Eq.(11) describes the voltage restriction relation after two diesel-generator sets enter into
parallel connection.
Eq.(9), Eq.(10) and Eq.(11) are the nonlinear mathematical model of two diesel-generator sets
on parallel connection, which reflects the relationship of interaction and mutual influence
between two diesel-generator sets, describing the variation law of power angle, speed and
voltage on two diesel-generator sets exactly.
3. Chaos oscillation analysis of diesel-generator set on parallel connection
This paper will make research on the stability of ship power system by the variation law of
power angle, speed and voltage on diesel-generator set. Due to the power transmission
between the diesel-generator sets on parallel connection, it is apt to produce power
oscillation. Power oscillation is the dynamic process of power of diesel-generator set
regulating repeatedly under the effect of some periodic interference.
Power oscillation is a chaos oscillation in nature from the point of view of chaos. The
fundamental feature of chaotic motion is highly sensitive to initial condition, the track which
is produced by two initial values which is very near each other, will separate according to
index pattern as time elapses, Lyapunov index is the quantity which describes the
phenomenon. Distinguishing methods of time series chaotic character include fix quantity
analysis and ocular analysis, first making numerical analysis for Lyapunov index and
judging the condition of chaos emerging, then determining if the chaos exists or not under
this condition by the method of ocular analysis. The methods of ocular analysis include time
course method, phase path chart method, strobe sampling method, Poincare cross section
method and power spectrum method. The methods of calculating Lyapunov index include
definition method, Wolf method and Jacobian method, Jacobian method is a method of
calculating Lyapunov index which develops in real application. This paper will use Jacobian
method to calculate Lyapunov index.
Considering following differential equation system
x F( x)
(12)
Robust Control Research of Chaos
Phenomenon for Diesel-Generator Set on Parallel Connection
61
dx
In the equation: x
,
m
x R . The evolution of tangent vector e of dot x( t) in the tangent
dt
space can be expressed by the equation as follow
F
e T( x( t)) e , T
(13)
x
In the equation: T is Jacobian matrix of F . The solution of equation can be expressed as
(
e t) U( t, (
e 0))
(14)
In the equation U : (0
e ) (
e t) is mapping of linear operator. The asymptotic behavior of
mapping U can be described by index as
1
(
e t)
( (
x 0), (
e 0)) lim ln
(15)
t t
(0
e )
So, the Lyapunov index of system (12) can be formulated as the mean of above repeat
process
1 k
((
e j 1) t
)
lim
ln
k k t
(
e j t
)
j 1
(16)
1
((
e k 1) t
)
(
e k t
)
(
e 2 t
)
lim
ln
k k t
(
e k t
)
(
e ( k 1) t
)
(
e t
)
For a phase space of n dimension, there will be n Lyapunov index, arranging them
according to the order from big to small, supposing
1
2
n ,
1 is called
maximum Lyapunov index. Generally speaking, having negative Lyapunov index
corresponds with contracting direction, the tracks which are near are stable in the part,
corresponding periodic motion. The positive Lyapunov index indicates that the tracks
which are near separate by index, the strange attractor is formed in phase space, the
Lyapunov index is bigger, the chaotic nature of system is stronger, vice versa. For a phase
space of n dimension, the maximum Lyapunov index is whether bigger than 0 or not is the
basis of judging the system if has chaos oscillation or not.
Computing Lyapunov index of two diesel-generator sets on parallel running with light load
separately, two diesel-generator sets all use conventional controllers. Fig.1 and Fig.2 give the
phase diagram of power angle, speed and voltage of two diesel-generator sets on parallel
running with 12.5% load separately.
Two diesel-generator sets run for 100 seconds on parallel connection with 12.5% load, the
initial va1ue of No.1 diesel-generator set is: ( ,, U) (0.1017,1.0662,0.9762) , Lyapunov
index is:
1
0.076789 , 2 0.035235 , 3
0.197558 ; the initial va1ue of No.2 diesel-
generator set is: ( ,, U) (0.1022,1.0662,0.9762) , Lyapunov index is:
1
0.076806 ,
2
0.035230 , 3
0.197571 .
Fig.3 and Fig.4 give the phase diagram of power angle, speed and voltage of two diesel-
generator sets on parallel running with 25% load separately.
62
Applications of Nonlinear Control
1.6
1.4
1.2
U1
1
0.8
-40
1.5
-30
1
-20
0.5
-10
0
0
10
-0.5
ω1
δ1
Fig. 1. Phase diagram of No.1 diesel-generator set when two sets load 12.5% on parallel
connection
1.6
1.4
1.2
U2
1
0.8
-40
1.5
-30
1
-20
0.5
-10
0
0
10
-0.5
ω2
δ2
Fig. 2. Phase diagram of No.2 diesel-generator set when two sets load 12.5% on parallel
connection
Robust Control Research of Chaos
Phenomenon for Diesel-Generator Set on Parallel Connection
63
1.3
1.2
1.1
U1
1
0.9
-15
1.1
-10
1
0.9
-5
0.8
0
0.7
5
ω1
δ1
Fig. 3. Phase diagram of No.1 diesel-generator set when two sets load 25% on parallel
connection
1.3
1.2
1.1
U2
1
0.9
-15
1.1
-10
1
0.9
-5
0.8
0
0.7
5
ω2
δ2
Fig. 4. Phase diagram of No.2 diesel-generator set when two sets load 25% on parallel
connection
Two diesel-generator sets run for 100 seconds on parallel connection with 25% load, the
initial va1ue of No.1 diesel-generator set is: ( ,, U) (0.1812,1.0607,0.9686) , Lyapunov
index is:
1
0.079251 , 2 0.034251 , 3
0.199955 ;the initial va1ue of No.2 diesel-
64
Applications of Nonlinear Control
generator set is: ( ,, U) (0.1820,1.0607,0.9686) , Lyapunov index is:
1
0.078311 ,
2
0.034953 , 3
0.199742 .
From Fig.1 to Fig.4 we can see, all maximum Lyapunov indexes of the system are greater than
0, showing that the system exists chaotic phenomenon. Two diesel-generator sets with light
load on parallel connection, enter into chaotic state after running a length of time, their specific
expression are the oscillation of power angle and speed. Two diesel-generator sets on parallel
connection load the lighter, the oscillation of power angle and speed is severer. The oscillation
of power angle means the oscillation of power, the reason is the nonlinearity of ship power
system and the power transmission between the two diesel-generator sets. The controller in
this paper is proportional controller, which is a linear controller. It can’t control the nonlinear
character of ship power system, it can’t average the load in parallel operation control, there is a
power angular difference between the diesel-generator sets, thus engendering the power
transmission between the sets, which results in the happening of chaotic phenomenon.
Fig.5 and Fig.6 give the phase diagram of power angle, speed and voltage of two diesel-
generator sets on parallel running with 25% load plus periodicity load separately. The
periodicity load usually appears in ship power system, it is widespread.
Two diesel-generator sets run for 100 seconds on parallel connection with 25% load increasing
periodicity load 0.01sint, the initial va1ue of No.1 diesel-generator set is:
( ,, U) (0.1812,1.0607,0.9686) , Lyapunov index is:
1
0.073257 ,
2
0.031824 ,
3
0.191497 ; the initial va1ue of No.2 diesel-generator set is:
( ,, U) (0.1820,1.0607,0.9686) , Lyapunov index is:
1
0.073393 ,
2
0.030161 ,
3
0.189995 .
1.6
1.4
1.2
U1
1
0.8
-30
1
-20
0.8
-10
0.6
0
0.4
10
0.2
ω1
δ1
Fig. 5. Phase diagram of No.1 diesel-generator set when two sets increase periodicity load
meanwhile load 25% on parallel connection
Robust Control Research of Chaos
Phenomenon for Diesel-Generator Set on Parallel Connection
65
1.6
1.4
1.2
U2
1
0.8
-30
1
-20
0.8
-10
0.6
0
0.4
10
0.2
ω2
δ2
Fig. 6. Phase diagram of No.2 diesel-generator set when two sets increase periodicity load
meanwhile load 25% on parallel connection
From Fig.5 and Fig.6 we can see, all maximum Lyapunov indexes of the system are greater
than 0, showing that the system exists chaotic phenomenon. The oscillation of power angle
and speed in two diesel-generator sets is severer than the state of not increasing periodicity
load. Because the periodicity load is nonlinear, increasing periodicity load intensifies the
nonlinearity of system, thus aggravating the power oscillation of system[4-6].
The computer simulation results show that it exists chaotic oscillation phenomenon when
two diesel-generator sets run on parallel connection with light load. Primary cause of
emerging this phenomenon is the nonlinearity of ship power system, minor cause is the
power transmission between the two diesel-generator sets. Besides this, using conventional
linear controller is also a key factor of generating the chaotic oscillation of system. Only
using nonlinear controller, making the nonlinear characteristic of ship power system offset
and compensate, can we solve the problem of system chaotic oscillation fundamentally. The
chaotic oscillation phenomenon is transition state between stable state and unstable state, it
must be prevented in order to ensure the stability of system.
4. Design of nonlinear robust synthetic controller
Mixed H-two/H-infinity control theory is a robust control theory that has speed
development from the eighties of 20 century, which can solve the problem of robust stability
and robust performance[7-10]. Because diesel-generator set control system is a nonlinear
control system, using the method of direct feedback linearization to linearize the nonlinear
system, the state feedback controller is designed for linearization system using mixed H-
two/H-infinity control theory, thus acquiring nonlinear robust control law in order to reach
the purpose of restraining the chaotic phenomenon of ship power system, improving the
stability of ship power system.
66
Applications of Nonlinear Control
Because of coupling action between speed and voltage, the nonlinear robust synthetic
controller is designed for diesel-generator set in order to control speed and voltage
synthetically, making the both interaction in minimum range, thus improving the stability of
frequency and voltage in ship power system.
The principle diagram of diesel-generator set synthetic control system based on nonlinear
robust synthetic controller is shown in Fig.7. The diesel-generator set synthetic control
system is made up of diesel engine, generator, nonlinear robust synthetic controller,
actuator, oil feeding mechanism and exciter. Nonlinear robust synthetic controller is made
up of two parts, one is nonlinear H-two/H-infinity speed controller, another is nonlinear H-
two/H-infinity voltage controller.
The differential equation of actuator is
d L
L
K
1
u
(17)
dt
1
1
T
1
T
The differential equation of exciter is
d E
E
K
fd
fd
2
u
(18)
dt
2
2
T
2
T
Fig. 7. Principle diagram of diesel-generator set synthetic control system
First step, we design nonlinear H-two/H-infinity speed controller.
Combining Eq.(1) with Eq.(17), we can get the nonliear mathemaical model of diesel engine
speed regulation system.
d
1,
d t
2
d
T
1
c
1
q
E U
1 U X
X
2
d
q
b
1
c
L
sin
sin 2 , (19)
d t
T
T
T
T X
T 2
a 0
a 0
a 0
a 0
d
a 0
XdXq
d L
L
K 1
u
dt
1
1
T
1
T
Since Eq.(19) has nonlinear feature, the method of direct feedback linearization is used to
linearize Eq.(19). Order x
1
, x
2
1 ,
Robust Control Research of Chaos
Phenomenon for Diesel-Generator Set on Parallel Connection
67
2
T
1
c
E U
U X X
2
1
q
1
d
q
b
x
, so Eq.(19) can be
3
1
c
L
sin
sin 2
T
T
T
T X
T 2
a 0
a 0
a 0
a 0
d
a 0
XdXq
written as
x 1 x 2
x
(20)
2
x 3
1
d w
2
T
c K
c
E U
U ( X X )
2 1
2
q
x
x
u
L
cos ( 1)
d
q
b
c os2
3
3
1
(
1)
a
T 0
a
T 1
T 0
a
T 1
T 0
a
T Xd 0
a
T XdXq 0
In the equation: 1
d w is the undesired signal which is assumed for using H-two/H-infinity
control method, including equivalence disturbance which is generated by disturbance
torque and modeling error.
Assigning virtual controlled variable
2
c K
c
E U
U ( X X )
2 1
2
q
v
u
L
cos ( 1)
d
q
c os2
(21)
1
(
1)