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Draft

Characterization of Surge Superposition following 2-Stage

Load Rejection in Hydroelectric Power Plant

Journal: Canadian Journal of Civil Engineering

Manuscript ID cjce-2015-0382.R1

Manuscript Type: Article

Date Submitted by the Author: 16-May-2016

Complete List of Authors: Chen, Sheng; Hohai University, Water Conservancy and Hydropower Engineering Zhang, jian; Hohai University Wang, Xichen; Hohai University, College of Water Conservancy and Hydropower Engineering

Keyword: Hydroelectric power plant, simultaneous and complete load rejection, 2-

stage load rejection, surge tank, surge superposition

https://mc06.manuscriptcentral.com/cjce-pubs

Canadian Journal of Civil Engineering

Draft

Characterization of Surge Superposition following 2-Stage Load

Rejection in Hydroelectric Power Plant

Sheng Chen; Jian Zhang; and Xichen Wang

Sheng Chen. College of Water Conservancy and Hydropower Engineering, Hohai University,

Nanjing 210098, P.R. China.

Jian Zhang. College of Water Conservancy and Hydropower Engineering, Hohai University,


Xichen Wang. College of Water Conservancy and Hydropower Engineering, Hohai University,


Corresponding author: Sheng Chen, 1st Xikang Road, Gulou District, Nanjing 210098, P.R.

China, +86 13914482300, e-mail: [email protected].

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Abstract: When calculating the maximum upsurge in surge tank due to load rejection in a

hydroelectric power plant, it has been natural and customary to believe that the maximum surge

amplitude occurs in simultaneous load rejection of all units at 100% load. As 2-stage load

rejection (2-stage LR), involving a step-wise reduction in load, is not considered since it is

assumed to produce less severe surge conditions. This study formulates the surge superposition

associated with 2-stage LR and shows, surprisingly but significantly, that such 2-stage LR

sometimes produces more severe surge conditions than simultaneous and complete load rejection

(SCLR). The results indicate that this unexpected phenomenon is ascribable to the resistant effect

of throttled surge tank, whose increase will lead to a greater difference in the maximum upsurges

between 2-stage LR and SCLR conditions. Different time intervals during 2-stage LR correspond

to different maximum upsurges. The analytical formula predicting the worst interval time is

derived exactly and verified with two numerical cases.

Key words: Hydroelectric power plant, simultaneous and complete load rejection, 2-stage load

rejection, surge tank, surge superposition, worst interval time

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1. Introduction

In order to reduce investment in hydroelectric power plant, combined or grouped supply is

widely used in practical terms, that is, all the turbines in one water conservancy system share a

common penstock. Therefore, bifurcations or trifurcations are provided for hydraulic connections

among the turbines. However, when one or several of the turbines reject full load by emergency

closure, the induced water hammer will transmit in the pipe system and finally affect other

properly working turbines, which is known as hydraulic disturbance. If the influence of the

disturbance exceeds the limit of relay protection, the working turbines will reject load immediately

as the consequence of the first load rejecting turbine(s). The whole process is called 2-stage LR

(shown in Fig. 1), first proposed by T. Yokoyama and K. Shimmei (1984) in the research into

dynamic behaviors of the reversible pump-turbine in pumped storage plants. Currently, 2-stage LR

is extensively investigated in pumped storage plants with concerns on the minimum draft tube

pressure (Zhang Jian et al. 2006, 2007, 2008a, 2008b; Fang and Koutnik 2012) due to the unique

S-shaped characteristics of reversible pump turbine.

A surge tank is usually provided in a diversion-type hydroelectric power plant to reduce

water hammer pressure and improve regulation characteristics of hydraulic turbines. The surge

tank should be of sufficient height to prevent overflow for all operation conditions (Moghaddam

2004), thus, the highest possible water level which can be anticipated during its operation is an

important issue for safety design of surge tank. Empirically, simultaneous and complete load

rejection (SCLR) is considered as the critical load case scenario for the highest upsurges without

considering some extreme cases, such as load acceptance followed by load rejection and load

rejection followed by load acceptance (Cheng et al. 2004; Yu et al. 2011). As mentioned

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previously, 2-stage LR is only used to deal with the extreme value of water hammer. To the best

of the author’s knowledge, none studies associated with 2-stage LR on surge superposition have

been conducted yet. This is due to the fact that 2-stage LR means increasing the total closure time

of the system and reducing the flow gradient in headrace tunnel, which finally results in relatively

uniform variation of water level in tank surge. Therefore, engineers and researchers are hardly

aware that 2-stage LR could be the critical load case scenario for the highest upsurges. In fact,

surge superposition of 2-stage LR sometimes could produce more severe surge amplitude. The

possibility of this surprising phenomenon could be partly verified by relevant studies for partial

load rejection (PLR) (Морозов 1947; Jaeger 1977; Wang 1983), conducted from the 1940s to the

1980s. It was concluded that the surge generated from a partial valve closure was potentially

greater than that from a fully open position. However, the generation of this surprising result

should be based on the premise of 0

0

1whx

S η= > , where 0

0w

k

hη = ( hw0 and k0 are head loss of

headrace tunnel and restricted orifice corresponding to initial discharge Q0, respectively)

and2

0

02 w

LfvS

gFh= ( L and f are length and cross-sectional area of headrace tunnel, respectively; and

F is cross-sectional area of surge chamber). The physical mechanism for this result of PLR can be

simply illustrated by the following Eq. [1].

[1] d

dw

L vZ h k

g t= − −

where Z is the height of the tank water level below the reservoir surface, measured positively

downwards; v is the velocity through headrace tunnel; hw and k are head loss through headrace

tunnel and restrict orifice, respectively; g is gravitational acceleration; and t is time. In fact, Eq. [1]

is the momentum equation which describes the relationship among elevation (Z), inertia head

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(d

d

L v

g t) and head loss (hw+k). During the load rejection, as v0 increases, inertia head increases to

yield higher surge amplitude Z. However, the head loss, which can weaken the mass oscillation,

increases as well. If 0

1x

η< ,

d

d

L v

g t is the dominant factor, and Z will increase with increasing

d

d

L v

g t and (hw+k). Whereas 0

1x

η> , (hw+k) becomes the key factor, and Z will decrease with

increasing d

d

L v

g t and (hw+k). In this case, PLR produces more severe maximum upsurge than

SCLR. So far, the assumption that increase of kinetic energy dominates surge amplitude is lack of

theoretical support. However, in engineering practice, the cross-sectional area of restricted orifice

is no less than 20% of the area of the headrace tunnel, so as to reduce the water hammer

transmitting through the surge tank (Mosonyi and Seth 1975). Hence, the premise of 0

1x

η> can

be rarely satisfied.

As 2-stage LR involving a step-wise reduction in load, PLR can be regarded as the second

stage of 2-stage LR, thus a changeable boundary condition for PLR can be regarded as the first

stage. After the first rejection of load, water level of the surge tank rises in the first 1/4 period of

surge oscillation. If the residual load is rejected at any instant during the 1/4 period, the

corresponding water level in 2-stage LR case will be definitely higher than that in PLR only. For a

hydraulic system with 0

1x

η< , this higher water level for PLR may overcome the limit of the

above premise and induce more severe surge conditions. Therefore, it is reasonable to infer that 2-

stage LR may produce more severe surge amplitude than that associated with SCLR.

The present study investigates the surge superposition following 2-stage LR and derives an

analytical formula to predict the worst interval time which may produce more severe surge

amplitude. Moreover, the influence of throttle on tank surge during 2-stage LR is discussed and

the influence law is given with case studies.

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2. Fundamental Equations and Analytical Solution

Rigid water column theory (Wylie and Streeter 1993) is introduced in the following analysis.

To simplify the derivation, two assumptions are made: (1) the friction coefficient during the

transient state is constant and independent of flow or Reynolds number (Bergant et al. 2001); and

(2) the residual flow after the first stage of load rejection stays constant.

The main difference between 2-Stage LR and SCLR is the instant rejecting load, and SCLR

is just the special case of 2-Stage LR. For a combined supply water conservancy system, assuming

2-Stage LR occurs and the full load is rejected twice. The demand discharge decreases from initial

Q0 to Q1, and finally to 0.

With reference to Fig. 2, for the first stage of load rejection, discharge is decreased from

initial Q0 to Q1, and the fundamental equations for the hydraulic system (Chaudhry 1979; Wylie

and Streeter 1993) are given by

[2] ( )1 1

d

d

L QZ Q Q Q Q Q Q

gf tα β= − − − +

[3] 1

d

d

ZQ Q F

t= +

where Q1 is the turbine demand; Q is the discharge through the tunnel at any instant, measured

positively in the water supply direction; F is the cross-sectional area of the surge tank; α is the

head loss coefficient of the tunnel including lengthwise and local components; and β is the head

loss coefficient across the restricted orifice.

The initial conditions of the differential Eqs. [2] and [3] are

[4] 2

0 00 0,

t tQ Q Z Qα

= == =

For the second stage of load rejection at Tc, discharge decreases from Q1 to 0, the

corresponding equations are as follows,

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[5] d

d

L QZ Q Q Q Q

gf tα β= + +

[6] d

0d

ZQ F

t= +

The initial conditions of the differential Eqs. [5] and [6] herein are

[7] ,c c

D Dt t t tQ Q Z Z

= == =

with Tc the specific instant when the residual load of the hydraulic system is rejected.

Taking Eqs. [2] to [7] simultaneously, by a series of transformations and simplifications

(referred to Appendix for derivation details), yields

[8] 1

2 2

1 1( ) ( )

Q QQ

Q Z Z Q Q Q Q Qα β α β−

=+ − − + − −

Eq. [8] is exactly what 2-stage LR should satisfy so as to produce more severe surge

amplitude. According to the discussion below, it shows better generality compared to the premise

for PLR to get higher maximum upsurge.

3. Discussion

From the previous work for PLR and Eq. [8], resistant coefficient of the orifice of surge tank

( β ) is one of the most important factor for surge superposition. According to the value of β, surge

superposition following 2-stage LR can be broadly lumped into two categories: β = 0 (simple

surge tank) and β ≠ 0 (throttled surge tank).

For the first case, substituting β = 0 into Eq. [8], yields

[9] 1 0Q =

As mentioned previously, Q1 represents the residual discharge of the turbine after the first

stage of load rejection. Q1 = 0 indicates that this manoeuvre actually is SCLR. Consequently, it is

believed that the maximum upsurge induced by SCLR is definitely higher than that caused by 2-

stage LR for a simple surge tank. This result is consistent with the achievement for PLR.

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As for the second one β ≠ 0, considering the practical water oscillation in surge tank, the

absolute value sign on the discharge term in Eq. [8] can be ignored, then

[10] 2

1

Z QQ Q

Q

αβ−

= +

Assume that the first stage of load rejection takes place at t = 0, and the total turbine demand

decreases instantaneously from Q0 to Q1. Meanwhile, the discharge through tunnel still remains Q0

as before at this instant due to large water inertia, and the water level stays the same as well. That

is to say, at t = 0,

[11] 2

1 0

0t

Z QQ Q Q

Q

αβ

=

−< + =

After the first stage of load rejection, the upsurge of the tank reaches the maximum value at t

= T / 4, with T the period of surge oscillation in a tunnel-tank system. Meanwhile, the discharge

through the tunnel is equal to what turbine demands. Notice that tank water level is measured

positively downwards, and the second term on the right side of Eq. [10] is negative, hence

[12] 22

1

1 1

14t T

Z QZ QQ Q Q

Q Q

ααβ β

=

−−> + = +

According to the continuity of the equation, along with Eqs. [11] and [12], there must be an

instant of time tc ∈ (0 , T / 4) satisfying Eq. [8] or Eq. [10]. If the residual load is rejected at the

instant of time tc, the upsurge will be maximum. Now that tc ≠ 0, it can be concluded that 2-stage

LR can produce higher upsurge for the throttled surge tank.

4. Case Study

To further validate the conclusions above, two examples listed below are taken. Case 1 is

performed by employing the fourth-order Runge-Kutta method, while Case 2 is performed by

method of characteristics.

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4.1. Case 1

The hydroelectric power plant employs the layout of one penstock with two turbine units and

a throttled surge tank. The main technical parameters are listed as follows: installed capacity is 2 ×

15 MW, headrace tunnel is 2,500 m in length with a diameter of 3.7 m and Manning’s roughness

coefficient is 0.014; rated head is 80.0 m, rated discharge is 2 × 21.7 m3s

-1, and the diameter of the

surge tank is 7.5 m. The period of surge oscillation T = 202.3 s.

The area ratios of orifice and tunnel RI are selected from 20% to 50%, which are commonly

used in engineering practice. The calculated results are shown in Table 1, Table 2, and Fig. 3 to

Fig. 5.

Case 1 studies 2-stage LR taking place in the layout of one penstock equipped with two units

and a surge tank. As referred to in Wang’s (1983) research, Table 1 calculates the system

parameters to determine whether the maximum upsurge occurs in PLR or not. Apparently, 0

1x

η<

holds for all the cases listed in Table 1, and it is impossible to produce more severe upsurge

following PLR.

However, when 2-stage LR is taken into consideration, the results are different. As shown in

Table 2, the upsurges produced by 2-stage LR are more severe for various orifice sizes. With the

increase of orifice area, the difference in the maximum upsurge decreases (shown in Fig. 3), so is

the worst interval time. And for the special case with RI ≥ 100% (simple surge tank), the

maximum upsurge is induced at tc = 0 s, which is in complete agreement with the discussion. All

of these indicate that the restrict orifice is the main factor making 2-stage LR more dangerous than

SCLR in upsurge.

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Fig. 4 shows the theoretical results of the worst interval time for 2-stage LR adopting Eq.

[10], which are presented by the intersection of straight line Q1 = 21.7 m3 and the other curves

calculated by Q + (Z – αQ2) / βQ. The analytical solutions agree well with the trial calculated

results listed in Table 2, which well verifie the correctness of Eq. [10]. More importantly, it

provides theoretical support to surge superposition under 2-stage LR.

Fig. 5 presents the surge history for different interval times following 2-stage LR when RI =

25%. As shown in Fig. 5, the curve labelled 1 is water level history for the first stage rejecting

50% load, tc = 0 s denotes SCLR, and the others are for 2-stage LR. Apparently, different interval

times correspond to different initial water levels for the second stage of load rejection. Within T /

4 (50.6 s), the initial water level is higher than that for SCLR, which becomes an important factor

on surge superposition for the second stage of load rejection. Before tc reaches the worst interval

time (27.5 s), the effect of higher initial water level is prominent; however, when tc > 27.5 s, this

effect is weakened. Eventually, surge superposition following 2-Stage LR leads to more severe

upsurge.

4.2. Case 2

Hydroelectric power plant employs the layout of one penstock with two turbine units and a

throttled surge tank. The main technical parameters are listed as follows: installed capacity is 2 ×

175 MW, headrace tunnel is 6091.10 m in length with a diameter of 8.5 m and Manning’s

roughness coefficient is 0.014; penstocks is 380.0 m in length with varied diameters of 8.5 m (35.0

m)→7.5 m (190.0 m)→5.0 m (130.0 m)→4.7 m (25.0 m) and roughness coefficient of 0.012;

rated head is 178.0 m, rated discharge is 2 × 110.2 m3s

-1, rated speed is 214.3 r/min, and the

diameter of the throttle and surge tank are 4.4 m and 25.0 m, respectively. The period of surge

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oscillation 1/4 T ≈ 120 s. The wicket gates are considered to close linearly in 12 s. The upstream

reservoir level is 1823.64 m, and the tail water level is 1610.35 m. In this situation, the initial

discharge of headrace tunnel is 2 × 93.10 m3s

-1, with initial head loss hw0 = 5.10 m.

In Case 1, many factors such as the elasticity of the pipe walls, the compressibility of water,

the influence of the penstock, the characteristics of a real turbine, as well as the closure law of

wicket gate are not taken into consideration. To explore the applicability of the formula in practice,

Eq. [10] is applied to a much more complex system based on an actual hydroelectric power plant

in China. The method of characteristics (MOC), which has been verified as an accurate numerical

approach (Karney and Ghidaoui 1997; Axworthy and Chabot 2004; Selek et al. 2004; Yu et al.

2011) and widely used, is applied to simulate hydraulic transients in Case 2. The basic boundary

conditions are referred to in the book Fluid Transients in Systems (Wylie and Streeter 1993).

As handled in Case 1, the maximum upsurge occurring in PLR should be excluded

beforehand. The detailed calculation process is shown in Table 3.

As shown in Fig. 6, the theoretical worst instant predicted by Eq. [10] is 49.7 s, which is

apparently in 1/4 T. Table 4 gives the calculation results of maximum upsurges with different

interval times adopting MOC, and the accurate worst instant is 55.0 s. There is discrepancy

between the predicted and actual worst instants because the MOC considers other dynamics

effects mentioned previously. The consideration of elasticity of the pipe walls, compressibility of

water, and the influence of the penstock is the requirement of elastic water column theory. The

characteristics of a real turbine and the closure law of wicket gate are accounted for so as to obtain

more accurate boundary conditions. These efforts make the numerical model fit the realistic

system better and the worst instant is delayed. However, the error is only 5.3s, and the

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corresponding difference in maximum upsurge is less than 0.01m, both of which are within

acceptable ranges. Therefore, Eq. [10] is successfully verified to predict the worst instant for 2-

stage LR with sufficient accuracy in the numerical simulation of maximum upsurge.

Moreover, if 2-stage LR occurs in the hydroelectric power plant, the residual load rejected at

tc =55 s will induce the highest maximum upsurge, 16.25 m. While SCLR happens (tc = 0 s), the

maximum upsurge is 15.3 m. The difference between these two conditions reaches nearly 1.0 m,

which may result in overtopping and constitute a potential hazard.

Through Case 2, the reliability and accuracy of Eq. [10] are further validated. It shows

objectively that 2-stage LR produces more severe surge conditions than SCLR, theoretically

supported by Eq. [10].

5. Conclusions

The maximum amplitude induced by surge superposition is a critical issue for surge tank design.

In this study, theoretical derivation is conducted to characterize the surge superposition following

2-stage LR, and the effects of the restrict orifice and interval time on surge superposition are

simulated and investigated.

In conclusion, the restrict orifice plays an important role in surge superposition of 2-stage LR,

and the increase in orifice area is beneficial to reducing the difference in the maximum upsurges

between 2-stage LR and SCLR. For simple surge tank, the maximum upsurge following 2-stage

LR is less severe; while for throttle surge tank, when the interval time is within T / 4, the

maximum upsurge following 2-stage LR is more severe compared to that in SCLR. And the worst

interval time can be predicted by the proposed analytical formula. A series of laboratory

experiment will be conducted to verify the present study in future research.

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6. Acknowledgements

The present project is financially supported by the National Natural Science Foundation of

China (No.51379064), the Fundamental Research Funds for the Central Universities (Grant

No.2015B00414 and No.2016B10814) and the Priority Academic Program Development of

Jiangsu Higher Education Institutions. The authors are grateful to everyone who has supported us

in technical and financial terms.

7. Appendix

The objective of this analytical derivation is to seek the extreme value of the differential Eqs.

[5] and [6], which represents the maximum water level Zm of the surge tank. The solutions of the

differential equations are influenced by the initial values. In other words, different initial values

give rise to different 'mZ . And the maximum Zm exists among a series of different 'mZ . The initial

values of Eqs. [5] and [6] are determined by Eqs. [2] and [3]. Thus, Zm can be obtained by solving

Eqs. [2] to [7] simultaneously.

The integral form of the solution to differential Eqs. [5] and [6] can be written as

[13] [ ]( ), dD

Q

DQ

Z Z f Z x x x= + ∫

in which [ ] d d( ),

d d

Z Zf Z x x

x Q= = ; and ZD, QD are instantaneous water level of surge tank and

instantaneous flow of headrace tunnel at tc, respectively. ZD and QD are also the initial conditions

for the second stage of load rejection.

After transformation, Eq. [13] takes the form

[14] [ ]1( , , , ) ( ), d 0D

Q

D D DQ

f Z Q Z Q Z Z f Z x x x= − − =∫

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In Eq. [14], the necessary condition for Z to get the extreme value is d0

d

Z

Q= with the premise

of ZD and QD are given as constants. Applying the derivation rule of implicit function (Stewart

2004), yields

1

1

d

d

f

ZQ

f Q

Z

∂∂

− =∂∂

Therefore, 1 0f

Q

∂=

∂. Setting

[15] 12 ( , , , ) 0D D

ff Z Q Z Q

Q

∂= =∂

Taking Eqs. [14] and [15] simultaneously gives

[16]

1

2

( , , , ) 0

( , , , ) 0

D D m m

D D m m

f Z Q Z Q

f Z Q Z Q

= =

By eliminating Qm in the system of Eqs. [16], one can obtain

[17] [ ],m m D DZ Z Z Q=

where Zm is the extreme value of the tank surge; and Qm is the corresponding discharge of the

headrace tunnel. Eq. [17] shows that the extreme value of the tank surge depends on the initial

conditions of ZD and QD. Since different instants tc rejecting residual load give rise to different ZD

and QD, how to determine the worst instant becomes the second problem.

To find the extreme value Zm among 'mZ , setting d 0mZ = , and then the total differentiation

of Zm is

[18] d d d 0m mm D D

D D

Z ZZ Z Q

Z Q

∂ ∂= + =∂ ∂

If make Zm equal to the extreme value 'mZ , the following condition should be satisfied

[19] d

d

m

D D

m D

D

Z

Q Z

Z Q

Z

∂−∂

=∂∂

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in which d

d

D

D

Z

Q is the slope of the curve of tank water lever-tunnel flow relationship for the first

stage of load rejection. Apparently, Eq. [19] is the necessary condition which should be satisfied

when the second stage of load rejection occurs.

Employing implicit function theorem of equations system (Stewart 2004), the term

m m

D D

Z Z

Q Z

∂ ∂−∂ ∂

in Eq. [19] can be obtained from Eq. [16] as follows

[20]

1 11 2

2 2

1 2 1 1

2 2

( , )1

( , )

( , )1

( , )

D mm

D mD mD

m D m

D mD D m

f Q f QZ f f

f Q f QJ Q QQ

Z f f f Z f Q

J Z QZ f Z f Q

∂ ∂ ∂ ∂∂ ∂−−

∂ ∂ ∂ ∂∂∂= − = −

∂ ∂ ∂ ∂ ∂ ∂−∂∂ ∂ ∂ ∂ ∂

where J is Jocobian determinant, taking the form

1 1

1 2

2 2

( , )

( , )

m m

m m

m m

f f

Z Qf fJ

f fZ Q

Z Q

∂ ∂

∂ ∂∂= =

∂ ∂∂∂ ∂

After simplification, Eq. [20] turns into

[21] 1

1

D D

m D

D

Z

Q f Q

Z f Z

Z

∂−∂ ∂ ∂

= −∂ ∂ ∂∂

From Eq. [14],

[ ]1 1d d( ), , 1

d dD

D D

x Qx QD DQ Q

f fZ Zf Z x x

Q x Q Z== =

∂ ∂= = = = −

∂ ∂

By substitution into Eq. [21], giving

[22] d

dD

D

m Q Q

D

Z

Q Z

Z Q

Z=

∂−∂

=∂∂

with d

dDQ Q

Z

Q=

the slope of the curve of tank water lever-tunnel flow relationship for the second

stage of load rejection. Substitution the Eq. [22] into Eq. [19] yields

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[23] d d

d dD

D

D Q Q

Z Z

Q Q=

=

Eq. [23] should be strictly complied with in 2-stage LR so as to produce more severe surge

amplitude. The physical interpretation is that the curves of surge tank water level-tunnel discharge

for the first and the second stages of load rejection are mutually tangent at the worst interval time.

For the hydraulic system involved in Fig. 2, Eq. [23] can be rewritten as

[24] 1

2 2

1 1( ) ( )

Q QQ

Q Z Z Q Q Q Q Qα β α β−

=+ − − + − −

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Yu, X.D., Zhang, J., and Hazrati, A. 2011. Critical oscillation superposition in a surge tank with a

long headrace tunnel. Canadian Journal of Civil Engineering, 38(3): 331-337.

Zhang, J., Hu, J.Y., Hu, M., Fang, J., and Chen, N. 2006. Study on the reversible pump-turbine

closing law and field test. In Proceedings of ASME Joint U.S.-European Fluids Engineering

Summer Meeting, FEDSM 2006, Miami, FL, 17-20 July 2006. pp. 931-936.

Zhang, J., Lu, W.H., Hu, J.Y., and Fan, B.Q. 2007. Study on the hydraulic transients of pumped-

turbines load successive rejection in pumped storage plant. In Proceedings of the 5th Joint

ASME/JSME Fluids Engineering Conference, FEDSM 2007, San Diego, California, 30 July-2

August. pp. 57-61.

Zhang, J., Lu, W.H., Fan, B.Q., and Hu, J.Y. 2008a. The influence of layout of water conveyance

system on the hydraulic transients for pumped-turbines 2-stage load rejection in pumped storage

station. Journal of Hydroelectric Engineering, 27(5): 158-162.

Zhang, J., and Suo, L.S. 2008b. Study on tailrace surge chamber installation and hydraulic

transients at pumped-storage plant. Water Resource Power, 26(3): 83-87.

List of symbols

f cross-sectional area of tunnel

F cross-sectional area of surge tank

g gravitational acceleration

hw head loss of tunnel

hw0 head loss corresponding to Q0

J Jocobian determinant

k head loss across restrict orifice

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k0 head loss corresponding to Q0

L length of tunnel

Q discharge through tunnel

Q0 initial discharge through tunnel

Q1 turbine demand

QD instantaneous flow at tc

Qm flow corresponding to Zm

RI area ratio of orifice and tunnel

S parameter indicates tunnel-tank characteristics, 2

0

02 w

LfV

gFh

T period of surge oscillation;

t time;

tc worst instant of time;

v velocity through tunnel;

v0 velocity corresponding to Q0;

x0 dimensionless parameters, 0wh

S;

Z height of water level;

ZD instantaneous water level at tc;

Zm extreme value of tank surge;

α head loss coefficient of tunnel;

β head loss coefficient across restrict orifice; and

η dimensionless parameter, 0

0w

k

h

.

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Table 1. Calculation processes to determine maximum upsurge to be occurred in PLR or SCLR.

RI

(%)

hw0

(m)

k0

(m)

S

(m)

1

η x0

Whether

0

1x

η>

Condition Inducing

Maximum Upsurge

20

7.6

57.0

65.7

0.13

0.12

No SCLR

25 36.3 0.21 No SCLR

30 25.2 0.30 No SCLR

35 18.5 0.41 No SCLR

40 14.2 0.54 No SCLR

45 11.2 0.68 No SCLR

50 9.1 0.84 No SCLR

≥100 0 ∞ No SCLR

Note: RI ≥ 100% represents a simple surge tank.

Table 2. Comparison of maximum upsurges between SCLR and 2-stage LR with different orifice

diameters.

Area Ratio of

Orifice and

Tunnel

RI (%)

Maximum

Upsurge Under

SCLR

-Zm (m)

Maximum

Upsurge Under

2-Stage LR

-Zm (m)

Worst Interval Time

tc (s) Difference

(m)

Eq. [10] Iteration

20 11.8 13.8 42.0 42.0 2.0

25 14.9 16.2 33.8 33.8 1.3

30 17.2 18.1 27.5 27.5 0.9

35 19.0 19.6 22.4 22.3 0.6

40 20.4 20.8 18.0 18.0 0.4

45 21.5 21.7 14.6 14.5 0.2

50 22.3 22.5 11.8 11.8 0.2

≥100 26.8 26.8 9.7 9.6 0.0

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Table 3. Calculation process to determine maximum upsurge to be occurred in PLR or SCLR.

RI

(%)

hw0

(m)

k0

(m)

S

(m)

1

η x0

Whether

0

1x

η>

Condition Inducing

Maximum Upsurge

26.8 5.1 21.2 26.2 0.24 0.20 No SCLR

Table 4. Maximum upsurges with different interval times using MOC.

Interval Time

tc

(s)

Maximum Upsurge

-Zm

(m)

Interval Time

tc

(s)

Maximum Upsurge

-Zm

(m)

0 15.3 60 16.2

10 15.5 70 16.1

20 15.8 80 15.9

30 16.0 90 15.6

40 16.2 100 15.2

50 16.24 110 14.7

55 16.25 120 14.1

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Fig. 1. Schematic diagram of 2-stage load rejection.

Fig. 2. Schematic of a typical reservoir-pipeline-surge tank system.

Fig. 3. Maximum upsurge with different interval time tc for 2-Stage LR.

Fig. 4. Worst instant for 2-stage LR based on Eq. [10].

Fig. 5. Surge history for different interval times of 2-stage LR ( RI = 25% ).

Fig. 6. Worst instant for 2-stage LR predicted by Eq. [10].

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Draft

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Draft

Upstream

Reservoir

Downstream

Reservoir

Penstock

Headrace

Tunnel

L

Z0=αQ02

Z

Throttled Surge Tank

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Draft

5

11

17

23

29

35

0 10 20 30 40 50

Maximum Upsurge -Zm

/ m

Interval time tc

/ s

RI=0.15 RI=0.20 RI=0.25 RI=0.30

RI=0.35 RI=0.40 RI=0.45 RI=0.50

RI = 0.20

RI = 0.40

RI = 0.25

RI = 0.45

RI = 0.30

RI = 0.50

RI = 0.35

RI ≥ 1.00

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Draft

0

9

18

27

36

45

0 10 20 30 40 50

Discharge

/ m

3/s

Time / s

RI=0.15 RI=0.20

RI=0.25 RI=0.30

RI=0.35 RI=0.40

RI=0.45 RI=0.50

RI = 0.20

RI = 0.30

RI = 0.40

RI = 0.50

RI = 0.25

RI = 0.35

RI = 0.45

RI ≥ 1.00

Residual discharge

Q1 = 21.7 m3/s

Q + ( Z - αQ2)/ βQ

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Draft

-10

-5

0

5

10

15

20

0 50 100 150 200 250

Tan

k S

urg

e /

m

Time / s

RI=0.15 RI=0.30

RI=0.35 RI=0.40

RI=0.45 RI=0.50

tc

= 0 s

tc

= 20 s

tc

= 35 s

tc

= 10 s

tc

= 30 s

tc

= 50 s

1

water level history for

fisrst stage of LR

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Draft

Q1 = 93.10 m3/s

0

40

80

120

160

200

0 20 40 60 80 100

Discharge/ m3/s

Time / s

49.7 s

Q + ( Z - αQ2 ) / βQ

Page 28 of 28



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