Assessment of Steady State Stability In MultiMachine Using Synchronous Power Coefficients.
Ayoade Benson OGUNDARE
Lagos State Polytechnic, Ikorodu, Lagos, Nigeria.
ayoadeogundare@yahoo.com
I. A. ADEJUMOBI
Federal University of Agriculture, Abokuta, Nigeria.
engradejumobi@yahoo.com
Abstract
The main aim of any electric power supply is to provide stable power supply at all times to all its consumers. Steady
state stability refers to the ability of the synchronous generators to remain in equilibrium after gradual load
changes. These will enable the power system engineers to make necessary planning. This paper presents steady state
stability in multi-machines by using synchronous power coefficients method. This method seeks change in active
power of generator with respect to change in systems load. In this method, the characteristics of transmission lines
that describe power systems network was retained by eliminating all buses except the generator internal buses of the
Y-admittance matrix by matrix partitioning. The reciprocal of the off-diagonal elements for the reduced admittance
matrix gives the transfer function. The voltages behind steady state reactance of the generators were determined.
The power transfer by the transmission network was obtained by steady state stability limit. An algorithm is
developed and implemented using MATLAB programming language. From the results, synchronous power
coefficients are all positive and therefore, the synchronism of the generators is not lost and hence the stability is
maintained.
KEYWORD Stability, synchronism, Elimination, Transmission,
1.0
Introduction
In order to achieve equilibrium state between the
One of the ways through which the development in a
production and consumption of electric energy, the
country can be assessed is the rate at which the
electrical performance and the operational features of
electricity being generated is utilized by the
the power system have to be studied. The most
consumers. As a result of high demand for electric
important aspect of this study is load-flow study.
energy in the country, an effort has to be made so as
to increase the level of production of electric energy
in order to meet the demand by the consumers.
Hence, the result of load-flow study can be used to
compute the losses in electrical network distribution
Ayoade Benson OGUNDARE & I. A. ADEJUMOBI
Page 1
as well as the flow of real and reactive powers for all
load changes, changes in excitation and prime mover
equipment-connecting buses. Load-flow analysis is
or line switching.[ 3]. This has made steady state
essential in planning the future development of the
stability of power system essential since minor
power system because satisfactory operation of the
disturbances occur continuously in the system.
system
Flexible
depends
on
knowing
the
effects
of
AC
transmission
system
(FACTS)
interconnections with other power systems of new
controllers have been mainly used for solving various
generating stations and new transmission lines before
power system steady state control problems [4].
they are installed.
The power flow through the AC transmission lines
2.0
mainly depends on line impedance, the magnitude of
Preliminary Theoretical Development
the sending end voltage, the magnitude of the
The continuous demand of electrical energy has
receiving end voltage and the phase angle between
caused the power system to be heavily loaded,
the sending and receiving end voltage. The power
leading to voltage instability. This has threaten the
flow through the transmission line is increased by
major objective of power system to effectively
providing suitable reactive power compensation [5].
generate, transmit and distribute electrical energy
In earlier days the transmittable power was increased
with
minimum
ecological
disturbance
over
by cancelling a portion of reactive line impedance by
transmission line from generating stations to load
using series capacitive compensation. Most of the
centres. Under heavy loaded conditions of power
power systems are subjected to electromechanical
system, there may be insufficient reactive power
oscillations. Power system stabilizers are used to
causing remarkable voltage drop at various buses.
eliminate these oscillations to some extent. Because
This result to loss in system stability hence voltage
of the implementation of fast acting power electronic
collapse which leads to blackout [1]. As demand for
based switching incorporated FACTS controllers,
electric power has been increased, power systems
these oscillations damp out in high speed and thus
have become more complex and modern control
improve the system stability. Many of the FACTS
systems are required for maintaining system stability
devices are connected in shunt or in series either
[ 2]. Steady state stability is concerned with the
without storage element or with storage element for
ability for the system generators to remain in
effective compensation [4, 5]. Changes in load affects
synchronism after minor disturbances such as gradual
Ayoade Benson OGUNDARE & I. A. ADEJUMOBI
Page 2
active power flow more than the reactive power flow
load demand, analysis of effectiveness of alternative
hence, the active power flow is used as a measure of
plans for future system expansion and continuous
stability in power systems. The study of power
monitoring of the current state of the system, such
system stability is therefore pre- requisite for normal
information is very essential. [10,11]
system operation. In multi-machine stability studies,
3.0
the only method that gives satisfactory result is the
Load-flow study in the power system parlance is the
step –by-step integration method, which is the most
steady state solution of the power system network.
reliable one[6]. It gives accurate result and can handle
The power system is modelled by an electric network
the complicated system model.
and solved for the steady-state powers and voltages at
It is of utmost importance to calculate the voltages
various buses in an electric network under existing or
and current at different parts that power system are
contemplated conditions of normal operations.
Newton-Raphson in Load Flow Studies
exposed to. This is essential not only to design the
For any network, there are two equations for each
different power system components [7] Such as
load bus, given by equations
generators, lines, transformers, shunt elements, etc
n
but also this can withstand the stress they are exposed
Pi = ∑ Vi Vk Yik cos(δ k − δ i + θik )
k =1
to during steady state operation without risk of
(1)
damages. Further more, for economical operation of
n
the system; the losses should be kept at low value.
Qi = − ∑ Vi Vk Yik sin (δ k − δ i + θ ik )
k =1
Taking various constraints into account, and risk that
the system enters into unstable modes of operation
(2)
Where i = 1, 2, ..., n
must be supervised. In other to do this satisfactorily,
And one equation for each voltage-controlled bus,
the state of the system must be known. The main
given by (1). These equations constitute a set of non
information
obtained
from
load
flow
studies
linear algebraic equations in terms of the independent
comprises the magnitudes and phase angles of load
variables, voltage magnitude in per unit, and phase
bus voltage, reactive power at generator buses, active
angle in radians.
The Jacobian matrix gives the
and reactive power flow in transmission lines, other
linearized relationship between small changes in
variable being specified [6, 9, 10]. To meet increase
Ayoade Benson OGUNDARE & I. A. ADEJUMOBI
Page 3
voltage angle and voltage magnitude with the small
where
change in real and reactive powers respectively.
Jacobian matrix respectively
3.1
H ik , N ik , J ik and Lik are the elements of the
3.2 Derivation of Power Transfer Index
Formation of Jacobian Matrix
Assuming a generator supplying power to a motor
The expressions to be used in evaluating the elements
through a transmission line as shown in figure 1
of Jacobian matrix are derived from power flow
equations (1) and (2).
The first derivative of a set of functions gives the
Jacobian elements. Power flow equations can be
Figure 1: Power flow modelling.
expressed as
∆C = − J ( x ) × ∆x
=
(3)
where
 Pi ( scheduled ) − Pi ( x )  ∆Pi ( x ) 
∆C = 
=

Qi (scheduled ) − Qi ( x ) ∆Qi ( x )
X=
(8)
+
(9)
(10)
(4)
And
∆x = J
−1
[ ∆C ]
=
(5)
correction vectors for both load and generator buses.
J (x) can be written as
 − J (x ) − J12 (x )
J ( x ) =  11

− J 21 ( x ) − J 22 (x )
)
(6)
"#
(11)
$
%
(12)
& ' &
"
(
(13)
%90 - #
# '
(14)
) 123 90
4
(7)
!
" 90, -
where
∂P( x )

− J11 ( x ) =
= H ik 
∂δ

∂P (x )
− J12 ( x) =
= N ik 

∂V


∂Q (x )
− J 21 (x ) =
= J ik 

∂δ

∂Q(x )
− J 22 (x ) =
= Lik 
∂V

∗
The Jacobian matrix can be re-arranged to get the
The Jacobian matrix
-
# -#
(
$cos 90
# -#
(15)
# -#
(16)
123#
)
$ -
561#
(
(17)
Ayoade Benson OGUNDARE & I. A. ADEJUMOBI
Page 4
Thus, the active power transfer through the
3.3 Network Reduction
93:
transmission line is;
123#
.
are the voltages calculated behind internal
steady state reactance.
(18)
The active power can be shown graphically as in
;<
figure 2
From load flow solution,
<
< 8> ?@>
A>∗
,
i = 1, 2……….m
m is the number of generators.
19
(20)
∗
< , !< , ;< Values
are
determined from the initial load flow solution.
Load equivalent admittances are given by;
C<D
8> ?@>
(21)
|A> |)
To include voltages behind steady state reactances, m
Figure 2: Representation of power flow
Any
change
in
system
loading
affects
the
buses are added.
n+1
G
n- bus network.
displacement of load angle. When the system is
operating within the region OA, any increase in
n+2
.
.
.
.
system load is matched by an increase in the active
power from the generator hence,
78
7&
(synchronous
power coefficient) is positive. At point A, the
generator active power is constant, then
78
7&
is zero.
Loads are converted
to constant
admittances.
G
n+m
G
Figure 3: Power system representation for stability
In the region AB, any increase in system load is
accompanied by a decrease in the active power from
the generator, therefore
78
7&
is negative.
78
7&
The load buses are eliminated by matrix partitioning
should be
greater than zero for the system to remain stable.
Ayoade Benson OGUNDARE & I. A. ADEJUMOBI
Page 5
FG H
;K
-JK P JK
I
JKK
J K
K
JK
L M AN O
N
J
(23)
(24)
Substitute (24) into (23)
$J
-J
P
K JKK JK
(
K
(25)
The resulted admittance matrix is J
- J K JKKP JK and the full characteristic of the original matrix is not lost.
The reciprocal of the off-diagonal elements of Y reduced gives the transfer function.
4.0 Data Acquisition
The method for this work is demonstrated using the parameters shown in the tables 1 and 2 below.
Table 1.0 Bus- data
Bus
No
Bus
type
Volt.
mag.
Ang.
Deg.
Pd
MW
Qd
Mvar
Pg
MW
Qg
Mvar
Qmin.
Qmax.
1
1
1.040
0
0
0
0
0
0
0
2
3
1.000
0
0.000
0.000
0.000
0.000
0
0
3
3
1.000
0
150
120
0
0
0
0
4
3
1.000
0
0.000
0.000
0.000
0.000
0
0
5
3
1.000
0
120
60
0.000
0.000
0
0
6
3
1.000
0
140
90
0.000
0.000
0
0
7
3
1.000
0
0.000
0.000
0.000
0.000
0
0
8
3
1.000
0
110
90
0.000
0.000
0
0
9
3
1.000
0
80
50
0.000
0.000
0
0
10
2
1.035
0
0.000
0.000
200
0.000
0
180
11
2
1.030
0
0.000
0.000
160
0.000
0
120
Ayoade Benson OGUNDARE & I. A. ADEJUMOBI
Page 6
Table 2.0 Line and Transformer data.
From
Bus No.
1
To
Bus No
2
R, PU
X, PU
P
0.000
0.006
0.000
Bus and Line data of tables 1 and 2 respectively, are
2
3
0.008
0.030
0.004
used to run the load flow using Newton-Ralphson
2
5
0.004
0.015
0.002
2
6
0.012
0.045
0.005
3
4
0.010
0.040
0.005
3
6
0.004
0.040
0.005
while table 5.0 shows the Voltage magnitude/angle
4
6
0.015
0.060
0.008
calculated behind steady state reactance.
4
9
0.018
0.070
0.009
4
10
0.000
0.008
0.000
5
7
0.005
0.043
0.003
6
8
0.006
0.048
0.000
7
8
0.006
0.035
0.004
7
11
0.000
0.010
0.000
8
9
0.005
0.048
0.000
Q
5.0
, PU
Result and Analysis.
method and the result displayed in table 3.0.The
reduced bus admittance matrix is shown in table 4.0
Table 3.0 Power Flow Solution by Newton-Ralphson Method.
Bus
no
1
Vol.
Mag
1.040
Ang.deg.
Pd (MW)
Qd (Mvar)
Pg (MW)
Qg (Mvar)
0.000
0.000
0.000
246.646
206.451
2
1.028
-0.793
0.000
0.000
0.000
0.000
3
0.997
-1.970
150.00
120.00
0.000
0.000
4
1.024
-0.608
0.000
0.000
0.000
0.000
5
1.017
-1.318
120.00
60.00
0.000
0.000
6
0.993
-2.277
140.00
90.00
0.000
0.000
7
1.021
-0.348
0.000
0.000
0.000
0.000
8
0.985
-2.414
110.00
90.00
0.000
90.00
9
0.981
-2.798
80.00
50.00
0.000
0.000
10
1.035
0.257
0.000
0.000
200.00
141.00
11
1.030
0.524
0.000
0.000
160.00
95.00
Ayoade Benson OGUNDARE & I. A. ADEJUMOBI
Page 7
78)
Table 4.0 Reduced Bus Admittance Matrix
0.4044
+
0.4006+1.2902i
7&)R
0.2718+0.8352i
= 1.5133 ×
1.2735√0.4006Q + 1.2902Q cos(13.4128-17.6434)
3.5157i
=2.596˃0
0.4006+1.2902i
0.6446-3.9705i
0.2718+0.8352i
0.3069+0.9567i
0.3069+0.9567i
0.2543-2.8759i
78)
7&QT
(31)
= 1.2735 × 1.3193
√0.3069Q + 0.9567Q cos(13.4128-17.6434)
=1.683˃0
Table 5.0 Voltage Magnitude/Angle.
78S
(32)
=
G(i)
E(i)
d0(i)
1
1.5133
18.2666
1.3193 × 1.5133√0.8352Q + 0.2718Q cos( 17.6434-
10
1.2735
13.4128
18.2666)
11
1.3193
17.6434
7&SR
=1.753˃0
(33)
78S
P
PQ
R )
123#PQ
Q
QP
QT
R)
) R
)R
T
TP
S R
SR
PQ
123#QP
R S
RS
123#PT ) S
)S
(26)
1.2735√0.9567Q + 0.3069Q cos(17.6434-13.4128)
=1.683˃0
123#QT 5.0
(34)
Conclusion:
(27)
coefficients (
TQ
123#TP
1.3193 ×
=
7&S)
7^
7&
The
synchronous
power
) between generators are calculated
and each value is greater than zero. This result shows
S )
S)
123#TQ (28)
78R
that any increase in system loading is matched by an
7&R)
increase in the active power from the generator
1.5133 × 1.2735√0.4006Q + 1.2902Q cos( 18.2666-
hence, steady state stability is maintained.
13.4128)=2.594˃0
78R
7&RS
(29)
=
1.5133 × 1.3193√0.2718Q + 0.8352Q cos( 18.266617.6434)
=1.753˃0
Ayoade Benson OGUNDARE & I. A. ADEJUMOBI
Page 8
References:
Advanced Technology (IJEAT) ISSN: 2249 – 8958,
[1] Bousmaha Bouchiba1 et-al: Control Of Multi-
Volume-2, Issue-4, April 2013.
Machine Using Adaptive Fuzzy, Serbian Journal Of
[7] Kiran Mishra, S. V Umredkar: ‘’ Transient
Electrical Engineering,
Stability Analysis
Vol. 8, No. 2, 111-126, May 2011.
International Journal of Science and Research (IJSR),
[2] Sangeetha C.N: Enhancement of stability in
India Online ISSN: 2319-7064.
multibus system using Static Synchronous Series
[8] Kanika Gupta, Ankit Pandey: ‘’Stabilization Of
Compensator (SSSC), The International Journal Of
Multi Machine System Connected To Infinite Bus,
Engineering And Science (IJES) || Volume || 2
International Journal Of Scientific & Technology
||Issue|| 12 || Pages || 48-53 ||| ISSN (e): 2319 – 1813
Research Volume 2, Issue 8, August 2013.
ISSN (p): 2319 – 1805, | 2013.
[9] Mohammad Abdul Baseer: ‘’ Transient Stability
[3] Sandeep Gupta R K Tripathi: Voltage stability
Improvement of Multi-machine Power System using
improvement in power system
using FACTS
Fuzzy Controlled TCSC, IOSR Journal of Electrical
controllers:
review,
and Electronics Engineering (IOSR-JEEE) e-ISSN:
State
of
the
art
IEEE
of
Multi
Machine
System,
Transactions on Power system ,pp 1-8
2278-1676,p-ISSN: 2320-3331, Volume 9, Issue 1
[4] Robert J. Davy and Ian A. Hiskens: Lyapunov
Ver. I PP 28-40., Jan. 2014.
Functions for Multimachine Power Systems with
[10] Ganiyu A. Ajenikoko, Anthony A. Olaomi: ‘’ A
Dynamic Loads, Ieee Transactions On Circuits And
Model For Assessment Of Transient Stability Of
Systems—I: Fundamental Theory And Applications,
Electrical Power System, International Journal of
Vol. 44, No. 9, September 1997.
Electrical and Computer Engineering (IJECE), Vol.
[5] Saadat, Hadi, Power System Analysis, New Delhi,
4, No. 4, pp. 498~511, 2014.
Tata McGraw-Hill
[11] A. B. Ogundare, A.S Alayande: ‘’Load Flow
Publishing Company Limited, 2002
Analysis and Simulation Using Newton-Raphson
[6] Devender Kumar, Balwinder Singh Surjan,
Iterative Method’’ Global jour. Of Engg. & Tech.,
‘’Transient Stability of a Multi Machine Power
Vol. 2 No. 4, 561-568. (2009).
System’’ International Journal of Engineering and
Ayoade Benson OGUNDARE & I. A. ADEJUMOBI
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