[american society of civil engineers world environmental and water resources congress 2014 -...

Fuzzy Analysis of Pressure-Deficient Water Distribution Networks

Rajesh Gupta1, Juhee Harkutiya2, Shilpa Dongre3 and Lindell Ormsbee4

1Professor of Civil Engineering, Visvesvaraya National Institute of Technology, Nagpur, M.S., INDIA, 440 010. e-mail: [email protected] 2Ex.PG Student, Department of Civil Engineering, Visvesvaraya National Institute of Technology, Nagpur, M.S., INDIA, 440 010. e-mail: 3Assistant Professor of Civil Engineering, Visvesvaraya National Institute of Technology, Nagpur, M.S., INDIA, 440 010. e-mail: [email protected] 4Director, Kentucky Water Resources Research Institute, 233 Mining & Mineral Resources Bldg., University of Kentucky, Lexington, KY 40506-0107. e-mail: [email protected]

ABSTRACT

Future water demands and pipe roughness coefficients used in the design of water distribution networks (WDNs) have a high degree of uncertainty. Fuzzy analysis of WDNs provide how the uncertainties in independent or basic parameters (such as nodal demands and pipe roughness coefficients) are propagated to dependent or derived parameters (such as pipe flows, pipe velocities and available pressure heads). Usually demand dependent analysis is used for such analysis. A WDN may be pressure-deficient or may become pressure-deficient because of high pipe roughness or inadequate pump pressure and may not be able to meet desired demands at all nodes. Thus, it is desirable to consider the nodal outflows as unknown and as a function of available pressure heads under pressure-deficient conditions. An approach for fuzzy analysis of a WDN which takes into account pressure-deficient condition using node flow analysis is presented in this paper. The proposed methodology is illustrated with example network taken from literature. The methodology is useful in identifying vulnerable zones in WDNs and can be extended for reliability analysis under uncertainty of nodal demands and pipe roughness coefficients. Comparison of results with usual demand-dependent analysis shows that proposed method is better in identifying vulnerable zones. INTRODUCTION

A water distribution network (WDN) is designed to supply anticipated demands with adequate residual heads at all the nodes of a network during the entire project life. The actual nodal demands may be more than the anticipated demands due to unanticipated accelerated growth over the entire area. Even though the overall growth is as anticipated, it may be imbalanced causing more demand in some zones and less in others. The withdrawal at some nodes may be more than design demands due to pilferage or excessive leakage of water. Thus, the projected demands are uncertain. Another uncertain parameter in the design of WDNs is the pipe roughness coefficient. Because of the natural aging process and deposition of tuberculation deposits on the inside surface of pipes, the pipe roughness coefficient changes with time and uncertainty creeps in due to the unknown magnitude of change.

435World Environmental and Water Resources Congress 2014: Water without Borders © ASCE 2014

World Environmental and Water Resources Congress 2014

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Several methodologies have been suggested in the recent past to show how parameter uncertainty is propagated to dependent parameters by considering uncertain parameters as fuzzy (Revelli and Ridolfi 2002, Branisavljevic and Ivetic 2006, Gupta and Bhave 2007, Shibu and Reddy 2011, Spiliotis and Tsakiris 2012). All these methods assume that uncertain demands are satisfied at all demand nodes which may not be true. Because of an increase in demands, the pressure heads at some of the nodes get reduced to an extent that these nodes may receive partial or no supply (Bhave 1981, Wagner et al., 1988, Kalungi and Tanyimboh 1999, Siew and Tanyimboh 2012, etc.). Thus, any analysis that assumes that such stochastic demands are always satisfied, (e.g. fuzzy demand dependent analysis (FDDA)) would be incorrect for pressure-deficient WDNs. Islam et al. (2013) realised this and calculated reduced flows based on available heads obtained through FDDA in an attempt to quantify reliability. It is necessary that available flows are determined by considering nodal heads and nodal flows simultaneously as suggested herein. In this paper, a new type of analysis called fuzzy node flow analysis (FNFA) is proposed for use in extending the FDDA methodology of Gupta and Bhave (2007) to include fuzzy analysis of WDNs under pressure deficient conditions. LITERATURE REVIEW Uncertainty in parameters arises owing to randomness and imprecise knowledge. Revelli and Redolfi (2002) suggested a fuzzy approach for analysis of pipe networks considering uncertainty in parameters owing to imprecise knowledge. Considering uncertainty in independent parameters (nodal demands, pipe roughness coefficients and reservoir levels), they obtained membership functions (MF) for dependent parameters (nodal head and pipe discharges) using a method based on interval algebra and optimization theory. The method required solving several nonlinear optimization problems. Branisavljevic and Ivetic (2006) used genetic algorithm for solving the nonlinear optimization models for finding the uncertainty in dependent parameters considering uncertainty in roughness of old pipes.

Bhave and Gupta (2006) Gupta and Bhave (2007) showed that dependent parameters change monotonically with independent parameters and suggested a simple method in which the impact of the change of independent parameters on a dependent parameter is determined in the first stage. Appropriate values of independent parameters are then selected based on the nature of impact to obtain a minimum and maximum value of dependent parameters and MF is generated.

Spiliolis and Tsakiris (2007) suggested an approach based on monotony of node flow continuity equations to develop MF for dependent parameters considering uncertainty in nodal demands. Shibu and Janga Reddy (2011) suggested a cross entropy approach for selection of the lowest/largest value of independent parameter for obtaining minimum/maximum value of dependent parameter. All the methods provide similar results. However, methods avoiding solving multiple optimization problems are simple. Further, any such method should be capable of considering uncertainties in both nodal demands and pipe roughness coefficients.

All the above fuzzy analysis approaches determine minimum and maximum values of the dependent parameters and their MFs by assuming that the imposed



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uncertain demands are satisfied. The nodal heads so obtained may be lower than the respective minimum required heads at one or more number of nodes and will result in a design unable to meet the required demands. However, as shown in this paper, it would be more appropriate to expand the definition of deficiency to include the impact of supply shortfalls. Thus, nodal outflows become another set of unknowns in this case. A methodology is suggested herein to obtain MFs of pipe discharges, nodal heads and available nodal flows under pressure deficient conditions that can be used to show determine the ability of a network to meet the required demands. PROPOSED THEORY AND PROCEDURES Membership functions for fuzzy parameters. The most common type of membership functions for fuzzy parameters are: (1) Triangular (2) Trapezoidal. A fuzzy nodal demand is shown in Fig. 1(a), stating the minimum, most likely (normal), and maximum values of nodal demands. A trapezoidal function for nodal demand as shown in Fig. 1(b) has upper and lower core values and the minimum and maximum support values (Xu and Goulter 1999, Bhave and Gupta 2004).

Fig 1. Membership functions for nodal demand (a) Triangular; (b) trapezoidal The triangular membership functions are given by

min , 0)( jjjA qqq ≤=μ (1)

norjjj

jnor

j

jjjA qqq

qq

qqq ≤≤

−−

= min

min

min

, )(μ (2)

max

max

max

, )( jjnor

j

jnor

j

jjjA qqq

qq

qqq ≤≤

−−

=μ (3)

maxjq , 0)( jjA qq ≥=μ

(4) in which Aμ is MF; min

jq , norjq and max

jq are the minimum, normal and maximum

demands at node j; and qj is the demand at node j lying between minjq and max

jq.

μA (qj) Core

Boundary Support

Boundary

qj qj

(b) (a)

0

1

μA

0

1

maxjqmin

jq

norjq



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Problem formulation for FDDA and FNFA. Consider a pipe network having M source nodes, labelled j = 1, …, M; N demand nodes labelled j = M + 1, …, M + N (= J); X pipe labelled x = 1, …, X; C basic loops or circuits labelled C = 1, …, C; and M - 1 pseudo labelled C = C + 1, …, C + M - 1. For this network, let us consider, the Hazen-William roughness coefficient (i.e. CHW) is fuzzy and cut level of α = α* for CHW of all pipes. Under FDDA, the node flow continuity equations for demand nodes and loop head loss equations for loops become respectively.

(5)

)11 , 0)(

)(68.1087.4

852.1

*

*

M- C, ..., CDC

QL

cx XxHW

XX +==∑∈ =

=

αα

αα

(6) However, for FNFA under pressure deficient conditions, qj in Eq. (5) changes to qj

avl (available nodal flow) and its value will be dependent on available head (Hj

avl). Using Wagner et al. (1988) head flow relationship qj can be given by ( ) ( ) min

** if ,0 j

avlj

avlj HHq ≤=

== αααα (7)

( ) ( ) ( ) if , *

minj

1

min

min

**

desj

avlj

n

jdesj

javl

jreqj

avlj HHH

HH

HHqq

j

<<⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−

−=

==

= αααα

αα (8)

( ) desj

avlj

reqj

avlj HHqq ≥= == **

)(if , αααα (9) Thus, to find the maximum and minimum values of dependent parameters Qx, Hj

avl and qj

avl, for α = α* the optimization problem is

JjXxqorHorQ avlj

avljx ...., ,1 , ,... ,1 ,) (Min or Max * ===αα (10)

Subject to equality constraint given by Eqs. (5) and (6), alternative constraint given by Eqs. (7), (8) and (9) and also a boundary value inequality constraint

XxCCC xHWHWxHW ,... ,1 ,)( )( *max

*min =≤≤ == αααα (11)

in which *min )( αα =xHWC and *

max )( αα =xHWC being the extreme of the interval

corresponding to cut-level α=α* of the membership function. Theory and Procedure. Gupta and Bhave (2007) observed a monotonic increase or a monotonic decrease of dependent parameters (pipe discharges and nodal heads) when independent parameters changes individually or collectively. Based on this observation, following inferences were drawn:

1. The maximum value of a dependent parameter occurred when the independent fuzzy parameters had their extreme values. This changes the boundary value inequality constraints of Eq. (11) to as alternate constraints.

*max

*min )( )( αααα == == xHWHWxHWxHW CCorCC (12)

2. If one extreme value of independent fuzzy parameter gave a maximum (minimum) value of a dependent parameter, the other extreme value of

1 , 0 )(*

, ...., J jqQ jjx

x ==+=∈

∑αα



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independent fuzzy parameter gave a minimum (maximum) value of the dependent parameter.

Fuzzy NFA incorporates nodal outflows as additional dependent parameters. The outflows at pressure-deficient nodes are also observed to change monotonically as a result of changing the nodal demands and pipe roughness coefficients. Using this observation of monotonic change, the fuzzy analysis methodology of Gupta and Bhave (2007) is extended to obtain a membership function of available nodal flows under pressure-deficient conditions. The fuzzy NFA procedure has the following steps: Step 1. Carry out a NFA of the network considering normal values of uncertain

parameters. Step 2. Change an uncertain parameter to its maximum value keeping other uncertain

parameters at their normal value; and obtain the values of dependent parameter. Note the impact of changing an uncertain parameter on the dependent parameter, i.e. whether increasing, decreasing or none. Step 2 is to be repeated for all uncertain parameters.

Step 3. To determine the maximum (or minimum) value of the dependent parameters, take an appropriate extreme value of an uncertain parameter based on its impact and carry out the analysis. Step 3 can be repeated for all dependent parameters and for different α-cuts.

ILLUSTRATIVE EXAMPLE A gravity looped network shown in Fig. 2, taken from Spiliotis and Tsakiris (2012) is used to illustrate the methodology.

Fig. 2. A looped water distribution network for illustrative example

Node 1 is a source node and nodes 2 to 11 are demand nodes. The water demand at each node is considered as a symmetrical fuzzy triangular number. The normal and the minimum and maximum demand values for each node together with the ground elevation of each node are presented in Table 1. The range of the water demand, which is represented by the width of the fuzzy number, is assumed to be 30% of the corresponding central value (Spiliotis and Tsakiris, 2012). The pipes are labelled from 1 to 15 and their diameter and length are given in Table 2. The CHW values for polyethylene pipes are taken as 140 (CPHEEO 1999) without any uncertainty. The total hydraulic head at the source node 1 is 200 m. The desired pressure at any demand node to meet the respective normal demand is 30 m above ground level (Spiliotis and Tsakiris 2007, 2012).

1 2 3 4 5

6 7 8 9 10

11

1 2 3 4 5

6

7 8 9 10

1112 13 14 15



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Table 1. Water demand and ground elevation at each node of the network

Node Water Demand, L/s Ground Elevation, m Minimum Normal Maximum

2 25.50 30 34.50 142 3 21.25 25 28.75 145 4 25.50 30 34.50 144 5 21.25 25 28.75 149 6 21.25 25 28.75 141 7 21.25 25 28.75 142 8 21.25 25 28.75 140 9 21.25 25 28.75 150

10 21.25 25 28.75 150 11 34.00 40 46.00 135

Table 2. Pipe diameter and length for each pipe of the network

Pipe From node To node Diameter, mm Length, m

1 1 2 352.6 800 2 2 3 312.8 800 3 3 4 277.6 800 4 4 5 198.2 800 5 5 11 176.2 1400 6 1 6 312.8 1200 7 6 7 312.8 800 8 7 8 246.8 800 9 8 9 220.4 800

10 9 10 198.2 800 11 10 11 176.2 1400 12 2 7 220.4 1200 13 3 8 123.4 1200 14 4 9 110.2 1200 15 5 10 110.2 1200

The network’s DDA and NFA. The DDA determines whether the network is satisfactory or not to meet the normal demands. The available HGL at demand nodes for normal demands are shown in Col. 4, Table 3. On comparing the available pressure (Col. 4) with the desired pressure (Col. 3), a deficiency in pressure head is observed at nodes 5, 9 and 10 and shown by bold faces. Thus, the network is pressure-deficient for normal demands. The NFA of the network is carried out to obtain the available nodal flows. The node head-flow relationship of Wagner et al. (1988) is used with exponent nj as 1.5 (Gupta and Bhave 1996). The available pressure and available nodal heads as obtained through NFA are given in Table 3 in Cols. 5 and 6, respectively. It can be observed that at pressure-deficient nodes 5, 9 and 10, demands are partially satisfied.



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Table 3. Analysis results of network for normal demands

Node Network data NHA Solution NFA Solution Demand,

L/s Desired HGL, m

Available HGL, m

Outflow, L/s

Available HGL, m

2 30 172 194.32 30.00 194.53 3 25 175 189.21 25.00 189.65 4 30 174 184.08 30.00 184.82 5 25 179 174.38 23.29 175.97 6 25 171 194.39 25.00 194.58 7 25 172 192.18 25.00 192.47 8 25 170 185.35 25.00 186.02 9 25 180 178.17 24.68 179.43

10 25 180 172.27 21.68 174.23 11 40 165 167.96 40.00 169.75

FDDA and FNFA. Initially, FDDA is carried out using Gupta and Bhave's (2007) approach by assuming the nodal demands to be completely satisfied irrespective of the available pressure head and the MFs of the nodal heads and the pipes discharges are then obtained. The MFs for only few typical nodal heads H5, H9, H10 and H11 are shown in Fig. 3 for brevity. Next, nodal outflows are considered pressure dependent and FNFA is carried out. For this example, a pressure of 30 m above ground level is considered to be sufficient for meeting normal demands. For additional outflow at any node, required pressure is increased (Agrawal et al. 2007). The required pressure head Hreq to meet the demand qreq (higher than qnor) is given by

( ) jnreqjjj

reqj qSHH += min (13)

( ) ( ) jnnorjj

desjj qHHS /min−= (14)

The MFs are obtained using the proposed methodology and MFs for few typical nodal heads H5, H9, H10 and H11 and nodal outflows q5, q10 and q11 are shown in Fig. 4. Following can be observed from Figs. 3 and 4:

1. The minimum extreme values of pressure heads at all the nodes are higher in Fig.4 as compared to that in Fig.3. For example, minimum value of H5 in Fig. 3 is 166.82 m, while the same in Fig. 4 is 171.74 m. The same is observed at other nodes (not shown in figures) also. It shows that FDDA under pressure-deficient conditions shows higher deficiency in pressure heads.

2. The deficiency in pressure heads is observed at nodes 5, 9 and 10 at α-cut=1 in both the Figs. 3 and 4. The minimum nodal heads values starts decreasing with the decrease in α-cut. FDDA (Fig. 3) starts showing deficiency in pressure head at node 11 (i.e. H11 < 165) at α-cut = 0.6. The FNFA (Fig. 4) shows that demands are satisfied at α-cut = 0.6 and also at α-cut = 0.4. It starts shows deficiency in pressure heads at α-cut=0.2.

3. The minimum demands at all the nodes except at node 10 (Fig. 4, minimum value of q10 = 21.04 L/s against required value of 21.25 L/s) are observed to



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be satisfied. It shows that network is not satisfactory even when all the nodes have lowest value of uncertain demands.

4. The FDDA and FNFA is also carried out by multiplying the normal demands by 1.5 and considering variability as 30%, as did by Spiliotis and Tsakiris (2012) to show the vulnerability of network. The FDDA showed a deficiency in pressure heads at six nodes (nodes 4, 5, 8, 9, 10 and 11) for α-cut=1, which increased to node 3 also for α-cut=0. However, the FNFA showed a deficiency in pressure heads (or supply shortfall) at only four nodes (nodes 5, 9, 10 and 11) for α-cut=1, which increased to only node 4 for α-cut=0. Thus, the FNFA showed network deficiency in a better way. The FDDA showed two extra nodes in vulnerable zone in both cases.

Fig. 3. Fuzzy MFs for nodal heads and pipe discharges considering FDDA. CONCLUSIONS Fuzzy node flow analysis is developed to determine the deficiency in nodal supplies under fuzzy nodal demands and pipe roughness coefficients. FNFA showing nodes with supply shortfalls along with pressure deficiency are observed to be better in identifying vulnerable zones as compared to FDDA that shows pressure deficiency

00.20.40.60.8

1

166 168 170 172 174 176 178 180 182

H5

μ(H

5)

00.20.40.60.8

1

170 172 174 176 178 180 182 184 186

H9

μ(H

9)

00.20.40.60.8

1

162 164 166 168 170 172 174 176 178 180 182

H10

μ(H

10)

00.20.40.60.81

155 160 165 170 175 180

H11

μ(H

11)



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for full demands. The use of supply shortfall could be also explored in quantifying the reliability and reliability-based design of WDNs.

Fig. 4. Fuzzy MFs for nodal heads and nodal flows considering FNFA.

00.20.40.60.81

170 172 174 176 178 180 182

H5

μ(H

5)

00.20.40.60.81

174 176 178 180 182 184 186

H9

μ(H

9)

00.20.40.60.81

168 170 172 174 176 178 180 182

H10

μ(H

10)

00.20.40.60.81

162 164 166 168 170 172 174 176 178

H11

μ(H11)

0

0.5

1

21 21.5 22 22.5 23 23.5 24 24.5

q5

μ(q 5

)

0

0.5

1

21 21.2 21.4 21.6 21.8 22

q10

μ(q 1

0)

0

0.5

1

30 32 34 36 38 40 42 44 46

q11

μ(q 1

1)



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REFERENCES Agrawal, M. L., Gupta, R., and Bhave, P. R. (2007). “Optimal design of level 1

redundant water distribution networks considering nodal storage.” J. Environ. Engg., 133(3), 319-330.

Bhave, P. R. (1981). “Node flow analysis of water distribution systems.” J. Transp. Engg., 107(4), 457-467.

Bhave, P. R., and Gupta, R. (2004). “Optimal design of water distribution networks for fuzzy demands.” J. Civil Engg. and Environ. Systems, 21(4), 229-245.

Bhave, P. R. and Gupta, R. (2006). Analysis of water distribution networks. Alpha Science International Limited, Oxford, UK.

Branisavljevic, N., and Ivetic, M. (2006). “Fuzzy approach in the uncertainty analysis of the water distribution network of Becej” J. Civil Engg. and Environ. Systems, 23(3), 221-236.

CPHEEO Manual (1999). Manual on water supply and treatment. Third Edition. Central Public Health and Environmental Engineering Org., Ministry of Urban Development, Govt. of India, New Delhi, India.

Gupta, R. and Bhave, P. R. (1996). “Comparison of methods for predicting deficient network performance.” J. Water Resour. Plan. and Manage., 122(3), 214-217.

Gupta, R., and Bhave, P. R. (2007). “Fuzzy parameters in pipe network analysis.” Civil Engg. and Environ. Systems, 24(1), 33-54.

Islam M. S., Sadiq R., Rodriguez, M. J., Najjaran, H., and Hoorfar, M. (2013) "Reliability assessment for water supply systems under uncertainties", J. Water Resour. Plan. and Manage. posted ahead of print January 18, 2013. doi:10.1061/(ASCE)WR.1943-5452.0000349.

Kalungi, P., and Tanyimboh, T. T. (2003), “Redundancy model for water distribution systems.” Reliab. Engg. and System Safety, 18(2), 275-286.

Revelli, R., and Ridolfi, L. (2002). “Fuzzy approach for analysis of pipe networks.” J. Hydraul. Engrg. 128(1), 93-101.

Siew, C., and Tanyimboh, T. T. (2012). “Pressure-dependent EPANET extension.” J. Water Resour. Manage., 26(6), 1477-1498.

Shibu, A., and Reddy, M. J. (2011). “Uncertainty analysis of water distribution networks by fuzzy– cross entropy approach.” World Academy of Science, Engg. and Tech., 59, 724-731.

Spiliotis, M., and Tsakiris G. (2007). “Minimum cost irrigation network design using interactive fuzzy integer programming.” J. Irrig. Drain. Engg., 133(3), 242-248.

Spiliotis, M., and Tsakiris G. (2012). “Water distribution network analysis under fuzzy demands.” J. Civil Engg. and Environ. Systems, 29 (2), 107-122.

Wagner, J., Shamir, U., and Marks, D. (1988). "Water distribution system reliability: Simulation Methods." J. Water Resour. Plan. Manage., 114(3), 276-293.

Xu, C., and Goulter, I. C. (1999). “Optimal design of water distribution networks using fuzzy optimization.” Civil Engg. and Environ. Systems, 16(4), 243-266.



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