2017 2nd International Conference on Energy, Power and Electrical Engineering (EPEE 2017) ISBN: 978-1-60595-514-8

Linear and Nonlinear Model of Hydro Generator System with Surge and its Simulation

Fan-nie KONG

School of Electrical Engineering, Guangxi University, Nanning, Guangxi Zhuang Autonomous Region 530004, China

Keywords: Hydro power system; Surge; Linear model; Nonlinear model; Unelastic water hammer effect.

Abstract. According to different hydraulic structure of practical hydraulic power system, combined

with hydro dynamic principle, analyzing and deducing unelastic water hammer effect linear model

and nonlinear model for system of tunnel-surge-penstock-generator unit (hydro generator system

with surge) by divided the hydro power system with surge into two subsystem:

reservoir-tunnel-surge and surge-penstock-generator unit in this paper. The simulation results of

models demonstrate that the established models can excellently reflect the inherent oscillation

performance for hydro power system with surge and it is not appropriate for substituting the

classical ideal hydro turbine model for hydro power system model with surge during the course of

investigation.

Introduction

According to the comprehensive research program recommended by the IEEE working group for

hydraulic power system, because constraints of specific geographical location information and

environment conditions, it shows difference of hydraulic structure for hydro power system. its

typical hydraulic structures are: reservoir-tunnel- turbine generator unit; reservoir- tunnel - surge

–penstock- turbine generator unit(hydro power system with surge). Therefore, the model of hydro

power system is mainly composed of hydraulic subsystem model and hydro generator subsystem

model [1]

.

From the point of view of the control theory, hydraulic power system model can be divided into

linear model and nonlinear model[2][3]

,in point of hydrodynamics, because the water is compressible

inherently, the hydraulics scholars will also divide the water transient process into unelastic water

hammer effect and elastic water hammer effect[3][4]

. The typical representative model mainly

include classical ideal hydro turbine model and comprehensive nonlinear hydro turbine model[4]-[6]

in use. Without consideration the differences of hydraulic structure and inherent water hammer

effect, so two kinds of model have limitations in application [1]

.

Based on the structure characteristics of actual hydraulic power system , synthesized the

hydraulic and electric discipline investigation characteristics, the linear model and unlinear model

of hydro power system hierarchy as reservoir- tunnel - surge –penstock- turbine generator unit are

deduced in conditions of considering the unelastic water hammer effect and hydraulic structure

characteristic, the investigation result can provide theoretical basis for regulation of hydro power

system with surge. ,

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Hydro Power System Model with Surge

Figure 1. Structure diagram of hydro power system with tunnel surge and penstock.

For hydro power system with surge, generation water first into tunnel, reach the surge, after

adjusted the pressure in surge, then spray turbine guide vane to drives turbine generator to rotate

and realize the water potential energy to mechanical energy , then to electric energy conversion. The

basic hydraulic structure for hydro power system with surge is shown in Fig.1.

A.Linear model According to hydraulic dynamics, the basic dynamic equation of a hydro power system with

surge can be written as[2][8]

:

flow equation: HGQ ⋅= (1)

Hydro turbine output mechanical power equation: HQPm ⋅= (2)

In equation (1) and (2), Q is water flow per unit,G is opening of the water gate per unit, H is

head pressure per unit.

Hydraulic dynamic equation: xHgtQ ∂

∂−=

∂

∂

(3)

In equation (3), x is distance between any two points for water movement direction; t is time

at which water flow between two points.

Fluid continuity equation: tHxQ ∂

∂−=

∂

∂α

(4)

combined equations (3) and (4) and taking into account fluid friction, the general solution of

the system of partial differential equations can be obtained by applying Laplasse transform theory: )sinh()1()cosh( 221 sTHZsTQQ ppp ⋅⋅⋅+⋅= (5) 22212 )tanh()(sec QQksTQZsThHH fppp ⋅−⋅⋅⋅−⋅⋅=

(6)

equation(5) and (6), 1Q , 2Q is flow for upstream starting point”1” and downstream end point "2"

respectively; 1H , 2H is water pressure corresponding to the point “1” and point “2” respectively;

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fk is pressure loss due to the friction ;s is the operator of Laplace transform.

To facilitate the analysis, hydro power system with surge was divided into two subsystem:

reservoir-tunnel-surge and surge-penstock-hydro turbine generator unit, combined with (1) - (6),

transfer function between flow deviation and water head for hydro turbine generator can be deduced

as[4][9]

: 00)( HH QQsF tttp−

−=

=)tanh()( )tanh()(1 sTZsG sTZsG eppp epp

++Φ

+

−

(7)

In equation(7), )(sG is transfer function between the water head deviation and flow deviation for

reservoir-tunnel-surge subsystem.it is described as: 00)( QQ HHsG p s−

−=

= secccs eccc CssTZsC sTZ⋅⋅+Φ+

+Φ )tanh(1 )tanh( (8)

In consideration of the unelastic water hammer effect, let 0=n ,the hyperbolic tangent function is sTsT epep ≈)tanh( , sTsT ecec ≈)tanh( , then the equation (7) and (8) can be written as: 00)( HH QQsF tttp−

−=

=sTZsG sTZsG eppp epp

++Φ

+

− )( )(1 (9) 00)( QQ HHsG p s

−

−=

= swccs wcc CTssC sT⋅⋅+Φ+

+Φ 21 (10)

In equation (9), (10), wpT is water staring time of penstock; wcT is water staring time of tunnel; epT

is elastic time of penstock; ecT is elastic time of tunnel; sC is storage constant of surge; pf is water

head losses coefficient of penstock; tf is water head losses coefficient of tunnel; sf is water head

losses coefficient of surge; pΦ is friction coefficient of penstock; cΦ is friction coefficient of

tunnel; pZ is hydraulic surge impedance of penstock; cZ is hydraulic surge impedance of tunnel.

Replace equation (10)for (9), the transfer function of unelastic water hammer effect between

hydro turbine output power and changes in guide vane position is[4][10]

: sTZsGsGsTZ sGsTZsGsTZGP eppeppp eppepppm⋅+⋅+⋅+Φ+

−⋅+⋅−Φ−

=∆

∆ )()(5.05.05.01 )()(1 (11)

Ignore pΦ and cΦ , equation (8) and (11) can be written as: swcwc CTs sTsG⋅⋅+

= 21)( (12)

let 79.5=wcT 、 22.138=sC [11][12], equation(10)can be written as: 252.9381 79.5)( sssG

+=

(13)

let 187.4=pz , 42.0=epT [11][12], equation(11)can be written as:

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GPsG mt∆

∆=)( = 32 32 9.34548026.153932187.4 1.69106055.313932187.4 sss sss

+++

−−+ = 17742.31.93915.825 15485.71.9394.1650 23 23

+++

+−+− sss sss (14)

B. Nonlinear model For Fig.1, according to basic flow dynamic characteristics and hydraulics principle. the hydraulic

dynamic equation for tunnel, surge and penstock can be established respectively. (1)Hydraulic dynamic equation of tunnel: 220 Qlr HHHH −−=

(15) 222 cpl QfH = (16) dtdQTH cwcQ =2

(17)

(2) Hydraulic dynamic equation of surge:

∫ −=21 ssssr QfdtQCH

(18)

(3) hydraulic dynamic equation of penstock: Qlrt HHHH −−= (19) 2tpl QfH =

(20) teppQ QsTZH )tanh(= (21)

The equations of opening and power output of hydro turbine are the same as equation (1) and (2). For surge node ,According to flow continuity principle, the flow balance equation of surge is: sct QQQ −=

(22)

In equation (15) - (22), 2lH is friction water head loss of tunnel; friction coefficient 2QH

is water hammer head loss of tunnel; 2pf is water head loss coefficient of tunnel; cQ is flow

of tunnel; rH is water head of surge; sQ is flow of surge.

Due to the dynamic hydraulic behavior of tunnel adjusted by reservoir and surge, so the dynamic

process of tunnel impacted on hydro turbine generator unit can be ignored compared to penstock,

and supposed the water hammer effect in tunnel is unelastic. For penstock, the water hammer effect

chosen as elastic hammer or unlastic hammer depends on the penstock length. In general, if

penstock length is more than 600m, then the water hammer effect may be taken for elastic water

hammer effect, otherwise for unelastic water hammer effect[3]

.

Supposed as unelastic water hammer effect, equation(21) can be written as: Qwpt HTdtdQ 1=

(23)

Then the model of hydroelectric power generation system can be obtained by per unit:

100

=+++

−−=

−−

=

−−=

−−=

ugTTgTT T hhhq qfC qqh hhhTq H Dppspsp wc lrc sss tcr ltrwpt motionloadm

1)( 2 )(120

�

�

�

�

�ω

ω

(24) Model Simulation

A.Linear model simulation For model of hydro power generation system with surge (11), supposed 79.5=wcT ,22.138=sC , 187.4=pz , 42.0=epT , the linear model of hydro power generation system with

surge shown as (14) ; for classical ideal hydro turbine model sTsTsG wpwpp 5.01 1)(+

−= ,supposed 79.5=wpT , the models step response and Nyquist curve are shown in Fig. 2 and Fig.3 respectively.

From Fig. 2, the step response curves are stable on 1,so the models are all stable. Compared to

the step response curve of classical ideal hydro turbine model(no surge), the step response curve of

the hydro power system model with surge is oscillation, the settle time is long. So it is not

appropriate that simplifying all the hydro power system model as classical ideal hydro turbine

model without consideration of actual hydraulic structure of hydro power station during the course

of research.

From Fig.3,the Nyquist curve of hydro power system model with surge surround the (-1,jo)

clockwisely. Because the model has one zero in the right half plane, so the system is stable. This

conclusion agrees with the time domain response.

Figure 2. Step response curve of hydro turbine Figure 3. Nyquist curve of hydro turbine model.

model with surge. with surge.

B.nonlinear model simulation

In order to analyze the stability of unelastic water hammer effect nonlinear model of hydro

generation system with surge (24), without applying external control signalu , supposed 1=D ,10=J , sTwp 82.0= , sTwc 15.9= , sCs 7.170= , ..17.1 upAt = , ..13.0 upqtnl = , sTs 5.0= , the dynamic

curve of state variable [ ]Tcrt gqhqx ω= shown as Fig. 4-Fig.8. 101

As shown in Fig.4-Fig.8, the unelastic water hammer effect nonlinear model of hydro generation

system with surge (23) is stable.

Figure 4. State variable ω curve. Figure 5. State variable tq curve.

Figure 6. State variable rh curve. Figure 7. State variable cq curve.

Figure 8. State variable g curve.

Conclusion

Based on the model analysis and model simulation curve, the following conclusions can be

obtained:

(1) For hydro power system with surge, because of the difference between hydraulic system

structure and ideal hydraulic structures, supposed unelastic water hammer effect, there is essential

differences between the dynamic characteristics of the linear model and the ideal hydro turbine

model. The step response of ideal hydro turbine model don’t oscillate but that of hydro power

system model with surge oscillates severely. Therefore, it is not appropriate that simplifying all the

hydro power generation system model as ideal hydro turbine model without consideration of actual

hydraulic structure of hydro power station.

(2) For hydro power system with surge, the five order nonlinear model is stable, compared to the

classical ideal hydro turbine model, the deduced model can offer dynamic process of key operation

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parameters of the hydro power generation system, very conducive to operation monitoring and

safety operation of the hydro power generation system, which provides a new research idea for

control design and operation monitoring.

Acknowledges

This work was supported by National Natural Science Foundation of China under granted

NO.51167003 and Guangxi Natural Science Foundation under granted 2014GXNSFAA118320.

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