Research ArticleHydraulic Transients in the Long Diversion-Type HydropowerStation with a Complex Differential Surge Tank
Xiaodong Yu1 Jian Zhang12 and Ling Zhou1
1 College of Water Conservancy and Hydropower Engineering Hohai University Nanjing 210098 China2 College of Hydraulic and Civil Engineering Xinjiang Agricultural University Urumqi 830052 China
Correspondence should be addressed to Jian Zhang jzhanghhueducn
Received 11 April 2014 Accepted 17 June 2014 Published 15 July 2014
Academic Editor Linni Jian
Copyright copy 2014 Xiaodong Yu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Based on the theory of hydraulic transients and the method of characteristics (MOC) a mathematic model of the differential surgetank with pressure-reduction orifices (PROs) and overflow weirs for transient calculation is proposed The numerical model ofhydraulic transients is established using the data of a practical hydropower station and the probable transients are simulated Theresults show that successive load rejection is critical for calculating the maximum pressure in spiral case and themaximum rotatingspeed of runner when the bifurcated pipe is converging under the surge tank in a diversion-type hydropower station the pressuredifference between two sides of breastwall is large during transient conditions and itwould bemore seriouswhen simultaneous loadrejections happen after load acceptance the reasonable arrangement of PROs on breast wall can effectively decrease the pressuredifference
1 Introduction
Various turbine operations such as load acceptance loadrejection and combination conditions produce transients inhydropower station which are directly related to the safetyof whole hydropower station and local power grid evenresulting in substantial damage and human loss in some cases[1ndash3] Study of hydraulic transients in hydropower stationsattracts many researchers because of its complexity andsignificance in practice Souza et al [4] simulated transientflow in hydropower plants by considering a nonlinear modelof the penstock and hydraulic turbine Selek et al [5] com-pared the computational results of transient pressures withmeasured data for the Catalan Power Plant in Turkey and theagreement is satisfactory Calamak and Bozkus [6] studiedthe performance of two run-of-river plants during transientconditions An et al [7] presented an effective theory for safecontrol of air cushion surge chambers based on the numericalsimulation of hydraulic transients in hydropower stationTheimpulse method was used to analyze hydraulic resonance inhydropower systems by Riasi et al [8] Except for the MOC[9] some new numericalmethods have been used to simulate
hydraulic transients in hydropower systems such as dynamicorificemodel [10] finite volumemethod [11] implicitmethodof characteristics [12] stochastic method [13] and 1-D-3-Dcoupling approach [14]
With the rapid exploitation of hydropower resources insome developing countries in recent years as well as theadvanced construction techniques the arrangement of thepipe systems and hydraulic devices is more complicatedSome diversion-type hydropower stations have a fairly longheadrace tunnel and the water inertia is very large In orderto reduce the amplitude of water level oscillations in surgetanks and accelerate the attenuation a new type of differentialsurge tank [15 16] is designed However some undesirabletransients may appear because of the complex arrangementsand new hydraulic devices [17]
This work investigates the transients in a practicalhydropower station with long headrace tunnel and com-plex differential surge tank Some undesirable transientsare mainly analyzed and the PROs are proposed to reducethe pressure difference on the breast wall The results canprovide reference for the design and operation of a similarproject
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 241868 11 pageshttpdxdoiorg1011552014241868
2 The Scientific World Journal
2 Mathematic Model and Calculation Method
21 Governing Equations and Solution Methods for Pres-surized Pipe Model The following equations describe one-dimensional transient flows [18]
Continuity equation
120597119867
120597119905+ 119881
120597119867
120597119909+1198862
119892
120597119881
120597119909+ 119881 sin 120579 = 0 (1)
Momentum equation
120597119867
120597119909+119881
119892
120597119881
120597119909+1
119892
120597119881
120597119905+119891 |119881|119881
2119892119863= 0 (2)
in which 119867 = piezometric head 119881 = flow velocity 119886 =
wave speed 119892 = acceleration due to gravity 119891 = Darcy-Weisbach friction factor 120579 = angle of the pipeline inclinedwith the horizontal119909 = distance along the pipelinemeasuredpositively in the downstream direction 119863 = diameter of thepipe and 119905 = time
The momentum and continuity equations governingtransient flow in closed conduits are classified as quasilinearhyperbolic partial differential equations (PDEs) for which noanalytical solutions are available for a general pipe systemThe method of characteristics (MOC) is widely employedto solve the set of PDEs and these two equations can betransformed into a couple of ordinary differential equations(ODEs) along the characteristic lines The friction item islinearized by using the trapezoidal rule which is of second-order accuracy and some small terms are dropped withoutloss in accuracy It is convenient to develop these equationswith the discharge 119876 instead of flow velocity 119881 as thedependent variable Then the equations can be transformedinto two algebraic equations for computer calculation afterintegration along the characteristic linesThe grid of pipelinesand the time interval follow Courant condition Δ119905 le Δ119909119886
22 Turbine Equations The equation for the acceleration ofthe rotating masses is [19]
119868119889120596
119889119905= 119879 minus 119879
119892 (3)
in which 119868 = the polar moment of inertia of rotating fluid andmechanical parts in the turbine-generator combination (119868 =1198821198772119892 where119882 = weight 119877 = radius of gyration) 120596 = the
angular speed of the unit 119879 = the torque produced by waterflowing through the unit and 119879
119892= the resistant torque from
the generatorWhen load rejection occurs the unit is instantly discon-
nected from the system and thus becomes isolated (ie119879119892=
0) Then the unit has to be quickly closed according to theemergency closure lawwhich is set to the governor in advanceto prevent extended periods of high overspeed which mayinduce severe spiral case pressure increments and draft tubepressure decrements This type of regulation is much severeand critical to the safety of the whole system which is
considered in this study Equation (3) can be transferredto algebraic equations by integration and Taylor expansionmethod which can be expressed as
119899119905= 1198991199050+Δ119905
119879119898
(151205731199050minus 05120573
1199050minusΔ119905) (4)
in which 119899 = the dimensionless rotating speed ratio 119873119873119903
120573 = the dimensionless torque ratio119879119879119903Δ119905 = time step and
119879119898= the mechanical starting time defined as 119879
119898= 1198681205962
119903119875119903
Equation (4) is explicit and the terms in the right-handside of the equation are known at any initial condition so therotating speed at each time step can be calculated Variationof the wicket gates (WG) opening during transients can beexpressed as
119910 = 1199100minusΔ119905
119879119888
(5)
where 119910 = the dimensionless WG opening 1199100= initial
dimensionless WG opening and 119879119888
= the closing timeconstant from full opening to closed
The equation for the head balance of turbine is
119867119879= (1198851+1199011
120574+12057211198811
2
2119892) minus (119885
2+1199012
120574+12057221198812
2
2119892) (6)
in which119867119879= head of turbine119885 = elevation of the turbine
119901 is pressure 119881 = flow velocity 120572 = the kinetic energycorrection coefficient which is generally equal to 1 120574 = unitweight of water 119892 = acceleration of gravity and the subscripts1 and 2 refer to the previous and latter section of the turbinerespectively
Positive compatibility equation of the inlet section is
1198671= 119862119875minus 119861119875119876 (7)
Negative compatibility equation of the outlet sectionis
1198672= 119862119872+ 119861119872119876 (8)
As is clear from the above equations the five parametersturbine head (119867
torque (119879) and dimensionlessWGopening (119910) are unknownFor the calculation of these unknown values at each time stepthe following procedure is used as well as the characteristiccurve of model turbine which is schematically shown inFigure 1 The computer model is encoded in the FORTRANprogramming language which has been used to simulatehydraulic transients in some practical hydropower stationsand the predictions agree well with the field test [20]
23 Boundary Conditions of Differential Surge Tankwith PROsand Overflow Weir The plan and profile of the differentialsurge tankwith overflowweir and PROs on the breast wall areshown in Figure 2119867 and119876 are the piezometric head and thedischargeThe subscripts 1 2 and 3 refer to the cross sections1 2 and 3 respectively 119885
119878119878and 119876
119878119878are the water level and
The Scientific World Journal 3
Yes
No
(y) from Eq (5)Calculate rotating speed (N) fromEq (4) with the known values of
the previous two time steps
Assume that turbine discharge (Q)is equal to the values of previoustime step (Q0)
Calculate turbine characteristics dataof the WG opening (y)
sections 1 2 and 3 respectively 119860119904119904 1198601199041 and 119860
1199042are the
cross-sectional areas of the main tank the riser 1 and theriser 2 respectively 119860TH1 119860TH2 and 119860TH3 are the cross-sectional areas of the throttle orifices at the bottom of themain tank the riser 1 and the riser 2 respectively 120585
1 1205852
and 1205853are the head loss coefficients of the throttle orifices
respectively which have different values for the flow into orout of the tank 119876
119884 1198761198841 and 119876
1198842are the discharge through
the overflow weir between the main tank and the risers119876119884= 1198761198841
+ 1198761198842 measured positively from the risers into
the main tank 119876119871 1198761198711 and 119876
1198712are the discharge through
the PROs on the breast wall 119876119871= 1198761198711
+ 1198761198712 measured
positively from the risers into themain tank Every parameterthat has a subscript 0 is a known value of a previous timestep
Discharge through the overflow weir can be given by theequation
(119895 = 1 minus 119870) are the cross-sectionalareas of the PROs and 119870 is the number of the PROs of eachriser 119885
119871119895(119895 = 1 minus 119870) are the elevations of the center of
the PROs and 1205831and 120583
2are the discharge coefficients of free
flow and submerged flow of the PROs respectively So thetotal discharge through the PROs on each breast wall may bewritten as
1198761198711=
119899
sum
119895=1
1198761198711119895 119876
1198712=
119899
sum
119895=1
1198761198712119895 (12)
Every variable of this boundary during transient processcan be derived with (9) to (12) employing the four-orderRunge-Kutta method
3 Case Study
The model is used to predict the hydraulic transients inthe practical hydropower station in China As shown inFigure 3 every two units share a common waterway systemwhich consists of the intake the gate shaft the headracetunnel the upstream differential surge tank penstock thetailrace tunnel and so on
The headrace tunnel has a length of 170 km with adiameter 119863 = 118m The differential surge tank with a ldquo119884rdquoshaped bifurcation pipe at the bottom is located at the end ofthe headrace tunnel The length of each penstock is 5300mand the diameter is 65mThe length of each tailrace tunnel is2600m and the diameter is 120mThe head loss coefficientsof the system are collected according to the project
The cross-sectional areas of the main tank and eachriser of the differential surge tank are 3464m2 and 346m2respectivelyThe cross-sectional areas of the throttle orifice atthe bottom of the main tank and each riser are 139m2 and190m2 respectively The elevation of the bottom of the maintank is 15752m while the elevation of the overflow weir atthe top of the riser is 16700m with the width 119861
119884= 60m
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0
30
60
90
120
150
0 10 20 30 40 50Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless WG openingDimensionless rotating speedPressure head in spiral case
(a) Values of turbine parameters versus time
Time (s)
minus400
minus200
0
200
400
0 200 400 600 800 1000
Discharge flow in or out of main tankDischarge flow in or out of 1 riserDischarge flow in or out of 2 riserDischarge through overflow weirsDischarge through PROs
Disc
harg
e (m
3middotsminus
1)
(b) Discharge of surge tank versus time
Time (s)
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(c) Water level in surge tank versus time
Time (s)
minus20
minus10
0
10
20
30
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(d) Water level difference on breast wall versus time
Figure 4 Numerical results of simultaneous load rejection
According to the laboratory experiments the values ofthe parameters are as follows the discharge coefficient of freeoverflow from the riser into the well 119896
11= 05 The discharge
coefficient of free overflow from the main tank into the riser1198961015840
11= 053 The discharge coefficients of submerged flow 119896
12
and 119896101584012
approximate to 80 of the discharge coefficients offree overflow The head loss coefficient of the orifice at thebottom of the main tank 120585
1= 199 for the flow into the main
tank while 1205851= 132 for the flow out of the main tank The
head loss coefficients of the orifice at the bottom of the risers1205852= 1205853= 172 for the flow into the risers while 120585
2= 1205853= 323
for the flow out of the risersThe discharge coefficients of freeflow and submerged flow of the PRO 120583
1= 06 and 120583
2= 05
respectivelyThe rated output of each Francis turbine is 610MW Its
rated head is 288m rated rotating speed is 1667 rpm andrated discharge is 2286m3s The diameter of the runner
is 656m The installation elevation of turbine is 13168mThe inertia (1198661198632) of the turbine and generator is about75800 tm2
31 Simulation of Simultaneous Load Rejection Based on themathematicalmodel and numericalmethods presented in theabove section the computermodel of the hydraulic transientsin the hydropower station is encoded in the FORTRANprogramming language A critical operation that is likelyto occur several times during the life of the project issimulated The upstream water level is at EL 16460m andthe downstream water level is at EL 13331m Full load isrejected at time 119905 = 0 and the WG are closed with the closingtime 119879
119888= 13 seconds under governor control It should be
noted that the effect of the PROs is ignored in this simulationthat is the cross-section areas of the PROs are considered as0 The calculation is shown in Figure 4
6 The Scientific World Journal
The changes of the dimensionless rotating speed of therunner the dimensionless WG opening and the pressurehead in the spiral case are shown in Figure 4(a) Whenthe units reject full load the rotating speed of the runneris increased rapidly The WG are quickly closed accordingto the emergency closure law which is set to the governorin advance to prevent extended periods of high overspeedleading to serious water hammer pressure in the spiral caseThe pressure in the spiral case increases quickly during theclosing process accompanying the phenomena of the wavepropagation and reflection After the closure of theWGmostof the water hammer wave has attenuated and the pressurein the spiral case changes slowly because of the water levelvariation in the surge tank
Figures 4(b) and 4(c) show the changes of the dischargeand the water levels of the surge tank After the closure of theWG the discharge in penstock decreases rapidly The waterin the headrace tunnel flows into the surge tank through theorifices at the bottom of the main tank and the risers sothe water levels in them increase fast As the cross-sectionalarea of the riser is smaller than the main tankrsquos the waterlevel in the riser rises more quickly When the water levelin the riser reaches the elevation of the overflow weir thewater freely spills into the main tank from the riser andthe increasing speed of the water level in the riser becomesslower When the water level in the main tank reaches theelevation of the overflow weir the free flow becomes thesubmerged flow Then the discharge that flows from theriser into the main tank decreases leading to the secondquick increase of water level in the riser in a short periodof time As the discharge into the surge tank decreases theflow direction reverses and the inflow becomes the outflowThe water levels in the main tank and the risers decreaseafter reaching the highest elevation Similarly the water levelin the riser falls more rapidly because of the small cross-sectional area When the water level in the riser is lowerthan that in the main tank the water flows from the maintank to the riser and the flow pattern changes from thesubmerged flow to the free flow with the variation of thewater levels in the main tank and the riser If the water levelin the main tank is lower than the elevation of the overflowweir the overflow stops and the water level in the riser fallsrapidly The discharge through the PROs is always equal to0 during the transient because the area is set to 0 in themodel
Figure 4(d) stands for the water level difference betweenthe two sides of the riserrsquos breast wall during the transientprocess When the load rejection happens the water levelin the riser rises quickly because of the small cross-sectionalarea while the water level in the main tank increases slowlyso the corresponding pressure differencemeasured positivelyin this situation increases quickly When the water level inthe riser reaches the elevation of the overflow weir it risesslowly because of the overflow and the pressure differencereduces gradually Then the pressure difference reverses withthe decrease of thewater level in the riserUnder this transientsimulation the positive pressure difference between the twosides of the riserrsquos breast wall is about 30m while the negativeone is about 20mThepressure difference is significant so the
breast wall in this kind of surge tankmust be of good physicalstrength
As the headrace tunnel of the hydropower station is verylong the water inertia is large so the surge tank periodis very long the amplitude of the oscillations is large andthe attenuation is very slow The transient process causedby previous condition has not disappeared the followingcondition may happen which may make the result of thetransient more serious
32 Simulation of Successive Load Rejection As the layoutof water diversion systems of hydropower stations becomesmore complex as well as the operation of grid systemscombination operating conditions become very commonsuch as load rejection after load acceptance and load rejectionone by one Although the probability of some extremeconditions is very small once they happen the consequencesare very serious Therefore it is necessary to consider thesefactors in the design of hydropower stations
With respect to the water hammer pressure in combina-tion conditions successive load rejection is studied beforeThis operation condition can make the negative pressurein the draft tube more serious than that in simultaneousload rejection in a pumped storage hydropower station [21]The tailrace tunnel is short in a diversion-type hydropowerstation so the draft tube pressure will not reduce too muchduring this condition But successive load rejection mayresult in high overspeed of the runner and larger pressure inthe spiral case in a long diversion-type hydropower stationwith the bifurcated pipe at the bottom of the surge tank thatis there are two independent penstocks for the units from thesurge tank as shown in Figure 3
The calculations are shown in Figure 5 The water levelsof the reservoirs are the same as the previous section Firstlythe 1 turbine rejects its full load and when the water level ofthe surge tank reaches the highest level the 2 turbine rejectsits full load
The dimensionlessWGopening the dimensionless rotat-ing speed of the runner and the pressure head in thespiral case during successive load rejection are shown inFigure 5(a) When the 1 turbine rejects full load the waterlevel of the surge tank begins to rise As shown in Figure 5(b)when the water level of the surge tank reaches the highestlevel it rises about 30m whichmakes the static head of the 2turbine increase by 30m Otherwise the units connect withlarge power grid under normal conditions and the speed ofthe 2 turbinewill not change due to the stable grid frequencyso the opening of theWG keeps constant as well As the statichead of the 2 turbine rises the demand discharge of turbinealso increases When the water level of the surge tank reachesthe highest level after the load rejection of the 1 turbinethe 2 turbine rejects full load Compared with simultaneousload rejection the maximum pressure in the spiral case andthe maximum rotating speed of the 2 turbine are increasedwhich are shown in Table 1Therefore if the water way systemof a hydropower station is arranged as this form especiallyas the headrace tunnel is very long successive load rejectioncondition will make the maximum pressure of the spiral caseand themaximum rotating speed of the runner more serious
The Scientific World Journal 7
0
30
60
90
120
150
0 40 80 120 160 200Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless 1 WG openingDimensionless 2 WG openingDimensionless 1 rotating speedDimensionless 2 rotating speedPressure head in 1 spiral casePressure head in 2 spiral case
(a) Values of turbine parameters versus time
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(b) Water level in surge tank versus time
Figure 5 Numerical results of successive load rejection
Table 1 Comparisons between simultaneous and successive loadrejection
33 Simulation of Simultaneous Load Rejection after LoadAcceptance Because of the small cross-sectional area thewater level in the riser rises or falls rapidly during transientprocess which creates an accelerating or decelerating headon the tunnel in a short period of time This effect reducesthe amplitude of water level oscillations in the surge tank andaccelerates the attenuation However due to rapid water levelvariations in the risers and slow variations in the main tankthe pressure difference between the two sides of the riserrsquosbreast wall is significant If the structure cannot bear thispressure difference it may cause collapse of the breast wall[22] Thus it is very important to find the critical operationconditions and to simulate the possible maximum pressuredifference on the breast wall in the design stage which arethe basis for the structure calculation of the breast wall
Simultaneous load rejection after load acceptance iscommon in the operation of hydropower stations Duringthis simulation the water levels of the reservoirs are thesame as the previous section As shown in Figure 6(a) twounits accept load one by one which results in the fall of thewater levels in the surge tank When the water levels of thesurge tank reach the lowest level two units reject full loadsimultaneously The water levels in the risers rise rapidly tothe elevation of the overflow weirs while the water level inthemain tank rises slowly and the elevation of the initial water
Table 2 Maximum pressure difference on breast wall underdifferent conditions
Operation conditions Maximum pressure difference onbreast wall
level is lower compared with the simultaneous load rejectioncase So the pressure difference on the breast wall is largerduring this combination operating condition As shown inFigure 6(b) the maximum pressure difference on the breastwall increases by 20m compared with simultaneous loadrejection
34 Control of the Pressure Difference on Breast Wall As itcan be seen from Table 2 the maximum pressure differencebetween two sides of the breast wall is close to 30m in simul-taneous load rejection while the difference can reach 50m incombination conditionsThis huge pressure difference bringsgreat challenge to the structural safety of the breast wall thushow to reduce the pressure difference between two sides of thebreast wall has become an issue to the design person
Guaranteeing adequate structural strength of the breastwall a row of the PROs can be set along height direction[23] When transient process occurs a large water leveldifference is created between the risers and the main tankin a differential surge tank The water level in the riser risesrapidly reaching the PROs then the water flows from theriser into the main tank through the orifices which slowingdown thewater level rise in the riser while speeding thewater
8 The Scientific World Journal
1580
1602
1624
1646
1668
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
Time (s)
minus20
0
20
40
60
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(b) Water level difference on breast wall
Figure 6 Numerical results of simultaneous load rejection after load acceptance
rise in the main tank Therefore the pressure difference onthe breast wall can be reduced The locations the quantityand the diameter of the PROs are fixed during this simulationfor easy comparison of the results In this case the elevationof the bottom floor is 15752m The first orifice is set at theelevation of 1585m on the breast wall with the rest PROssetting every 12m upwards and the breast wall of each riser isinstalled with 6 orifices namely 1ndash6 in turn and the diameterof each PRO is 10m The calculation condition is the sameas the former section and the numerical results are shown inFigure 7
Figure 7(a) shows the variation of the water levels inthe surge tank after setting the PROs on the breast wallCompared with the results in Figure 6(a) that no PROs areset on the breast wall the amplitude of water level oscillationsand surge attenuations in the surge tank is almost the samewhile the rising speed of the water level in the riser is slowingdown obviously
Figure 7(b) shows the discharge through each PRO in the1 riser during the transients When the units accept load thewater level falls quickly in the riser but slowly in the maintank which forms negative water level difference on the twosides of PROs and causes the water to flow from the maintank into the riser As the initial water levels in both the maintank and the risers are above the elevation of the highest PROthat is the 6 PRO submerged flow occurs in the PROs 1ndash6 when the water level difference appears When the waterlevels in the risers fall below the elevation of the 6 PROthe flow pattern at the 6 PRO turns from the submergedflow to the free flow with the water level in the main tankcontinuing to fall the discharge decreases gradually whenthewater level in themain tank falls below the elevation of the6 PRO the discharge through the 6 PRO turns to 0 Similarphenomenon could be seen in other PROs when the waterlevels continue to fall In addition the flow pattern is alwaysthe submerged flow at 1 PRO because of its low elevation
When the water level reaches its lowest elevation in thesurge tank two units reject full load at the same time In
this condition the water level in the riser rises quickly andthe positive water level difference is formed in the 1 PROcausing the water to flow from the riser into the main tank inthe type of submerged flow When the water level in the riserrises to the 2 PRO the water flows from the riser to the maintank in the type of free flow with the water level in the risercontinuing to rise the discharge through the PROs increasesgradually When the water in the main tank reaches the 2PRO the free flow here turns to a submerged one with thewater level in the main tank continuing to rise the dischargedecreases gradually A similar phenomenon occurs in otherPROs subsequently When the water level in the main tank isover the 6 PRO submergedflowoccurs in every PROand thedischarge is the same because of the equal pressure differenceon the two sides of each PRO
Figure 7(c) shows the water level difference on the twosides of the breast wall of the riser Because of the PROsthe water level changes slower in the riser but quicker inthe main tank which reduces the pressure difference on thebreast wall of the riser As shown in Table 2 the maximumwater level difference between the two sides of the breast wallis reduced almost by 20m compared with the results withoutPROs on the breast wall Therefore setting appropriate PROscan effectively reduce the pressure difference between the twosides of the breast wall It should be noted that more PROsand bigger diameters are effective to reduce the pressuredifference on the breast wall but too many or too bigPROs would affect the differential effect of the surge tankthereby increasing the maximum surge and reducing thesurge attenuation in the surge tank
4 Conclusions
This paper provides a mathematical model for the differentialsurge tank with PROs and overflow weirs for transientcalculations The numerical model of hydraulic transientsis established using the data of a practical hydropower
The Scientific World Journal 9
0 10008006004002001580
1602
1624
1646
1668
1690
Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
minus10
minus5
0
5
10
15
0 1000800600400200
Discharge through 1 PRODischarge through 2 PRODischarge through 3 PRODischarge through 4 PRODischarge through 5 PRODischarge through 6 PRO
Time (s)
Disc
harg
e (m
3middotsminus
1)
(b) Discharge through PROs versus time
0 1000800600400200Time (s)
minus20
0
20
40
60
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(c) Water level difference on breast wall versus time
Figure 7 Numerical results of simultaneous load rejection after load acceptance with PROs on breast wall
station and the probable operation conditions are simulatedand analyzed The proposed mathematical model and thevalues of some coefficients used in the simulation canprovide reference for the simulation of hydraulic transientsin this type of hydropower station In a long diversion-typehydropower station with the bifurcated pipe at the bottomof the surge tank successive load rejection condition canmake the maximum pressure in the spiral case and themaximum rotating speedmore serious comparedwith simul-taneous load rejection Additionally the pressure differenceon the breast wall is significant during transients especiallyduring the combination condition that simultaneous load
rejection after load acceptance while setting appropriatePROs can reduce the pressure difference effectively Note thatthe present mathematical model and numerical applicationsneed field test verification which will be conducted in theadditional investigation
Notation
MOC Method of characteristicsODE Ordinary differential equationPDE Partial differential equation
10 The Scientific World Journal
PRO Pressure-reduction OrificeWG Wicket gates119886 Wave velocity119860 Cross-sectional area of pipe119860119878 Cross-sectional area of riser
119860119878119878 Cross-sectional area of main tank
119860TH Cross-sectional area of throttle orifice119861119872 119861119875 Known constants in compatibility equa-tions
119861119884 Width of overflow weir
119862119872 119862119875 Known constants in compatibility equa-tions
119891 Darcy-Weisbach friction factor1198661198632 Moment of inertia of unit
119867 Piezometric head119867119879 Turbine head
11989611 Discharge coefficient of free overflow
through overflow weir11989612 Discharge coefficient of submerged over-
flow through overflow weir119899 Dimensionless rotating speed119876 Discharge through pipe119876119871 Discharge through PROs
119876119878 Discharge in riser
119876119878119878 Discharge in main tank
119876119884 Discharge through overflow weirs
119879 Torque on turbine119879119888 Closing time constant from full opening to
closed119879119898 Mechanical starting time
119910 Dimensionless WG opening119885119871 Elevation of the center of PRO
119885119878 Water level in riser
119885119878119878 Water level in main tank
119885119884 Elevation of overflow weir
120573 Dimensionless torque120585 Head loss coefficient of throttle orifice1205831 Discharge coefficients of free flow of PRO
1205832 Discharge coefficients of submerged flowof
PRO
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported by the National Natural Sci-ence Foundation of China (Grant no 51379064) the Nat-ural Science Foundation of Jiangsu Province (Grant noBK20130839) the Open Research Fund Program of State KeyLaboratory ofWater Resources andHydropower EngineeringScience (Grant no 2013B116) and the Fundamental ResearchFunds for the Central Universities of China (grant no2013B06114)
References
[1] A Adamkowski ldquoCase study lapino powerplant penstockfailurerdquo Journal of Hydraulic Engineering vol 127 no 7 pp 547ndash555 2001
[2] J Yang K Zhao L Li and PWu ldquoAnalysis on the causes of units7 and 9 accidents at Sayano-Shushenskaya hydropower stationrdquoJournal of Hydroelectric Engineering vol 30 no 4 pp 226ndash2342011 (Chinese)
[3] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006
[4] O H Souza Jr N Barbieri and A H M Santos ldquoStudy ofhydraulic transients in hydropower plants through simulationof nonlinear model of penstock and hydraulic turbine modelrdquoIEEE Transactions on Power Systems vol 14 no 4 pp 1269ndash1272 1999
[5] B Selek M S Kirkgoz and Z Selek ldquoComparison of computedwater hammer pressures with test results for the Catalan powerplant in Turkeyrdquo Canadian Journal of Civil Engineering vol 31no 1 pp 78ndash85 2004
[6] M Calamak and Z Bozkus ldquoComparison of performance oftwo run-of-river plants during transient conditionsrdquo Journal ofPerformance of Constructed Facilities vol 27 no 5 pp 624ndash6322013
[7] J F An J Zhang and A Hazrati ldquoSafe control of air cushionsurge chambers in hydropower systemsrdquo Scientia Iranica vol20 no 6 pp 1605ndash1611 2013
[8] A Riasi A Nourbakhsh and M Raisee ldquoNumerical modelingfor hydraulic resonance in hydropower systems using impulseresponserdquo Journal of Hydraulic Engineering vol 136 no 11 pp929ndash934 2010
[9] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005
[10] H Ramos and A B Almeida ldquoDynamic orifice model onwaterhammer analysis of high or medium heads of smallhydropower schemesrdquo Journal of Hydraulic Research vol 39 no4 pp 429ndash436 2001
[11] T Kolsek J Duhovnik and A Bergant ldquoSimulation of unsteadyflow and runner rotation during shut-down of an axial waterturbinerdquo Journal of Hydraulic Research vol 44 no 1 pp 129ndash137 2006
[12] M H Afshar M Rohani and R Taheri ldquoSimulation oftransient flow in pipeline systems due to load rejection and loadacceptance by hydroelectric power plantsrdquo International Journalof Mechanical Sciences vol 52 no 1 pp 103ndash115 2010
[13] Q K Zhang B Karney L Suo and A F Colombo ldquoStochasticanalysis of water hammer and applications in reliability-basedstructural design for hydro turbine penstocksrdquo Journal ofHydraulic Engineering vol 137 no 11 pp 1509ndash1521 2011
[14] X X Zhang and Y G Cheng ldquoSimulation of hydraulictransients in hydropower systems using the 1-D-3-D couplingapproachrdquo Journal of Hydrodynamics vol 24 no 4 pp 595ndash604 2012
[15] S Y Wu G Wang and J Wang ldquoOptimization of the typeof upstream surge chamber in Jinping Hydropower StationrdquoSichuan Water Power no 6 pp 93ndash96 104 2008 (Chinese)
[16] S R Wang T X Liu and W L Zou ldquoThe advantage ofnew differential surge chamber and its applicationrdquo Journal ofTsinghua University vol 2 pp 73ndash84 1988 (Chinese)
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
2 The Scientific World Journal
2 Mathematic Model and Calculation Method
21 Governing Equations and Solution Methods for Pres-surized Pipe Model The following equations describe one-dimensional transient flows [18]
Continuity equation
120597119867
120597119905+ 119881
120597119867
120597119909+1198862
119892
120597119881
120597119909+ 119881 sin 120579 = 0 (1)
Momentum equation
120597119867
120597119909+119881
119892
120597119881
120597119909+1
119892
120597119881
120597119905+119891 |119881|119881
2119892119863= 0 (2)
in which 119867 = piezometric head 119881 = flow velocity 119886 =
wave speed 119892 = acceleration due to gravity 119891 = Darcy-Weisbach friction factor 120579 = angle of the pipeline inclinedwith the horizontal119909 = distance along the pipelinemeasuredpositively in the downstream direction 119863 = diameter of thepipe and 119905 = time
The momentum and continuity equations governingtransient flow in closed conduits are classified as quasilinearhyperbolic partial differential equations (PDEs) for which noanalytical solutions are available for a general pipe systemThe method of characteristics (MOC) is widely employedto solve the set of PDEs and these two equations can betransformed into a couple of ordinary differential equations(ODEs) along the characteristic lines The friction item islinearized by using the trapezoidal rule which is of second-order accuracy and some small terms are dropped withoutloss in accuracy It is convenient to develop these equationswith the discharge 119876 instead of flow velocity 119881 as thedependent variable Then the equations can be transformedinto two algebraic equations for computer calculation afterintegration along the characteristic linesThe grid of pipelinesand the time interval follow Courant condition Δ119905 le Δ119909119886
22 Turbine Equations The equation for the acceleration ofthe rotating masses is [19]
119868119889120596
119889119905= 119879 minus 119879
119892 (3)
in which 119868 = the polar moment of inertia of rotating fluid andmechanical parts in the turbine-generator combination (119868 =1198821198772119892 where119882 = weight 119877 = radius of gyration) 120596 = the
angular speed of the unit 119879 = the torque produced by waterflowing through the unit and 119879
119892= the resistant torque from
the generatorWhen load rejection occurs the unit is instantly discon-
nected from the system and thus becomes isolated (ie119879119892=
0) Then the unit has to be quickly closed according to theemergency closure lawwhich is set to the governor in advanceto prevent extended periods of high overspeed which mayinduce severe spiral case pressure increments and draft tubepressure decrements This type of regulation is much severeand critical to the safety of the whole system which is
considered in this study Equation (3) can be transferredto algebraic equations by integration and Taylor expansionmethod which can be expressed as
119899119905= 1198991199050+Δ119905
119879119898
(151205731199050minus 05120573
1199050minusΔ119905) (4)
in which 119899 = the dimensionless rotating speed ratio 119873119873119903
120573 = the dimensionless torque ratio119879119879119903Δ119905 = time step and
119879119898= the mechanical starting time defined as 119879
119898= 1198681205962
119903119875119903
Equation (4) is explicit and the terms in the right-handside of the equation are known at any initial condition so therotating speed at each time step can be calculated Variationof the wicket gates (WG) opening during transients can beexpressed as
119910 = 1199100minusΔ119905
119879119888
(5)
where 119910 = the dimensionless WG opening 1199100= initial
dimensionless WG opening and 119879119888
= the closing timeconstant from full opening to closed
The equation for the head balance of turbine is
119867119879= (1198851+1199011
120574+12057211198811
2
2119892) minus (119885
2+1199012
120574+12057221198812
2
2119892) (6)
in which119867119879= head of turbine119885 = elevation of the turbine
119901 is pressure 119881 = flow velocity 120572 = the kinetic energycorrection coefficient which is generally equal to 1 120574 = unitweight of water 119892 = acceleration of gravity and the subscripts1 and 2 refer to the previous and latter section of the turbinerespectively
Positive compatibility equation of the inlet section is
1198671= 119862119875minus 119861119875119876 (7)
Negative compatibility equation of the outlet sectionis
1198672= 119862119872+ 119861119872119876 (8)
As is clear from the above equations the five parametersturbine head (119867
torque (119879) and dimensionlessWGopening (119910) are unknownFor the calculation of these unknown values at each time stepthe following procedure is used as well as the characteristiccurve of model turbine which is schematically shown inFigure 1 The computer model is encoded in the FORTRANprogramming language which has been used to simulatehydraulic transients in some practical hydropower stationsand the predictions agree well with the field test [20]
23 Boundary Conditions of Differential Surge Tankwith PROsand Overflow Weir The plan and profile of the differentialsurge tankwith overflowweir and PROs on the breast wall areshown in Figure 2119867 and119876 are the piezometric head and thedischargeThe subscripts 1 2 and 3 refer to the cross sections1 2 and 3 respectively 119885
119878119878and 119876
119878119878are the water level and
The Scientific World Journal 3
Yes
No
(y) from Eq (5)Calculate rotating speed (N) fromEq (4) with the known values of
the previous two time steps
Assume that turbine discharge (Q)is equal to the values of previoustime step (Q0)
Calculate turbine characteristics dataof the WG opening (y)
sections 1 2 and 3 respectively 119860119904119904 1198601199041 and 119860
1199042are the
cross-sectional areas of the main tank the riser 1 and theriser 2 respectively 119860TH1 119860TH2 and 119860TH3 are the cross-sectional areas of the throttle orifices at the bottom of themain tank the riser 1 and the riser 2 respectively 120585
1 1205852
and 1205853are the head loss coefficients of the throttle orifices
respectively which have different values for the flow into orout of the tank 119876
119884 1198761198841 and 119876
1198842are the discharge through
the overflow weir between the main tank and the risers119876119884= 1198761198841
+ 1198761198842 measured positively from the risers into
the main tank 119876119871 1198761198711 and 119876
1198712are the discharge through
the PROs on the breast wall 119876119871= 1198761198711
+ 1198761198712 measured
positively from the risers into themain tank Every parameterthat has a subscript 0 is a known value of a previous timestep
Discharge through the overflow weir can be given by theequation
(119895 = 1 minus 119870) are the cross-sectionalareas of the PROs and 119870 is the number of the PROs of eachriser 119885
119871119895(119895 = 1 minus 119870) are the elevations of the center of
the PROs and 1205831and 120583
2are the discharge coefficients of free
flow and submerged flow of the PROs respectively So thetotal discharge through the PROs on each breast wall may bewritten as
1198761198711=
119899
sum
119895=1
1198761198711119895 119876
1198712=
119899
sum
119895=1
1198761198712119895 (12)
Every variable of this boundary during transient processcan be derived with (9) to (12) employing the four-orderRunge-Kutta method
3 Case Study
The model is used to predict the hydraulic transients inthe practical hydropower station in China As shown inFigure 3 every two units share a common waterway systemwhich consists of the intake the gate shaft the headracetunnel the upstream differential surge tank penstock thetailrace tunnel and so on
The headrace tunnel has a length of 170 km with adiameter 119863 = 118m The differential surge tank with a ldquo119884rdquoshaped bifurcation pipe at the bottom is located at the end ofthe headrace tunnel The length of each penstock is 5300mand the diameter is 65mThe length of each tailrace tunnel is2600m and the diameter is 120mThe head loss coefficientsof the system are collected according to the project
The cross-sectional areas of the main tank and eachriser of the differential surge tank are 3464m2 and 346m2respectivelyThe cross-sectional areas of the throttle orifice atthe bottom of the main tank and each riser are 139m2 and190m2 respectively The elevation of the bottom of the maintank is 15752m while the elevation of the overflow weir atthe top of the riser is 16700m with the width 119861
119884= 60m
The Scientific World Journal 5
0
30
60
90
120
150
0 10 20 30 40 50Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless WG openingDimensionless rotating speedPressure head in spiral case
(a) Values of turbine parameters versus time
Time (s)
minus400
minus200
0
200
400
0 200 400 600 800 1000
Discharge flow in or out of main tankDischarge flow in or out of 1 riserDischarge flow in or out of 2 riserDischarge through overflow weirsDischarge through PROs
Disc
harg
e (m
3middotsminus
1)
(b) Discharge of surge tank versus time
Time (s)
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(c) Water level in surge tank versus time
Time (s)
minus20
minus10
0
10
20
30
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(d) Water level difference on breast wall versus time
Figure 4 Numerical results of simultaneous load rejection
According to the laboratory experiments the values ofthe parameters are as follows the discharge coefficient of freeoverflow from the riser into the well 119896
11= 05 The discharge
coefficient of free overflow from the main tank into the riser1198961015840
11= 053 The discharge coefficients of submerged flow 119896
12
and 119896101584012
approximate to 80 of the discharge coefficients offree overflow The head loss coefficient of the orifice at thebottom of the main tank 120585
1= 199 for the flow into the main
tank while 1205851= 132 for the flow out of the main tank The
head loss coefficients of the orifice at the bottom of the risers1205852= 1205853= 172 for the flow into the risers while 120585
2= 1205853= 323
for the flow out of the risersThe discharge coefficients of freeflow and submerged flow of the PRO 120583
1= 06 and 120583
2= 05
respectivelyThe rated output of each Francis turbine is 610MW Its
rated head is 288m rated rotating speed is 1667 rpm andrated discharge is 2286m3s The diameter of the runner
is 656m The installation elevation of turbine is 13168mThe inertia (1198661198632) of the turbine and generator is about75800 tm2
31 Simulation of Simultaneous Load Rejection Based on themathematicalmodel and numericalmethods presented in theabove section the computermodel of the hydraulic transientsin the hydropower station is encoded in the FORTRANprogramming language A critical operation that is likelyto occur several times during the life of the project issimulated The upstream water level is at EL 16460m andthe downstream water level is at EL 13331m Full load isrejected at time 119905 = 0 and the WG are closed with the closingtime 119879
119888= 13 seconds under governor control It should be
noted that the effect of the PROs is ignored in this simulationthat is the cross-section areas of the PROs are considered as0 The calculation is shown in Figure 4
6 The Scientific World Journal
The changes of the dimensionless rotating speed of therunner the dimensionless WG opening and the pressurehead in the spiral case are shown in Figure 4(a) Whenthe units reject full load the rotating speed of the runneris increased rapidly The WG are quickly closed accordingto the emergency closure law which is set to the governorin advance to prevent extended periods of high overspeedleading to serious water hammer pressure in the spiral caseThe pressure in the spiral case increases quickly during theclosing process accompanying the phenomena of the wavepropagation and reflection After the closure of theWGmostof the water hammer wave has attenuated and the pressurein the spiral case changes slowly because of the water levelvariation in the surge tank
Figures 4(b) and 4(c) show the changes of the dischargeand the water levels of the surge tank After the closure of theWG the discharge in penstock decreases rapidly The waterin the headrace tunnel flows into the surge tank through theorifices at the bottom of the main tank and the risers sothe water levels in them increase fast As the cross-sectionalarea of the riser is smaller than the main tankrsquos the waterlevel in the riser rises more quickly When the water levelin the riser reaches the elevation of the overflow weir thewater freely spills into the main tank from the riser andthe increasing speed of the water level in the riser becomesslower When the water level in the main tank reaches theelevation of the overflow weir the free flow becomes thesubmerged flow Then the discharge that flows from theriser into the main tank decreases leading to the secondquick increase of water level in the riser in a short periodof time As the discharge into the surge tank decreases theflow direction reverses and the inflow becomes the outflowThe water levels in the main tank and the risers decreaseafter reaching the highest elevation Similarly the water levelin the riser falls more rapidly because of the small cross-sectional area When the water level in the riser is lowerthan that in the main tank the water flows from the maintank to the riser and the flow pattern changes from thesubmerged flow to the free flow with the variation of thewater levels in the main tank and the riser If the water levelin the main tank is lower than the elevation of the overflowweir the overflow stops and the water level in the riser fallsrapidly The discharge through the PROs is always equal to0 during the transient because the area is set to 0 in themodel
Figure 4(d) stands for the water level difference betweenthe two sides of the riserrsquos breast wall during the transientprocess When the load rejection happens the water levelin the riser rises quickly because of the small cross-sectionalarea while the water level in the main tank increases slowlyso the corresponding pressure differencemeasured positivelyin this situation increases quickly When the water level inthe riser reaches the elevation of the overflow weir it risesslowly because of the overflow and the pressure differencereduces gradually Then the pressure difference reverses withthe decrease of thewater level in the riserUnder this transientsimulation the positive pressure difference between the twosides of the riserrsquos breast wall is about 30m while the negativeone is about 20mThepressure difference is significant so the
breast wall in this kind of surge tankmust be of good physicalstrength
As the headrace tunnel of the hydropower station is verylong the water inertia is large so the surge tank periodis very long the amplitude of the oscillations is large andthe attenuation is very slow The transient process causedby previous condition has not disappeared the followingcondition may happen which may make the result of thetransient more serious
32 Simulation of Successive Load Rejection As the layoutof water diversion systems of hydropower stations becomesmore complex as well as the operation of grid systemscombination operating conditions become very commonsuch as load rejection after load acceptance and load rejectionone by one Although the probability of some extremeconditions is very small once they happen the consequencesare very serious Therefore it is necessary to consider thesefactors in the design of hydropower stations
With respect to the water hammer pressure in combina-tion conditions successive load rejection is studied beforeThis operation condition can make the negative pressurein the draft tube more serious than that in simultaneousload rejection in a pumped storage hydropower station [21]The tailrace tunnel is short in a diversion-type hydropowerstation so the draft tube pressure will not reduce too muchduring this condition But successive load rejection mayresult in high overspeed of the runner and larger pressure inthe spiral case in a long diversion-type hydropower stationwith the bifurcated pipe at the bottom of the surge tank thatis there are two independent penstocks for the units from thesurge tank as shown in Figure 3
The calculations are shown in Figure 5 The water levelsof the reservoirs are the same as the previous section Firstlythe 1 turbine rejects its full load and when the water level ofthe surge tank reaches the highest level the 2 turbine rejectsits full load
The dimensionlessWGopening the dimensionless rotat-ing speed of the runner and the pressure head in thespiral case during successive load rejection are shown inFigure 5(a) When the 1 turbine rejects full load the waterlevel of the surge tank begins to rise As shown in Figure 5(b)when the water level of the surge tank reaches the highestlevel it rises about 30m whichmakes the static head of the 2turbine increase by 30m Otherwise the units connect withlarge power grid under normal conditions and the speed ofthe 2 turbinewill not change due to the stable grid frequencyso the opening of theWG keeps constant as well As the statichead of the 2 turbine rises the demand discharge of turbinealso increases When the water level of the surge tank reachesthe highest level after the load rejection of the 1 turbinethe 2 turbine rejects full load Compared with simultaneousload rejection the maximum pressure in the spiral case andthe maximum rotating speed of the 2 turbine are increasedwhich are shown in Table 1Therefore if the water way systemof a hydropower station is arranged as this form especiallyas the headrace tunnel is very long successive load rejectioncondition will make the maximum pressure of the spiral caseand themaximum rotating speed of the runner more serious
The Scientific World Journal 7
0
30
60
90
120
150
0 40 80 120 160 200Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless 1 WG openingDimensionless 2 WG openingDimensionless 1 rotating speedDimensionless 2 rotating speedPressure head in 1 spiral casePressure head in 2 spiral case
(a) Values of turbine parameters versus time
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(b) Water level in surge tank versus time
Figure 5 Numerical results of successive load rejection
Table 1 Comparisons between simultaneous and successive loadrejection
33 Simulation of Simultaneous Load Rejection after LoadAcceptance Because of the small cross-sectional area thewater level in the riser rises or falls rapidly during transientprocess which creates an accelerating or decelerating headon the tunnel in a short period of time This effect reducesthe amplitude of water level oscillations in the surge tank andaccelerates the attenuation However due to rapid water levelvariations in the risers and slow variations in the main tankthe pressure difference between the two sides of the riserrsquosbreast wall is significant If the structure cannot bear thispressure difference it may cause collapse of the breast wall[22] Thus it is very important to find the critical operationconditions and to simulate the possible maximum pressuredifference on the breast wall in the design stage which arethe basis for the structure calculation of the breast wall
Simultaneous load rejection after load acceptance iscommon in the operation of hydropower stations Duringthis simulation the water levels of the reservoirs are thesame as the previous section As shown in Figure 6(a) twounits accept load one by one which results in the fall of thewater levels in the surge tank When the water levels of thesurge tank reach the lowest level two units reject full loadsimultaneously The water levels in the risers rise rapidly tothe elevation of the overflow weirs while the water level inthemain tank rises slowly and the elevation of the initial water
Table 2 Maximum pressure difference on breast wall underdifferent conditions
Operation conditions Maximum pressure difference onbreast wall
level is lower compared with the simultaneous load rejectioncase So the pressure difference on the breast wall is largerduring this combination operating condition As shown inFigure 6(b) the maximum pressure difference on the breastwall increases by 20m compared with simultaneous loadrejection
34 Control of the Pressure Difference on Breast Wall As itcan be seen from Table 2 the maximum pressure differencebetween two sides of the breast wall is close to 30m in simul-taneous load rejection while the difference can reach 50m incombination conditionsThis huge pressure difference bringsgreat challenge to the structural safety of the breast wall thushow to reduce the pressure difference between two sides of thebreast wall has become an issue to the design person
Guaranteeing adequate structural strength of the breastwall a row of the PROs can be set along height direction[23] When transient process occurs a large water leveldifference is created between the risers and the main tankin a differential surge tank The water level in the riser risesrapidly reaching the PROs then the water flows from theriser into the main tank through the orifices which slowingdown thewater level rise in the riser while speeding thewater
8 The Scientific World Journal
1580
1602
1624
1646
1668
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
Time (s)
minus20
0
20
40
60
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(b) Water level difference on breast wall
Figure 6 Numerical results of simultaneous load rejection after load acceptance
rise in the main tank Therefore the pressure difference onthe breast wall can be reduced The locations the quantityand the diameter of the PROs are fixed during this simulationfor easy comparison of the results In this case the elevationof the bottom floor is 15752m The first orifice is set at theelevation of 1585m on the breast wall with the rest PROssetting every 12m upwards and the breast wall of each riser isinstalled with 6 orifices namely 1ndash6 in turn and the diameterof each PRO is 10m The calculation condition is the sameas the former section and the numerical results are shown inFigure 7
Figure 7(a) shows the variation of the water levels inthe surge tank after setting the PROs on the breast wallCompared with the results in Figure 6(a) that no PROs areset on the breast wall the amplitude of water level oscillationsand surge attenuations in the surge tank is almost the samewhile the rising speed of the water level in the riser is slowingdown obviously
Figure 7(b) shows the discharge through each PRO in the1 riser during the transients When the units accept load thewater level falls quickly in the riser but slowly in the maintank which forms negative water level difference on the twosides of PROs and causes the water to flow from the maintank into the riser As the initial water levels in both the maintank and the risers are above the elevation of the highest PROthat is the 6 PRO submerged flow occurs in the PROs 1ndash6 when the water level difference appears When the waterlevels in the risers fall below the elevation of the 6 PROthe flow pattern at the 6 PRO turns from the submergedflow to the free flow with the water level in the main tankcontinuing to fall the discharge decreases gradually whenthewater level in themain tank falls below the elevation of the6 PRO the discharge through the 6 PRO turns to 0 Similarphenomenon could be seen in other PROs when the waterlevels continue to fall In addition the flow pattern is alwaysthe submerged flow at 1 PRO because of its low elevation
When the water level reaches its lowest elevation in thesurge tank two units reject full load at the same time In
this condition the water level in the riser rises quickly andthe positive water level difference is formed in the 1 PROcausing the water to flow from the riser into the main tank inthe type of submerged flow When the water level in the riserrises to the 2 PRO the water flows from the riser to the maintank in the type of free flow with the water level in the risercontinuing to rise the discharge through the PROs increasesgradually When the water in the main tank reaches the 2PRO the free flow here turns to a submerged one with thewater level in the main tank continuing to rise the dischargedecreases gradually A similar phenomenon occurs in otherPROs subsequently When the water level in the main tank isover the 6 PRO submergedflowoccurs in every PROand thedischarge is the same because of the equal pressure differenceon the two sides of each PRO
Figure 7(c) shows the water level difference on the twosides of the breast wall of the riser Because of the PROsthe water level changes slower in the riser but quicker inthe main tank which reduces the pressure difference on thebreast wall of the riser As shown in Table 2 the maximumwater level difference between the two sides of the breast wallis reduced almost by 20m compared with the results withoutPROs on the breast wall Therefore setting appropriate PROscan effectively reduce the pressure difference between the twosides of the breast wall It should be noted that more PROsand bigger diameters are effective to reduce the pressuredifference on the breast wall but too many or too bigPROs would affect the differential effect of the surge tankthereby increasing the maximum surge and reducing thesurge attenuation in the surge tank
4 Conclusions
This paper provides a mathematical model for the differentialsurge tank with PROs and overflow weirs for transientcalculations The numerical model of hydraulic transientsis established using the data of a practical hydropower
The Scientific World Journal 9
0 10008006004002001580
1602
1624
1646
1668
1690
Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
minus10
minus5
0
5
10
15
0 1000800600400200
Discharge through 1 PRODischarge through 2 PRODischarge through 3 PRODischarge through 4 PRODischarge through 5 PRODischarge through 6 PRO
Time (s)
Disc
harg
e (m
3middotsminus
1)
(b) Discharge through PROs versus time
0 1000800600400200Time (s)
minus20
0
20
40
60
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(c) Water level difference on breast wall versus time
Figure 7 Numerical results of simultaneous load rejection after load acceptance with PROs on breast wall
station and the probable operation conditions are simulatedand analyzed The proposed mathematical model and thevalues of some coefficients used in the simulation canprovide reference for the simulation of hydraulic transientsin this type of hydropower station In a long diversion-typehydropower station with the bifurcated pipe at the bottomof the surge tank successive load rejection condition canmake the maximum pressure in the spiral case and themaximum rotating speedmore serious comparedwith simul-taneous load rejection Additionally the pressure differenceon the breast wall is significant during transients especiallyduring the combination condition that simultaneous load
rejection after load acceptance while setting appropriatePROs can reduce the pressure difference effectively Note thatthe present mathematical model and numerical applicationsneed field test verification which will be conducted in theadditional investigation
Notation
MOC Method of characteristicsODE Ordinary differential equationPDE Partial differential equation
10 The Scientific World Journal
PRO Pressure-reduction OrificeWG Wicket gates119886 Wave velocity119860 Cross-sectional area of pipe119860119878 Cross-sectional area of riser
119860119878119878 Cross-sectional area of main tank
119860TH Cross-sectional area of throttle orifice119861119872 119861119875 Known constants in compatibility equa-tions
119861119884 Width of overflow weir
119862119872 119862119875 Known constants in compatibility equa-tions
119891 Darcy-Weisbach friction factor1198661198632 Moment of inertia of unit
119867 Piezometric head119867119879 Turbine head
11989611 Discharge coefficient of free overflow
through overflow weir11989612 Discharge coefficient of submerged over-
flow through overflow weir119899 Dimensionless rotating speed119876 Discharge through pipe119876119871 Discharge through PROs
119876119878 Discharge in riser
119876119878119878 Discharge in main tank
119876119884 Discharge through overflow weirs
119879 Torque on turbine119879119888 Closing time constant from full opening to
closed119879119898 Mechanical starting time
119910 Dimensionless WG opening119885119871 Elevation of the center of PRO
119885119878 Water level in riser
119885119878119878 Water level in main tank
119885119884 Elevation of overflow weir
120573 Dimensionless torque120585 Head loss coefficient of throttle orifice1205831 Discharge coefficients of free flow of PRO
1205832 Discharge coefficients of submerged flowof
PRO
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported by the National Natural Sci-ence Foundation of China (Grant no 51379064) the Nat-ural Science Foundation of Jiangsu Province (Grant noBK20130839) the Open Research Fund Program of State KeyLaboratory ofWater Resources andHydropower EngineeringScience (Grant no 2013B116) and the Fundamental ResearchFunds for the Central Universities of China (grant no2013B06114)
References
[1] A Adamkowski ldquoCase study lapino powerplant penstockfailurerdquo Journal of Hydraulic Engineering vol 127 no 7 pp 547ndash555 2001
[2] J Yang K Zhao L Li and PWu ldquoAnalysis on the causes of units7 and 9 accidents at Sayano-Shushenskaya hydropower stationrdquoJournal of Hydroelectric Engineering vol 30 no 4 pp 226ndash2342011 (Chinese)
[3] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006
[4] O H Souza Jr N Barbieri and A H M Santos ldquoStudy ofhydraulic transients in hydropower plants through simulationof nonlinear model of penstock and hydraulic turbine modelrdquoIEEE Transactions on Power Systems vol 14 no 4 pp 1269ndash1272 1999
[5] B Selek M S Kirkgoz and Z Selek ldquoComparison of computedwater hammer pressures with test results for the Catalan powerplant in Turkeyrdquo Canadian Journal of Civil Engineering vol 31no 1 pp 78ndash85 2004
[6] M Calamak and Z Bozkus ldquoComparison of performance oftwo run-of-river plants during transient conditionsrdquo Journal ofPerformance of Constructed Facilities vol 27 no 5 pp 624ndash6322013
[7] J F An J Zhang and A Hazrati ldquoSafe control of air cushionsurge chambers in hydropower systemsrdquo Scientia Iranica vol20 no 6 pp 1605ndash1611 2013
[8] A Riasi A Nourbakhsh and M Raisee ldquoNumerical modelingfor hydraulic resonance in hydropower systems using impulseresponserdquo Journal of Hydraulic Engineering vol 136 no 11 pp929ndash934 2010
[9] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005
[10] H Ramos and A B Almeida ldquoDynamic orifice model onwaterhammer analysis of high or medium heads of smallhydropower schemesrdquo Journal of Hydraulic Research vol 39 no4 pp 429ndash436 2001
[11] T Kolsek J Duhovnik and A Bergant ldquoSimulation of unsteadyflow and runner rotation during shut-down of an axial waterturbinerdquo Journal of Hydraulic Research vol 44 no 1 pp 129ndash137 2006
[12] M H Afshar M Rohani and R Taheri ldquoSimulation oftransient flow in pipeline systems due to load rejection and loadacceptance by hydroelectric power plantsrdquo International Journalof Mechanical Sciences vol 52 no 1 pp 103ndash115 2010
[13] Q K Zhang B Karney L Suo and A F Colombo ldquoStochasticanalysis of water hammer and applications in reliability-basedstructural design for hydro turbine penstocksrdquo Journal ofHydraulic Engineering vol 137 no 11 pp 1509ndash1521 2011
[14] X X Zhang and Y G Cheng ldquoSimulation of hydraulictransients in hydropower systems using the 1-D-3-D couplingapproachrdquo Journal of Hydrodynamics vol 24 no 4 pp 595ndash604 2012
[15] S Y Wu G Wang and J Wang ldquoOptimization of the typeof upstream surge chamber in Jinping Hydropower StationrdquoSichuan Water Power no 6 pp 93ndash96 104 2008 (Chinese)
[16] S R Wang T X Liu and W L Zou ldquoThe advantage ofnew differential surge chamber and its applicationrdquo Journal ofTsinghua University vol 2 pp 73ndash84 1988 (Chinese)
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
sections 1 2 and 3 respectively 119860119904119904 1198601199041 and 119860
1199042are the
cross-sectional areas of the main tank the riser 1 and theriser 2 respectively 119860TH1 119860TH2 and 119860TH3 are the cross-sectional areas of the throttle orifices at the bottom of themain tank the riser 1 and the riser 2 respectively 120585
1 1205852
and 1205853are the head loss coefficients of the throttle orifices
respectively which have different values for the flow into orout of the tank 119876
119884 1198761198841 and 119876
1198842are the discharge through
the overflow weir between the main tank and the risers119876119884= 1198761198841
+ 1198761198842 measured positively from the risers into
the main tank 119876119871 1198761198711 and 119876
1198712are the discharge through
the PROs on the breast wall 119876119871= 1198761198711
+ 1198761198712 measured
positively from the risers into themain tank Every parameterthat has a subscript 0 is a known value of a previous timestep
Discharge through the overflow weir can be given by theequation
(119895 = 1 minus 119870) are the cross-sectionalareas of the PROs and 119870 is the number of the PROs of eachriser 119885
119871119895(119895 = 1 minus 119870) are the elevations of the center of
the PROs and 1205831and 120583
2are the discharge coefficients of free
flow and submerged flow of the PROs respectively So thetotal discharge through the PROs on each breast wall may bewritten as
1198761198711=
119899
sum
119895=1
1198761198711119895 119876
1198712=
119899
sum
119895=1
1198761198712119895 (12)
Every variable of this boundary during transient processcan be derived with (9) to (12) employing the four-orderRunge-Kutta method
3 Case Study
The model is used to predict the hydraulic transients inthe practical hydropower station in China As shown inFigure 3 every two units share a common waterway systemwhich consists of the intake the gate shaft the headracetunnel the upstream differential surge tank penstock thetailrace tunnel and so on
The headrace tunnel has a length of 170 km with adiameter 119863 = 118m The differential surge tank with a ldquo119884rdquoshaped bifurcation pipe at the bottom is located at the end ofthe headrace tunnel The length of each penstock is 5300mand the diameter is 65mThe length of each tailrace tunnel is2600m and the diameter is 120mThe head loss coefficientsof the system are collected according to the project
The cross-sectional areas of the main tank and eachriser of the differential surge tank are 3464m2 and 346m2respectivelyThe cross-sectional areas of the throttle orifice atthe bottom of the main tank and each riser are 139m2 and190m2 respectively The elevation of the bottom of the maintank is 15752m while the elevation of the overflow weir atthe top of the riser is 16700m with the width 119861
119884= 60m
The Scientific World Journal 5
0
30
60
90
120
150
0 10 20 30 40 50Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless WG openingDimensionless rotating speedPressure head in spiral case
(a) Values of turbine parameters versus time
Time (s)
minus400
minus200
0
200
400
0 200 400 600 800 1000
Discharge flow in or out of main tankDischarge flow in or out of 1 riserDischarge flow in or out of 2 riserDischarge through overflow weirsDischarge through PROs
Disc
harg
e (m
3middotsminus
1)
(b) Discharge of surge tank versus time
Time (s)
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(c) Water level in surge tank versus time
Time (s)
minus20
minus10
0
10
20
30
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(d) Water level difference on breast wall versus time
Figure 4 Numerical results of simultaneous load rejection
According to the laboratory experiments the values ofthe parameters are as follows the discharge coefficient of freeoverflow from the riser into the well 119896
11= 05 The discharge
coefficient of free overflow from the main tank into the riser1198961015840
11= 053 The discharge coefficients of submerged flow 119896
12
and 119896101584012
approximate to 80 of the discharge coefficients offree overflow The head loss coefficient of the orifice at thebottom of the main tank 120585
1= 199 for the flow into the main
tank while 1205851= 132 for the flow out of the main tank The
head loss coefficients of the orifice at the bottom of the risers1205852= 1205853= 172 for the flow into the risers while 120585
2= 1205853= 323
for the flow out of the risersThe discharge coefficients of freeflow and submerged flow of the PRO 120583
1= 06 and 120583
2= 05
respectivelyThe rated output of each Francis turbine is 610MW Its
rated head is 288m rated rotating speed is 1667 rpm andrated discharge is 2286m3s The diameter of the runner
is 656m The installation elevation of turbine is 13168mThe inertia (1198661198632) of the turbine and generator is about75800 tm2
31 Simulation of Simultaneous Load Rejection Based on themathematicalmodel and numericalmethods presented in theabove section the computermodel of the hydraulic transientsin the hydropower station is encoded in the FORTRANprogramming language A critical operation that is likelyto occur several times during the life of the project issimulated The upstream water level is at EL 16460m andthe downstream water level is at EL 13331m Full load isrejected at time 119905 = 0 and the WG are closed with the closingtime 119879
119888= 13 seconds under governor control It should be
noted that the effect of the PROs is ignored in this simulationthat is the cross-section areas of the PROs are considered as0 The calculation is shown in Figure 4
6 The Scientific World Journal
The changes of the dimensionless rotating speed of therunner the dimensionless WG opening and the pressurehead in the spiral case are shown in Figure 4(a) Whenthe units reject full load the rotating speed of the runneris increased rapidly The WG are quickly closed accordingto the emergency closure law which is set to the governorin advance to prevent extended periods of high overspeedleading to serious water hammer pressure in the spiral caseThe pressure in the spiral case increases quickly during theclosing process accompanying the phenomena of the wavepropagation and reflection After the closure of theWGmostof the water hammer wave has attenuated and the pressurein the spiral case changes slowly because of the water levelvariation in the surge tank
Figures 4(b) and 4(c) show the changes of the dischargeand the water levels of the surge tank After the closure of theWG the discharge in penstock decreases rapidly The waterin the headrace tunnel flows into the surge tank through theorifices at the bottom of the main tank and the risers sothe water levels in them increase fast As the cross-sectionalarea of the riser is smaller than the main tankrsquos the waterlevel in the riser rises more quickly When the water levelin the riser reaches the elevation of the overflow weir thewater freely spills into the main tank from the riser andthe increasing speed of the water level in the riser becomesslower When the water level in the main tank reaches theelevation of the overflow weir the free flow becomes thesubmerged flow Then the discharge that flows from theriser into the main tank decreases leading to the secondquick increase of water level in the riser in a short periodof time As the discharge into the surge tank decreases theflow direction reverses and the inflow becomes the outflowThe water levels in the main tank and the risers decreaseafter reaching the highest elevation Similarly the water levelin the riser falls more rapidly because of the small cross-sectional area When the water level in the riser is lowerthan that in the main tank the water flows from the maintank to the riser and the flow pattern changes from thesubmerged flow to the free flow with the variation of thewater levels in the main tank and the riser If the water levelin the main tank is lower than the elevation of the overflowweir the overflow stops and the water level in the riser fallsrapidly The discharge through the PROs is always equal to0 during the transient because the area is set to 0 in themodel
Figure 4(d) stands for the water level difference betweenthe two sides of the riserrsquos breast wall during the transientprocess When the load rejection happens the water levelin the riser rises quickly because of the small cross-sectionalarea while the water level in the main tank increases slowlyso the corresponding pressure differencemeasured positivelyin this situation increases quickly When the water level inthe riser reaches the elevation of the overflow weir it risesslowly because of the overflow and the pressure differencereduces gradually Then the pressure difference reverses withthe decrease of thewater level in the riserUnder this transientsimulation the positive pressure difference between the twosides of the riserrsquos breast wall is about 30m while the negativeone is about 20mThepressure difference is significant so the
breast wall in this kind of surge tankmust be of good physicalstrength
As the headrace tunnel of the hydropower station is verylong the water inertia is large so the surge tank periodis very long the amplitude of the oscillations is large andthe attenuation is very slow The transient process causedby previous condition has not disappeared the followingcondition may happen which may make the result of thetransient more serious
32 Simulation of Successive Load Rejection As the layoutof water diversion systems of hydropower stations becomesmore complex as well as the operation of grid systemscombination operating conditions become very commonsuch as load rejection after load acceptance and load rejectionone by one Although the probability of some extremeconditions is very small once they happen the consequencesare very serious Therefore it is necessary to consider thesefactors in the design of hydropower stations
With respect to the water hammer pressure in combina-tion conditions successive load rejection is studied beforeThis operation condition can make the negative pressurein the draft tube more serious than that in simultaneousload rejection in a pumped storage hydropower station [21]The tailrace tunnel is short in a diversion-type hydropowerstation so the draft tube pressure will not reduce too muchduring this condition But successive load rejection mayresult in high overspeed of the runner and larger pressure inthe spiral case in a long diversion-type hydropower stationwith the bifurcated pipe at the bottom of the surge tank thatis there are two independent penstocks for the units from thesurge tank as shown in Figure 3
The calculations are shown in Figure 5 The water levelsof the reservoirs are the same as the previous section Firstlythe 1 turbine rejects its full load and when the water level ofthe surge tank reaches the highest level the 2 turbine rejectsits full load
The dimensionlessWGopening the dimensionless rotat-ing speed of the runner and the pressure head in thespiral case during successive load rejection are shown inFigure 5(a) When the 1 turbine rejects full load the waterlevel of the surge tank begins to rise As shown in Figure 5(b)when the water level of the surge tank reaches the highestlevel it rises about 30m whichmakes the static head of the 2turbine increase by 30m Otherwise the units connect withlarge power grid under normal conditions and the speed ofthe 2 turbinewill not change due to the stable grid frequencyso the opening of theWG keeps constant as well As the statichead of the 2 turbine rises the demand discharge of turbinealso increases When the water level of the surge tank reachesthe highest level after the load rejection of the 1 turbinethe 2 turbine rejects full load Compared with simultaneousload rejection the maximum pressure in the spiral case andthe maximum rotating speed of the 2 turbine are increasedwhich are shown in Table 1Therefore if the water way systemof a hydropower station is arranged as this form especiallyas the headrace tunnel is very long successive load rejectioncondition will make the maximum pressure of the spiral caseand themaximum rotating speed of the runner more serious
The Scientific World Journal 7
0
30
60
90
120
150
0 40 80 120 160 200Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless 1 WG openingDimensionless 2 WG openingDimensionless 1 rotating speedDimensionless 2 rotating speedPressure head in 1 spiral casePressure head in 2 spiral case
(a) Values of turbine parameters versus time
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(b) Water level in surge tank versus time
Figure 5 Numerical results of successive load rejection
Table 1 Comparisons between simultaneous and successive loadrejection
33 Simulation of Simultaneous Load Rejection after LoadAcceptance Because of the small cross-sectional area thewater level in the riser rises or falls rapidly during transientprocess which creates an accelerating or decelerating headon the tunnel in a short period of time This effect reducesthe amplitude of water level oscillations in the surge tank andaccelerates the attenuation However due to rapid water levelvariations in the risers and slow variations in the main tankthe pressure difference between the two sides of the riserrsquosbreast wall is significant If the structure cannot bear thispressure difference it may cause collapse of the breast wall[22] Thus it is very important to find the critical operationconditions and to simulate the possible maximum pressuredifference on the breast wall in the design stage which arethe basis for the structure calculation of the breast wall
Simultaneous load rejection after load acceptance iscommon in the operation of hydropower stations Duringthis simulation the water levels of the reservoirs are thesame as the previous section As shown in Figure 6(a) twounits accept load one by one which results in the fall of thewater levels in the surge tank When the water levels of thesurge tank reach the lowest level two units reject full loadsimultaneously The water levels in the risers rise rapidly tothe elevation of the overflow weirs while the water level inthemain tank rises slowly and the elevation of the initial water
Table 2 Maximum pressure difference on breast wall underdifferent conditions
Operation conditions Maximum pressure difference onbreast wall
level is lower compared with the simultaneous load rejectioncase So the pressure difference on the breast wall is largerduring this combination operating condition As shown inFigure 6(b) the maximum pressure difference on the breastwall increases by 20m compared with simultaneous loadrejection
34 Control of the Pressure Difference on Breast Wall As itcan be seen from Table 2 the maximum pressure differencebetween two sides of the breast wall is close to 30m in simul-taneous load rejection while the difference can reach 50m incombination conditionsThis huge pressure difference bringsgreat challenge to the structural safety of the breast wall thushow to reduce the pressure difference between two sides of thebreast wall has become an issue to the design person
Guaranteeing adequate structural strength of the breastwall a row of the PROs can be set along height direction[23] When transient process occurs a large water leveldifference is created between the risers and the main tankin a differential surge tank The water level in the riser risesrapidly reaching the PROs then the water flows from theriser into the main tank through the orifices which slowingdown thewater level rise in the riser while speeding thewater
8 The Scientific World Journal
1580
1602
1624
1646
1668
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
Time (s)
minus20
0
20
40
60
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(b) Water level difference on breast wall
Figure 6 Numerical results of simultaneous load rejection after load acceptance
rise in the main tank Therefore the pressure difference onthe breast wall can be reduced The locations the quantityand the diameter of the PROs are fixed during this simulationfor easy comparison of the results In this case the elevationof the bottom floor is 15752m The first orifice is set at theelevation of 1585m on the breast wall with the rest PROssetting every 12m upwards and the breast wall of each riser isinstalled with 6 orifices namely 1ndash6 in turn and the diameterof each PRO is 10m The calculation condition is the sameas the former section and the numerical results are shown inFigure 7
Figure 7(a) shows the variation of the water levels inthe surge tank after setting the PROs on the breast wallCompared with the results in Figure 6(a) that no PROs areset on the breast wall the amplitude of water level oscillationsand surge attenuations in the surge tank is almost the samewhile the rising speed of the water level in the riser is slowingdown obviously
Figure 7(b) shows the discharge through each PRO in the1 riser during the transients When the units accept load thewater level falls quickly in the riser but slowly in the maintank which forms negative water level difference on the twosides of PROs and causes the water to flow from the maintank into the riser As the initial water levels in both the maintank and the risers are above the elevation of the highest PROthat is the 6 PRO submerged flow occurs in the PROs 1ndash6 when the water level difference appears When the waterlevels in the risers fall below the elevation of the 6 PROthe flow pattern at the 6 PRO turns from the submergedflow to the free flow with the water level in the main tankcontinuing to fall the discharge decreases gradually whenthewater level in themain tank falls below the elevation of the6 PRO the discharge through the 6 PRO turns to 0 Similarphenomenon could be seen in other PROs when the waterlevels continue to fall In addition the flow pattern is alwaysthe submerged flow at 1 PRO because of its low elevation
When the water level reaches its lowest elevation in thesurge tank two units reject full load at the same time In
this condition the water level in the riser rises quickly andthe positive water level difference is formed in the 1 PROcausing the water to flow from the riser into the main tank inthe type of submerged flow When the water level in the riserrises to the 2 PRO the water flows from the riser to the maintank in the type of free flow with the water level in the risercontinuing to rise the discharge through the PROs increasesgradually When the water in the main tank reaches the 2PRO the free flow here turns to a submerged one with thewater level in the main tank continuing to rise the dischargedecreases gradually A similar phenomenon occurs in otherPROs subsequently When the water level in the main tank isover the 6 PRO submergedflowoccurs in every PROand thedischarge is the same because of the equal pressure differenceon the two sides of each PRO
Figure 7(c) shows the water level difference on the twosides of the breast wall of the riser Because of the PROsthe water level changes slower in the riser but quicker inthe main tank which reduces the pressure difference on thebreast wall of the riser As shown in Table 2 the maximumwater level difference between the two sides of the breast wallis reduced almost by 20m compared with the results withoutPROs on the breast wall Therefore setting appropriate PROscan effectively reduce the pressure difference between the twosides of the breast wall It should be noted that more PROsand bigger diameters are effective to reduce the pressuredifference on the breast wall but too many or too bigPROs would affect the differential effect of the surge tankthereby increasing the maximum surge and reducing thesurge attenuation in the surge tank
4 Conclusions
This paper provides a mathematical model for the differentialsurge tank with PROs and overflow weirs for transientcalculations The numerical model of hydraulic transientsis established using the data of a practical hydropower
The Scientific World Journal 9
0 10008006004002001580
1602
1624
1646
1668
1690
Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
minus10
minus5
0
5
10
15
0 1000800600400200
Discharge through 1 PRODischarge through 2 PRODischarge through 3 PRODischarge through 4 PRODischarge through 5 PRODischarge through 6 PRO
Time (s)
Disc
harg
e (m
3middotsminus
1)
(b) Discharge through PROs versus time
0 1000800600400200Time (s)
minus20
0
20
40
60
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(c) Water level difference on breast wall versus time
Figure 7 Numerical results of simultaneous load rejection after load acceptance with PROs on breast wall
station and the probable operation conditions are simulatedand analyzed The proposed mathematical model and thevalues of some coefficients used in the simulation canprovide reference for the simulation of hydraulic transientsin this type of hydropower station In a long diversion-typehydropower station with the bifurcated pipe at the bottomof the surge tank successive load rejection condition canmake the maximum pressure in the spiral case and themaximum rotating speedmore serious comparedwith simul-taneous load rejection Additionally the pressure differenceon the breast wall is significant during transients especiallyduring the combination condition that simultaneous load
rejection after load acceptance while setting appropriatePROs can reduce the pressure difference effectively Note thatthe present mathematical model and numerical applicationsneed field test verification which will be conducted in theadditional investigation
Notation
MOC Method of characteristicsODE Ordinary differential equationPDE Partial differential equation
10 The Scientific World Journal
PRO Pressure-reduction OrificeWG Wicket gates119886 Wave velocity119860 Cross-sectional area of pipe119860119878 Cross-sectional area of riser
119860119878119878 Cross-sectional area of main tank
119860TH Cross-sectional area of throttle orifice119861119872 119861119875 Known constants in compatibility equa-tions
119861119884 Width of overflow weir
119862119872 119862119875 Known constants in compatibility equa-tions
119891 Darcy-Weisbach friction factor1198661198632 Moment of inertia of unit
119867 Piezometric head119867119879 Turbine head
11989611 Discharge coefficient of free overflow
through overflow weir11989612 Discharge coefficient of submerged over-
flow through overflow weir119899 Dimensionless rotating speed119876 Discharge through pipe119876119871 Discharge through PROs
119876119878 Discharge in riser
119876119878119878 Discharge in main tank
119876119884 Discharge through overflow weirs
119879 Torque on turbine119879119888 Closing time constant from full opening to
closed119879119898 Mechanical starting time
119910 Dimensionless WG opening119885119871 Elevation of the center of PRO
119885119878 Water level in riser
119885119878119878 Water level in main tank
119885119884 Elevation of overflow weir
120573 Dimensionless torque120585 Head loss coefficient of throttle orifice1205831 Discharge coefficients of free flow of PRO
1205832 Discharge coefficients of submerged flowof
PRO
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported by the National Natural Sci-ence Foundation of China (Grant no 51379064) the Nat-ural Science Foundation of Jiangsu Province (Grant noBK20130839) the Open Research Fund Program of State KeyLaboratory ofWater Resources andHydropower EngineeringScience (Grant no 2013B116) and the Fundamental ResearchFunds for the Central Universities of China (grant no2013B06114)
References
[1] A Adamkowski ldquoCase study lapino powerplant penstockfailurerdquo Journal of Hydraulic Engineering vol 127 no 7 pp 547ndash555 2001
[2] J Yang K Zhao L Li and PWu ldquoAnalysis on the causes of units7 and 9 accidents at Sayano-Shushenskaya hydropower stationrdquoJournal of Hydroelectric Engineering vol 30 no 4 pp 226ndash2342011 (Chinese)
[3] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006
[4] O H Souza Jr N Barbieri and A H M Santos ldquoStudy ofhydraulic transients in hydropower plants through simulationof nonlinear model of penstock and hydraulic turbine modelrdquoIEEE Transactions on Power Systems vol 14 no 4 pp 1269ndash1272 1999
[5] B Selek M S Kirkgoz and Z Selek ldquoComparison of computedwater hammer pressures with test results for the Catalan powerplant in Turkeyrdquo Canadian Journal of Civil Engineering vol 31no 1 pp 78ndash85 2004
[6] M Calamak and Z Bozkus ldquoComparison of performance oftwo run-of-river plants during transient conditionsrdquo Journal ofPerformance of Constructed Facilities vol 27 no 5 pp 624ndash6322013
[7] J F An J Zhang and A Hazrati ldquoSafe control of air cushionsurge chambers in hydropower systemsrdquo Scientia Iranica vol20 no 6 pp 1605ndash1611 2013
[8] A Riasi A Nourbakhsh and M Raisee ldquoNumerical modelingfor hydraulic resonance in hydropower systems using impulseresponserdquo Journal of Hydraulic Engineering vol 136 no 11 pp929ndash934 2010
[9] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005
[10] H Ramos and A B Almeida ldquoDynamic orifice model onwaterhammer analysis of high or medium heads of smallhydropower schemesrdquo Journal of Hydraulic Research vol 39 no4 pp 429ndash436 2001
[11] T Kolsek J Duhovnik and A Bergant ldquoSimulation of unsteadyflow and runner rotation during shut-down of an axial waterturbinerdquo Journal of Hydraulic Research vol 44 no 1 pp 129ndash137 2006
[12] M H Afshar M Rohani and R Taheri ldquoSimulation oftransient flow in pipeline systems due to load rejection and loadacceptance by hydroelectric power plantsrdquo International Journalof Mechanical Sciences vol 52 no 1 pp 103ndash115 2010
[13] Q K Zhang B Karney L Suo and A F Colombo ldquoStochasticanalysis of water hammer and applications in reliability-basedstructural design for hydro turbine penstocksrdquo Journal ofHydraulic Engineering vol 137 no 11 pp 1509ndash1521 2011
[14] X X Zhang and Y G Cheng ldquoSimulation of hydraulictransients in hydropower systems using the 1-D-3-D couplingapproachrdquo Journal of Hydrodynamics vol 24 no 4 pp 595ndash604 2012
[15] S Y Wu G Wang and J Wang ldquoOptimization of the typeof upstream surge chamber in Jinping Hydropower StationrdquoSichuan Water Power no 6 pp 93ndash96 104 2008 (Chinese)
[16] S R Wang T X Liu and W L Zou ldquoThe advantage ofnew differential surge chamber and its applicationrdquo Journal ofTsinghua University vol 2 pp 73ndash84 1988 (Chinese)
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
sections 1 2 and 3 respectively 119860119904119904 1198601199041 and 119860
1199042are the
cross-sectional areas of the main tank the riser 1 and theriser 2 respectively 119860TH1 119860TH2 and 119860TH3 are the cross-sectional areas of the throttle orifices at the bottom of themain tank the riser 1 and the riser 2 respectively 120585
1 1205852
and 1205853are the head loss coefficients of the throttle orifices
respectively which have different values for the flow into orout of the tank 119876
119884 1198761198841 and 119876
1198842are the discharge through
the overflow weir between the main tank and the risers119876119884= 1198761198841
+ 1198761198842 measured positively from the risers into
the main tank 119876119871 1198761198711 and 119876
1198712are the discharge through
the PROs on the breast wall 119876119871= 1198761198711
+ 1198761198712 measured
positively from the risers into themain tank Every parameterthat has a subscript 0 is a known value of a previous timestep
Discharge through the overflow weir can be given by theequation
(119895 = 1 minus 119870) are the cross-sectionalareas of the PROs and 119870 is the number of the PROs of eachriser 119885
119871119895(119895 = 1 minus 119870) are the elevations of the center of
the PROs and 1205831and 120583
2are the discharge coefficients of free
flow and submerged flow of the PROs respectively So thetotal discharge through the PROs on each breast wall may bewritten as
1198761198711=
119899
sum
119895=1
1198761198711119895 119876
1198712=
119899
sum
119895=1
1198761198712119895 (12)
Every variable of this boundary during transient processcan be derived with (9) to (12) employing the four-orderRunge-Kutta method
3 Case Study
The model is used to predict the hydraulic transients inthe practical hydropower station in China As shown inFigure 3 every two units share a common waterway systemwhich consists of the intake the gate shaft the headracetunnel the upstream differential surge tank penstock thetailrace tunnel and so on
The headrace tunnel has a length of 170 km with adiameter 119863 = 118m The differential surge tank with a ldquo119884rdquoshaped bifurcation pipe at the bottom is located at the end ofthe headrace tunnel The length of each penstock is 5300mand the diameter is 65mThe length of each tailrace tunnel is2600m and the diameter is 120mThe head loss coefficientsof the system are collected according to the project
The cross-sectional areas of the main tank and eachriser of the differential surge tank are 3464m2 and 346m2respectivelyThe cross-sectional areas of the throttle orifice atthe bottom of the main tank and each riser are 139m2 and190m2 respectively The elevation of the bottom of the maintank is 15752m while the elevation of the overflow weir atthe top of the riser is 16700m with the width 119861
119884= 60m
The Scientific World Journal 5
0
30
60
90
120
150
0 10 20 30 40 50Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless WG openingDimensionless rotating speedPressure head in spiral case
(a) Values of turbine parameters versus time
Time (s)
minus400
minus200
0
200
400
0 200 400 600 800 1000
Discharge flow in or out of main tankDischarge flow in or out of 1 riserDischarge flow in or out of 2 riserDischarge through overflow weirsDischarge through PROs
Disc
harg
e (m
3middotsminus
1)
(b) Discharge of surge tank versus time
Time (s)
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(c) Water level in surge tank versus time
Time (s)
minus20
minus10
0
10
20
30
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(d) Water level difference on breast wall versus time
Figure 4 Numerical results of simultaneous load rejection
According to the laboratory experiments the values ofthe parameters are as follows the discharge coefficient of freeoverflow from the riser into the well 119896
11= 05 The discharge
coefficient of free overflow from the main tank into the riser1198961015840
11= 053 The discharge coefficients of submerged flow 119896
12
and 119896101584012
approximate to 80 of the discharge coefficients offree overflow The head loss coefficient of the orifice at thebottom of the main tank 120585
1= 199 for the flow into the main
tank while 1205851= 132 for the flow out of the main tank The
head loss coefficients of the orifice at the bottom of the risers1205852= 1205853= 172 for the flow into the risers while 120585
2= 1205853= 323
for the flow out of the risersThe discharge coefficients of freeflow and submerged flow of the PRO 120583
1= 06 and 120583
2= 05
respectivelyThe rated output of each Francis turbine is 610MW Its
rated head is 288m rated rotating speed is 1667 rpm andrated discharge is 2286m3s The diameter of the runner
is 656m The installation elevation of turbine is 13168mThe inertia (1198661198632) of the turbine and generator is about75800 tm2
31 Simulation of Simultaneous Load Rejection Based on themathematicalmodel and numericalmethods presented in theabove section the computermodel of the hydraulic transientsin the hydropower station is encoded in the FORTRANprogramming language A critical operation that is likelyto occur several times during the life of the project issimulated The upstream water level is at EL 16460m andthe downstream water level is at EL 13331m Full load isrejected at time 119905 = 0 and the WG are closed with the closingtime 119879
119888= 13 seconds under governor control It should be
noted that the effect of the PROs is ignored in this simulationthat is the cross-section areas of the PROs are considered as0 The calculation is shown in Figure 4
6 The Scientific World Journal
The changes of the dimensionless rotating speed of therunner the dimensionless WG opening and the pressurehead in the spiral case are shown in Figure 4(a) Whenthe units reject full load the rotating speed of the runneris increased rapidly The WG are quickly closed accordingto the emergency closure law which is set to the governorin advance to prevent extended periods of high overspeedleading to serious water hammer pressure in the spiral caseThe pressure in the spiral case increases quickly during theclosing process accompanying the phenomena of the wavepropagation and reflection After the closure of theWGmostof the water hammer wave has attenuated and the pressurein the spiral case changes slowly because of the water levelvariation in the surge tank
Figures 4(b) and 4(c) show the changes of the dischargeand the water levels of the surge tank After the closure of theWG the discharge in penstock decreases rapidly The waterin the headrace tunnel flows into the surge tank through theorifices at the bottom of the main tank and the risers sothe water levels in them increase fast As the cross-sectionalarea of the riser is smaller than the main tankrsquos the waterlevel in the riser rises more quickly When the water levelin the riser reaches the elevation of the overflow weir thewater freely spills into the main tank from the riser andthe increasing speed of the water level in the riser becomesslower When the water level in the main tank reaches theelevation of the overflow weir the free flow becomes thesubmerged flow Then the discharge that flows from theriser into the main tank decreases leading to the secondquick increase of water level in the riser in a short periodof time As the discharge into the surge tank decreases theflow direction reverses and the inflow becomes the outflowThe water levels in the main tank and the risers decreaseafter reaching the highest elevation Similarly the water levelin the riser falls more rapidly because of the small cross-sectional area When the water level in the riser is lowerthan that in the main tank the water flows from the maintank to the riser and the flow pattern changes from thesubmerged flow to the free flow with the variation of thewater levels in the main tank and the riser If the water levelin the main tank is lower than the elevation of the overflowweir the overflow stops and the water level in the riser fallsrapidly The discharge through the PROs is always equal to0 during the transient because the area is set to 0 in themodel
Figure 4(d) stands for the water level difference betweenthe two sides of the riserrsquos breast wall during the transientprocess When the load rejection happens the water levelin the riser rises quickly because of the small cross-sectionalarea while the water level in the main tank increases slowlyso the corresponding pressure differencemeasured positivelyin this situation increases quickly When the water level inthe riser reaches the elevation of the overflow weir it risesslowly because of the overflow and the pressure differencereduces gradually Then the pressure difference reverses withthe decrease of thewater level in the riserUnder this transientsimulation the positive pressure difference between the twosides of the riserrsquos breast wall is about 30m while the negativeone is about 20mThepressure difference is significant so the
breast wall in this kind of surge tankmust be of good physicalstrength
As the headrace tunnel of the hydropower station is verylong the water inertia is large so the surge tank periodis very long the amplitude of the oscillations is large andthe attenuation is very slow The transient process causedby previous condition has not disappeared the followingcondition may happen which may make the result of thetransient more serious
32 Simulation of Successive Load Rejection As the layoutof water diversion systems of hydropower stations becomesmore complex as well as the operation of grid systemscombination operating conditions become very commonsuch as load rejection after load acceptance and load rejectionone by one Although the probability of some extremeconditions is very small once they happen the consequencesare very serious Therefore it is necessary to consider thesefactors in the design of hydropower stations
With respect to the water hammer pressure in combina-tion conditions successive load rejection is studied beforeThis operation condition can make the negative pressurein the draft tube more serious than that in simultaneousload rejection in a pumped storage hydropower station [21]The tailrace tunnel is short in a diversion-type hydropowerstation so the draft tube pressure will not reduce too muchduring this condition But successive load rejection mayresult in high overspeed of the runner and larger pressure inthe spiral case in a long diversion-type hydropower stationwith the bifurcated pipe at the bottom of the surge tank thatis there are two independent penstocks for the units from thesurge tank as shown in Figure 3
The calculations are shown in Figure 5 The water levelsof the reservoirs are the same as the previous section Firstlythe 1 turbine rejects its full load and when the water level ofthe surge tank reaches the highest level the 2 turbine rejectsits full load
The dimensionlessWGopening the dimensionless rotat-ing speed of the runner and the pressure head in thespiral case during successive load rejection are shown inFigure 5(a) When the 1 turbine rejects full load the waterlevel of the surge tank begins to rise As shown in Figure 5(b)when the water level of the surge tank reaches the highestlevel it rises about 30m whichmakes the static head of the 2turbine increase by 30m Otherwise the units connect withlarge power grid under normal conditions and the speed ofthe 2 turbinewill not change due to the stable grid frequencyso the opening of theWG keeps constant as well As the statichead of the 2 turbine rises the demand discharge of turbinealso increases When the water level of the surge tank reachesthe highest level after the load rejection of the 1 turbinethe 2 turbine rejects full load Compared with simultaneousload rejection the maximum pressure in the spiral case andthe maximum rotating speed of the 2 turbine are increasedwhich are shown in Table 1Therefore if the water way systemof a hydropower station is arranged as this form especiallyas the headrace tunnel is very long successive load rejectioncondition will make the maximum pressure of the spiral caseand themaximum rotating speed of the runner more serious
The Scientific World Journal 7
0
30
60
90
120
150
0 40 80 120 160 200Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless 1 WG openingDimensionless 2 WG openingDimensionless 1 rotating speedDimensionless 2 rotating speedPressure head in 1 spiral casePressure head in 2 spiral case
(a) Values of turbine parameters versus time
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(b) Water level in surge tank versus time
Figure 5 Numerical results of successive load rejection
Table 1 Comparisons between simultaneous and successive loadrejection
33 Simulation of Simultaneous Load Rejection after LoadAcceptance Because of the small cross-sectional area thewater level in the riser rises or falls rapidly during transientprocess which creates an accelerating or decelerating headon the tunnel in a short period of time This effect reducesthe amplitude of water level oscillations in the surge tank andaccelerates the attenuation However due to rapid water levelvariations in the risers and slow variations in the main tankthe pressure difference between the two sides of the riserrsquosbreast wall is significant If the structure cannot bear thispressure difference it may cause collapse of the breast wall[22] Thus it is very important to find the critical operationconditions and to simulate the possible maximum pressuredifference on the breast wall in the design stage which arethe basis for the structure calculation of the breast wall
Simultaneous load rejection after load acceptance iscommon in the operation of hydropower stations Duringthis simulation the water levels of the reservoirs are thesame as the previous section As shown in Figure 6(a) twounits accept load one by one which results in the fall of thewater levels in the surge tank When the water levels of thesurge tank reach the lowest level two units reject full loadsimultaneously The water levels in the risers rise rapidly tothe elevation of the overflow weirs while the water level inthemain tank rises slowly and the elevation of the initial water
Table 2 Maximum pressure difference on breast wall underdifferent conditions
Operation conditions Maximum pressure difference onbreast wall
level is lower compared with the simultaneous load rejectioncase So the pressure difference on the breast wall is largerduring this combination operating condition As shown inFigure 6(b) the maximum pressure difference on the breastwall increases by 20m compared with simultaneous loadrejection
34 Control of the Pressure Difference on Breast Wall As itcan be seen from Table 2 the maximum pressure differencebetween two sides of the breast wall is close to 30m in simul-taneous load rejection while the difference can reach 50m incombination conditionsThis huge pressure difference bringsgreat challenge to the structural safety of the breast wall thushow to reduce the pressure difference between two sides of thebreast wall has become an issue to the design person
Guaranteeing adequate structural strength of the breastwall a row of the PROs can be set along height direction[23] When transient process occurs a large water leveldifference is created between the risers and the main tankin a differential surge tank The water level in the riser risesrapidly reaching the PROs then the water flows from theriser into the main tank through the orifices which slowingdown thewater level rise in the riser while speeding thewater
8 The Scientific World Journal
1580
1602
1624
1646
1668
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
Time (s)
minus20
0
20
40
60
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(b) Water level difference on breast wall
Figure 6 Numerical results of simultaneous load rejection after load acceptance
rise in the main tank Therefore the pressure difference onthe breast wall can be reduced The locations the quantityand the diameter of the PROs are fixed during this simulationfor easy comparison of the results In this case the elevationof the bottom floor is 15752m The first orifice is set at theelevation of 1585m on the breast wall with the rest PROssetting every 12m upwards and the breast wall of each riser isinstalled with 6 orifices namely 1ndash6 in turn and the diameterof each PRO is 10m The calculation condition is the sameas the former section and the numerical results are shown inFigure 7
Figure 7(a) shows the variation of the water levels inthe surge tank after setting the PROs on the breast wallCompared with the results in Figure 6(a) that no PROs areset on the breast wall the amplitude of water level oscillationsand surge attenuations in the surge tank is almost the samewhile the rising speed of the water level in the riser is slowingdown obviously
Figure 7(b) shows the discharge through each PRO in the1 riser during the transients When the units accept load thewater level falls quickly in the riser but slowly in the maintank which forms negative water level difference on the twosides of PROs and causes the water to flow from the maintank into the riser As the initial water levels in both the maintank and the risers are above the elevation of the highest PROthat is the 6 PRO submerged flow occurs in the PROs 1ndash6 when the water level difference appears When the waterlevels in the risers fall below the elevation of the 6 PROthe flow pattern at the 6 PRO turns from the submergedflow to the free flow with the water level in the main tankcontinuing to fall the discharge decreases gradually whenthewater level in themain tank falls below the elevation of the6 PRO the discharge through the 6 PRO turns to 0 Similarphenomenon could be seen in other PROs when the waterlevels continue to fall In addition the flow pattern is alwaysthe submerged flow at 1 PRO because of its low elevation
When the water level reaches its lowest elevation in thesurge tank two units reject full load at the same time In
this condition the water level in the riser rises quickly andthe positive water level difference is formed in the 1 PROcausing the water to flow from the riser into the main tank inthe type of submerged flow When the water level in the riserrises to the 2 PRO the water flows from the riser to the maintank in the type of free flow with the water level in the risercontinuing to rise the discharge through the PROs increasesgradually When the water in the main tank reaches the 2PRO the free flow here turns to a submerged one with thewater level in the main tank continuing to rise the dischargedecreases gradually A similar phenomenon occurs in otherPROs subsequently When the water level in the main tank isover the 6 PRO submergedflowoccurs in every PROand thedischarge is the same because of the equal pressure differenceon the two sides of each PRO
Figure 7(c) shows the water level difference on the twosides of the breast wall of the riser Because of the PROsthe water level changes slower in the riser but quicker inthe main tank which reduces the pressure difference on thebreast wall of the riser As shown in Table 2 the maximumwater level difference between the two sides of the breast wallis reduced almost by 20m compared with the results withoutPROs on the breast wall Therefore setting appropriate PROscan effectively reduce the pressure difference between the twosides of the breast wall It should be noted that more PROsand bigger diameters are effective to reduce the pressuredifference on the breast wall but too many or too bigPROs would affect the differential effect of the surge tankthereby increasing the maximum surge and reducing thesurge attenuation in the surge tank
4 Conclusions
This paper provides a mathematical model for the differentialsurge tank with PROs and overflow weirs for transientcalculations The numerical model of hydraulic transientsis established using the data of a practical hydropower
The Scientific World Journal 9
0 10008006004002001580
1602
1624
1646
1668
1690
Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
minus10
minus5
0
5
10
15
0 1000800600400200
Discharge through 1 PRODischarge through 2 PRODischarge through 3 PRODischarge through 4 PRODischarge through 5 PRODischarge through 6 PRO
Time (s)
Disc
harg
e (m
3middotsminus
1)
(b) Discharge through PROs versus time
0 1000800600400200Time (s)
minus20
0
20
40
60
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(c) Water level difference on breast wall versus time
Figure 7 Numerical results of simultaneous load rejection after load acceptance with PROs on breast wall
station and the probable operation conditions are simulatedand analyzed The proposed mathematical model and thevalues of some coefficients used in the simulation canprovide reference for the simulation of hydraulic transientsin this type of hydropower station In a long diversion-typehydropower station with the bifurcated pipe at the bottomof the surge tank successive load rejection condition canmake the maximum pressure in the spiral case and themaximum rotating speedmore serious comparedwith simul-taneous load rejection Additionally the pressure differenceon the breast wall is significant during transients especiallyduring the combination condition that simultaneous load
rejection after load acceptance while setting appropriatePROs can reduce the pressure difference effectively Note thatthe present mathematical model and numerical applicationsneed field test verification which will be conducted in theadditional investigation
Notation
MOC Method of characteristicsODE Ordinary differential equationPDE Partial differential equation
10 The Scientific World Journal
PRO Pressure-reduction OrificeWG Wicket gates119886 Wave velocity119860 Cross-sectional area of pipe119860119878 Cross-sectional area of riser
119860119878119878 Cross-sectional area of main tank
119860TH Cross-sectional area of throttle orifice119861119872 119861119875 Known constants in compatibility equa-tions
119861119884 Width of overflow weir
119862119872 119862119875 Known constants in compatibility equa-tions
119891 Darcy-Weisbach friction factor1198661198632 Moment of inertia of unit
119867 Piezometric head119867119879 Turbine head
11989611 Discharge coefficient of free overflow
through overflow weir11989612 Discharge coefficient of submerged over-
flow through overflow weir119899 Dimensionless rotating speed119876 Discharge through pipe119876119871 Discharge through PROs
119876119878 Discharge in riser
119876119878119878 Discharge in main tank
119876119884 Discharge through overflow weirs
119879 Torque on turbine119879119888 Closing time constant from full opening to
closed119879119898 Mechanical starting time
119910 Dimensionless WG opening119885119871 Elevation of the center of PRO
119885119878 Water level in riser
119885119878119878 Water level in main tank
119885119884 Elevation of overflow weir
120573 Dimensionless torque120585 Head loss coefficient of throttle orifice1205831 Discharge coefficients of free flow of PRO
1205832 Discharge coefficients of submerged flowof
PRO
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported by the National Natural Sci-ence Foundation of China (Grant no 51379064) the Nat-ural Science Foundation of Jiangsu Province (Grant noBK20130839) the Open Research Fund Program of State KeyLaboratory ofWater Resources andHydropower EngineeringScience (Grant no 2013B116) and the Fundamental ResearchFunds for the Central Universities of China (grant no2013B06114)
References
[1] A Adamkowski ldquoCase study lapino powerplant penstockfailurerdquo Journal of Hydraulic Engineering vol 127 no 7 pp 547ndash555 2001
[2] J Yang K Zhao L Li and PWu ldquoAnalysis on the causes of units7 and 9 accidents at Sayano-Shushenskaya hydropower stationrdquoJournal of Hydroelectric Engineering vol 30 no 4 pp 226ndash2342011 (Chinese)
[3] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006
[4] O H Souza Jr N Barbieri and A H M Santos ldquoStudy ofhydraulic transients in hydropower plants through simulationof nonlinear model of penstock and hydraulic turbine modelrdquoIEEE Transactions on Power Systems vol 14 no 4 pp 1269ndash1272 1999
[5] B Selek M S Kirkgoz and Z Selek ldquoComparison of computedwater hammer pressures with test results for the Catalan powerplant in Turkeyrdquo Canadian Journal of Civil Engineering vol 31no 1 pp 78ndash85 2004
[6] M Calamak and Z Bozkus ldquoComparison of performance oftwo run-of-river plants during transient conditionsrdquo Journal ofPerformance of Constructed Facilities vol 27 no 5 pp 624ndash6322013
[7] J F An J Zhang and A Hazrati ldquoSafe control of air cushionsurge chambers in hydropower systemsrdquo Scientia Iranica vol20 no 6 pp 1605ndash1611 2013
[8] A Riasi A Nourbakhsh and M Raisee ldquoNumerical modelingfor hydraulic resonance in hydropower systems using impulseresponserdquo Journal of Hydraulic Engineering vol 136 no 11 pp929ndash934 2010
[9] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005
[10] H Ramos and A B Almeida ldquoDynamic orifice model onwaterhammer analysis of high or medium heads of smallhydropower schemesrdquo Journal of Hydraulic Research vol 39 no4 pp 429ndash436 2001
[11] T Kolsek J Duhovnik and A Bergant ldquoSimulation of unsteadyflow and runner rotation during shut-down of an axial waterturbinerdquo Journal of Hydraulic Research vol 44 no 1 pp 129ndash137 2006
[12] M H Afshar M Rohani and R Taheri ldquoSimulation oftransient flow in pipeline systems due to load rejection and loadacceptance by hydroelectric power plantsrdquo International Journalof Mechanical Sciences vol 52 no 1 pp 103ndash115 2010
[13] Q K Zhang B Karney L Suo and A F Colombo ldquoStochasticanalysis of water hammer and applications in reliability-basedstructural design for hydro turbine penstocksrdquo Journal ofHydraulic Engineering vol 137 no 11 pp 1509ndash1521 2011
[14] X X Zhang and Y G Cheng ldquoSimulation of hydraulictransients in hydropower systems using the 1-D-3-D couplingapproachrdquo Journal of Hydrodynamics vol 24 no 4 pp 595ndash604 2012
[15] S Y Wu G Wang and J Wang ldquoOptimization of the typeof upstream surge chamber in Jinping Hydropower StationrdquoSichuan Water Power no 6 pp 93ndash96 104 2008 (Chinese)
[16] S R Wang T X Liu and W L Zou ldquoThe advantage ofnew differential surge chamber and its applicationrdquo Journal ofTsinghua University vol 2 pp 73ndash84 1988 (Chinese)
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World Journal 5
0
30
60
90
120
150
0 10 20 30 40 50Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless WG openingDimensionless rotating speedPressure head in spiral case
(a) Values of turbine parameters versus time
Time (s)
minus400
minus200
0
200
400
0 200 400 600 800 1000
Discharge flow in or out of main tankDischarge flow in or out of 1 riserDischarge flow in or out of 2 riserDischarge through overflow weirsDischarge through PROs
Disc
harg
e (m
3middotsminus
1)
(b) Discharge of surge tank versus time
Time (s)
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(c) Water level in surge tank versus time
Time (s)
minus20
minus10
0
10
20
30
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(d) Water level difference on breast wall versus time
Figure 4 Numerical results of simultaneous load rejection
According to the laboratory experiments the values ofthe parameters are as follows the discharge coefficient of freeoverflow from the riser into the well 119896
11= 05 The discharge
coefficient of free overflow from the main tank into the riser1198961015840
11= 053 The discharge coefficients of submerged flow 119896
12
and 119896101584012
approximate to 80 of the discharge coefficients offree overflow The head loss coefficient of the orifice at thebottom of the main tank 120585
1= 199 for the flow into the main
tank while 1205851= 132 for the flow out of the main tank The
head loss coefficients of the orifice at the bottom of the risers1205852= 1205853= 172 for the flow into the risers while 120585
2= 1205853= 323
for the flow out of the risersThe discharge coefficients of freeflow and submerged flow of the PRO 120583
1= 06 and 120583
2= 05
respectivelyThe rated output of each Francis turbine is 610MW Its
rated head is 288m rated rotating speed is 1667 rpm andrated discharge is 2286m3s The diameter of the runner
is 656m The installation elevation of turbine is 13168mThe inertia (1198661198632) of the turbine and generator is about75800 tm2
31 Simulation of Simultaneous Load Rejection Based on themathematicalmodel and numericalmethods presented in theabove section the computermodel of the hydraulic transientsin the hydropower station is encoded in the FORTRANprogramming language A critical operation that is likelyto occur several times during the life of the project issimulated The upstream water level is at EL 16460m andthe downstream water level is at EL 13331m Full load isrejected at time 119905 = 0 and the WG are closed with the closingtime 119879
119888= 13 seconds under governor control It should be
noted that the effect of the PROs is ignored in this simulationthat is the cross-section areas of the PROs are considered as0 The calculation is shown in Figure 4
6 The Scientific World Journal
The changes of the dimensionless rotating speed of therunner the dimensionless WG opening and the pressurehead in the spiral case are shown in Figure 4(a) Whenthe units reject full load the rotating speed of the runneris increased rapidly The WG are quickly closed accordingto the emergency closure law which is set to the governorin advance to prevent extended periods of high overspeedleading to serious water hammer pressure in the spiral caseThe pressure in the spiral case increases quickly during theclosing process accompanying the phenomena of the wavepropagation and reflection After the closure of theWGmostof the water hammer wave has attenuated and the pressurein the spiral case changes slowly because of the water levelvariation in the surge tank
Figures 4(b) and 4(c) show the changes of the dischargeand the water levels of the surge tank After the closure of theWG the discharge in penstock decreases rapidly The waterin the headrace tunnel flows into the surge tank through theorifices at the bottom of the main tank and the risers sothe water levels in them increase fast As the cross-sectionalarea of the riser is smaller than the main tankrsquos the waterlevel in the riser rises more quickly When the water levelin the riser reaches the elevation of the overflow weir thewater freely spills into the main tank from the riser andthe increasing speed of the water level in the riser becomesslower When the water level in the main tank reaches theelevation of the overflow weir the free flow becomes thesubmerged flow Then the discharge that flows from theriser into the main tank decreases leading to the secondquick increase of water level in the riser in a short periodof time As the discharge into the surge tank decreases theflow direction reverses and the inflow becomes the outflowThe water levels in the main tank and the risers decreaseafter reaching the highest elevation Similarly the water levelin the riser falls more rapidly because of the small cross-sectional area When the water level in the riser is lowerthan that in the main tank the water flows from the maintank to the riser and the flow pattern changes from thesubmerged flow to the free flow with the variation of thewater levels in the main tank and the riser If the water levelin the main tank is lower than the elevation of the overflowweir the overflow stops and the water level in the riser fallsrapidly The discharge through the PROs is always equal to0 during the transient because the area is set to 0 in themodel
Figure 4(d) stands for the water level difference betweenthe two sides of the riserrsquos breast wall during the transientprocess When the load rejection happens the water levelin the riser rises quickly because of the small cross-sectionalarea while the water level in the main tank increases slowlyso the corresponding pressure differencemeasured positivelyin this situation increases quickly When the water level inthe riser reaches the elevation of the overflow weir it risesslowly because of the overflow and the pressure differencereduces gradually Then the pressure difference reverses withthe decrease of thewater level in the riserUnder this transientsimulation the positive pressure difference between the twosides of the riserrsquos breast wall is about 30m while the negativeone is about 20mThepressure difference is significant so the
breast wall in this kind of surge tankmust be of good physicalstrength
As the headrace tunnel of the hydropower station is verylong the water inertia is large so the surge tank periodis very long the amplitude of the oscillations is large andthe attenuation is very slow The transient process causedby previous condition has not disappeared the followingcondition may happen which may make the result of thetransient more serious
32 Simulation of Successive Load Rejection As the layoutof water diversion systems of hydropower stations becomesmore complex as well as the operation of grid systemscombination operating conditions become very commonsuch as load rejection after load acceptance and load rejectionone by one Although the probability of some extremeconditions is very small once they happen the consequencesare very serious Therefore it is necessary to consider thesefactors in the design of hydropower stations
With respect to the water hammer pressure in combina-tion conditions successive load rejection is studied beforeThis operation condition can make the negative pressurein the draft tube more serious than that in simultaneousload rejection in a pumped storage hydropower station [21]The tailrace tunnel is short in a diversion-type hydropowerstation so the draft tube pressure will not reduce too muchduring this condition But successive load rejection mayresult in high overspeed of the runner and larger pressure inthe spiral case in a long diversion-type hydropower stationwith the bifurcated pipe at the bottom of the surge tank thatis there are two independent penstocks for the units from thesurge tank as shown in Figure 3
The calculations are shown in Figure 5 The water levelsof the reservoirs are the same as the previous section Firstlythe 1 turbine rejects its full load and when the water level ofthe surge tank reaches the highest level the 2 turbine rejectsits full load
The dimensionlessWGopening the dimensionless rotat-ing speed of the runner and the pressure head in thespiral case during successive load rejection are shown inFigure 5(a) When the 1 turbine rejects full load the waterlevel of the surge tank begins to rise As shown in Figure 5(b)when the water level of the surge tank reaches the highestlevel it rises about 30m whichmakes the static head of the 2turbine increase by 30m Otherwise the units connect withlarge power grid under normal conditions and the speed ofthe 2 turbinewill not change due to the stable grid frequencyso the opening of theWG keeps constant as well As the statichead of the 2 turbine rises the demand discharge of turbinealso increases When the water level of the surge tank reachesthe highest level after the load rejection of the 1 turbinethe 2 turbine rejects full load Compared with simultaneousload rejection the maximum pressure in the spiral case andthe maximum rotating speed of the 2 turbine are increasedwhich are shown in Table 1Therefore if the water way systemof a hydropower station is arranged as this form especiallyas the headrace tunnel is very long successive load rejectioncondition will make the maximum pressure of the spiral caseand themaximum rotating speed of the runner more serious
The Scientific World Journal 7
0
30
60
90
120
150
0 40 80 120 160 200Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless 1 WG openingDimensionless 2 WG openingDimensionless 1 rotating speedDimensionless 2 rotating speedPressure head in 1 spiral casePressure head in 2 spiral case
(a) Values of turbine parameters versus time
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(b) Water level in surge tank versus time
Figure 5 Numerical results of successive load rejection
Table 1 Comparisons between simultaneous and successive loadrejection
33 Simulation of Simultaneous Load Rejection after LoadAcceptance Because of the small cross-sectional area thewater level in the riser rises or falls rapidly during transientprocess which creates an accelerating or decelerating headon the tunnel in a short period of time This effect reducesthe amplitude of water level oscillations in the surge tank andaccelerates the attenuation However due to rapid water levelvariations in the risers and slow variations in the main tankthe pressure difference between the two sides of the riserrsquosbreast wall is significant If the structure cannot bear thispressure difference it may cause collapse of the breast wall[22] Thus it is very important to find the critical operationconditions and to simulate the possible maximum pressuredifference on the breast wall in the design stage which arethe basis for the structure calculation of the breast wall
Simultaneous load rejection after load acceptance iscommon in the operation of hydropower stations Duringthis simulation the water levels of the reservoirs are thesame as the previous section As shown in Figure 6(a) twounits accept load one by one which results in the fall of thewater levels in the surge tank When the water levels of thesurge tank reach the lowest level two units reject full loadsimultaneously The water levels in the risers rise rapidly tothe elevation of the overflow weirs while the water level inthemain tank rises slowly and the elevation of the initial water
Table 2 Maximum pressure difference on breast wall underdifferent conditions
Operation conditions Maximum pressure difference onbreast wall
level is lower compared with the simultaneous load rejectioncase So the pressure difference on the breast wall is largerduring this combination operating condition As shown inFigure 6(b) the maximum pressure difference on the breastwall increases by 20m compared with simultaneous loadrejection
34 Control of the Pressure Difference on Breast Wall As itcan be seen from Table 2 the maximum pressure differencebetween two sides of the breast wall is close to 30m in simul-taneous load rejection while the difference can reach 50m incombination conditionsThis huge pressure difference bringsgreat challenge to the structural safety of the breast wall thushow to reduce the pressure difference between two sides of thebreast wall has become an issue to the design person
Guaranteeing adequate structural strength of the breastwall a row of the PROs can be set along height direction[23] When transient process occurs a large water leveldifference is created between the risers and the main tankin a differential surge tank The water level in the riser risesrapidly reaching the PROs then the water flows from theriser into the main tank through the orifices which slowingdown thewater level rise in the riser while speeding thewater
8 The Scientific World Journal
1580
1602
1624
1646
1668
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
Time (s)
minus20
0
20
40
60
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(b) Water level difference on breast wall
Figure 6 Numerical results of simultaneous load rejection after load acceptance
rise in the main tank Therefore the pressure difference onthe breast wall can be reduced The locations the quantityand the diameter of the PROs are fixed during this simulationfor easy comparison of the results In this case the elevationof the bottom floor is 15752m The first orifice is set at theelevation of 1585m on the breast wall with the rest PROssetting every 12m upwards and the breast wall of each riser isinstalled with 6 orifices namely 1ndash6 in turn and the diameterof each PRO is 10m The calculation condition is the sameas the former section and the numerical results are shown inFigure 7
Figure 7(a) shows the variation of the water levels inthe surge tank after setting the PROs on the breast wallCompared with the results in Figure 6(a) that no PROs areset on the breast wall the amplitude of water level oscillationsand surge attenuations in the surge tank is almost the samewhile the rising speed of the water level in the riser is slowingdown obviously
Figure 7(b) shows the discharge through each PRO in the1 riser during the transients When the units accept load thewater level falls quickly in the riser but slowly in the maintank which forms negative water level difference on the twosides of PROs and causes the water to flow from the maintank into the riser As the initial water levels in both the maintank and the risers are above the elevation of the highest PROthat is the 6 PRO submerged flow occurs in the PROs 1ndash6 when the water level difference appears When the waterlevels in the risers fall below the elevation of the 6 PROthe flow pattern at the 6 PRO turns from the submergedflow to the free flow with the water level in the main tankcontinuing to fall the discharge decreases gradually whenthewater level in themain tank falls below the elevation of the6 PRO the discharge through the 6 PRO turns to 0 Similarphenomenon could be seen in other PROs when the waterlevels continue to fall In addition the flow pattern is alwaysthe submerged flow at 1 PRO because of its low elevation
When the water level reaches its lowest elevation in thesurge tank two units reject full load at the same time In
this condition the water level in the riser rises quickly andthe positive water level difference is formed in the 1 PROcausing the water to flow from the riser into the main tank inthe type of submerged flow When the water level in the riserrises to the 2 PRO the water flows from the riser to the maintank in the type of free flow with the water level in the risercontinuing to rise the discharge through the PROs increasesgradually When the water in the main tank reaches the 2PRO the free flow here turns to a submerged one with thewater level in the main tank continuing to rise the dischargedecreases gradually A similar phenomenon occurs in otherPROs subsequently When the water level in the main tank isover the 6 PRO submergedflowoccurs in every PROand thedischarge is the same because of the equal pressure differenceon the two sides of each PRO
Figure 7(c) shows the water level difference on the twosides of the breast wall of the riser Because of the PROsthe water level changes slower in the riser but quicker inthe main tank which reduces the pressure difference on thebreast wall of the riser As shown in Table 2 the maximumwater level difference between the two sides of the breast wallis reduced almost by 20m compared with the results withoutPROs on the breast wall Therefore setting appropriate PROscan effectively reduce the pressure difference between the twosides of the breast wall It should be noted that more PROsand bigger diameters are effective to reduce the pressuredifference on the breast wall but too many or too bigPROs would affect the differential effect of the surge tankthereby increasing the maximum surge and reducing thesurge attenuation in the surge tank
4 Conclusions
This paper provides a mathematical model for the differentialsurge tank with PROs and overflow weirs for transientcalculations The numerical model of hydraulic transientsis established using the data of a practical hydropower
The Scientific World Journal 9
0 10008006004002001580
1602
1624
1646
1668
1690
Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
minus10
minus5
0
5
10
15
0 1000800600400200
Discharge through 1 PRODischarge through 2 PRODischarge through 3 PRODischarge through 4 PRODischarge through 5 PRODischarge through 6 PRO
Time (s)
Disc
harg
e (m
3middotsminus
1)
(b) Discharge through PROs versus time
0 1000800600400200Time (s)
minus20
0
20
40
60
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(c) Water level difference on breast wall versus time
Figure 7 Numerical results of simultaneous load rejection after load acceptance with PROs on breast wall
station and the probable operation conditions are simulatedand analyzed The proposed mathematical model and thevalues of some coefficients used in the simulation canprovide reference for the simulation of hydraulic transientsin this type of hydropower station In a long diversion-typehydropower station with the bifurcated pipe at the bottomof the surge tank successive load rejection condition canmake the maximum pressure in the spiral case and themaximum rotating speedmore serious comparedwith simul-taneous load rejection Additionally the pressure differenceon the breast wall is significant during transients especiallyduring the combination condition that simultaneous load
rejection after load acceptance while setting appropriatePROs can reduce the pressure difference effectively Note thatthe present mathematical model and numerical applicationsneed field test verification which will be conducted in theadditional investigation
Notation
MOC Method of characteristicsODE Ordinary differential equationPDE Partial differential equation
10 The Scientific World Journal
PRO Pressure-reduction OrificeWG Wicket gates119886 Wave velocity119860 Cross-sectional area of pipe119860119878 Cross-sectional area of riser
119860119878119878 Cross-sectional area of main tank
119860TH Cross-sectional area of throttle orifice119861119872 119861119875 Known constants in compatibility equa-tions
119861119884 Width of overflow weir
119862119872 119862119875 Known constants in compatibility equa-tions
119891 Darcy-Weisbach friction factor1198661198632 Moment of inertia of unit
119867 Piezometric head119867119879 Turbine head
11989611 Discharge coefficient of free overflow
through overflow weir11989612 Discharge coefficient of submerged over-
flow through overflow weir119899 Dimensionless rotating speed119876 Discharge through pipe119876119871 Discharge through PROs
119876119878 Discharge in riser
119876119878119878 Discharge in main tank
119876119884 Discharge through overflow weirs
119879 Torque on turbine119879119888 Closing time constant from full opening to
closed119879119898 Mechanical starting time
119910 Dimensionless WG opening119885119871 Elevation of the center of PRO
119885119878 Water level in riser
119885119878119878 Water level in main tank
119885119884 Elevation of overflow weir
120573 Dimensionless torque120585 Head loss coefficient of throttle orifice1205831 Discharge coefficients of free flow of PRO
1205832 Discharge coefficients of submerged flowof
PRO
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported by the National Natural Sci-ence Foundation of China (Grant no 51379064) the Nat-ural Science Foundation of Jiangsu Province (Grant noBK20130839) the Open Research Fund Program of State KeyLaboratory ofWater Resources andHydropower EngineeringScience (Grant no 2013B116) and the Fundamental ResearchFunds for the Central Universities of China (grant no2013B06114)
References
[1] A Adamkowski ldquoCase study lapino powerplant penstockfailurerdquo Journal of Hydraulic Engineering vol 127 no 7 pp 547ndash555 2001
[2] J Yang K Zhao L Li and PWu ldquoAnalysis on the causes of units7 and 9 accidents at Sayano-Shushenskaya hydropower stationrdquoJournal of Hydroelectric Engineering vol 30 no 4 pp 226ndash2342011 (Chinese)
[3] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006
[4] O H Souza Jr N Barbieri and A H M Santos ldquoStudy ofhydraulic transients in hydropower plants through simulationof nonlinear model of penstock and hydraulic turbine modelrdquoIEEE Transactions on Power Systems vol 14 no 4 pp 1269ndash1272 1999
[5] B Selek M S Kirkgoz and Z Selek ldquoComparison of computedwater hammer pressures with test results for the Catalan powerplant in Turkeyrdquo Canadian Journal of Civil Engineering vol 31no 1 pp 78ndash85 2004
[6] M Calamak and Z Bozkus ldquoComparison of performance oftwo run-of-river plants during transient conditionsrdquo Journal ofPerformance of Constructed Facilities vol 27 no 5 pp 624ndash6322013
[7] J F An J Zhang and A Hazrati ldquoSafe control of air cushionsurge chambers in hydropower systemsrdquo Scientia Iranica vol20 no 6 pp 1605ndash1611 2013
[8] A Riasi A Nourbakhsh and M Raisee ldquoNumerical modelingfor hydraulic resonance in hydropower systems using impulseresponserdquo Journal of Hydraulic Engineering vol 136 no 11 pp929ndash934 2010
[9] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005
[10] H Ramos and A B Almeida ldquoDynamic orifice model onwaterhammer analysis of high or medium heads of smallhydropower schemesrdquo Journal of Hydraulic Research vol 39 no4 pp 429ndash436 2001
[11] T Kolsek J Duhovnik and A Bergant ldquoSimulation of unsteadyflow and runner rotation during shut-down of an axial waterturbinerdquo Journal of Hydraulic Research vol 44 no 1 pp 129ndash137 2006
[12] M H Afshar M Rohani and R Taheri ldquoSimulation oftransient flow in pipeline systems due to load rejection and loadacceptance by hydroelectric power plantsrdquo International Journalof Mechanical Sciences vol 52 no 1 pp 103ndash115 2010
[13] Q K Zhang B Karney L Suo and A F Colombo ldquoStochasticanalysis of water hammer and applications in reliability-basedstructural design for hydro turbine penstocksrdquo Journal ofHydraulic Engineering vol 137 no 11 pp 1509ndash1521 2011
[14] X X Zhang and Y G Cheng ldquoSimulation of hydraulictransients in hydropower systems using the 1-D-3-D couplingapproachrdquo Journal of Hydrodynamics vol 24 no 4 pp 595ndash604 2012
[15] S Y Wu G Wang and J Wang ldquoOptimization of the typeof upstream surge chamber in Jinping Hydropower StationrdquoSichuan Water Power no 6 pp 93ndash96 104 2008 (Chinese)
[16] S R Wang T X Liu and W L Zou ldquoThe advantage ofnew differential surge chamber and its applicationrdquo Journal ofTsinghua University vol 2 pp 73ndash84 1988 (Chinese)
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
6 The Scientific World Journal
The changes of the dimensionless rotating speed of therunner the dimensionless WG opening and the pressurehead in the spiral case are shown in Figure 4(a) Whenthe units reject full load the rotating speed of the runneris increased rapidly The WG are quickly closed accordingto the emergency closure law which is set to the governorin advance to prevent extended periods of high overspeedleading to serious water hammer pressure in the spiral caseThe pressure in the spiral case increases quickly during theclosing process accompanying the phenomena of the wavepropagation and reflection After the closure of theWGmostof the water hammer wave has attenuated and the pressurein the spiral case changes slowly because of the water levelvariation in the surge tank
Figures 4(b) and 4(c) show the changes of the dischargeand the water levels of the surge tank After the closure of theWG the discharge in penstock decreases rapidly The waterin the headrace tunnel flows into the surge tank through theorifices at the bottom of the main tank and the risers sothe water levels in them increase fast As the cross-sectionalarea of the riser is smaller than the main tankrsquos the waterlevel in the riser rises more quickly When the water levelin the riser reaches the elevation of the overflow weir thewater freely spills into the main tank from the riser andthe increasing speed of the water level in the riser becomesslower When the water level in the main tank reaches theelevation of the overflow weir the free flow becomes thesubmerged flow Then the discharge that flows from theriser into the main tank decreases leading to the secondquick increase of water level in the riser in a short periodof time As the discharge into the surge tank decreases theflow direction reverses and the inflow becomes the outflowThe water levels in the main tank and the risers decreaseafter reaching the highest elevation Similarly the water levelin the riser falls more rapidly because of the small cross-sectional area When the water level in the riser is lowerthan that in the main tank the water flows from the maintank to the riser and the flow pattern changes from thesubmerged flow to the free flow with the variation of thewater levels in the main tank and the riser If the water levelin the main tank is lower than the elevation of the overflowweir the overflow stops and the water level in the riser fallsrapidly The discharge through the PROs is always equal to0 during the transient because the area is set to 0 in themodel
Figure 4(d) stands for the water level difference betweenthe two sides of the riserrsquos breast wall during the transientprocess When the load rejection happens the water levelin the riser rises quickly because of the small cross-sectionalarea while the water level in the main tank increases slowlyso the corresponding pressure differencemeasured positivelyin this situation increases quickly When the water level inthe riser reaches the elevation of the overflow weir it risesslowly because of the overflow and the pressure differencereduces gradually Then the pressure difference reverses withthe decrease of thewater level in the riserUnder this transientsimulation the positive pressure difference between the twosides of the riserrsquos breast wall is about 30m while the negativeone is about 20mThepressure difference is significant so the
breast wall in this kind of surge tankmust be of good physicalstrength
As the headrace tunnel of the hydropower station is verylong the water inertia is large so the surge tank periodis very long the amplitude of the oscillations is large andthe attenuation is very slow The transient process causedby previous condition has not disappeared the followingcondition may happen which may make the result of thetransient more serious
32 Simulation of Successive Load Rejection As the layoutof water diversion systems of hydropower stations becomesmore complex as well as the operation of grid systemscombination operating conditions become very commonsuch as load rejection after load acceptance and load rejectionone by one Although the probability of some extremeconditions is very small once they happen the consequencesare very serious Therefore it is necessary to consider thesefactors in the design of hydropower stations
With respect to the water hammer pressure in combina-tion conditions successive load rejection is studied beforeThis operation condition can make the negative pressurein the draft tube more serious than that in simultaneousload rejection in a pumped storage hydropower station [21]The tailrace tunnel is short in a diversion-type hydropowerstation so the draft tube pressure will not reduce too muchduring this condition But successive load rejection mayresult in high overspeed of the runner and larger pressure inthe spiral case in a long diversion-type hydropower stationwith the bifurcated pipe at the bottom of the surge tank thatis there are two independent penstocks for the units from thesurge tank as shown in Figure 3
The calculations are shown in Figure 5 The water levelsof the reservoirs are the same as the previous section Firstlythe 1 turbine rejects its full load and when the water level ofthe surge tank reaches the highest level the 2 turbine rejectsits full load
The dimensionlessWGopening the dimensionless rotat-ing speed of the runner and the pressure head in thespiral case during successive load rejection are shown inFigure 5(a) When the 1 turbine rejects full load the waterlevel of the surge tank begins to rise As shown in Figure 5(b)when the water level of the surge tank reaches the highestlevel it rises about 30m whichmakes the static head of the 2turbine increase by 30m Otherwise the units connect withlarge power grid under normal conditions and the speed ofthe 2 turbinewill not change due to the stable grid frequencyso the opening of theWG keeps constant as well As the statichead of the 2 turbine rises the demand discharge of turbinealso increases When the water level of the surge tank reachesthe highest level after the load rejection of the 1 turbinethe 2 turbine rejects full load Compared with simultaneousload rejection the maximum pressure in the spiral case andthe maximum rotating speed of the 2 turbine are increasedwhich are shown in Table 1Therefore if the water way systemof a hydropower station is arranged as this form especiallyas the headrace tunnel is very long successive load rejectioncondition will make the maximum pressure of the spiral caseand themaximum rotating speed of the runner more serious
The Scientific World Journal 7
0
30
60
90
120
150
0 40 80 120 160 200Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless 1 WG openingDimensionless 2 WG openingDimensionless 1 rotating speedDimensionless 2 rotating speedPressure head in 1 spiral casePressure head in 2 spiral case
(a) Values of turbine parameters versus time
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(b) Water level in surge tank versus time
Figure 5 Numerical results of successive load rejection
Table 1 Comparisons between simultaneous and successive loadrejection
33 Simulation of Simultaneous Load Rejection after LoadAcceptance Because of the small cross-sectional area thewater level in the riser rises or falls rapidly during transientprocess which creates an accelerating or decelerating headon the tunnel in a short period of time This effect reducesthe amplitude of water level oscillations in the surge tank andaccelerates the attenuation However due to rapid water levelvariations in the risers and slow variations in the main tankthe pressure difference between the two sides of the riserrsquosbreast wall is significant If the structure cannot bear thispressure difference it may cause collapse of the breast wall[22] Thus it is very important to find the critical operationconditions and to simulate the possible maximum pressuredifference on the breast wall in the design stage which arethe basis for the structure calculation of the breast wall
Simultaneous load rejection after load acceptance iscommon in the operation of hydropower stations Duringthis simulation the water levels of the reservoirs are thesame as the previous section As shown in Figure 6(a) twounits accept load one by one which results in the fall of thewater levels in the surge tank When the water levels of thesurge tank reach the lowest level two units reject full loadsimultaneously The water levels in the risers rise rapidly tothe elevation of the overflow weirs while the water level inthemain tank rises slowly and the elevation of the initial water
Table 2 Maximum pressure difference on breast wall underdifferent conditions
Operation conditions Maximum pressure difference onbreast wall
level is lower compared with the simultaneous load rejectioncase So the pressure difference on the breast wall is largerduring this combination operating condition As shown inFigure 6(b) the maximum pressure difference on the breastwall increases by 20m compared with simultaneous loadrejection
34 Control of the Pressure Difference on Breast Wall As itcan be seen from Table 2 the maximum pressure differencebetween two sides of the breast wall is close to 30m in simul-taneous load rejection while the difference can reach 50m incombination conditionsThis huge pressure difference bringsgreat challenge to the structural safety of the breast wall thushow to reduce the pressure difference between two sides of thebreast wall has become an issue to the design person
Guaranteeing adequate structural strength of the breastwall a row of the PROs can be set along height direction[23] When transient process occurs a large water leveldifference is created between the risers and the main tankin a differential surge tank The water level in the riser risesrapidly reaching the PROs then the water flows from theriser into the main tank through the orifices which slowingdown thewater level rise in the riser while speeding thewater
8 The Scientific World Journal
1580
1602
1624
1646
1668
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
Time (s)
minus20
0
20
40
60
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(b) Water level difference on breast wall
Figure 6 Numerical results of simultaneous load rejection after load acceptance
rise in the main tank Therefore the pressure difference onthe breast wall can be reduced The locations the quantityand the diameter of the PROs are fixed during this simulationfor easy comparison of the results In this case the elevationof the bottom floor is 15752m The first orifice is set at theelevation of 1585m on the breast wall with the rest PROssetting every 12m upwards and the breast wall of each riser isinstalled with 6 orifices namely 1ndash6 in turn and the diameterof each PRO is 10m The calculation condition is the sameas the former section and the numerical results are shown inFigure 7
Figure 7(a) shows the variation of the water levels inthe surge tank after setting the PROs on the breast wallCompared with the results in Figure 6(a) that no PROs areset on the breast wall the amplitude of water level oscillationsand surge attenuations in the surge tank is almost the samewhile the rising speed of the water level in the riser is slowingdown obviously
Figure 7(b) shows the discharge through each PRO in the1 riser during the transients When the units accept load thewater level falls quickly in the riser but slowly in the maintank which forms negative water level difference on the twosides of PROs and causes the water to flow from the maintank into the riser As the initial water levels in both the maintank and the risers are above the elevation of the highest PROthat is the 6 PRO submerged flow occurs in the PROs 1ndash6 when the water level difference appears When the waterlevels in the risers fall below the elevation of the 6 PROthe flow pattern at the 6 PRO turns from the submergedflow to the free flow with the water level in the main tankcontinuing to fall the discharge decreases gradually whenthewater level in themain tank falls below the elevation of the6 PRO the discharge through the 6 PRO turns to 0 Similarphenomenon could be seen in other PROs when the waterlevels continue to fall In addition the flow pattern is alwaysthe submerged flow at 1 PRO because of its low elevation
When the water level reaches its lowest elevation in thesurge tank two units reject full load at the same time In
this condition the water level in the riser rises quickly andthe positive water level difference is formed in the 1 PROcausing the water to flow from the riser into the main tank inthe type of submerged flow When the water level in the riserrises to the 2 PRO the water flows from the riser to the maintank in the type of free flow with the water level in the risercontinuing to rise the discharge through the PROs increasesgradually When the water in the main tank reaches the 2PRO the free flow here turns to a submerged one with thewater level in the main tank continuing to rise the dischargedecreases gradually A similar phenomenon occurs in otherPROs subsequently When the water level in the main tank isover the 6 PRO submergedflowoccurs in every PROand thedischarge is the same because of the equal pressure differenceon the two sides of each PRO
Figure 7(c) shows the water level difference on the twosides of the breast wall of the riser Because of the PROsthe water level changes slower in the riser but quicker inthe main tank which reduces the pressure difference on thebreast wall of the riser As shown in Table 2 the maximumwater level difference between the two sides of the breast wallis reduced almost by 20m compared with the results withoutPROs on the breast wall Therefore setting appropriate PROscan effectively reduce the pressure difference between the twosides of the breast wall It should be noted that more PROsand bigger diameters are effective to reduce the pressuredifference on the breast wall but too many or too bigPROs would affect the differential effect of the surge tankthereby increasing the maximum surge and reducing thesurge attenuation in the surge tank
4 Conclusions
This paper provides a mathematical model for the differentialsurge tank with PROs and overflow weirs for transientcalculations The numerical model of hydraulic transientsis established using the data of a practical hydropower
The Scientific World Journal 9
0 10008006004002001580
1602
1624
1646
1668
1690
Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
minus10
minus5
0
5
10
15
0 1000800600400200
Discharge through 1 PRODischarge through 2 PRODischarge through 3 PRODischarge through 4 PRODischarge through 5 PRODischarge through 6 PRO
Time (s)
Disc
harg
e (m
3middotsminus
1)
(b) Discharge through PROs versus time
0 1000800600400200Time (s)
minus20
0
20
40
60
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(c) Water level difference on breast wall versus time
Figure 7 Numerical results of simultaneous load rejection after load acceptance with PROs on breast wall
station and the probable operation conditions are simulatedand analyzed The proposed mathematical model and thevalues of some coefficients used in the simulation canprovide reference for the simulation of hydraulic transientsin this type of hydropower station In a long diversion-typehydropower station with the bifurcated pipe at the bottomof the surge tank successive load rejection condition canmake the maximum pressure in the spiral case and themaximum rotating speedmore serious comparedwith simul-taneous load rejection Additionally the pressure differenceon the breast wall is significant during transients especiallyduring the combination condition that simultaneous load
rejection after load acceptance while setting appropriatePROs can reduce the pressure difference effectively Note thatthe present mathematical model and numerical applicationsneed field test verification which will be conducted in theadditional investigation
Notation
MOC Method of characteristicsODE Ordinary differential equationPDE Partial differential equation
10 The Scientific World Journal
PRO Pressure-reduction OrificeWG Wicket gates119886 Wave velocity119860 Cross-sectional area of pipe119860119878 Cross-sectional area of riser
119860119878119878 Cross-sectional area of main tank
119860TH Cross-sectional area of throttle orifice119861119872 119861119875 Known constants in compatibility equa-tions
119861119884 Width of overflow weir
119862119872 119862119875 Known constants in compatibility equa-tions
119891 Darcy-Weisbach friction factor1198661198632 Moment of inertia of unit
119867 Piezometric head119867119879 Turbine head
11989611 Discharge coefficient of free overflow
through overflow weir11989612 Discharge coefficient of submerged over-
flow through overflow weir119899 Dimensionless rotating speed119876 Discharge through pipe119876119871 Discharge through PROs
119876119878 Discharge in riser
119876119878119878 Discharge in main tank
119876119884 Discharge through overflow weirs
119879 Torque on turbine119879119888 Closing time constant from full opening to
closed119879119898 Mechanical starting time
119910 Dimensionless WG opening119885119871 Elevation of the center of PRO
119885119878 Water level in riser
119885119878119878 Water level in main tank
119885119884 Elevation of overflow weir
120573 Dimensionless torque120585 Head loss coefficient of throttle orifice1205831 Discharge coefficients of free flow of PRO
1205832 Discharge coefficients of submerged flowof
PRO
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported by the National Natural Sci-ence Foundation of China (Grant no 51379064) the Nat-ural Science Foundation of Jiangsu Province (Grant noBK20130839) the Open Research Fund Program of State KeyLaboratory ofWater Resources andHydropower EngineeringScience (Grant no 2013B116) and the Fundamental ResearchFunds for the Central Universities of China (grant no2013B06114)
References
[1] A Adamkowski ldquoCase study lapino powerplant penstockfailurerdquo Journal of Hydraulic Engineering vol 127 no 7 pp 547ndash555 2001
[2] J Yang K Zhao L Li and PWu ldquoAnalysis on the causes of units7 and 9 accidents at Sayano-Shushenskaya hydropower stationrdquoJournal of Hydroelectric Engineering vol 30 no 4 pp 226ndash2342011 (Chinese)
[3] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006
[4] O H Souza Jr N Barbieri and A H M Santos ldquoStudy ofhydraulic transients in hydropower plants through simulationof nonlinear model of penstock and hydraulic turbine modelrdquoIEEE Transactions on Power Systems vol 14 no 4 pp 1269ndash1272 1999
[5] B Selek M S Kirkgoz and Z Selek ldquoComparison of computedwater hammer pressures with test results for the Catalan powerplant in Turkeyrdquo Canadian Journal of Civil Engineering vol 31no 1 pp 78ndash85 2004
[6] M Calamak and Z Bozkus ldquoComparison of performance oftwo run-of-river plants during transient conditionsrdquo Journal ofPerformance of Constructed Facilities vol 27 no 5 pp 624ndash6322013
[7] J F An J Zhang and A Hazrati ldquoSafe control of air cushionsurge chambers in hydropower systemsrdquo Scientia Iranica vol20 no 6 pp 1605ndash1611 2013
[8] A Riasi A Nourbakhsh and M Raisee ldquoNumerical modelingfor hydraulic resonance in hydropower systems using impulseresponserdquo Journal of Hydraulic Engineering vol 136 no 11 pp929ndash934 2010
[9] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005
[10] H Ramos and A B Almeida ldquoDynamic orifice model onwaterhammer analysis of high or medium heads of smallhydropower schemesrdquo Journal of Hydraulic Research vol 39 no4 pp 429ndash436 2001
[11] T Kolsek J Duhovnik and A Bergant ldquoSimulation of unsteadyflow and runner rotation during shut-down of an axial waterturbinerdquo Journal of Hydraulic Research vol 44 no 1 pp 129ndash137 2006
[12] M H Afshar M Rohani and R Taheri ldquoSimulation oftransient flow in pipeline systems due to load rejection and loadacceptance by hydroelectric power plantsrdquo International Journalof Mechanical Sciences vol 52 no 1 pp 103ndash115 2010
[13] Q K Zhang B Karney L Suo and A F Colombo ldquoStochasticanalysis of water hammer and applications in reliability-basedstructural design for hydro turbine penstocksrdquo Journal ofHydraulic Engineering vol 137 no 11 pp 1509ndash1521 2011
[14] X X Zhang and Y G Cheng ldquoSimulation of hydraulictransients in hydropower systems using the 1-D-3-D couplingapproachrdquo Journal of Hydrodynamics vol 24 no 4 pp 595ndash604 2012
[15] S Y Wu G Wang and J Wang ldquoOptimization of the typeof upstream surge chamber in Jinping Hydropower StationrdquoSichuan Water Power no 6 pp 93ndash96 104 2008 (Chinese)
[16] S R Wang T X Liu and W L Zou ldquoThe advantage ofnew differential surge chamber and its applicationrdquo Journal ofTsinghua University vol 2 pp 73ndash84 1988 (Chinese)
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World Journal 7
0
30
60
90
120
150
0 40 80 120 160 200Time (s)
Dim
ensio
nles
s WG
ope
ning
dim
ensio
nles
s rot
atin
g sp
eed
()
200
250
300
350
400
Pres
sure
hea
d (m
)
Dimensionless 1 WG openingDimensionless 2 WG openingDimensionless 1 rotating speedDimensionless 2 rotating speedPressure head in 1 spiral casePressure head in 2 spiral case
(a) Values of turbine parameters versus time
1600
1618
1636
1654
1672
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(b) Water level in surge tank versus time
Figure 5 Numerical results of successive load rejection
Table 1 Comparisons between simultaneous and successive loadrejection
33 Simulation of Simultaneous Load Rejection after LoadAcceptance Because of the small cross-sectional area thewater level in the riser rises or falls rapidly during transientprocess which creates an accelerating or decelerating headon the tunnel in a short period of time This effect reducesthe amplitude of water level oscillations in the surge tank andaccelerates the attenuation However due to rapid water levelvariations in the risers and slow variations in the main tankthe pressure difference between the two sides of the riserrsquosbreast wall is significant If the structure cannot bear thispressure difference it may cause collapse of the breast wall[22] Thus it is very important to find the critical operationconditions and to simulate the possible maximum pressuredifference on the breast wall in the design stage which arethe basis for the structure calculation of the breast wall
Simultaneous load rejection after load acceptance iscommon in the operation of hydropower stations Duringthis simulation the water levels of the reservoirs are thesame as the previous section As shown in Figure 6(a) twounits accept load one by one which results in the fall of thewater levels in the surge tank When the water levels of thesurge tank reach the lowest level two units reject full loadsimultaneously The water levels in the risers rise rapidly tothe elevation of the overflow weirs while the water level inthemain tank rises slowly and the elevation of the initial water
Table 2 Maximum pressure difference on breast wall underdifferent conditions
Operation conditions Maximum pressure difference onbreast wall
level is lower compared with the simultaneous load rejectioncase So the pressure difference on the breast wall is largerduring this combination operating condition As shown inFigure 6(b) the maximum pressure difference on the breastwall increases by 20m compared with simultaneous loadrejection
34 Control of the Pressure Difference on Breast Wall As itcan be seen from Table 2 the maximum pressure differencebetween two sides of the breast wall is close to 30m in simul-taneous load rejection while the difference can reach 50m incombination conditionsThis huge pressure difference bringsgreat challenge to the structural safety of the breast wall thushow to reduce the pressure difference between two sides of thebreast wall has become an issue to the design person
Guaranteeing adequate structural strength of the breastwall a row of the PROs can be set along height direction[23] When transient process occurs a large water leveldifference is created between the risers and the main tankin a differential surge tank The water level in the riser risesrapidly reaching the PROs then the water flows from theriser into the main tank through the orifices which slowingdown thewater level rise in the riser while speeding thewater
8 The Scientific World Journal
1580
1602
1624
1646
1668
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
Time (s)
minus20
0
20
40
60
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(b) Water level difference on breast wall
Figure 6 Numerical results of simultaneous load rejection after load acceptance
rise in the main tank Therefore the pressure difference onthe breast wall can be reduced The locations the quantityand the diameter of the PROs are fixed during this simulationfor easy comparison of the results In this case the elevationof the bottom floor is 15752m The first orifice is set at theelevation of 1585m on the breast wall with the rest PROssetting every 12m upwards and the breast wall of each riser isinstalled with 6 orifices namely 1ndash6 in turn and the diameterof each PRO is 10m The calculation condition is the sameas the former section and the numerical results are shown inFigure 7
Figure 7(a) shows the variation of the water levels inthe surge tank after setting the PROs on the breast wallCompared with the results in Figure 6(a) that no PROs areset on the breast wall the amplitude of water level oscillationsand surge attenuations in the surge tank is almost the samewhile the rising speed of the water level in the riser is slowingdown obviously
Figure 7(b) shows the discharge through each PRO in the1 riser during the transients When the units accept load thewater level falls quickly in the riser but slowly in the maintank which forms negative water level difference on the twosides of PROs and causes the water to flow from the maintank into the riser As the initial water levels in both the maintank and the risers are above the elevation of the highest PROthat is the 6 PRO submerged flow occurs in the PROs 1ndash6 when the water level difference appears When the waterlevels in the risers fall below the elevation of the 6 PROthe flow pattern at the 6 PRO turns from the submergedflow to the free flow with the water level in the main tankcontinuing to fall the discharge decreases gradually whenthewater level in themain tank falls below the elevation of the6 PRO the discharge through the 6 PRO turns to 0 Similarphenomenon could be seen in other PROs when the waterlevels continue to fall In addition the flow pattern is alwaysthe submerged flow at 1 PRO because of its low elevation
When the water level reaches its lowest elevation in thesurge tank two units reject full load at the same time In
this condition the water level in the riser rises quickly andthe positive water level difference is formed in the 1 PROcausing the water to flow from the riser into the main tank inthe type of submerged flow When the water level in the riserrises to the 2 PRO the water flows from the riser to the maintank in the type of free flow with the water level in the risercontinuing to rise the discharge through the PROs increasesgradually When the water in the main tank reaches the 2PRO the free flow here turns to a submerged one with thewater level in the main tank continuing to rise the dischargedecreases gradually A similar phenomenon occurs in otherPROs subsequently When the water level in the main tank isover the 6 PRO submergedflowoccurs in every PROand thedischarge is the same because of the equal pressure differenceon the two sides of each PRO
Figure 7(c) shows the water level difference on the twosides of the breast wall of the riser Because of the PROsthe water level changes slower in the riser but quicker inthe main tank which reduces the pressure difference on thebreast wall of the riser As shown in Table 2 the maximumwater level difference between the two sides of the breast wallis reduced almost by 20m compared with the results withoutPROs on the breast wall Therefore setting appropriate PROscan effectively reduce the pressure difference between the twosides of the breast wall It should be noted that more PROsand bigger diameters are effective to reduce the pressuredifference on the breast wall but too many or too bigPROs would affect the differential effect of the surge tankthereby increasing the maximum surge and reducing thesurge attenuation in the surge tank
4 Conclusions
This paper provides a mathematical model for the differentialsurge tank with PROs and overflow weirs for transientcalculations The numerical model of hydraulic transientsis established using the data of a practical hydropower
The Scientific World Journal 9
0 10008006004002001580
1602
1624
1646
1668
1690
Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
minus10
minus5
0
5
10
15
0 1000800600400200
Discharge through 1 PRODischarge through 2 PRODischarge through 3 PRODischarge through 4 PRODischarge through 5 PRODischarge through 6 PRO
Time (s)
Disc
harg
e (m
3middotsminus
1)
(b) Discharge through PROs versus time
0 1000800600400200Time (s)
minus20
0
20
40
60
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(c) Water level difference on breast wall versus time
Figure 7 Numerical results of simultaneous load rejection after load acceptance with PROs on breast wall
station and the probable operation conditions are simulatedand analyzed The proposed mathematical model and thevalues of some coefficients used in the simulation canprovide reference for the simulation of hydraulic transientsin this type of hydropower station In a long diversion-typehydropower station with the bifurcated pipe at the bottomof the surge tank successive load rejection condition canmake the maximum pressure in the spiral case and themaximum rotating speedmore serious comparedwith simul-taneous load rejection Additionally the pressure differenceon the breast wall is significant during transients especiallyduring the combination condition that simultaneous load
rejection after load acceptance while setting appropriatePROs can reduce the pressure difference effectively Note thatthe present mathematical model and numerical applicationsneed field test verification which will be conducted in theadditional investigation
Notation
MOC Method of characteristicsODE Ordinary differential equationPDE Partial differential equation
10 The Scientific World Journal
PRO Pressure-reduction OrificeWG Wicket gates119886 Wave velocity119860 Cross-sectional area of pipe119860119878 Cross-sectional area of riser
119860119878119878 Cross-sectional area of main tank
119860TH Cross-sectional area of throttle orifice119861119872 119861119875 Known constants in compatibility equa-tions
119861119884 Width of overflow weir
119862119872 119862119875 Known constants in compatibility equa-tions
119891 Darcy-Weisbach friction factor1198661198632 Moment of inertia of unit
119867 Piezometric head119867119879 Turbine head
11989611 Discharge coefficient of free overflow
through overflow weir11989612 Discharge coefficient of submerged over-
flow through overflow weir119899 Dimensionless rotating speed119876 Discharge through pipe119876119871 Discharge through PROs
119876119878 Discharge in riser
119876119878119878 Discharge in main tank
119876119884 Discharge through overflow weirs
119879 Torque on turbine119879119888 Closing time constant from full opening to
closed119879119898 Mechanical starting time
119910 Dimensionless WG opening119885119871 Elevation of the center of PRO
119885119878 Water level in riser
119885119878119878 Water level in main tank
119885119884 Elevation of overflow weir
120573 Dimensionless torque120585 Head loss coefficient of throttle orifice1205831 Discharge coefficients of free flow of PRO
1205832 Discharge coefficients of submerged flowof
PRO
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported by the National Natural Sci-ence Foundation of China (Grant no 51379064) the Nat-ural Science Foundation of Jiangsu Province (Grant noBK20130839) the Open Research Fund Program of State KeyLaboratory ofWater Resources andHydropower EngineeringScience (Grant no 2013B116) and the Fundamental ResearchFunds for the Central Universities of China (grant no2013B06114)
References
[1] A Adamkowski ldquoCase study lapino powerplant penstockfailurerdquo Journal of Hydraulic Engineering vol 127 no 7 pp 547ndash555 2001
[2] J Yang K Zhao L Li and PWu ldquoAnalysis on the causes of units7 and 9 accidents at Sayano-Shushenskaya hydropower stationrdquoJournal of Hydroelectric Engineering vol 30 no 4 pp 226ndash2342011 (Chinese)
[3] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006
[4] O H Souza Jr N Barbieri and A H M Santos ldquoStudy ofhydraulic transients in hydropower plants through simulationof nonlinear model of penstock and hydraulic turbine modelrdquoIEEE Transactions on Power Systems vol 14 no 4 pp 1269ndash1272 1999
[5] B Selek M S Kirkgoz and Z Selek ldquoComparison of computedwater hammer pressures with test results for the Catalan powerplant in Turkeyrdquo Canadian Journal of Civil Engineering vol 31no 1 pp 78ndash85 2004
[6] M Calamak and Z Bozkus ldquoComparison of performance oftwo run-of-river plants during transient conditionsrdquo Journal ofPerformance of Constructed Facilities vol 27 no 5 pp 624ndash6322013
[7] J F An J Zhang and A Hazrati ldquoSafe control of air cushionsurge chambers in hydropower systemsrdquo Scientia Iranica vol20 no 6 pp 1605ndash1611 2013
[8] A Riasi A Nourbakhsh and M Raisee ldquoNumerical modelingfor hydraulic resonance in hydropower systems using impulseresponserdquo Journal of Hydraulic Engineering vol 136 no 11 pp929ndash934 2010
[9] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005
[10] H Ramos and A B Almeida ldquoDynamic orifice model onwaterhammer analysis of high or medium heads of smallhydropower schemesrdquo Journal of Hydraulic Research vol 39 no4 pp 429ndash436 2001
[11] T Kolsek J Duhovnik and A Bergant ldquoSimulation of unsteadyflow and runner rotation during shut-down of an axial waterturbinerdquo Journal of Hydraulic Research vol 44 no 1 pp 129ndash137 2006
[12] M H Afshar M Rohani and R Taheri ldquoSimulation oftransient flow in pipeline systems due to load rejection and loadacceptance by hydroelectric power plantsrdquo International Journalof Mechanical Sciences vol 52 no 1 pp 103ndash115 2010
[13] Q K Zhang B Karney L Suo and A F Colombo ldquoStochasticanalysis of water hammer and applications in reliability-basedstructural design for hydro turbine penstocksrdquo Journal ofHydraulic Engineering vol 137 no 11 pp 1509ndash1521 2011
[14] X X Zhang and Y G Cheng ldquoSimulation of hydraulictransients in hydropower systems using the 1-D-3-D couplingapproachrdquo Journal of Hydrodynamics vol 24 no 4 pp 595ndash604 2012
[15] S Y Wu G Wang and J Wang ldquoOptimization of the typeof upstream surge chamber in Jinping Hydropower StationrdquoSichuan Water Power no 6 pp 93ndash96 104 2008 (Chinese)
[16] S R Wang T X Liu and W L Zou ldquoThe advantage ofnew differential surge chamber and its applicationrdquo Journal ofTsinghua University vol 2 pp 73ndash84 1988 (Chinese)
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
8 The Scientific World Journal
1580
1602
1624
1646
1668
1690
0 200 400 600 800 1000Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
Time (s)
minus20
0
20
40
60
0 200 400 600 800 1000
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(b) Water level difference on breast wall
Figure 6 Numerical results of simultaneous load rejection after load acceptance
rise in the main tank Therefore the pressure difference onthe breast wall can be reduced The locations the quantityand the diameter of the PROs are fixed during this simulationfor easy comparison of the results In this case the elevationof the bottom floor is 15752m The first orifice is set at theelevation of 1585m on the breast wall with the rest PROssetting every 12m upwards and the breast wall of each riser isinstalled with 6 orifices namely 1ndash6 in turn and the diameterof each PRO is 10m The calculation condition is the sameas the former section and the numerical results are shown inFigure 7
Figure 7(a) shows the variation of the water levels inthe surge tank after setting the PROs on the breast wallCompared with the results in Figure 6(a) that no PROs areset on the breast wall the amplitude of water level oscillationsand surge attenuations in the surge tank is almost the samewhile the rising speed of the water level in the riser is slowingdown obviously
Figure 7(b) shows the discharge through each PRO in the1 riser during the transients When the units accept load thewater level falls quickly in the riser but slowly in the maintank which forms negative water level difference on the twosides of PROs and causes the water to flow from the maintank into the riser As the initial water levels in both the maintank and the risers are above the elevation of the highest PROthat is the 6 PRO submerged flow occurs in the PROs 1ndash6 when the water level difference appears When the waterlevels in the risers fall below the elevation of the 6 PROthe flow pattern at the 6 PRO turns from the submergedflow to the free flow with the water level in the main tankcontinuing to fall the discharge decreases gradually whenthewater level in themain tank falls below the elevation of the6 PRO the discharge through the 6 PRO turns to 0 Similarphenomenon could be seen in other PROs when the waterlevels continue to fall In addition the flow pattern is alwaysthe submerged flow at 1 PRO because of its low elevation
When the water level reaches its lowest elevation in thesurge tank two units reject full load at the same time In
this condition the water level in the riser rises quickly andthe positive water level difference is formed in the 1 PROcausing the water to flow from the riser into the main tank inthe type of submerged flow When the water level in the riserrises to the 2 PRO the water flows from the riser to the maintank in the type of free flow with the water level in the risercontinuing to rise the discharge through the PROs increasesgradually When the water in the main tank reaches the 2PRO the free flow here turns to a submerged one with thewater level in the main tank continuing to rise the dischargedecreases gradually A similar phenomenon occurs in otherPROs subsequently When the water level in the main tank isover the 6 PRO submergedflowoccurs in every PROand thedischarge is the same because of the equal pressure differenceon the two sides of each PRO
Figure 7(c) shows the water level difference on the twosides of the breast wall of the riser Because of the PROsthe water level changes slower in the riser but quicker inthe main tank which reduces the pressure difference on thebreast wall of the riser As shown in Table 2 the maximumwater level difference between the two sides of the breast wallis reduced almost by 20m compared with the results withoutPROs on the breast wall Therefore setting appropriate PROscan effectively reduce the pressure difference between the twosides of the breast wall It should be noted that more PROsand bigger diameters are effective to reduce the pressuredifference on the breast wall but too many or too bigPROs would affect the differential effect of the surge tankthereby increasing the maximum surge and reducing thesurge attenuation in the surge tank
4 Conclusions
This paper provides a mathematical model for the differentialsurge tank with PROs and overflow weirs for transientcalculations The numerical model of hydraulic transientsis established using the data of a practical hydropower
The Scientific World Journal 9
0 10008006004002001580
1602
1624
1646
1668
1690
Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
minus10
minus5
0
5
10
15
0 1000800600400200
Discharge through 1 PRODischarge through 2 PRODischarge through 3 PRODischarge through 4 PRODischarge through 5 PRODischarge through 6 PRO
Time (s)
Disc
harg
e (m
3middotsminus
1)
(b) Discharge through PROs versus time
0 1000800600400200Time (s)
minus20
0
20
40
60
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(c) Water level difference on breast wall versus time
Figure 7 Numerical results of simultaneous load rejection after load acceptance with PROs on breast wall
station and the probable operation conditions are simulatedand analyzed The proposed mathematical model and thevalues of some coefficients used in the simulation canprovide reference for the simulation of hydraulic transientsin this type of hydropower station In a long diversion-typehydropower station with the bifurcated pipe at the bottomof the surge tank successive load rejection condition canmake the maximum pressure in the spiral case and themaximum rotating speedmore serious comparedwith simul-taneous load rejection Additionally the pressure differenceon the breast wall is significant during transients especiallyduring the combination condition that simultaneous load
rejection after load acceptance while setting appropriatePROs can reduce the pressure difference effectively Note thatthe present mathematical model and numerical applicationsneed field test verification which will be conducted in theadditional investigation
Notation
MOC Method of characteristicsODE Ordinary differential equationPDE Partial differential equation
10 The Scientific World Journal
PRO Pressure-reduction OrificeWG Wicket gates119886 Wave velocity119860 Cross-sectional area of pipe119860119878 Cross-sectional area of riser
119860119878119878 Cross-sectional area of main tank
119860TH Cross-sectional area of throttle orifice119861119872 119861119875 Known constants in compatibility equa-tions
119861119884 Width of overflow weir
119862119872 119862119875 Known constants in compatibility equa-tions
119891 Darcy-Weisbach friction factor1198661198632 Moment of inertia of unit
119867 Piezometric head119867119879 Turbine head
11989611 Discharge coefficient of free overflow
through overflow weir11989612 Discharge coefficient of submerged over-
flow through overflow weir119899 Dimensionless rotating speed119876 Discharge through pipe119876119871 Discharge through PROs
119876119878 Discharge in riser
119876119878119878 Discharge in main tank
119876119884 Discharge through overflow weirs
119879 Torque on turbine119879119888 Closing time constant from full opening to
closed119879119898 Mechanical starting time
119910 Dimensionless WG opening119885119871 Elevation of the center of PRO
119885119878 Water level in riser
119885119878119878 Water level in main tank
119885119884 Elevation of overflow weir
120573 Dimensionless torque120585 Head loss coefficient of throttle orifice1205831 Discharge coefficients of free flow of PRO
1205832 Discharge coefficients of submerged flowof
PRO
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported by the National Natural Sci-ence Foundation of China (Grant no 51379064) the Nat-ural Science Foundation of Jiangsu Province (Grant noBK20130839) the Open Research Fund Program of State KeyLaboratory ofWater Resources andHydropower EngineeringScience (Grant no 2013B116) and the Fundamental ResearchFunds for the Central Universities of China (grant no2013B06114)
References
[1] A Adamkowski ldquoCase study lapino powerplant penstockfailurerdquo Journal of Hydraulic Engineering vol 127 no 7 pp 547ndash555 2001
[2] J Yang K Zhao L Li and PWu ldquoAnalysis on the causes of units7 and 9 accidents at Sayano-Shushenskaya hydropower stationrdquoJournal of Hydroelectric Engineering vol 30 no 4 pp 226ndash2342011 (Chinese)
[3] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006
[4] O H Souza Jr N Barbieri and A H M Santos ldquoStudy ofhydraulic transients in hydropower plants through simulationof nonlinear model of penstock and hydraulic turbine modelrdquoIEEE Transactions on Power Systems vol 14 no 4 pp 1269ndash1272 1999
[5] B Selek M S Kirkgoz and Z Selek ldquoComparison of computedwater hammer pressures with test results for the Catalan powerplant in Turkeyrdquo Canadian Journal of Civil Engineering vol 31no 1 pp 78ndash85 2004
[6] M Calamak and Z Bozkus ldquoComparison of performance oftwo run-of-river plants during transient conditionsrdquo Journal ofPerformance of Constructed Facilities vol 27 no 5 pp 624ndash6322013
[7] J F An J Zhang and A Hazrati ldquoSafe control of air cushionsurge chambers in hydropower systemsrdquo Scientia Iranica vol20 no 6 pp 1605ndash1611 2013
[8] A Riasi A Nourbakhsh and M Raisee ldquoNumerical modelingfor hydraulic resonance in hydropower systems using impulseresponserdquo Journal of Hydraulic Engineering vol 136 no 11 pp929ndash934 2010
[9] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005
[10] H Ramos and A B Almeida ldquoDynamic orifice model onwaterhammer analysis of high or medium heads of smallhydropower schemesrdquo Journal of Hydraulic Research vol 39 no4 pp 429ndash436 2001
[11] T Kolsek J Duhovnik and A Bergant ldquoSimulation of unsteadyflow and runner rotation during shut-down of an axial waterturbinerdquo Journal of Hydraulic Research vol 44 no 1 pp 129ndash137 2006
[12] M H Afshar M Rohani and R Taheri ldquoSimulation oftransient flow in pipeline systems due to load rejection and loadacceptance by hydroelectric power plantsrdquo International Journalof Mechanical Sciences vol 52 no 1 pp 103ndash115 2010
[13] Q K Zhang B Karney L Suo and A F Colombo ldquoStochasticanalysis of water hammer and applications in reliability-basedstructural design for hydro turbine penstocksrdquo Journal ofHydraulic Engineering vol 137 no 11 pp 1509ndash1521 2011
[14] X X Zhang and Y G Cheng ldquoSimulation of hydraulictransients in hydropower systems using the 1-D-3-D couplingapproachrdquo Journal of Hydrodynamics vol 24 no 4 pp 595ndash604 2012
[15] S Y Wu G Wang and J Wang ldquoOptimization of the typeof upstream surge chamber in Jinping Hydropower StationrdquoSichuan Water Power no 6 pp 93ndash96 104 2008 (Chinese)
[16] S R Wang T X Liu and W L Zou ldquoThe advantage ofnew differential surge chamber and its applicationrdquo Journal ofTsinghua University vol 2 pp 73ndash84 1988 (Chinese)
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World Journal 9
0 10008006004002001580
1602
1624
1646
1668
1690
Time (s)
Wat
er le
vel i
n su
rge t
ank
(m)
Water level in main tankWater level in 1 riserWater level in 2 riser
(a) Water level in surge tank versus time
minus10
minus5
0
5
10
15
0 1000800600400200
Discharge through 1 PRODischarge through 2 PRODischarge through 3 PRODischarge through 4 PRODischarge through 5 PRODischarge through 6 PRO
Time (s)
Disc
harg
e (m
3middotsminus
1)
(b) Discharge through PROs versus time
0 1000800600400200Time (s)
minus20
0
20
40
60
Wat
er le
vel d
iffer
ence
(m)
Water level difference on 1 breast wallWater level difference on 2 breast wall
(c) Water level difference on breast wall versus time
Figure 7 Numerical results of simultaneous load rejection after load acceptance with PROs on breast wall
station and the probable operation conditions are simulatedand analyzed The proposed mathematical model and thevalues of some coefficients used in the simulation canprovide reference for the simulation of hydraulic transientsin this type of hydropower station In a long diversion-typehydropower station with the bifurcated pipe at the bottomof the surge tank successive load rejection condition canmake the maximum pressure in the spiral case and themaximum rotating speedmore serious comparedwith simul-taneous load rejection Additionally the pressure differenceon the breast wall is significant during transients especiallyduring the combination condition that simultaneous load
rejection after load acceptance while setting appropriatePROs can reduce the pressure difference effectively Note thatthe present mathematical model and numerical applicationsneed field test verification which will be conducted in theadditional investigation
Notation
MOC Method of characteristicsODE Ordinary differential equationPDE Partial differential equation
10 The Scientific World Journal
PRO Pressure-reduction OrificeWG Wicket gates119886 Wave velocity119860 Cross-sectional area of pipe119860119878 Cross-sectional area of riser
119860119878119878 Cross-sectional area of main tank
119860TH Cross-sectional area of throttle orifice119861119872 119861119875 Known constants in compatibility equa-tions
119861119884 Width of overflow weir
119862119872 119862119875 Known constants in compatibility equa-tions
119891 Darcy-Weisbach friction factor1198661198632 Moment of inertia of unit
119867 Piezometric head119867119879 Turbine head
11989611 Discharge coefficient of free overflow
through overflow weir11989612 Discharge coefficient of submerged over-
flow through overflow weir119899 Dimensionless rotating speed119876 Discharge through pipe119876119871 Discharge through PROs
119876119878 Discharge in riser
119876119878119878 Discharge in main tank
119876119884 Discharge through overflow weirs
119879 Torque on turbine119879119888 Closing time constant from full opening to
closed119879119898 Mechanical starting time
119910 Dimensionless WG opening119885119871 Elevation of the center of PRO
119885119878 Water level in riser
119885119878119878 Water level in main tank
119885119884 Elevation of overflow weir
120573 Dimensionless torque120585 Head loss coefficient of throttle orifice1205831 Discharge coefficients of free flow of PRO
1205832 Discharge coefficients of submerged flowof
PRO
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported by the National Natural Sci-ence Foundation of China (Grant no 51379064) the Nat-ural Science Foundation of Jiangsu Province (Grant noBK20130839) the Open Research Fund Program of State KeyLaboratory ofWater Resources andHydropower EngineeringScience (Grant no 2013B116) and the Fundamental ResearchFunds for the Central Universities of China (grant no2013B06114)
References
[1] A Adamkowski ldquoCase study lapino powerplant penstockfailurerdquo Journal of Hydraulic Engineering vol 127 no 7 pp 547ndash555 2001
[2] J Yang K Zhao L Li and PWu ldquoAnalysis on the causes of units7 and 9 accidents at Sayano-Shushenskaya hydropower stationrdquoJournal of Hydroelectric Engineering vol 30 no 4 pp 226ndash2342011 (Chinese)
[3] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006
[4] O H Souza Jr N Barbieri and A H M Santos ldquoStudy ofhydraulic transients in hydropower plants through simulationof nonlinear model of penstock and hydraulic turbine modelrdquoIEEE Transactions on Power Systems vol 14 no 4 pp 1269ndash1272 1999
[5] B Selek M S Kirkgoz and Z Selek ldquoComparison of computedwater hammer pressures with test results for the Catalan powerplant in Turkeyrdquo Canadian Journal of Civil Engineering vol 31no 1 pp 78ndash85 2004
[6] M Calamak and Z Bozkus ldquoComparison of performance oftwo run-of-river plants during transient conditionsrdquo Journal ofPerformance of Constructed Facilities vol 27 no 5 pp 624ndash6322013
[7] J F An J Zhang and A Hazrati ldquoSafe control of air cushionsurge chambers in hydropower systemsrdquo Scientia Iranica vol20 no 6 pp 1605ndash1611 2013
[8] A Riasi A Nourbakhsh and M Raisee ldquoNumerical modelingfor hydraulic resonance in hydropower systems using impulseresponserdquo Journal of Hydraulic Engineering vol 136 no 11 pp929ndash934 2010
[9] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005
[10] H Ramos and A B Almeida ldquoDynamic orifice model onwaterhammer analysis of high or medium heads of smallhydropower schemesrdquo Journal of Hydraulic Research vol 39 no4 pp 429ndash436 2001
[11] T Kolsek J Duhovnik and A Bergant ldquoSimulation of unsteadyflow and runner rotation during shut-down of an axial waterturbinerdquo Journal of Hydraulic Research vol 44 no 1 pp 129ndash137 2006
[12] M H Afshar M Rohani and R Taheri ldquoSimulation oftransient flow in pipeline systems due to load rejection and loadacceptance by hydroelectric power plantsrdquo International Journalof Mechanical Sciences vol 52 no 1 pp 103ndash115 2010
[13] Q K Zhang B Karney L Suo and A F Colombo ldquoStochasticanalysis of water hammer and applications in reliability-basedstructural design for hydro turbine penstocksrdquo Journal ofHydraulic Engineering vol 137 no 11 pp 1509ndash1521 2011
[14] X X Zhang and Y G Cheng ldquoSimulation of hydraulictransients in hydropower systems using the 1-D-3-D couplingapproachrdquo Journal of Hydrodynamics vol 24 no 4 pp 595ndash604 2012
[15] S Y Wu G Wang and J Wang ldquoOptimization of the typeof upstream surge chamber in Jinping Hydropower StationrdquoSichuan Water Power no 6 pp 93ndash96 104 2008 (Chinese)
[16] S R Wang T X Liu and W L Zou ldquoThe advantage ofnew differential surge chamber and its applicationrdquo Journal ofTsinghua University vol 2 pp 73ndash84 1988 (Chinese)
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
10 The Scientific World Journal
PRO Pressure-reduction OrificeWG Wicket gates119886 Wave velocity119860 Cross-sectional area of pipe119860119878 Cross-sectional area of riser
119860119878119878 Cross-sectional area of main tank
119860TH Cross-sectional area of throttle orifice119861119872 119861119875 Known constants in compatibility equa-tions
119861119884 Width of overflow weir
119862119872 119862119875 Known constants in compatibility equa-tions
119891 Darcy-Weisbach friction factor1198661198632 Moment of inertia of unit
119867 Piezometric head119867119879 Turbine head
11989611 Discharge coefficient of free overflow
through overflow weir11989612 Discharge coefficient of submerged over-
flow through overflow weir119899 Dimensionless rotating speed119876 Discharge through pipe119876119871 Discharge through PROs
119876119878 Discharge in riser
119876119878119878 Discharge in main tank
119876119884 Discharge through overflow weirs
119879 Torque on turbine119879119888 Closing time constant from full opening to
closed119879119898 Mechanical starting time
119910 Dimensionless WG opening119885119871 Elevation of the center of PRO
119885119878 Water level in riser
119885119878119878 Water level in main tank
119885119884 Elevation of overflow weir
120573 Dimensionless torque120585 Head loss coefficient of throttle orifice1205831 Discharge coefficients of free flow of PRO
1205832 Discharge coefficients of submerged flowof
PRO
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
This paper was supported by the National Natural Sci-ence Foundation of China (Grant no 51379064) the Nat-ural Science Foundation of Jiangsu Province (Grant noBK20130839) the Open Research Fund Program of State KeyLaboratory ofWater Resources andHydropower EngineeringScience (Grant no 2013B116) and the Fundamental ResearchFunds for the Central Universities of China (grant no2013B06114)
References
[1] A Adamkowski ldquoCase study lapino powerplant penstockfailurerdquo Journal of Hydraulic Engineering vol 127 no 7 pp 547ndash555 2001
[2] J Yang K Zhao L Li and PWu ldquoAnalysis on the causes of units7 and 9 accidents at Sayano-Shushenskaya hydropower stationrdquoJournal of Hydroelectric Engineering vol 30 no 4 pp 226ndash2342011 (Chinese)
[3] A Bergant A R Simpson and A S Tijsseling ldquoWater hammerwith column separation a historical reviewrdquo Journal of Fluidsand Structures vol 22 no 2 pp 135ndash171 2006
[4] O H Souza Jr N Barbieri and A H M Santos ldquoStudy ofhydraulic transients in hydropower plants through simulationof nonlinear model of penstock and hydraulic turbine modelrdquoIEEE Transactions on Power Systems vol 14 no 4 pp 1269ndash1272 1999
[5] B Selek M S Kirkgoz and Z Selek ldquoComparison of computedwater hammer pressures with test results for the Catalan powerplant in Turkeyrdquo Canadian Journal of Civil Engineering vol 31no 1 pp 78ndash85 2004
[6] M Calamak and Z Bozkus ldquoComparison of performance oftwo run-of-river plants during transient conditionsrdquo Journal ofPerformance of Constructed Facilities vol 27 no 5 pp 624ndash6322013
[7] J F An J Zhang and A Hazrati ldquoSafe control of air cushionsurge chambers in hydropower systemsrdquo Scientia Iranica vol20 no 6 pp 1605ndash1611 2013
[8] A Riasi A Nourbakhsh and M Raisee ldquoNumerical modelingfor hydraulic resonance in hydropower systems using impulseresponserdquo Journal of Hydraulic Engineering vol 136 no 11 pp929ndash934 2010
[9] M S Ghidaoui M Zhao D A McInnis and D H AxworthyldquoA review of water hammer theory and practicerdquo AppliedMechanics Reviews vol 58 no 1ndash6 pp 49ndash75 2005
[10] H Ramos and A B Almeida ldquoDynamic orifice model onwaterhammer analysis of high or medium heads of smallhydropower schemesrdquo Journal of Hydraulic Research vol 39 no4 pp 429ndash436 2001
[11] T Kolsek J Duhovnik and A Bergant ldquoSimulation of unsteadyflow and runner rotation during shut-down of an axial waterturbinerdquo Journal of Hydraulic Research vol 44 no 1 pp 129ndash137 2006
[12] M H Afshar M Rohani and R Taheri ldquoSimulation oftransient flow in pipeline systems due to load rejection and loadacceptance by hydroelectric power plantsrdquo International Journalof Mechanical Sciences vol 52 no 1 pp 103ndash115 2010
[13] Q K Zhang B Karney L Suo and A F Colombo ldquoStochasticanalysis of water hammer and applications in reliability-basedstructural design for hydro turbine penstocksrdquo Journal ofHydraulic Engineering vol 137 no 11 pp 1509ndash1521 2011
[14] X X Zhang and Y G Cheng ldquoSimulation of hydraulictransients in hydropower systems using the 1-D-3-D couplingapproachrdquo Journal of Hydrodynamics vol 24 no 4 pp 595ndash604 2012
[15] S Y Wu G Wang and J Wang ldquoOptimization of the typeof upstream surge chamber in Jinping Hydropower StationrdquoSichuan Water Power no 6 pp 93ndash96 104 2008 (Chinese)
[16] S R Wang T X Liu and W L Zou ldquoThe advantage ofnew differential surge chamber and its applicationrdquo Journal ofTsinghua University vol 2 pp 73ndash84 1988 (Chinese)
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
The Scientific World Journal 11
[17] X D Yu J Zhang and A Hazrati ldquoCritical superpositioninstant of surge waves in surge tank with long headrace tunnelrdquoCanadian Journal of Civil Engineering vol 38 no 3 pp 331ndash3372011
[18] E B Wylie V L Streeter and L S Suo Fluid Transients inSystems Prentice-Hall Englewood Cliffs NJ USA 1993
[19] M H Chaudhry Applied Hydraulic Transients Springer NewYork NY USA 2013
[20] J ZhangDWang J Hu J Zhou and J Fang ldquoStudy on field testand simulating calculation following load rejections of tongbaipumped storage power stationrdquo in Proceedings of the ASMEFluids Engineering Division Summer Conference vol 2 pp 349ndash354 August 2008
[21] J Zhang W Lu B Fan and J Hu ldquoThe influence of layout ofwater conveyance system on the hydraulic transients of pump-turbines load successive rejection in pumped storage stationrdquoJournal of Hydroelectric Engineering vol 27 no 5 pp 158ndash1622008 (Chinese)
[22] N S Wang D Q Zheng and Y C Fan ldquoStudy on the differ-ential surge tank in a power plant with a long approach tunnelunder the most unfavouiable operation conditionrdquo Journal ofHydraulic Engineering no 6 pp 23ndash29 1995 (Chinese)
[23] J X Zhou andDY Liu ldquoDifferential pressure and surge analysisof differential surge tank with interconnecting holesrdquo Journalof Hohai University (Natural Sciences) no 5 pp 587ndash591 2007(Chinese)
