modeling and simulation of stirling engine fuel supply system based on bond graph theory
TRANSCRIPT
Modeling and Simulation of Stirling Engine Fuel Supply System based
on Bond Graph Theory
Jianxin Liu1,a, Kaibin Zhao1,b, Huiyong Du1,c, Min Li1,d, Zhancheng Wang1,e
1Henan University of Science and Technology, Luoyang, Henan, China
[email protected], [email protected], [email protected], [email protected], [email protected]
Keywords: Stirling Engine; Bond Graph; Fuel pressure control; Simulink; Algebraic loop
Abstract. Bond Graph theory is a powerful tool in the field of dynamics and multiple-energy
domains analyses of complex engineering systems. On the purpose of analyses the dynamic
characters of Stirling Engine fuel supply system, the author built a Stirling Engine fuel supply system
model based on Bond Graph theory, derived the state equations including pump system and hydraulic
system equations and built the Simulink model for simulation, finally verified the accuracy of the
model through experiment.
Introduction
The Stirling Engine is an external heating (or combustion) piston engine, working in the way of
closed regenerative cycle [1]. Fuel nozzles of the burner are cyclone atomizing nozzles which work in
the mode of open and continuous injection. To improve stability of engine operation, we need to
control the injection quantity accurately within per unit time [2]. The control mode of Stirling Engine
fuel supply system in this paper is based on speed regulating [3]. To research the system dynamic
characteristics and improve the designing of electric control system, in this paper, the author built a
model and derived system state equations of the Stirling Engine fuel supply system based on Bond
Graph theory, built a Simulink model for system digital simulation and finally verified the accuracy
of the model through an experiment.
Basic of Bond Graph Theory
Bond Graph theory was initiated by professor Paynter in the early 1960s. Based on basic principle of
energy conservation, the theory convert the machinery, circuit, hydraulic, thermal, and other physical
quantities into four kinds of generalized variables which are effort, flow, displacement and
momentum depending on the similarity relationship of the power transport, storage, dissipation and
transformation [4]. When starting to build a model in Bond Graph theory, we should first replace the
practical components by standard Bond Graph elements. Secondly, connect the multiports of the
elements by bonds, indicate the power orientations and causal stroke. Finally, assign numbers to the
bonds to accomplish the computable Bond Graph model.
Applied Mechanics and Materials Vols. 143-144 (2012) pp 74-78Online available since 2011/Dec/08 at www.scientific.net© (2012) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.143-144.74
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Stirling Engine Fuel Supply System
Figure 1 is a schematic diagram of the Stirling Engine fuel supply system. Low pressure was
maintained at 0.25MPa. A high pressure plunger pump is driven by a servo-motor, each piston pumps
a certain quantity of fuel in turn every rotation of the eccentric shaft. At inlet and outlet of each
compressive cylinder, there was a check valve. After the high pressure fuel flows through the valve,
the fuel will first flow through a short pipeline, then it flows into the accumulator which can be a
buffer, finally, the fuel goes into a long fuel pipeline, and is sprayed by the cyclone atomizing nozzle
into burner. According to energy domain, this system can be divided into mechanical systems and
hydraulic system.
Fig.1. Stirling Engine fuel supply system
Bond Graph Model of the Fuel Supply System
High Pressure Plunger Pumps Model. The internal and external fuel leakage, the friction between
moving parts and the elastic deformation of this hydraulic system were ignored. Because this research
is not on the details of the valve's dynamic characteristics, we simplify the check valve as a modulated
nonlinear resistive element R. According to the Bond Graph theory rules, we built a single piston
pump as shown in figure 2. Each bond was assigned a number, and bi-directional signals include
variables effort (ei), variables flow (fi) could be transferred through each bond. The high pressure
fuel pump can be expressed as combinations of three parallel single piston pumps.
In figure 2, flow source Sf1 represents constant pump rotation speed, effort source Se6 represents
constant fuel pressure, C3 represents liquid capacity effect of the piston compressive cylinder. The
liquid capacity is nonlinear because the cylinder changes from closed system to open system in the
compression stroke. R5 and R8 are used to simulate the nonlinear liquid resistance of the inlet and
outlet of cylinder.
Fig.2. Single piston pump in Bond Graph Fig.3. Hydraulic unit in Bond Graph
Hydraulic System Model. Hydraulic models can be divided into three levels: ideal model, static
model and dynamic model which will result in different simulation accuracy. Because the hydraulic
dynamic characteristics had a significant impact on Stirling Engine performance, it is necessary for us
check valve
cyclone atomizing
nozzle
plunger pump
accumulator
fuel pipeline
eccentric cam
pressure sensor
low pressure
fuel tank
Applied Mechanics and Materials Vols. 143-144 75
to build its dynamic model for analysis and optimization. Stirling Engine hydraulic system included
pipeline, accumulator and cyclone atomizing nozzle. Although each of these three parts has its own
geometric dimension, in Bond Graph theory, they can be described by the same hydraulic unit with
specific resistive, capacitive and inertial parameters. In addition, considering the pipeline's length and
the pump's high frequency movement, we need to built a sectional lumped parameter model which
basic idea was to divide the whole pipe into some limited length segments, each segment was a
dynamic cell. Based on experience, the number of segments (n) was proportional to the length of pipe
(L), if n≥L/L0 (L0=1m, n≥2), the pressure wave and attenuation of simulation results were similar to
the experimental result [5]. Based on Bond Graph theory, a hydraulic unit is built as shown in figure
3. Each unit has two segments.
System State Equations
Physical effects, components connections and power transmissions were described in the model
clearly. According to Bond Graph theory, the model also contains state equations which described
the dynamic performance of the system. We used displacement qi and momentum pi as the system
state variables. According to the Bond Graph rules [6], we got a basic systems state equations as
shown in the following:
State Equations of Single Piston Pump. Equations (1) and (2) show the single piston pump
state equations and output equation:
3 3 3
3 1 6 3 5 3 9 8
/
( ) / ( ) /
e q C
q Sf A Se e R e e R
=
= + − − −i
(1)
9 3 9 8( ) /f e e R= − (2)
Where:0
/iC V B= ,
0V was initial volume, B was bulk modulus;
State Equations of Hydraulic Unit. Equations (3) and (4) showed the hydraulic unit state
equations and output equation:
3
1
3
3 61
3 4
1 3 6
3 8
6
3 8
6 8
8 9
6 8
_
_
pq Q in
I
p qqp R
C I C
p pq
I I
q pp R p out
C I
= −
= − − = − = − −
i
i
i
i
(3)
1
1
8
8
_
_
qp in
C
pQ out
I
=
=
(4)
Where: /iI l Aρ= , 4128 /
iR l dµ π= ,
0/
iC V B= , l and A was length and cross-sectional area of the
segment.
76 Electrical Information and Mechatronics and Applications
Simulation Model in Simulink
There are three main simulation algorithms for hydraulic system: signal flow method, based on
equation method and energy ports method. The model which was built in the Simulink environment
in this paper belongs to signal flow method. In Simulink, we built each functional module according
to state equation, connected the function modules in accordance with the signal flow and adopted
scope blocks to monitor simulation process in real-time at key points. The simulation parameters
which were used in this paper are shown in table 1.
Table 1. Simulation parameters
Cam
Eccentricity
Plunger
Diameter
Compression
Clearance
Pipeline
Diameter
Short
Pipeline
Length
Long
Pipeline
Length
Accumulator
Diameter
Accumulator
Length
Nozzle
Orifice
Diameter
2.5 mm 6.5 mm 0.002 mm 2 mm 335 mm 1913 mm 22.28 mm 440 mm 0.38 mm
Figure 4 shows the Simulink model according to the system state equation. As shown, feedback
loops are used widely in the system. Because of the simulation time sequence of the computer, if the
input signal directly depends on the output signal and output signal directly depends on the input
signal at the same time, there would be algebraic loops in some of the feedback loops. Algebraic
loops would seriously reduce the speed and precision of simulation. To remove algebraic loops,
there are two methods that are transform-method and split-method [7], but we could not use the
transform-method if there are nonlinear units in functions. Alternatively, we have to use some non-
direct-through modules to replace the direct-through modules in same function or insert Memory
Block. Because the latter treatment adds some extra delay, the system do not have much stability
margin would become unstable, so it need more test to validate its rationality.
Fig. 4. Simulation model in Simulink
Comparison of Simulation and Experiment
The Simulink solution used in this model was variable step-size ode45 algorithm. With the increasing
of motor speed, the system disturbance frequency increase too, so the simulation maximum step and
error threshold should be decreased to avoid reduced precision. By comparison, set them to 1e-6 will
be suitable.
Applied Mechanics and Materials Vols. 143-144 77
Time (s)
Pre
ssu
re (
Bar
)
The high pressure fuel pump was driven by a three-phase servo motor, and the servo system was
controlled by a particular ECU. When receiving a voltage signal from external controller, it would
stable the motor speed through a closed-loop control. The pressure sensor which was used to get the
accumulator's pressure was Kistler4262a, the data acquisition facility is consist of WaveBook/512
and DasyLab. The Sampling frequency was 100 KHz, and a low-pass Butterworth Filter to reform
the waveform for denoise was used in the experiment. Figure 5 shows the waveform at speed of 800
r/min,in which, the volatile blue continuous line is measured pressure and the smooth red dash line
is simulation result. As shown, the simulation model based on Bond Graph theory can reflect the
dynamic characteristics of the system.
Fig.5. Measured and simulated pressure waveform at 800 r/min
Conclusion
Compared with the other dynamic analysis methods, the Bond Graph theory could solve multiple
energy domains systems using a simple structure containing a great deal of information. Based on
this theory, Bond Graph model which reveals the interaction between components and energy
conversion was built in this paper, state equations of the Stirling Engine fuel supply system and a
Simulink model were built. Finally, we simulated the dynamic characteristic of the system and
verified the accuracy of the model through experiment. The work will provide a new way for system
optimization.
References
[1] Jin Donghan: Stirling Engine Technology (Harbin Engineering University publishing, China
2009).
[2] Guo Yanjie. Stirling Engine and Energy Saving. Jiangxi Energy, 04 (1997). p. 1
[3] Liu Jianxin, Zhu Jiang, et al. Stirling Engine Fuel System Pressure Fluctuation Simulation.
Journal of Henan University of science and technology, 05 (2009). p. 24
[4] H.M.Panynter. Analysis and Design of Engineering Systems. MIT Press, Cambridge, Mass, 1961
[5] Tian Shujun, Zhang Hong. Simulink Simulation Study on Dynamic Characteristics of Hydraulic
Pressure Pipeline. Journal of system simulation, 05 (2006). p. 1136
[6] Wang Zhongshuang. The Bond Graph Theory and Its Application in System Dynamics. (Harbin
Engineering University publishing. China 2007).
[7] Qiu Jie, Yuan Weilan. Study on remove algebraic loop problems in digital computer simulation.
Computer Simulation, 07 (2003). p. 33
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.259.5
60
60.5
61
61.5
62
62.5
63
时时（ ）s
78 Electrical Information and Mechatronics and Applications
Electrical Information and Mechatronics and Applications 10.4028/www.scientific.net/AMM.143-144 Modeling and Simulation of Stirling Engine Fuel Supply System Based on Bond Graph Theory 10.4028/www.scientific.net/AMM.143-144.74