water
Article
Numerical Modeling and Hydraulic Optimization of
a Surge Tank Using Particle Swarm Optimization
Khem Prasad Bhattarai , Jianxu Zhou * , Sunit Palikhe , Kamal Prasad Pandey
and Naresh Suwal
College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China;
kpb63@outlook.com (K.P.B.); sunit.palikhe@hhu.edu.cn (S.P.); lotuskamal44@hotmail.com (K.P.P.);
narcissussuwal@gmail.com (N.S.)
* Correspondence: jianxuzhou@163.com
Received: 25 February 2019; Accepted: 4 April 2019; Published: 6 April 2019
Abstract: In a pressurized water conveyance system, such as a hydropower system, during hydraulic
transients, maximum and minimum pressures at various controlling sections are of prime concern
for designing a safe and efficient surge tank. Similarly, quick damping of surge waves is also very
helpful for the sound functioning of the hydro-mechanical system. Several parameters like diameter
of the surge tank, diameter of the orifice, operating discharge, working head, etc., influence the
maximum/minimum surge, damping of surge waves in the surge tank, and the difference of maximum
pressure head at the bottom tunnel and maximum water level in the surge tank. These transient
behaviors are highly conflicting in nature, especially for different diameters of orifices (DO ) and
diameters of surge tanks (DS ). Hence, a proper optimization method is necessary to investigate
the best values of DO and DS to enhance the safety and efficiency of the surge tank. In this paper,
these variables are accurately determined through numerical analysis of the system by the Method
of Characteristics (MOC). Furthermore, the influence on the transient behavior with changing DO
and DS is investigated and finally, optimum values of DO and DS are determined using Particle
Swarm Optimization (PSO) to minimize the effects of hydraulic transients on the system without
compromising the stability and efficiency of the surge tank. The obtained results show significant
improvements over the contemporary methods of finding DO and DS for surge tank design.
Keywords: numerical analysis; method of characteristics; hydraulic transients; surge analysis; surge
tank; particle swarm optimization
1. Introduction
Hydraulic transients occur whenever there is a disturbance in the steady state condition of a
system, causing variations in pressure and discharge in the system. The control of hydraulic transients
in pressurized systems like water supply, wastewater, and hydropower systems, is a major concern
for engineers regarding the safety and effective operation of the system throughout the life span of
the project. Uncontrolled hydraulic transients in the system have led to several cases of accidents in
hydropower stations in the past [1–3]. A number of methods and models have been developed for the
analysis of hydraulic transients, like the graphical method, the algebraic method, the implicit method,
the wave characteristic method, the Method of Characteristics (MOC), the Zielke model, the Brunone
model, Surge2012, and much more commercial software [4–9]. Calamak et al. [10] developed a
computer program using MOC, which accurately predicted pressures for various simulations in case
studies having a Pelton and Francis turbine. Martin et al. [11] performed 3-D and 1-D simulations for
various unsteady friction formulations and discovered that the convolution-based unsteady friction
models, which consider a set of previous time steps in simulation, perform better than the instantaneous
Water 2019, 11, 715; doi:10.3390/w11040715
www.mdpi.com/journal/water
Water 2019, 11, 715
2 of 19
acceleration-based models. Among these methods, MOC is a very popular method, as it gives accurate
results with a shorter computational time for analyzing the hydraulic transients in a pipeline system,
and can be conveniently used in computer modeling [12,13]. In this study, MOC is used, considering
the steady-state friction along the pipe, for numerical modeling and hydraulic transient analysis of
the given generalized hydropower system, where the transient is generated by gradual closing and
opening of the valve.
Researchers have extensively studied hydraulic behaviors of the pressurized system to discover
methods for improvements and successfully tackle hydraulic transient complications in the system.
Yu et al. [14] performed 1-D analysis of a complex differential surge tank for a long diversion-type
hydropower station using MOC and concluded that the high pressure difference produced between
two sides of breast wall during transient conditions can be effectively reduced by setting appropriate
pressure reduction orifices. Guo et al. [15] reviewed transient processes and safety measures for
hydropower stations with long headrace tunnels, and revealed that the maximum pressure in
hydropower stations with long headrace tunnels depends on the maximum water level in the surge
tank, rather than the closing mechanism of the guide vane. Zhou et al. [16] introduced the addition of
interconnecting holes in a differential surge tank, which significantly reduced the pressure on the breast
wall and made gate maneuver much easier and more effective. Covas and Ramos [17] used inverse
transient analysis for the identification of the location of a leak in a water pipeline system through
a transient generated by fast change in flow conditions. Balacco et al. [18] carried out laboratory
experiments on an undulating pipeline consisting of an orifice for air venting, creating a complex
air-water environment. They analyzed the pressure surge build during rapid filling of the pipeline and
found that the maximum pressure is reached during the mass oscillation phase. A plethora of research
could be found on the sensitivity analysis of parameters like diameter of the orifice (DO ), diameter of
the surge tank (DS ), working head, operating discharge, valve closure mechanism, pipe material and
diameter, etc., for various transient effects [19–23]
Hydraulic transients in pressurized systems are complex phenomena which are interdependent
of several parameters, and also highly site-specific. Thus, designing an effective and well-optimized
hydraulic system is always a challenging work. Several attempts of optimizing the orifice in a surge tank
have been made using 1-D and 3-D simulations [24–27]. Similarly, different optimizing tools have been
successfully employed to obtain the most advantageous design of hydraulic systems. Afshar et al. [28]
applied the NSGA-II optimization algorithm for developing an optimized closing-rule curve for valves
to control the effects of a water hammer in pressurized pipelines. Kim et al. [29] implemented the
genetic algorithm (GA) with the impulse response method, and then optimized the location and
dimensions of a surge tank considering the safety and cost of the system. Fathi-Moghadam et al. [30]
employed GA to optimize the diameters of a headrace tunnel, penstocks, and surge tank considering
the benefit-cost ratio as the objective function. Their model successfully optimized the diameters in a
system consisting of branching tunnels and penstocks with a surge tank. Jung et al. [31] integrated
both GA and PSO simultaneously with hydraulic transient analysis to discover the best location
and combination of transient protection devices in a pipe network. Similarly, Moghaddas et al. [32]
compared central force optimization (CFO) and GA for optimizing the pipeline protection cost of a water
transfer project considering hydraulic transients induced during pump maneuver. They optimized
the volume of the air chamber along with the location and type of air-inlet valves with minimum
cost as the objective function. Generally, researchers have performed the optimization of a surge tank
considering the cost of the project as the objective function and including some hydraulic transient
characteristics, while optimization dedicated to reducing the severe effects of the extreme transient
conditions is very limited. This study solely emphasizes the hydraulic transient characteristics of the
system and uses PSO for optimization of the diameter of the surge tank (DS ) and diameter of the
orifice (DO ). Generally, researchers have used GA for optimization, while PSO has been proved to
be more convenient and effective. GA is more suitable for discrete optimization and has been more
rigorously studied and applied in diverse fields due to its much earlier introduction (in 1989) than other
Water 2019, 11, 715
3 of 19
modern optimization techniques, while PSO is a much more recent technique (in 1995) and suitable for
problems and has a higher converging speed towards the solution3[33–36].
of 19
Hassan et al. [37] compared PSO and GA and discovered that the computational effort and efficiency
solution
is for
significantly
less athan
that of GA
to reach
the same high-quality
solution.
Similarly,
required
PSO to reach
high-quality
solution
is significantly
less than that
of GA to
reach the
Kecskes
et al. [38] and
Duan etSimilarly,
al. [39] also
compared
PSOand
andDuan
GA on
their
models
and
same high-quality
solution.
Kecskes
et al. [38]
et al.
[39]respective
also compared
PSO
and
discovered
that
PSO
was
better
and
has
a
high
convergence
ratio
compared
to
GA.
On
the
other
hand,
GA on their respective models and discovered that PSO was better and has a high convergence ratio
every
engineering
has their
own
unique
characteristics
which
demand
in
compared
to GA.problem
On the other
hand,
every
engineering
problem
has their
owncertain
uniquealterations
characteristics
general
optimization
procedures.
In
this
study,
we
have
attempted
to
optimize
the
surge
tank
based
which demand certain alterations in general optimization procedures. In this study, we have attempted
onto
itsoptimize
hydraulicthe
transient
characteristics
with
a highlytransient
non-linear
solution andwith
constrained
surge tank
based on its
hydraulic
characteristics
a highlyvariables.
non-linear
In solution
these conditions,
PSO is much
easierIntothese
formulate
and apply
formuch
optimization
it acquires
the
and constrained
variables.
conditions,
PSO is
easier to and
formulate
and apply
global
optimization
value
quickly
and
conveniently
with
fairly
agreeable
results.
for optimization and it acquires the global optimization value quickly and conveniently with fairly
In this study,
agreeable
results.a generalized system of a high-head hydropower station is selected for analysis
during In
extreme
casesa generalized
of hydraulicsystem
transients.
The system
consists ofstation
a constant-head
this study,
of a high-head
hydropower
is selected upstream
for analysis
reservoir,
headrace
tunnel,
orifice
surge
tank,
penstock
pipe,
and
turbine
at
the
end
of
the
penstock.
during extreme cases of hydraulic transients. The system consists of a constant-head upstream reservoir,
Because
thistunnel,
study mainly
focuses
on the
surge pipe,
analysis
hydraulic
of the surge
tank,
headrace
orifice surge
tank,
penstock
andand
turbine
at the optimization
end of the penstock.
Because
this
thestudy
downstream
turbine
can
be
simplified
into
a
valve
boundary,
as
shown
in
Figure
1.
First,
1-D
mainly focuses on the surge analysis and hydraulic optimization of the surge tank, the downstream
numerical
modeling
and analysis
the boundary,
hydraulic transients,
occurring
to 1-D
valve
operationmodeling
of the
turbine can
be simplified
into a of
valve
as shown in
Figure 1.due
First,
numerical
system,
is carried
outhydraulic
by using MOC.
Gradual
closing
of to
thevalve
valveoperation
during the
maximum
levelout
and analysis
of the
transients,
occurring
due
of the
system, water
is carried
in by
theusing
reservoir
and
gradual
opening
of
the
valve
during
the
minimum
water
level
in
the
reservoir
MOC. Gradual closing of the valve during the maximum water level in the reservoir and
aregradual
considered
as the
twovalve
worstduring
cases, the
which
give the
maximum
water
level in as
thethe
opening
of the
minimum
water
level inand
the minimum
reservoir are
considered
surge
tank,
respectively.
Secondly,
major
hydraulic
transient
behaviors
in
the
system
are
studied
and
two worst cases, which give the maximum and minimum water level in the surge tank, respectively.
analyzed.
Secondly, major hydraulic transient behaviors in the system are studied and analyzed.
continuous
Water
2019, 11, 715optimization
Figure 1. General layout of the hydropower system.
Figure 1. General layout of the hydropower system.
Finally, the necessary objective functions and constraints are developed regarding the transient
Finally, the necessary objective functions and constraints are developed regarding the transient
behavior of the system, and optimization is performed using the Particle Swarm Optimization (PSO)
behavior of the system, and optimization is performed using the Particle Swarm Optimization (PSO)
method to find the optimum diameters of the surge tank (DS ) and orifice (DO ). The results obtained
method to find the optimum diameters of the surge tank (DS) and orifice (DO). The results obtained
are compared with the contemporary methods of finding DO and DS , and discussions are presented.
are compared with the contemporary methods of finding DO and DS, and discussions are presented.
2. Material and Methods
2. Material and Methods
2.1. Governing Equations
2.1. Governing Equations
The 1-D governing equations for the physics of hydraulic transients are the equations of motion
The
1-D governing
equations
for the physics
of hydraulic
transients
are the equations
of motion
and
continuity,
also called
the Saint-Venant
equations
[12]. These
are hyperbolic
quasilinear
partial
and
continuity,
also called
thehaving
Saint-Venant
equations
[12]. they
These
hyperbolic
quasilinear
differential
equations,
and,
no analytical
solution,
areare
most
accurately
solved bypartial
MOC.
differential equations, and, having no analytical solution, they are most accurately solved by MOC.
fV V
∂H 𝜕𝑉
∂V 𝑓𝑉ǀ𝑉ǀ
𝜕𝐻
(1) (1)
𝑔g ∂x ++ ∂t++ 2D= 0,= 0,
𝜕𝑥 𝜕𝑡
2𝐷
2 ∂V 𝜕𝐻∂H
𝑐 c𝜕𝑉
+ + = 0,
= 0,
𝑔 g𝜕𝑥∂x 𝜕𝑡 ∂t
(2) (2)
where H is the piezometric head; V is the mean flow velocity; f is the Darcy-Weisbach friction factor;
D is the diameter of the pipe; g is the acceleration due to gravity; and c is the wave speed.
Water 2019, 11, 715
4 of 19
where H is the piezometric head; V is the mean flow velocity; f is the Darcy-Weisbach friction factor;
D is the diameter of the pipe; g is the acceleration due to gravity; and c is the wave speed.
WaterMethod
2019, 11,of715
2.2.
Characteristics
4 of 19
For 1-Dofnumerical
modeling of the hydropower system, the most widely used method called
2.2. Method
Characteristics
Method of Characteristics (MOC) is applied, as proposed by Wylie et al. [12], which converts the
For 1-D numerical modeling of the hydropower system, the most widely used method called
equations of motion and continuity into two ordinary differential equations that are then solved as a
Method of Characteristics (MOC) is applied, as proposed by Wylie et al. [12], which converts the
simple form, Equations (3) and (4), by the finite-difference method.
equations of motion and continuity into two ordinary differential equations that are then solved as a
simple form, Equations (3) and (4), by the
: HP = CP − BQmethod.
(3)
C+finite-difference
P
i
i
𝐶 : 𝐻 = 𝐶 − 𝐵𝑄
(3)
C− : HPi = CM + BQPi
(4)
𝐶 : 𝐻 = 𝐶 + 𝐵𝑄
(4)
CP = Hi−1 + BQi−1 − RQi−1 |Qi−1 |
(5)
(5)
𝐶 = 𝐻 + 𝐵𝑄 − 𝑅𝑄 |𝑄 |
CM = Hi+1 − BQi+1 + RQi+1 Qi+1
(6)
(6)
𝐶 = 𝐻 − 𝐵𝑄 + 𝑅𝑄 |𝑄 |
c
B= 𝑐
(7)
gA
𝐵=
(7)
𝑔𝐴
f ∆x
R = 𝑓∆𝑥2
(8)
𝑅 = 2gDA
(8)
2𝑔𝐷𝐴
where Hi and Qi are the piezometric head and discharge, respectively, at any time and location; A is
where Hi and Qi are the piezometric head and discharge, respectively, at any time and location; A is
the area of the pipe; and f is the coefficient of friction.
the area of the pipe; and f is the coefficient of friction.
As shown in Figure 2, the characteristic lines C+ and C− intersect at point P. Thus, with the known
As shown in Figure 2, the characteristic lines C+ and C− intersect at point P. Thus, with the known
initial values during a steady state condition, Equations (3)–(8) give the values of head and discharge
initial values during a steady state condition, Equations (3)–(8) give the values of head and discharge
at a particular time and position in the x-t grid.
at a particular time and position in the x-t grid.
Figure
Figure 2.
2. Characteristic
Characteristic lines
lines in
in x-t
x-t grid.
grid.
2.3. Boundary Conditions
2.3. Boundary Conditions
In this study, we considered a simple model of a high-head hydropower system for hydraulic
In this study, we considered a simple model of a high-head hydropower system for hydraulic
transient analysis and optimization of its hydraulic characteristics. 1-D numerical analysis was carried
transient analysis and optimization of its hydraulic characteristics. 1-D numerical analysis was
out using MATLAB. Based on the layout of the hydropower system shown in Figure 1, the following
carried out using MATLAB. Based on the layout of the hydropower system shown in Figure 1, the
boundary conditions were established, as described by Wylie et al. [12].
following boundary conditions were established, as described by Wylie et al. [12].
2.3.1. Reservoir
2.3.1. Reservoir
The system consists of an upstream reservoir and it is assumed that water level of the reservoir
Theconstant
system consists
of an upstream
reservoir and it is assumed that water level of the reservoir
remains
during transient
evaluation.
remains constant during transient evaluation.
H𝐻P1 =
=Z
𝑍res
(9)
(9)
thepiezometric
piezometrichead
headatatthe
thereservoir
reservoirand
andZZ
thewater
waterlevel
levelofofthe
thereservoir.
reservoir.
where H𝐻P1 isisthe
where
resresisisthe
2.3.2. In-Line Surge Tank
In the pressurized water conveyance system, the surge tank is a very important structure to
effectively control the hazardous effects of hydraulic transients. In this study, the restricted orifice
surge tank, as shown in Figure 3,is located at the end of the headrace tunnel. The equations defining
Water 2019, 11, 715
5 of 19
2.3.2. In-Line Surge Tank
In the pressurized water conveyance system, the surge tank is a very important structure to
effectively control the hazardous effects of hydraulic transients. In this study, the restricted orifice
surge tank, as shown in Figure 3, is located at the end of the headrace tunnel. The equations defining
the boundary conditions of the restricted orifice surge tank are presented below [12]:
HP = ZS + RS |QS |QS
(10)
ZS = W (QS + QS0 ) + ZS0
(11)
Q1 = Q2 + QS
(12)
RS =
1
2gCd 2 AO 2
∆t
2AS
s
Lt AS
T = 2π
gAt
W=
(13)
(14)
(15)
where Q1 and Q2 are the discharges in pipe 1 and pipe 2, respectively, at the point of surge tank; QS is
the discharge flowing into the surge tank; HP is the piezometric head in the pipe at the bottom of the
surge tank; ZS is the water level in the surge tank, also referred to as surge head in further sections;
QS0 and ZS0 are the previous discharge and water level in the surge tank, respectively; AO and AS
are the area of orifice and area of surge tank, respectively; Cd is the coefficient of discharge; ∆t is the
iteration time period considered for simulation; T is the time period for mass oscillation in the surge
tank; and Lt and At are the length and area of the headrace tunnel, respectively. Equations (3), (4),
Water(10)–(12)
2019, 11, 715
and
are solved together to get the values of the variables at the surge tank boundary. 5 of 19
Figure 3. Restricted orifice surge tank.
Figure 3. Restricted orifice surge tank.
2.3.3. Free Discharge Valve at the End
(10)
𝐻 and
= 𝑍 closing
+ 𝑅 |𝑄of|𝑄the valve causes transients. For gradual
In this model, linear gradual opening
closure, the time period of closure of the valve is more than the pipe characteristic time, i.e., Tc > 2L
.
(11)c
𝑍 = 𝑊(𝑄 + 𝑄 ) + 𝑍
The time of closure/opening of the valve for general surge analysis and the optimization procedure
(12)
𝑄 = 𝑄 used
+ 𝑄 closing time of guide vanes in hydropower
is considered to be 10 s according to the practically
stations. In surge analysis, the closing/opening of valve
1 is started at 5 s and the valve is completely
= in Section 3.5, an elaborated analysis is conducted
(13)
closed/opened at 15 s in linear progression. 𝑅Later,
2𝑔𝐶 𝐴
to demonstrate the transient changes for various closing/opening times of the valve, including both
Δ𝑡
𝑊=
(14)
2𝐴
𝑇 = 2𝜋
𝐿𝐴
𝑔𝐴
(15)
where Q1 and Q2 are the discharges in pipe 1 and pipe 2, respectively, at the point of surge tank; QS
is the discharge flowing into the surge tank; HP is the piezometric head in the pipe at the bottom of
the surge tank; ZS is the water level in the surge tank, also referred to as surge head in further sections;
Water 2019, 11, 715
6 of 19
fast and gradual closure/opening. The equations determining the valve boundary conditions are as
follows [12].
q
QPN+1 = −BCV +
CV =
(BCV )2 + 2CV CP
( Q0 τ ) 2
2H0
(16)
(17)
where QPN+1 is the discharge at the valve; τ is the valve opening; Q0 is the steady state flow; and H0 is
the operating head across the valve, and for a free discharge valve, H0 is the piezometric head at the
inlet of the valve.
2.4. Stability Criteria
The hydraulic transient causes pressure and discharge fluctuations in the surge tank. During
such conditions, the surge tank is vulnerable to various instabilities. For a surge tank to be stable and
effective, the following criteria must be fulfilled:
2.4.1. Minimum Area of Surge Tank
During hydraulic transients, fluctuation of the water level occurs in the surge tank, which must
be dampened in a reasonably short time to ensure the stability of the system. In addition to this,
the maximum and minimum water level in the surge tank should be properly analyzed so that the
system stays safe in the worst condition. For this, Thoma derived an equation called the Thoma
equation, which gives the minimum area of a stable surge tank [40]. However, Jaeger later proved
that this area is not sufficient for the stability of a surge tank and introduced a safety coefficient in the
Thoma equation [41].
Lt At
Ath = e
(18)
2αgH1
e = 1 + 0.482
α=
Zmax
Ho
H w o At 2
Q2
H1 = Ho − Hwo − 3Hwm
(19)
(20)
(21)
where Ath is the Thoma area; e is the safety coefficient; Lt and At are the length and area of the headrace
tunnel, respectively; Ho is the gross head in the turbine; Hwo is the head loss in the headrace tunnel;
Hwm is the head loss in the penstock; Zmax is the maximum water level in the surge tank; and Q is the
discharge in the tunnel.
2.4.2. Vortex Control
In the case of start-up of the pressurized water conveyance system, the surge tank acts as a
temporary intake, supplying discharge to the system. This causes hydraulic transients in the system
and the water level in the surge tank falls down to its minimum level. During such conditions,
if the water level falls below a certain elevation, there is a very high risk of vortex formation or air
entrainment, which can have devastating effects on the system. The following steps were followed to
check the vortex formation in the system after the optimized values of DO and DS were obtained:
1.
2.
The minimum possible down surge that can be created in the system is calculated, which is
defined as objective function 2 in Section 2.5.3;
From this minimum water surface elevation, the bottom surface elevation of the surge tank is
subtracted to obtain the height of the minimum water column in the tank;
Water 2019, 11, 715
3.
7 of 19
Then, the critical submergence for the possible suction vortex is calculated using the Gordon
equation [42]:
√
(22)
Scr = cV d
where Scr is the critical submergence depth; c is an empirical coefficient whose value ranges from
0.55 to 0.73 (c = 0.55 for symmetrical intake); V is the velocity at intake; and d is the diameter of
the pipe at intake, and for this case, the velocity and the diameter of penstock pipe are considered
to check vortex formation in the surge tank.
The minimum water column depth in the surge tank must be greater than the critical submergence
depth (Scr ) to prevent the formation of a harmful vortex in the system.
2.5. Optimization
First, 1-D numerical modeling and analysis of the hydropower system were conducted for linear
gradual closing and gradual opening of the valve, respectively, using MATLAB and the values of the
piezometric head and discharge at the surge tank were obtained using MOC. From these obtained
values, the following objective functions and constraints were developed for optimizing the diameter
of the surge tank (DS ) and the diameter of the orifice (DO ).
2.5.1. Objective Function 1
The maximum water level in the surge tank (Max1) is reached during full load rejection, as shown
in Figure 4a, when the system is working with full capacity and the reservoir is at the highest water
level. This value is obtained in this study by 1-D numerical modeling of the system using MOC.
This value gives the maximum possible water level in the surge tank and determines the height and
performance of the surge tank during the worst transient condition and hence, plays a vital role in
surge tank design. Minimization of this maximum value is taken as the first objective function in this
study. Many researchers have derived empirical or stochastic equations for simplicity in the calculation
of the maximum surge head in the surge tank [43–46], but MOC gives the most accurate results for
surge analysis from which the maximum value can be easily obtained.
Minimize: F1 (x1 , x2 )
where F1 (x1 , x2 ) is the function for the maximum water level for different diameters of the orifice
(x1 ) and diameters of the surge tank (x2 ).
2.5.2. Objective Function 2
The minimum water level in the surge tank (Min1) is reached during full load acceptance, as shown
in Figure 4b, when the system is put into operation with minimum capacity and the reservoir is
at the lowest water level. This gives the value of the lowest possible water level in the surge tank,
or the lowest submergence depth, which essentially determines the safety of the system from harmful
vortices or air entrainments. This minimum value, obtained after surge analysis of the worst scenario
of transients for the lowest possible water level in the surge tank using MOC, is assigned as objective
function 2. The value of the lowest submergence depth should be increased to ensure effective working
of the system, even in severe hydraulic transients.
Minimize: −F2 (x1 , x2 )
where F2 (x1 , x2 ) is the function for minimum surge pressure for different diameters of the orifice
(x1 ) and diameters of the surge tank (x2 ) (negative sign is for minimization).
2.5.3. Objective Function 3
Whenever hydraulic transients occur in the system, oscillation of pressure and discharge is
observed, which is gradually dampened by friction and various surge controlling devices, like the
air chamber, surge tank, pressure release valves, etc. Damping of surge waves is a very crucial
characteristic of the surge tank and effective surge tanks are required to dampen the surge waves as
Water 2019, 11, 715
8 of 19
quickly as possible. In this study, we performed surge analysis for the worst condition of transients
that gives the maximum water level in the surge tank, as explained in Section 2.5.1, extracted two
successive values of maximum water level as shown in Figure 4a, and calculated the percentage of
damping of surge waves using Equation (23).
D=
Max1 − Max2
× 100%
Max1
(23)
where D is the percentage of surge pressure damping; and Max1 and Max2 are the first and second
maximum water level during full load rejection, respectively, as shown in Figure 4a.
Minimize: −F3 (x1 , x2 )
3 (x1 , x2 ) is the function for the percentage of damping obtained for different values
Water where
2019, 11,F715
8 of of
19
diameter of the orifice (x1 ) and diameter of the surge tank (x2 ) (negative sign is for minimization).
These three
functions
are combined
into a single
usingfunction
normalization.
threeobjective
objective
functions
are combined
into objective
a singlefunction
objective
using
normalization.
Figure 4. Water level vs time graph for a restricted orifice surge tank during linear gradual closure of
the valve (a) and linear gradual opening
opening of
of the
the valve
valve (b).
(b).
2.6.
Normalization
2.6. Normalization
The
objective
functions
chosen
in this
study
is highly
conflicting
and the
range
The nature
natureofofthe
thethree
three
objective
functions
chosen
in this
study
is highly
conflicting
and
the
of
their
values
is
also
very
diverse,
as
shown
in
Section
3.2.
Hence,
normalization
was
done
so
that
range of their values is also very diverse, as shown in Section 3.2. Hence, normalization was done all
so
objective
functions
fell
into
a
similar
range,
and
could
be
combined
under
a
single
objective
function.
that all objective functions fell into a similar range, and could be combined under a single objective
The
unity-based
normalization
was used,was
which
brings
all brings
the values
in the
range
1]. of [0, 1].
function.
The unity-based
normalization
used,
which
all the
values
in of
the[0,
range
=
F𝐹iN ((𝑥
x1 ,,x𝑥2 )) =
(x1 ,, 𝑥x2 )) −
𝐹
− F𝐹i,min
Fi (𝑥
,
𝐹
−
𝐹
Fi,max
−
F
,
,
i,min
(24)
(24)
(25)
𝐹(𝑥 , 𝑥 ) =X
∑𝐹 (𝑥 , 𝑥 )
F(x1 , x2 ) =
FiN (x1 , x2 )
(25)
where F(x1,x2) is the normalized objective function; FiN(x1,x2) represents the individual objective
where F(x1 ,x2 ) is the normalized objective function; FiN (x1 ,x2 ) represents the individual objective
functions; and Fi,max and Fi,min are the maximum and minimum values of each objective function within
functions; and Fi,max and Fi,min are the maximum and minimum values of each objective function
the given range, respectively.
within the given range, respectively.
2.7. Variables and Constraints
2.7. Variables and Constraints
This study intends to find a well-optimized surge tank which effectively overcomes all the
This study intends to find a well-optimized surge tank which effectively overcomes all the
hydraulic transient complications. The diameter of the orifice (DO) and the diameter of the surge tank
hydraulic transient complications. The diameter of the orifice (DO ) and the diameter of the surge
(DS) are considered as variables during the optimization. The lower and upper limits of these
tank (DS ) are considered as variables during the optimization. The lower and upper limits of these
diameters are determined as follows:
diameters are determined as follows:
According to the “Chinese design code for surge chamber of hydropower stations”, the
According to the “Chinese design code for surge chamber of hydropower stations”, the minimum
minimum and maximum area of the orifice must be 25% and 45% of the area of the headrace tunnel,
and maximum area of the orifice must be 25% and 45% of the area of the headrace tunnel, respectively,
respectively, for optimum performance of the surge tank [19,47]. Hence, we obtained,
for optimum performance of the surge tank [19,47]. Hence, we obtained,
•
3.1 𝑚 ≤ 𝐷 ≤ 4.1 𝑚
For the diameter of surge tank (Ds), the minimum value is obtained by the Thoma area, as
discussed in Section 2.4.1, and the maximum value is taken to be 12 m based on the preliminary
observations, as presented in Section 2.8, and the results obtained during optimization. Thus, we
have,
Water 2019, 11, 715
9 of 19
3.1 m ≤ DO ≤ 4.1 m
For the diameter of surge tank (Ds), the minimum value is obtained by the Thoma area, as discussed
in Section 2.4.1, and the maximum value is taken to be 12 m based on the preliminary observations, as
presented in Section 2.8, and the results obtained during optimization. Thus, we have,
6.3 m ≤ DS ≤ 12 m
Besides these, the difference between the maximum piezometric head at the bottom tunnel of the
surge tank and the maximum water level of the surge tank plays a crucial role in designing an efficient
surge tank. In this study, the DO and DS are accepted only when the difference between the maximum
head at the bottom tunnel and the maximum water level in the surge tank is less than 1 m. When the
difference is higher, the surge tank is not effective and the headrace tunnel becomes more vulnerable to
Water 2019, effects
11, 715 [47].
9 of 19
transient
max(HP ) − max(ZS ) ≤ 1 m
where
𝐻 is
piezometric
head
at the
bottom
tunnel
of of
the
surge
where
HPthe
is the
piezometric
head
at the
bottom
tunnel
the
surgetank
tankand
and𝑍ZS isis the
the water
water level
level
in
the
surge
tank.
in the surge tank.
2.8. Acceptance
2.8.
Acceptance Region
Region
The constraints
constraints described
Section 2.7
2.7 define
define the
the total
total population
population for
for optimization
optimization and
and also
also
The
described in
in Section
distinguish
valid
and
invalid
regions
based
on
the
difference
of
the
maximum
piezometric
head
at
distinguish valid and invalid regions based on the difference of the maximum piezometric head at
the bottom
bottom tunnel
tunnel of
of the
the surge
surge tank
tank and
and the
the maximum
maximum water
water level
level in
in the
the surge
surge tank.
tank. Figure
5 shows
shows
the
Figure 5
various
values
of
D
O
and
D
S
within
the
range
of
optimization
discussed
earlier.
The
green
area
various values of DO and DS within the range of optimization discussed earlier. The green area in
in the
the
graph shows
shows the
the valid
validregion,
region,where
wherethe
thevalue
valueofofthe
thedifference
difference
maximum
heads
is less
than
1
graph
ofof
thethe
maximum
heads
is less
than
1 m,
m, and
the area
red area
the graph
the invalid
the difference
of the maximum
and
the red
in theingraph
showsshows
the invalid
region,region,
havinghaving
the difference
of the maximum
heads of
headsthan
of more
than eventually
1 m. This makes
eventually
makes the optimization
problem
highly
and
more
1 m. This
the optimization
problem highly
non-linear
andnon-linear
discontinuous
discontinuous
and objective
the individual
objective
functions
are highly
conflicting in nature.
and
the individual
functions
are highly
conflicting
in nature.
Figure 5.
5. Acceptable
Acceptable (green)
(green) and
and unacceptable
unacceptable (red)
(red) region
region in
in the
Figure
the total
total population.
population.
2.9. Particle Swarm Optimization
2.9. Particle Swarm Optimization
The Particle Swarm Optimization (PSO) is a nature-inspired, metaheuristic stochastic optimization
The Particle Swarm Optimization (PSO) is a nature-inspired, metaheuristic stochastic
tool developed by Kennedy et al. [34]. PSO imitates the social behavior of swarm in which, initially,
optimization tool developed by Kennedy et al. [34]. PSO imitates the social behavior of swarm in
a random set of the population is selected, and then, in each iteration, each particle explores and moves
which, initially, a random set of the population is selected, and then, in each iteration, each particle
to a new location based on the best solution of that individual particle, and the best solution of all the
explores and moves to a new location based on the best solution of that individual particle, and the
particles. A parameter called velocity is added to each particle’s position in each iteration, which moves
best solution of all the particles. A parameter called velocity is added to each particle’s position in
it towards the global minimum or maximum. Due to its simplicity and robust nature, PSO can easily
each iteration, which moves it towards the global minimum or maximum. Due to its simplicity and
optimize complex problems and has been widely used in engineering optimization works.
robust nature, PSO can easily optimize complex problems and has been widely used in engineering
n
o
n
o
optimization works.
(V ) = θ × (V ) + c r (P
) − (X )
+ c r (G ) − (X )
(26)
j i
i
j i−1
(𝑉 ) = 𝜃 × (𝑉 )
1 1
+ 𝑐 𝑟 (𝑃
𝜃 =𝜃
,
) − (𝑋 )
−
2 2
j i−1
best,j i
𝜃
(𝑋 ) = (𝑋 )
+ 𝑐 𝑟 (𝐺
−𝜃
𝑁
𝑖
+ (𝑉 )
𝑖 = 1, 2, . . . . , 𝑁; 𝑎𝑛𝑑 𝑗 = 1, 2, . . . . , 𝑀
best i
j i−1
) − (𝑋 )
(26)
(27)
(28)
Water 2019, 11, 715
10 of 19
θi = θMax −
θMax − θMin
i
N
(X j )i = (X j )i−1 + (V j )i
(27)
(28)
i = 1, 2, . . . , N; and j = 1, 2, . . . , M
where (Xj )i and (Vj )i are the position and velocity of the jth particle in the ith iteration; c1 and c2 are
cognitive and social learning rates; r1 and r2 are random numbers in the range of 0 and 1; (Pbest,j )i is the
best result obtained for the jth particle till the ith iteration; (Gbest )i is the best result among all particles
till the ith iteration; θi is the inertia weight for the ith iteration, developed by Shi et al. [48], to dampen
Water
2019, 11, 715over each iterations for efficient optimization; θMax and θMin are the maximum
10 of
19
the velocities
and
minimum values of inertia weight, respectively; and M and N are the total number of population and
Figure
6 shows
therespectively.
flowchart for PSO used for the optimization of the hydraulic characteristics
iterations
used
in PSO,
of theFigure
generalized
system.
Inused
this study,
the difference
of the maximum
head at
6 showshydropower
the flowchart
for PSO
for the when
optimization
of the hydraulic
characteristics
of
the
bottom
tunnel
at
the
point
of
the
surge
tank
and
the
maximum
water
level
in
the
surge
tank
is
the generalized hydropower system. In this study, when the difference of the maximum head at the
less
than
or
equal
to
1
m
(𝐻
≤
1
m),
the
solution
and
corresponding
D
O
and
D
S
are
considered
bottom tunnel at the point of the surge tank and the maximum water level in the surge tank is less
acceptable,
more
than and
1 m,
they are considered
If this head
than or equalwhile,
to 1 mwhen
(Hd ≤𝐻1 m),isthe
solution
corresponding
DO and Dunacceptable.
S are considered acceptable,
difference
is
smaller,
it
ensures
efficient
working
of
the
surge
tank
[47].
During
optimization,
avoid
while, when Hd is more than 1 m, they are considered unacceptable. If this head difference istosmaller,
the
solutions
of theworking
unacceptable
a technique
of reducing
the position
is applied,
as shown
in
it ensures
efficient
of theregion,
surge tank
[47]. During
optimization,
to avoid
the solutions
of the
Figure
6.
Whenever
the
new
position
of
any
particle
is
calculated
using
Equation
(28),
if
the
new
unacceptable region, a technique of reducing the position is applied, as shown in Figure 6. Whenever
position
intoof
the
unacceptable
region, then
theEquation
average of
theif previous
position falls
and into
current
the new falls
position
any
particle is calculated
using
(28),
the new position
the
position
is taken
as the
new
again,
the region
of the
position
is checked.
unacceptable
region,
then
theposition,
average and
of the
previous
position
andnew
current
position
is taken Every
as the
time,
the velocity
the new
position
reduced
by halfisuntil
it fallsEvery
into the
acceptance
region.
the
new position,
andof
again,
the region
of is
the
new position
checked.
time,
the velocity
of theIfnew
new
position
does
not
lie
in
the
acceptance
region
for
50
iterations,
a
random
new
position
is
assumed
position is reduced by half until it falls into the acceptance region. If the new position does not lie in the
within
the range
the
reduction
new positions
reduces
the
acceptance
regionoffor
50variables.
iterations,This
a random
new in
position
is assumed
withinthe
thespeed
rangeofofacquiring
the variables.
global
minimum
or
maximum,
but
more
importantly,
it
enables
the
program
to
stay
in
the
acceptable
This reduction in new positions reduces the speed of acquiring the global minimum or maximum,
region
during
all calculations.
but more
importantly,
it enables the program to stay in the acceptable region during all calculations.
Figure 6. Flowchart of particle swarm optimization.
Figure 6. Flowchart of particle swarm optimization.
3. Results and Discussions
3.1. Surge Analysis
Water 2019, 11, 715
11 of 19
3. Results and Discussions
3.1. Surge Analysis
The surge behavior in the surge tank reveals important characteristics of transients occurring in a
pressurized water conveyance system. The given generalized hydropower system is first analyzed
numerically for gradual closure and opening of the valve respectively using the MOC. This method has
been experimentally validated by several researchers in real world problems [6,11,14,49–51]. In recent
years, many model tests for the hydropower systems have been conducted in the laboratory, mainly
including the hydraulic characteristics of the surge tank [49,50]. The models provided complete head
loss coefficients for water flowing into or out of the surge tank, which revealed the complete hydraulic
transient processes during load rejection or load acceptance, and also display great agreement with
the results obtained by MOC. Thus, MOC is widely accepted as a reliable tool for accurate numerical
analysis involving computer modeling, especially for surge analysis.
The system consists of a headrace pipe of length 570 m (head loss coefficient R = 6.81687 × 10−6 )
and a penstock pipe of length 1000 m (head loss coefficient R = 9.3451 × 10−6 ). It is divided into
L
157 sections, each of a 10 m length, with 158 nodal points. Applying Courant’s condition, i.e., ∆t = cN
,
the time step of analysis is taken to be 0.008 s. An upstream reservoir is located at node 1 in the system.
The maximum and minimum operational water level of reservoir are 520 m and 500 m, respectively,
and the valve discharge is 132.4 m3 /s. The restricted orifice surge tank with DS = 9 m and DO = 4.3 m
is at the end of the headrace tunnel at node 58 and a free discharge valve at the end of the penstock at
node 158. The time of closure and opening of the valve was 10 s and the total time of analysis was 200 s.
Initially, a steady state condition was modeled, and then the closing/opening of valve was started
at 5 s and the valve was completely closed at 15 s in linear progression. Surge analysis was carried
out to reveal important hydraulic characteristics occurring due to gradual closure and opening of the
valve. Later, PSO was used to optimize the diameter of the orifice and the diameter of the surge tank to
improve the transient controlling capacity, and design a more efficient and effective surge tank.
Figure 7 depicts the most important characteristics of the hydraulic transient in the hydropower
system due to valve maneuver, which plays a vital role in surge tank design. Figure 7a shows the
water level in the surge tank, during the maximum water level in the reservoir and full load rejection
condition. Hence, the maximum value of surge head shown by the graph indicates the maximum
possible surge head reached for the given condition of valve closure. Initially, the water level is constant
for 5 s, which represents the steady state condition. After the start of valve closure at 5 s, the water level
gradually rises to its maximum value and starts oscillating to consecutive maximum and minimum
values due to water hammer action. Damping of surge waves in this condition occurs gradually and its
value, as given by Equation (23), is calculated to be 1.378%. Similarly, Figure 7b shows the water level
in the surge tank during the minimum water level in the reservoir and full load acceptance condition
and hence it reveals the minimum possible water level for the given time of valve opening. The initial
horizontal line represents the steady state for 5 s, valve opening is started at 5 s, and the water level
drops to its lowest value and then starts oscillating. Discharge is kept constant, regardless of the water
level in the reservoir. Damping of surge waves during the valve opening condition occurs very fast,
while during valve closing, surge waves dampen gradually and hence, damping of surge waves during
the valve closing condition should be properly analyzed. The theoretical value of the time period of
surge waves, given by Equation (15), is calculated to be 69.524 s, and from this analysis, this value is
found to be 69.888 s. This shows that the modeling and analysis results are fairly accurate.
3.2. Effects of Changing Do and Ds
The maximum and minimum water level in the surge tank as shown in Figure 8 and damping of
surge waves is the most important characteristic, which determines the efficient design and stability
of the surge tank. The sensitivity of these three parameters is analyzed with varying diameters of
orifice (Do) and diameters of surge tank (Ds), as shown in Figure 8. As the DO is increased, the value of
system due to valve maneuver, which plays a vital role in surge tank design. Figure 7a shows the
water level in the surge tank, during the maximum water level in the reservoir and full load rejection
condition. Hence, the maximum value of surge head shown by the graph indicates the maximum
possible surge head reached for the given condition of valve closure. Initially, the water level is
Water
2019, for
11, 715
19
constant
5 s, which represents the steady state condition. After the start of valve closure at 512s,ofthe
water level gradually rises to its maximum value and starts oscillating to consecutive maximum and
minimum values due to water hammer action. Damping of surge waves in this condition occurs
maximum head gradually increases and minimum head gradually decreases. Thus, the lower value of
gradually and its value, as given by Equation (23), is calculated to be 1.378%. Similarly, Figure 7b
DO is more favorable, which was also concluded by Diao et al. [19]. Contrary to that, when the DS
shows the water level in the surge tank during the minimum water level in the reservoir and full load
is increased, the maximum water level gradually decreases and the minimum water level gradually
acceptance condition and hence it reveals the minimum possible water level for the given time of
increases. Thus, it can be concluded that a higher value of DS is favorable to minimize hydraulic
valve opening. The initial horizontal line represents the steady state for 5 s, valve opening is started
transient effects in the system. The percentage of damping of surge waves during gradual valve closure
at 5 s, and the water level drops to its lowest value and then starts oscillating. Discharge is kept
first increases, reaches a maximum, and then decreases with increasing DO , and damping gradually
constant, regardless of the water level in the reservoir. Damping of surge waves during the valve
decreases with an increase in DS . In addition to this, the difference of the maximum piezometric head
opening condition occurs very fast, while during valve closing, surge waves dampen gradually and
at the bottom tunnel of the surge tank and the maximum water level in the surge tank is required to
hence, damping of surge waves during the valve closing condition should be properly analyzed. The
be below 1 m for an effective surge tank. All these features make the system highly conflicting and
theoretical value of the time period of surge waves, given by Equation (15), is calculated to be 69.524
discontinuous,
thus, a robust optimizing tool is required to obtain global optimum values12ofofthe
Water
11,this
715 and
19
s, and2019,
from
analysis, this value is found to be 69.888 s. This shows that the modeling and analysis
variables within the defined constraints.
results are fairly accurate.
3.2. Effects of Changing Do and Ds
The maximum and minimum water level in the surge tank as shown in Figure 8 and damping
of surge waves is the most important characteristic, which determines the efficient design and
stability of the surge tank. The sensitivity of these three parameters is analyzed with varying
diameters of orifice (Do) and diameters of surge tank (Ds), as shown in Figure 8. As the DO is
increased, the value of maximum head gradually increases and minimum head gradually decreases.
Thus, the lower value of DO is more favorable, which was also concluded by Diao et al. [19]. Contrary
to that, when the DS is increased, the maximum water level gradually decreases and the minimum
water level gradually increases. Thus, it can be concluded that a higher value of DS is favorable to
minimize hydraulic transient effects in the system. The percentage of damping of surge waves during
gradual valve closure first increases, reaches a maximum, and then decreases with increasing DO, and
damping gradually decreases with an increase in DS. In addition to this, the difference of the
maximum piezometric head at the bottom tunnel of the surge tank and the maximum water level in
the surge tank is required to be below 1 m for an effective surge tank. All these features make the
system highly conflicting and discontinuous, and thus, a robust optimizing tool is required to obtain
Figure 7. Surge analysis in surge tank during transients caused by gradual closing (a) and opening (b)
Figure
7. Surge
analysis
in surge
tank within
during the
transients
caused
by gradual closing (a) and opening
global
optimum
values
of the
variables
defined
constraints.
of valve.
(b) of valve.
Figure
8. Maximum
Maximum water
water level
level(a),(d);
(a,d); minimum
and
percentage
of of
damping
(c,f)
for
Figure 8.
minimumwater
waterlevel
level(b,e);
(b),(e);
and
percentage
damping
(c),(f)
different
diameters
of
orifice
(Do)
and
diameters
of
surge
tank
(Ds).
for different diameters of orifice (Do) and diameters of surge tank (Ds).
Three
inin
thethe
given
system,
as discussed
in Section
2.5,
Three major
majorcharacteristics
characteristicsofofhydraulic
hydraulictransients
transients
given
system,
as discussed
in Section
are
taken
as
the
objective
functions
for
optimizing
D
and
D
.
For
normalizing
the
three
individual
O
2.5, are taken as the objective functions for optimizing
DSO and DS. For normalizing the three
individual objective functions into a single objective function, the maximum and minimum values of
each individual function are required, which are calculated using the PSO in each objective function
within the given range of variables. Table 1 shows the obtained minimum and maximum values,
which can also be verified by analyzing the nature of graphs presented in Figure 8.
Water 2019, 11, 715
13 of 19
objective functions into a single objective function, the maximum and minimum values of each
individual function are required, which are calculated using the PSO in each objective function within
the given range of variables. Table 1 shows the obtained minimum and maximum values, which can
also be verified by analyzing the nature of graphs presented in Figure 8.
Table 1. Maximum and minimum values of each objective function.
For Maximum Values
For Minimum Values
Objective
Functions
Values of Functions
Do (m)
Ds (m)
Values of Functions
Do (m)
Ds (m)
Function 1
Function 2
Function 3
541.550 m
488.740 m
2.038 %
4.1
3.1
3.552
6.3
12
6.3
524.594 m
473.056 m
0.562 %
3.1
4.1
3.1
12
6.3
12
Water 2019, 11, 715
13 of 19
3.3. Optimization with PSO
3.3. Optimization with PSO
As described earlier, the maximum water level, the minimum water level, and damping are chosen
As described earlier, the maximum water level, the minimum water level, and damping are
as objective functions for optimization, with diameter of the orifice (DO ) and diameter of the surge
chosen as objective functions for optimization, with diameter of the orifice (DO) and diameter of the
tank (DS ) as variables. All three objective functions are reduced to minimization problem and a single
surge tank (DS) as variables. All three objective functions are reduced to minimization problem and
objective function is designed by using normalization. The cognitive and social learning rates are taken
a single objective function is designed by using normalization. The cognitive and social learning rates
to be 1 (c1 = c2 = 1). Initially, random values of the variables are provided, and then, 500 iterations
are taken to be 1 (c1 = c2 = 1). Initially, random values of the variables are provided, and then, 500
are carried out with a population size of 10. Due to the small range of variation of variables, a small
iterations are carried out with a population size of 10. Due to the small range of variation of variables,
population is considered and the discontinuous nature of the problem, as discussed in Section 2.8,
a small population is considered and the discontinuous nature of the problem, as discussed in Section
required a large number of iterations before the optimized values were obtained. The following
2.8, required a large number of iterations before the optimized values were obtained. The following
graph in Figure 9 shows the convergence of the values towards the optimum minimal solution after
graph in Figure 9 shows the convergence of the values towards the optimum minimal solution after
optimization with PSO.
optimization with PSO.
Figure
Figure 9.
9. Graph
Graph showing
showing the
the minimization
minimization of
of results
results in
in each
each iteration
iteration using
using PSO.
PSO.
After
After performing
performingPSO
PSOfor
forthe
thegiven
givensystem,
system,the
theoptimized
optimizedvalues
valuesofofDDOO and
and D
DSS were
were obtained
obtained to
to
be
4.1
m
and
11.362
m,
respectively.
These
optimized
dimensions
of
the
surge
tank
produced
significant
be 4.1 m and 11.362 m, respectively. These optimized dimensions of the surge tank produced
improvements
in the maximum
minimum
level in
the surge
tank,
damping
of
significant improvements
in theand
maximum
andwater
minimum
water
level in
the while
surge the
tank,
while the
surge
waves
was
not
improved,
as
shown
in
Table
2.
In
order
to
improve
the
damping
of
surge
damping of surge waves was not improved, as shown in Table 2. In order to improve the damping
waves,
a weighting
factor of 0.25,
0.25,
wasand
multiplied
each objective
function,
respectively,
of surge
waves, a weighting
factor
of and
0.25,0.5
0.25,
0.5 wasin
multiplied
in each
objective
function,
and
optimization
performed
again.
The optimized
values
were obtained
as Do
= 3.684as
m Do
and=
respectively,
and was
optimization
was
performed
again. The
optimized
values were
obtained
Ds
=
6.434
m
and
better
damping
of
surge
waves
was
obtained,
while
the
values
of
the
maximum
3.684 m and Ds = 6.434 m and better damping of surge waves was obtained, while the values ofand
the
minimum
water
level
in
the
surge
tank
were
worse
than
the
original
case.
For
these
two
optimization
maximum and minimum water level in the surge tank were worse than the original case. For these
cases
and the original
case,
carried
outanalysis
and theisresults
areout
presented
Figure are
10.
two optimization
cases
andsurge
the analysis
original is
case,
surge
carried
and theinresults
The
values
of
each
objective
function
for
all
three
cases
are
tabulated
in
Table
2
and
improvements
presented in Figure 10. The values of each objective function for all three cases are tabulated in Table
obtained
are analyzed.obtained are analyzed.
2 and improvements
Table 2. Comparison and improvements in each objective function after optimization.
Objective
Without
Optimization
Normalized Optimization
Normalized Optimization with
Weighting
Water 2019, 11, 715
14 of 19
Table 2. Comparison and improvements in each objective function after optimization.
Objective
Functions
Function 1
Function 2
Function
3 715
Water
2019, 11,
Water 2019, 11, 715
Without Optimization
(Original)
Normalized Optimization
Do = 4.1 m and Ds = 11.362 m
Normalized Optimization
with Weighting
Do = 3.684 m and Ds = 6.434 m
Do = 4.3 m and
Ds = 9 m
Obtained Values
Improvement
Obtained Values
533.643 m
480.609 m
1.378%
528.937 m
485.095 m
0.973%
4.706 m
4.486 m
−0.405%
539.142 m
474.618 m
2.009%
Improvement
−4.706 m
−5.991 m
0.631%
14 of 19
14 of 19
Figure
10.
Comparison of
of surge
surge analysis
for
different
Figure 10.
10. Comparison
Comparison
analysis for
for different
different cases
cases of
of valve
valve closing
closing (a)
(a) and
and opening
opening (b).
(b).
Figure
of surge
analysis
cases
of
valve
closing
(a)
and
opening
(b).
After examining the results shown in Figure 10 and Table 2, the first case of optimization is
After examining
examining the
the results
results shown
shown in
in Figure
Figure 10
10 and
and Table
Table 2,
2, the
the first
first case of
of optimization
optimization is
is
After
concluded
to be better than
the other one.
An improvement
of 4.706
m andcase
4.486 m
in maximum
concluded to
to be better
better than
than the
the other
other one.
one. An improvement
improvement of
of 4.706
4.706 m
m and
and 4.486
4.486 m in maximum
maximum and
and
concluded
and
minimumbepossible
water levels
in theAn
surge tank, respectively,
were
obtainedmininthis
case for the
minimum
possible
water
levels
in
the
surge
tank,
respectively,
were
obtained
in
this
case
for
the
given
minimum
possible water
levels
in the
surge
tank, respectively,
were as
obtained
in m
this
case
given
given
hydropower
system.
Hence,
the
optimized
values are taken
Do = 4.1
and
Dsfor
= the
11.362
m
hydropower
system.
Hence,
the
optimized
values
are
taken
as
Do
=
4.1
m
and
Ds
=
11.362
m
and
hydropower
system.
the showing
optimizedpiezometric
values are head
taken at
asthe
Do bottom
= 4.1 mtunnel
and Dsof=the
11.362
m tank
and
and
surge analysis
is Hence,
conducted
surge
surge analysis
analysis is conducted showing
showing piezometric
piezometric head
head at
at the
the bottom
bottom tunnel of
of the
the surge
surge tank
tank and
and
surge
and
water levelisinconducted
the surge tank referring
to these values,
as shown intunnel
Figure 11.
The difference
of
water
level
in
the
surge
tank
referring
to
these
values,
as
shown
in
Figure
11.
The
difference
of
water levelhead
in the
tankcases
referring
tothe
these
values,
as condition
shown inisFigure
The
difference
of
maximum
for surge
these two
during
valve
closure
found 11.
to be
0.999
m and the
maximum head
head for these
these two
two cases
cases during
during the
the valve
valve closure
closure condition
condition is
is found
found to
to be
be 0.999
0.999 m
m and
and the
the
maximum
graphs
for thesefor
two heads
are almost
overlapping,
which indicates
sound
functioning
of the
surge
graphs
for
these
two
heads
are
almost
overlapping,
which
indicates
sound
functioning
of
the
surge
graphs
for these
heads are almost
overlapping,
which
functioning
thebottom
surge
tank.
Thus,
in thistwo
optimization,
both water
level in the
surgeindicates
tank andsound
piezometric
head atofthe
tank.
Thus,
in
this
optimization,
both
water
level
in
the
surge
tank
and
piezometric
head
at
the
bottom
tank. Thus,
in this optimization,
water
level inperformance
the surge tank
andsurge
piezometric
tunnel
are effectively
controlled both
for the
optimum
of the
tank. head at the bottom
tunnel are
are effectively
effectively controlled
controlled for
for the
the optimum
optimum performance
performance of
of the
the surge
surge tank.
tank.
tunnel
Figure 11.
11. Piezometric
Piezometric head
head at
at the
the bottom
bottom tunnel
tunnel of
of surge
surge tank
tank and
and water
water level
level in
in surge
surge tank
tank during
during
Figure
gradual
closing
(a)
and
opening
(b)
of
valve.
gradual closing (a) and opening (b) of valve.
3.4. Vortex
Vortex Formation
Formation Check
Check
3.4.
The minimum
minimum possible
possible surge
surge head
head in
in the
the surge
surge tank
tank during
during hydraulic
hydraulic transients
transients in
in the
the given
given
The
system
after
optimization
was
found
to
be
485.095
m.
Elevation
of
the
bottom
of
the
surge
tank
was
system after optimization was found to be 485.095 m. Elevation of the bottom of the surge tank was
450.00 m and hence the minimum water column height in this system became 35.095 m. From
Water 2019, 11, 715
15 of 19
3.4. Vortex Formation Check
The minimum possible surge head in the surge tank during hydraulic transients in the given system
after optimization was found to be 485.095 m. Elevation of the bottom of the surge tank was 450.00 m
and hence the minimum water column height in this system became 35.095 m. From Equation (22),
the critical submergence depth of flow was calculated to be 6.638 m, which is way below the minimum
water column height in the surge tank. Therefore, the optimized surge tank effectively prevents the
formation
of715
a vortex in the system during the worst hydraulic transient condition.
Water 2019, 11,
15 of 19
3.5.
3.5. Time
Time of
of Closure
Closure and
and Opening
Opening of
of Valve
Valve
The
have significant
significant effects
The closure
closure and
and opening
opening time
time of
of the
the valve
valve can
can have
effects on
on the
the transients
transients occurring
occurring
in
the
system
[23–25,28].
Generally,
in
hydropower
projects,
gradual
closure
and
opening
valve
in the system [23–25,28]. Generally, in hydropower projects, gradual closure and opening ofof
valve
is
is
adapted
for
adjusting
the
condition
of
flow
in
the
system
and
power
demands.
In
this
study,
adapted for adjusting the condition of flow in the system and power demands. In this study, all the
all
the analysis
is conducted
withclosing
linear closing
and opening
of the
in 10 s.12
Figures
andthe
13
analysis
is conducted
with linear
and opening
of the valve
in valve
10 s. Figures
and 1312
show
show
the
sensitivity
analysis
for
valve
closing/opening
time
in
linear
progression
for
the
optimized
sensitivity analysis for valve closing/opening time in linear progression for the optimized values of
values
DS .closing/opening
The valve closing/opening
time
is changed
10interval
s to 60 sofat10an
interval
O and
DO andof
DSD
. The
valve
time is changed
from
10 s to 60from
s at an
s. For
time
of
s. For
time10ofs to
closure
from
10 s to 30water
s, thelevel
maximum
water
level and
is almost
the same
and it
of 10
closure
from
30 s, the
maximum
is almost
the same
it gradually
decreases
gradually
decreases
a further
increase
in closing
time,
shown in
12. Similarly,
for the
for a further
increasefor
in closing
time,
as shown
in Figure
12.as
Similarly,
forFigure
the valve
opening case,
the
valve
opening
case,
the
minimum
water
level
gradually
increases
with
an
increase
in
valve
opening
minimum water level gradually increases with an increase in valve opening time, as shown in Figure
time,
as shown
in Figure
13. Hence,
as valvethe
maneuver
time
increases,
the maximum
level
13. Hence,
as valve
maneuver
time increases,
maximum
water
level decreases
and thewater
minimum
decreases
and
the
minimum
water
level
increases,
and
thus
the
effects
of
transients
are
slightly
reduced.
water level increases, and thus the effects of transients are slightly reduced. Generally, the effect of
Generally,
the effectclosing/opening
of changing gradual
of the effect
valve in
hassurge
negligible
effect
in but
surge
changing gradual
time closing/opening
of the valve has time
negligible
analysis
[15],
in
analysis
[15],
but
in
this
case,
the
effect
is
significant.
Hence,
valve
or
guide
vanes’
closing/opening
this case, the effect is significant. Hence, valve or guide vanes’ closing/opening time is also an
time
is alsoparameter
an important
for the
hydraulic optimization
the for
surge
for and
this study
important
forparameter
the hydraulic
optimization
of the surge of
tank
thistank
study
shouldand
be
should
be considered
future optimization
cases.
An elaborate
study
the effects
of changing
closure
considered
in future in
optimization
cases. An
elaborate
study on
the on
effects
of changing
closure
time
time
for maximum
piezometric
at bottom
the bottom
tunnel
of the
surge
tank
maximum
water
level
for maximum
piezometric
headhead
at the
tunnel
of the
surge
tank
andand
maximum
water
level
in
in
the
surge
tank
is
performed,
as
shown
in
Figure
14.
The
pipe
characteristic
time
for
this
case
is
1.667
s
the surge tank is performed, as shown in Figure 14. The pipe characteristic time for this case is 1.667
and
closing
s and
closingofofthe
thevalve
valvebelow
belowthis
thistime
timeindicates
indicatesfast
fastclosure
closureofofthe
thevalve.
valve.The
The maximum
maximum pressure
pressure
head
at
the
bottom
tunnel
of
the
surge
tank
is
very
high
for
fast
closure
and
it
quickly
decreases
as soon
head at the bottom tunnel of the surge tank is very high for fast closure and it quickly
decreases
as
as
gradual
closureclosure
occursoccurs
and gradually
becomes
almost almost
constant
after theafter
valve
reaches
soon
as gradual
and gradually
becomes
constant
theclosing
valve time
closing
time
15
s. Contrary
to that, the
maximum
water level
inlevel
the surge
during
fast closure
is lower
it
reaches
15 s. Contrary
to that,
the maximum
water
in thetank
surge
tank during
fast closure
is and
lower
increases
linearly
till
2.5
s
of
valve
closing
time,
and
afterwards,
the
changes
in
the
maximum
water
and it increases linearly till 2.5 s of valve closing time, and afterwards, the changes in the maximum
level
the surge
are
very
small.
difference
of theseoftwo
pressure
heads for
a closing
of
waterinlevel
in thetank
surge
tank
are
very The
small.
The difference
these
two pressure
heads
for a time
closing
more
than
10
s
is
less
than
1
m,
which
indicates
effective
functioning
of
the
surge
tank.
The
effects
time of more than 10 s is less than 1 m, which indicates effective functioning of the surge tank. The
of
fast closing
of the valve
more
in tunnel
pressure
changes,
while while
for thefor
surge
tank,
effects
of fast closing
of theare
valve
aresignificant
more significant
in tunnel
pressure
changes,
the surge
they
comparatively
less significant.
tank,are
they
are comparatively
less significant.
Figure 12.
12. Surge
Surgeanalysis
analysisfor
forgradual
gradualclosure
closure
valve
with
varying
of closure
ofvalve
the valve
Figure
of of
thethe
valve
with
varying
timetime
of closure
of the
(τc).
(τc).
Water 2019,
2019, 11,
11, 715
715
Water
Water 2019, 11, 715
16
16of
of 19
19
16 of 19
Figure 13. Surge analysis for gradual opening of the valve with varying time of opening of the valve
Figure
Surgeanalysis
analysisfor
forgradual
gradual
opening
valve
with
varying
of opening
the
Figure 13.
13. Surge
opening
of of
thethe
valve
with
varying
timetime
of opening
of theofvalve
(τo).
valve (τo).
(τo).
Figure 14. Maximum piezometric head at the bottom tunnel
tunnel of
of the
the surge
surgetank
tank(H
(HP.max
P.max)) and maximum
Figurelevel
14. Maximum
head
at
the bottom
of the
surge
tank (HP.max) and maximum
water
in the surgepiezometric
tank (ZS,max
fordifferent
different
timestunnel
ofclosure
closure
ofthe
thevalve.
valve.
S,max) )for
times
of
of
water level in the surge tank (ZS,max) for different times of closure of the valve.
4. Conclusions
4. Conclusions
4. Conclusions
Surge tanks are very important structures for a pressurized water conveyance system,
Surge tanks are very important structures for a pressurized water conveyance system, which
whichSurge
prevents the
effects of
hydraulic
in the
system.
Valve or guide which
vane
areharmful
very important
structures
fortransients
pressurized
water
prevents thetanks
harmful
effects
of hydraulic
transients
ina the
system. Valve
orconveyance
guide vanesystem,
operation is a
operation
is aharmful
very frequent
in a hydropower
or otherValve
water conveyance
withisaa
prevents
the
effectsphenomenon
ofahydraulic
transients
in the
system.
guide with
vanesystem
very
frequent
phenomenon
in
hydropower
or other
water
conveyanceorsystem
aoperation
surge tank,
surge
tank,
which
causes
harmful
transients
and
the
system
should
be
robust
enough
to
cope
with
very frequent
in a hydropower
or other
waterbeconveyance
systemtowith
tank,
which
causes phenomenon
harmful transients
and the system
should
robust enough
copea surge
with such
such
hazardous
conditions.
The efficient
design
of a should
surge tank
is
vital to
ensuretomaximum
benefit
which
causes
harmful
transients
and
the
system
be
robust
enough
cope
with
hazardous conditions. The efficient design of a surge tank is vital to ensure maximum benefit such
and
and
safety in
the system.
In this study,
surge
is carried
out toensure
examine
vital parameters
of
hazardous
conditions.
ofanalysis
a surge
is vital
maximum
benefit and
safety
in the
system. InThe
thisefficient
study, design
surge analysis
is tank
carried
out to
to examine
vital parameters
of
hydraulic
transient
caused
by
opening
and
closing
of
the
valve.
Further,
optimization
is
done
using
safety in transient
the system.
In this
study, surge
analysis
to examine
vital parameters
of
hydraulic
caused
by opening
and closing
of is
thecarried
valve. out
Further,
optimization
is done using
PSO
to investigate
the
best values
of DO and
Dclosing
and enhance
the performance
of the surgeistank.
Effects
hydraulic
transient
caused
by
opening
and
of
the
valve.
Further,
optimization
done
using
S
PSO to investigate the best values of DO and DS and enhance the performance of the surge tank. Effects
of
various
parameters
hydraulic
transients
are
studied
and
favorable conditions
are
discussed.
PSO
to investigate
the during
best
values
of DO and
D
S and
enhance
performance
of the surge
tank.
Effects
of
various
parameters
during
hydraulic
transients
are the
studied
and favorable
conditions
are
The
following
are
the
conclusions
made
in
this
study:
of
various
parameters
during
hydraulic
transients
are
studied
and
favorable
conditions
are
discussed. The following are the conclusions made in this study:
discussed. The following are the conclusions made in this study:
modelofofaageneralized
generalizedhydropower
hydropowersystem
systemconsisting
consistingof
of an
an upstream
upstream reservoir,
reservoir, an
an orifice
orifice
1.1. AAmodel
1. surge
Asurge
model
of
a
generalized
hydropower
system
consisting
of
an
upstream
reservoir,
an
tank,
conduit
pipes,
and
an
outlet
valve
was
designed
and
numerically
analyzed
using
tank, conduit pipes, and an outlet valve was designed and numerically analyzed orifice
using
surge
tank,
conduit
pipes,
and
an
outlet
valve
was
designed
and
numerically
analyzed
MOCininMATLAB.
MATLAB.Surge
Surgeanalysis
analysisby
byMOC
MOCrevealed
revealed the
the variation
variation of
of water
water level
level in
in the
the surge
surgeusing
tank
MOC
tank
MOC
in
MATLAB.
Surge
analysis
by
MOC
revealed
the
variation
of
water
level
in
the
surge
tank
in
different
conditions
and
assisted
in
defining
and
modeling
the
required
hydraulic
parameters
in different conditions and assisted in defining and modeling the required hydraulic parameters
in
different
conditions
and
assisted
in
defining
and
modeling
the
required
hydraulic
parameters
for
optimization
of
the
surge
tank;
for optimization of the surge tank;
for optimization of the surge tank;
Water 2019, 11, 715
2.
3.
4.
17 of 19
The maximum and minimum possible water level in the surge tank and damping of surge waves
were considered as the important parameters for surge tank optimization. Analyses in this study
show that these transient behaviors are highly conflicting in nature for different values of DO
and DS . In addition, for certain values of DO and DS , the difference of maximum piezometric
head at the bottom tunnel of the surge tank and maximum water level in the surge tank becomes
unacceptable. Hence, a proper optimization method is necessary to investigate the best values of
DO and DS to enhance the efficiency of the surge tank and minimize the effects of transients in the
worst conditions;
The Particle Swarm Optimization successfully optimized the values of DO and DS with a significant
improvement in the maximum and minimum water level in the surge tank with reasonable
damping of surge waves and the recommended surge tank was free from vortex formation.
The overall performance of the surge tank is better than the contemporary methods;
Based on the sensitivity analysis of the valve’s closing and opening time, significant changes were
observed in surge analysis for different closing and opening times of the valve. This indicates
that valve or guide vanes’ closing/opening time should also be considered during the hydraulic
optimization of the surge tank in further cases. The difference of maximum piezometric head
at the bottom tunnel of the surge tank and maximum water level in the surge tank was studied
for various times of closure of the valve and it is concluded that the closing time of the valve
results in more significant pressure changes in the penstock pipe and headrace tunnels, while in
the surge tank, the effect is comparatively lower.
This study establishes a foundation for the design of a hydraulically optimized restricted orifice
surge tank using PSO. In future research, more parameters like location of the surge tank, valve closure
time and mechanism, bifurcation of penstocks, etc., could be included in optimization. In addition to
that, optimization of other types of orifices or other surge controlling devices could also be done.
Author Contributions: K.P.B. and J.Z. conceptualized the research. K.P.B. defined the methodology and carried
out the analysis and S.P. helped in numerical modeling. J.Z. supervised the work and K.P.P. and N.S. assisted in
preparing MATLAB codes. All the authors reviewed, edited, and approved the manuscript.
Funding: The paper was completed within the research projects funded by the National Natural Science
Foundation of China (Grant Nos. 51079051 and 51479071, and also financially supported by the Priority Academic
Program Development of Jiangsu Higher Education Institutions (PAPD, SYS1401).
Conflicts of Interest: The authors declare no conflict of interest.
References
1.
2.
3.
4.
5.
6.
7.
8.
Adamkowski, A. Case study: Lapino powerplant penstock failure. J. Hydraul. Eng. 2001, 127, 547–555.
[CrossRef]
Bonin, C.C. Water-hammer damage to Oigawa power station. J. Eng. Power 1960, 82, 111–119. [CrossRef]
Arrington, R.M. Failure of water-operated needle valves at Bartlett dam and Oneida station hydroelectric
plant. In Proceedings of the 3rd ASME/JSME Joint Fluids Engineering Conference; American Society of Mechanical
Engineers: New York, NY, USA, 1999; pp. 18–22.
Boulos, P.; Karney, B.; Wood, D.; Lingireddy, S. Hydraulic Transient Guidelines. J. Am. Water Resour. Assoc.
2005, 97, 111–124. [CrossRef]
Sam Ani, H.M.V.; Khayatzadeh, A. Transient flow in pipe networks. J. Hydraul. Res. 2002, 40, 637–644.
[CrossRef]
Shani, T.; Gupta, T. Hydraulic transient flow analysis using method of characteristics. Int. J. Innov. Res. Sci.
Eng. Technol. 2017, 6, 14812–14827.
Izquierdo, J.; Iglesias, P.L. Mathematical modelling of hydraulic transients in complex systems. Math. Comput.
Model. 2004, 39, 529–540. [CrossRef]
Salmanzadeh, M. Numerical method for modeling transient flow in distribution systems. Int. J. Comput. Sci.
Netw. Secur. 2013, 13, 72–78.
Water 2019, 11, 715
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
18 of 19
Chaichan, M.T.; Al-Zubaidi, D.S.M. Control of hydraulic transients in the water piping system in
Badra-Pumping Station No. 5. Al-Nahrain Univ. Coll. Eng. J. 2015, 18, 229–239.
Calamak, M.; Bozkus, Z. Comparison of performance of two run-of-river plants during transient conditions.
J. Perform. Constr. Facil. 2012, 27, 624–632. [CrossRef]
Martins, N.M.C.; Brunone, B.; Meniconi, S.; Ramos, H.M.; Covas, D.I.C. CFD and 1D approaches for the
unsteady friction analysis of low Reynolds number turbulent flows. J. Hydraul. Eng. 2017, 143, 4017050.
[CrossRef]
Wylie, E.B.; Streeter, V.L.; Suo, L. Fluid Transients in Systems; Prentice Hall: Englewood Cliffs, NJ, USA, 1993.
Chaudhry, M.H. Applied Hydraulic Transients; Van Nostrand Reinhold: New York, NY, USA, 1979.
Yu, X.; Zhang, J.; Zhou, L. Hydraulic transients in the long diversion-type hydropower station with a complex
differential surge tank. Sci. World J. 2014, 2014, 241868. [CrossRef] [PubMed]
Guo, W.; Zhu, D. A review of the transient process and control for a hydropower station with a super long
headrace tunnel. Energies 2018, 11, 2994. [CrossRef]
Zhou, J.X.; Liu, D.Y. Differential pressure and surge analysis of differential surge tank with interconnecting
holes. J. Hohai Univ. Nat. Sci. 2007, 5, 587–591.
Covas, D.; Ramos, H. Case studies of leak detection and location in water pipe systems by inverse transient
analysis. J. Water Resour. Plan. Manag. 2010, 136, 248–257. [CrossRef]
Balacco, G.; Fontana, N.; Apollonio, C.; Giugni, M.; Marini, G.; Piccinni, A.F. Pressure surges during filling of
partially empty undulating pipelines. ISH J. Hydraul. Eng. 2018, 1–9. [CrossRef]
Diao, X.F.; Zhang, X.H.; Yang, Y.Y. Sensitivity analysis of restricted orifice in the hydropower station.
In Proceedings of the 2016 5th International Conference on Civil, Architectural and Hydraulic Engineering
(ICCAHE 2016), Zhuhai, China, 30–31 July 2016.
Ramos, H.; Covas, D.; Borga, A.; Loureiro, D. Surge damping analysis in pipe systems: Modelling and
experiments. J. Hydraul. Res. 2004, 42, 413–425. [CrossRef]
Emadi, J.; Solemani, A. Maximum water hammer sensitivity analysis. World Acad. Sci. Eng. Technol. 2011, 49,
416–419.
Wood, D.J.; Jones, S.E. Water-hammer charts for various types of valves. J. Hydraul. Div. 1973, 99, 167–178.
Wan, W.; Li, F. Sensitivity analysis of operational time differences for a pump–valve system on a water
hammer response. J. Press. Vessel Technol. 2016, 138, 11303. [CrossRef]
Richter, W.; Dobler, W.; Knoblauch, H. Hydraulic and numerical modelling of an asymmetric orifice within
a surge tank. In Proceedings of the 4th IAHR Interrnational Symposium on Hydraulic Structures, Porto,
Portugal, 9–11 February 2012.
Gabl, R.; Achleitner, S.; Neuner, J.; Gotsch, H.; Aufleger, M. 3D-numerical optimisation of an asymmetric
orifice in the surge tank of a high-head power plant. In Proceedings of the 34th World Congress of the
International Association for Hydro-Environment Research and Engineering: 33rd Hydrology and Water
Resources Symposium and 10th Conference on Hydraulics in Water Engineering Engineers, Brisbane,
Australia, 26 June–1 July 2011; pp. 2428–2435.
Hachem, F.; Nicolet, C.; Duarte, R.; Cesare, G. De Hydraulic design of the diaphragm’s orifice at the entrance
of the surge shaft of FMHL pumped-storage power plant. In Proceedings of the 2013 IAHR World Congress,
Chengdu, China, 8–13 September 2013; pp. 1–10.
Steyrer, P. Economic surge tank design by sofisticated hydraulic throttling. In Proceedings of the 28th IAHR
Congress, Graz, Austria, 22–27 August 1999.
Afshar, H.; Kerachian, R.; Bazargan-Lari, M.R.; Niktash, A.R. Developing a closing rule curve for valves in
pipelines to control the water hammer impacts: Application of the NSGA-II optimization model. In Pipelines
2008: Pipeline Asset Management: Maximizing Performance of our Pipeline Infrastructure; American Society of
Civil Engineers: Atlanta, GA, USA, 2008; pp. 1–10.
Kim, S. Design of surge tank for water supply systems using the impulse response method with the GA
algorithm. J. Mech. Sci. Technol. 2010, 24, 629–636. [CrossRef]
Fathi-Moghadam, M.; Haghighipour, S.; Samani, H.M.V. Design-variable optimization of hydropower
tunnels and surge tanks using a genetic algorithm. J. Water Resour. Plan. Manag. 2013, 139, 200–208.
[CrossRef]
Jung, B.S.; Karney, B.W.; Asce, M. Hydraulic optimization of transient protection devices using GA and PSO
approaches. J. Water Resour. Plan. Manag. 2006, 9496, 44–52. [CrossRef]
Water 2019, 11, 715
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
19 of 19
Moghaddas, S.M.J.; Samani, H.M.V. Application of central force optimization method to design transient
protection devices for water transmission pipelines. Mod. Appl. Sci. 2017, 11, 76–85. [CrossRef]
Golberg, D.E. Genetic algorithms in search, optimization, and machine learning. Addion Wesley 1989, 1989, 36.
Eberhart, R.; Kennedy, J. Particle swarm optimization. In Proceedings of the IEEE International Conference
on Neural Networks, Perth, Australia, 27 November–1 December 1995.
Kachitvichyanukul, V. Comparison of three evolutionary algorithms: GA, PSO, and DE. Ind. Eng. Manag. Syst.
2012, 11, 215–223. [CrossRef]
Bai, Q. Analysis of particle swarm optimization algorithm. Comput. Inf. Sci. 2010, 3, 180. [CrossRef]
Hassan, R.; Cohanim, B.; De Weck, O.; Venter, G. A comparison of particle swarm optimization and the
genetic algorithm. In Proceedings of the 46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics
and Materials Conference, Austin, TX, USA, 18–21 April 2005; p. 13.
Kecskés, I.; Székács, L.; Fodor, J.C.; Odry, P. PSO and GA optimization methods comparison on simulation
model of a real hexapod robot. In Proceedings of the 2013 IEEE 9th International Conference on Computational
Cybernetics (ICCC), Tihany, Hungary, 8–10 July 2013; IEEE: Tihany, Hungary; pp. 125–130.
Duan, Y.; Harley, R.G.; Habetler, T.G. Comparison of particle swarm optimization and genetic algorithm in
the design of permanent magnet motors. In Proceedings of the 2009 IEEE 6th International Power Electronics
and Motion Control Conference, Wuhan, China, 17–20 May 2009; pp. 822–825.
Thoma, D. Zur Theorie des Wasserschlosses bei Selbsttatig Geregelten Turbinenanlagen; Oldenbourg: Munchen,
Germany, 1910. (In German)
Jaeger, C. Present trends in surge tank design. Proc. Inst. Mech. Eng. 1954, 168, 91–124. [CrossRef]
Gordon, J.L. Vortices at intakes. Water Power 1970, 137–138. Available online: https://ci.nii.ac.jp/naid/
10027805517/ (accessed on 6 April 2019).
Li, G.H.; Jiang, H.J.; Zhang, J.; Chen, S. The explicit solution to calculate maximum upsurge in water chamber
surge tank under superposed condition. Appl. Mech. Mater. 2013, 427–429, 392–395. [CrossRef]
Jian, Z.; Suo, L. Explicit formula for calculating surges in throttled surge tank following load rejection.
J. Hydraul. Eng. 1999, 2, 70–74.
Yu, X.; Zhang, J.; Hazrati, A. Critical superposition instant of surge waves in surge tank with long headrace
tunnel. Can. J. Civ. Eng. 2011, 38, 331–337. [CrossRef]
Guo, W.; Yang, J.; Teng, Y. Surge wave characteristics for hydropower station with upstream series double
surge tanks in load rejection transient. Renew. Energy 2017, 108, 488–501. [CrossRef]
National Energy Administration of China. Design code for surge chamber of hydropower stations. China Electr.
Power Press 2014, NB/T 35021. Available online: www.gong123.com/dianli/2016-09-16/1182918.html#edown
(accessed on 12 Ocotober 2018).
Eberhart, R.C.; Shi, Y. Empirical study of particle swarm optimization. In Proceedings of the 1999 Congress on
Evolutionary Computation, Washington, DC, USA, 6–9 July 1999; IEEE: Washington, DC, USA; pp. 1945–1950.
[CrossRef]
Zhang, W.; Cai, F.; Zhou, J.; Hua, Y. Experimental Investigation on Air-Water Interaction in a Hydropower
Station Combining a Diversion Tunnel with a Tailrace Tunnel. Water 2017, 9, 274. [CrossRef]
Zhou, J.; Cai, F.; Hu, M. Stability analysis of hydropower stations with complex flow patterns in the tailrace
tunnel. In Proceedings of the ASME 2013 Fluids Engineering Division Summer Meeting, Incline Village, NV, USA,
July 7–11, 2013; American Society of Mechanical Engineers: New York, NY, USA, 2013; p. V002T11A011.
[CrossRef]
Thanapandi, P.; Prasad, R. Centrifugal pump transient characteristics and analysis using the method of
characteristics. Int. J. Mech. Sci. 1995, 37, 77–89. [CrossRef]
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC BY) license (http://creativecommons.org/licenses/by/4.0/).