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Looped water distribution networks design using a resilience index based heuristic approach
Urban Water 2 (2000) 115±122
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Looped water distribution networks design using a resilience index based heuristic approach Ezio Todini * Department of Earth and Geo-Environmental Sciences, University of Bologna, Via Zamboni, 67, I-40128 Bologna, Italy Received 9 December 1999; received in revised form 19 April 2000; accepted 11 September 2000
Abstract A technique, based upon the original de®nition of a ``resilience'' index to account for the fact that water distribution networks are designed as ``looped systems'' in order to increase the hydraulic reliability and the availability of water during pipe failures, is introduced. The problem is formulated as a vector optimisation problem with two objective functions: cost and resilience. The solution of the vector optimisation problem, in the space spanned by the objective functions is the non-dominated or Pareto set, the edge of which is rapidly approximated by the proposed heuristic technique even in the case of large real world networks. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Water distribution; Looped networks; Resilience; Reliability; Multi-objective; Optimisation; Heuristic
1. Introduction The modern history of water distribution network design goes back to the beginning of the nineteenth century when industrialisation created the conditions for welfare in the cities. Initially the economical designs were mainly oriented to minimising the cost of bringing the water to the city in terms of the cost of one or more reservoirs, the pipe(s) bringing the water to the city, and the cost of the water distribution network. Most of the nineteenth century systems were designed in this way and were generally schematised as a main ring pipe of large diameter (for instance 1000 mm) with several smaller radial pipes (say 400 mm) reaching into the centre of the ring in order to guarantee the delivery of water at any point, even if one of the pipes had a failure. Although the radial pipes were hydraulically connected in the centre, due to computational diculties they were considered disconnected during hydraulic computation. Under these circumstances an analytical solution that allowed the engineer to provide a standardised design for the main pipes could be produced. The very small pipes (100±200 mm) were then introduced without real economical analysis.
*
Tel.: +39-051-209-4537; fax: +39-051-209-4522. E-mail address: [email protected] (E. Todini).
With the advent of digital computers, several approaches were developed for minimum cost design of water distribution networks. In the case of irrigation distribution systems, which are generally designed as trees, the problem can be easily solved by means of Dynamic Programming or Linear Programming; unfortunately these approaches cannot be easily used for urban water distribution systems because they are generally designed not as a tree but as a series of interconnected closed loops, owing to the need to be able to guarantee delivery of water to the public under pipe failure conditions. In these pipe pressure scenarios, the ¯ow direction may vary as a function of the selected pipe diameters. In order to allow for two possible ¯ow conditions per pipe each pipe must be represented with two arches with opposite ¯ow direction. This requires the variables to be separable and increases the computational eort enormously. Several direct optimisation approaches have been tried in the past. However these were generally limited to the case of few pipes due to the computational eort required. An interesting approach by Alperovits and Shamir (1977) linearised the cost function taking as the decision variable, not the diameter of pipes, but rather the length for which a given diameter had to be used. This approach reduced the original non-linear problem into a linear one for which Linear Programming could provide a solution with reasonable computer time requirements even with very large water distribution systems.
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E. Todini / Urban Water 2 (2000) 115±122
Regardless to the optimisation approach taken for the optimal design of the looped water distribution systems, one basic question which is rarely asked explicitly in the literature is: why are we designing a looped network topology? The answer is obviously simple: looped topology allows for redundancy, which assists in ensuring that there is sucient capability in the system to allow to overcome local failures and to guarantee the distribution of water to users. This capability of overcoming stress or failure conditions is generally called ``resilience''. Unfortunately all ``minimum cost'' objective functions inevitably lead to a spanning tree with no redundancies unless speci®c constraints are imposed. An interesting example was recently proposed by Stanic, Avakumovic, Kapelan, and Larsen (1998) who used an evolutionary algorithm for the selection of the most appropriate tree shaped network which achieved high levels of system reliability with respect to pipe failure. This outcome is shown clearly in Abebe and Solomatine (1998) who compared several solutions for the simple two-loop network used as the ®rst example in this paper (Fig. 1). The ``best'' solutions (Goulter, Lussier, & Morgan, 1986; Kessler & Shamir, 1989; Eiger, Shamir, & Ben-Tal, 1994; Savic & Walters, 1997) for this network reduce the diameter of pipes 4 and 8 to an allowed minimum. Removing these two pipes, the looped network obviously degenerates into the essential and minimum cost tree. But what happens if one of the pipes, for example pipe 3, fails: users of nodes 4, 6 and 7 cannot be served. This situation would also happen if pipes 4 and 8 have a very small diameter as in the above mentioned optimal solutions. This simple example shows that the choice of the ``minimum cost function'' is not the one that allows for correctly designing a looped water distribution network because the objective function does not incorporate the concept of ``reliability'' adequately, which is the main reason for avoiding a tree shaped network. According to Mays (1996), failures may be mechanical (such as pipe breakage, pump failure, power outages, control valve failure, etc.) and hydraulic (such as changes in demand or in pressure head, ageing of pipes, inadequate pipe seizing, insucient pumping capacity, insucient storage capability). Several authors have tried to combine the two types of failures in order to de®ne a measure of reliability. Explicit consideration of reliability is in fact one of the most dicult tasks faced in water distribution network design (Goulter & Bouchart, 1990). Moreover, for water distribution systems there are no universally accepted de®nitions for risk and reliability which has resulted into a large number of publications describing the dierent approaches (see Mays, 1996). The formalisation of the reliability concept in a more comprehensive multi-objective function remains an area
requiring further research. However, in this paper the problem of accounting for reliability is overcome by the introduction of a new concept, the ``resilience'', which is strongly related to the intrinsic capability of the system to overcome failures and which does not require the statistical analysis of the dierent types of uncertainty which are required for the de®nition of the reliability constraints. The resilience concept was then used in this paper for the development of a heuristic optimisation approach which approximately describes, in the cost vs resilience space, the edge of the non-dominated solutions, the Pareto set, thus allowing the designer to identify reasonable solutions with limited computational requirements. The proposed formulation, is based upon the following physical and hydraulic considerations and does not require the statistical inference on the probability distributions of the dierent types of failures. Although used to increase reliability, a looped network topology, does not per se guarantee the delivery of water at the dierent nodes under modi®ed or stress conditions. Failures or modi®ed and increased demand conditions will in fact tend to increase the internal energy dissipation and if a surplus of energy is not available, a failure in delivery will be inevitable, regardless to the cause and to its probability of occurrence. Therefore, in order to increase reliability, regardless from the mechanisms or wrong assumptions causing stresses to the system, one would like to increase the resilience, namely the capability of the designed system to react and to overcome stress conditions. This increase in resilience, for a given topology, is here expressed as an increase of the energetic redundancy which means a decrease of the internal energy dissipation. The designer must then trade o resilience with cost and since there are wide ranges for which a small increase in cost lead to a large increase in resilience this paper introduces a heuristic technique able to approximate the edge of the non-dominated solutions, thus allowing for rapidly ®nding an ensemble of trade o solutions. 2. The concept of resilience In a water distribution system, the cost of energy appears explicitly only if the water must be pumped. In those cases where pumping is required the optimisation problem may be posed by formulating a tradeo between the cost of pumping and the cost of the pipe network. If the pipes are too small the pumping (capital and energy) cost increases due to the high head losses. Installation and running costs may together become too large to allow for economical delivery in terms of demand of water and of sucient head. When dealing with gravity driven water distribution systems, the cost used in an optimisation scheme is generally limited to the cost
E. Todini / Urban Water 2 (2000) 115±122
117
Fig. 1. Flow chart of the heuristic optimisation process.
of pipes and inevitably its minimisation produces a tree shaped network. If the water distribution network is tree shaped and optimised in cost, any failure, and in particular a pipe failure, could have severe consequences in terms of reliability: some of the nodes would not be served or poorly served. In order to reduce the risk of failure,
designers have used the concept of topological redundancy by adding pipes and closing loops so that the ¯ow could reach a given node from alternative routes in case of failures. Nevertheless it is possible to make the following considerations: if the water is delivered at each node, satisfying the demand in terms of design ¯ow and head
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E. Todini / Urban Water 2 (2000) 115±122
exactly, whenever the demand will increase or a pipe fails the water ¯ow will change and the original network is transformed into a new one with higher internal energy losses. In this situation, it might again be impossible to deliver the desired ¯ow rate at a minimum delivery pressure. The immediate consequence is that, in a looped network, we would like to provide at each node more power (energy per unit time) than that required, in order to have a sucient surplus to be dissipated internally in case of failures. This surplus can be used to characterise the resilience of the looped network, namely its intrinsic capability of overcoming sudden failures. It should be noted that, although the concept of resilience does not involve statistical considerations on failures, its increase will lead to improved network reliability. If we denote with Ptot c
nr X
Qk Hk ;
1
k1
the total available power at the entrance in the water distribution network, where c is the speci®c weight of water, Qk and Hk are the discharge and the head, respectively, relevant to each reservoir k, while nk is the number of reservoirs, the following simple relationship exists: Ptot Pint Pext ;
2
where PP int is the power dissipated in the pipes while nn qi hi is the power that is delivered to the Pext c i1 users in terms of ¯ow qi and head hi at each node i, with nn the number of nodes. A resilience index Ir may be de®ned as: P n =Pmax , where Pint Ptot ÿ c ni1 qi hi is Ir 1 ÿ Pint the amount of power dissipated in the network Pnn to satisfy Ptot ÿ c i1 qi hi the the total demand and Pmax maximum power that would be dissipated internally in order to satisfy the constraints in terms of demand and head at the nodes. After appropriate substitutions, the resilience index Ir can be written as: Pnn ÿ q hi ÿ h 3 Ir Pnr i1 i Pnn i : k1 Qk Hk ÿ i1 qi hi Eq. (3) represents the index on which the proposed heuristic optimisation is based. It should be noted that the resilience index can be easily modi®ed in order to account for the presence of pumps by modifying Ptot , to give: Ptot c
nr X k1
Qk Hk
np X
Pj ;
4
j1
where Pj is the power introduced into the network by the jth pump and np the number of pumps. Consequently the resilience index becomes:
Pn n ÿ i1 qi hi ÿ hi Pnp Pnn : Ir Pnr j1 Pj =c ÿ k1 Qk Hk i1 qi hi
5
Another useful index is the failure index If de®ned as: Pnn Ifi P If nni1 ; q i1 i hi where 0ÿ Ifi qi hi ÿ hi
8i : hi P hi ; 8i : hi < hi :
6
This index can be used both to identify infeasibilities during the optimisation process and to evaluate and compare the eect of pipe failures. The third index is the available surplus of head in the most depressed node de®ned as: 8i 1; nn : 7 Is min hi ÿ hi This quantity, which is not used in the optimisation process, gives insight in the hydraulic behaviour of the network, and can also be used to discriminate among the solutions, as in the third example (Fig. 5).
3. The heuristic approach The topology of an urban water distribution system is generally imposed by the structure of the urban context: roads, buildings, industrial areas, hospitals, etc. and a layout is generally available. Therefore, the optimal design of the water distribution network to be solved assumes a pre-de®ned topology and a set of constraints in terms of amount of water to be delivered at the nodes, where the demand is assumed to be concentrated, and of minimum head necessary to draw water from the tap. The design demands in terms of quantity of water to be delivered are preliminarily established on the basis of population consumption and growth, industrial development, ®re hydrants, etc., while the minimum head distribution is de®ned on the basis of the delivery pressure needs taking into account the topographical elevation. The available commercial diameters and the relevant costs (which would include additional costs such as for instance the excavation cost and the cost of special parts) complete the required information. The proposed procedure starts from an initial set of diameters determined by the designer on the basis of his experience. This is not essential, since any set of diameters can be used. Nevertheless a reasonable initial solution can help in the search. The following process is then applied. (i) Fix a value for the minimum required resilience index. (ii) All the pipes are set at a given diameter. The initial diameter for the pipes can be selected on the basis
E. Todini / Urban Water 2 (2000) 115±122
of engineering considerations or more simply can be set to the largest class of commercial diameters. (iii) Network hydraulic analysis, which is performed using the ``global gradient'' developed by Todini (1979), Todini and Pilati (1988) which has recently gained some credibility after its inclusion in EPANET (Rossman, 1993). From the results of the analysis, if the failure index (Eq. (6)) is null the procedure starts reducing the pipe diameters, otherwise the pipe diameters must be increased. (iv) Reduction of diameters is performed for the pipe connecting nodes i and j for which the largest decrement in cost per unit of power dissipation occurs for one step in commercial diameters Dkij ! Dijkÿ1 of the generic pipe connecting nodes i; j, as " # Cijkÿ1 ÿ Cijk 8i; j; 8 fi ; j g max ÿ kÿ1 fi;jg Pij ÿ Pijk where fi ; j g is the pipe connecting nodes i and j while fi; jg represents the set of all pipes. The ®rst pipes to be reduced in diameter tend to be the longest and largest. Before applying the reduction in diameter three controls are made. The ®rst is a velocity control: the maximum velocity in the modi®ed pipe should not exceed a pre-determined value (generally set to 2 m/s). The second check is made on the basis of the resilience index, given as a function of an approximate estimate of the overall increase in internal power dissipation, which should be larger of a speci®ed value. The third control is made on the basis of an estimate of the failure index, given in terms of the resilience index, which must not be positive. This procedure is followed iteratively, without recourse to a new network analysis at each diameter change, until no more moves are possible. At this point, a network analysis is performed (iii) given the new diameter con®guration. (v) The increase of diameters proceeds according to the largest decrement of the internal power dissipation per unit cost: " # Pijk ÿ Pijkÿ1 8i; j; 9 fi ; j g max ÿ k fi;jg Cij ÿ Cijkÿ1 as a function of the change in the diameter Dijkÿ1 ! Dkij of the generic pipe connecting nodes i; j. This process of diameter increase will tend to increase the size of pipes in the present con®guration for which power can be gained at smaller increases of cost. Following the ¯ow chart of the heuristic optimisation process given in Fig. 1, the increase in diameters stops either according to a maximum number of moves or when the estimate of the reliability index is higher than the given value. The objective of the proposed approach, which tends to increase the resilience as a function of the
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looped structure of the pipe networks, is in fact to try to distribute the ¯ow more evenly among all the pipes, which is the opposite of concentrating the ¯ow in a spanning tree. Again, at the end of the step, a new network analysis is performed (iii) and both the resilience and the failure indexes are computed before starting again from (iv). (vi) When no more moves are possible, or after a ®xed number of steps, a new value for the minimum acceptable resilience index is given (i) and the procedure starts again. After a number of steps, generally less than 50, it is possible to construct and analyse a cost vs resilience function that gives indication on the set of optimal possible solutions. It must be emphasised that, due to the heuristic nature of the approach, this function is not the mathematically exact Pareto limiting curve, but it is a very good approximation of it, which can be used to identify a set of acceptable solutions. It is worthwhile noticing that the inclusion of pumps can be treated similarly to the change in diameter. The introduction of a pump will increase the total available power, thus increasing the resilience index. However, at the same time introducing pumps will increase the overall cost of the network.
4. Examples of application Three examples are reported in the paper in order to illustrate the concepts and the possibilities oered by the approach. The ®rst example is derived from the Alperovits and Shamir (1977) problem and was reported in Abebe and Solomatine (1998). This network is the extremely simpli®ed two-loop network shown in Fig. 2 for which nodal head and demands are given as in Table 1 and all pipes 1000 m long. The cost of the dierent diameter classes (which were kept in inches to be consistent with the previous papers), in terms of unspeci®ed cost units, is given in Table 2.
Fig. 2. The two-loop network.
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E. Todini / Urban Water 2 (2000) 115±122
Table 1 Head and demand values for the two-loop network Node
Head (m)
Demand m3 =h
1 2 3 4 5 6 7
210 150 160 155 150 165 160
ÿ1120.00 100.00 100.00 120.00 270.00 330.00 200.00 Fig. 3. The limitig curve of the Pareto set of solutions in the two objectives space.
Table 2 Diameter classes and cost of pipes Diameter (inches)
Cost (units)
1 2 3 4 6 8 10 12 14 16 18 20 22 24
2 5 8 11 16 23 32 50 60 90 130 170 300 550
The analysis based on the heuristic method, using the Hazen±Williams formula with a friction coecient of 130 to be consistent with the previous papers, produced the edge of the Pareto set shown in Fig. 3. The information in this ®gure provides insights into tradeos between the two objectives, namely cost and the resilience. It can be seen from the ®rst part of the limiting curve in Fig. 3 that the resilience index can be doubled with very small increases in cost. The set of four solutions (A, B, C and D) in the range of 0.4±0.5 of the resilience
index is reported in Table 3 and compared to the optimal cost solution in terms of pipe diameters, cost, resilience index and head surplus of the worst case node. It can be seen that with an increase in cost from 0:419 106 to 0:450 106 units both, the resilience index and the head surplus (Sol. A) are eectively doubled. Moreover the designer can decide to spend a little more and obtain solution D with a resilience index further increased to 0.48 and a head surplus increased to 3.44 m. The second example is based upon the schematic looped main described in Fig. 4(a) and (b), using the same diameter classes and cost data of the previous example with the relevant nodal heads and demands as given in Table 4. The layouts shown in Fig. 4(a) and (b) are only topological. The pipe actual lengths are as follows: the pipe connecting nodes 0±1 is 2000 m long, the contour pipes are all 1000 m long and all the internal pipes connected to node 9 are 1210 m long. Table 4 gives the head and the resulting ¯ow in node 0 together with the prescribed minimum head and demand for nodes 1±9. The results of the application of the methodology to this example are shown in Table 5 where the classical design of the main loop is compared to the design obtained using the heuristic approach described in this paper. The cost of the classical design is higher
Table 3 Comparison of optimal and alternative solutions for the two-loop network Pipe n
Cost opt.
Sol. A
Sol. B
Sol. C
Sol. D
1 2 3 4 5 6 7 8 Res. ind. Head surp. (m) Cost (units)
18 10 16 4 16 10 10 1 0.22 0.50 0:419 106
18 16 14 6 14 1 14 10 0.41 1.08 0:450 106
20 14 14 6 14 1 14 10 0.47 1.90 0:460 106
20 14 14 8 14 1 14 10 0.48 2.84 0:467 106
20 14 14 6 14 1 14 12 0.48 3.44 0:478 106
E. Todini / Urban Water 2 (2000) 115±122
Fig. 4. (a) A classical main loop network design. (b) An improved design for the same system.
Table 4 Head and demand values for the second example Node
Head (m)
Demand m3 =h
0 1 2 3 4 5 6 7 8 9
200 197 193 192 191 191 191 192 193 189
ÿ180.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00
Table 5 Comparison between a classical main loop network design and an interesting solution provided by the heuristic approach Pipe
Main loop
Improved
0±1 1±2 2±3 3±4 4±5 5±6 6±7 7±8 8±1 1±9 2±9 3±9 4±9 5±9 6±9 7±9 8±9 Res. ind. Head surp. (m) Cost (units)
16 12 12 12 12 12 12 12 12 3 3 3 3 3 3 3 3 0.20 0.70 0:657 106
18 14 10 8 1 1 8 10 14 10 1 1 1 8 1 1 1 0.52 7.08 0:565 106
(0:657 106 ) but produces a low resilience (Ir 0:20 with only 0.7 m of surplus head available in the most depressed node. The design resulting from the use of the proposed method is less expensive (0:565 106 ), more
121
resilient ( Ir 0:52 and has a surplus head of more than 7 m in the most depressed node. The ®nal example is presented in order to show the eciency of the proposed approach in the case of a relatively complex network (230 pipes, 167 nodes and three reservoirs). The boundary curve of the Pareto set for this example was found with a computational eort of less than 10 on a portable PC (266 Mh). The resilience index, the cost and the minimum surplus head for the dierent alternative solutions for this example are given in Table 6. The edge of the Pareto boundary is shown in the space of cost and reliability index (Fig. 5) and in the space of cost and minimum surplus head (Fig. 6). In this example, the complex looped topological structure of the pipe network seems to guarantee a minimum for the resilience index of 0.59 Table 6 Comparison among several solutions found for the third example in terms of resilience index, cost and minimum surplus head Solution
Resilience index
Cost (units)
Head surp. (m)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.593 0.607 0.635 0.658 0.671 0.681 0.700 0.705 0.727 0.737 0.748 0.750 0.756 0.763 0.765 0.767 0.806 0.815 0.826 0.874
4263 4279 4326 4318 4357 4381 4397 4382 4433 4479 4484 4493 4499 4525 4580 4585 4678 4719 4766 4984
0.59 0.56 0.51 0.03 1.07 1.83 1.67 1.85 1.81 3.54 3.66 3.56 3.43 3.81 4.94 5.06 5.18 5.24 5.25 6.37
Fig. 5. The limitig curve of the Pareto set of solutions in the cost-resilience index space for the third example.
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E. Todini / Urban Water 2 (2000) 115±122
Fig. 6. The limitig curve of the Pareto set of solutions in the costminimum surplus head space for the third example.
for all feasible solutions. However, the various solutions can now be considered not only on the basis of the tradeos between cost and resilience but also on the basis of tradeos in the minimum surplus of head in the most depressed node.
5. Conclusions The concept of minimum cost optimisation for looped municipal water distribution systems is reviewed under a new multi-objective approach. This approach introduced in the paper employs a simple heuristic technique that tries to emulate the reasoning of an engineer who is aiming on the one hand at reducing the cost while, on the other hand, at preserving a sucient degree of resilience in the system to cope with possible failures. After de®ning the problem into a two objective space, it was shown that the technique is capable of rapidly describing the optimal Pareto set which allows satisfactory solutions to be found as a tradeo between system resilience and cost. From three examples used in the paper it shows that the satisfactory solutions do not correspond either to the minimum cost solutions or to the classical main loop solutions. The classical loop solutions are in general more reliable than the minimum cost solutions, but can be further improved under a more systematic multiobjective analysis. The eciency of the proposed technique is not limited to the case of extremely simpli®ed networks. The eciency also prevails in the case of complex networks for
which a long computational time can be expected when using more rigorous optimisation techniques. The considerations introduced in this paper need to be further developed and expanded by the scholars and designers who deal with optimisation of water distribution systems with the aim of producing both ecient and practically usable tools. In particular the proposed technique could be easily tested on the basis of reliability considerations and included into a knowledge based Decision Support System aimed at helping the engineer in selecting the most appropriate design for water distribution networks.
References Abebe, A. J., & Solomatine, D. P. (1998). Application of global optimization to the design of pipe networks. In Babovic & Larsen (Eds.), Hydroinformatics '98 (pp. 989±996). Rotterdam: Balkema. Alperovits, E., & Shamir, U. (1977). Design of optimal water distribution systems. Water Resource Research, 13(6), 885±900. Eiger, G. U., Shamir, U., & Ben-Tal, A. (1994). Optimal design of water distribution networks. Water Resource Research, 30(9), 2637±2646. Goulter, I. C., Lussier, B. M., & Morgan, D. R. (1986). Implications of head loss path choice in the optimisation of water distribution networks. Water Resource Research, 22(5), 819±822. Goulter, I. C., & Bouchart, F. (1990). Reliability constrained pipe networks model. Journal of Hydraulic Engineering ASCE, 116(2), 221±229. Kessler, A., & Shamir, U. (1989). Analysis of the linear programming gradient method for optimal design of water supply networks. Water Resource Research, 25(7), 1469±1480. Mays, L. W., (1996). Review of reliability analysis of water distribution systems. In Tikle, Goulter, Xu, Wasimi, & Bouchart (Eds.), Stochastic hydraulics '96 (pp. 53±62). Rotterdam: Balkema. Rossman, L. A., (1993). EPANET, Users manual. Risk Reduction Engineering Laboratory, Oce of Research & Development, US Environmental Protection Agency, Cincinnati, Ohio. Savic, D. A., & Walters, G. A. (1997). Genetic algorithms for least-cost design of water distribution networks. Water Resource Planning and Management, 123(2), 66±77. Stanic, M., Avakumovic, D., & Kapelan, Z., 1998. Evolutionary algorithm for determining optimal layout of water distribution networks. In Babovic, & Larsen (Eds.), Hydroinformatics '98 (pp. 901±908). Rotterdam: Balkema. Todini, E., (1979). Un metodo del gradiente per la veri®ca delle reti idrauliche (in Italian). Bollettino deglli Ingegneri della Toscana n.11. Todini, E., & Pilati, S. (1988). A gradient algorithm for the analysis of pipe networks. In B. Coulbeck & Chun-Hou Orr (Eds.), Computer applications in water supply (pp. 1±20). Wiley: Research Studies Press.
www.elsevier.com/locate/urbwat
Looped water distribution networks design using a resilience index based heuristic approach Ezio Todini * Department of Earth and Geo-Environmental Sciences, University of Bologna, Via Zamboni, 67, I-40128 Bologna, Italy Received 9 December 1999; received in revised form 19 April 2000; accepted 11 September 2000
Abstract A technique, based upon the original de®nition of a ``resilience'' index to account for the fact that water distribution networks are designed as ``looped systems'' in order to increase the hydraulic reliability and the availability of water during pipe failures, is introduced. The problem is formulated as a vector optimisation problem with two objective functions: cost and resilience. The solution of the vector optimisation problem, in the space spanned by the objective functions is the non-dominated or Pareto set, the edge of which is rapidly approximated by the proposed heuristic technique even in the case of large real world networks. Ó 2000 Elsevier Science Ltd. All rights reserved. Keywords: Water distribution; Looped networks; Resilience; Reliability; Multi-objective; Optimisation; Heuristic
1. Introduction The modern history of water distribution network design goes back to the beginning of the nineteenth century when industrialisation created the conditions for welfare in the cities. Initially the economical designs were mainly oriented to minimising the cost of bringing the water to the city in terms of the cost of one or more reservoirs, the pipe(s) bringing the water to the city, and the cost of the water distribution network. Most of the nineteenth century systems were designed in this way and were generally schematised as a main ring pipe of large diameter (for instance 1000 mm) with several smaller radial pipes (say 400 mm) reaching into the centre of the ring in order to guarantee the delivery of water at any point, even if one of the pipes had a failure. Although the radial pipes were hydraulically connected in the centre, due to computational diculties they were considered disconnected during hydraulic computation. Under these circumstances an analytical solution that allowed the engineer to provide a standardised design for the main pipes could be produced. The very small pipes (100±200 mm) were then introduced without real economical analysis.
*
Tel.: +39-051-209-4537; fax: +39-051-209-4522. E-mail address: [email protected] (E. Todini).
With the advent of digital computers, several approaches were developed for minimum cost design of water distribution networks. In the case of irrigation distribution systems, which are generally designed as trees, the problem can be easily solved by means of Dynamic Programming or Linear Programming; unfortunately these approaches cannot be easily used for urban water distribution systems because they are generally designed not as a tree but as a series of interconnected closed loops, owing to the need to be able to guarantee delivery of water to the public under pipe failure conditions. In these pipe pressure scenarios, the ¯ow direction may vary as a function of the selected pipe diameters. In order to allow for two possible ¯ow conditions per pipe each pipe must be represented with two arches with opposite ¯ow direction. This requires the variables to be separable and increases the computational eort enormously. Several direct optimisation approaches have been tried in the past. However these were generally limited to the case of few pipes due to the computational eort required. An interesting approach by Alperovits and Shamir (1977) linearised the cost function taking as the decision variable, not the diameter of pipes, but rather the length for which a given diameter had to be used. This approach reduced the original non-linear problem into a linear one for which Linear Programming could provide a solution with reasonable computer time requirements even with very large water distribution systems.
1462-0758/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved. PII: S 1 4 6 2 - 0 7 5 8 ( 0 0 ) 0 0 0 4 9 - 2
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E. Todini / Urban Water 2 (2000) 115±122
Regardless to the optimisation approach taken for the optimal design of the looped water distribution systems, one basic question which is rarely asked explicitly in the literature is: why are we designing a looped network topology? The answer is obviously simple: looped topology allows for redundancy, which assists in ensuring that there is sucient capability in the system to allow to overcome local failures and to guarantee the distribution of water to users. This capability of overcoming stress or failure conditions is generally called ``resilience''. Unfortunately all ``minimum cost'' objective functions inevitably lead to a spanning tree with no redundancies unless speci®c constraints are imposed. An interesting example was recently proposed by Stanic, Avakumovic, Kapelan, and Larsen (1998) who used an evolutionary algorithm for the selection of the most appropriate tree shaped network which achieved high levels of system reliability with respect to pipe failure. This outcome is shown clearly in Abebe and Solomatine (1998) who compared several solutions for the simple two-loop network used as the ®rst example in this paper (Fig. 1). The ``best'' solutions (Goulter, Lussier, & Morgan, 1986; Kessler & Shamir, 1989; Eiger, Shamir, & Ben-Tal, 1994; Savic & Walters, 1997) for this network reduce the diameter of pipes 4 and 8 to an allowed minimum. Removing these two pipes, the looped network obviously degenerates into the essential and minimum cost tree. But what happens if one of the pipes, for example pipe 3, fails: users of nodes 4, 6 and 7 cannot be served. This situation would also happen if pipes 4 and 8 have a very small diameter as in the above mentioned optimal solutions. This simple example shows that the choice of the ``minimum cost function'' is not the one that allows for correctly designing a looped water distribution network because the objective function does not incorporate the concept of ``reliability'' adequately, which is the main reason for avoiding a tree shaped network. According to Mays (1996), failures may be mechanical (such as pipe breakage, pump failure, power outages, control valve failure, etc.) and hydraulic (such as changes in demand or in pressure head, ageing of pipes, inadequate pipe seizing, insucient pumping capacity, insucient storage capability). Several authors have tried to combine the two types of failures in order to de®ne a measure of reliability. Explicit consideration of reliability is in fact one of the most dicult tasks faced in water distribution network design (Goulter & Bouchart, 1990). Moreover, for water distribution systems there are no universally accepted de®nitions for risk and reliability which has resulted into a large number of publications describing the dierent approaches (see Mays, 1996). The formalisation of the reliability concept in a more comprehensive multi-objective function remains an area
requiring further research. However, in this paper the problem of accounting for reliability is overcome by the introduction of a new concept, the ``resilience'', which is strongly related to the intrinsic capability of the system to overcome failures and which does not require the statistical analysis of the dierent types of uncertainty which are required for the de®nition of the reliability constraints. The resilience concept was then used in this paper for the development of a heuristic optimisation approach which approximately describes, in the cost vs resilience space, the edge of the non-dominated solutions, the Pareto set, thus allowing the designer to identify reasonable solutions with limited computational requirements. The proposed formulation, is based upon the following physical and hydraulic considerations and does not require the statistical inference on the probability distributions of the dierent types of failures. Although used to increase reliability, a looped network topology, does not per se guarantee the delivery of water at the dierent nodes under modi®ed or stress conditions. Failures or modi®ed and increased demand conditions will in fact tend to increase the internal energy dissipation and if a surplus of energy is not available, a failure in delivery will be inevitable, regardless to the cause and to its probability of occurrence. Therefore, in order to increase reliability, regardless from the mechanisms or wrong assumptions causing stresses to the system, one would like to increase the resilience, namely the capability of the designed system to react and to overcome stress conditions. This increase in resilience, for a given topology, is here expressed as an increase of the energetic redundancy which means a decrease of the internal energy dissipation. The designer must then trade o resilience with cost and since there are wide ranges for which a small increase in cost lead to a large increase in resilience this paper introduces a heuristic technique able to approximate the edge of the non-dominated solutions, thus allowing for rapidly ®nding an ensemble of trade o solutions. 2. The concept of resilience In a water distribution system, the cost of energy appears explicitly only if the water must be pumped. In those cases where pumping is required the optimisation problem may be posed by formulating a tradeo between the cost of pumping and the cost of the pipe network. If the pipes are too small the pumping (capital and energy) cost increases due to the high head losses. Installation and running costs may together become too large to allow for economical delivery in terms of demand of water and of sucient head. When dealing with gravity driven water distribution systems, the cost used in an optimisation scheme is generally limited to the cost
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Fig. 1. Flow chart of the heuristic optimisation process.
of pipes and inevitably its minimisation produces a tree shaped network. If the water distribution network is tree shaped and optimised in cost, any failure, and in particular a pipe failure, could have severe consequences in terms of reliability: some of the nodes would not be served or poorly served. In order to reduce the risk of failure,
designers have used the concept of topological redundancy by adding pipes and closing loops so that the ¯ow could reach a given node from alternative routes in case of failures. Nevertheless it is possible to make the following considerations: if the water is delivered at each node, satisfying the demand in terms of design ¯ow and head
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exactly, whenever the demand will increase or a pipe fails the water ¯ow will change and the original network is transformed into a new one with higher internal energy losses. In this situation, it might again be impossible to deliver the desired ¯ow rate at a minimum delivery pressure. The immediate consequence is that, in a looped network, we would like to provide at each node more power (energy per unit time) than that required, in order to have a sucient surplus to be dissipated internally in case of failures. This surplus can be used to characterise the resilience of the looped network, namely its intrinsic capability of overcoming sudden failures. It should be noted that, although the concept of resilience does not involve statistical considerations on failures, its increase will lead to improved network reliability. If we denote with Ptot c
nr X
Qk Hk ;
1
k1
the total available power at the entrance in the water distribution network, where c is the speci®c weight of water, Qk and Hk are the discharge and the head, respectively, relevant to each reservoir k, while nk is the number of reservoirs, the following simple relationship exists: Ptot Pint Pext ;
2
where PP int is the power dissipated in the pipes while nn qi hi is the power that is delivered to the Pext c i1 users in terms of ¯ow qi and head hi at each node i, with nn the number of nodes. A resilience index Ir may be de®ned as: P n =Pmax , where Pint Ptot ÿ c ni1 qi hi is Ir 1 ÿ Pint the amount of power dissipated in the network Pnn to satisfy Ptot ÿ c i1 qi hi the the total demand and Pmax maximum power that would be dissipated internally in order to satisfy the constraints in terms of demand and head at the nodes. After appropriate substitutions, the resilience index Ir can be written as: Pnn ÿ q hi ÿ h 3 Ir Pnr i1 i Pnn i : k1 Qk Hk ÿ i1 qi hi Eq. (3) represents the index on which the proposed heuristic optimisation is based. It should be noted that the resilience index can be easily modi®ed in order to account for the presence of pumps by modifying Ptot , to give: Ptot c
nr X k1
Qk Hk
np X
Pj ;
4
j1
where Pj is the power introduced into the network by the jth pump and np the number of pumps. Consequently the resilience index becomes:
Pn n ÿ i1 qi hi ÿ hi Pnp Pnn : Ir Pnr j1 Pj =c ÿ k1 Qk Hk i1 qi hi
5
Another useful index is the failure index If de®ned as: Pnn Ifi P If nni1 ; q i1 i hi where 0ÿ Ifi qi hi ÿ hi
8i : hi P hi ; 8i : hi < hi :
6
This index can be used both to identify infeasibilities during the optimisation process and to evaluate and compare the eect of pipe failures. The third index is the available surplus of head in the most depressed node de®ned as: 8i 1; nn : 7 Is min hi ÿ hi This quantity, which is not used in the optimisation process, gives insight in the hydraulic behaviour of the network, and can also be used to discriminate among the solutions, as in the third example (Fig. 5).
3. The heuristic approach The topology of an urban water distribution system is generally imposed by the structure of the urban context: roads, buildings, industrial areas, hospitals, etc. and a layout is generally available. Therefore, the optimal design of the water distribution network to be solved assumes a pre-de®ned topology and a set of constraints in terms of amount of water to be delivered at the nodes, where the demand is assumed to be concentrated, and of minimum head necessary to draw water from the tap. The design demands in terms of quantity of water to be delivered are preliminarily established on the basis of population consumption and growth, industrial development, ®re hydrants, etc., while the minimum head distribution is de®ned on the basis of the delivery pressure needs taking into account the topographical elevation. The available commercial diameters and the relevant costs (which would include additional costs such as for instance the excavation cost and the cost of special parts) complete the required information. The proposed procedure starts from an initial set of diameters determined by the designer on the basis of his experience. This is not essential, since any set of diameters can be used. Nevertheless a reasonable initial solution can help in the search. The following process is then applied. (i) Fix a value for the minimum required resilience index. (ii) All the pipes are set at a given diameter. The initial diameter for the pipes can be selected on the basis
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of engineering considerations or more simply can be set to the largest class of commercial diameters. (iii) Network hydraulic analysis, which is performed using the ``global gradient'' developed by Todini (1979), Todini and Pilati (1988) which has recently gained some credibility after its inclusion in EPANET (Rossman, 1993). From the results of the analysis, if the failure index (Eq. (6)) is null the procedure starts reducing the pipe diameters, otherwise the pipe diameters must be increased. (iv) Reduction of diameters is performed for the pipe connecting nodes i and j for which the largest decrement in cost per unit of power dissipation occurs for one step in commercial diameters Dkij ! Dijkÿ1 of the generic pipe connecting nodes i; j, as " # Cijkÿ1 ÿ Cijk 8i; j; 8 fi ; j g max ÿ kÿ1 fi;jg Pij ÿ Pijk where fi ; j g is the pipe connecting nodes i and j while fi; jg represents the set of all pipes. The ®rst pipes to be reduced in diameter tend to be the longest and largest. Before applying the reduction in diameter three controls are made. The ®rst is a velocity control: the maximum velocity in the modi®ed pipe should not exceed a pre-determined value (generally set to 2 m/s). The second check is made on the basis of the resilience index, given as a function of an approximate estimate of the overall increase in internal power dissipation, which should be larger of a speci®ed value. The third control is made on the basis of an estimate of the failure index, given in terms of the resilience index, which must not be positive. This procedure is followed iteratively, without recourse to a new network analysis at each diameter change, until no more moves are possible. At this point, a network analysis is performed (iii) given the new diameter con®guration. (v) The increase of diameters proceeds according to the largest decrement of the internal power dissipation per unit cost: " # Pijk ÿ Pijkÿ1 8i; j; 9 fi ; j g max ÿ k fi;jg Cij ÿ Cijkÿ1 as a function of the change in the diameter Dijkÿ1 ! Dkij of the generic pipe connecting nodes i; j. This process of diameter increase will tend to increase the size of pipes in the present con®guration for which power can be gained at smaller increases of cost. Following the ¯ow chart of the heuristic optimisation process given in Fig. 1, the increase in diameters stops either according to a maximum number of moves or when the estimate of the reliability index is higher than the given value. The objective of the proposed approach, which tends to increase the resilience as a function of the
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looped structure of the pipe networks, is in fact to try to distribute the ¯ow more evenly among all the pipes, which is the opposite of concentrating the ¯ow in a spanning tree. Again, at the end of the step, a new network analysis is performed (iii) and both the resilience and the failure indexes are computed before starting again from (iv). (vi) When no more moves are possible, or after a ®xed number of steps, a new value for the minimum acceptable resilience index is given (i) and the procedure starts again. After a number of steps, generally less than 50, it is possible to construct and analyse a cost vs resilience function that gives indication on the set of optimal possible solutions. It must be emphasised that, due to the heuristic nature of the approach, this function is not the mathematically exact Pareto limiting curve, but it is a very good approximation of it, which can be used to identify a set of acceptable solutions. It is worthwhile noticing that the inclusion of pumps can be treated similarly to the change in diameter. The introduction of a pump will increase the total available power, thus increasing the resilience index. However, at the same time introducing pumps will increase the overall cost of the network.
4. Examples of application Three examples are reported in the paper in order to illustrate the concepts and the possibilities oered by the approach. The ®rst example is derived from the Alperovits and Shamir (1977) problem and was reported in Abebe and Solomatine (1998). This network is the extremely simpli®ed two-loop network shown in Fig. 2 for which nodal head and demands are given as in Table 1 and all pipes 1000 m long. The cost of the dierent diameter classes (which were kept in inches to be consistent with the previous papers), in terms of unspeci®ed cost units, is given in Table 2.
Fig. 2. The two-loop network.
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Table 1 Head and demand values for the two-loop network Node
Head (m)
Demand m3 =h
1 2 3 4 5 6 7
210 150 160 155 150 165 160
ÿ1120.00 100.00 100.00 120.00 270.00 330.00 200.00 Fig. 3. The limitig curve of the Pareto set of solutions in the two objectives space.
Table 2 Diameter classes and cost of pipes Diameter (inches)
Cost (units)
1 2 3 4 6 8 10 12 14 16 18 20 22 24
2 5 8 11 16 23 32 50 60 90 130 170 300 550
The analysis based on the heuristic method, using the Hazen±Williams formula with a friction coecient of 130 to be consistent with the previous papers, produced the edge of the Pareto set shown in Fig. 3. The information in this ®gure provides insights into tradeos between the two objectives, namely cost and the resilience. It can be seen from the ®rst part of the limiting curve in Fig. 3 that the resilience index can be doubled with very small increases in cost. The set of four solutions (A, B, C and D) in the range of 0.4±0.5 of the resilience
index is reported in Table 3 and compared to the optimal cost solution in terms of pipe diameters, cost, resilience index and head surplus of the worst case node. It can be seen that with an increase in cost from 0:419 106 to 0:450 106 units both, the resilience index and the head surplus (Sol. A) are eectively doubled. Moreover the designer can decide to spend a little more and obtain solution D with a resilience index further increased to 0.48 and a head surplus increased to 3.44 m. The second example is based upon the schematic looped main described in Fig. 4(a) and (b), using the same diameter classes and cost data of the previous example with the relevant nodal heads and demands as given in Table 4. The layouts shown in Fig. 4(a) and (b) are only topological. The pipe actual lengths are as follows: the pipe connecting nodes 0±1 is 2000 m long, the contour pipes are all 1000 m long and all the internal pipes connected to node 9 are 1210 m long. Table 4 gives the head and the resulting ¯ow in node 0 together with the prescribed minimum head and demand for nodes 1±9. The results of the application of the methodology to this example are shown in Table 5 where the classical design of the main loop is compared to the design obtained using the heuristic approach described in this paper. The cost of the classical design is higher
Table 3 Comparison of optimal and alternative solutions for the two-loop network Pipe n
Cost opt.
Sol. A
Sol. B
Sol. C
Sol. D
1 2 3 4 5 6 7 8 Res. ind. Head surp. (m) Cost (units)
18 10 16 4 16 10 10 1 0.22 0.50 0:419 106
18 16 14 6 14 1 14 10 0.41 1.08 0:450 106
20 14 14 6 14 1 14 10 0.47 1.90 0:460 106
20 14 14 8 14 1 14 10 0.48 2.84 0:467 106
20 14 14 6 14 1 14 12 0.48 3.44 0:478 106
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Fig. 4. (a) A classical main loop network design. (b) An improved design for the same system.
Table 4 Head and demand values for the second example Node
Head (m)
Demand m3 =h
0 1 2 3 4 5 6 7 8 9
200 197 193 192 191 191 191 192 193 189
ÿ180.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00
Table 5 Comparison between a classical main loop network design and an interesting solution provided by the heuristic approach Pipe
Main loop
Improved
0±1 1±2 2±3 3±4 4±5 5±6 6±7 7±8 8±1 1±9 2±9 3±9 4±9 5±9 6±9 7±9 8±9 Res. ind. Head surp. (m) Cost (units)
16 12 12 12 12 12 12 12 12 3 3 3 3 3 3 3 3 0.20 0.70 0:657 106
18 14 10 8 1 1 8 10 14 10 1 1 1 8 1 1 1 0.52 7.08 0:565 106
(0:657 106 ) but produces a low resilience (Ir 0:20 with only 0.7 m of surplus head available in the most depressed node. The design resulting from the use of the proposed method is less expensive (0:565 106 ), more
121
resilient ( Ir 0:52 and has a surplus head of more than 7 m in the most depressed node. The ®nal example is presented in order to show the eciency of the proposed approach in the case of a relatively complex network (230 pipes, 167 nodes and three reservoirs). The boundary curve of the Pareto set for this example was found with a computational eort of less than 10 on a portable PC (266 Mh). The resilience index, the cost and the minimum surplus head for the dierent alternative solutions for this example are given in Table 6. The edge of the Pareto boundary is shown in the space of cost and reliability index (Fig. 5) and in the space of cost and minimum surplus head (Fig. 6). In this example, the complex looped topological structure of the pipe network seems to guarantee a minimum for the resilience index of 0.59 Table 6 Comparison among several solutions found for the third example in terms of resilience index, cost and minimum surplus head Solution
Resilience index
Cost (units)
Head surp. (m)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.593 0.607 0.635 0.658 0.671 0.681 0.700 0.705 0.727 0.737 0.748 0.750 0.756 0.763 0.765 0.767 0.806 0.815 0.826 0.874
4263 4279 4326 4318 4357 4381 4397 4382 4433 4479 4484 4493 4499 4525 4580 4585 4678 4719 4766 4984
0.59 0.56 0.51 0.03 1.07 1.83 1.67 1.85 1.81 3.54 3.66 3.56 3.43 3.81 4.94 5.06 5.18 5.24 5.25 6.37
Fig. 5. The limitig curve of the Pareto set of solutions in the cost-resilience index space for the third example.
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Fig. 6. The limitig curve of the Pareto set of solutions in the costminimum surplus head space for the third example.
for all feasible solutions. However, the various solutions can now be considered not only on the basis of the tradeos between cost and resilience but also on the basis of tradeos in the minimum surplus of head in the most depressed node.
5. Conclusions The concept of minimum cost optimisation for looped municipal water distribution systems is reviewed under a new multi-objective approach. This approach introduced in the paper employs a simple heuristic technique that tries to emulate the reasoning of an engineer who is aiming on the one hand at reducing the cost while, on the other hand, at preserving a sucient degree of resilience in the system to cope with possible failures. After de®ning the problem into a two objective space, it was shown that the technique is capable of rapidly describing the optimal Pareto set which allows satisfactory solutions to be found as a tradeo between system resilience and cost. From three examples used in the paper it shows that the satisfactory solutions do not correspond either to the minimum cost solutions or to the classical main loop solutions. The classical loop solutions are in general more reliable than the minimum cost solutions, but can be further improved under a more systematic multiobjective analysis. The eciency of the proposed technique is not limited to the case of extremely simpli®ed networks. The eciency also prevails in the case of complex networks for
which a long computational time can be expected when using more rigorous optimisation techniques. The considerations introduced in this paper need to be further developed and expanded by the scholars and designers who deal with optimisation of water distribution systems with the aim of producing both ecient and practically usable tools. In particular the proposed technique could be easily tested on the basis of reliability considerations and included into a knowledge based Decision Support System aimed at helping the engineer in selecting the most appropriate design for water distribution networks.
References Abebe, A. J., & Solomatine, D. P. (1998). Application of global optimization to the design of pipe networks. In Babovic & Larsen (Eds.), Hydroinformatics '98 (pp. 989±996). Rotterdam: Balkema. Alperovits, E., & Shamir, U. (1977). Design of optimal water distribution systems. Water Resource Research, 13(6), 885±900. Eiger, G. U., Shamir, U., & Ben-Tal, A. (1994). Optimal design of water distribution networks. Water Resource Research, 30(9), 2637±2646. Goulter, I. C., Lussier, B. M., & Morgan, D. R. (1986). Implications of head loss path choice in the optimisation of water distribution networks. Water Resource Research, 22(5), 819±822. Goulter, I. C., & Bouchart, F. (1990). Reliability constrained pipe networks model. Journal of Hydraulic Engineering ASCE, 116(2), 221±229. Kessler, A., & Shamir, U. (1989). Analysis of the linear programming gradient method for optimal design of water supply networks. Water Resource Research, 25(7), 1469±1480. Mays, L. W., (1996). Review of reliability analysis of water distribution systems. In Tikle, Goulter, Xu, Wasimi, & Bouchart (Eds.), Stochastic hydraulics '96 (pp. 53±62). Rotterdam: Balkema. Rossman, L. A., (1993). EPANET, Users manual. Risk Reduction Engineering Laboratory, Oce of Research & Development, US Environmental Protection Agency, Cincinnati, Ohio. Savic, D. A., & Walters, G. A. (1997). Genetic algorithms for least-cost design of water distribution networks. Water Resource Planning and Management, 123(2), 66±77. Stanic, M., Avakumovic, D., & Kapelan, Z., 1998. Evolutionary algorithm for determining optimal layout of water distribution networks. In Babovic, & Larsen (Eds.), Hydroinformatics '98 (pp. 901±908). Rotterdam: Balkema. Todini, E., (1979). Un metodo del gradiente per la veri®ca delle reti idrauliche (in Italian). Bollettino deglli Ingegneri della Toscana n.11. Todini, E., & Pilati, S. (1988). A gradient algorithm for the analysis of pipe networks. In B. Coulbeck & Chun-Hou Orr (Eds.), Computer applications in water supply (pp. 1±20). Wiley: Research Studies Press.