Modeling and simulation of a passive variable inertia flywheel for diesel generator

Energy Reports xxx (xxxx) xxx

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Research paper

Modeling and simulation of a passive variable inertia flywheel for diesel generator ∗

Yi ping Zhang a , , Xingwang Zhang a , Ting Qian a,b , Rui Hu a a b

School of Mechanical and Electrical Engineering, Nanchang Institute of Technology, Nanchang 330039, China No.1 Branch of Wuhan Eng Middle School

article

info

Article history: Received 13 August 2019 Received in revised form 12 December 2019 Accepted 4 January 2020 Available online xxxx Keywords: Variable inertia flywheel Diesel generator Micro-grid Speed stability

a b s t r a c t The power supply system of diesel generators is isolated. Sudden loading and unloading will lead to fluctuation of the engine speed and exert an impact on the power supply quality. In order to improve the speed stability of diesel generators against the loading impact, this paper proposes a flywheel with variable moment of inertia, through the adjustment of which, the speed stability can be improved. Imbalance between the output torque and the load torque of the diesel engine can cause changes in the moment of inertia of variable inertia flywheel. The variable inertia flywheel forms the internal feedback loop of the speed control system, it takes the Angular acceleration of engine as the input signal and its output reduces the pulse impact of loading. SIMULINK simulation results under sudden loading and unloading condition have verified that, the variable inertia flywheel makes the diesel generator less sensitive to the loading impact. Meanwhile, variable inertia flywheel changes the maximum magnitude and bandwidth of closed-loop system, and endows the speed control system with a smoother response and better anti-disturbance ability. © 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction The power supply system of Diesel generator (DG) is isolated, Sudden loading and unloading will cause the fluctuation of DG speed and affect the power quality. Flywheel energy storage system (FESS) is an efficient device to decrease the speed fluctuation of DG and improve power quality of micro-grid (Yuan et al., 2010; Li et al., 2011; Pullen, 2019; Arani et al., 2017). When system energy is unbalanced, it can charge and discharge for numerous cycles without any depreciation, consequently it is characterized by long durability and less maintenance. FESS, as a developing technology, will play an important role in the future power supply system and micro-grid. It has been applied to hybrid vehicle, railway, wind power, hybrid power generation, power network, and other fields, and it can efficiently enhance the quality of power system (Mousavi et al., 2017; Spiryagin et al., 2015; Sebastián and PeA-Alzola, 2015). The flywheel of FESS can be classified into the fixed inertia flywheel (FIF) and the variable inertia flywheel (VIF). The current flywheels of DG are all FIF, during the running of the generators, their inertia stays constant. VIF can change the inertia of rotating machinery and control the inertia of the power system. The inertia control can reduce the instantaneous frequency deviation (Chamorro et al., 2019; Hajiakbari Fini and ∗ Corresponding author. E-mail address: [email protected] (Y.p. Zhang).

Hamedani Golshan, 2019; Mehrasa et al., 2019). Environmental preservation has integrated more generator micro-grid (Hosseinnia and Tousi, 2019), and the most significant technical problem lies in achieving the frequency stability. The inertia control will overcome the frequency stability on hybrid micro-grid (Dreidy et al., 2017; Mishra et al., 2015). Compared with FIF, VIF has a bigger energy storage capacity, consequently, it performs better in stabilizing the system frequency. The application of VIF to the wind turbine can achieve the stability of system frequency. The VIF can store energy when the wind turbine encounters a gust and it will release energy when there is no wind (Jauch, 2015; Hamzaoui et al., 2016). VIF can be applied to the hydraulic motor to absorb fluctuating energy, it can improve the stability of the hydraulic system and achieve passive vibration control of the hydraulic system (Yang et al., 2016; Mahato et al., 2019). VIF applied to DG will absorb excess kinetic energy generated by diesel engine during unloading, and discharge energy while loading, the principle is shown in (1). JG

dω dt

= Me − Mc

(1)

Due to the response delay in the speed control system, when the load torque changes, the torque of DG cannot be adjusted in time, so the angular acceleration will be not zero. If the inertia JG of DG can be adjusted when load changes, the fluctuation of speed can be reduced.

https://doi.org/10.1016/j.egyr.2020.01.001 2352-4847/© 2020 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/).

Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.

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Abbreviations DG FESS VIF FIF O

Diesel generator Flywheel Energy Storage System Variable inertia flywheel Fixed inertia flywheel Flywheel center

Constant JG G d D T k C r1 l1

m1

µ ω0 x1

ωn ξ a, k1 , d1 kp , ki , kd g K x0 K1 T1

Inertia of diesel generator without variable inertia flywheel (kg m2 ) The modulus of rigidity (GPa) The wire diameter of spring (mm) Spring diameter (mm) The effective number of coils of spring Spring rate (N/m) Coefficient of viscous damping The radius of the central axis of flywheel (m) The distance from the centroid of the mass block to the connection point of spring (m) Mass of mass block (kg) Coefficient of Coulomb damping Angular speed of flywheel at rated speed Displacement of mass block at rated speed (m) System resonance frequency System damping ratio Coefficient of engine torque Parameters of PID controller Gravity acceleration (m/s2 ) Damping coefficient of generator damper winding Spring length when the flywheel is stop (m) Gain of speed actuator Time constant of speed actuator

Variables

ω y Me Mc E F1 F2 f1 x Jflywheel Jtotal Jmass Jframe L

Angular speed of flywheel and diesel generator (rad/s) Speed of diesel generator (RPM) Torque of diesel generator (N m) Load torque (N m) Kinetic energy (J) Centrifugal force (N) Spring tension (N) Friction (N) Displacement of mass block (m) Inertia of flywheel (kg m2 ) Total inertia of diesel generator with variable inertia flywheel (kg m2 ) Inertia of mass block (kg m2 ) Inertia of variable inertia flywheel frame (kg m2 ) Displacement of the fuel pump rack (mm)

u Fd n N

Input of actuator The stability factor Numbers of flywheel mass blocks Normal force on the friction surface (N)

A kind of VIF was proposed in Elliott and Lapen (2009), the centrifugal force can change the displacement of the mass block and the control system can lock the position of mass block with hydraulic fluid and control valve. The VIF can start with small inertia and rotate at high speed with big inertia, it can improve stability of machinery in high speed, but it need additional control system and its inertia was locked by hydraulic fluid. Dugas (2015) proposed a kind of VIF, which can acquire variable inertia through the valve’s control over the fluid flow. The inertia of VIF in Jayakar and Das (2012) can be changed by filling and draining. By controlling the fluid in the chamber, the VIF can minimize the variations in engine speed caused by changes in the power load, but this becomes complicated due to the filling and controlling of fluid. Yuan et al. (2010) and Li et al. (2011) have proposed the application of VIF to DG and verified that the VIF can stabilize the speed of DG, but did not establish a complete mathematical model and control system model. A complete system model is the premise of control theory analysis and research, it is the basis for the precise motion control and smart hybrid microgrid control with variable inertia. In this paper, a flywheel with variable inertia for DG is proposed. The control system mode of VIF is developed and it is integrated into the speed control system of DG. The simulation result shows that the VIF makes the DG less sensitive to the loading impact and improves the power quality. 2. Introduction of VIF VIFs in Dugas (2015) and Jayakar and Das (2012) are not suitable for DG since it need the additional control and filling of fluid, and when the valve has locked the fluid flow, the moment of inertia of VIF in Elliott et al. (2009) will not change. Based on the VIF in Jauch (2015) applied to wind turbine and the VIF in Hamzaoui et al. (2016) and Yang et al. (2016) applied to hydraulic system, we proposed a VIF for DG. Active VIF need a driving system and its structure is complicated. However, the passive VIF has a simpler structure and it suits DG. The VIF proposed in this paper is a passive one. As shown in Fig. 1. The VIF proposed here is composed by mass block, spring, sliding rod and frame. When the flywheel is stopped, all the mass blocks are located near the center of flywheel due to the tension of the spring (Fig. 1A). When the flywheel rotates, the mass blocks are simultaneously subjected to the tension of spring and centrifugal force. As the speed accumulates, the mass blocks slide from the center of flywheel to the rim of flywheel (Fig. 1B). The sliding of mass blocks causes the inertia of the flywheel to change. 2.1. Energy storage in flywheel The kinetic energy E of a mass block spinning at ω is given by Eq. (2).

∫ E=

J ω dω =

1 2

J ω2

(2)

The moment of inertiaJ of mass block is determined from its mass distribution dm along its radiusr, given by Eq. (2).

∫ J =

r 2 dm

(3)

Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.

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The total spring rate for each mass block is determined by the mass of the block and the rated speed of flywheel. The spring rate is calculated by Eq. (4) (Shigley and Mischke, 1996; Chironis and associate, 1961; Richard and Budynas, 2015; Ugural, 2015). k=

Gd4

(4) 8D3 T (G is the modulus of rigidity, d is the wire diameter of spring, D is the coil diameter of spring, and T is the effective number of coils) 3.2. Force analysis of mass block

Fig. 1. VIF in charged state (B) and in discharged state (A), (1—mass block, 2—spring, 3—sliding rod, and 4—rim of flywheel).

When the flywheel rotates, the mass block is subjected to centrifugal force F1 , spring tension F2 and frictionf 1 . When the flywheel accelerates, the mass block will move toward the flywheel rim due to increasing centrifugal force, and if the flywheel decelerates, mass block will move toward the flywheel center. Fig. 4 is the free-body diagram (A, B) and spring–mass–damper model (C) of mass block. And (5) is the force balance equation of mass block. F1 = F2 + f1 + m1 x¨

Fig. 2. Installation layout of diesel engine unit.

(5)

When the flywheel stops, the mass blocks will gather around the center of the flywheel under the pulling force of the spring. When the flywheel rotates, the mass blocks are subject to a centrifugal force in the direction to the flywheel rim. As the rotational speed reaches a certain degree, the centrifugal force will overcome the friction and the spring tension, and push the mass blocks to slide toward the flywheel rim. When the mass blocks slides, the spring will be stretched, and the spring tension will increase until the forces of the mass block reaches a balance. When the speed decreases, the mass blocks will slide back to the center of the flywheel and accordingly the tension of the spring will reduce until the forces of the mass blocks reach a balance again. Fig. 5A is the diagram when the flywheel stops, and Fig. 5B is the diagram when the flywheel rotates. During the rotation, the flywheel is subjected to the centrifugal force which can be written as (6). F1 = m1 ω2 (r1 + l1 + x0 + x)

(6)

The tension of spring is (7). Fig. 3. prototype (note: 1—mass block, 2—spring, 3—sliding rod, 4—rim of flywheel, 5—airtight plate, 6—side cover, 7—side cover, 8—air hole).

2.2. Installation of VIF The VIF (VIF) can be installed between the cooling fan and the diesel engine. Fig. 2 shows the installation layout for VIF on DG. 3. Modeling of VIF Fig. 3 is the prototype of VIF. There are sixteen sliding rods installed around the center of the flywheel, a mass block on every two sliding rods and a spring on each sliding rod. The spring links the mass block to the flywheel center. The sliding rod is inserted into the holes of mass block, and the mass block can move smoothly on the sliding rod. The airtight plate forms the viscous damper of mass block, and the air holes on the side cover are used for adjusting the viscous coefficient of mass block. 3.1. Design of VIF The flywheel is equipped with 8 mass blocks, each mass block is linked by cylindrical extension springs to the center of flywheel.

F2 = kx

(7)

During the rotation, when the centroid of the mass is below the center of the flywheel, the gravity of mass block will enhance the centrifugal force, however, when the centroid of the mass is above the center of the flywheel, it weakens the centrifugal force. It can be calculated by Eq. (8). m1 g cos(ωt)

(8)

When moving in the closed flywheel, the mass block will suffer from the air resistance, consequently, coulomb friction and viscous damping will exist in the process of moving. The friction can be obtained by (9). dx ) × µN + C (9) dt dt (f1 is frictional force, µ is Coulomb friction factor, N is normal force on friction surface, C is the viscous damping coefficient, x is displacement of the mass block) The normal force N is related to the gravity of the mass block and the inertial force of the mass block during rotation. Since the friction surface changes in 360 degree, the normal force generated by the gravity of mass block is a cosine function and when the flywheel speed changes, the mass block will produce an inertial force on the friction surface, which is proportional to flywheel f1 = sign(

dx

Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.

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Fig. 4. The force schematic diagram of mass block. (note: F1 —centrifugal force, F2 —spring tension, f1 —friction, k —spring rate, C —damping coefficient, x —displacement of mass block, ω—angular velocity of flywheel, O—flywheel center)

speeds. In the second column, the gravity impact on the centrifugal force and the friction are positive and negative alternating high frequency signals, so the effect of gravity can be ignored. The coulomb friction of mass block is produced by the normal force from gravity of mass block and the friction surface changes in 360 degree, so it is a positive and negative alternating high frequency signal. When the flywheel uniformly accelerates to 1500 RPM, the angular acceleration is a constant value, so the friction generated by inertia of mass block is a constant value as shown in the third picture of second column. 3.3. Linearization of motion equation The Taylor series may be utilized in Eqs. (5), then we have Fig. 5. The mass block status when the flywheel stops and rotates.

angular acceleration. The normal force is composed of mass block gravity and inertial force, so the friction force is obtained as (10).

f1 = (m1 (l1 + r1 + x0 + x)

dω

+ m1 g cos(ωt))µ + C

dx

F1 (ω0 , x1 ) − F2 (x1 ) − f1 (ω ˙ 0 , x˙ 1 ) = m1 x¨ 1 (10)

dt dt When the speed of the flywheel is constant, the mass block reaches a balance, the force f1 and m1 x¨ in Eq. (5) is zero and the centrifugal force F1 equals the tension of spring F2 . The displacement of the mass block in steady speed can be calculated by Eq. (11). x=

m1 ω2 (r1 + l1 + x0 ) k − m1 ω2

(11)

Compared with the centrifugal force, the influence of the gravity of the mass blocks is negligible. And the mass block displacement x in the Coulomb friction in Eq. (10) can be replaced by Eq. (11) as a simplification. The motion equation of mass block can be written as (12). m1 ω (r1 + l1 + x0 + x) − kx − m1 µ(l1 + r1 + x0 2

dx d2 x m1 ω2 (r1 + l1 + x0 ) dω + ) − C = m 1 k − m1 ω 2 dt dt dt 2

∂ F1 ∂ F2 ∂ F1 ∆ω + ∆x + F1 (ω0 , x1 ) − ∆x − F2 (x1 ) ∂ω ∂x ∂x ∂ f1 ∂ f1 ∂ (m1 x¨ ) − ∆x˙ − ∆ω˙ − f1 (ω˙ 0 , x˙ 1 ) = ∆x¨ + m1 x¨ 1 ˙ ∂x ∂ ω˙ ∂ x¨ (ω0 , x1 ) is the Taylor expansion point, And

(12)

In Eq. (12), the first item is centrifugal force, the second item is spring tension, the third item is friction force and the fourth one is viscous damping. Fig. 6 shows the centrifugal force, spring tension, displacement of the mass block and curve in which the inertia changes with the rotational speed. In the first column, when the speed is lower than 1000 RPM, the centrifugal force, displacement and spring tension change slowly. However, when the speed exceeds 1000 RPM, their values increase in quadratic equation, which indicates that the flywheel has a larger energy storage capacity at high

(13)

(14)

The linearization equation is

∂ F1 ∂ F2 ∂ f1 ∂ f1 ∂ (m1 x¨ ) ∂ F1 ∆ω + ∆x − ∆x − ∆x˙ − ∆ω˙ = ∆x¨ (15) ˙ ∂ω ∂x ∂x ∂x ∂ ω˙ ∂ x¨ Combined with (12) and (15), following equation can be obtained. m1 ∆x¨ + C ∆x˙ + (k − m1 ω02 )∆x = (r1 + l1 + x0 + x1 )

× (2m1 ω0 ∆ω − m1 µ∆ω˙ )

(16)

3.4. Mass block displacement control system The transfer function can be obtained from (16) as X (s)

ω(s)

=

(r1 + l1 + x0 + x1 )(2ω0 − µs) s2 +

C s m1

(17)

+ ( mk − ω02 ) 1

The transfer function (17) is used to investigate the dynamic performance of the system around the generic equilibrium point (ω0 , x1 ). The state–space representation is (18) and (19).

⎡ 0 [ ] z˙1 ⎢ =⎣ k z˙2 ω02 −

m1

1

−

⎤

⎥ C ⎦ m1

x = (r1 + l1 + x0 + x1 ) 2ω0

[

z1

0

[ ]

[ ] ω

+ z2

(18)

1

[ ] ] z1 −µ z2

(19)

(z1 and z2 are the state variables, ω is system input, x is system output which is the displacement of mass block).

Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.

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Fig. 6. Simulation diagram of flywheel.

After the size of the flywheel being determined, there are three changeable parameters (C, k, m1 ) in the system that can be changed. However, there is a one-to-one constraint relationship between k and m1 in Eq. (11), so the system has only two free parameters. The nature frequency and damping ratio of the system can be obtained as (20).

⎧ √ k ⎪ ⎪ ⎪ − ω02 ω = n ⎪ ⎪ m1 ⎨

(20)

C

⎪ ξ= √ ⎪ ⎪ ⎪ k ⎪ ⎩ 2m1 − ω02 m1

By considering the constraint relation in Eq. (11), Eq. (20) becomes

√ ⎧ ⎪ ω02 (r1 + l1 + x0 + x1 ) ⎪ ⎪ − ω02 ωn = ⎪ ⎪ ⎪ x1 ⎨ C

ξ= √ ⎪ ⎪ ⎪ ⎪ ω02 (r1 + l1 + x0 + x1 ) ⎪ ⎪ ⎩ 2m1 − ω02

(21)

In (21), the system nature frequency ωn is determined byr 1 , l1 , x1 , so it is decided by the size of flywheel ξ is inversely proportional to the mass m1 and proportional to the viscous damping coefficient C. To get the best damping ratio (ξ = 0.707) for the system, Eq. (22) should be obeyed. C = m1

2ω02 (r1 + l1 + x0 + x1 ) x1

K1 =

(r1 + l1 + x0 + x1 )2ω0 k m1

− ω02

(23)

The viscous damping can be changed by increasing the air tightness of the flywheel. The system step response and Bode diagram with different viscous damping coefficient C can be shown in Fig. 7. When the viscous damping between the mass block and the sliding rod increases, the oscillation amplitude of the system and the overshoot decrease. Meanwhile, the maximum value of the frequency response also reduces. In conclusion, the viscous damping coefficient is important, small viscous damping will lead to the vibration of VIF. 3.5. Calculation of inertia The total inertia of the flywheel consists of fixed parts and variable parts. The fixed part is mainly produced by the flywheel frame, the sliding rod and the spring while the variable part is generated by the mass blocks. The formula for calculating the total inertia of the DG is (24).

⎧ ⎨

x1

√

gain of the system is constant.

⎩J

mass

Jtotal = JG + Jflywheel Jflywheel = Jframe + Jmass = 8 × m1 × (l1 + r1 + x0 + x)2

(24)

(Jtotal is the total inertia, JG is the inertia of diesel generator, Jflywheel is the inertia of flywheel, Jframe is the inertia of flywheel frame, Jmass is the inertia of mass block) 4. DG speed control system without VIF

− 2ω

2 0

(22)

The system gain is given by Eq. (23), it is determined by the parameter of flywheel size. When r1 , l1 , x1 are determined, the

4.1. Model of diesel engine In terms of the static characteristics of the generator, the following differential equation can be obtained (Wang et al., 2018;

Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.

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Fig. 7. Step response and Bode diagram of mass block displacement control system with different viscous damping.

Thomas Harding et al., 2017).

The differential equation of speed control system is (30).

dω

+ K ω = Me − Mc (25) dt (JG is the inertia of DG without VIF, ω is the angular speed of DG, Me is the torque of diesel engine, Mc is the torque of Load, K is the damping coefficient of generator damper winding) In Eq. (25), K ω is the damping torque produced by damper winding. The torque Me of diesel engine is in a non-linear relationship with the position of the fuel pump rack L and the rotating speed ω, the linear expression around the rated speed is as (26) (Huang, 2013; Huang et al., 2011). JG

Me = f (L, ω) Me = d1 + k1 ∆ω + a∆L

{

(26)

⎧ dω k1 − K a d1 − k1 ω0 − aL0 Mc ⎪ ⎪ ω+ L+ − ⎨ dt = J J J JG G G G ⎪ L K1 dL ⎪ ⎩ =− + u dt

⎡ [ ] L˙ ω˙

−

1

⎢ = ⎣ aT1 JG

(a, k1 , d1 are coefficients of engine torque, ω = ω0 + ∆ω, L = L0

[ ×

0

dt

k1 − K JG

ω+

a JG

L+

d1 − k1 ω0 − aL0 JG

−

Mc JG

(27)

4.2. Model of speed control system The governor compares the actual speed with the reference speed and operates the fuel rack of the diesel engine according to the deviation. The control object of the governor is the fuel rack lever of diesel engine. By adjusting the fuel rack lever, fuel injection will be adjusted to control the engine speed (Pagán Rubio Jose et al., 2018). Fig. 8 is the block diagram of DG speed control system (Guo et al., 2017; Shen and Su, 2012). Controller is a PID controller that converts the deviation signal into the input signal of actuator. Gpid (s) = kp +

ki

dt

=−

L T1

+

K1 T1

y= 0

[ ]

⎥ L

d1 − k1 ω0 − aL0 Mc

60 2π

]

⎤

⎡

0 +⎣1 JG

⎡

K1

0 1⎦

−

JG

⎤

+ ⎣ T1 ⎦ u

(31)

0

][ ] L

ω

(32)

(u is system input, y is system output which is the speed increment of DG) Fig. 9 is the step response of speed control system, u and Mc are system inputs and the ω is system output.

5. Speed control system of DG with VIF

5.1. Block diagram model

+ kd s

(28)

s The actuator converts the signal u outputs by the PID controller into the displacement signal L of the throttle rack, and the differential equation is dL

[

⎤

k1 − K ⎦ ω JG

Combined with (25) and (26), following equation can be obtained.

=

T1

The state-space representation of DG speed control system is (29) and (30).

+ ∆L) dω

T1

(30)

u

Fig. 8. Speed control system of DG.

(29)

The block diagram of the speed control system without VIF is shown in Fig. 10 (Iwanski et al., 2018). The controller and actuator adjust the torque of diesel engine to balance the load torque. The imbalance of the torque will affect the speed of the diesel generators. The system takes the change of speed as feedback, and the inertia JG of DG in this system is constant. In the speed control system of DG with VIF, VIF is an energy storage device, it forms the internal feedback loop. When the load torque increases, VIF compensates the system by releasing stored kinetic energy. When load torque reduces, VIF will absorb excessive energy from the system. The formed feedback loop takes the Angular acceleration of engine as the input signal, its output reduces the pulse impact of loading (see Fig. 11).

Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.

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Fig. 9. Step response of DG speed control system.

Fig. 10. DG speed control system without VIF.

Fig. 11. DG speed control system with VIF. (F (X) is the function of Jtotal in Eq. (31), Gc (s) is the transfer function of VIF).

5.2. Speed control system with VIF Combining Eqs. (12), (24) and (30), the equations of speed control system can be obtained as (33)

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

dω dt

=

k1 − K Jtotal

ω+

dt 2

d x dt 2

+

C dx m1 dt

a

Jtotal dL

+(

L+

=− k

m1

−µ(l1 + r1 + x0 +

L T1

d1 − k1 ω0 − aL0 Jtotal

+

K1 T1

−

Mc Jtotal

u

− ω2 )x = ω2 (r1 + l1 + x0 )

speed control system without VIF from (30), so the VIF does not change the steady state of the system. From the following simulation result, the system is stable with the parameters in this paper, but the instability boundary is unknown, and it is useful for flywheel design. To evaluate the stability of the speed control system (33), we can find a Lyapunov function to verify the stability of the system, and also, we can deduce the linear model by linear approximation and calculate system gain margin and phase margin to determine the stability of the system. For the deeper investigate of system stability, we can consult (Kamaleddin and Nikravesh, 2013). (k1 − K )ω + aK1 u + d1 − k1 ω0 − aL0 = Mc

m1 ω2 (r1 + l1 + x0 ) dω ) k − m1 ω2 dt

(34)

5.3. Simulation

Jtotal = JG + Jflywheel = JG + Jframe + 8m1 × (r1 + l1 + x0 + x)2 (33)

In Eq. (33), Suppose that dω/dt = 0, dL/dt = 0, d2 x/dt 2 = 0, dx/dt = 0, then we obtain the equilibrium state equation (34). And we can obtain the same equilibrium state equation of DG

The parameters used in this simulation are shown in Table 1, the rated power and speed of DG is 1250 kW and 1500 RPM, it is a marine generator (Huang, 2013; Huang et al., 2011). Using (33), the speed control system in which DG is combined with VIF, to simulate the sudden load-changing process. Fig. 12

Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.

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Y.p. Zhang, X. Zhang, T. Qian et al. / Energy Reports xxx (xxxx) xxx Table 1 Parameters. Parameter

Symbol

Value

Parameter

Symbol

Value

Inertia of DG

JG

71.8 kg m2

Viscous coefficient

C

30

Damping coefficient

K

11.08

Spring rate

k

200 N/m

Gain of speed actuator

K1

0.2

Mass of mass block

m1

4 kg

Time constant of actuator

T1

0.05

Coulomb coefficient

µ

0.15

Coefficient of engine torque

a

1245

Radius of flywheel central axis

r1

0.13 m

Coefficient of engine torque

d1

15 042

Spring length when flywheel stops

x0

0.2 m

Coefficient of engine torque

k1

−2.095

Displacement at rated speed

x1

0.3695 m

Numbers of mass blocks

n

8

Angular velocity at rated speed

ω0

157 rad/s

Inertia of flywheel frame

Jframe

20 kg m2

Centroid to spring (mass block)

l1

0.05 m

Fuel pump rack position

L0

5 mm

shows the mass block displacement, the inertia of VIF and the total inertia of the DG during the simulation period. The engine subject to sudden loading at the 3rd, and subject to sudden unloading at 10th second, Fig. 12 shows that the displacement and inertia change with the rotational speed of the engine. In Fig. 13, the subplots A, B and C respectively show the angular velocity, the load torque and the torque absorbed by the VIF. During second 0 to 3, since the engine speed in the subplot A stabilizes at the rated speed, the torque absorbed by the VIF in the subplot C is zero. At the 3rd second, the load torque of subplot B suddenly increases by 3000 N m, causing the engine speed in subplot A to decrease, and the VIF in subplot C discharges a torque and releases energy due to the change in the rotating speed. The torque discharged by the VIF compensates the torque of engine, and the compensation reduces the fluctuation of the engine speed. At the 10th second, the load torque suddenly decreases to 0, and the engine speed in subplot A increases. The sudden increase in the speed causes the VIF in subplot C to generate a damping torque and absorb energy from engine. Since the VIF absorbs the transient excess torque generated by DG, the fluctuation of engine speed is reduced. In Fig. 13C, since the flywheel and the shaft of diesel engine are connected rigidly, the response delay of the flywheel is zero. As a conclusion, in the torque balance system of the DG, when the input energy in the system outweighs the output energy, the VIF will absorb excessive energy. When the input energy is lower than the output, the VIF will discharge energy. Fig. 14 demonstrates the comparison of rotating speeds under sudden loading impact. Subplot A shows that, when the load is suddenly increased in the first second, the speed fluctuation of engine with VIF is smaller than that with fixed inertia. Subplot B shows that as the load increases, the inertia of the VIF decreases. From the flywheel kinetic energy formula (2), it is known that the flywheel is releasing energy to compensate for the temporary lacking of the engine torque. Fig. 15 shows the comparison of the speed when the engine suddenly reduces the load. In subplot A, the speed of engine with VIF is more stable than the engine with fixed inertia. As shown in subplot B, when the load of DG suddenly reduces, the inertia of VIF increases and stores kinetic energy which absorbs excessive engine torque. VIF changes the close-loop characteristics of the speed control system. It reduces the maximum magnitude, resonant frequency and bandwidth of the close-loop frequency response. The maximum magnitude of close-loop frequency response represents the smoothness of the system response, and the bandwidth represents the disturbance rejection performance. As shown in Fig. 16, subplot A and B are the frequency response of the transfer function ω(s)/ωref (s) and ω(s)/MC (s) respectively, the magnitude of frequency response is decreased in high frequency, and low gain in high frequency reduces the sensitivity to load sudden changing.

With the application of VIF, the magnitude curve changes to

−40 dB/decade at 1.5 rad/s instead of 2 rad/s. Then, the slope becomes 0 dB/decade and changes to −40 dB/decade at the frequency band 100 rad/s–200 rad/s. Since the VIF changes the slope of magnitude curve to 0 dB/decade, the phase characteristic curve gets a peak wave at 170 rad/s. The final slope of magnitude curve is −40 dB/decade, and the phase characteristic curve goes to −180 degree. In frequency-domain, the magnitude of the control system represents the ability of the system to track and reproduce the input signal, so the magnitude of the system reflects the system’s response for input and interfering signals. As shown in Fig. 17, subplot A and B are the Nyquist contour of the transfer function ω(s)/ωref (s) and ω(s)/M C (s) respectively, As the quantity of the mass block increases, amplitude of the two transfer function gradually decreases. In conclusion, VIF equips the system with a more stable reaction to the input and disturbance. The amplitude of the system is on the decline, then the speed control system will show less sensitivity toward the impact of loads. In Fig. 17A and B, the Nyquist contour of system will not encircle the −1 point no matter with or without VIF, according to the Nyquist criterion, if there is no pole in right-hand s-plane, the system will be stable. 6. Conclusion The main contribution in this paper is summarized in three parts. Firstly, a VIF applied to DG is proposed, the mathematical model as well as control system are deduced and its performance is analyzed. Secondly, the mode of VIF is integrated into the speed control system of DG. Thirdly, the hybrid speed control system is simulated. The simulation result shows that, when the system energy is unbalanced, the VIF can absorb the excessive energy of the system, it can make the DG speed more stable and improve its robustness. Compared with FIF, VIF has a larger storage and performs better in stabilizing the rotational speed of DG. (1) In the displacement control system of the mass block, the viscous damping coefficient between the mass block and the sliding rod is important. If the viscous damping is too small, the control system will oscillate which will cause vibrations in the flywheel. It can be enhanced by increasing the air resistance. The system gain is determined by the parameter of flywheel size, and the system damping ratio ˆı is proportional to the viscous damping coefficient and inversely proportional to the mass of mass block. (2) The VIF proposed in this paper is a passive one which does not require driving mechanism. The energy which drives the VIF to change the inertia is excessive in the system. When the load torque increases, the VIF can discharge the kinetic energy to compensate for the transient shortcoming of the engine torque, conversely, the VIF absorbs the transient excess energy. (3) In terms of the inherent characteristics of DG, such as the inertia and friction of the speed actuator, the space for improving

Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.

Y.p. Zhang, X. Zhang, T. Qian et al. / Energy Reports xxx (xxxx) xxx

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Fig. 12. Mass block displacement and inertia during sudden load changing process.

Fig. 13. Engine speed, load torque and flywheel torque in simulation.

Fig. 14. Sudden increase in load torque.

the performance of speed control control algorithm is limited, and from modern control method is 2007; Larguech et al., 2016; Gu

system through optimizing the high order controller obtained also hard to realize (Pedrycz, et al., 2005). The structure of

passive VIF is simple and suitable for DG. VIF represents another solution to the precise control. Besides, combining VIF with control Algorithm optimization can better improve the system.

Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.

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Y.p. Zhang, X. Zhang, T. Qian et al. / Energy Reports xxx (xxxx) xxx

Fig. 15. Sudden reduction in load torque.

Fig. 16. Closed-loop frequency response (A-ω(s)/ωref (s), B-ω(s)/M C (s)).

Fig. 17. Nyquist contour of closed-loop (A-ω(s)/ωref (s); B-ω(s)/M C (s)).

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

CRediT authorship contribution statement

Yi ping Zhang: Conceptualization, Methodology, Supervision, Writing - original draft, Methodology. Xingwang Zhang: Resources, Investigation. Ting Qian: Formal analysis, Software, Visualization, Writing - review & editing. Rui Hu: Data Curation, Funding acquisition.

Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.

Y.p. Zhang, X. Zhang, T. Qian et al. / Energy Reports xxx (xxxx) xxx

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Please cite this article as: Y.p. Zhang, X. Zhang, T. Qian et al., Modeling and simulation of a passive variable inertia flywheel for diesel generator. Energy Reports (2020), https://doi.org/10.1016/j.egyr.2020.01.001.