Reliability-Based Optimal Design of Water Distribution Networks under ...

International Review of Mechanical Engineering (I.RE.M.E.), Vol. 3, N. 4 July 2009

Reliability-Based Optimal Design of Water Distribution Networks under Steady and Transient Conditions Berge O. Djebedjian Abstract – The problem of water distribution system under steady and transient conditions is formulated in this study as an optimization problem under hydraulic reliability. The hydraulic reliability is the probability that a water distribution system can supply consumers’ demands over a specified time interval under specified conditions. The chance constraints formulation is used for the evaluation of the network hydraulic reliability. The approach integrates a genetic algorithm (GA) as an optimization tool, Newton-Raphson as a hydraulic analysis solver, a transient analysis program combined with chance constraint model to evaluate the network reliability. The approach was applied on a network for both steady state and transient conditions. The latter was introduced to the water system by pump power failure and sudden valve closure. The single objective is to minimize the pipe cost for a predefined nodal pressure heads requirements and prescribed level of uncertainty. The application of approach to the case study shows the capability of the chance constraints and the genetic algorithm to find the optimal pipe diameters. This reliability-based optimization technique is an important tool for design and operation of water networks without using hydraulic devices for water hammer control. Copyright © 2009 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Water Distribution Systems, Genetic Algorithms, Optimization, Chance Constraints, Reliability

Nomenclature

H

a Wave speed A Cross-sectional area of the pipe ci , j Di , j Cost of pipe per unit length

(

)

CP-SS Penalty cost in case of steady state CP-WH Penalty cost in case of water hammer CP-WH-MAX Penalty cost in case of water hammer when H j , max > H max, TR CP-WH-MIN Penalty cost in case of water hammer when H j , min < H min, TR CT Di , j

Network total cost Diameter of the pipe connecting nodes i and j

D max

Maximum diameter

D min

Minimum diameter

Ep

Energy supplied by a pump

f i, j

Friction factor of pipe

g hf

Gravitational acceleration Head loss due to friction in a pipe

H Hi

Piezometric head above arbitrary datum Pressure head at node i

Hj

Pressure head at node j

H

Maximum pressure head at node j under

j , max

j , min

H max, TR

hammer Maximum allowable pressure head for transient

H min, ST

conditions Minimum allowable pressure head for steady

H min, TR

state Minimum allowable pressure head for

Hp

transient conditions Head delivered by pump

Li , j

Length of pipe

M Mp

Total number of nodes in the network Number of parts into which the pipe is

N Ns P[] qi , j

Manuscript received and revised June 2009, accepted July 2009

water hammer Minimum pressure head at node j under water

divided Total number of pipes Total number of simulations Probability Flow rate in the pipe connecting nodes i and j

Qj

External demand at node j

Qp

Pump discharge

R t Vi , j

Friction term Time Flow velocity

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382

B. O. Djebedjian

x Z

αj Qj

directly from the parameters which define the network demands and the ability of the network to meet these demands, such as minimum cut set method, frequency and duration analysis and chance constraint method (e.g. Lansey et al. [4]; Xu and Goulter [5]; Kapelan et al. [6]; Abdel-Gawad [7]). (b) Simulation approach: the network is evaluated using different user defined scenarios or during extended period simulations, such as Monte Carlo simulations (e.g. Bao and Mays [8]; Gupta and Bhave [9]). A major weakness of the Monte Carlo simulations is that the number of hydraulic simulations that is required is exceedingly large, otherwise the accuracy of the results cannot be guaranteed. The comparison of the two approaches for dealing with uncertainty in water distribution systems optimization is presented in Djebedjian et al. [10]. Numerous optimization techniques are used in water distribution systems under steady state to identify the optimal cost and optimal pipe diameters. These include the deterministic optimization techniques such as linear programming (e.g. Bhave and Sonak [11]; Berghout and Kuczera [12]), and non-linear programming (e.g. Sakarya and Mays [13]), and the stochastic optimization techniques such as the genetic algorithms and simulated annealing (e.g. Yu et al. [14]; Djebedjian et al. [15]). Genetic Algorithms (GAs), in general, have an advantage over classical optimization techniques in that they may be used to find optimal or near optimal solutions to nonlinear, discrete, and discontinuous problems. Also, they use a population of evolving solutions and identify several solutions from which the decision maker can select, rather than a single optimum. The main disadvantage lies in the high computational intensity. The concept of water distribution network optimization under transient conditions was examined for a simple pipeline (Laine and Karney [16]), a network (Zhang [17]), sprinkler irrigation systems (Kaya and Güney [18]), selection of hydraulic devices for water hammer control (Jung and Karney [19], [20]), pipeline optimization with sudden valve opening (Jung and Karney [2]), pump power failure (Djebedjian et al. [21]), and pump power failure and sudden valve closure (Djebedjian et al. [22]). The use of reliability-based optimization techniques in the least-cost design of water distribution networks under transient conditions has received little attention. Djebedjian [23] studied the reliability-based optimization of water distribution networks under water hammer. The genetic algorithm was used for the optimization and the Monte Carlo simulation for incorporating the demand uncertainty in the network design. The purpose of the present paper is to obtain a reliability-based optimal design of a water distribution system considering both steady and transient states. The chance constraint method is used for the reliability, and GA is used for the optimization method. A formulation using GA is developed for the optimal design of a water

Distance along the pipe axis Objective function Probability level for the demand at node j Mean nodal demand

σQj

Standard deviation of nodal demand

φ φ[

Cumulative distribution function

] Standard normal distribution function Subscript i,j connection between nodes i and j Abbreviation COVQ Coefficient of variation of nodal demand GA Genetic Algorithm µGA Micro-Genetic Algorithm I.

Introduction

Water hammer is the pressure wave created by the sudden flow changes generated by rapid valve switching or unexpected pump shut down such as that encountered at power failure [1]. The pressure waves in water hammer are of such high amplitudes that they cause damage to pipes, pumps, valves and other fittings. Transient flow pressure can also be generated by water hammer to cause vapor columns and cavities that can be damaging to pipelines. A good understanding of the characteristics of water hammer is essential to the design of a safe and reliable pipeline system [1]. Numerous techniques are used in controlling transients in water distribution systems including design considerations, operational considerations, and employing surge control devices [2]. Reducing or preventing water hammer damages is achieved by examining numerically the initial water system design under transient conditions by integrating water hammer analysis with the design of an optimum pipe size for a pumping associated system. A satisfactory water distribution system should provide water in the required quantities at desired nodal pressure heads throughout its design period. This goal can be determined from water supply reliability. The reliability of a water distribution system can be defined as the probability of the system being able to provide the required flow at the required pressure at any node. Evaluation of water distribution systems reliability is extremely complex because reliability depends on a large number of parameters, some of which are quality and quantity of water available at source; failure rates of supply pumps; power outages; flow capacity of transmission mains; roughness characteristics influencing the flow capacity of the various links of the distribution networks; pipe breaks and valve failures; variation in daily, weekly, and seasonal demands; as well as demand growth over the years. Two main approaches are available for assessment of reliability, (Goulter et al. [3]): (a) Analytical approach: a closed form of solution for the reliability is derived Copyright © 2009 Praise Worthy Prize S.r.l. - All rights reserved

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Dmin ≤ Di , j ≤ Dmax

distribution system and applied to a case study with both steady state and transient conditions. The transient flow is introduced to the water distribution system by the pump power failure and sudden valve closure. The methodology used in this paper differs from that used in Djebedjian [23]. Both methodologies couple a genetic algorithm with a hydraulic simulator for steady and water hammer analysis to identify the optimal diameters for the water distribution network. However, they differ in the handling of the demand uncertainty. Djebedjian [23] used the simulation approach in which the Monte Carlo simulation utilizes a sufficiently large number of realizations of the demand for adequate uncertainty representation, and the genetic algorithm identifies an optimal network design for each realization. The present study uses the analytical approach by applying the chance constraint method.

II.

Hj

j

minimum probability level of compliance. Namely, the probability, P [ ] , of a constraint being satisfied is greater than or equal to a pre-specified value α j :

P

Therefore, the optimization problem can be expressed as, Lansey et al. [3]: Objective function Z is to minimize the total design cost: Z = min. C T =

(1)

hf = Ep

Di , j 2 g

2

π g

Di5, j

(

)

the average value µQ j from both sides of the inequality

(2)

= Hi − H j

(8)

deviation, σ Q j , as: Q j ~ N µQ j , σ Q j . By subtracting of Eq. (7) and dividing both sides by the standard deviation ( σ Q j ) then the constraint becomes:

expressed by the Darcy-Weisbach formula:

=

( )

ci , j Di , j ⋅ Li , j

i , j∈N

Subject to the constraints: Eqs. (4), (5) and (7). The future nodal demand is represented by normal random variables with mean, Q j , and standard

The head loss due to friction in a pipe, h f , is

h f = fi , j

(7)

j

The conservation of energy states that the total head loss around any loop must equal to zero or is equal to the energy delivered by a pump, E p , if there is any:

qi2, j

qi , j ≤ Q j ≤ 1 − α j

or

j

8 f i , j Li , j

(6)

j

The layout, nodal demand, and minimum head requirement of the water distribution network pipe are assumed to be known. Under steady state conditions, the conservation of mass states that the discharge into each node must be equal to that leaving the node, except for storage nodes (tanks and reservoirs). For a total number of nodes M in the network, it is written as:

Li , j Vi 2,j

qi , j ≥ Q j ≥ α j

P

Steady State Analysis

j = 1,..., M

(5)

The chance-constrained formulation was introduced by Charnes and Cooper [24] and offers an efficient framework to model uncertainties in numerous applications in water resource management. It deals with uncertain RHS's (right hand side) assuming the decision maker is willing to make a probabilistic statement about the frequency with which constraints need to be satisfied. The application of the chance constrained programming for the optimization problem with the objective function Z and inequality constraint of Eq. (1): qi , j ≥ Q j , can limit such constraint to a desired

The reliability-based optimal design of water distribution networks is a least cost optimization problem under uncertainty, where the source of uncertainty is the future nodal demands. The objective function is to design a water distribution network while minimizing the cost and meeting the pressure requirements in terms of given reliability level under uncertain demands.

qi , j = Q j

j = 1,..., M

H j , min ST

Chance Constraint Method

II.2.

Problem Formulation

II.1.

(4)

P

(3)

The design constraints and the hydraulic constraints are given respectively as:

j

qi , j − µQ j

σQj

≤

Q j − µQ j

σQj

≤ 1−α j

(9)

Eq. (9) can be rewritten in a simplified form as:

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B. O. Djebedjian

φ

Q j − µQ j

≤ 1−α j

σQj

transformed by the method of characteristics into pairs of ordinary equations which can be integrated to yield finite difference equations. More details of the water hammer analysis can be found in Larock et al. [26]. The hydraulic constraint is given as:

(10)

where φ is the cumulative distribution function and φ [ ] is the standard normal distribution function. The final deterministic form of the constraint Eq. (7) is now written as:

Q j − µQ j

σ Qj

H min, TR ≤ H k ≤ H max, TR k = 1,..., M p II.4.

≤φ

−1

(1 − α j )

Therefore, the nodal demand constraint can be written as a simple bound constraint as:

(

)

(12)

For a given α j , the nodal demand is calculated using the equality of Eq. (12). The final deterministic chance constraint model for water distribution networks is given by the objective function Eq. (8) subject to the constraints Eqs. (4), (5) and (12). The model is nonlinear because of the nonlinear objective function, Eq. (8), and the non linear constraint, Eq. (12), for every node. The other constraints given by Eq. (4) for every pipe and Eq. (5) for every node are considered to be simple bound. The coefficient of variation of nodal demand is defined as the standard deviation of demand divided by the mean of demand:

COVQ =

II.3.

σQ

Z = C T + C P − SS + C P −WH

if H min, ST − H j ≤ 0

0 C P − SS =

CT M

(H min, ST

)

(19)

(20)

The penalty costs C P −WH - MAX and C P −WH - MIN are given respectively as, Djebedjian et al. [22]:

if H j , max − H max,Tr ≤ 0

0 C P −WH - MAX =

(15)

CT

M j =1

(H j, max − H max,Tr )

(21)

if H j , max − H max, Tr > 0

For the Darcy-Weisbach formula, the friction term R is represented as:

R = f ∆ x /(2 g D A 2 ) (15)

if H min, Tr − H

0

(16) and

j

C P −WH = C P −WH - MAX + C P −WH - MIN

(14)

R 1 ∂q ∂H + + q q =0 g A ∂t ∂ x ∆ x

(14)

−H

if H min, ST − H j > 0

Water Hammer Analysis

equations

j =1

The total penalty cost in case of water hammer is described as follows:

∂ H a2 ∂ q + =0 ∂t gA ∂x

governing

M

(13)

µQ

(18)

The penalty cost in case of steady state is presented by the adaptive penalty method given as, Djebedjian et al. [15]:

The water hammer computations in closed conduits are based on the continuity equation and the equation of motion, Wylie et al. [25]:

The

Objective Function

The objective of the optimum design model presented in Eq. (8) is modified to minimize the total design costs under the constraint of minimum head requirements in steady state condition and minimum and maximum heads requirements in transient condition (water hammer). The later is included in order to protect the system from negative or positive transient pressures. More specifically, the optimization problem is to minimize the objective function Z. It is the summation of the network cost, Eq. (8), and penalty cost in both cases: steady state, CP − SS , and water hammer, CP −WH :

(11)

Q j ≤ µ Q j + σ Q j φ −1 1 − α j

(17)

C P −WH - MIN =

CT

M j =1

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≤0

(H min, Tr − H j , min )

if H min, Tr − H

are

j , min

j , min

(22)

>0

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III. Genetic Algorithm

pressure constraints. The pressure violation at the node, at which the pressure deficit is maximum, is used as the basis for computation of the penalty cost. The maximum pressure deficit is multiplied by a penalty factor ( CT / M ), [15]. 6. Transient analysis of each network. A transient analysis solver computes the transient pressure heads resulting from the pump power failure or sudden valve closure. The minimum and maximum pressure heads are computed in each pipe of the network and compared with the minimum and maximum allowable pressure heads, and any pressure deficits are noted. 7. Computation of penalty cost for transient state. The GA assigns a penalty cost if a pipe design does not satisfy the minimum and maximum allowable pressure heads constraints. The penalty cost is estimated as the pressure violation multiplied by a penalty factor equals to the cost of the specified network. 8. Computation of total network cost. The total cost of each network in the current population is taken as the sum of the network cost (Step 2), the penalty cost for steady state (Step 5), and the penalty cost for transient state (Step 7). 9. Computation of the fitness. The fitness of the coded string is taken as some function of the total network cost. For each proposed pipe network in the current population, it is computed as the negative value of the total network cost. 10. Generation of a new population using the selection operator. The GA generates new members of the next generation by a selection scheme. 11. The crossover operator. Crossover occurs with some specified probability of crossover for each pair of parent strings selected in Step 10. 12. The mutation operator. Mutation occurs with some specified probability of mutation for each bit in the strings, which have undergone crossover. 13. Production of successive generations. The use of the three operators described above produces a new generation of pipe network designs using Steps 2 to 13. The GA repeats the process to generate successive generations. The last cost strings (e.g., the best 20) are stored and updated as cheaper cost alternatives are generated. A flow chart of the reliability-based optimization program GACCWHnet for both steady state and transient conditions is depicted in Fig. 1. In an early research; Djebedjian [23] used the Monte Carlo simulation for the reliability but it takes considerable computer time. The reduction of the computational time is achieved by the application of chance constraint formulation which reduces computer time significantly. The developed GACCWHnet program permits the reliability-based optimization of water network subjected to several events that can cause rapid transients. These

Genetic algorithms are nature based stochastic computational techniques. The major advantages of these algorithms are their broad applicability, flexibility and their ability to find optimal or near optimal solutions with relatively modest computational requirements. GAs have proven useful in a variety of search and optimization problems in engineering, science and commerce (Goldberg [27]). The algorithms are based on the principle of the survival of the fittest, which tries to retain genetic information from generation to generation. The GA approach does not require certain restrictive conditions (e.g., continuity, differentiability to the second order, etc.); properties that can seldom be guaranteed for water distribution problems, especially under transient conditions. The brief idea of GA is to select population of initial solution points scattered randomly in the optimized space, then converge to better solutions by applying in iterative manner the following three processes (reproduction/selection, crossover and mutation) until desired criteria for stopping is achieved. The micro-Genetic Algorithm (µGA), Krishnakumar [28], is a "small population" GA. In contrast to the Simple Genetic Algorithm, which requires a large number of individuals in each population (i.e., 30 - 200); the µGA uses a small population size. The optimization program is written in FORTRAN language and it links the µGA, the Newton simulation technique for the steady state hydraulic simulation, and the transient analysis. A brief description of the steps in using GA for pipe network optimization, [29], and including water hammer and reliability is as follows: 1. Generation of initial population. The GA randomly generates an initial population of coded strings representing pipe network solutions of population size Npopsiz. Each of the Npopsiz strings represents a possible combination of pipe sizes. 2. Computation of network cost. For each Npopsiz string in the population, the GA decodes each substring into the corresponding pipe size and computes the total material cost. The GA determines the costs of each trial pipe network design in the current population. 3. Computation of demands. For the given nodal demand of the network design, the application of the chance constraints method for the given value of α computes the new values of demands at nodes. 4. Steady state analysis of each network. A steady state hydraulic network solver computes the heads and discharges under the specified demands for each of the network designs in the population. The actual nodal pressures are compared with the minimum allowable pressure heads, and any pressure deficits are noted. 5. Computation of penalty cost for steady state. The GA assigns a penalty cost for each demand if a pipe network design does not satisfy the minimum Copyright © 2009 Praise Worthy Prize S.r.l. - All rights reserved

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include the power failure at any number of pump stations, sudden valve closure at either end of any number of pipes, staged valve closure at the downstream end of any number of pipes and the sudden demand

changes at any number of junctions. In this study the transients resulted from the pump power failure and sudden valve closure are investigated. Change Demands According to α

Convert Optimized Diameters to Commercial Diameters

Produce Optimized Diameters

Steady State Analysis Analyze Given Network Get Pressure Heads & Velocities

Material Cost, CT Penalty Cost, CP-SS

Yes

If : H ≤ Hmin-ST

No Transient Analysis Get Minimum and Maximum Pressure Heads

No

Maximum Generation

Penalty Cost, CP-WH

Yes

If : Hmin ≤ Hmin-TR Hmax ≥ Hmax-TR

Yes Comprise between produced groups of diameters to select the group that has the least cost

No Fitness, −Z

Best Solution Fig. 1. Flow chart of the GACCWHnet program

IV.

Case Study

The case study is based on a pre-defined water supply piping network, Fig. 2; Larock et al. [26]; and previously used in Djebedjian et al. [22] and Djebedjian [23]. The system comprises eleven pipes, nine nodes, one pump, one valve at the downstream end of pipe 2, and two reservoirs at nodes 1 and 6 which have heads of 20 ft. The demands at nodes and the lengths of the pipes are given in Table I. The total network demand for the steady-state simulation is 12 cubic feet per second. In order to introduce transient conditions into the case study, a variety of possible causes could be selected. For convenience, a pump power failure and a sudden valve closure are chosen to characterize the transient

performance of the system. The set of commercially available pipe diameters (in inches) is (6, 8, 10, 12, and 15) and the corresponding cost per foot length is (15, 25, 35, 45, and 65 units), respectively. N For all the pipes, the roughness height and wave speed are 0.007 inch and 3300 ft/sec, respectively. The pump performance curve for the pump in pipe 9 is defined by: H p = −0.5 Q p2 − 0.3 Q p + 90 , with Q p in ft3/sec and H p in ft, and the brake horsepower (in hp) is defined by: Power = 1.2 Q p + 30 . The pump runs at 1750 rev/min, and the moment of inertia of pump and motor is 40 lb.ft2.

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Fig. 2. Typical Piping Network, Larock et al. [26]

demands, i.e. optimization. Higher values of α refer to more stringent performance requirements so that the likelihood of not meeting future demands is reduced. The pressure head restrictions for the case study which are to be fulfilled are the required minimum pressure head at all nodes is given the value of 80 ft for the steady state whereas for the transient conditions, the minimum and maximum pressure heads are given the values of 80 ft and 180 ft, respectively. The GACCWHnet optimization program is a binarycoded µGA and is applied to the network using the following values for µGA parameters: the population size of a GA run, Npopsiz = 5, the maximum number of generations to run by the GA, Maxgen = 2000, and the array of integer number of possibilities per parameter Nposibl = 16 and different values for the initial random number seed for the GA run, Idum, were tested to achieve the optimal cost. The mutation and crossover rates were set to 0.2 and 0.5, respectively. For steady state calculations, the accuracy was 0.0001 ft3/s. The time of the transient flow simulation was taken as 40 s and the hydraulic time step t was 0.04 s. The network containing 11 pipes and with 5 available commercial pipe sizes has a total solution space of 511 = 4.88 107 different network designs. Using the GA optimization techniques, and for all 48 runs used in the following calculations, the maximum number of function evaluations was 7327 to reach the optimal solution and this is only a very small fraction of the total search space (0.015%).

TABLE I DATA FOR THE PIPING NETWORK Node Elevation Demand ID (ft) (cfs) 1 2 3 4 5 6 7 8 9

1460 1300 1290 1310 1280 1380 1320 1310 1260

V.

0 3 2 4 0 1 2

Pipe ID

Start Node

End Node

Length (ft)

1 2 3 4 5 6 7 8 9 10 11

1 2 2 3 4 4 5 5 6 7 8

2 4 3 5 5 7 8 9 7 8 9

800 1000 600 1200 800 1200 1500 1800 1000 400 800

Results and Discussion

The reliability-based optimization of pipe sizing under steady state and water hammer event caused by pump power failure and sudden valve closure is studied using the genetic algorithms for optimization and the chance constraint formulation as the reliability method. Three coefficients of variation (COVQ) equal 10%, 20% and 30% of nodal demand are used in the calculations. For each COVQ, a least-cost strategy with target reliabilities (uncertainty of the future demand) α j = 50, 60, 70, 80, 90 and 95% were created. Values of the cumulative distribution function φ (z ) for a standard normal variable z are given for negative values of the standard normal deviate ranging from 0 to −8.99 (e.g. Nowak and Collins [30]). For the values of α (0.5, 0.6, 0.7, 0.8, 0.9, 0.95) the corresponding values of φ −1 (1 − α ) , Eq. (12), are (0, −0.2525, −0.525, −0.84,

V.1.

Steady State Optimization

The reliability-based optimization under steady state condition provides the optimal operational behavior of the system which is the required minimum pressure head at all nodes. The optimal cost and pipe diameters of best solutions of the case study network obtained by GACCWHnet program under steady state conditions

−1.28, −1.645). A standard deviation equal to zero refers to the case of no uncertainty, and the larger the standard deviation, the greater the uncertainty. Using α = 0.5 is equivalent to using mean values of the nodal Copyright © 2009 Praise Worthy Prize S.r.l. - All rights reserved

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and for different reliability requirements, when the coefficients of variation are 10%, 20% and 30% of nodal demand are summarized in Tables II, III and IV, respectively. In the three tables, the optimization (i.e. with zero uncertainty) is given by α = 0.5 and the optimal cost is 220,500 units. It is worth mentioning that the two pipes 1 and 9 are excluded from the application of the penalty cost in the steady state condition, Eq. (19), as they are connected to the two reservoirs of heights of 20 m which is much less than the allowable minimum pressure. The optimal solutions for COVQ = 10%, Table II, illustrate two exceptional equal results at reliabilities α = 0.5 and α = 0.6. The optimal costs and diameters are identical. This may be attributed to the suitability of that optimal design to support the increased nodal demand when α = 0.6. Also, it can be observed that six pipes have 6 inches diameter which is the minimum available diameter, Dmin . The unavailability of smaller diameter may be the reason of that equality of optimal diameters as the GA is restricted by that minimum diameter. The obtained designs of the reliability-based optimization by GACCWHnet program reveals that for each COVQ, there are many pipes' optimal diameters which remain unchangeable with variation of α. The number of pipes unaffected by the value of α decreases with the increase of COVQ. For COVQ = 10%, Table II, the optimal diameters of pipes 2 to 8 and 11 remains constant. The number of pipes unaffected by the value of α is 8 which decreases to 7 pipes for COVQ = 20% (Table III) and to 5 pipes for COVQ = 30% (Table IV). This phenomenon is attributed to the fact that for higher values of COVQ, the pipe diameters must be increased to fulfill the performance requirements.

reliability of network increases the performance of network at normal conditions. TABLE III OPTIMAL COSTS AND PIPE DIAMETERS (IN INCHES) FOR DIFFERENT REQUIRED NETWORK RELIABILITIES AT COVQ = 20% FOR THE STEADY STATE CONDITION

α 0.50 0.60 0.70 0.80 0.90 0.95

0.50 0.60 0.70 0.80 0.90 0.95

Pipe ID 2

3

4

5

6

7

8

9

10

11

Cost (Units)

12 12 10 10 12 12

8 8 8 8 8 8

8 8 8 8 8 8

6 6 6 6 6 6

6 6 6 6 6 6

6 6 6 6 6 6

6 6 6 6 6 6

6 6 6 6 6 6

8 8 10 10 10 10

8 8 8 10 8 10

6 6 6 6 6 6

220,500 220,500 222,500 226,500 230,500 234,500

3

4 5

6

7

8

9

10

11

Cost (Units)

12 12 12 12 12 12

8 8 8 8 8 8

8 8 8 8 8 10

6 6 6 6 6 6

6 6 6 6 6 6

6 6 6 6 6 6

6 6 6 6 6 6

8 8 10 10 10 10

8 10 8 10 10 10

6 6 6 6 8 8

220,500 224,500 230,500 234,500 242,500 248,500

6 6 6 6 6 6

FOR THE STEADY STATE CONDITION

α 0.50 0.60 0.70 0.80 0.90 0.95

Pipe ID 1

2

3

4 5

6

7

8

9

10

11

Cost (Units)

12 10 10 12 12 12

8 8 8 8 8 8

8 8 8 8 10 10

6 6 6 6 8 8

6 6 6 6 6 6

6 6 6 6 6 6

6 6 6 6 6 6

8 10 10 10 10 10

8 10 10 10 10 12

6 6 8 8 6 8

220,500 226,500 234,500 242,500 252,500 264,500

6 6 6 6 6 6

270000

COVQ = 10% COVQ = 20% COVQ = 30%

Cost (Units)

260000

250000

240000

230000

220000 0.5

0.6

0.7

0.8

0.9

1

α

Fig. 3. Total cost versus uncertainty α for COVQ = 10%, 20% and 30% for the steady state condition

FOR THE STEADY STATE CONDITION

1

2

TABLE IV OPTIMAL COSTS AND PIPE DIAMETERS (IN INCHES) FOR DIFFERENT REQUIRED NETWORK RELIABILITIES AT COVQ = 30%

TABLE II OPTIMAL COSTS AND PIPE DIAMETERS (IN INCHES) FOR DIFFERENT REQUIRED NETWORK RELIABILITIES AT COVQ = 10% α

Pipe ID 1

V.2.

Pump Power Failure

Water hammer is developed by a rapid change in the flow rate in a hydraulic system. Many events may result in pressure oscillations under normal operating conditions. In this study, two transient causes were modeled for the network: pump power failure and sudden valve closure. The optimal steady state solution (at α = 0.5, Table II) does not satisfy the transient condition caused by pump power failure. The pressure heads at all pipes except pipes 1, 3 and 9 don’t satisfy the minimum pressure head requirements. Similar to the previous results for the reliability-based optimization for the steady state condition, the results obtained for the water hammer event caused by the pump power failure are mentioned. Tables V, VI and VII give the optimal cost and pipe diameters of best solutions for

Fig. 3 shows these results of optimal cost and network reliability at coefficients of variation COVQ = 10%, 20% and 30%. It is evident that at constant coefficient of variation, the optimal cost increases with the increase of required network reliability. Also, for the same required reliability, the optimal cost increases with the increase of coefficient of variation. This is expected due to the fact that the higher the reliability requirement, the greater the cost of design. The high Copyright © 2009 Praise Worthy Prize S.r.l. - All rights reserved

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COVQ = 10%, 20% and 30% of nodal demand. The optimal cost of optimization (α = 0.5) is 250,500 units which is higher than that for steady state by 13.6%. The reliability-based optimal designs for this case demonstrates that the number of pipes unaffected by the value of α are 7, 6 and 6 for COVQ = 10%, Table V, COVQ = 20%, Table VI, COVQ = 30%, Table VII. The previous observation of optimal costs equality for COVQ = 10% at α = 0.5 and 0.6 are noted also in Table V. The results of the optimal costs given in Tables VVII are shown in Fig. 4. The two trends of optimal cost vs. reliability level α for constant COVQ and optimal cost vs. COVQ for constant α are the same as in Fig. 3. Generally, the optimal costs are greater than the steady state indicating that the design of water distribution networks to be reliable under water hammer event caused by pump power failure is more expensive. It is worth mentioning that the decision maker can prefer the employment of specialized surge control devices or the utilization of the optimal pipe design with pressure heads in the allowable range.

320000

Cost (Units)

300000

240000 0.5

Pipe ID

0.50 0.60 0.70 0.80 0.90 0.95

2

3

4

5

6 7 8 9

10

11

Cost (Units)

15 15 15 15 15 15

12 12 12 12 12 12

8 8 10 8 10 10

6 6 6 6 6 8

6 6 6 10 8 6

6 6 6 6 6 6

10 10 10 8 10 10

6 6 6 6 6 6

250,500 250,500 256,500 262,500 264,500 268,500

6 6 6 6 6 6

6 6 6 6 6 6

6 6 6 6 6 6

2

3

4

5 6 7 8 9

10

11

0.50 0.60 0.70 0.80 0.90 0.95

15 15 15 15 15 15

12 12 12 12 12 12

8 8 10 8 12 10

6 6 8 6 8 8

6 8 6 8 6 6

10 10 8 10 10 10

6 6 6 6 6 6

250,500 258,500 264,500 270,500 274,500 280,500

6 6 6 8 6 8

6 6 6 6 6 6

6 6 6 6 6 6

6 6 6 6 6 6

0.50 0.60 0.70 0.80 0.90 0.95

COVQ = 10% COVQ = 20%

0

10

20

30

40

Time (s)

Fig. 5(a) 200

160

Pressure Head (ft)

2

3

4

5

6 7 8 9

10

11

15 15 15 15 15 15

12 12 12 15 15 15

8 10 12 8 10 12

6 6 8 6 6 6

6 8 6 8 10 12

6 6 6 6 6 6

10 8 8 10 8 8

6 6 6 6 6 6

250,500 260,500 270,500 278,500 288,500 302,500

6 6 6 6 6 6

Node 2

COVQ = 30%

1

6 6 6 6 6 6

80

0

Cost (Units)

6 6 6 6 6 6

120

40

FOR THE PUMP POWER FAILURE

Pipe ID

1

160

TABLE VII OPTIMAL COSTS AND PIPE DIAMETERS (IN INCHES) FOR DIFFERENT REQUIRED NETWORK RELIABILITIES AT COVQ = 30% α

0.9

200

Pressure Head (ft)

1

Cost (Units)

0.8

Figs. 5 illustrate the nodal pressure head variation during the pump power failure event for α = 0.95 and COVQ = 10%, 20% and 30%. The choice of this high value of α indicates the situation of more reliable network in comparison with the other low values. The optimal design fulfills the minimum and maximum nodal pressure heads of 80 and 180 ft, respectively as shown in the figure. Each simulation continues until t = 40 s to ensure the satisfaction of nearly steady state condition. Node 6, which is located geometrically at the reservoir upstream of the pump, is used to present the numerical pressure head results at the pump discharge during the water hammer event. The pressure head variation at this node reveals that it decreases firstly to 20 ft which is the head of reservoir 6 then followed by pressure waves created by the backflow to the pump under the effect of high elevated reservoir 1.

FOR THE PUMP POWER FAILURE

Pipe ID

0.7

Fig. 4. Total cost versus uncertainty α for COVQ = 10%, 20% and 30% for the pump power failure

TABLE VI OPTIMAL COSTS AND PIPE DIAMETERS (IN INCHES) FOR DIFFERENT REQUIRED NETWORK RELIABILITIES AT COVQ = 20% α

0.6

α

FOR THE PUMP POWER FAILURE

1

280000

260000

TABLE V OPTIMAL COSTS AND PIPE DIAMETERS (IN INCHES) FOR DIFFERENT REQUIRED NETWORK RELIABILITIES AT COVQ = 10% α

COVQ = 10% COVQ = 20% COVQ = 30%

Copyright © 2009 Praise Worthy Prize S.r.l. - All rights reserved

120

80

Node 3

COVQ = 10%

40

COVQ = 20% COVQ = 30%

0

0

10

20

30

40

Time (s)

Fig. 5(b)

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200

200

160

160

Pressure Head (ft)

Pressure Head (ft)

B. O. Djebedjian

120

80

Node 4

120

80

Node 9

COVQ = 10%

40

COVQ = 10%

40

COVQ = 20%

COVQ = 20%

COVQ = 30%

0

0

10

20

30

COVQ = 30%

0

40

Time (s)

10

20

30

40

Time (s)

Fig. 5(c)

200

0

Fig. 5(h) Figs. 5. Nodal pressure head versus time at α = 0.95 and COVQ = 10%, 20% and 30% for the pump power failure

Pressure Head (ft)

160

V.3.

120

Water hammer is introduced into the case study by suddenly closing the valve which is located at the downstream end of pipe 2. Upon the sudden closure of the valve the velocity of water at the valve is forced suddenly to zero. Consequently, the pressure head at the valve increases suddenly to change the momentum of the water at the valve to zero. The optimal costs and optimal pipe diameters resulted from the GACCWHnet program for the sudden valve closure for COVQ = 10%, 20% and 30% at different values of α are mentioned in Tables VIII, IX and X, respectively. The optimal diameters demonstrate that the number of pipes unaffected by the value of α for COVQ = 10%, 20%, and 30% are 3, 3 and 4 pipes with the optimal diameters of pipes 7 and 8 having constant diameters. Also, as in the previous cases, there is equality of optimal costs for COVQ = 10% at α = 0.5 and 0.6, Table VIII. Fig. 6 illustrates the results of the optimal costs given in Tables VIII-X.

80

Node 5

COVQ = 10%

40

COVQ = 20% COVQ = 30%

0

0

10

20

30

40

Time (s)

Fig. 5(d)

200

Node 6 (Pump discharge) COVQ = 10%

Pressure Head (ft)

160

COVQ = 20% COVQ = 30%

120

80

40

0

0

10

20

30

40

Time (s)

Fig. 5(e)

Pressure Head (ft)

200

160

TABLE VIII OPTIMAL COSTS AND PIPE DIAMETERS (IN INCHES) FOR DIFFERENT REQUIRED NETWORK RELIABILITIES AT COVQ = 10%

120

FOR THE SUDDEN VALVE CLOSURE

α

80

Node 7

COVQ = 10%

40

0.50 0.60 0.70 0.80 0.90 0.95

COVQ = 20% COVQ = 30%

0

0

10

20

30

40

Time (s)

Fig. 5(f)

200

160

Pressure Head (ft)

Sudden Valve Closure

80

Node 8

COVQ = 10% COVQ = 20% COVQ = 30%

0

0

10

20

30

2

3

4

5

6

7

8

9

10

11

Cost (Units)

8 8 8 8 8 8

6 6 8 6 6 8

10 10 10 12 12 10

8 6 8 6 8 6

6 6 6 6 8 6

10 10 10 10 12 12

6 6 6 6 6 6

6 6 6 6 6 6

12 15 12 15 12 15

8 8 8 8 10 12

8 6 8 8 6 10

264,500 264,500 274,500 278,500 286,500 310,500

The general trends for the relationship between the cost and the reliability are in agreement with level α as in Figs. 3 and 4 but the curves are not smooth as in the previous ones indicating the difficulties faced by the GA to find other better solutions for this more complicated case. The comparison of the reliabilitybased optimization for water hammer event caused by sudden valve closure with that of pump power failure indicates that the optimal costs in the case of sudden valve closure are higher than that in the case of pump

120

40

Pipe ID 1

40

Time (s)

Fig. 5(g)

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International Review of Mechanical Engineering, Vol. 3, n. 4

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B. O. Djebedjian

power failure, e.g. at α = 0.5 the optimal cost of optimization is 264,500 units with an increase of 5.6% than that for pump power failure. Also, the variation in the optimal diameters in the case of sudden valve closure is vast compared to that of the case of pump power failure. The nodal pressure heads after the sudden valve closure are shown in Figs. 7 for α = 0.95 and COVQ = 10%, 20% and 30%.

first 10 seconds. Also, relative to the previous case of pump power failure, it can be observed that there is a rapid decay and smoothing of pressure peaks after 20 s and quite smooth curves after that. However, more pressure wave cycles are observed in the sudden valve closure than in the case of pump power failure. 200

TABLE IX OPTIMAL COSTS AND PIPE DIAMETERS (IN INCHES) FOR DIFFERENT REQUIRED NETWORK RELIABILITIES AT COVQ = 20%

0.50 0.60 0.70 0.80 0.90 0.95

1

2

3

4

5

6

7

8

9

10

11

Cost (Units)

8 8 8 10 10 10

6 6 6 6 6 6

10 10 12 15 12 12

8 6 8 10 12 10

6 6 8 6 6 6

10 10 12 10 10 12

6 6 6 6 6 6

6 6 6 6 6 6

12 15 12 12 15 15

8 10 10 6 8 10

8 6 6 6 6 6

264,500 268,500 286,500 290,500 314,500 318,500

0.50 0.60 0.70 0.80 0.90 0.95

Node 2

COVQ = 10% COVQ = 20% COVQ = 30%

0

0

10

1

2

3

4

5

6

7

8

9

10

11

Cost (Units)

8 8 10 10 10 12

6 6 6 6 6 6

10 10 15 12 12 15

8 8 12 10 12 15

6 6 6 6 6 6

10 12 10 12 15 10

6 6 6 6 6 6

6 6 6 6 6 6

12 12 10 15 12 15

8 12 6 12 10 6

8 8 6 6 6 6

264,500 284,500 292,500 322,500 334,500 354,500

30

40

Fig. 7(a)

120

80

Node 3

COVQ = 10%

40

COVQ = 20% COVQ = 30%

0

0

10

The choice of simulation time t = 40 s is sufficient and all the nodal pressure heads are within the minimum and maximum nodal pressure heads requirements of 80 and 180 ft, respectively.

20

30

40

Time (s)

Fig. 7(b)

200

Pressure Head (ft)

160

360000

COVQ = 10% COVQ = 20% COVQ = 30%

340000

20

Time (s)

160

FOR THE SUDDEN VALVE CLOSURE

Pipe ID

80

200

TABLE X OPTIMAL COSTS AND PIPE DIAMETERS (IN INCHES) FOR DIFFERENT REQUIRED NETWORK RELIABILITIES AT COVQ = 30% α

120

40

Pressure Head (ft)

Pipe ID

α

Pressure Head (ft)

160

FOR THE SUDDEN VALVE CLOSURE

120

80

Node 4

COVQ = 10%

40

COVQ = 20%

Cost (Units)

COVQ = 30% 320000

0

0

10

20

30

40

Time (s)

300000

Fig. 7(c)

200 280000

0.6

0.7

0.8

0.9

Pressure Head (ft)

160 260000 0.5

1

α

Fig. 6. Total cost versus uncertainty α for COVQ = 10%, 20% and 30% for the sudden valve closure

120

80

Node 5

COVQ = 10%

40

The comparison of transient severity for the two causes of water hammer is clarified even if the two optimal designs are not identical. Compared to the pump power failure; the sudden valve closure results in rapid transients in nodal pressure heads especially in the

COVQ = 20% COVQ = 30%

0

0

10

20

30

40

Time (s)

Fig. 7(d)

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International Review of Mechanical Engineering, Vol. 3, n. 4

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B. O. Djebedjian

200

Fig. 8. The cost ratio is defined as the ratio between the optimal cost for the studied case and the optimal cost for the steady state at α = 0.5 (220,500 units). The figure indicates generally that the optimal cost increases gradually from the steady state to the pump power failure to the sudden valve closure. The cost ratios for optimization (α = 0.5) for steady state, pump power failure and sudden valve closure are 1, 1.136 and 1.2, respectively. For COV = 30% and α = 0.95, the cost ratios for these three cases are 1.2, 1.372 and 1.608, respectively. Whereas, the average cost ratios calculated for 18 values of different COV and α for each of steady state, pump power failure and sudden valve closure are 1.057, 1.210 and 1.329, respectively.

Node 6 (Pump discharge) COVQ = 10%

Pressure Head (ft)

160

COVQ = 20% COVQ = 30%

120

80

40

0

0

10

20

30

40

Time (s)

Fig. 7(e)

200

2 120

80 COVQ = 10%

40

COVQ = 20% COVQ = 30%

0

10

20

30

1.6

Steady, COVQ = 30% Steady, COVQ = 20% Steady, COVQ = 10%

1.4

1 0.5

0.6

0.7

0.8

0.9

1

α

160

Pressure Head (ft)

Pump, COVQ = 10%

1.2

Fig. 7(f)

200

Fig. 8. Cost ratio versus uncertainty α for COVQ = 10%, 20% and 30% for the steady, pump power failure and sudden valve closure

120

80

The results shown in Fig. 8 indicate the effectiveness of the selection of pipe diameters for surge protection and the related increase in total cost to respond to the increase in nodal demands. Although that this study is limited to two causes of transients but the GACCWHnet program has the capability of studying the other causes like sudden increase in a nodal demand and gradual close of valve with different rates. This study is mainly limited to the selection of optimal pipe diameters but other design criteria as pipe material and pipe thickness could also be taken into consideration in the reliability-based optimization of water distribution systems under steady and transient conditions.

Node 8

COVQ = 10%

40

COVQ = 20% COVQ = 30%

0

10

20

30

40

Time (s)

Fig. 7(g)

200

160

Pressure Head (ft)

Pump, COVQ = 30%

40

Time (s)

0

Valve, COVQ = 10% Pump, COVQ = 20%

Node 7

0

Valve, COVQ = 30% Valve, COVQ = 20%

1.8

Cost Ratio

Pressure Head (ft)

160

120

80

Node 9

COVQ = 10%

40

COVQ = 20% COVQ = 30%

0

0

10

20

30

V.5.

40

Time (s)

The technique used for the reliability-based optimization of water distribution systems integrates the steady-state hydraulic analysis, chance constraint formulation, water hammer analysis and the genetic algorithm. Repetitive calculations for a number of evaluations (= Npopsiz x Maxgen) require substantial computational effort - even for small networks. In this study, completion of an optimization run based on chance constraint requires a total run time of only 18 minutes on a computer with an Intel Pentium® M (2798 MHz) processor.

Fig. 7(h)

Fig. 7. Nodal pressure head versus time at α = 0.95 and COVQ = 10%, 20% and 30% for the sudden valve closure

V.4.

Computational Efforts

Comparison of Optimal Costs

The comparison of the optimal costs resulted from the reliability-based optimization for steady state, pump power failure and sudden valve closure is shown in Copyright © 2009 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Mechanical Engineering, Vol. 3, n. 4

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B. O. Djebedjian

VI.

Conclusion

[11] P. R. Bhave, V. V. Sonak, A Critical Study of the Linear Programming Gradient Method for Optimal Design of Water Supply Networks, Water Resources Research, Vol. 28, n. 6, pp. 1577-1584, 1992. [12] B. L. Berghout, G. Kuczera, Network Linear Programming as Pipe Network Hydraulic Analysis Tool, Journal of Hydraulic Engineering, ASCE, Vol. 123, n. 6, pp. 549-559, 1997. [13] B. A. Sakarya, L. W. Mays, Optimal Operation of Water Distribution Pumps Considering Water Quality, Journal of Water Resources Planning and Management, ASCE, Vol. 126, n. 4, pp. 210-220, 2000. [14] T. C. Yu, T. Q. Zhang, X. Li, Optimal Operation of Water Supply Systems with Tanks Based on Genetic Algorithm, Journal of Zhejiang University SCIENCE, Vol. 6A(8), pp. 886893, 2005. [15] B. Djebedjian, A. Yaseen, M. A. Rayan, A New Adaptive Penalty Method for Constrained Genetic Algorithm and its Application to Water Distribution Systems, International Pipeline Conference and Exposition 2006, September 25-29, 2006, Calgary, Alberta, Canada, IPC2006-10235. [16] D. A. Laine, B. W. Karney, Transient Analysis and Optimization in Pipeline – A Numerical Exploration, 3rd International Conference on Water Pipeline Systems, Edited by R. Chilton, 1997, pp. 281-296. [17] Z. Zhang, Fluid Transients and Pipeline Optimization using Genetic Algorithms, Master Thesis, University of Toronto, Canada, 1999. [18] B. Kaya, M. S. Güney, An Optimization Model and WaterHammer for Sprinkler Irrigation Systems, Turkish Journal of Engineering and Environmental Sciences, Vol. 24, pp. 203-215, 2000. [19] B. S. Jung, B. W. Karney, Optimum Selection of Hydraulic Devices for Water Hammer Control in the Pipeline Systems using Genetic Algorithm, FEDSM’03 4TH ASME_JSME Joint Fluids Engineering Conference, Honolulu, Hawaii, USA, July 611, 2003, Paper FEDSM2003-5262. [20] B. S. Jung, B. W. Karney, Transient State Control in Pipelines using GAs and Particle Swarm Optimization, Proc. of 6th International Conference on Hydroinformatics, Singapore, 2004. [21] B. Djebedjian, M. S. Mohamed, A. Mondy, M. A. Rayan, Network Optimization for Steady Flow and Water Hammer Using Genetic Algorithms, Proceedings of Ninth International Water Technology Conference, IWTC 2005, Sharm El-Sheikh, Egypt, 17-20 March, 2005, pp. 1101-1115. [22] B. Djebedjian, M. S. Mohamed, A. Mondy, M. A. Rayan, Cost Optimization of Water Distribution Systems Subjected to Water Hammer, Proceedings of IWTC'2009, Thirteenth International Water Technology Conference, March 12-15, 2009, Hurghada, Egypt, pp. 491-513. [23] B. Djebedjian, Reliability-Based Water Network Optimization for Steady State Flow and Water Hammer, International Pipeline Conference and Exposition 2006 (IPC2006), September 25-29, 2006, Calgary, Alberta, Canada, Paper IPC2006-10234. [24] A. Charnes, W. W. Cooper, Chance Constrained Programming, Management Science, Vol. 6, n. 1, 1959, pp. 73-79. [25] E. B. Wylie, V. L. Streeter, L. Suo, Fluid Transients in Systems (Englewood Cliffs, New Jersey, U.S.A., 1993). [26] B. E. Larock, R. W. Jeppson, G. Z. Watters, Hydraulics of Pipeline Systems (CRC Press LLC, New York, 2000). [27] D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley Reading, Mass., 1989). [28] K. Krishnakumar, Micro-Genetic Algorithms for Stationary and Non-Stationary Function Optimization, Proc. Soc. Photo-Opt. Instrum. Eng. (SPIE) on Intelligent Control and Adaptive Systems, Vol. 1196, Philadelphia, PA, 1989, pp. 289-296. [29] A. R. Simpson, L. J. Murphy, G. C. Dandy, Pipe Network Optimization using Genetic Algorithms, Paper presented at ASCE, Water Resources Planning and Management Specialty Conference, ASCE, Seattle, USA, 1993. [30] A. S. Nowak, K. R. Collins, Reliability of Structures (McGrawHill, Inc., 2000, p. 338).

The reliability-based optimization of water distribution systems under transient state is achieved in this paper by obtaining the optimal pipe diameters considering both steady state and water hammer. The following conclusions can be drawn: 1) The capability of the reliability-based optimization program GACCWHnet to find the optimal pipe diameters for cases with and without uncertainty in nodal demands under steady and transient conditions. 2) The selection of pipe diameters to make the nodal pressures between the maximum and minimum allowable pressures is an effective control strategy for hydraulic transients. 3) For the case study, the optimal costs in the case of sudden valve closure are higher than that in the case of pump power failure. The cost ratios for optimization (α = 0.5) for steady state, pump power failure and sudden valve closure are 1, 1.136 and 1.2, respectively. 4) The average cost ratios calculated for 18 values of different COV and α for each of steady state, pump power failure and sudden valve closure are 1.057, 1.210 and 1.329, respectively.

References [1]

H. Liu, Pipeline Engineering (Lewis Publishers, CRC Press LLC, 2003). [2] B. S. Jung, B. W. Karney, Fluid Transients and Pipeline Optimization using GA and PSO: the Diameter Connection, Urban Water Journal, Vol. 1, n. 2, pp. 167-176, 2004. [3] I. Goulter, M. Thomas, L. W. Mays, B. Sakarya, F. Bouchart, Y. K. Tung, Reliability Analysis for Design, in Water Distribution Systems Handbook (Larry W. Mays, Editor in Chief, McGrawHill, 2000, pp. 18.1-18.52). [4] K. E. Lansey, N. Duan, L. W. Mays, Y-K. Tung, Water Distribution System Design Under Uncertainties, Journal of Water Resources Planning and Management, ASCE, Vol. 115, n. 5, pp. 630-645, 1989. [5] C. Xu, I. C. Goulter, Reliability-Based Optimal Design of Water Distribution Networks, ASCE Journal of Water Resources Planning and Management, Vol. 125, n. 6, pp. 352-362, 1999. [6] Z. S. Kapelan, D. A. Savic, G. A. Walters, Multiobjective Sampling Design for Water Distribution Model Calibration, Journal of Water Resources Planning and Management, ASCE, Vol. 29, n. 6, pp. 466-479, 2003. [7] H. A. A. Abdel-Gawad, Optimal Design of Water Distribution Networks under a Specific Level of Reliability, Proceedings of Ninth International Water Technology Conference IWTC9, Sharm El-Sheikh, Egypt, March 17-20, 2005, pp. 641-654. [8] Y. Bao, L. W. Mays, Model for Water Distribution System Reliability, Journal of Hydraulic Engineering, Vol. 116, n. 9, pp. 1119-1137, 1990. [9] R. Gupta, P. R. Bhave, Reliability Analysis of Water Distribution Systems, Journal of Environmental Engineering, ASCE, Vol. 120, n. 2, pp. 447-460, 1994. [10] B. Djebedjian, H. A. A. Abdel-Gawad, R. Ezzeldin, A. Yaseen M. A. Rayan, Evaluation of Capacity Reliability-Based and Uncertainty-Based Optimization of Water Distribution Systems, Proceedings of IWTC'2007, Eleventh International Water Technology Conference, March 15-18, 2007, Sharm El-Sheikh, Egypt, pp. 565-587.

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Authors’ information Faculty of Engineering, Mansoura University, Egypt, tel.: 0020502244403, fax: 0020502244690, [email protected], www.iwtc.info/CV-Djebedjian.htm Berge Djebedjian was born in Alexandria (Egypt) in 1959. He is associate professor at Mechanical Power Engineering Department, Mansoura University, Egypt where he teaches hydraulic machines and hydraulic control. He received the engineer degree in mechanics in 1981 and the Ph.D. degree in 1997. He has authored over 45 papers in various journals and conference proceedings. His research interests are in the areas of water distribution networks, computational fluid dynamics and desalination. Dr. Djebedjian serves in organizing committees of several conferences.

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