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ANALYSIS OF WATER HAMMER BASED ON NETWORK THEORY
K.FUJINO
Electric Power Development Co. Ltd. 6-15-1
Ginza, Chuo-Ku,
Tokyo 104-8165, Japan
Phone : +81-3-3546-2211; Fax : +81-3-3544-1385;
E-mail : fujino@epdc.co.jp
N.TAMAI
University of Tokyo 7-3-1 Hongo, Bunkyo-Ku,
Tokyo 113-8656, Japan
Phone : +81-3-5841-6105; Fax : +81-3-5841-6130;
E-mail : tamai@hydra.t.u-tokyo.ac.jp
ABSTRACT
In hydropower projects, it is very popular to
investigate several alternatives of waterway at a design stage, in point of not
only alignment but also connectivity of conduits, number of wye branches and
turbines. However, there are few satisfactory methods to analyze transient
hydraulic phenomena such as surging and water hammer for arbitrary waterways. In
this paper, a method based on a network theory is proposed to analyze such
hydraulic phenomena, and its validity is proven by comparison with real data
obtained at a pumped storage power station in service.
INTRODUCTION
Generally a waterway system may have plural
intakes, surgetanks, wye branches and power units. They are optimized at a
design stage in point of construction cost, operation and maintenance,
including an alignment of waterways. However degree of freedom in choice of
alternative layouts is often restricted due to limited functions of computer
codes to analyze transient hydraulic phenomena such as surging and water
hammer. In order to resolve limitation described above, authors applied a
network theory to generalize an arbitrary waterway system in the water hammer analysis.
Generally a waterway is decomposed into elements such as reservoirs, unit
conduits, surgetanks, branches and power units from viewpoint of function. A
network theory can formulate transient hydraulic phenomena of entire waterway
system by composing simultaneous linear equations that express characteristics
of waterway units and boundary conditions.
EXPRESSION OF WATER HAMMER BY A NETWORK MODEL
Concept of Network
When a set of lines {Bk} (k=1..K) and a set of
nodes {Na} (a=1..A) exist, if an incidence function [Bk:Na], which expresses
incidence relationship between lines and nodes, is defined, the entirety of two
sets and the incidence relationship is called a "linear graph". Value
of [Bk:Na] is as follows:
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If Na is
the initial point of Bk, If Na is the terminal point
of Bk, If Na is neither initial
nor terminal point of Bk, |
[Bk:Na] = 1 [Bk:Na] = - 1 [Bk:Na] = 0 |
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(1) |
When some physical quantities are given to the lines or nodes of a linear graph, the linear graph is called a "network". Therefore, water hammer in a unit pipe of Fig.1 is a network, which is composed of {Bk} (k=1), {Na} (a=1,2), [Bk:Na] (=1 or -1), and physical quantities H(t) (head) and QU(t) (flow discharge), given at two nodes.
Arbitrary Waterway System of a Pumped Storage
Transient phenomenon of a conduit network is
determined by obtaining the value of head and discharge at every end point of
component conduits at each time. A conduit network of a pumped storage station
is composed of an upper reservoir, surgetanks, pump-turbines, a lower
reservoir, and pressurized conduits connecting them. In a conduit network
containing these components, equations of water hammer and continuity of head
and discharge are obtained as follows.
Basic Equation of Water Hammer
In a conduit shown in Fig.1, upstream end is
named "1", downstream end is "2". Distance between 1 and 2
is "L", and a sectional area of the conduit is "a". When
head, flow velocity and velocity of pressure wave in a conduit are written as
"H", "V" and "c" respectively, basic equations of
water hammer are as follows:
H1(t) - SQ1(t) = H2(t-L/c) -
SQ2(t-L/c) (2)
H2(t) + SQ2(t) = H1(t-L/c) +
SQ1(t-L/c) (3)
, where
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Fig.1 : Water Hammer in a Conduit |
When the coefficient of total head loss between
nodes 1 and 2 is written "R", considering the direction of the flow,
equations (2) and (3) are modified as follows.
As shown in eqs.(4) and (5), if H and Q are
known at certain time and point in the pipe, H and Q in an interval L and at
time (L/c) later are determined. The right-hand sides of (4) and (5) are
thereafter written "C1,2(t-L/c)" and "C2,1(t-L/c)",
respectively:
H1(t) - SQ1(t) = H2(t-L/c) - SQ2(t-L/c) + R ´ Q2(t-L/c) 1/2Q2(t-L/c) 1/2 (4)
H2(t) + SQ2(t) = H1(t-L/c) + SQ1(t-L/c) - R ´ Q2(t-Dt) 1/2Q2(t-Dt) 1/2 (5)
, where Dt shows the time interval for actual
calculation.
Equations of continuity of head and discharge
Total numbers of unit pipes and nodes of a
conduit network are K and A, respectively. When one end of the k-th pipe Bk (k=
1,..,K) is the a-th node Na (a= 1,..,A), head and discharge at this end is
written Hk,a(t), Qk,a(t) respectively. Using these notation, equations of
continuity of head and discharge are written as follows.
Type 1. Continuity of head at the ends of unit
pipes converging at one node (Fig.2)
[Bki:Na] Hki,a(t) + [Bkj:Na] Hkj,a(t) = 0 (6)
, where ki and kj are numbers of two among the
unit pipes converging at node Na.
Type 2. Continuity of head in open water
surface (Figs. 3 and 4)
Hk,a(t) - HR = 0 (Reservoir) (7)
, where HR (constant) is static pressure head
at the intake or outlet.
Hk,a(t) - 
= 0 (Surgetank) (8)
Reflection of the pressure wave is assumed to
occur in Na, bottom of the tank, so that L may be constant.
Type 3. Continuity of discharge in a node and a
pump-turbine (Fig.2 and 5)
kn
å[Bk:Na]Qk,a(t) = 0 (Conversion
of Unit Pipes) (9)
k=k1
, where k1 through kn are
number of unit pipes converging at node Na.
[Bki:Nai]Qki,ai(t) = [Bkj:Naj]Qkj,aj(t) = 0 (Pump-Turbine in stoppage) (10)
[Bki:Nai]Qki,ai(t) + [Bkj:Naj]Qkj,aj(t) = 0 (Pump-Turbine in operation) (11)
, where ki and kj are numbers of unit pipes connected
to entrance and outlet of a
pump-turbine.
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Fig.2 : Conversion of Unit Pipes |
Fig.3 : Reservoir |
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Fig.4 : Surgetank |
Fig.5 : Pump-Turbine |
RELATIONSHIP between
Discharge and Head in Pump-Turbines
A linear equation (12) is derived for the
pump-turbine in operation.
(12)
Equation (12) expresses the relationship
between head at the entrance and outlet of a pump-turbine and discharge passing
it. Its detail is explained in section 3.
EXPRESSION of Water Hammer by a Network Model
Total number of the equations of continuity is
2K. Combining with the basic equation of water hammer, of which number is also
2K, simultaneous equations (13) are obtained. Hence S, H and C denote a
coefficient matrix, a variable vector and a constant vector respectively. As
shown in eq. (13), vector H is composed only of quantities at time t, vector C
is of quantities at time t-Dt, and matrix S does not contain any element of H and S. Therefore eq. (13)
is a linear network model, H is calculated as follows.
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(13) |
H = S-1 x C (14)
Equation (14) means that, if condition of
transient phenomenon at time t-Dt is known, condition at time t is calculated from it. Initial condition
(at t=0) is always determined by a calculation method such as Hardy Cross
Method. Therefore, a network model determines a transient phenomenon of an
arbitrary conduit network.
DEVELOPMENT OF A COMPUTER CODE
Based on the method explained above, the
authors developed a computer code for analysis of transient phenomenon in
arbitrary conduit system. In this code, a routine for linearization of head-discharge
characteristics of pump-turbines in every time interval Dt is important. Its structure is as
follows. Similarity law of pump-turbine is expressed by equations (15) through
(17).
(15) 
(16) 
(17)
, where n(t), N(t); q(t), Q(t); t (t), T(t) are rotational speed,
discharge and torque of model and real turbine, respectively. H(t) is head
difference between entrance and outlet of the real turbine. M is scale of the model. Guide vane opening f(t) is known at any time. In the
pump-turbine of Fig.5, if Qki,ai(t-Dt) is known, model flow q(t-Dt) is obtained from (16). By the n-q
diagram (Fig.6-A), model rotational speed n(t-Dt) and locally linearized equation
(18) is obtained. As Dt is small,
(19) is assumed to be valid. From (15), (16) and (19), (20) is obtained. Linearizing (20), (12) is
obtained. In equation (12), y and Z is expressed
by (21) and (22) respectively.
(18)
(19)
(20)
(21)
(22)
As Z contains N(t),
it must be determined. From the value n(t-Dt) and n-t diagram (Fig.6-B), model torque t(t-Dt) is determined. T(t-Dt) and N(t) are determined by (17)
and (23), respectively, where I is inertia moment of rotating part.
(23)
SIMULATION OF THE OBSERVED PERFORMANCE
Conduit System of Okukiyotsu No.2 Power Station and its Modelling
Okukiyotsu No.2 Power Station is a pumped
storage owned by EPDC in Japan. Maximum discharge, standard effective head,
installed capacity are 154m3/s, 470m, 600MW (2 units @300MW), respectively. The
profile and typical cross sections of the conduit are shown in Fig.7. The
conduit is single, but the upstream and downstream parts of two pump-turbines
are bifurcated. Two surge tanks are equipped in upstream end of the penstock
and downstream of the bifurcation of tailrace tunnel, respectively. Calculation
model of this conduit system is shown in Fig.8.
Fig.6 : Pump-Turbine Characteristics and
Determination of Pump-Turbine Condition
Fig.7 : Profile and Cross Sections of the
Conduit System
Comparison of Observed and Calculated Water Hammer
During the test operation of Okukiyotsu No.2
Pumped Storage from 1995 to 1996, water hammer was observed just upstream and
downstream of the pump-turbines.
Figs.9-A and 9-C show the pressure change for a
full load rejection observed just upstream and downstream of the pump-turbine
No.1 respectively. Figs.9-B and 9-D also show the pressure change calculated
under the same conditions as in the observation, such as guide vane closing
mode f(t), reservoir water level etc. This
computer code simulates the actual pressure change well.
Fig.8 : Calculation Model
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A B C D |
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Fig.9 : Observed and Calculated Water Hammer
Conclusion
Transient hydraulic phenomenon in arbitrary
conduit system can be analyzed universally by a network model. A computer code
for this model was developed. The calculated predicted output of the model
agreed well with observation.
ACKNOWLEDGEMENTS
The authors would like to thank Dr. Yoshihisa
Kawahara of the Public Works Research Institute of the Ministry of Construction
for suggestions on the computer code.
REFERENCE
CHACHRA,V., GHARE,P.M., MOORE,J.M. Applications of Graph Theory Algorithms. New York, Elsevier North Holland, 1979, 421p. ( ISBN 0-444-00268-5 )
