to 14.7001 10 4 and 4.2209 10 4 at disturbance 0 = 0.5 rad/s for algorithm
backpropagation adaptive learning rate (traingda) and backpropagation learning rate algorithm
with momentum (traingdx),respectively. Moreover,performances were obtained to 16.8361 10 4
and 4.6115 10 4 at disturbance 0 1.3825 rad/s. Furthermore, performances were obtained to
17.4185 10 4 and 4.9442 10 4at the disturbance 0at the value of 1.7003 rad/s. During the
training process the best performancewas obtained to 4.2209 10 4 at the disturbance of 0.5
rad/s.

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Figure 7. The chaotic behavior of the at 0 =1.7003 rad/s
(a). Blue = exact model; red = RNN model (b). Error signal ofthe

Figs.7-9 show the time responses of an exact and Elman recurrent neural network (RNN)
model. Fig.7(a) shows rotor speed deviation ( ) time response which was oscillated due to the
disturbance occurred at 01.7003 rad/s. Rotor speed oscillations exist in range from 1.6052 to
1.5679 rad/s and from 1.511 to 1.6045 for the exact and RNN, respectively. Fig. 7(b) shows
error signal of the rotor speed deviation; where the error signal is the difference of the exact and
RNN model of the rotor speed deviation.

Voltage angle ( L) at Bus 2 is affected by disturbing of energy (DE) at generator bus ( 00.5
rad/s). The oscillation on voltage angle occurred at generator bus in a few second,then this
oscillation decreased gradually and route to equilibrium point (fixed point) at point of 0.1128and
0.1116 rad for exact and RNN models, respectively. The error signal of the voltage angle was
measured by mean square error (MSE = 3.8193%), and these results are shown in Table 4.

Figure 8.The chaotic behavior of the voltage angle when 0 at 1.7003 rad/s.
(a). Blue = exact; red = RNN (b). Error signal of the L

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Figure 9. The voltage magnitude (VL) time response at 0 = 1.7003 rad/s
(a). Blue = exact model; red = RNN model (b). Error signalof the VL

The voltage angle oscillation increased at the disturbance 1.3825, 1.600 and 1.7003 rad/s for
exact model with amplitude in ranges (0.0600 to 0.1995 rad), (0.0351 to 0.2730rad), (0.0345 to
0.2748rad) and (0.0340 to 0.2756rad), respectively. And the oscillation for RNN model arefrom
0.0501 to 0.1879 rad, from 0.0460 to 0.2644rad, from 0.0332 to 0.2618radand from 0.0342 to
0.2613rad, respectively. This oscillation occurred in a longer time. Voltage angle time response
occurring at disturbance 01.7003 rad/s can be shown in Fig. 8.

When the disturbance ( 0) at the value of 0.5 rad/s,the voltage magnitude oscillated in a few
seconds. Furthermore, its decreased gradually route to equilibrium state (fixed point) at point
1.095 pu and 1.008 for exact and RNN model, respectively. By increasing disturbance at 0
1.3824 rad/s voltage magnitude is oscillated in a longer time in ranges (0.9967 to 1.1207pu) and
then amplitude reduced and fixed point at 1.1095 pu (1520 s).

On the opposite, when the disturbing of energy was increased up to 1.3825, 1.600 and 1.7003
rad/s, voltage magnitude oscillated for the exact model where the amplitude increased
from0.8307 to 1.1220pu, from0.8285 to 1.1118puand from 0.8290 to 1.1119pu, respectively.
And the oscillation for RNN model was in the ranges from 0.8497 to 1.1158pu, from 0.8580 to
1.1235puand from 0.8642 to 1.1185pu, respectively.In Fig.9, we can show that the voltage
magnitude of the exact and RNN modelsexhibit chaotic behavior.

Table 3.Performance of training algorithm using learningrate momentum

0
(rad/s)

Training Times (s)
102

Performances
MSE ( 10-4)

traingda traingdx traingda traingdx
0.5 69.3861 37.403 14.7001 4.2209

1.3824 68.3250 42.342 17.2014 4.9080
1.3825 67.3329 36.750 16.8361 4.6115
1.7003 70.5781 41.840 17.4185 4.9442

State trajectory(orbit) of the against is shown in Fig.10, where many circlesare made by
themselves with boundary ranges from 1.6011 to +1.5535 rad/sandfrom 0.1165 to +0.7583 rad
for the min- maxand min- max, respectively. Thestate trajectoriesofthe RNN model are made in
rangesfrom 1.6020 to +1.5524 rad/sandfrom 0.1145 to +0.7598 rad, respectively. The
attractive form of the - is known as strange attractor (chaotic attractor).The strange
attractorsof the LagainstVLare shown in Fig.11. The strange attractor coordinateswere from
0.0345to 0.2748 rad and from 0.8285 to 1.1118pu for Lmax- Lmin and VLmax-VLmin, respectively.

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Meanwhile, the RNN model of the L-VLwas from 0.0332to 0.2618 rad and from 0.8280 to
1.1235pu for Lmax- Lmin and VLmax-VLmin, respectively.

Table4.Power system state when variation of the DE was applied.
0&Model (rad) (rad/s) L (rad/s) VL (pu)

0.5Exact eq 0.3095 osc 0.2104 to
0.2123

eq
0.1128

eq 1.095

RNN eq 0.3194 osc 0.2008 to
0.2010

eq
0.1116

eq 1.008

MSE (%) 0.2636 11.1792 3.8193 8.7051
1.3824Exact osc 0.0245

to 0.6160
osc 1.1546 to

1.1049
osc0.0600 to

0.1995
osc0.9967 to

1.1207
RNN osc 0.0256

to 0.6165
osc 1.0246 to

1.0049
osc0.0501 to

0.1879
osc0.9970 to

1.1135
MSE (%) 3.9625 6.3023 0.2040 0.1154
1.3425Exact osc 0.1156

to 0.7578
osc 1.5711 to

1.5142
osc0.0351 to

0.2730
osc0.8307 to

1.1220
RNN osc 0.1148

to 0.7510
osc 1.5734 to

1.5165
osc0.0460 to

0.2644
osc0.8497 to

1.1158
MSE (%) 0.68 0.23 1.09 1.90
1.6000Exact osc 0.1165

to 0.7583
osc 1.6011 to

1.5535
osc 0. 0345
to 0. 2748

osc 0.8285 to
1. 1118

RNN osc 0.1645
to 0.7598

osc 1.6020 to
1.5524

osc0.0332 to
0. 2618

osc0.8580 to
1. 1235

MSE(%) 0.2163 2.8779 0.0460 0.0407
1.7003Exact osc .1157 to

0.7601
osc 1.6052 to

1.5679
osc 0. 0340
to 0. 2756

osc 0.8290 to
1. 1119

RNN osc 0.1345
to 0.7457

osc 1.511 to
1.6045

osc 0.0342 to
0. 2613

osc 0.8642 to
1. 1185

MSE(%) 1.0522 17.8296 0.1284 0.1470
Note: eq = equilibrium point (fixed point); osc = oscillation.

Figure 10. State trajectory of the - when disturbance was applied at 0 = 1.600

rad/s

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Furthermore, existence of the chaotic attractors can also be depicted in Figs.12 and 13 for the
01.7003 rad/s. Fig.12 was produced by the againsts state trajectories at coordinates from
1.6052 to +1.5679 rad/sandfrom 0.1157 to +0.7601 rad for the min- max and min- max,

respectively. The results of theRNN model are depicted by red circles at coordinates from
1.5110 to +1.6045 rad/sandfrom 0.1345 to +0.7457 rad for the min- max and min- max,

respectively.

Fig.13 shows the LagainstVL state trajectories at coordinates from 0.0351to 0.2756 rad and from
0.8290 to 1.1119 pu for the Lmax- Lmin and the VLmax-VLmin, respectively. State trajectories of the
RNN model can be depicted by red points at coordinates from 0.0342to 0.2613 rad and from
0.8642 to 1.1185 pu for the Lmax- Lmin and the VLmax-VLmin, respectively. The complete simulation
results are tabulated in Table 4.

Figure 11. The L-VL state trajectory when the DE at 0 = 1.6 rad/s was applied

Figure 12. The - state trajectory when the DE at the value of 1.7003 rad/s was

applied to a power system

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Figure 13. The L-VLstate trajectory when the DE at the value of 1.7003 rad/s was applied

to a power system

Based on the in Table4that the largest MSE was 17.8296, where the largest MSE was obtained
onthe speed rotor deviation ( ) at the value of 1.7003 rad/s. Simulation results show that
chaotic behavior of power systems can be modeled by the Elman recurrent neural network.

6. Conclusion

Chaotic oscillationsin power systems using exact and RNN models are deeply studied in this
research. The exact model was obtained using mathematical model. Then, the RNN model is
obtained by training process using the data from exact model simulation. The training of the
RNN model using adaptive learning rate both with and without momentum is compared. The
performace of the adaptive learning rate with momentum is better than the other one. Chaotic
behaviors are detected in power systems by appearing chaotic attractors both at power angle-
rotor speed and at magnitude-angle voltage state trajectories in phase-plane.

7. Future Works

Chaotic behavior of power systems was an interest topic research in recent years. In the future,
thechaotic behavior of power systems should be reduced and vanished by applying control
strategy properly.

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