A Water Distribution System (WDS) that is structurally inadequate needs to be rehabilitated in order to restore the required performance. The latter is a function ...
th
10 International Conference on Hydroinformatics HIC 2012, Hamburg, GERMANY
THE INFLUENCE OF NETWORK TOPOLOGY ON WATER DISTRIBUTION SYSTEM PERFORMANCE C. TRICARICO (1), R. GARGANO (1), G. DE MARINIS (1), M. S. MORLEY (2), Z. KAPELAN(2), D. A. SAVIC (2)
(1): Dipartimento di Ingegneria Civile e Meccanica, Università degli Studi di Cassino e del Lazio Meridionale, via Di Biasio, 43, Cassino, (Italy) (2): Centre for Water Systems, University of Exeter, Harrison Building, North Park Road, Exeter, Devon (UK) A Water Distribution System (WDS) that is structurally inadequate needs to be rehabilitated in order to restore the required performance. The latter is a function mainly of the water requested by users supplied and by the topological scheme of the network. In order to contribute to identifying which rehabilitation solution could lead to a lower systemic risk of not supplying the water required, together with a lower rehabilitation cost, in this work a study of the influence of the topologic scheme on the performance of the network is analyzed. In particular the loop redundancy and the robustness of the network scheme is investigated by applying an optimization technique based on evolutionary algorithms and taking into account the water required at each node by the users. The objectives of the optimization problem are the minimization of the total rehabilitation costs (sum of the structural costs and lost revenue costs) and the minimization of system risk. The latter is defined as being the product of the probability of failure in satisfying the water demand and the consequence of this failure, i.e. the total volume of water not delivered. Water demands are considered as uncertain parameters due to the inherent difficulties involved in their estimation and they are modelled by means of probability density functions estimated in experimental studies conducted on real water distribution networks. The methodology presented has been applied to the same network characterized by four configurations with different redundancy and relevant conclusions are drawn considering the influence of the topological network on system reliability/risk. INTRODUCTION In a Water Distribution System rehabilitation/design, the topology of the network is usually assumed to be an input datum. All of the possible design/rehabilitation solutions derived by optimization methodologies, thus, consider different configurations of the same topological scheme in which solely the existing components of the network are assumed as decision variables: pipe diameters, electromechanical components, etc (e.g. [1];[2];[3];[4]). The aim of this work is to analyze the influence of the network topology on the different rehabilitation configurations obtained by varying pipe diameters (decision variable) in a cost/reliability optimization problem.
Under this assumption, on the basis of some case studies derived from the same water network, the efficiency of the rehabilitation solutions is obtained by comparison of both the topological scheme and the diameter variations. METHODOLOGY The WDS rehabilitation problem is formulated here as a multiobjective optimization method under uncertain demands. The characterization and quantification of uncertainty in water demands is achieved by means of probability analysis. The water consumption required at each network node is modelled as an independent random variable distributed with a pre-defined Probability Density Function (PDF). The parameters and the type of probability distribution adopted are derived through experimental analysis performed on a real-life distribution network in southern Italy [5];[6]. The optimization problem has been solved by applying the robust non-repeating nondominated Sorted Genetic Algorithm II (rnrNSGAII) introduced by Morley [7]. The objectives considered are the minimization of the total rehabilitation cost CTot – the sum of the structural costs CST and the lost revenue costs CLR [4] - and the minimization of the network risk, KNet - estimated herein as the maximum of the risks evaluated at each node of the network, Ki [8]. The latter is defined as the product of the probability of failure in satisfying the water demand (1 – Ri) - where Ri is the hydraulic reliability of the system node ith, defined as the probability of meeting the water requirements for that network node - and the consequence of this failure. The risk for each node i of the system is thus estimated by the following expression: NS NS Q − ∑ REQi s ∑ Q DELi s s =1 K i = (1 − Ri ) s =1 N S Q REQi s ∑ s =1
∆t ∀i ∈ {1,.., N n }
(1)
where NS represents the number of samples generated by means of the probabilistic approach and QREQi and QDELi are respectively the water required and the effectively water delivered at each node i. The Ri relationship is evaluated as a function of the supply coefficient αi(Hi), i.e. considering the water effectively supplied to the customers as a function of the available nodal heads Hi:
NS
Ri =
∑α s =1
NS
is
∀i ∈ {1,.., N n } (2)
where : 0 H −z i α i (H i ) = i H i , min − z i 1
H i < zi
,
, z i ≤ H i ≤ H i , min H i , min < H i
,
The stochastic multiple objective optimization problem formulation is thus as follows:
Minimize : CTot = C ST + C LR
Minimize : K Net = max{K i }
(3)
Subject to: Hydraulic equation constraints: Nj ,i
∑q
j
− Q REQi = 0
j =1
(4) ε
H j ,u − H j ,d = r j ⋅ q j
( j = 1,..., N l )
(5)
Decision variable constraint:
Dk ∈ D (k = 1,..., N d )
(6)
where: qj is the flow in the jth pipe; Hj,u - head at upstream node of the jth pipe; Hj,d - head at downstream node of the jth pipe; rj - coefficient of the jth pipe (headloss formula, function of pipe length, diameter and roughness coefficient); ε is the flow exponent function of the headloss formula used; Nj,i is the number of pipes connected to the ith network node; Nl is the number of network links; Nn the number of network nodes. Dk is the value of the kth discrete decision variable (rehabilitation option index); D is the discrete set of available rehabilitation options, herein considered solely in changing the existing pipes diameters; Nd is the number of decision variables; In order to estimate the objectives of the optimization problem and thus the potential failure of the network in supplying the water required by users, a pressure-driven hydraulic solver, in which the outflow from the network is assumed to be a function of the heads available at each of the nodes, has been adopted in order to model the system behaviour [9].
CASE STUDY AND RESULTS The methodology presented has been applied to several synthetic examples, all derived from the same network (Network A), which is characterized by a different value of robustness of the topological scheme. In particular, the same number of nodes have been kept in the different configurations, reducing the number of links in order to make it less redundant in terms of the number of loops until the branched configuration is reached. The original network analyzed (Net A3) has 13 pipes, 10 nodes and a single reservoir at node 9 as shown in Figure 1. This network is derived from one of the example networks provided with the EPANET2 hydraulic solver [10]. The possible rehabilitation solutions for each of the potential 13 new pipes in the network have been considered: do not substitute that pipe or substitute the pipe with a new pipe having one of the 15 available diameters. The total number of possible solutions, i.e. network configurations, is thus equal to 1613. The head of the reservoir at the node 9 has been assumed of 278m. The input characteristics considered in the optimization model are reported in Table 1: Table 1. Node and pipe characteristics - Network A3 Node 110 111 112 113 114
Elevation [m] 205 205 202 200 202
Q [l/s] 35.0 35.0 22.5 33.75
Node 115 116 117 118 2
Elevation [m] 200 199 202 202 215
Q [l/s] 45 33.75 22.5 22.5 35.0
Pipe 1 2 3 4 5 6 7
L [m] 3209 1609 1609 1609 1609 1609 61
D [mm] 457 356 254 254 305 152 457
Pipe 8 9 10 11 12 13
L [m] 1609 1609 1609 1609 1609 100
D [mm] 254 305 203 203 152 152
The Net A3 has been varied in schemes that are always less redundant by progressively removing the links. In particular, the number of nodes (and thus the water requirement at each node) is kept invariant while in a second configuration pipe 4 has been cut off leading to a structure with 2 loops (Net A2). Subsequently, pipe 9 has been eliminated in order to have a single loop network (Net A1) and at the end the branched configuration has been obtained by deleting pipe 5 (Net A0). Pipes eliminated in order to produce the different looped schemes are highlighted in table 1.
2
2
7
7
9
9
112 2
13
110
3
111
13
9
1
112 2
113
110
4
10 115
8 5
5
116
114
12
12 11
117
6
117
118
Net A3 (3 loops)
6
2
7
9
118
Net A2 (2 loops) 2
7
9
112 2 110
116
114
11
13
113
9
1
10 115
8
3
111
3
113
111 1
13
110
3
10 115
8 5
114
113
111 1
10 115
8
112 2
116
116
114
12
12
11
11
117
6
Net A1 (1 loop)
118
117
6
118
Net A0 (branched)
Figure 1. Networks A topology The input characteristics considered in the optimization model are herein reported. Multiple rnrNSGAII runs were performed using several different initial populations (i.e. different random seeds), employing 20 Latin Hypercube (LH) samples and a minimum chromosome age of 20 generations. In all runs, a population of 100 chromosomes was used and run for 1000 generations. Following the optimization runs, all of the solutions obtained were reevaluated using 100,000 Monte Carlo samples. In order to apply the pressuredriven methodology, a minimum pressure requirement of 20m has been adopted for all cases. The lost revenue estimation has been undertaken using as an example a unit cost of water of 1€/m3 and, in order to compare these costs with those of the infrastructure, a system lifespan of 50 years and an interest rate of 3.2% (an average value at the European level) were used in the relevant calculations. A selection of the solutions obtained through this process for all the four network configurations considered are illustrated in figure 2.
4,000,000
CTot [€] 3,500,000
Net A3
Net A2
Net A1
Net A0
3,000,000
2,500,000
2,000,000
1,500,000
1,000,000 0.00
KNet 0.05
0.10
0.15
0.20
0.25
0.30
0.35
Figure 2. Pareto optimal solutions for network A problems As one might expect, in all cases examined, as the total costs of rehabilitation increase, the potential risk of not supplying the water required to the customers reduces. By means of the extension to the risk of the cost of reliability optimization [4] it is possible to note that the optimal configuration which has been circled in the relevant figures is related to that particular solution of the Pareto Front which corresponds the minimum total rehabilitation cost, i.e. minimum structural cost and minimum lost revenue for water companies, with the minimum risk of failure in supplying the water requested. This solution, the Economic Level of Reliability ELR, according to social and technical considerations (e.g. [11]), can be seen as the optimal configuration to be adopted in a rehabilitation programme or at least as a threshold solution which helps the decision maker in choosing the rehabilitation configuration to adopt among those solutions having a lower risk values than that previously identified. Through the comparison of the four Pareto Optimal Fronts determined (Figure 2), it is possible to note how the redundancy of the looped system leads to solutions with a lower risk of not supplying the water demanded by customers compared to the less looped or branched network and with, at the same time, lower total rehabilitation costs. In particular, referring to the ELR solution identified for Net A3, it is reported in table 4 how solutions with the same risk value could be more costly relative to less redundant configurations.
Table 4. Cost analysis for the same network performance
Net A3 (3 loops) ELR Net A2 (2 loops) Net A1 (1 loop) Net A0 (branched)
K 0.089 0.084 0.094 0.088
CTot [€] 1,117,260 1,254,230 2,849,660 3,255,430
The pipes eliminated from the Net A3 configuration to obtain the other topology configurations are all of the same length and have diameters ranging between 254mm 305mm, as reported in table 1. In the range of the costs considered in the optimization procedure, the total replacement cost for each pipe is about 400,000 - 450,000 euro respectively for D equal to 254mm or 305mm. On this basis, it is possible to note that switching from the 2 loops configuration to the Net A1 or Net A0 configuration, the increase in the total rehabilitation cost for the same risk value could be drastically reduced if the rehabilitation solution is considering to replace the pipes eliminated. The same consideration cannot be applied instead for the Net A2 towards the Net A3 configuration, leading to the assumption that maybe the original network was particularly redundant. CONCLUSION In order to analyze the influence of the topological scheme of a network on the system rehabilitation configuration and thus on its performance, four different networks derived from the same scheme have been analyzed. These networks present the same number of nodes but are characterized by a differing number of loops. The results obtained by applying the multiobjective optimization procedure based on a cost-risk methodology have shown the remarkable influence of the topological scheme on the rehabilitation solution, at least for the network examples considered herein. The increase of the number of loops, through the addition of some links, could lead to a significant decrease of the risk in not supplying the customers with water, whilst at the same time, realizing savings in rehabilitation cost. From the comparison of the different topological configurations adopted for the same network, moreover, it is possible to note how the rehabilitation options selected by the optimization are more costly for the less redundant configurations compared to the more looped designs, considering the same risk of failure, i.e. the same reliability. This result highlights the importance of the topological scheme on system reliability that could have a greater influence with respect to the pipe diameter selections in network rehabilitation/design. This could further lead to attempts to optimize the optimal rehabilitation/design configuration consider not just the duplication/substitution of existing pipes, but also the possibility of making the actual network scheme more redundant.
REFERENCES [1] Lansey, K.E. and Mays, L.W. (1989) “Optimization Model for Water Distribution System Design”, Journal of Hydraulic Engineering - ASCE, 115(10), 1401-1418.
[2] Kapelan, Z., Savić, D.A. and Walters, G.A. (2005) “Multiobjective Design of Water Distribution Systems under Uncertainty”, Water Resources Research, 41(11), W11407
[3] Su, Y.C., Mays, L.W., Duan, N. and Lansey, K.E. (1987) “Reliability-based optimization model for water distribution system”, Journal of Hydraulic Engineering - ASCE, 114(12), 1539-1556.
[4] Tricarico, C., Gargano, R., Kapelan, Z., Savić, D.A. and de Marinis, G. (2006) “Economic Level of Reliability for the Rehabilitation of Hydraulic Networks”, Journal of Civil Engineering and Environmental Systems , 23(3), 191-207 [5] de Marinis, G., Gargano, R. and Tricarico C. (2006) “Water demand models for a small number of users”, Proceedings of the 8th Annual Water Distribution Systems Analysis Symposium WDSA2006, Buchberger, S.G. (ed.), August 27-30, Cincinnati, Ohio, (accessed April 1, 2008). [6] Tricarico, C., de Marinis, G., Gargano, R. and Leopardi, A. (2007) “Peak Residential Water Demand”, Proceedings of the Institution of Civil Engineers, Water Management, 160(2), 115121. [7] Morley, M.S. (2008) A Framework for Evolutionary Optimization Applications in Water Distribution Systems, PhD Thesis, Centre for Water Systems, University of Exeter, UK. 278pp. [8] de Marinis G., Gargano, R., Kapelan Z., Morley M., Savic D. and Tricarico C., (2008) "RiskCost based Decision Support System for the Rehabilitation of Water Distribution Networks", Proceedings of the Annual Water Distribution Systems Analysis Symposium WDSA2008, Kobus, Van Zyl (ed.), August 17-20, Kruger National Park, South Africa, ASCE, ISBN 978-0-78441024-0 [9] Morley, M.S. and Tricarico, C. (2008) Pressure Driven Demand Extension for EPANET (EPANETpdd) – Technical Report 2008-02, Centre for Water Systems, University of Exeter, UK. 10pp. [10] Rossman, L.A. (2000) EPANET 2 Users Manual, United States Environmental Protection Agency, Cincinnati, USA. [11] Kumar, A. and Kansal, M.L. (1995) “Discussion of ’Reliability Analysis of Water Distribution Systems’ by Gupta, R. and Bhave, P.R.”, Journal of Environmental Engineering - ASCE, 121(9), 674-677.
