-
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;
[email protected]
Received 17 July 2014; Accepted 20 October 2014; Published 19
November 2014
Academic Editor: Hongbin Zhang
Copyright 2014 Wencheng Guo et al.This is an open access article
distributed under the Creative Commons Attribution License,which
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.
Firstly,the 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 [25] and
proportional-integral-derivative (PID) type governor[610]. The
adjustment of governor parameters was also
researched [11]. As for penstock of HPP without surge tank,Ruud
[12] investigated the instability of a hydraulic turbinewith a very
long penstock;Murty 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
performance;Souza 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
pageshttp://dx.doi.org/10.1155/2014/349086
http://dx.doi.org/10.1155/2014/349086
-
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 Turbinegenerator
Gridload
Feedback element
Measurement element
Point elementAmplification elementCorrection element
Actuator
Turbine control system
+
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 deeply.Secondly,
there is only little research onHPPwith surge tank.It 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 follows.In Section 2, the complete
linear mathematical model ofturbine regulating system that includes
all subsystems (i.e.,headrace 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.
2.1. 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.
2.1.1. Turbine Regulating System of Isolated HPP withSurge
Tank
(1) Controlled System [19, 20]. Momentum equation of head-race
tunnel:
=
+
20
0
. (1)
Continuity equation of surge tank:
=
. (2)
Momentum equation of penstock:
=
20
0
. (3)
-
Mathematical Problems in Engineering 3
1++
++
+
+
++
+
Kp
Ki/s
eqx
eqy
eqh
ey
Gs(s)
ex
eh
Mg(s)
eg
X(s)1/Tas
(a) Overall block diagram of turbine regulating system
+
++
Twts
2ht0/H0
Twys
2hy0/H0
TFs
Qt(s) H(s)
(b) Block diagram of pipeline system (with surgetank)
2ht0/H0
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:
=
+
+
,
=
+
+
.
(4)
First derivative differential equation of generator:
= (
+
) . (5)
(2) Turbine Control System [19, 20]. Equation of governor:
=
. (6)
The nomenclatures in (1)(6) are presented inAppendix A.
2.1.2. Turbine Regulating System of Isolated HPPwithout
SurgeTank. Delete (1) and (2) and reformulate (3) to the
followingform:
=
20
0
. (7)
Then (7), (4)(6) are the complete mathematical model ofturbine
regulating system of isolatedHPPwithout surge tank.Note that this
model (see (7), (4)(6)) can as well be obtainedfrom (1)(6) in the
conditions of
= 0,
0= 0, and
= 0.
2.2. Overall Transfer Function. For the situation of
loaddisturbance, the block diagram of turbine regulating system
is determined by the basic equations in Section 2.1 and shownin
Figure 2, where
() = ()/
() is the transfer function
of pipeline system and can be derived from the Laplacetransforms
of (1)(3) forHPPwith surge tank and (7) forHPPwithout surge tank.
is complex variable.
According to Figures 2(a) and 2(b) and the Laplacetransforms of
(1)(6), the following overall transfer functionof turbine
regulating system of isolated HPP with surge tankis obtained:
() =
()
()
=
(03
+ 12
+ 2 +
3) /
05+
14+
23+
32+
4 +
5
,
(8)
where () and () are the Laplace transforms of load
disturbance
and time response of the frequency x,respectively, 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)(6) are as follows:
() =
()
()
=
(2 +
3) /
23+
32+
4 +
5
. (9)
Note that (9) can also be obtained from (8) by letting
= 0, 0
= 0, and = 0. The expressions of coefficients
in (9) are the special cases of those in (8) when ,
0, and
are both 0.
-
4 Mathematical Problems in Engineering
Instability region
Stability region
Stability boundary
Z
X
YKp/K
i(s
)
1/Kp
(a) Stability region in the plane of 1/and
/
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 oscillation,and 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 unstable.However, the stable
oscillation is only restricted to dampedoscillation in practical
projects. This paper uses the latterdefinition.
3.1. Free Oscillation Equation and Stability Criterion. The
sta-bility of regulating system is described by free
oscillationequation and discriminated by stability criterion.
3.1.1. 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:
2
3
3+
3
2
2+
4
+ 5= 0, (10)
0
5
5+
1
4
4+
2
3
3+
3
2
2+
4
+ 5= 0. (11)
3.1.2. 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 discriminants
1> 0 and
2> 0 simultaneously, the system without surge
tank is stable. Similarly, the system with surge tank is
stablein the conditions of
1> 0,
2> 0, and
4> 0.
3.2. 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)
1=
> 0 ( = 2, 3, 4, 5)
1=
> 0 ( = 0, 1, 2, 3, 4, 5)
2=
34
25> 0
2=
12
03> 0
4= (
12
03)(
34
25)
(14
05)2
> 0
the abscissa and ordinate of coordinate plane are selected
as1/
and
/
, 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 system,the variation ranges of
and
0are selected as 04 s (4 s is
the limit value of) and 010%
, respectively. In addition,
the sensitivity analysis of net head is carried out under
largeamplitude of variation (0.67
1.33
) so that the effects of
different operating conditions can be revealed.Aiming at two
cases ofHPPA (with surge tank of real case
and without surge tank of assumed case), the investigationof the
effects of
,
0, and
0on stability is proceeded
by controlling variable method. The default values of ,
0, and
0are 2.0s, 4.0m (4.4%
), and 90m (1.00
),
respectively. The values of other parameters are as follows:=
1.5,
= 1,
= 1,
= 0.5,
= 0,
= 1,
=
8.34 s, = 0, and = 0.9, in which = /
is amplification
coefficient of sectional area of surge tank, and is critical
stable sectional area.
-
Mathematical Problems in Engineering 5
0.0 0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
0.0 0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
0.25 0.30 0.353
4
5
0.0 0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
0.25 0.30 0.353
4
5
Kp/K
i(s
)Kp/K
i(s
)Kp/K
i(s
)
1/Kp
1/Kp
1/Kp
Twt = 1.0 sTwt = 2.0 s
Twt = 3.0 sTwt = 4.0 s
ht0 = 0.0 mht0 = 2.0 mht0 = 4.0 m
ht0 = 6.0 mht0 = 8.0 m
H0 = 60 mH0 = 90 mH0 = 120 m
(a1) Twt
(a2) ht0
(a3) H0
(a) System without surge tank
30
0.0 0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
0.0 0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
S
0.0 0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
Kp/K
i(s
)Kp/K
i(s
)Kp/K
i(s
)
1/Kp
1/Kp
1/Kp
Twt = 0.0 sTwt = 1.0 sTwt = 2.0 s
Twt = 3.0 sTwt = 4.0 s
ht0 = 0.0 mht0 = 2.0 mht0 = 4.0 m
ht0 = 6.0 mht0 = 8.0 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 tank,
has significant effect on stability while the effectsof
0and
0are relatively small. When
increases
from 1 s to 4 s, the stability region reduces obviously;that is,
the stability of system notably worsens. Withthe rise of
0from 0.0m to 8.0m (8.9%
), the
stability region enlarges slightly. On the contrary,
thestability region diminishes slightly if
0increases
from 60m (0.67) to 120m (1.33
).
(2) For the turbine regulating system with surge tank,,
0, and
0all have bigger effects on stability,
especially, 0. The stability region enlarges with
the decrease of
and increase of 0. When
0
decreases from 8.0m (8.9%) to 2.0m (2.2%
), the
stability region enlarges dramatically. It is importantto note
that there is an intersection point between thestability boundary
curves of
0= 0.0m and
0=
2.0m (Point in Figure 4(b2)). In the right sideof Point the
stability region diminishes with theincrease of
0while the change law is just opposite
to the left side of Point .(3) By comparing the turbine
regulating system without
surge tank and that with surge tank, the followingresults can be
obtained. The effect laws of
on
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
0as well as
0on
these two systems are contrary and the influence onsystem with
surge tank is far more sensitive than thatwithout surge tank.
When
,
0, and
0change,
the variation amplitude of stability boundary curvesin the
domain of small 1/
is much greater than that
of big 1/for 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.
4.1. Time Response of the Frequency. The input signal for astep
load disturbance can be computed from
() =
0/,
in which0
is relative value of the load step. Substitution of
() =
0/ 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:
() =
3
=23
5
=25
0
, (12)
() =
3
=03
5
=05
0
. (13)
4.2. Regulation Quality Analysis. By proceeding in a
similarmanner with stability analysis in Section 3.2, 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,
0= 0.1, the time responses of the
frequency of the two turbine regulating systems are shown
inFigure 5, in which
and
are 2.0 and 0.1 s1, respectively,
and other parameters are the same as those in Section 3.2.Note
that the formula of the period of water-level fluctu-
ation in surge tank in frictional headrace tunnel, surge
tanksystem 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 tank.The 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 period,small 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 tank,like
the effects on stability,
has significant effect
on time response of the frequency while the effectsof
0and
0are relatively small. When
rises,
the maximum amplitude, overshoot, and numberof oscillation
enlarge dramatically and regulationquality worsen obviously. The
maximum amplitudeand overshoot increase with the rising of
0and
-
Mathematical Problems in Engineering 7
Table 2: Characteristic parameters for time responses of the
frequency of turbine regulating system with surge tank or not
under0= 0.1.
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)
(s)0 / 0.0318 0.0140 0.0007 323.87 313.911 0.0337 0.0344 0.0141
0.0007 322.21 313.912 0.0416 0.0419 0.0142 0.0007 322.21 313.913
0.0529 0.0534 0.0145 0.0006 322.21 313.914 0.0666 0.0670 0.0145
0.0006 322.21 313.91
0(m)0 0.0395 0.0398 0.0122 0.0010 317.33 310.722 0.0404 0.0409
0.0132 0.0008 318.94 312.214 0.0416 0.0419 0.0142 0.0007 322.21
313.916 0.0427 0.0429 0.0156 0.0006 325.55 315.878 0.0439 0.0443
0.0172 0.0004 328.96 318.15
0(m)60 0.0427 0.0426 0.0227 0.0005 356.99 342.1190 0.0416 0.0419
0.0142 0.0007 322.21 313.91120 0.0410 0.0415 0.0105 0.0013 286.90
305.00
Table 3: Basic information of actual examples of HPP.
HPP Rated power output(MW) Rated head
(m) Rated discharge
(m3/s)
(s)
(s)
0(m)
0(m)
A 51.28 90.00 62.70 17.75 2.33 7.57 5.53B 118.56 177.00 72.50
39.73 1.82 20.53 5.12C 610.00 288.00 228.60 23.84 1.26 12.92
2.91
regulation quality worsens as a consequence. If 0
increases, the maximum amplitude and overshootdecrease and then
regulation quality is improved.
(3) For the turbine regulating systemwith surge tank, theeffect
laws of
,
0, and
0on head wave are the
samewith those on the time response of the frequencyof system
without surge tank;
has almost no
influence on tail wave while the influences of 0and
0are significant.With the increase of
, the period
of tail wave reduces and the maximum amplitudeand attenuation
rate of tail wave enlarge. As a result,regulation quality will get
better or worse. When
0
rises, regulation quality notably worsens because ofthe increase
of period and maximum amplitude andthe decrease of attenuation
rate. In contrast with
0,
the period and the maximum amplitude reduce andattenuation rate
enlarges with the rising of
0, and the
regulation quality notably gets better.In 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 and net
head of turbine regulating
system with surge tank or not under0
= 0.1 correspond-ing to time response of the frequency .
Figure 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 response,and they are larger than that
of time response of thefrequency, especially in the system with
surge tank.This 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 same,while
their regulation qualities are of significant dif-ferences.
-
8 Mathematical Problems in Engineering
0 30 60 90 120 1500.02
0.00
0.02
0.04
0.06
0.08
x
0 30 60 90 120 1500.02
0.00
0.02
0.04
0.06
0.08
x
0 30 60 90 120 1500.02
0.00
0.02
0.04
0.06
0.08
x
t (s)
t (s)
t (s)
Twt = 1.0 sTwt = 2.0 s
Twt = 3.0 sTwt = 4.0 s
(a1) Twt
(a2) ht0
(a3) H0
ht0 = 0.0 mht0 = 2.0 mht0 = 4.0 m
ht0 = 6.0 mht0 = 8.0 m
H0 = 60 mH0 = 90 mH0 = 120 m
(a) System without surge tank
0 200 400 600 800 10000.02
0.00
0.02
0.04
0.06
0.08
x 0 20 40 60 80 1000.000.020.040.060.08
Tail wave
Head wave
0 200 400 600 800 10000.02
0.00
0.02
0.04
0.06
0.08
x 0 20 40 60 801000.000.020.040.06
0 200 400 600 800 10000.02
0.00
0.02
0.04
0.06
0.08
x 0 20 40 60 80 1000.000.020.040.06
t (s)
t (s)
t (s)
Twt = 0.0 sTwt = 1.0 sTwt = 2.0 s
Twt = 3.0 sTwt = 4.0 s
(b1) Twt
(b2) ht0
(b3) H0
ht0 = 0.0 mht0 = 2.0 mht0 = 4.0 m
ht0 = 6.0 mht0 = 8.0 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
5.1. 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 asfollows.The stability and
regulation quality of systemwithoutsurge tank are determined by the
dynamic response (e.g.,time 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
0.3
0.2
0.1
0.0
0.1
0.2Re
spon
ses
t (s)
Frequency xGuide vane opening yNet head h
(a) System without surge tank
0 200 400 600 800 1000
0.3
0.2
0.1
0.0
0.1
0.2
Resp
onse
s
Frequency xGuide vane opening yNet 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
inertia.However,water hammer wave which is low frequency
fluctuationhas little influence on water-level fluctuation in surge
tank.Therefore, 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 surge tank-penstocksubsystem. 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),0is
represented in the form of
0/
0which indicates that
the effect of 0is actualized by serving as the amplification
coefficient of 0(i.e., 1/
0). This result reveals the internal
cause of the opposite effects of 0and
0.
5.2. 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
,
0,
0,
, and
to guarantee preferable stability and regulation
quality.
5.3. 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.
5.3.1. 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
Twts
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
0be 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
(i.e., 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.
5.3.2. Equivalent Model for Regulation Quality of System
withSurge Tank. For regulation quality of system with surge
tank,the head loss and water inertia of penstock are the
principalfactor and secondary factor, respectively. Proceeding
simi-larly as Section 5.3.1, 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:
() =
3
=13
5
=15
0
. (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
is 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 (i.e., =
()) from original fifth order model directly, but alsodifficult to
carry out theoretical analysis. This paper realizes
-
10 Mathematical Problems in Engineering
0.0 0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
Original third order modelEquivalent third order model
Kp/K
i(s
)
1/Kp
(a) HPP B
0.0 0.2 0.4 0.6 0.8 1.00
5
10
15
20
25
30
Original third order modelEquivalent third order model
Kp/K
i(s
)
1/Kp
(b) HPP C
Figure 8: Comparison of stability regions between equivalent
third order model and original third order model.
+
+
+
2ht0/H0
Twys
2hy0/H0
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 solved.Themethod 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 tank.However, 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 wave.Theeffect of
0on stability and regulation quality which
is opposite to that of 0is actualized by serving as the
amplification coefficient of 0(i.e. 1/
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 equation.For regulation quality of system with surge
tank,the fourth order response obtained by neglecting
-
Mathematical Problems in Engineering 11
0 400 800 1200 1600 2000
0.02
0.01
0.00
0.01
0.02
0.03
0.04
x
Original fifth order response modelEquivalent fourth order
response model
t (s)
(a) HPP B
0 400 800 1200 1600 2000
0.02
0.01
0.00
0.01
0.02
0.03
0.04
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 Section.Note the following.
(1) = /0, = (
0)/
0,
= (
0)/
0,
= (
0)/
0, = (
0)/
0, = (
0)/
0,
= (
0)/
0, and
= (
0)/
0are
the relative deviations of corresponding variables.Thesubscript
0 refers to the initial value:
0=
0=
0,
=
0/
0.
(2) The six transfer coefficients are defined as follows:=
/,
=
/,
=
/,
=
/
,
= /, and
=
/.
(3) is actually equal to the relative deviation of load in
the isolated operation. Hence, is regarded as the
load disturbance.
B. Expressions of Coefficients
The expressions of coefficients in overall transfer function(see
(8)) are as follows:
0=
19,
1=
110+
29+
512,
2=
111+
210+
39+
513+
612,
3=
211+
310+
49+
613+
712,
4=
311+
410+
713+
812,
5=
411+
813,
0=
1,
1=
2,
2=
3,
3=
4,
1=
,
2=
[
(1 +
20
0
) +
20
0
] ,
3=
(
+
) +
20
0
(1 +
20
0
) ,
4= 1 +
2 (0
+ 0)
0
,
5=
,
6=
(
20
0
+
20
0
) ,
7=
+
+
20
0
20
0
,
8=
2 (0
+ 0)
0
,
9=
,
10
=
(
)
+
,
11
= ,
12
=
,
13
= .
(B.1)
-
12 Mathematical Problems in Engineering
C. Basic Information of ActualExamples of HPP
See Table 3.
D. Period of Water-Level Fluctuation inSurge Tank in
FrictionalHeadrace Tunnel-Surge Tank System
Based on reference [1], the free oscillation equation of
water-level fluctuation in surge tank in frictional headrace
tunnel-surge tank system is derived as follows:
2
2+ 2
+ 2
= 0, (D.1)
where = (V0/2)[2/
/(
0 2
0)], = (
/
)(1 (2
0/(
0 2
0))), =
0/V2
0, and V
0is flow
velocity in headrace tunnel.The period of water-level
fluctuation in surge tank is
obtained according to (D.1):
= 2/2
2. If the fric-tion is neglected, the formula of period is
simplified to
=
2/
.
Nomenclature
: Change of surge tank water level (positivedirection is
downward)
: Headrace tunnel discharge
: Unit frequency
: Kinetic moment
: Length of headrace tunnel
: Sectional area of headrace tunnel
0: Head loss of headrace tunnel
: Water inertia time constant of headrace
tunnel: Sectional area of surge tank,
,
: Moment transfer coefficients of turbine
: Unit inertia time constant
: Proportional gain
: Net head
: Penstock discharge
: Guide vane opening
: Resisting moment
: Length of penstock
: Sectional area of penstock
0: Head loss of penstock
: Water inertia time constant of penstock
: Time constant of surge tank
,
,
: Discharge transfer coefficients of turbine
: Load self-regulation coefficient
: 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
Nostrand,New York, NY, USA, 2014.
[2] H. M. Paynter, A Palimpsest on the Electronic Analog Art,
GA.Philbrick Researche, Boston, Mass, USA, 1955.
[3] L. M. Hovey, Optimum adjustment of hydro governors
onmanitoba hydro system, IEEETransactions on Power Apparatusand
Systems, vol. 81, no. 3, pp. 581587, 1962.
[4] M. H. Chaudhry, Governing stability of a hydroelectric
powerplant,Water Power, vol. 22, no. 4, pp. 131136, 1970.
[5] H. V. Pico and J. McCalley, Modeling and analysis of
speedcontrols in hydro-turbines for frequency performance,
inProceedings of the North American Power Symposium, pp. 17,Boston,
Mass, USA, August 2011.
[6] D. H. Thorne and E. F. Hill, Extensions of stability
boundariesof a hydraulic turbine generating unit, IEEE Transactions
onPower Apparatus and Systems, vol. 94, no. 4, pp. 14011409,
1975.
[7] D. T. Phi, E. J. Bourque, D. H. Thorne, and E. F. Hill,
Analysisand application of the stability limits of a
hydro-generatingunit, IEEE Transactions on Power Apparatus and
Systems, vol.100, no. 7, pp. 32013211, 1981.
[8] N. S. Dhaliwal and H. E.Wichert, Analysis of PID governors
inmulti-machine system, IEEE Transactions on Power Apparatusand
Systems, vol. 97, no. 2, pp. 456463, 1978.
[9] S. Hagihara, H. Yokota, K. Goda, and K. Isobe, Stability of
ahydraulic turbine generating unit controlled by PID governor,IEEE
transactions on power apparatus and systems, vol. 98, no.6, pp.
22942298, 1979.
[10] K. A. Naik, P. Srikanth, and A. K. Chande, A novel
governorcontrol for stability enhancement of hydro power plant
withwater hammer effect, in International Conference on
EmergingTrends in Electrical and Computer Technology, pp. 4045,
TamilNadu, India, March 2011.
[11] T. Stein, Frequency control under isolated network
conditions,Water Power, vol. 22, no. 9, pp. 320324, 1970.
[12] F. O. Ruud, Instability of a hydraulic turbine with a very
longpenstock, Journal of Engineering for Gas Turbines and
Power,vol. 87, no. 3, pp. 290294, 1965.
[13] M. S. R. Murty and M. V. Hariharan, Analysis and
improve-ment of the stability of a hydro-turbine generating unit
withlong penstock, IEEE Transactions on Power Apparatus andSystems,
vol. 103, no. 2, pp. 360367, 1984.
[14] O. H. Souza and N. Barbieri, Study of hydraulic transients
inhydropower plants through simulation of nonlinear model
ofpenstock and hydraulic turbine model, IEEE Transactions onPower
Systems, vol. 14, no. 4, pp. 12691272, 1999.
[15] C. K. Sanathanan, Accurate low order model for
hydraulicturbine-penstock, IEEE Transactions on Energy
Conversion,vol. 2, no. 2, pp. 196200, 1966.
[16] G. I. Krivehenko, E. V. Kwyatkovskaya, A. E. Lyubitsky,
andS. V. Ostroumov, Some special conditions of unit operationin
hydropower plant with long penstocks, in Proceedingsof 8th
Symposium of IAHR Section for Hydraulic Machinery,Equipment and
Cavitation, pp. 465475, Leningrad, Russia,September 1976.
-
Mathematical Problems in Engineering 13
[17] L. Fu, J.-D. Yang, H.-Y. Bao, and J.-P. Li, Generator
unitfrequency fluctuations of hydropower station with surge
tankunder load disturbance, Journal of Hydraulic Engineering,
vol.39, no. 11, pp. 11901196, 2008.
[18] L. Fu, J. P. Li, J. D. Yang, and H. Y. Bao, Research on
dynamicquality of governing system of hydropower station with
tailracesurge tank, Journal of Hydroelectric Engineering, vol. 29,
no. 2,pp. 163176, 2010.
[19] V. L. Streeter and E. B. Wylie, Fluid Transients,
McGraw-Hill,New York, NY, USA, 1978.
[20] IEEE Working Group, Hydraulic turbine and turbine
controlmodels for system dynamic studies, IEEE Transactions onPower
Systems, vol. 7, no. 1, pp. 167179, 1992.
[21] G. F. Franklin, D. Powell, and A. E. Naeini, Feedback
Control ofDynamic Systems, Prentice Hall, Upper Saddle River, NJ,
USA,2009.
[22] S. P.Wei,Hydraulic 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 athttp://www.hindawi.com
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttp://www.hindawi.com
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Probability and StatisticsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
OptimizationJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
CombinatoricsHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing
Corporationhttp://www.hindawi.com Volume 2014
International Journal of Mathematics and Mathematical
Sciences
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
The Scientific World JournalHindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume
2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014
Stochastic AnalysisInternational Journal of
