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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
Generating unitReservoir
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
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
Twt = 30 sTwt = 40 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
Twt = 30 sTwt = 40 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
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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)
Twt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(a) System without surge tank
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)
Twt = 00 sTwt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(b) System with surge tank
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
Mathematical Problems in Engineering 9
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
(a) System without surge tank
0 200 400 600 800 1000
minus03
minus02
minus01
00
01
02
Resp
onse
s
Frequency x
Guide vane opening y
Net head h
(b) System with surge tank
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
10 Mathematical Problems in Engineering
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
Original third order modelEquivalent third order model
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
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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
Generating unitReservoir
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
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
Twt = 30 sTwt = 40 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
Twt = 30 sTwt = 40 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
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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)
Twt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(a) System without surge tank
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)
Twt = 00 sTwt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(b) System with surge tank
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
Mathematical Problems in Engineering 9
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
(a) System without surge tank
0 200 400 600 800 1000
minus03
minus02
minus01
00
01
02
Resp
onse
s
Frequency x
Guide vane opening y
Net head h
(b) System with surge tank
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
10 Mathematical Problems in Engineering
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
Original third order modelEquivalent third order model
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
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
Twt = 30 sTwt = 40 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
Twt = 30 sTwt = 40 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
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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)
Twt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(a) System without surge tank
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)
Twt = 00 sTwt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(b) System with surge tank
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
Mathematical Problems in Engineering 9
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
(a) System without surge tank
0 200 400 600 800 1000
minus03
minus02
minus01
00
01
02
Resp
onse
s
Frequency x
Guide vane opening y
Net head h
(b) System with surge tank
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
10 Mathematical Problems in Engineering
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
Original third order modelEquivalent third order model
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
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
Twt = 30 sTwt = 40 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
Twt = 30 sTwt = 40 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
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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)
Twt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(a) System without surge tank
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)
Twt = 00 sTwt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(b) System with surge tank
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
Mathematical Problems in Engineering 9
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
(a) System without surge tank
0 200 400 600 800 1000
minus03
minus02
minus01
00
01
02
Resp
onse
s
Frequency x
Guide vane opening y
Net head h
(b) System with surge tank
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
10 Mathematical Problems in Engineering
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
Original third order modelEquivalent third order model
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
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
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
Twt = 30 sTwt = 40 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
Twt = 30 sTwt = 40 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
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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)
Twt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(a) System without surge tank
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)
Twt = 00 sTwt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(b) System with surge tank
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
Mathematical Problems in Engineering 9
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
(a) System without surge tank
0 200 400 600 800 1000
minus03
minus02
minus01
00
01
02
Resp
onse
s
Frequency x
Guide vane opening y
Net head h
(b) System with surge tank
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
10 Mathematical Problems in Engineering
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
Original third order modelEquivalent third order model
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
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
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Differential EquationsInternational Journal of
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Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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)
Twt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(a) System without surge tank
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)
Twt = 00 sTwt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(b) System with surge tank
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
Mathematical Problems in Engineering 9
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
(a) System without surge tank
0 200 400 600 800 1000
minus03
minus02
minus01
00
01
02
Resp
onse
s
Frequency x
Guide vane opening y
Net head h
(b) System with surge tank
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
10 Mathematical Problems in Engineering
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
Original third order modelEquivalent third order model
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
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
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
8 Mathematical Problems in Engineering
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)
Twt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(a) System without surge tank
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)
Twt = 00 sTwt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(b) System with surge tank
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
Mathematical Problems in Engineering 9
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
(a) System without surge tank
0 200 400 600 800 1000
minus03
minus02
minus01
00
01
02
Resp
onse
s
Frequency x
Guide vane opening y
Net head h
(b) System with surge tank
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
10 Mathematical Problems in Engineering
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
Original third order modelEquivalent third order model
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
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
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)
Twt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(a) System without surge tank
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)
Twt = 00 sTwt = 10 sTwt = 20 s
Twt = 30 sTwt = 40 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
(b) System with surge tank
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
Mathematical Problems in Engineering 9
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
(a) System without surge tank
0 200 400 600 800 1000
minus03
minus02
minus01
00
01
02
Resp
onse
s
Frequency x
Guide vane opening y
Net head h
(b) System with surge tank
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
10 Mathematical Problems in Engineering
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
Original third order modelEquivalent third order model
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
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
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
(a) System without surge tank
0 200 400 600 800 1000
minus03
minus02
minus01
00
01
02
Resp
onse
s
Frequency x
Guide vane opening y
Net head h
(b) System with surge tank
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
10 Mathematical Problems in Engineering
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
Original third order modelEquivalent third order model
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
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
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
Original third order modelEquivalent third order model
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
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
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
Original fifth order response modelEquivalent fourth order response model
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)
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 13
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 13
[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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of