research article effect mechanism of penstock on stability

Research ArticleEffect Mechanism of Penstock on Stability and RegulationQuality of Turbine Regulating System

Wencheng Guo Jiandong Yang Jieping Chen and Yi Teng

State Key Laboratory of Water Resources and Hydropower Engineering Science Wuhan University Wuhan 430072 China

Correspondence should be addressed to Wencheng Guo wench24126com

Received 17 July 2014 Accepted 20 October 2014 Published 19 November 2014

Academic Editor Hongbin Zhang

Copyright copy 2014 Wencheng Guo et alThis 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

This paper studies the effect mechanism of water inertia and head loss of penstock on stability and regulation quality of turbineregulating system with surge tank or not and proposes the construction method of equivalent model of regulating system Firstlythe complete linear mathematical model of regulating system is established Then the free oscillation equation and time responseof the frequency that describe stability and regulation quality respectively are obtained Finally the effects of penstock are analysedby using stability region and response curves The results indicate that the stability and regulation quality of system without surgetank are determined by time response of frequency which only depends on water hammer wave in penstock while for system withsurge tank the time response of frequency depending on water hammer wave in penstock and water-level fluctuation in surge tankjointly determines the stability and regulation quality Water inertia of penstock mainly affects the stability and time response offrequency of system without surge tank as well as the stability and head wave of time response of frequency with surge tank Headloss of penstock mainly affects the stability and tail wave of time response of frequency with surge tank

1 Introduction

Turbine regulating system is the core component of load fre-quency control (LFC) of hydropower system When hydro-electric power plant (HPP) operates under isolated modeor becomes isolated from the grid the regulating systemshould maintain adequate stability margins as well as certainregulation quality Stability and regulation quality are twosides which are the unity of opposites of regulating systemand are influenced by hydraulic mechanical and electricalfactors [1] Pipeline network is the foundation of hydraulicfactors As a key link of pipeline network penstock hassignificant and unique effects on stability and regulationquality Hence a thorough and detailed understanding ofeffects of penstock is necessary for proper control of stabilityand regulation quality of turbine regulating system

For the subject of stability and regulation quality previousresearch centres upon governor Many authors studied theworking performance of temporary droop type governor [2ndash5] and proportional-integral-derivative (PID) type governor[6ndash10] The adjustment of governor parameters was also

researched [11] As for penstock of HPP without surge tankRuud [12] investigated the instability of a hydraulic turbinewith a very long penstockMurty andHariharan [13] analysedthe influence of water column elasticity on the stabilitylimits of hydroturbine generating unit with long penstockand proposed a modified water column compensator toenhance the stability regions and dynamic performanceSouza and Barbieri [14] discussed hydraulic transients inhydropower plants based on the nonlinearmodel of penstockand hydraulic turbine model Sanathanan [15] proposed amethod for obtaining accurate low order model for hydraulicturbine penstock Krivehenko et al [16] studied some specialconditions of unit operation in hydropower plant with longpenstocks If a HPP has surge tank its penstock is usuallyneglected to simplify the mathematical model of turbineregulating system [17 18]

It can be found from the above summary that the previousresearch on the effects of penstock on stability and regulationquality forms two major limitations Firstly the researchobject is mainly the HPP which has long penstock and doesnot have surge tank However the most important factors

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 349086 13 pageshttpdxdoiorg1011552014349086

2 Mathematical Problems in Engineering

Penstock

Draft tube

Generating unitReservoir

Qt Lt ft ht0 Twt

(a) Pipeline and power generating system (withoutsurge tank)

Z

Penstock

Headrace tunnel

Draft tube


Qt Lt ft ht0 Twt

Qy Ly fy hy0 Twy

Surge tank TF

(b) Pipeline and power generating system (with surge tank)

Headrace tunnelSurge tank Turbine

generatorGridload

Feedback element

Measurement element

Point elementAmplification elementCorrection element

Actuator

Turbine control system

+minus

minus

Controlled system

Penstock

(c) Turbine regulating system

Figure 1 Turbine regulating system of isolated HPP with surge tank or not

of penstock are water inertia and head loss The effects ofthese two factors are not investigated and compared deeplySecondly there is only little research onHPPwith surge tankIt is well recognized that surge tank is indeed an importantmeasure of pressure reduction Since the influence of water-level fluctuation in surge tank the dynamic response ofregulating system with surge tank is significantly differentfrom the case without surge tank

This paper aims to overcome the above two limitationsand thoroughly study the effect mechanism of water inertiaand head loss of penstock on stability and regulation qualityof turbine regulating system with surge tank or not It isassumed that the system operates on an isolated load andthe water column is rigid This paper is organized as followsIn Section 2 the complete linear mathematical model ofturbine regulating system that includes all subsystems (ieheadrace tunnel surge tank penstock turbine generator andgovernor) is established and the overall transfer functionsof systems without surge tank and with surge tank arederived from the complete mathematical model under stepload disturbance In Sections 3 and 4 based on the freeoscillation equation and time response of the frequency ofsystem derived from overall transfer function the effectsof water inertia and head loss of penstock on stability andregulation quality are analysed by using stability regionand response curves In Section 5 the effect mechanism ofpenstock is epurated and summarized Then according tothis effect mechanism the improvement methods of stabilityand regulation quality and constructionmethod of equivalentmodel of regulating system are proposed

2 Mathematical Model

Theturbine regulating systemof isolatedHPPwith surge tankor not is illustrated in Figure 1

21 Basic Equations The HPP without surge tank can beregarded as a special case of HPP with surge tank when thelength of headrace tunnel and the sectional area of surge tankare both 0 Hence in this section the complete mathematicalmodel of turbine regulating systemof isolatedHPPwith surgetank is first established and then the model of that withoutsurge tank can be obtained as a special case

211 Turbine Regulating System of Isolated HPP withSurge Tank

(1) Controlled System [19 20] Momentum equation of head-race tunnel

119911 = 119879119908119910

119889119902119910

119889119905

+

2ℎ1199100

1198670

119902119910 (1)

Continuity equation of surge tank

119902119910= 119902119905minus 119879119865

119889119911

119889119905

(2)

Momentum equation of penstock

ℎ = minus119879119908119905

119889119902119905

119889119905

minus

2ℎ1199050

1198670

119902119905minus 119911 (3)

Mathematical Problems in Engineering 3

1++minus

++

+

+minus

minus++

+

Kp

Kis

eqx

eqy

eqh

ey

Gs(s)

ex

eh

Mg(s)

eg

X(s)1Tas

(a) Overall block diagram of turbine regulating system

+minus

++ minus

minus minus

Twts

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)

(b) Block diagram of pipeline system (with surgetank)

minus

minus

2ht0H0

Qt(s)Twts

H(s)

(c) Block diagram of pipeline sys-tem (without surge tank)

Figure 2 Block diagram of turbine regulating system

Moment equation and discharge equation of turbine

119898119905= 119890ℎℎ + 119890119909119909 + 119890119910119910

119902119905= 119890119902ℎℎ + 119890119902119909119909 + 119890119902119910119910

(4)

First derivative differential equation of generator

119879119886

119889119909

119889119905

= 119898119905minus (119898119892+ 119890119892119909) (5)

(2) Turbine Control System [19 20] Equation of governor

119889119910

119889119905

= minus119870119901

119889119909

119889119905

minus 119870119894119909 (6)

The nomenclatures in (1)ndash(6) are presented inAppendix A

212 Turbine Regulating System of Isolated HPPwithout SurgeTank Delete (1) and (2) and reformulate (3) to the followingform

ℎ = minus119879119908119905

119889119902119905

119889119905

minus

2ℎ1199050

1198670

119902119905 (7)

Then (7) (4)ndash(6) are the complete mathematical model ofturbine regulating system of isolatedHPPwithout surge tankNote that this model (see (7) (4)ndash(6)) can as well be obtainedfrom (1)ndash(6) in the conditions of 119879

119908119910= 0 ℎ1199100

= 0 and 119879119865= 0

22 Overall Transfer Function For the situation of loaddisturbance the block diagram of turbine regulating system

is determined by the basic equations in Section 21 and shownin Figure 2 where119866

119904(119904) = 119867(119904)119876

119905(119904) is the transfer function

of pipeline system and can be derived from the Laplacetransforms of (1)ndash(3) forHPPwith surge tank and (7) forHPPwithout surge tank 119904 is complex variable

According to Figures 2(a) and 2(b) and the Laplacetransforms of (1)ndash(6) the following overall transfer functionof turbine regulating system of isolated HPP with surge tankis obtained

119866 (119904) =

119883 (119904)

119872119892(119904)

= minus

119904 (11988701199043

+ 11988711199042

+ 1198872119904 + 1198873) 119870119894

11988601199045+ 11988611199044+ 11988621199043+ 11988631199042+ 1198864119904 + 1198865

(8)

where 119872119892(119904) and 119883(119904) are the Laplace transforms of load

disturbance 119898119892

and time response of the frequency xrespectively and the former is input signal and the lateris output signal The expressions of coefficients in (8) arepresented in Appendix B

By proceeding in a similar manner the overall transferfunction of turbine regulating systemof isolatedHPPwithoutsurge tank is derived from Figures 2(a) and 2(c) and theLaplace transforms of (7) (4)ndash(6) are as follows

119866 (119904) =

119883 (119904)

119872119892(119904)

= minus

119904 (1198872119904 + 1198873) 119870119894

11988621199043+ 11988631199042+ 1198864119904 + 1198865

(9)

Note that (9) can also be obtained from (8) by letting119879119908119910

= 0 ℎ1199100

= 0 and 119879119865= 0 The expressions of coefficients

in (9) are the special cases of those in (8) when 119879119908119910 ℎ1199100 and

119879119865are both 0


Instability region

Stability region

Stability boundary

Z

X

YKpK

i(s

)

1Kp

(a) Stability region in the plane of 1119870119901and

119870119901119870119894

x

X

Y

Z

t (s)

(b) Types of free oscillations corresponding to different regions

Figure 3 Stability region of turbine regulating system

3 Effect of Penstock on Stability

Stability reflects the performance of free oscillation ofdynamic system that restores to a new equilibrium state afterinput disturbance vanishes The free oscillation is dividedinto three types damped oscillation persistent oscillationand divergent oscillation (shown in Figure 3(b)) On the basisof the definition of Lyapunov on stability [21] the first twotypes of oscillation are stable and the third one is unstableHowever the stable oscillation is only restricted to dampedoscillation in practical projects This paper uses the latterdefinition

31 Free Oscillation Equation and Stability Criterion The sta-bility of regulating system is described by free oscillationequation and discriminated by stability criterion

311 Free Oscillation Equation The following third orderand fifth order linear homogeneous differential equationsobtained from (9) and (8) are the free oscillation equations ofturbine regulating system without surge tank and with surgetank respectively

1198862

1198893

119909

1198891199053+ 1198863

1198892

119909

1198891199052+ 1198864

119889119909

119889119905

+ 1198865= 0 (10)

1198860

1198895

119909

1198891199055+ 1198861

1198894

119909

1198891199054+ 1198862

1198893

119909

1198891199053+ 1198863

1198892

119909

1198891199052+ 1198864

119889119909

119889119905

+ 1198865= 0 (11)

312 Stability Criterion By applying Routh-Hurwitz crite-rion [21] the stability criterions of turbine regulating systemrepresented by (10) and (11) are listed in Table 1

When the coefficients in (10) satisfy the discriminantsΔ1gt 0 and Δ

2gt 0 simultaneously the system without surge

tank is stable Similarly the system with surge tank is stablein the conditions of Δ1015840

1gt 0 Δ1015840

2gt 0 and Δ1015840

4gt 0

32 Stability Analysis The stability region is the region thatsatisfies stability criterion of regulating system In this paper

Table 1 Stability Criterion

System without surge tank(10) System with surge tank (11)

Δ1= 119886119894gt 0 (119894 = 2 3 4 5) Δ

1015840

1= 119886119894gt 0 (119894 = 0 1 2 3 4 5)

Δ2= 11988631198864minus 11988621198865gt 0 Δ

1015840

2= 11988611198862minus 11988601198863gt 0

Δ1015840

4= (11988611198862minus 11988601198863)(11988631198864minus 11988621198865)

minus (11988611198864minus 11988601198865)2

gt 0

the abscissa and ordinate of coordinate plane are selected as1119870119901and119870

119901119870119894 respectively and the stability region is illus-

trated in Figure 3(a)The corresponding relation between theregions in coordinate plane and the types of free oscillationsis shown in Figure 3(b)

This paper takes HPP A as example (basic informationis shown in Table 3 of Appendix C) to analyse the effectmechanism of water inertia and head loss of penstock onstability of turbine regulating system with surge tank ornot In order to make sure that the results have universalsignificance and can be applied to any hydroelectric systemthe variation ranges of 119879

119908119905and ℎ1199050are selected as 0sim4 s (4 s is

the limit value of119879119908119905) and 0sim10119867

119903 respectively In addition

the sensitivity analysis of net head is carried out under largeamplitude of variation (067119867

119903sim133119867

119903) so that the effects of

different operating conditions can be revealedAiming at two cases ofHPPA (with surge tank of real case

and without surge tank of assumed case) the investigationof the effects of 119879

119908119905 ℎ1199050 and 119867

0on stability is proceeded

by controlling variable method The default values of 119879119908119905

ℎ1199050 and 119867

0are 20s 40m (44119867

119903) and 90m (100119867

119903)

respectively The values of other parameters are as follows119890ℎ= 15 119890

119909= minus1 119890

119910= 1 119890

119902ℎ= 05 119890

119902119909= 0 119890

119902119910= 1 119879

119886=

834 s 119890119892= 0 and 119899 = 09 in which 119899 = 119865119865

119905ℎis amplification

coefficient of sectional area of surge tank and 119865119905ℎis critical

stable sectional area


00 02 04 06 08 100

5

10

15

20

25

30

00 02 04 06 08 100

5

10

15

20

25

30

025 030 0353

4

5

00 02 04 06 08 100

5

10

15

20

25

30

025 030 0353

4

5

KpK

i(s

)KpK

i(s

)KpK

i(s

)

1Kp

1Kp

1Kp

Twt = 10 sTwt = 20 s


ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m

(a1) Twt

(a2) ht0

(a3) H0

(a) System without surge tank

30

00 02 04 06 08 100

5

10

15

20

25

00 02 04 06 08 100

5

10

15

20

25

30

S

00 02 04 06 08 100

5

10

15

20

25

30

KpK

i(s

)KpK

i(s

)KpK

i(s

)

1Kp

1Kp

1Kp

Twt = 00 sTwt = 10 sTwt = 20 s


ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m

(b1) Twt

(b2) ht0

(b3) H0

(b) System with surge tank

Figure 4 Effects of penstock and net head on stability regions


The stability regions of turbine regulating system withsurge tank or not are shown in Figure 4

Figure 4 shows the following

(1) For the turbine regulating system without surge tank119879119908119905

has significant effect on stability while the effectsof ℎ1199050and119867

0are relatively small When 119879

119908119905increases

from 1 s to 4 s the stability region reduces obviouslythat is the stability of system notably worsens Withthe rise of ℎ

1199050from 00m to 80m (89119867

119903) the

stability region enlarges slightly On the contrary thestability region diminishes slightly if 119867

0increases

from 60m (067119867119903) to 120m (133119867

119903)

(2) For the turbine regulating system with surge tank119879119908119905 ℎ1199050 and 119867

0all have bigger effects on stability

especially ℎ1199050 The stability region enlarges with

the decrease of 119879119908119905

and increase of 1198670 When ℎ

1199050

decreases from 80m (89119867119903) to 20m (22119867

119903) the

stability region enlarges dramatically It is importantto note that there is an intersection point between thestability boundary curves of ℎ

1199050= 00m and ℎ

1199050=

20m (Point 119878 in Figure 4(b2)) In the right sideof Point 119878 the stability region diminishes with theincrease of ℎ

1199050while the change law is just opposite

to the left side of Point 119878(3) By comparing the turbine regulating system without

surge tank and that with surge tank the followingresults can be obtained The effect laws of 119879

119908119905on

the stability of these two systems are consistent andthe difference is that the influence on system withoutsurge tank is more sensitive than that with surgetank while the effect laws of ℎ

1199050as well as 119867

0on

these two systems are contrary and the influence onsystem with surge tank is far more sensitive than thatwithout surge tank When 119879

119908119905 ℎ1199050 and 119867

0change

the variation amplitude of stability boundary curvesin the domain of small 1119870

119901is much greater than that

of big 1119870119901for system without surge tank however

the variation amplitudes of stability boundary curvesin these two domains are close for system with surgetank

4 Effect of Penstock on Regulation Quality

Regulation quality reflects the rapidity and stationarity ofdynamic response of regulating system The common useddynamic performance indexes that evaluate regulation qual-ity are peak time settling time overshoot and number ofoscillation Regulation quality depends on its own oscilla-tion characteristic of dynamic response [22] The dynamicresponse of turbine regulating system is represented by thetime response of the frequency of hydroelectric generatingunit Hence the regulation quality is determined by oscilla-tion characteristic of time response of the frequency in timedomain

41 Time Response of the Frequency The input signal for astep load disturbance can be computed from119872

119892(119904) = 119898

1198920119904

in which1198981198920

is relative value of the load step Substitution of119872119892(119904) = 119898

1198920119904 into (9) and (8) yields the following output

signals of time response of the frequency for system withoutsurge tank and system with surge tank respectively

119883(119904) = minus

sum3

119894=21198871198941199043minus119894

sum5

119894=21198861198941199045minus119894

1198981198920

119870119894

(12)

119883(119904) = minus

sum3

119894=01198871198941199043minus119894

sum5

119894=01198861198941199045minus119894

1198981198920

119870119894

(13)

42 Regulation Quality Analysis By proceeding in a similarmanner with stability analysis in Section 32 HPP A is alsotaken as an example to analyse the effect mechanism of waterinertia and head loss of penstock on regulation quality ofturbine regulating system with surge tank or not For the caseof 10 load step reduction when the unit operates at ratedpower output that is 119898

1198920= minus01 the time responses of the

frequency of the two turbine regulating systems are shown inFigure 5 in which 119870

119901and 119870

119894are 20 and 01 sminus1 respectively

and other parameters are the same as those in Section 32Note that the formula of the period of water-level fluctu-

ation in surge tank in frictional ldquoheadrace tunnel surge tankrdquosystem is shown in Appendix D

Figure 5 and Table 2 show the following

(1) Under step change in load there are obvious differ-ences of time responses of the frequency betweensystem without surge tank and that with surge tankThe time response of the frequency of system withoutsurge tank is a single property oscillation which iscaused by water hammer wave in penstock and thisresponse has the characteristics of short period largeamplitude and fast attenuation The time response ofthe frequency of systemwith surge tank is superposedby twooscillations of different properties In these twooscillations the one in the beginning time intervalof time response of the frequency is called headwave and the other in the follow-up time interval iscalled tail wave (shown in Figure 5(b1)) of which theformer has the same property with the oscillation ofsystem without surge tank and the latter belongs tolow frequency forced oscillation caused bywater-levelfluctuation in surge tank The period of tail wave isconsistent with that of water-level fluctuation in surgetank Tail wave has the characteristics of long periodsmall amplitude and slow attenuation and it is themain body of time response of the frequency andthe principal factor that determines the regulationquality

(2) For the turbine regulating system without surge tanklike the effects on stability 119879

119908119905has significant effect

on time response of the frequency while the effectsof ℎ1199050

and 1198670are relatively small When 119879

119908119905rises

the maximum amplitude overshoot and numberof oscillation enlarge dramatically and regulationquality worsen obviously The maximum amplitudeand overshoot increase with the rising of ℎ

1199050and


Table 2 Characteristic parameters for time responses of the frequency of turbine regulating system with surge tank or not under1198981198920= minus01

Types of fluctuationSystem without surge tank System with surge tank Water-level fluctuation

in surge tankHead wave Tail waveMaximum Maximum Amplitude Attenuation rate Period (s) Period (s)

119879119908119905

(s)0 00318 00140 00007 32387 313911 00337 00344 00141 00007 32221 313912 00416 00419 00142 00007 32221 313913 00529 00534 00145 00006 32221 313914 00666 00670 00145 00006 32221 31391

ℎ1199050(m)0 00395 00398 00122 00010 31733 310722 00404 00409 00132 00008 31894 312214 00416 00419 00142 00007 32221 313916 00427 00429 00156 00006 32555 315878 00439 00443 00172 00004 32896 31815

1198670(m)60 00427 00426 00227 00005 35699 3421190 00416 00419 00142 00007 32221 31391120 00410 00415 00105 00013 28690 30500

Table 3 Basic information of actual examples of HPP

HPP Rated power output119873119903(MW) Rated head119867

119903(m) Rated discharge 119876

119903(m3s) 119879

119908119910(s) 119879

119908119905(s) ℎ

1199100(m) ℎ

1199050(m)

A 5128 9000 6270 1775 233 757 553B 11856 17700 7250 3973 182 2053 512C 61000 28800 22860 2384 126 1292 291

regulation quality worsens as a consequence If 1198670

increases the maximum amplitude and overshootdecrease and then regulation quality is improved

(3) For the turbine regulating systemwith surge tank theeffect laws of 119879

119908119905 ℎ1199050 and 119867

0on head wave are the

samewith those on the time response of the frequencyof system without surge tank 119879

119908119905has almost no

influence on tail wave while the influences of ℎ1199050and

1198670are significantWith the increase of119879

119908119905 the period

of tail wave reduces and the maximum amplitudeand attenuation rate of tail wave enlarge As a resultregulation quality will get better or worse When ℎ

1199050

rises regulation quality notably worsens because ofthe increase of period and maximum amplitude andthe decrease of attenuation rate In contrast with ℎ

1199050

the period and the maximum amplitude reduce andattenuation rate enlarges with the rising of119867

0 and the

regulation quality notably gets betterIn the high head HPP pressure fluctuation in penstock

and limited speed of guide vane movement are importantfor the stable and secure operation at the changes of poweror frequency especially at load rejections and emergencyshut-down functions Figure 6 gives the time responses of theguide vane opening 119910 and net head ℎ of turbine regulating

system with surge tank or not under1198981198920= minus01 correspond-

ing to time response of the frequency 119909Figure 6 shows the following

(1) The change laws of guide vane opening response andnet head response of system without surge tank arethe same with those of system with surge tank Whenthe frequency increases (or decreases) the guide vaneopening decreases (or increases) to reduce (or rise)the discharge and output power and then the net headincreases (or decreases) because of the reduction (orrising) of discharge

(2) The amplitude of variation of guide vane openingresponse is larger than that of net head responseand they are larger than that of time response of thefrequency especially in the system with surge tankThis result indicates that guide vane opening responseand net head response are more sensitive than timeresponse of the frequency to load disturbance

(3) The stability of these three responses is the samewhile their regulation qualities are of significant dif-ferences


0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

t (s)

t (s)

t (s)



(a1) Twt

(a2) ht0

(a3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m


0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006008

Tail wave

Head wave

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80100000002004006

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006

t (s)

t (s)

t (s)



(b1) Twt

(b2) ht0

(b3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m


Figure 5 Effects of penstock and net head on time responses of the frequency

5 Effect Mechanism of Water Inertia andHead Loss of Penstock and Its Applications

51 Effect Mechanism of Water Inertia and Head Loss ofPenstock Based on the analyses in Sections 3 and 4 theeffect mechanism of penstock is epurated and summarized asfollowsThe stability and regulation quality of systemwithoutsurge tank are determined by the dynamic response (egtime response of the frequency) which only depends onwater

hammer wave in penstock However for system with surgetank the dynamic response depending on water hammerwave in penstock and water-level fluctuation in surge tankjointly determines the stability and regulation quality Specificto the effects of water inertia and head loss of penstock

Water inertia of penstock is the principal aspect thatinfluences water hammer wave Hence the stability and timeresponse of the frequency of system without surge tank aswell as the stability and head wave of system with surge


0 10 20 30 40 50

minus03

minus02

minus01

00

01

02Re

spon

ses

t (s)

Frequency x

Guide vane opening y

Net head h


0 200 400 600 800 1000

minus03

minus02

minus01

00

01

02

Resp

onse

s

Frequency x


Net head h


Figure 6 Time responses of the guide vane opening and net head

tank are significantly impacted by thewater inertiaHoweverwater hammer wave which is low frequency fluctuationhas little influence on water-level fluctuation in surge tankTherefore there is almost no effect of the water inertia on thetail wave of system with surge tank

Head loss of penstock is the damping of turbine regulatingsystem and influences the water-level fluctuation charac-teristic in surge tank mainly by impacting the water flowmovement and energy consumption of ldquosurge tank-penstockrdquosubsystem Hence the head loss almost has no effect onthe stability and time response of the frequency of systemwithout surge tank while the stability and tail wave of systemwith surge tank are notably affected In addition in themathematicalmodel of turbine regulating system (Section 2)ℎ1199050is represented in the form of ℎ

11990501198670which indicates that

the effect of 1198670is actualized by serving as the amplification

coefficient of ℎ1199050(ie 1119867

0) This result reveals the internal

cause of the opposite effects of ℎ1199050and119867

0

52 Application I Improvements of Stability and RegulationQuality According to the effect mechanism of water inertiaand head loss of penstock the stability and regulation qualitycan be improved specifically The methods of improvementare the results in Sections 3 and 4 Reversely in the design ofHPP the effect mechanism can provide theoretical founda-tion and guidance for reasonable selections of 119879

119908119905 ℎ1199050 1198670

119870119901 and 119870

119894to guarantee preferable stability and regulation

quality

53 Application II Construction of Equivalent Model Byneglecting secondary factors in the complete mathematicalmodel of turbine regulating system based on the effectmechanism of penstock some equivalent simplified modelscan be constructed

531 Equivalent Model for Stability of System without SurgeTank For stability of system without surge tank the waterinertia and head loss of penstock are the principal factor andsecondary factor respectively Hence if the head loss item

minusTwts

Qt(s) H(s)

Figure 7 Block diagramof pipeline systemwithout surge tankwhenhead loss of penstock is neglected

is neglected the original block diagram of pipeline systemwithout surge tank shown in Figure 2(c) is simplified to theblock diagram shown in Figure 7 Then the equivalent freeoscillation equation of (10) can be obtained by letting ℎ

1199050be 0

This equivalent equation is also third order HPP B and HPPC (shown in Table 3 of Appendix C assumed cases withoutheadrace tunnel and surge tank) are taken as examples toverify the stability of this equivalent third order model andoriginal third ordermodel (ie see (10))The stability regionsof these twomodels are shown in Figure 8 It can be seen thatthe stability regions of equivalent model and original modelare nearly overlapped

532 Equivalent Model for Regulation Quality of System withSurge Tank For regulation quality of system with surge tankthe head loss and water inertia of penstock are the principalfactor and secondary factor respectively Proceeding simi-larly as Section 531 Figure 9 and (14) obtained by neglectingthe water inertia item are the equivalent simplified blockdiagram of pipeline systemwith surge tank of Figure 2(b) andtime response of the frequency of (13) respectively

119883(119904) = minus

sum3

119894=11198871198941199043minus119894

sum5

119894=11198861198941199045minus119894

1198981198920

119870119894

(14)

Equation (14) is a fourth order response model and itscoefficients are the special cases of those in original fifth orderresponse model (see (13)) when 119879

119908119905is 0 According to Galois

theory [23] original fifth order model has no extract rootsformulas Therefore it is not only impossible to solve thefluctuation equation of time response of the frequency (ie119909 = 119909(119905)) from original fifth order model directly but alsodifficult to carry out theoretical analysis This paper realizes


00 02 04 06 08 100

5

10

15

20

25

30

Original third order modelEquivalent third order model

KpK

i(s

)

1Kp

(a) HPP B

00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(b) HPP C

Figure 8 Comparison of stability regions between equivalent third order model and original third order model

+minus

+

+minus

minus

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)

Figure 9 Block diagram of pipeline system with surge tank whenwater inertia of penstock is neglected

order reduction by using the effect mechanism of penstockand obtains an equivalent fourth order responsemodel whichcan be theoretically solvedThemethod and result have greatapplication values

Figure 10 compares the time responses of the frequencybetween equivalent fourth order model and original fifthorder model using HPP B and HPP C There is a satisfactoryagreement between the time responses of the frequency ofthese two models This result indicates that the equivalentfourth order model can represent and replace the originalfifth order model

6 Conclusions

Aiming at the turbine regulating system of isolated HPPwithout surge tank and that with surge tank this paperstudies the effect mechanism of water inertia and headloss of penstock on stability and regulation quality underload disturbance based on the free oscillation equation andtime response of the frequency of system The constructionmethods of equivalent models for stability and regulation

quality are proposed according to the effect mechanism Themajor conclusions are summarized as follows

(1) The stability and regulation quality of system withoutsurge tank are determined by time response of the fre-quency which only depends on water hammer wavein penstock while for system with surge tank thetime response of the frequency depending on waterhammer wave in penstock and water-level fluctuationin surge tank jointly determines the stability andregulation quality

(2) Water inertia of penstock mainly affects the stabilityand time response of the frequency of system withoutsurge tank as well as the stability and head waveof time response of the frequency with surge tankHowever it has almost no effect on the tail wave oftime response of the frequency with surge tank

(3) Head loss of penstock mainly affects the stability andtail wave of time response of the frequency with surgetank rather than the stability and time response ofthe frequency without surge tank and head waveTheeffect of119867

0on stability and regulation quality which

is opposite to that of ℎ1199050is actualized by serving as the

amplification coefficient of ℎ1199050(ie 1119867

0)

(4) The effect mechanism of penstock can be appliedas theoretical foundation and guidance to improvestability and regulation quality

(5) For stability of system without surge tank the thirdorder free oscillation equation obtained by neglectingthe head loss item of penstock is the equivalentmodel of original third order free oscillation equationFor regulation quality of system with surge tankthe fourth order response obtained by neglecting


0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x

Original fifth order response modelEquivalent fourth order response model

t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C

Figure 10 Comparison of time responses of the frequency between equivalent fourth order model and original fifth order model

the water inertia item of penstock is the equivalentmodel of original fifth order response

Appendices

A Definitions of Parameters

See Nomenclature SectionNote the following

(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are

the relative deviations of corresponding variablesThesubscript ldquo0rdquo refers to the initial value 119876

0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760

(2) The six transfer coefficients are defined as follows119890ℎ= 120597119898

119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910

(3) 119898119892is actually equal to the relative deviation of load in

the isolated operation Hence 119898119892is regarded as the

load disturbance

B Expressions of Coefficients

The expressions of coefficients in overall transfer function(see (8)) are as follows

1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)


C Basic Information of ActualExamples of HPP

See Table 3

D Period of Water-Level Fluctuation inSurge Tank in FrictionallsquolsquoHeadrace Tunnel-Surge Tankrsquorsquo System

Based on reference [1] the free oscillation equation of water-level fluctuation in surge tank in frictional ldquoheadrace tunnel-surge tankrdquo system is derived as follows

1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow

velocity in headrace tunnelThe period of water-level fluctuation in surge tank is

obtained according to (D1) 119879119904119905= 2120587radic120596

2minus 1205752 If the fric-

tion is neglected the formula of period is simplified to 119879119904119905=

2120587radic119871119910119865119892119891119910

Nomenclature

Δ119885 Change of surge tank water level (positivedirection is downward)

119876119910 Headrace tunnel discharge

119899 Unit frequency119872119905 Kinetic moment

119871119910 Length of headrace tunnel

119891119910 Sectional area of headrace tunnel

ℎ1199100 Head loss of headrace tunnel

119879119908119910 Water inertia time constant of headrace

tunnel119865 Sectional area of surge tank119890ℎ 119890119909 119890119910 Moment transfer coefficients of turbine

119879119886 Unit inertia time constant

119870119901 Proportional gain

119867 Net head119876119905 Penstock discharge

119884 Guide vane opening119872119892 Resisting moment

119871119905 Length of penstock

119891119905 Sectional area of penstockℎ1199050 Head loss of penstock

119879119908119905 Water inertia time constant of penstock

119879119865 Time constant of surge tank

119890119902ℎ 119890119902119909 119890119902119910 Discharge transfer coefficients of turbine

119890119892 Load self-regulation coefficient

119870119894 Integral gain

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Projects nos 51379158 and 51039005)

References

[1] M H Chaudhry Applied Hydraulic Transienis Van NostrandNew York NY USA 2014

[2] H M Paynter A Palimpsest on the Electronic Analog Art GAPhilbrick Researche Boston Mass USA 1955

[3] L M Hovey ldquoOptimum adjustment of hydro governors onmanitoba hydro systemrdquo IEEETransactions on Power Apparatusand Systems vol 81 no 3 pp 581ndash587 1962

[4] M H Chaudhry ldquoGoverning stability of a hydroelectric powerplantrdquoWater Power vol 22 no 4 pp 131ndash136 1970

[5] H V Pico and J McCalley ldquoModeling and analysis of speedcontrols in hydro-turbines for frequency performancerdquo inProceedings of the North American Power Symposium pp 1ndash7Boston Mass USA August 2011

[6] D H Thorne and E F Hill ldquoExtensions of stability boundariesof a hydraulic turbine generating unitrdquo IEEE Transactions onPower Apparatus and Systems vol 94 no 4 pp 1401ndash1409 1975

[7] D T Phi E J Bourque D H Thorne and E F Hill ldquoAnalysisand application of the stability limits of a hydro-generatingunitrdquo IEEE Transactions on Power Apparatus and Systems vol100 no 7 pp 3201ndash3211 1981

[8] N S Dhaliwal and H EWichert ldquoAnalysis of PID governors inmulti-machine systemrdquo IEEE Transactions on Power Apparatusand Systems vol 97 no 2 pp 456ndash463 1978

[9] S Hagihara H Yokota K Goda and K Isobe ldquoStability of ahydraulic turbine generating unit controlled by PID governorrdquoIEEE transactions on power apparatus and systems vol 98 no6 pp 2294ndash2298 1979

[10] K A Naik P Srikanth and A K Chande ldquoA novel governorcontrol for stability enhancement of hydro power plant withwater hammer effectrdquo in International Conference on EmergingTrends in Electrical and Computer Technology pp 40ndash45 TamilNadu India March 2011

[11] T Stein ldquoFrequency control under isolated network conditionsrdquoWater Power vol 22 no 9 pp 320ndash324 1970

[12] F O Ruud ldquoInstability of a hydraulic turbine with a very longpenstockrdquo Journal of Engineering for Gas Turbines and Powervol 87 no 3 pp 290ndash294 1965

[13] M S R Murty and M V Hariharan ldquoAnalysis and improve-ment of the stability of a hydro-turbine generating unit withlong penstockrdquo IEEE Transactions on Power Apparatus andSystems vol 103 no 2 pp 360ndash367 1984

[14] O H Souza and N Barbieri ldquoStudy of hydraulic transients inhydropower plants through simulation of nonlinear model ofpenstock and hydraulic turbine modelrdquo IEEE Transactions onPower Systems vol 14 no 4 pp 1269ndash1272 1999

[15] C K Sanathanan ldquoAccurate low order model for hydraulicturbine-penstockrdquo IEEE Transactions on Energy Conversionvol 2 no 2 pp 196ndash200 1966

[16] G I Krivehenko E V Kwyatkovskaya A E Lyubitsky andS V Ostroumov ldquoSome special conditions of unit operationin hydropower plant with long penstocksrdquo in Proceedingsof 8th Symposium of IAHR Section for Hydraulic MachineryEquipment and Cavitation pp 465ndash475 Leningrad RussiaSeptember 1976


[17] L Fu J-D Yang H-Y Bao and J-P Li ldquoGenerator unitfrequency fluctuations of hydropower station with surge tankunder load disturbancerdquo Journal of Hydraulic Engineering vol39 no 11 pp 1190ndash1196 2008

[18] L Fu J P Li J D Yang and H Y Bao ldquoResearch on dynamicquality of governing system of hydropower station with tailracesurge tankrdquo Journal of Hydroelectric Engineering vol 29 no 2pp 163ndash176 2010

[19] V L Streeter and E B Wylie Fluid Transients McGraw-HillNew York NY USA 1978

[20] IEEE Working Group ldquoHydraulic turbine and turbine controlmodels for system dynamic studiesrdquo IEEE Transactions onPower Systems vol 7 no 1 pp 167ndash179 1992

[21] G F Franklin D Powell and A E Naeini Feedback Control ofDynamic Systems Prentice Hall Upper Saddle River NJ USA2009

[22] S PWeiHydraulic Turbine Regulation HuazhongUniversity ofScience and Technology Press Wuhan China 2009

[23] J A Gallian Contemporary Abstract Algebra Cengage Learn-ing Boston Mass USA 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Penstock

Draft tube


Qt Lt ft ht0 Twt

(a) Pipeline and power generating system (withoutsurge tank)

Z

Penstock

Headrace tunnel

Draft tube


Qt Lt ft ht0 Twt

Qy Ly fy hy0 Twy

Surge tank TF

(b) Pipeline and power generating system (with surge tank)

Headrace tunnelSurge tank Turbine

generatorGridload

Feedback element

Measurement element

Point elementAmplification elementCorrection element

Actuator

Turbine control system

+minus

minus

Controlled system

Penstock

(c) Turbine regulating system

Figure 1 Turbine regulating system of isolated HPP with surge tank or not

of penstock are water inertia and head loss The effects ofthese two factors are not investigated and compared deeplySecondly there is only little research onHPPwith surge tankIt is well recognized that surge tank is indeed an importantmeasure of pressure reduction Since the influence of water-level fluctuation in surge tank the dynamic response ofregulating system with surge tank is significantly differentfrom the case without surge tank

This paper aims to overcome the above two limitationsand thoroughly study the effect mechanism of water inertiaand head loss of penstock on stability and regulation qualityof turbine regulating system with surge tank or not It isassumed that the system operates on an isolated load andthe water column is rigid This paper is organized as followsIn Section 2 the complete linear mathematical model ofturbine regulating system that includes all subsystems (ieheadrace tunnel surge tank penstock turbine generator andgovernor) is established and the overall transfer functionsof systems without surge tank and with surge tank arederived from the complete mathematical model under stepload disturbance In Sections 3 and 4 based on the freeoscillation equation and time response of the frequency ofsystem derived from overall transfer function the effectsof water inertia and head loss of penstock on stability andregulation quality are analysed by using stability regionand response curves In Section 5 the effect mechanism ofpenstock is epurated and summarized Then according tothis effect mechanism the improvement methods of stabilityand regulation quality and constructionmethod of equivalentmodel of regulating system are proposed

2 Mathematical Model

Theturbine regulating systemof isolatedHPPwith surge tankor not is illustrated in Figure 1

21 Basic Equations The HPP without surge tank can beregarded as a special case of HPP with surge tank when thelength of headrace tunnel and the sectional area of surge tankare both 0 Hence in this section the complete mathematicalmodel of turbine regulating systemof isolatedHPPwith surgetank is first established and then the model of that withoutsurge tank can be obtained as a special case

211 Turbine Regulating System of Isolated HPP withSurge Tank

(1) Controlled System [19 20] Momentum equation of head-race tunnel

119911 = 119879119908119910

119889119902119910

119889119905

+

2ℎ1199100

1198670

119902119910 (1)

Continuity equation of surge tank

119902119910= 119902119905minus 119879119865

119889119911

119889119905

(2)

Momentum equation of penstock

ℎ = minus119879119908119905

119889119902119905

119889119905

minus

2ℎ1199050

1198670

119902119905minus 119911 (3)


1++minus

++

+

+minus

minus++

+

Kp

Kis

eqx

eqy

eqh

ey

Gs(s)

ex

eh

Mg(s)

eg

X(s)1Tas


+minus

++ minus

minus minus

Twts

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)


minus

minus

2ht0H0

Qt(s)Twts

H(s)




119898119905= 119890ℎℎ + 119890119909119909 + 119890119910119910

119902119905= 119890119902ℎℎ + 119890119902119909119909 + 119890119902119910119910

(4)


119879119886

119889119909

119889119905

= 119898119905minus (119898119892+ 119890119892119909) (5)


119889119910

119889119905

= minus119870119901

119889119909

119889119905

minus 119870119894119909 (6)



ℎ = minus119879119908119905

119889119902119905

119889119905

minus

2ℎ1199050

1198670

119902119905 (7)


119908119910= 0 ℎ1199100

= 0 and 119879119865= 0



119904(119904) = 119867(119904)119876




119866 (119904) =

119883 (119904)

119872119892(119904)

= minus

119904 (11988701199043

+ 11988711199042

+ 1198872119904 + 1198873) 119870119894

11988601199045+ 11988611199044+ 11988621199043+ 11988631199042+ 1198864119904 + 1198865

(8)





119866 (119904) =

119883 (119904)

119872119892(119904)

= minus

119904 (1198872119904 + 1198873) 119870119894

11988621199043+ 11988631199042+ 1198864119904 + 1198865

(9)


= 0 ℎ1199100



119879119865are both 0


Instability region

Stability region

Stability boundary

Z

X

YKpK

i(s

)

1Kp


119870119901119870119894

x

X

Y

Z

t (s)







1198862

1198893

119909

1198891199053+ 1198863

1198892

119909

1198891199052+ 1198864

119889119909

119889119905

+ 1198865= 0 (10)

1198860

1198895

119909

1198891199055+ 1198861

1198894

119909

1198891199054+ 1198862

1198893

119909

1198891199053+ 1198863

1198892

119909

1198891199052+ 1198864

119889119909

119889119905

+ 1198865= 0 (11)





1gt 0 Δ1015840

2gt 0 and Δ1015840

4gt 0




Δ1= 119886119894gt 0 (119894 = 2 3 4 5) Δ

1015840

1= 119886119894gt 0 (119894 = 0 1 2 3 4 5)

Δ2= 11988631198864minus 11988621198865gt 0 Δ

1015840

2= 11988611198862minus 11988601198863gt 0

Δ1015840

4= (11988611198862minus 11988601198863)(11988631198864minus 11988621198865)

minus (11988611198864minus 11988601198865)2

gt 0









119903sim133119867




119908119905 ℎ1199050 and 119867



ℎ1199050 and 119867

0are 20s 40m (44119867

119903) and 90m (100119867

119903)


119909= minus1 119890

119910= 1 119890

119902ℎ= 05 119890

119902119909= 0 119890

119902119910= 1 119879

119886=

834 s 119890119892= 0 and 119899 = 09 in which 119899 = 119865119865





00 02 04 06 08 100

5

10

15

20

25

30

00 02 04 06 08 100

5

10

15

20

25

30

025 030 0353

4

5

00 02 04 06 08 100

5

10

15

20

25

30

025 030 0353

4

5

KpK

i(s

)KpK

i(s

)KpK

i(s

)

1Kp

1Kp

1Kp



ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m

(a1) Twt

(a2) ht0

(a3) H0


30

00 02 04 06 08 100

5

10

15

20

25

00 02 04 06 08 100

5

10

15

20

25

30

S

00 02 04 06 08 100

5

10

15

20

25

30

KpK

i(s

)KpK

i(s

)KpK

i(s

)

1Kp

1Kp

1Kp



ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m

(b1) Twt

(b2) ht0

(b3) H0











1199050from 00m to 80m (89119867

119903) the


0increases

from 60m (067119867119903) to 120m (133119867

119903)




the decrease of 119879119908119905


1199050


119903) the


1199050= 00m and ℎ

1199050=





119908119905on


1199050as well as 119867

0on


119908119905 ℎ1199050 and 119867

0change








119892(119904) = 119898

1198920119904

in which1198981198920




119883(119904) = minus

sum3

119894=21198871198941199043minus119894

sum5

119894=21198861198941199045minus119894

1198981198920

119870119894

(12)

119883(119904) = minus

sum3

119894=01198871198941199043minus119894

sum5

119894=01198861198941199045minus119894

1198981198920

119870119894

(13)




119901and 119870










119908119905rises


1199050and





119879119908119905

(s)0 00318 00140 00007 32387 313911 00337 00344 00141 00007 32221 313912 00416 00419 00142 00007 32221 313913 00529 00534 00145 00006 32221 313914 00666 00670 00145 00006 32221 31391

ℎ1199050(m)0 00395 00398 00122 00010 31733 310722 00404 00409 00132 00008 31894 312214 00416 00419 00142 00007 32221 313916 00427 00429 00156 00006 32555 315878 00439 00443 00172 00004 32896 31815

1198670(m)60 00427 00426 00227 00005 35699 3421190 00416 00419 00142 00007 32221 31391120 00410 00415 00105 00013 28690 30500




119903(m3s) 119879

119908119910(s) 119879

119908119905(s) ℎ

1199100(m) ℎ

1199050(m)

A 5128 9000 6270 1775 233 757 553B 11856 17700 7250 3973 182 2053 512C 61000 28800 22860 2384 126 1292 291




119908119905 ℎ1199050 and 119867






119908119905 the period


1199050


1199050


0 and the









0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

t (s)

t (s)

t (s)



(a1) Twt

(a2) ht0

(a3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m


0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006008

Tail wave

Head wave

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80100000002004006

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006

t (s)

t (s)

t (s)



(b1) Twt

(b2) ht0

(b3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m








0 10 20 30 40 50

minus03

minus02

minus01

00

01

02Re

spon

ses

t (s)

Frequency x


Net head h


0 200 400 600 800 1000

minus03

minus02

minus01

00

01

02

Resp

onse

s

Frequency x


Net head h










0


119908119905 ℎ1199050 1198670

119870119901 and 119870


quality



minusTwts

Qt(s) H(s)



1199050be 0



119883(119904) = minus

sum3

119894=11198871198941199043minus119894

sum5

119894=11198861198941199045minus119894

1198981198920

119870119894

(14)





00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(a) HPP B

00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(b) HPP C


+minus

+

+minus

minus

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)




6 Conclusions









0)




0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C



Appendices



(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are


0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760


119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910



load disturbance



1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)



See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1++minus

++

+

+minus

minus++

+

Kp

Kis

eqx

eqy

eqh

ey

Gs(s)

ex

eh

Mg(s)

eg

X(s)1Tas


+minus

++ minus

minus minus

Twts

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)


minus

minus

2ht0H0

Qt(s)Twts

H(s)




119898119905= 119890ℎℎ + 119890119909119909 + 119890119910119910

119902119905= 119890119902ℎℎ + 119890119902119909119909 + 119890119902119910119910

(4)


119879119886

119889119909

119889119905

= 119898119905minus (119898119892+ 119890119892119909) (5)


119889119910

119889119905

= minus119870119901

119889119909

119889119905

minus 119870119894119909 (6)



ℎ = minus119879119908119905

119889119902119905

119889119905

minus

2ℎ1199050

1198670

119902119905 (7)


119908119910= 0 ℎ1199100

= 0 and 119879119865= 0



119904(119904) = 119867(119904)119876




119866 (119904) =

119883 (119904)

119872119892(119904)

= minus

119904 (11988701199043

+ 11988711199042

+ 1198872119904 + 1198873) 119870119894

11988601199045+ 11988611199044+ 11988621199043+ 11988631199042+ 1198864119904 + 1198865

(8)





119866 (119904) =

119883 (119904)

119872119892(119904)

= minus

119904 (1198872119904 + 1198873) 119870119894

11988621199043+ 11988631199042+ 1198864119904 + 1198865

(9)


= 0 ℎ1199100



119879119865are both 0


Instability region

Stability region

Stability boundary

Z

X

YKpK

i(s

)

1Kp


119870119901119870119894

x

X

Y

Z

t (s)







1198862

1198893

119909

1198891199053+ 1198863

1198892

119909

1198891199052+ 1198864

119889119909

119889119905

+ 1198865= 0 (10)

1198860

1198895

119909

1198891199055+ 1198861

1198894

119909

1198891199054+ 1198862

1198893

119909

1198891199053+ 1198863

1198892

119909

1198891199052+ 1198864

119889119909

119889119905

+ 1198865= 0 (11)





1gt 0 Δ1015840

2gt 0 and Δ1015840

4gt 0




Δ1= 119886119894gt 0 (119894 = 2 3 4 5) Δ

1015840

1= 119886119894gt 0 (119894 = 0 1 2 3 4 5)

Δ2= 11988631198864minus 11988621198865gt 0 Δ

1015840

2= 11988611198862minus 11988601198863gt 0

Δ1015840

4= (11988611198862minus 11988601198863)(11988631198864minus 11988621198865)

minus (11988611198864minus 11988601198865)2

gt 0









119903sim133119867




119908119905 ℎ1199050 and 119867



ℎ1199050 and 119867

0are 20s 40m (44119867

119903) and 90m (100119867

119903)


119909= minus1 119890

119910= 1 119890

119902ℎ= 05 119890

119902119909= 0 119890

119902119910= 1 119879

119886=

834 s 119890119892= 0 and 119899 = 09 in which 119899 = 119865119865





00 02 04 06 08 100

5

10

15

20

25

30

00 02 04 06 08 100

5

10

15

20

25

30

025 030 0353

4

5

00 02 04 06 08 100

5

10

15

20

25

30

025 030 0353

4

5

KpK

i(s

)KpK

i(s

)KpK

i(s

)

1Kp

1Kp

1Kp



ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m

(a1) Twt

(a2) ht0

(a3) H0


30

00 02 04 06 08 100

5

10

15

20

25

00 02 04 06 08 100

5

10

15

20

25

30

S

00 02 04 06 08 100

5

10

15

20

25

30

KpK

i(s

)KpK

i(s

)KpK

i(s

)

1Kp

1Kp

1Kp



ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m

(b1) Twt

(b2) ht0

(b3) H0











1199050from 00m to 80m (89119867

119903) the


0increases

from 60m (067119867119903) to 120m (133119867

119903)




the decrease of 119879119908119905


1199050


119903) the


1199050= 00m and ℎ

1199050=





119908119905on


1199050as well as 119867

0on


119908119905 ℎ1199050 and 119867

0change








119892(119904) = 119898

1198920119904

in which1198981198920




119883(119904) = minus

sum3

119894=21198871198941199043minus119894

sum5

119894=21198861198941199045minus119894

1198981198920

119870119894

(12)

119883(119904) = minus

sum3

119894=01198871198941199043minus119894

sum5

119894=01198861198941199045minus119894

1198981198920

119870119894

(13)




119901and 119870










119908119905rises


1199050and





119879119908119905

(s)0 00318 00140 00007 32387 313911 00337 00344 00141 00007 32221 313912 00416 00419 00142 00007 32221 313913 00529 00534 00145 00006 32221 313914 00666 00670 00145 00006 32221 31391

ℎ1199050(m)0 00395 00398 00122 00010 31733 310722 00404 00409 00132 00008 31894 312214 00416 00419 00142 00007 32221 313916 00427 00429 00156 00006 32555 315878 00439 00443 00172 00004 32896 31815

1198670(m)60 00427 00426 00227 00005 35699 3421190 00416 00419 00142 00007 32221 31391120 00410 00415 00105 00013 28690 30500




119903(m3s) 119879

119908119910(s) 119879

119908119905(s) ℎ

1199100(m) ℎ

1199050(m)

A 5128 9000 6270 1775 233 757 553B 11856 17700 7250 3973 182 2053 512C 61000 28800 22860 2384 126 1292 291




119908119905 ℎ1199050 and 119867






119908119905 the period


1199050


1199050


0 and the









0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

t (s)

t (s)

t (s)



(a1) Twt

(a2) ht0

(a3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m


0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006008

Tail wave

Head wave

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80100000002004006

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006

t (s)

t (s)

t (s)



(b1) Twt

(b2) ht0

(b3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m








0 10 20 30 40 50

minus03

minus02

minus01

00

01

02Re

spon

ses

t (s)

Frequency x


Net head h


0 200 400 600 800 1000

minus03

minus02

minus01

00

01

02

Resp

onse

s

Frequency x


Net head h










0


119908119905 ℎ1199050 1198670

119870119901 and 119870


quality



minusTwts

Qt(s) H(s)



1199050be 0



119883(119904) = minus

sum3

119894=11198871198941199043minus119894

sum5

119894=11198861198941199045minus119894

1198981198920

119870119894

(14)





00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(a) HPP B

00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(b) HPP C


+minus

+

+minus

minus

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)




6 Conclusions









0)




0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C



Appendices



(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are


0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760


119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910



load disturbance



1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)



See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Instability region

Stability region

Stability boundary

Z

X

YKpK

i(s

)

1Kp


119870119901119870119894

x

X

Y

Z

t (s)







1198862

1198893

119909

1198891199053+ 1198863

1198892

119909

1198891199052+ 1198864

119889119909

119889119905

+ 1198865= 0 (10)

1198860

1198895

119909

1198891199055+ 1198861

1198894

119909

1198891199054+ 1198862

1198893

119909

1198891199053+ 1198863

1198892

119909

1198891199052+ 1198864

119889119909

119889119905

+ 1198865= 0 (11)





1gt 0 Δ1015840

2gt 0 and Δ1015840

4gt 0




Δ1= 119886119894gt 0 (119894 = 2 3 4 5) Δ

1015840

1= 119886119894gt 0 (119894 = 0 1 2 3 4 5)

Δ2= 11988631198864minus 11988621198865gt 0 Δ

1015840

2= 11988611198862minus 11988601198863gt 0

Δ1015840

4= (11988611198862minus 11988601198863)(11988631198864minus 11988621198865)

minus (11988611198864minus 11988601198865)2

gt 0









119903sim133119867




119908119905 ℎ1199050 and 119867



ℎ1199050 and 119867

0are 20s 40m (44119867

119903) and 90m (100119867

119903)


119909= minus1 119890

119910= 1 119890

119902ℎ= 05 119890

119902119909= 0 119890

119902119910= 1 119879

119886=

834 s 119890119892= 0 and 119899 = 09 in which 119899 = 119865119865





00 02 04 06 08 100

5

10

15

20

25

30

00 02 04 06 08 100

5

10

15

20

25

30

025 030 0353

4

5

00 02 04 06 08 100

5

10

15

20

25

30

025 030 0353

4

5

KpK

i(s

)KpK

i(s

)KpK

i(s

)

1Kp

1Kp

1Kp



ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m

(a1) Twt

(a2) ht0

(a3) H0


30

00 02 04 06 08 100

5

10

15

20

25

00 02 04 06 08 100

5

10

15

20

25

30

S

00 02 04 06 08 100

5

10

15

20

25

30

KpK

i(s

)KpK

i(s

)KpK

i(s

)

1Kp

1Kp

1Kp



ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m

(b1) Twt

(b2) ht0

(b3) H0











1199050from 00m to 80m (89119867

119903) the


0increases

from 60m (067119867119903) to 120m (133119867

119903)




the decrease of 119879119908119905


1199050


119903) the


1199050= 00m and ℎ

1199050=





119908119905on


1199050as well as 119867

0on


119908119905 ℎ1199050 and 119867

0change








119892(119904) = 119898

1198920119904

in which1198981198920




119883(119904) = minus

sum3

119894=21198871198941199043minus119894

sum5

119894=21198861198941199045minus119894

1198981198920

119870119894

(12)

119883(119904) = minus

sum3

119894=01198871198941199043minus119894

sum5

119894=01198861198941199045minus119894

1198981198920

119870119894

(13)




119901and 119870










119908119905rises


1199050and





119879119908119905

(s)0 00318 00140 00007 32387 313911 00337 00344 00141 00007 32221 313912 00416 00419 00142 00007 32221 313913 00529 00534 00145 00006 32221 313914 00666 00670 00145 00006 32221 31391

ℎ1199050(m)0 00395 00398 00122 00010 31733 310722 00404 00409 00132 00008 31894 312214 00416 00419 00142 00007 32221 313916 00427 00429 00156 00006 32555 315878 00439 00443 00172 00004 32896 31815

1198670(m)60 00427 00426 00227 00005 35699 3421190 00416 00419 00142 00007 32221 31391120 00410 00415 00105 00013 28690 30500




119903(m3s) 119879

119908119910(s) 119879

119908119905(s) ℎ

1199100(m) ℎ

1199050(m)

A 5128 9000 6270 1775 233 757 553B 11856 17700 7250 3973 182 2053 512C 61000 28800 22860 2384 126 1292 291




119908119905 ℎ1199050 and 119867






119908119905 the period


1199050


1199050


0 and the









0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

t (s)

t (s)

t (s)



(a1) Twt

(a2) ht0

(a3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m


0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006008

Tail wave

Head wave

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80100000002004006

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006

t (s)

t (s)

t (s)



(b1) Twt

(b2) ht0

(b3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m








0 10 20 30 40 50

minus03

minus02

minus01

00

01

02Re

spon

ses

t (s)

Frequency x


Net head h


0 200 400 600 800 1000

minus03

minus02

minus01

00

01

02

Resp

onse

s

Frequency x


Net head h










0


119908119905 ℎ1199050 1198670

119870119901 and 119870


quality



minusTwts

Qt(s) H(s)



1199050be 0



119883(119904) = minus

sum3

119894=11198871198941199043minus119894

sum5

119894=11198861198941199045minus119894

1198981198920

119870119894

(14)





00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(a) HPP B

00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(b) HPP C


+minus

+

+minus

minus

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)




6 Conclusions









0)




0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C



Appendices



(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are


0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760


119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910



load disturbance



1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)



See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 02 04 06 08 100

5

10

15

20

25

30

00 02 04 06 08 100

5

10

15

20

25

30

025 030 0353

4

5

00 02 04 06 08 100

5

10

15

20

25

30

025 030 0353

4

5

KpK

i(s

)KpK

i(s

)KpK

i(s

)

1Kp

1Kp

1Kp



ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m

(a1) Twt

(a2) ht0

(a3) H0


30

00 02 04 06 08 100

5

10

15

20

25

00 02 04 06 08 100

5

10

15

20

25

30

S

00 02 04 06 08 100

5

10

15

20

25

30

KpK

i(s

)KpK

i(s

)KpK

i(s

)

1Kp

1Kp

1Kp



ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m

(b1) Twt

(b2) ht0

(b3) H0











1199050from 00m to 80m (89119867

119903) the


0increases

from 60m (067119867119903) to 120m (133119867

119903)




the decrease of 119879119908119905


1199050


119903) the


1199050= 00m and ℎ

1199050=





119908119905on


1199050as well as 119867

0on


119908119905 ℎ1199050 and 119867

0change








119892(119904) = 119898

1198920119904

in which1198981198920




119883(119904) = minus

sum3

119894=21198871198941199043minus119894

sum5

119894=21198861198941199045minus119894

1198981198920

119870119894

(12)

119883(119904) = minus

sum3

119894=01198871198941199043minus119894

sum5

119894=01198861198941199045minus119894

1198981198920

119870119894

(13)




119901and 119870










119908119905rises


1199050and





119879119908119905

(s)0 00318 00140 00007 32387 313911 00337 00344 00141 00007 32221 313912 00416 00419 00142 00007 32221 313913 00529 00534 00145 00006 32221 313914 00666 00670 00145 00006 32221 31391

ℎ1199050(m)0 00395 00398 00122 00010 31733 310722 00404 00409 00132 00008 31894 312214 00416 00419 00142 00007 32221 313916 00427 00429 00156 00006 32555 315878 00439 00443 00172 00004 32896 31815

1198670(m)60 00427 00426 00227 00005 35699 3421190 00416 00419 00142 00007 32221 31391120 00410 00415 00105 00013 28690 30500




119903(m3s) 119879

119908119910(s) 119879

119908119905(s) ℎ

1199100(m) ℎ

1199050(m)

A 5128 9000 6270 1775 233 757 553B 11856 17700 7250 3973 182 2053 512C 61000 28800 22860 2384 126 1292 291




119908119905 ℎ1199050 and 119867






119908119905 the period


1199050


1199050


0 and the









0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

t (s)

t (s)

t (s)



(a1) Twt

(a2) ht0

(a3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m


0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006008

Tail wave

Head wave

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80100000002004006

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006

t (s)

t (s)

t (s)



(b1) Twt

(b2) ht0

(b3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m








0 10 20 30 40 50

minus03

minus02

minus01

00

01

02Re

spon

ses

t (s)

Frequency x


Net head h


0 200 400 600 800 1000

minus03

minus02

minus01

00

01

02

Resp

onse

s

Frequency x


Net head h










0


119908119905 ℎ1199050 1198670

119870119901 and 119870


quality



minusTwts

Qt(s) H(s)



1199050be 0



119883(119904) = minus

sum3

119894=11198871198941199043minus119894

sum5

119894=11198861198941199045minus119894

1198981198920

119870119894

(14)





00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(a) HPP B

00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(b) HPP C


+minus

+

+minus

minus

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)




6 Conclusions









0)




0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C



Appendices



(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are


0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760


119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910



load disturbance



1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)



See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















1199050from 00m to 80m (89119867

119903) the


0increases

from 60m (067119867119903) to 120m (133119867

119903)




the decrease of 119879119908119905


1199050


119903) the


1199050= 00m and ℎ

1199050=





119908119905on


1199050as well as 119867

0on


119908119905 ℎ1199050 and 119867

0change








119892(119904) = 119898

1198920119904

in which1198981198920




119883(119904) = minus

sum3

119894=21198871198941199043minus119894

sum5

119894=21198861198941199045minus119894

1198981198920

119870119894

(12)

119883(119904) = minus

sum3

119894=01198871198941199043minus119894

sum5

119894=01198861198941199045minus119894

1198981198920

119870119894

(13)




119901and 119870










119908119905rises


1199050and





119879119908119905

(s)0 00318 00140 00007 32387 313911 00337 00344 00141 00007 32221 313912 00416 00419 00142 00007 32221 313913 00529 00534 00145 00006 32221 313914 00666 00670 00145 00006 32221 31391

ℎ1199050(m)0 00395 00398 00122 00010 31733 310722 00404 00409 00132 00008 31894 312214 00416 00419 00142 00007 32221 313916 00427 00429 00156 00006 32555 315878 00439 00443 00172 00004 32896 31815

1198670(m)60 00427 00426 00227 00005 35699 3421190 00416 00419 00142 00007 32221 31391120 00410 00415 00105 00013 28690 30500




119903(m3s) 119879

119908119910(s) 119879

119908119905(s) ℎ

1199100(m) ℎ

1199050(m)

A 5128 9000 6270 1775 233 757 553B 11856 17700 7250 3973 182 2053 512C 61000 28800 22860 2384 126 1292 291




119908119905 ℎ1199050 and 119867






119908119905 the period


1199050


1199050


0 and the









0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

t (s)

t (s)

t (s)



(a1) Twt

(a2) ht0

(a3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m


0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006008

Tail wave

Head wave

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80100000002004006

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006

t (s)

t (s)

t (s)



(b1) Twt

(b2) ht0

(b3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m








0 10 20 30 40 50

minus03

minus02

minus01

00

01

02Re

spon

ses

t (s)

Frequency x


Net head h


0 200 400 600 800 1000

minus03

minus02

minus01

00

01

02

Resp

onse

s

Frequency x


Net head h










0


119908119905 ℎ1199050 1198670

119870119901 and 119870


quality



minusTwts

Qt(s) H(s)



1199050be 0



119883(119904) = minus

sum3

119894=11198871198941199043minus119894

sum5

119894=11198861198941199045minus119894

1198981198920

119870119894

(14)





00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(a) HPP B

00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(b) HPP C


+minus

+

+minus

minus

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)




6 Conclusions









0)




0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C



Appendices



(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are


0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760


119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910



load disturbance



1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)



See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119879119908119905

(s)0 00318 00140 00007 32387 313911 00337 00344 00141 00007 32221 313912 00416 00419 00142 00007 32221 313913 00529 00534 00145 00006 32221 313914 00666 00670 00145 00006 32221 31391

ℎ1199050(m)0 00395 00398 00122 00010 31733 310722 00404 00409 00132 00008 31894 312214 00416 00419 00142 00007 32221 313916 00427 00429 00156 00006 32555 315878 00439 00443 00172 00004 32896 31815

1198670(m)60 00427 00426 00227 00005 35699 3421190 00416 00419 00142 00007 32221 31391120 00410 00415 00105 00013 28690 30500




119903(m3s) 119879

119908119910(s) 119879

119908119905(s) ℎ

1199100(m) ℎ

1199050(m)

A 5128 9000 6270 1775 233 757 553B 11856 17700 7250 3973 182 2053 512C 61000 28800 22860 2384 126 1292 291




119908119905 ℎ1199050 and 119867






119908119905 the period


1199050


1199050


0 and the









0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

t (s)

t (s)

t (s)



(a1) Twt

(a2) ht0

(a3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m


0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006008

Tail wave

Head wave

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80100000002004006

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006

t (s)

t (s)

t (s)



(b1) Twt

(b2) ht0

(b3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m








0 10 20 30 40 50

minus03

minus02

minus01

00

01

02Re

spon

ses

t (s)

Frequency x


Net head h


0 200 400 600 800 1000

minus03

minus02

minus01

00

01

02

Resp

onse

s

Frequency x


Net head h










0


119908119905 ℎ1199050 1198670

119870119901 and 119870


quality



minusTwts

Qt(s) H(s)



1199050be 0



119883(119904) = minus

sum3

119894=11198871198941199043minus119894

sum5

119894=11198861198941199045minus119894

1198981198920

119870119894

(14)





00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(a) HPP B

00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(b) HPP C


+minus

+

+minus

minus

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)




6 Conclusions









0)




0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C



Appendices



(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are


0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760


119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910



load disturbance



1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)



See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

0 30 60 90 120 150minus002

000

002

004

006

008

x

t (s)

t (s)

t (s)



(a1) Twt

(a2) ht0

(a3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m


0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006008

Tail wave

Head wave

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80100000002004006

0 200 400 600 800 1000minus002

000

002

004

006

008

x 0 20 40 60 80 100000002004006

t (s)

t (s)

t (s)



(b1) Twt

(b2) ht0

(b3) H0

ht0 = 00 mht0 = 20 mht0 = 40 m

ht0 = 60 mht0 = 80 m

H0 = 60 mH0 = 90 mH0 = 120 m








0 10 20 30 40 50

minus03

minus02

minus01

00

01

02Re

spon

ses

t (s)

Frequency x


Net head h


0 200 400 600 800 1000

minus03

minus02

minus01

00

01

02

Resp

onse

s

Frequency x


Net head h










0


119908119905 ℎ1199050 1198670

119870119901 and 119870


quality



minusTwts

Qt(s) H(s)



1199050be 0



119883(119904) = minus

sum3

119894=11198871198941199043minus119894

sum5

119894=11198861198941199045minus119894

1198981198920

119870119894

(14)





00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(a) HPP B

00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(b) HPP C


+minus

+

+minus

minus

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)




6 Conclusions









0)




0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C



Appendices



(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are


0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760


119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910



load disturbance



1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)



See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50

minus03

minus02

minus01

00

01

02Re

spon

ses

t (s)

Frequency x


Net head h


0 200 400 600 800 1000

minus03

minus02

minus01

00

01

02

Resp

onse

s

Frequency x


Net head h










0


119908119905 ℎ1199050 1198670

119870119901 and 119870


quality



minusTwts

Qt(s) H(s)



1199050be 0



119883(119904) = minus

sum3

119894=11198871198941199043minus119894

sum5

119894=11198861198941199045minus119894

1198981198920

119870119894

(14)





00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(a) HPP B

00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(b) HPP C


+minus

+

+minus

minus

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)




6 Conclusions









0)




0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C



Appendices



(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are


0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760


119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910



load disturbance



1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)



See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(a) HPP B

00 02 04 06 08 100

5

10

15

20

25

30


KpK

i(s

)

1Kp

(b) HPP C


+minus

+

+minus

minus

2ht0H0

Twys

2hy0H0

TFs

Qt(s) H(s)




6 Conclusions









0)




0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C



Appendices



(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are


0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760


119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910



load disturbance



1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)



See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(a) HPP B

0 400 800 1200 1600 2000

minus002

minus001

000

001

002

003

004

x


t (s)

(b) HPP C



Appendices



(1) 119911 = Δ1198851198670 ℎ = (119867 minus 119867

0)1198670 119902119910= (119876119910minus 1198760)1198760

119902119905= (119876119905minus1198760)1198760 119909 = (119899 minus 119899

0)1198990 119884 = (119884 minus 119884

0)1198840

119898119905= (119872119905minus1198721199050)1198721199050 and119898

119892= (119872119892minus1198721199050)1198721198920are


0= 1198761199100=

1198761199050 119879119865= 119865119867

01198760


119905120597ℎ 119890

119909= 120597119898

119905120597119909 119890

119910= 120597119898

119905120597119910 119890

119902ℎ= 120597119902119905

120597ℎ 119890119902119909= 120597119902119905120597119909 and 119890

119902119910= 120597119902119905120597119910



load disturbance



1198860= 11989111198919

1198861= 119891111989110+ 11989121198919+ 119891511989112

1198862= 119891111989111+ 119891211989110+ 11989131198919+ 119891511989113+ 119891611989112

1198863= 119891211989111+ 119891311989110+ 11989141198919+ 119891611989113+ 119891711989112

1198864= 119891311989111+ 119891411989110+ 119891711989113+ 119891811989112

1198865= 119891411989111+ 119891811989113

1198870= 1198911

1198871= 1198912

1198872= 1198913

1198873= 1198914

1198911= 119890119902ℎ119879119865119879119908119910119879119908119905

1198912= 119879119865[119879119908119910(1 + 119890

119902ℎ

2ℎ1199050

1198670

) + 119879119908119905119890119902ℎ

2ℎ1199100

1198670

]

1198913= 119890119902ℎ(119879119908119910+ 119879119908119905) + 119879119865

2ℎ1199100

1198670

(1 + 119890119902ℎ

2ℎ1199050

1198670

)

1198914= 1 + 119890

119902ℎ

2 (ℎ1199100+ ℎ1199050)

1198670

1198915= 119879119865119879119908119910119879119908119905

1198916= 119879119865(119879119908119910

2ℎ1199050

1198670

+ 119879119908119905

2ℎ1199100

1198670

)

1198917= 119879119908119910+ 119879119908119905+ 119879119865

2ℎ1199100

1198670

2ℎ1199050

1198670

1198918=

2 (ℎ1199100+ ℎ1199050)

1198670

1198919=

119879119886

119870119894

11989110=

(119890119892minus 119890119909)

119870119894

+

119890119910119870119901

119870119894

11989111= 119890119910

11989112=

119890ℎ119890119902119909

119870119894

minus

119890ℎ119890119902119910119870119901

119870119894

11989113= minus119890ℎ119890119902119910

(B1)



See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











See Table 3



1198892

119911

1198891199052+ 2120575

119889119911

119889119905

+ 1205962

119911 = 0 (D1)

where 120575 = (V11991002)[2120572119892119871

119910minus 119891119910119865(1198670minus 2ℎ1199050)] 120596 = (119892119891

119910

119871119910119865)(1 minus (2ℎ

1199100(1198670minus 2ℎ1199050))) 120572 = ℎ

1199100V21199100 and V

1199100is flow





2120587radic119871119910119865119892119891119910

Nomenclature






















Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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