Stability and Sensitivity Analysis and OptimizationControl of the Hydro-turbine Generator UnitYousong Shi  ( 1432317459@qq.com )

Huazhong University of Science and Technology - Main Campus: Huazhong University of Science andTechnologyJianzhong Zhou 

Huazhong University of Science and Technology - Main Campus: Huazhong University of Science andTechnology

Research Article

Keywords: Sensitivity, Stability, Hydro-turbine governing system, Shafting system, Multi-objectiveoptimization control, Nonlinear dynamic

Posted Date: April 16th, 2021

DOI: https://doi.org/10.21203/rs.3.rs-348512/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License.  Read Full License

1

* Corresponding Author: Jianzhong Zhou Tel.: +86 13607174132. E-mail address: 2730106235@qq.com

Stability and sensitivity analysis and multi-objective optimization control of the hydro-turbine generator unit 1

Yousong Shi Jianzhong Zhou* Jiapeng Ren Baonan Liu Yuxin Li Wei Liu 2

(School of civil and hydraulic engineering, Hua Zhong university of science and technology, Wuhan 430074, China) 3

Abstract 4

The hydro-turbine governing system (HTGS) and shafting system are mutually coupled. However, the interaction 5

between them has always been neglected. This paper aims to explore the stability and sensitivity of the governor control 6

parameters to the HTGS and shafting system and make the optimal control of the stable operation for the hydro-turbine 7

generator unit(HTGU). First, a novel HTGU motion equation is proposed, which can make connections between the 8

HTGS and the shafting system of the HTGU. And on this basis, the nonlinear coupling mathematical model of the 9

HTGS and the shafting system is established. According to the nonlinear mathematical model, the sensitivity of the 10

governor control parameters on the operating stability of the HTGU is obtained. Then, a multi-objective governor 11

control parameters optimization strategy is proposed. Furthermore, the chaotic-dominated sorting genetic algorithm 12

II(NSGA-II) and multi-objective evolutionary algorithm based on decomposition(MOEAD) were introduced to obtain 13

the optimal control parameter and mutually verify the effectiveness of the optimization effect. Finally, the nonlinear 14

dynamic characteristics of HTGU under optimal control were revealed. The simulation results show that the rotation 15

speed deviation and shafting system vibrations are sensitive on the PID parameters in some ranges and the stable region 16

will be decreased when considering the shafting system vibrations. The multi-objective PID parameter optimization 17

strategy shows good control performance on the nonlinear dynamic characteristics of the HTGU. The shafting system 18

vibrations excited by the coupled vibration sources are quasi-period in 3D space. In addition to this, these results and 19

the optimization strategy can provide some bases for the design and stable operation of the HTGU.20

2

keywords 1

Sensitivity, Stability, Hydro-turbine governing system, Shafting system, Multi-objective optimization control, Nonlinear 2

dynamic. 3

List of symbols 4

Symbol Symbolic meaning

h Hydro-turbine working head ( H )

tq Flows of the penstock ( Q )

0th Head loss of penstock

wtT Penstock flows inertia time constant

x The HTGU speed( n )relative deviation

y Guide vane relative opening ( Y )

tm The relative dynamic torque( t

M )

gm The relative resistance torque( g

M )

he , x

e , ye Hydro-turbine torque transfer coefficient

qxe , qy

e , qhe Hydro-turbine flow transfer coefficient

abT The inertia time constant of the HTGU

pk The proportional gain

ik The integral gain

dk The differential gain

yT Main servomotor time constant

u Governor regulating output

ge Load self-regulation coefficient

a Center distance between upper guide bearing and generator

b Center distance between lower guide bearing and generator

c Center distance between lower guide bearing and hydro-turbine guide bearing

d Distance between hydro-turbine guide bearing and center of the hydro-turbine runner

l Length of spindle

L Generator rotor height

D Generator rotor diameter

B The magnetic density

General coefficient

fe The dynamic eccentricity distance of the generator

1 The normal clearance of generator rotor and stator

K The equivalent stiffness of the sealing forces

sD The equivalent damping coefficient

f The nonlinear function of displacement perturbation

te The relative eccentricity of the hydro-turbine runner

2 The sealing clearance of hydro-turbine runner

zK The undetermined coefficient of axial hydrodynamic thrust

1D The hydro-turbine runner diameter

11m The equivalent mass of the generator spindle and generator

22m The equivalent mass of the hydro-turbine spindle and hydro-turbine

g The gravitational acceleration

1r Radial displacement of the generator

3

2r Radial displacement of the hydro-turbine runner

3r Radial displacement of the upper guide bearing

4r Radial displacement of the lower guide bearing

5r Radial displacement of the hydro-turbine guide bearing

2m The mass of the hydro-turbine runner

1m

The mass of the generator rotor

1e The generator mass eccentricity distance

2e The hydro-turbine runner mass eccentricity distance

1j Rotary inertia of the generator

2j Rotary inertia of the hydro-turbine runner

1 Phase position of the generator

2 Phase position of the hydro-turbine runner

1 1 1, ,x y z The central coordinates of the generator

2 2 2, ,x y z The central coordinates of the hydro-turbine runner

1K The stiffness of the upper guide bearing

2K The stiffness of the lower guide bearing

3K The stiffness of the hydro-turbine guide bearing

yK The elastic coefficient of the spindle

zk The thrust guide bearing stiffness

lk The axial stiffness

Hd The inner diameter of the spindle

Bd The external diameter of the spindle

l The total length of the generator lower spindle and the hydro-turbine spindle

E Elastic modulus of the spindle

1c The damping coefficient of the generator

2c The damping coefficient of the hydro-turbine runner

tc The torsional damping coefficient of the spindle

xiF

The excitation force in the horizontal x direction

yiF The excitation force in the horizontal y direction

ziF The excitation force in the horizontal z direction

T The total kinetic energy of the HTGU

U The elastic potential energy of the HTGU

iQ The corresponding generalized force

gM The generator magnetic torque

tM The hydro-turbine torque

1 The initial phase of the generator

2 The initial phase of the hydro-turbine runner

Torsion angle of the spindle

The angular speed of the HTGU

PID-MOO PID parameters multi-objective optimized

PID-Stein PID parameters obtained by Stein’s formula

1. Introduction 1

Water energy is a renewable energy source that can be efficiently converted into electricity [1]. At present, there 2

4

are conventional hydropower generation, pumped storage power generation, tidal power generation, ocean current 1

power generation. They mainly play the role of peak and frequency regulation in the power grid, especially 2

conventional hydropower generation and pumped storage power generation [2-3]. Therefore, the efficient, safe, 3

and stable operation of the HTGU has always been the hotspot and difficulty in the research, which mainly 4

involves the stability, the opening and stopping strategy, the equipment fault diagnosis, and the status evaluation, 5

etc. Of course, stability is the most important one that is concerned with the safety and stability of the power grid 6

and the HTGU. 7

In recent years, the research on the stability of HTGU has been divided into two main directions: the coupling 8

shafting system of the hydro-turbine governing system and the coupling power network of the hydro-turbine 9

governing system [4-5]. They are all complex nonlinear systems, and the study of their complex nonlinear 10

dynamic behavior is of great significance for the realization of optimal economic operation control [6]. In 11

previous works, the stability and dynamic characteristics of the HTGS were studied using the Routh-Hurwitz 12

criterion [7], root locus method [8], state-space method [9], and Hopf bifurcation theory [10-12]. In [13], Hopf 13

bifurcation theory is applied to analyze the influence of the structural parameters of the sloping ceiling tailrace 14

tunnel on the stability domain of the PID control parameters. In [11,14], the simulation results show that the delay 15

and saturation of the servo-system have a certain degree of impact on the stability of the HTGS. At present, in the 16

practical application of engineering, PID is the most important controller. Some scholars have also proposed 17

fractional order PID controller, sliding mode controller, state feedback controller, and so on. In[15], a novel 18

mathematical model of Francis HTGS with a straight-tube surge tank is proposed, where the unstable oscillations 19

of this model are studied extensively and presented in the forms of the bifurcation diagram, time waveform plot, 20

phase trajectories, and power spectrum. To eliminate these undesirable behaviors, a specified fuzzy sliding mode 21

controller is designed. In[16], the state equation of HTGS with the super long headrace tunnel(SLHT) was derived, and 22

the robust H∞ control strategy is designed based on the exact linearization of HTGS with SLHT and the construction of 23

5

nominal output function (NOF). In[17], it was focused on the design of the fractional-order PID (FOPID) controller for 1

the HTGS and the optimal control parameters were obtained using NSGA-II. In[18], the fuzzy fractional-order PID 2

(FFOPID) controller is designed to improve the stability of the nonlinear pumped storage unit(PSU), and the control 3

parameters of the FOPID are optimized based on the multi-objective gravitational search algorithm (MOGSA). 4

As far as shafting system vibration is concerned, the research on vibration characteristics gradually develops from 5

single vibration under a single excitation to multi-directional vibration under the influence of the coupled vibration 6

sources. Such as the differential equation of lateral vibration of the shafting is established, and the bifurcated 7

characteristics of the vibration caused by rotor mass eccentricity and rotor speed of the HTGU are revealed [19, 20]. 8

The torsional vibration characteristics of the HTGU are analyzed, meanwhile, the influences of the UMP [21], the fluid 9

seal excitation [22], the rotors Rub-impact [23], the hydraulic unbalance [24], the guide bearing loose [25], the 10

gyroscopic effect [26], the shaft misalignment [27], and pressure pulsation of the draft tube [28] effect on shafting 11

system vibrations are investigated. With the further development of the research, the hydraulic-mechanical-electrical 12

coupling factors are considered in the shafting system to explore the nonlinear vibration dynamic behavior. A 13

mathematical model considering the coupling relationship between hydraulic system, electrical system, and guide vane 14

mechanical system was established, the sensitivity of various parameters was revealed [29-31]. In [32], the generalized 15

Hamiltonian model of the shafting system is established which provides a new idea for exploring shafting vibration. In 16

[33], the fractional-order dynamic model of the shafting system and the HTGS is established to investigate the shafting 17

vibration nonlinear dynamic behavior. Besides, The 3D FEA model for the HTGU coupling foundation of hydropower 18

station is established by taking the guide bearing as the bridge connecting the shafting and the structure of the 19

hydropower house, and then the influence of the shafting system vibration on hydropower house vibration and its 20

response mechanism is studied numerically [34, 35]. 21

To sum up, the stability and dynamic characteristics of HTGS are obtained by different methods, and some 22

controllers are proposed to achieve effective control. On the other hand, the shafting system vibrations dynamic 23

6

characteristics under different excitation sources are verified, which lays a theoretical foundation for vibration control 1

and fault diagnosis of the HTGU. However, the stability and sensitivity of the PID parameters to the HTGS and the 2

shafting system have not been revealed. The nonlinear dynamic characteristics of the HTGS and shafting system 3

vibration under the coupling effects of the mass eccentricity, the UMP, the arcuate whirl of the rotor, and the seal 4

excitation under the optimal PID control haven’t been explored which seriously restricted the secure and stable 5

operation of the HTGU, and brought huge economic losses. Therefore, it is urgent to explore the stability and sensitivity 6

of the PID parameter under the action of hydraulic-mechanical-electrical coupling vibration sources and realize optimal 7

control to ensure the safe and stable operation of the HTGU. 8

As summarized above, the novelty and contributions of this study are: (1) A novel and practical nonlinear 9

mathematical model coupling HTGS and shafting system is established. (2) The stability and sensitivity of ration speed 10

deviation and shafting system vibrations to the PID parameters are revealed. (3) The parameters chosen of the PID 11

governor are formulated as a multi-objective optimization problem that considers the HTGS and shafting system 12

vibrations simultaneously for the first time, in which the objective functions are composed by the ITSE. (4)The 13

influence of the self-regulating coefficient( ge ) on the nonlinear dynamic characteristics of the HTGU is revealed. (5) 14

The nonlinear dynamic characteristics of the shafting system vibrations in the 3D space, which are generated by the 15

mass eccentricity, the UMP, the arcuate whirl of the rotor, and the seal excitation coupled vibration sources under the 16

optimal control are revealed. Of course, there are still some shortcomings in this paper, the gyroscopic effect for 17

shafting vibration is ignored. Meanwhile, the idealized models are adopted for external excitation sources of hydraulic, 18

mechanical, and electrical systems, which cannot fully reflect the external excitation action on the HTGU. 19

The remainder of this paper is organized as follows. In section 2, the nonlinear mathematical modeling. In section 3, 20

the stability and sensitivity of the governor PID parameters to the HTGU. In section 4, the governor PID parameter 21

multi-objective parameter optimization control. In section 5, the nonlinear dynamic characteristics of the HTGU. Finally, 22

in section 6, the conclusions are presented. 23

7

2. Nonlinear mathematical modeling 1

The layout of a conventional hydroelectric power plant and a pumping and storage station is shown in Fig. 1. In this 2

paper, we mainly involve the water diversion system, the speed regulation system, and the shafting system. 3

Upstream

reservoir

Governor

Ball

valve

Volute

Draft tubeDownstrea

m reservoir

Shafting

4

Fig. 1 Structure of a hydropower station 5

2.1 Hydro-turbine governing system equation 6

Ignoring the complex layout of the hydraulic water diversion system, only consider the upstream reservoir, the 7

penstock, the governor, the hydro-turbine, the downstream reservoir, and the generator. The universal mathematical 8

model of the HTGS is established as follows: 9

(1) Dynamic equation of linear penstock [13]: 10

0

0

d 2

d

t t

wt t

q hh T q

t H (1)

in Eq (1), 0 0/h H H H , 0 0/t

q Q Q Q are the working head ( H ), and the flows of the penstock ( Q ) of 11

the HTGU, respectively. The subscript "0" represents the initial value, 0th is the head loss of penstock, wtT is the 12

inertia time constant of flows in the penstock. 13

8

(2) Hydro-turbine torque and flows equation [13]: 1

t h x y

t qh qx qy

m e h e x e y

q e h e x e y

(2)

(3) Generator motion equation [13]: 2

d 1

dt g g

ab

xm m e

t Tx (3)

(4) Electro-hydraulic servo system differential equation [36]: 3

d

d y

y u y

t T

(4)

(5) PID controller equation[36]: 4 2

2

d d d

d d dp i d

u x xk k x k

t t t (5)

in Eq (2)–(5), 0 0/x n n n , 0 0/y Y Y Y , 0 0/t t t t

m M M M , 0 0/g g g g

m M M M are the 5

relative values of the deviation of HTGU speed ( n ), guide vane opening (Y ), dynamic torque ( tM ), and resistance 6

torque ( gM ), respectively. The subscript "0" represents the initial value. h

e , xe , and y

e are the torque transfer 7

coefficient of the hydro-turbine. qxe , qy

e , and qhe are the flow transfer coefficient of the hydro-turbine. ab

T is the 8

inertia time constant of the HTGU. pk , i

k , and dk are the proportional gain, the integral gain, and the differential 9

gain, respectively. yT is the time constant of the main servomotor. u is the regulating output for the governor. g

e is 10

load self-regulation coefficient 11

Combining the Eq(1)–Eq(5), the HTGS equation can be written as follows: 12

0

0

2/

1

t qx qy t

t t wt

qh

h qx h qyh t

x g y

qh qh qh

t qx qy h qxh

h x y g t x g

qh

g

ab

y

p d

g i

ab h qab q

x e mT

u yy

T

k ku m e k x

q e x e y hq q T

e H

e e e ee qe x e y

e e e

q e x e y e eee e x e y x q e

ee

eT T e

&

&

&

&

&h qy

y

h qh

e ex e y

e

& &

(6)

2.2 Shafting Transverse, longitudinal, and torsional coupled vibration equations 13

9

As can be seen from Fig. 2, ( 1,2,3)i

B i represents the upper, lower, and hydro-turbine guide 1

bearings. 1,2,3,4,5i

O i and 1,2,3,4,5i

r i represent the geometric centers and radial radii of the generator, 2

hydro-turbine runner, and upper, lower, and hydro-turbine guide bearings, respectively. We have 2 2

i i ir x y , and 3

we denote 1 3O O a , 1 4O O b , 4 5O O c , 2 5O O d . From the geometrical relation in Fig. 2, the following 4

equations can be obtained [34]: 5

1 2

3( )

a b b c d a c d r abrr

b b c d

(7)

1 2

4

c d r brr

b c d

(8)

1 2

5

dr b c rr

b c d

(9)

According to, Eq (7), Eq (8), and Eq (9), the radial vibration radius of the generator, the hydro-turbine runner, and the 6

guide bearings are mutually mapped. 7

a

b

c

d

Upper guide bearings

1r

2r

3r

4r

5r5o

2o

3o

4o

1oGenerator

Lower guide bearings

Hydro-turbine guide

bearings

Hydro-turbine runner

8

Fig. 2 Shafting structure diagram 9

As we know the shafting system vibration is mainly caused by hydraulic-mechanical-electrical coupled factors. The 10

mechanical vibration sources are mainly divided into mechanical unbalance forces and rotary unbalance forces, which 11

10

are mainly distributed on the generator and the hydro-turbine runner, the calculation formula can be expressed as[23]: 1

2F me (10)

The electrical vibration source is the UMP, which is calculated as follows [19]: 2

2

10.5

f

ump

eBF DL

(11)

in Eq (11), L , and D are the generator rotor height and diameter, B is the magnetic density, is a general 3

coefficient ( 0.2 ~ 0.5 ), fe is the dynamic eccentricity distance of generator ( 2 2

1 1fe x y ), and 1 is the 4

normal clearance of generator rotor and stator. 5

The water flowing to the flange seals forms circumference flow causing the sealing forces, which can be written 6

as[37]: 7

2 2

02 02 02 02 02

2 2

02 02 02 02 02

2

2

x lf f f f s s f f f

y lf f f f s s f f f

F K m x D y D x m y m x

F K m y D x D y m x m y

& & &&

& & && (12)

where K , sD , and f

are the equivalent stiffness of the sealing forces, the equivalent damping coefficient, and the 8

nonlinear function of displacement perturbation, respectively. Their corresponding expressions are written as follows: 9

0

2

0

2

0

0

1

1

(1 )

j

t

j

s t

b

f t

K K e

D D e

e

(13)

in Eq(13), we have 1

32

j , 00 1b . The te is the relative eccentricity of the runner ( 2 2

2 2 2/t

e x y ,2 is 10

sealing clearance). Where j , 0b , and 0 were used to describe the specific sealing, generally, we have 0 0.5 . 11

The sealing force was calculated as follows： 12

2

0 3 0 0 1 3 2 3, , fK D T m T (14)

The operation is accompanied by arcuate whirl centrifugal force of the generator and hydro-turbine, which were 13

calculated as follows[38]: 14

11

2

1 1 1 3 4

2

2

2G G

m ejF

g

r r

(15)

22

2 2 5

-

22T G

m ej rF

g

(16)

1

The HTGU is always accompanied by axial hydrodynamic thrust during operation, which is determined by the 2

empirical formula as follow[39]: 3

2

1 0.254z z

n QF K D

H

(17)

where zK is the undetermined coefficient (generally 0.7 0.8

zK ). n is the speed of the runner under specific 4

working conditions. Q is the overflow rate of the runner under specific working conditions. H is the working head 5

under specific working conditions. 1D is the hydro-turbine runner diameter. 6

Assuming that the hydro-turbine and generator spindles are connected rigidly[32], and when axial vibration is 7

considered, the mass of the generator spindle is concentrated on the rotor equivalently, which was written as 11m . The 8

mass of the turbine spindle is concentrated on the runner equivalently, which was written as 22m . 9

The axial force at the generator is mainly provided by the thrust bearing, which is approximately expressed as the 10

combined force of the unit's gravity and axial hydraulic thrust, so we have: 11

11 22G zF m m g F (18)

Based on the above supposes, the kinetic energy of the SCV can be written as: 12

22 2 2 2 2 2

1 1 1 1 1 1 1 1 1 1 111 1 1 1 1 1 1

2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2

1

2 2 2 2 2 2

2

22 2

1 12 cos 2 sin

2 2

1 12 cos 2 sin

2 2

1

2

1

2

T m x y e e y e x J m e

m x y e e y e x J m e

m z

m z

& & & & &

& & &

& & & &

& & & & &&

(19)

in Eq (19), , ,i i i

x y z , iJ , im , ie , i , and i i & 1,2i are the centroid coordinates, the rotational inertia, the 13

equivalent concentrated masses, the eccentricity distance, the rotation angle, and the rotational angular velocity of the 14

generator rotor and the hydro-turbine runner, respectively. 15

Considering transverse-longitudinal and flexural-torsional coupled vibrations, the potential energy of the shafting for 16

12

the HTGU can be expressed as follows： 1

2

2 2 2 2 2 2 2 2

1 1 11 2 2 22 1

2

1 11 1 2

1 2 2 12

2

1 2 22 2

1 1 1

2 22z l ly

k k z

U x y K x y K x y x y K

K m gz k z m gz

(20)

where 11K , 22K , and 12K is the equivalent stiffness of the shafting system of the HTGU, yK is the elastic coefficient 2

of the spindle, lk is the axial stiffness, z

k is the thrust guide bearing stiffness. 1 and 2 are the rotation angles of the 3

generator and the hydro-turbine runner, respectively [19]. 4

22 2

111 1 2 32 22

22 2

222 1 2 32 22

1 212 1 2 32 22

c dA dK k k k a

B b c d b c d

b cA bK k k k b

B b c d b c d

b c d d b cA AK k k k c

B b c d b c d

(21)

in Eq (21), 1A a b b c d , 2A ab , ( )B b b c d 1k , 2k , and 3k are the supporting stiffness of 5

the upper guide bearing, the lower guide bearing, and the hydro-turbine guide bearing, respectively. 6

2 2

32

H B

y

E d dK

l

(22)

32l y

k K (23)

in Eq(22), E is the elastic modulus of the spindle. Hd and B

d are the internal and external diameters of the spindle. 7

l is the total length of the generator lower spindle and the hydro-turbine spindle. 8

Assuming that the shafting damping is simplified as linear damping applied to the generator and hydro-turbine runner, 9

and the external excitation force is also considered, then the generalized force of the HTGU can be formulated as 10

follows [32]: 11

13

z

xi i xi

yi i

i

yi

i

i

ii zi

Q c F

Q c Fy

Q F

x

zc

(24)

in Eq (24), i

c is the damping coefficient, xi

F is the excitation force in the horizontal x direction, yiF is the excitation 1

force in the horizontal y direction, and ziF is the excitation force in the horizontal z direction. 2

Lagrange equation can be expressed as[32]: 3

d

di

i i i

T T UQ

t q q q

&

(25)

where T is the kinetic energy expressed by various generalized coordinates and generalized velocities of the system. 4

U is the potential energy represented by the generalized coordinates. iQ is the corresponding generalized force. 5

Since the quality of the guide bearing is much lower than those of the generator and the hydro-turbine runner, the 6

vibration of the guide bearing can be ignored. Substituting Eq (19), Eq (20), and Eq (24) into Eq (25), the vibration 7

motion equations of the shafting can be written as: 8

2 2

2 2 2

1 1 1 1 1 1 1 1 1 11 1 12 1 1 1 1 12 2

11 1 1 1

1 1

2 2

2

1 11

2 2

1 1 1 1 1 1 1 1 1 11 1 12 1 1 1 1 12 2

1 1

2

sin 2 cos cos

cos 2 sin sin

G G u

G G

z z

m

p

G

p

um

l

x ym x c x m e x K x K m e F

x y

x ym y c y

F a

F b

m z c z k k z m g F

m e y K y K m e Fx y

m

c

&& &

&& &&&

&

&

&& &

& &

&

2 2

1 1 2

2 2 2 2 2 2 2 2 22 2 12 2 2 2 22 2

2 2

2 2

1 1 2

2 2 2 2 2 2 2 2 2

- 2

- 2

22 2

22 2 12 2

2 2 2 22

2 2 22 2

2 2

1

cos

s

sin 2 cos

cos 2 insin

2

( )

T G

T

x lf

y

z

f G

z l

l

xF d

F e

m z c

yx c x m e x K x K m e F

x y

x ym y c y m e y K y K m e F

x y

z k z m f

m

g

J

F

&& &

&& &

&& && &

&& &

2

1 1 1 1 1 1 1 1 1 1 1

2

2 2 2 2 2 2 2 2 2 2 2 2

cos sin

2 cos sin

y g

y t

e m e y m e x g

h

K M

J m e m e y m e x K M

&& && &&

&& && &&

(26)

in Eq (26), t t gBM m M , g g gB

M m M are the hydro-turbine torque, and the generator torque of the HTGU, 9

respectively. 1 2 1 2= = = = & & is the inherent coaxial rotation, 1 is the rotational phase of the generator rotor, 2 is 10

14

the rotational phase of the hydro-turbine runner. we have 1 1d t and 2 2d t , it can be further 1

obtained that the phase difference between the generator and the hydro-turbine runner is 1 2 1 2 . 2

2.3 Nonlinear equation coupling the HTGS and the shafting system 3

In Eq(26), the shafting system we have 2 2

2 2 2 1 1 12 ( ) 2 ( )J m e g J m e h and ( ) ( )g h which can be obtained: 4

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

2 2

1 1 1 2 2 2

2 2

2 2 2 1 1 1

2 2 2 2

2 2 2 1 1 1 1 1 1 2 2 2

cos sin cos sin

2 2

2 2

2 2 2 2

gB g t

m

y

B

m e y m e x m e y m e x

J m e J m e

J m e J m eJK

J m e J m e J m e J m e

S m m

&& && && &&&&

(27)

1 1 1 1 1 1 2 2 2 2 2 2cos sin cos sind

t gM M m e y x m e yJ

dx

t && && && && (28)

where 2 2

2 2 2 1 1 12 2 eJ J m e J m . Meanwhile, considering the change of angular velocity( ) in the transient 5

process of perturbation factor, the Eq (28) can be written as: 6

1 1 1 1 1 1 2 2 2 2 2 2cos sid

/d

n cos sinmt Bg

M M m e y x m eJt

y x && && && && (29)

Due to the 0 0/x n n n , and 0 0/ 30 / 30 / / 30n n n , so we haved d

d d

x

t t

. gB

M is the basic value 7

of the generator torque, mB is the basic value of rotation speed (Where /

gB gB mBM S ,

30

r

mB

n 8

/t t gB t gB mB

M m M m S , and /g g gB g gB mB

M m M m S ). 9

The inertia time constant of the HTGU is 2 2 2

2 2 2 1 1 12 2

ab

gB

mBJ m e J m e

TS

. Further, Substituting J , ab

T 10

into Eq (29), and rewritten tM and g

M in Eq(29), the motion equation of the HTGU can be written as: 11

1 1 11 11 22 2 2 2 2sin cos sin cod

d

smBt g

ab ab gB

e em mx

t T T

m x y m x y

S

&& && && && (30)

When the load self-regulation factor is considered, and Substituting Eq (2) into Eq (30) the HTGU rotation equation 12

can be obtained as follows: 13

15

1 1 1 1 1 2 2 21 2 22sin cos sin cod

d

s

t qx qy

h x y g g

qh

ab gB

mB

ab

m x y m

q e x e ye e x e y m e x

e e xex

t T S

y

T

&& && && && (31)

in Eq (31), 1 1mB

x t , 2 1mB

x t . 1

According to Eq(31), the HTGS and shafting system were connected as a new nonlinear complex coupled system, 2

substituted Eq (31) into Eq (6) replace x , and substituted Eq (27) into Eq (26) to replace Eq(g) and Eq(h). the novel 3

nonlinear equation is written as follows: 4

1 1 1 1 1

0

2 2 2 2 2

0

1 2sin cos sin co

2/

smB

y

p d

i

a

t qx qy t

t t wt

qh

t qx qy

h x y g g

qh

ab ab gB

t qx qy

h x y g g

b qh

q e x e y hq q T

e H

q e x e ye e x e y m e x

e ee m x y m x yx

T T S

q e x e ye e x e y m e x

u yy

T

k ku k x

T e

&& && && &&

&

&

&

&

2 2

2 2

1 1 1 1 1 1 1 1 11 1 12 1 1 1 12 2

1 1

1 1 1 1 1 1 1 1

2

cos sin 2 1 cos /

sin cos 2

h qx h qyh

t x g

ab

ump

y

qh qh

G G mB

ump G

qh

G

eT

x yF F m e c x x K x

e e e ee

K m e x m

q e x e y

x

e

y

y F F m

e e

K

x x

xe c y y

&

& & &

&& &

&&& &

2

1

1 1 1 1 11 11

2

- 2

2 2

2 2

11 1 12 1 1 12 2

1 1

2 2

1 1

2 2 2 2 2 2 2 22 2 12 2 2 2 22 2

2 2

2 2 2- 2

1 sin /

sin 2 1 c

/

co

i

s /

n

os

s

G z z l

T

mB

x lf mB

l Ty f

G

G

x yy K m e x m

x y

x yx F m e c x x K x K m e x m

x y

z F c z k k z m g m

F x

Fy F m e

&& &

&&& &

&& &

2 2

1 1

2 2 2 2 22 2 12 2 2 2 22 2

2 2

1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

2 2

1 1 1 2 2 2

2

2 2 2 1 1 1

2

2 2 2 2 22 22

cos 2 1 sin /

cos sin cos sin

2

2

/

2

2

mB

y

z z l

t

x

z F c z k z m g m

c

x yc y y K y K m e x m

x y

m e y m e x m e y m e x

J m e J m e

J m e J m eK

&& &

&

&& && &&& &

&&&

2 222 2 2 1 1 1

2 2 2 2

2 2 2 1 1 1 1 1 1 2 2 2

2 2

2 2 2 2

gB g

mB

tJ m e J m e

J m e J m e J m e J m

m m

e

S

(32)

in Eq (32), tc& is the linear torsional damping force. 5

3. Stability and sensitivity of the governor PID parameters to the hydro-turbine generator unit 6

16

The governor is one of the most important parts of the HTGU. The control parameters of the governor have a very 1

important influence on the regulation performance and stability of the unit. At the same time, PID governor is widely 2

used in engineering practice. Therefore, we focus on the analysis of PID parameters on the stability and sensitivity of 3

the unit. A hydropower station is selected for analyzing the stability and sensitivity of the governor parameter. The 4

classical fourth-order Runge-Kutta method is adopted to solve the nonlinear equation of the coupling HTGS with the 5

shafting system in MATLAB. The actual parameters of a hydropower station are shown in Table 1. 6

Table 1 Actual parameters of a hydropower station 7

parameter value unit parameter value unit

1m 54.91 10 kg

a 2.54 m

2m 45.6 10

kg b 2.04 m

11m 55.65432 10

kg c 8.19 m

22m 51.17 10

kg d 1.765 m

1c 64.1 10 /N s m 1e 0.0005 m

2c 62.52 10 /N s m 2e

0.0005 m

1zc

63.6 10 /N s m Hd

1.15 m

2zc

66.31 10 /N s m Bd 0.3 m

tc 1000 /N s rad 1d

7.44 m

1k 81.85 10 /N m L 2.927 m

2k 81.45 10 /N m D 7.44 m

3k 71.15 10 /N m 1 0.0004 m

zk

71.27 10 /N m 2 0.0021 m

1j 64.625 10

2kg m l 10.3 m

2j 52 10

2kg m r

n 250 / minr

E 9206 10 a

P gBS 306 MW

0H 195 2m H O 0th 3.2723 m

Q 176.1 3 /m s yT 0.02 s

xe -1 ab

T 10.8072 s

qhe 0.5 wt

T 2.74 s

qye 1 0 0.23

he 1.5 1D

5.34 m

ye 1 B 1 T

g 9.81 2/m s

0.5

2 0.11 0n 0.066

0 0.6 0m -0.25

p 60.5 10 a

P 0b 3.5

zK 0.8 0T

0.0657

0Ra 2703 ar 1.86 m

In this section, Stein's formula[40] was adopted to obtain the combination of PID parameters. While the relative value 8

17

of load disturbance is 0.1 or 0.1 , the stability and sensitivity of the PID parameters to the HTGS and shafting 1

system vibrations are obtained using the bifurcation diagram. Since the deviation of the characteristic quantity of the 2

HTGS is consistent, the deviation of the rotation speed is selected as the typical characteristic variable of the HTGS. 3

3.1 The stability and sensitivity of pk to the rotation speed deviation and shafting system vibrations 4

The stability and sensitivity of the rotation speed deviation and shafting system vibrations to pk are shown in Fig.3. 5

From Fig.3(a), it is seen that when pk is 0.1,1.65 , the rotation speed deviation is bifurcated which shows a strong 6

sensitivity. When pk is 1.65,5.4 , the rotation speed deviation is stable. When p

k is 5.4,5.6 , the rotation speed 7

deviation is chaotic that shows a strong sensitivity. As can be seen from Fig.3(b), When pk is 0.1,5.6 , the negative 8

torsion vibration angles are stable. From Fig.3(c), it is seen that when pk is 0.1,2 , the generator vibration of 9

x direction is chaotic that shows obvious vibration and strong sensitivity. When pk is 2,4.05 , the generator vibration 10

of x direction is quasi-periodic which shows slight vibration，and when pk is 4.05,5.35 , the generator vibration of 11

x direction still shows quasi-periodic but vibration amplitude increase obviously. When pk is 5.35,5.6 , the 12

generator vibration of x direction is chaotic that shows obvious vibration and strong sensitivity. As can be seen 13

from Fig.3(d), when pk is 0.1,2.15 , the generator vibration of y direction is chaotic that shows obvious vibration 14

and strong sensitivity. When pk is 2.15,3.2 , the generator vibration of y direction is quasi-periodic which shows 15

slight vibration, and When pk is 3.2,5.3 , the generator vibration of y direction still shows quasi-periodic but 16

vibration amplitude increase obviously. When pk is 5.3,5.6 , the generator vibration of y direction is chaotic that 17

shows obvious vibration and strong sensitivity. As can be seen from Fig.3(e), when pk is 0.1,1.35 , the generator 18

vibration of z direction is bifurcated which shows slight vibration and slight sensitivity. When pk is 1.35,5.35 , the 19

generator vibration of z direction is periodic which shows slight vibration. When pk is 5.35,5.6 , the generator 20

vibration of z direction is chaotic that shows obvious vibration and strong sensitivity. As can be seen from Fig.3(f) 21

and (g), when pk is 0.1,1 , the hydro-turbine vibration of x and y direction is chaotic that shows obvious vibration 22

and strong sensitivity. When pk is 1,5.4 , the hydro-turbine vibration of x and y direction is quasi-periodic which 23

18

shows slight vibration. When pk is 5.4,5.6 , the hydro-turbine vibration of x and y direction is chaotic that shows 1

obvious vibration and strong sensitivity. From Fig.3(h), it is seen that when pk is 0.1,1.35 , the hydro-turbine 2

vibration of z direction is bifurcated which shows obvious vibration and strong sensitivity. When pk is 1.35,5.3 , the 3

hydro-turbine vibration of y direction is periodic which shows slight vibration. When pk is 5.3,5.6 , the 4

hydro-turbine vibration of y direction is chaotic that shows obvious vibration and strong sensitivity. Meanwhile, 5

under the different load disturbances, the shafting radial vibration shows almost identical stability, However, the axial 6

vibration shows the same stability of pk but the larger amplitude was generated by smaller the load. 7

(a) Rotation speed deviation bifurcation diagram of pk (b) Torsion angle bifurcation diagram of p

k

(c) Generator vibration ( 1x ) bifurcation diagram of pk (d) Generator vibration ( 1y ) bifurcation diagram of p

k

19

(e) Generator vibration ( 1z ) bifurcation diagram of pk (f) Hydro-turbine vibration ( 2x ) bifurcation diagram of p

k

(g) Hydro-turbine vibration ( 2y ) bifurcation diagram of pk (h) Hydro-turbine vibration ( 2z ) bifurcation diagram of p

k

Fig.3 The rotation speed deviation and SCV dynamic characteristics to pk

3.2 The Stability and sensitivity of ik to the rotation speed deviation and shafting systems vibrations 1

The stability and sensitivity of ik to the rotation speed deviation and shafting system vibrations are shown in Fig.4. 2

From Fig.4(a), it is seen that when ik is 0.1,0.77 , the rotation speed deviation is stable. When i

k is 0.77,1.15 , the 3

rotation speed deviation is bifurcated which shows strong sensitivity. As can be seen from Fig.4(b), 4

When ik is 0.1,1.15 , the negative torsion vibration of HTGU is stable. From Fig.4(c), it is seen that 5

when ik is 0.1,0.21 , the generator vibration of x direction is chaotic that shows obvious vibration and strong 6

sensitivity. When ik is 0.21,0.72 , the generator vibration of x direction is quasi-periodic which shows slight 7

vibration. When ik is 0.72,1.15 , the generator vibration of x direction is chaotic that shows obvious vibration and 8

strong sensitivity. As can be seen from Fig.4(d), when ik is 0.1,0.26 , the generator vibration of y direction is chaotic 9

that shows obvious vibration and strong sensitivity. When ik is 0.26,0.59 , the generator vibration of y direction is 10

quasi-periodic which shows slight vibration. When ik is 0.59,1.15 , the generator vibration of y direction is chaotic 11

20

that shows obvious vibration and strong sensitivity. As can be seen from Fig.4(e), when ik is 0.1,0.83 , the 1

generator vibration of z direction is periodic which shows slight vibration. When ik is 0.83,1.15 , the generator 2

vibration of z direction is bifurcated which shows obvious vibration and strong sensitivity. As can be seen from 3

Fig.4(f) and (g), when ik is 0.1,0.89 , the hydro-turbine vibration of x and y direction is quasi-periodic which shows 4

slight vibration. When ik is 0.89,1.15 , the hydro-turbine vibration of x and y direction is chaotic that shows 5

obvious vibration and strong sensitivity. From Fig.4(h), it is seen that when ik is 0.1,0.86 , the hydro-turbine 6

vibration of y direction is periodic which shows slight vibration. When ik is 0.86,1.15 , the hydro-turbine vibration 7

of y direction is chaotic that shows obvious vibration and strong sensitivity. Obviously, under the different load 8

disturbances, the shafting radial vibration shows almost identical stability, However, the axial vibration shows the same 9

bifurcated characteristics of ik but the smaller amplitude was generated by a larger load. 10

(a) Rotation speed deviation bifurcation diagram of ik (b) Torsion angle bifurcation diagram of i

k

(c) Generator vibration ( 1x ) bifurcation diagram of ik (d) Generator vibration ( 1y ) bifurcation diagram of i

k

21

(e) Generator vibration ( 1z ) bifurcation diagram of ik (f) Hydro-turbine vibration ( 2x ) bifurcation diagram of i

k

(g) Hydro-turbine vibration ( 2y ) bifurcation diagram of ik (h) Hydro-turbine vibration ( 2z ) bifurcation diagram of i

k

Fig. 4 The rotation speed deviation and SCV dynamic characteristics of ik

3.3 The stability and sensitivity of dk to the rotation speed deviation and shafting system vibrations 1

The stability and sensitivity of dk to the rotation speed deviation and shafting system vibrations are shown in Fig.5. 2

From Fig.5(a), it is seen that when dk is 0.1,5.48 , the rotation speed deviation is stable. When d

k is 5.48,5.5 , the 3

rotation speed deviation is chaotic that shows a strong sensitivity. As can be seen from Fig.5(b), When dk is 0.1,5.5 , 4

the negative torsion vibration of HTGU is stable. From Fig.5(c), it is seen that when dk is 0.1,5.46 , the generator 5

vibration of x direction is quasi-periodic that shows slight vibration. When dk is 5.46,5.5 , the generator vibration 6

of x direction is chaotic that shows obvious vibration and strong sensitivity. As can be seen from Fig.5(d), 7

when dk is 0.1,5.46 , the generator vibration of y direction is quasi-periodic which shows slight vibration, but when 8

dk is 0.1,1.78 and 3.62,5.46 , it shows the slight sensitivity. When d

k is 5.46,5.5 , the generator vibration 9

of y direction is chaotic that shows obvious vibration and strong sensitivity. As can be seen from Fig.5(e), 10

when dk is 0.1,5.44 , the generator vibration of z direction is periodic which shows slight vibration. When d

k 11

22

is 5.44,5.5 , the generator vibration of z direction is chaotic that shows obvious vibration and strong sensitivity. 1

As can be seen from Fig.5(f) and (g), when dk is 0.1,0.5.46 , the hydro-turbine vibration of x and y direction is 2

quasi-periodic that shows slight vibration. When dk is 5.46,5.5 , the hydro-turbine vibration of x and y direction is 3

chaotic that shows obvious vibration and strong sensitivity. From Fig.5(h), it is seen that when dk is 0.1,5.48 , the 4

hydro-turbine vibration of y direction is periodic which shows slight vibration. When dk is 5.48,5.5 , the 5

hydro-turbine vibration of y direction is chaotic that shows obvious vibration and strong sensitivity. Obviously, 6

under the different load disturbances, the shafting vibration shows almost identical stability of dk , However, the axial 7

vibration's larger amplitude was generated by a smaller load. 8

(a) Rotation speed deviation bifurcation diagram of dk (b) Torsion angle bifurcation diagram of d

k

(c) Generator vibration ( 1x ) bifurcation diagram of dk (d) Generator vibration ( 1y ) bifurcation diagram of d

k

23

(e) Generator vibration ( 1z ) bifurcation diagram of dk (f) Hydro-turbine vibration ( 2x ) bifurcation diagram of d

k

(g) Hydro-turbine vibration ( 2y ) bifurcation diagram of dk (h) Hydro-turbine vibration ( 2z ) bifurcation diagram of d

k

Fig. 5 The rotation speed deviation and SCV dynamic characteristics to dk

4. The governor PID parameter multi-objective optimization control 1

Based on the coupling of HTGS with shafting system nonlinear mathematical model, a multi-objective 2

optimization strategy of PID parameters is proposed with the following innovations: (1) The objective functions 3

consider the rotation speed deviation, the opening of the guide vane, and the shafting system vibrations simultaneously. 4

(2) The objective function is composed of the ITSE.（3）The NSGA-II[41] and MOEAD[42] which have shown high 5

efficiency and competitive performance in solving the multi-objective optimization problems compared with other 6

evolutionary algorithms were introduced. 7

4.1 Objective functions 8

In the process of load regulation, the rotation speed, guide vane opening, and shafting system vibration will be 9

volatile. Under small load disturbance, to make the complex coupled system to obtain the response output overshoot 10

24

minimize and the shortest regulation time. The ITSE ( 2

0dITSE te t t

) of rotation speed deviation, guide vane 1

opening deviation, vibration radius deviation of the generator and hydro-turbine in 3D space are selected to establishes 2

the objective functions, which was written as follows: 3

1 1 11 22

2 2 11 22

x r r

y r r

J W ITSE ITSE ITSE

J W ITSE ITSE ITSE

(33)

In Eq(33), 1 2 2W W ， 2 2 2

11 1 1 1r x y z ，and 2 2 2

22 2 2 2r x y z . 4

4.2 Decision variables 5

When load disturbance occurs to the HTGU, improper PID parameter combination may lead to the whole system 6

slow stable response, or even goes into an unstable or chaotic state, which will lead to control failure of the whole 7

system and lead to power grid oscillation and abnormal vibration of the HTGU. Therefore, we select the parallel PID 8

parameters pk , ik , and dk as the decision variables. In section 3, the rotation speed deviation and shafting system 9

vibration stability and sensitivity of pk , ik , and dk are given, which can be concluded that the system has different 10

stability domains for different characteristic variables of pk , ik , and dk . There, we define the periodic and 11

quasi-periodic vibration regions of shafting system vibrations and the stable region of HTGS to form the relative 12

stability region. Therefore, the intersection principle was used to obtain the common relative stable domain consider the 13

shafting system vibration of the parameters of pk , ik , and dk , and then the boundary of the common stability domain 14

is taken as the boundary of the decision variables. so we have: 15

1 1 1 2 2 2

1 1 1 2 2 2

1 1 1 2 2 2

ps p xs p s p x s p y s p z s p x s p y s p z s

is i xs i s i x s i y s i z s i x s i y s i z s

ds d xs d s d x s d y s d z s d x s d y s d z s

k k k k k k k k k

k k k k k k k k k

k k k k k k k k k

(34)

Thus, the relative stable domain that considers the HTGS and the shafting system of the decision variables 16

are 2.15,3.2psk ， 0.26,0.59isk , and 1.78,3.62dsk , respectively. 17

4.3 Multi-objective optimization 18

According to the above determined objective functions and decision variable boundary, the NSGA-II and MOEAD 19

25

multi-objective optimization algorithms were used to establish the optimization process as shown in Fig.6, and the 1

optimization results of the dual-objective and parameter were obtained as shown in Fig.7. When the load disturbance 2

occurs to the HTGU, since the current focus is on the overregulation amount and regulation time of the rotation speed 3

deviation of the unit, so the smaller 1J is selected after optimized. The detailed results are shown in Table 2. 4

Start

Initial

population

Initial calculation

objective function

Gen<maxGen

End

NSGA-II or MOEAD

multi-objective optimize

Yes

No

5

Fig 6 Optimization process diagram 6

(a) Mg=-0.1 (b) Mg=0.1

Fig 7. Pareto optimal solution obtained by different algorithms

Table 2 the optimal results 7

algorithms Load disturbance 1J 2J pk ik dk

NSGA-II 0.1gm 34.9 3585.5 3.2 0.49 2.69

MOEAD 0.1gm

26

5 Nonlinear dynamic characteristics of the HTGU 1

In this section, the nonlinear dynamic characteristics of the HTGS, generator vibration, hydro-turbine vibration, and 2

torsion vibration were revealed using the time domain feature, phase trajectory, Poincare map, and frequency spectrum 3

while the small load disturbance happened with the optimal PID control parameters. 4

5.1 Nonlinear dynamic characteristics of the HTGU to the self-regulating coefficient 5

The nonlinear dynamic characteristics of the HTGS are shown in Fig.8. As can be seen from Fig.8(a) and (b), while 6

the load disturbance is 0.1gm , the optimized PID parameters make the adjustment time of tq , x , and y were 7

significantly shortened, whereas the overshoot of tq , x and y was increased obviously. Meanwhile, it is seen that 8

the overshoot of tq , x , and y was decreased gradually with the self-regulation coefficient increased. 9

(a) 0.1gm (b) 0.1gm

Fig.8 The nonlinear dynamic characteristics of the HTGS

The dynamic characteristics of generator vibration radii in 3D space are shown in Fig.9. As can be seen from Fig.9(a) 10

and (b), while the load disturbance is 0.1gm the optimized PID parameters make the adjustment time was 11

significantly shortened and the vibration amplitude decreased slightly. Meanwhile, it is seen that the dithering 12

amplitude of 11r was decreased gradually with the self-regulation coefficient increased, however, the dithering time 13

is increased slightly. 14

27

(a) 0.1gm (b) 0.1gm

Fig 9 Robustness of optimized PID controller for 11r to the load-self regulation coefficient

The dynamic characteristics of hydro-turbine vibration radii in 3D space are shown in Fig.10. As can be seen from 1

Fig.10(a) and (b), while the load disturbance is 0.1gm the optimized PID parameters make the adjustment time 2

was significantly shortened and the vibration amplitude decreased slightly. Meanwhile, it is seen that the dithering 3

amplitude of 22r was decreased gradually with the self-regulation coefficient increased, however, the dithering time 4

is increased slightly. 5

0.1gm 0.1gm

Fig 10 Robustness of optimized PID controller for 22r to the load-self regulation coefficient

The dynamic characteristics of spindle torsion vibration are shown in Fig.11. As can be seen from Fig.11(a) and (b), 6

while the load disturbance is 0.1gm the shafting torsion vibration is stable. Meanwhile, it is seen that the 7

torsion vibration remains same with the self-regulation coefficient increased. 8

9

28

0.1gm 0.1gm

Fig.11 Robustness of optimized PID controller for torsion angle to the load-self regulation coefficient

5.2 Nonlinear dynamic characteristics of shafting system vibrations 1

According to the above research, the nonlinear dynamic characteristics of the shafting system vibrations excited by 2

the mass eccentricity, the UMP, the arcuate whirl of the rotor, and the seal excitation coupled vibration sources, 3

controlled with the optimal PID parameters, and self-regulation coefficient is 1.05 are analyzed. 4

（1）The nonlinear dynamic characteristics of generator vibration 5

The generator vibration nonlinear dynamic characteristics are shown in Fig.12. From Fig.12(a), the generator 6

vibration axis locus is multiple closed curves in 3D space, which shows quasi-periodic vibration. As can be seen from 7

Fig.12 (b), (c), (d), (e), (f), and (g), the generator vibration axis locus are multiple closed curves in x - y , 8

x - z , y - z 2D space, and the Poincare section show the attractors are a finite number, which can qualitatively analyze 9

that the hydro-turbine vibration of x - y , x - z , y - z 2D space are quasi-periodic. As can be seen from Fig.12 (h) and 10

(l), during load disturbance, the generation vibration of x and y direction go through three fluctuations: first, it 11

decreases rapidly and then increases rapidly; second, it decreases slightly and then increases; finally, it decreases 12

rapidly and then increases rapidly and stable in a certain area. As can be seen from Fig.12 (i), (j), (m), and (n) the 13

generator vibration of x and y direction velocity displacement phase plane are multiple closed curves and the 14

velocity displacement Poincare section show the attractors are a finite number, which can qualitatively analyze that the 15

generator vibration of x and y direction are quasi-periodic. As can be seen from Fig.12(k) and (o), the generator 16

vibration of x and y directions vibration is accompanied by three main vibration frequency components, namely, 17

29

0.08 octaves, 0.18 octaves, and 0.24octaves. As can be seen from Fig.12(p), during load disturbance, the generator 1

vibration of z direction changes with the corresponding change of load and stable in a certain area, which can be 2

analyzed qualitatively the vibration and the load of the HTGU stay the positive correlation. As can be seen from 3

Fig.12 (q) and (r) the generator vibration of z direction velocity displacement phase plan are multiple closed 4

curves and the velocity displacement Poincare section shows the attractors are a finite number, which can 5

qualitatively analyze that the generator vibration of z direction is quasi-periodic. As can be seen from Fig.12(s), 6

the generator vibration of z direction is accompanied by 0.1 octaves. To sum up the above, it can be seen that the 7

vibration of the generator of x and y directions has almost the same vibration characteristics. 8

(a) Three-dimensional axis locus of generator vibration

(b) The axis locus of x - y direction (c)The Poincare section of x - y direction

30

(d) The axis locus of x - z direction (e) The Poincare section of x - z direction

(f) Thel axis locus of y - z direction (g) The Poincare section of y - z direction

(h) The time-domain diagram of the x direction (i) The velocity displacement phase plan of x direction

31

(j) The velocity displacement Poincare section of x direction

(k) The frequency spectrum of x direction

(l) The time-domain diagram of y direction (m) The velocity displacement phase plan of y

direction

(n) The velocity displacement Poincare section of y

direction (o) The frequency spectrum of y direction

32

(p)The time-domain diagram of z direction (q) The velocity displacement phase plan of z

direction

(r) The velocity displacement Poincare section of z

direction (s) The frequency spectrum of the z direction

Fig. 12 The nonlinear dynamic characteristics of generator vibration

（2）The nonlinear dynamic characteristics of hydro-turbine vibration 1

The hydro-turbine vibration nonlinear dynamic characteristics are shown in Fig.13. From Fig.13(a), the hydro-turbine 2

vibration axis locus is multiple closed curves in 3D space, which shows quasi-periodic vibration. As can be seen from 3

Fig.13 (b), (c), (d), (e), (f), and (g), the hydro-turbine vibration axis locus are multiple closed curves in x - y , 4

x - z , y - z 2D space, and the Poincare section show the attractors are a finite number, which can qualitatively analyze 5

that the hydro-turbine vibration of x - y , x - z , y - z 2D space are quasi-periodic. As can be seen from Fig.13 (h) and 6

(l), during load disturbance, the hydro-turbine vibration of x and y direction go through small fluctuations and 7

steadily in a certain area. As can be seen from Fig.13 (i), (j), (m), and (n), the hydro-turbine vibration of x and 8

33

y direction velocity displacement phase plane are multiple closed curves and the velocity displacement Poincare 1

section show the attractors are a finite number, which can qualitatively analyze that the hydro-turbine vibration of x and 2

y direction are quasi-periodic. As can be seen from Fig.13 (k) and (o), the hydro-turbine vibration of x and 3

y directions vibration is accompanied by three main vibration frequency components, namely, 0.1 octaves, 4

0.34octaves, 0.56 octaves, 0.78 octaves, and 0.96 octaves. As can be seen from Fig.13 (p), during load disturbance 5

the hydro-turbine vibration of z direction changes with the corresponding change of load and stable in a certain area, 6

which can be analyzed qualitatively the vibration and the load of the HTGU stay the positive correlation. As can 7

be seen from Fig.13 (q) and (r), the hydro-turbine vibration of z direction velocity displacement phase plan are 8

multiple closed curves, and the velocity displacement Poincare section shows the attractors are a finite number, 9

which can qualitatively analyze that the hydro-turbine vibration of z direction is quasi-periodic. As can be seen 10

from Fig.13 (s), the hydro-turbine vibration of z direction is accompanied by 0.04 octaves. 11

(a) Three-dimensional axis locus of hydro-turbine vibration

34

(b) The axis locus of x - y direction (c)The Poincare section of x - y direction

(d) The axis locus of x - z direction (e)The Poincare section of x - z direction

(f) The axis locus of y - z direction (g)The Poincare section of y - z direction

35

(h) The time-domain diagram of x direction (i) The velocity displacement phase plan of x

direction

(j) The velocity displacement Poincare section of x direction

(k) The frequency spectrum of x direction

(l) The time-domain diagram of y direction (m) The velocity displacement phase plan of y

direction

36

(n) The velocity displacement Poincare section of y direction

(o) The frequency spectrum of y direction

(p) The time-domain diagram of z direction (q) The velocity displacement phase plan of z

direction

(r) The velocity displacement Poincare section of

z direction (s) The frequency spectrum of z direction

Fig.13 The nonlinear dynamic characteristics of hydro-turbine vibration

（3）The nonlinear dynamic characteristics of torsion vibration 1

37

According to Fig. 14(a), it can be seen that the torsional vibration angle quickly stabilizes at 0.013 rad. From Fig.14(b) 1

and (c), the Poincare section is approximately linear, which can qualitatively analyze that the torsional vibration is 2

stable. It can be seen from Fig.14(d) that the spindle has smaller torsional vibration spectrum components. 3

4

(a) Torsion vibration time-domain diagram (b) Torsion vibration velocity displacement phase plan

(c) Torsion vibration velocity displacement Poincare

section (d) Torsion vibration frequency spectrum

Fig.14 The nonlinear dynamic characteristics of torsion vibration

6. Conclusions 5

In this study, first, we establish a nonlinear mathematical model that considers the HTGS and the shafting system. 6

Meanwhile，the stability and sensitivity of the rotation speed deviation and shafting system vibrations to the PID 7

parameters were revealed by the bifurcation diagram. second, an innovational optimized strategy that considers the 8

ITSE of the HTGS and shafting system vibrations was proposed, and then the NSGA-II and MOEAD were introduced 9

to obtain the optimal PID control parameter for the multi-objective function. Finally, the nonlinear dynamic 10

38

characteristics of the HTGS and the shafting system vibrations which are controlled by the optimized PID parameter 1

were analyzed. The major conclusions are summarized as follows: 2

(1) The simulation results show that the nonlinear mathematical model coupling the HTGS and the shafting system 3

is effective. It can reveal the interaction characteristics between the shafting system and the HTGS and reveal the 4

stability and sensitivity of some key parameters to the HTGS and shafting system vibrations, and these results will 5

provide some bases for the design and stable operation of the HTGU. 6

(2) The multi-objective PID parameter optimization strategy shows good performance, it makes the adjusting-time 7

shorten 12 seconds of the HTGS and the shafting system vibrations. The self-regulating coefficient can reduce the 8

overshoot of the HTGS effectively, and suppressed the fluctuation amplitude. 9

(3) The rotation speed deviation and shafting system vibrations have different stability and sensitivity to the PID 10

parameters. The relative stable region of pk ， i

k ，and dk will decrease when considering the shafting system vibrations, 11

the specific region are 2.1,3.85psk ， 0.26,0.59isk ，and 0.1,5.4dsk , respectively. 12

(4) The generator and hydro-turbine vibrations in the 3D space that excited by the mass eccentricity, the UMP, the 13

arcuate whirl of the rotor, and the seal excitation coupled vibration sources are quasi-periodic composed of several 14

frequency components, whereas the torsion vibration is stable with a tiny frequency component, 15

acknowledgment 16

This work was supported by the National Key R&D Program of China (GrantNo.2016YFC0402205), National 17

Natural Science Foundation of China(U185202), National Natural Science Foundation of China(52039004) 18

Declaration of interests 19

The authors declare that they have no known competing financial interests or personal relationships that could have 20

appeared to influence the work reported in this paper. 21

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Figures

Figure 1

Structure of a hydropower station

Figure 2

Shafting structure diagram

Figure 3

The rotation speed deviation and SCV dynamic characteristics to Kp

Figure 4

The rotation speed deviation and SCV dynamic characteristics of Kl

Figure 5

The rotation speed deviation and SCV dynamic characteristics to Kd

Figure 6

Optimization process diagram

Figure 7

Pareto optimal solution obtained by different algorithms

Figure 8

The nonlinear dynamic characteristics of the HTGS

Figure 9

Robustness of optimized PID controller for to the load-self regulation coe�cient

Figure 10

Robustness of optimized PID controller for to the load-self regulation coe�cient

Figure 11

Robustness of optimized PID controller for torsion angle to the load-self regulation coe�cient

Figure 12

The nonlinear dynamic characteristics of generator vibration

Figure 13

The nonlinear dynamic characteristics of hydro-turbine vibration

Figure 14

The nonlinear dynamic characteristics of torsion vibration