MULTIOBJECTIVE OPTIMIZATION OF WATER DISTRIBUTION ...

33 downloads 16984 Views 134KB Size Report
cost of the network design/rehabilitation; b) probability of network failure due to uncertainty in input parameters. The sources of uncertainty analyzed here.
MULTIOBJECTIVE OPTIMIZATION OF WATER DISTRIBUTION SYSTEM DESIGN UNDER UNCERTAIN DEMAND AND PIPE ROUGHNESS Artem V. Babayan ∗ Dragan A. Savic ∗ Godfrey A. Walters ∗



Department of Engineering, School of Engineering, Computer Science and Mathematics, University of Exeter, North Park Road, Exeter, EX4 4QF, UK;

Abstract: The problem of the stochastic (i.e. robust) water distribution system (WDS) design is formulated and solved here as a multiobjective optimization problem under uncertainty. Two parameters are subject to minimization - a) cost of the network design/rehabilitation; b) probability of network failure due to uncertainty in input parameters. The sources of uncertainty analyzed here are future water consumption and pipe roughnesses. All uncertain model input variables are assumed to be independent random variables following some prespecified probability density function (PDF). To avoid using a computationally demanding sampling-based technique for uncertainty quantification the original stochastic formulation is replaced by a deterministic one. After some simplifications, a fast numerical integration method is used to quantify the uncertainties. The optimization problem is solved using a multiobjective genetic algorithm (GA). The proposed robust design method is applied to the New York Tunnels rehabilitation case study. The trade-off curve between cost and robustness is found. Keywords: Robust design, uncertainty quantification, multiobjective optimization

1. INTRODUCTION The vast majority of mathematical models use deterministic approaches to describe various processes and systems. In contrast all real life problems incorporate uncertainty in one way or another. It could be uncertainty in measurement, uncertainty in estimation of parameters, uncertainty as to which processes one should include into the model etc. Such contradiction between ”mathematical determinism” and ”natural uncertainty” can seriously affect the reliability of the results of mathematical modelling. So developing methodologies which allow us to take into account uncertainty when predicting the behavior of the system is of great practical interest.

In the process of design of water distribution networks the most uncertain quantities are, probably, water consumption and pipe roughness coefficients. While it is possible to estimate the present water demand reasonably well (Obradovic and Lonsdale (1998)), the situation becomes much worse when future demands need to be predicted. Also, the pipe friction factor can change significantly with age - depending on pipe material and water corrosive characteristic. Given the above, the need for considering the design of WDS under uncertainty in input parameters within an optimisation framework is obvious. Water distribution system design optimization is one of the most heavily researched areas in the

hydraulics profession (see Goulter (1992), Lansey (2000) for detailed review). Recently, genetic algorithms (GA) have become the preferred water system design optimization technique for many researchers (Dandy et al. (1993), Savic and Walters (1997)) because GAs demonstrate good ability to deal with complex, nonlinear and discrete optimization problems. When uncertainty is taken into account, it is usually incorporated into the problem formulation as a constraint on minimal system robustness or penalty for fitness function (Lansey et al. (1989), Xu and Goulter (1999), Kapelan et al. (2003), Tolson et al. (2004), Babayan et al. (2004)). In this way the result of each run of optimization process is one minimalcost WDS which provides at least this predefined level of robustness (here under robustness we understand the ability of the system to satisfy the specifications despite the possible errors in input parameters). Nevertheless, a decision-maker is often interested in getting several alternatives i.e. trade-off curve between cost and robustness. One of advantages of genetic algorithms is their ability to be applied to multiobjective problems and to provide such a curve in one run. Therefore it is natural to consider robustness as the second objective (after cost), rather than as a constraint. In general, including uncertainty into the problem formulation requires from the engineer application of one of the methodologies for quantification its influence on the system. These methods can be divided into two groups - one using stochastic simulation (sampling), the other - replacing stochastic formulation with deterministic one. The simplest way is to use one of the sampling techniques (Monte Carlo simulation and its modifications) (Bao and Mays (1990)), however such a straightforward approach requires an unreasonable amount of computational effort especially when using GA as one needs to calculate the fitness function for a large number of network configurations. The main area of usage of sampling-based methods is checking robustness of the final solution and verification of deterministic approaches (Xu and Goulter (1998),Tolson et al. (2004)Babayan et al. (2004)). Still, it is possible to exploit the stochastic nature of the GA and to incorporate stochastic approach into optimization process as was done in Kapelan et al. (2003). Analytically-based methods usually allow the engineer to get results much (up to several orders of magnitude) faster, which makes them the preferable instrument for optimization problems. Today several techniques are available (see Haldar and Mahadevan (2000) and Zhao and Ono (2001), for review), some of them were applied to WDS optimization problem. In (Xu and Goulter (1998)) a probabilistic hydraulic model was used for the first time in the water distribution system

design optimization. The WDS hydraulic model uncertainties were quantified using the analytical technique known as the first-order second-moment (FOSM) reliability method. This method assumes that relationship between uncertain and response variables is very close to linear, which is often not the case for water distribution systems. In addition to this it can not use information about probability distribution function (PDF) of uncertain parameters when such information is available. The same authors (Xu and Goulter (1999)) later used the more accurate the first-order reliability method (FORM). To calculate the uncertainties, the FORM requires repetitive calculation of the first-order derivatives and matrix inversions, which is computationally very demanding even in the case of small networks and may lead to a number of numerical problems. Also when using this method it is difficult to determine, how uncertainty in different parameters affects the system’s robustness. The least cost design problem was, in both cases, solved using the generalized reduced gradient optimization algorithm. Being a local search method, it can easily be trapped in the local minimum (Savic and Walters (1997)). Recently, Tolson et al. (2004) overcame the problem associated with the use of gradient search methods by using GAs to solve the optimal water distribution system design problem. However, the authors still used the first order reliability method to quantify uncertainties which has its own drawbacks. To overcome the limitations of all the aforementioned WDS design approaches, a new robust design methodology was developed recently (Babayan et al. (2004)), based on numerical integration. In this paper the improved methodology is used to solve the multiobjective problem. The paper is organized as follows: after this introduction a robust least-cost design problem is formulated. This is followed by presentation of the methodology used to solve the aforementioned problem and its application to a case study. At the end, relevant conclusions are drawn.

2. PROBLEM DEFINITION The objectives of the robust least cost design model presented here are to minimize total design (i.e. rehabilitation) costs and maximize the level of design robustness. More specifically, the optimisation problem is formulated as follows: M inimize Cost(D1 , D2 , , DN d )

(1)

M aximize min (P (Hi ≥ Hi,min ), i = 1, .., Nn ) (2) subject to: (a) Mass and energy balance constraints:

Ni X

Qm − Qd,i = 0

(i = 1, ..., Nn )

(3)

ξHi

  +∞ Z M Y Hi (X) = ηj (Xj ) dX

m=1

−∞

Hi,u − Hi,d − ∆Hi = 0

(i = 1, ..., Nl )

(4) σHi

(b) Decision variables constraint: Di ∈ D (i = 1, ..., Nd )

j=1

  +∞ Z M Y (Hi (X) − ξHi )2 ηj (Xj ) dX = j=1

−∞

(5)

where: Cost - total design/rehabilitation cost; Di - value of the i-th discrete decision variable (in general, design/rehabilitation option index); D discrete set of available design/rehabilitation options; P - probability that head (Hi ) at networks node is equal to or above the corresponding minimum requirements for that node (Hi,min ); Qm flows in all Ni pipes connected to the i-th network node; Qd,i - demand at i-th node; Hi,u head at upstream node of the i-th pipe; Hi,d head at downstream node of the i-th pipe; ∆Hi difference between i-th pipe’s total headloss and pumping head; Nd - number of decision variables; Nl - number of network links; Nn - number of network nodes. In the model presented here it is assumed that nodal demands and pipe roughness coefficients are the only sources of uncertainty, i.e. it is assumed that all other WDS simulation model inputs are deterministic variables. In addition to this, all uncertain parameters are assumed to be independent random variables, each of them following some PDF. Note, that, when using GA to solve the optimization problem (1)–(5), constraints (3)–(4) can be automatically satisfied by linking GA to the deterministic WDS solver, i.e. by using the latter to calculate nodal heads for each demand sample. Constraint (5) can also be automatically satisfied by using the appropriate GA coding.

where ηj (Xj ) is the known probability distribution function for jth of M uncertain parameters. Because of the implicit relationship between demands and heads it is impossible to calculate the above integrals directly. Straightforward numerical evaluation requires an unreasonable amount of computational effort, therefore we will use a simplified method of evaluating ξHi and σHi which is based on some assumptions. First of all assume the validity of superposition principle: Hi (ξX1 + t1 , ξX2 + t2 , . . . , ξXM + tM ) − Hi (ξX1 , ξX2 , . . . , ξXM ) ≈ M X

(Hi (ξX1 , . . . , ξXi + ti , . . . , ξXM )−

i=1

Hi (ξX1 , ξX2 , . . . , ξXM ))

where ti - random fluctuation in the ith uncertain variable. Note that PDF of uncertain variable is non-zero in some area only (e.g. uniform distribution) or decreases exponentially with distance from the mean value (e.g. normal distribution). Hence, all we need is that a superposition principle is satisfied in some area around ξX - about two standard deviations is enough in most cases. It is worth to note also, that assumption (7) is less strict than requirement of linear relationship between uncertain and response variables. For example, equality (7) is satisfied if Hi (X1 , X2 ) = X12 + X22 . Taking into account (7) and assumption about the nodal demands and pipe roughness independence we have the following approximation for nodal head means and standard deviations:

3. SOLUTION TECHNIQUE ξHi ≈ Hi (ξX ) +

The main idea behind the approach proposed is to replace the stochastic objective (2) with the deterministic one:

 min

M aximize α =  ξHi − Hi,min , i = 1, ..., Nn σHi

(7)

M X

αij

(8)

j=1

σHi

+∞ M Z X ≈ (Hi (Xj ) − H(ξX ) j=1−∞

(6)

where α - parameter which determines the level of system’s robustness, ξHi and σHi - mean and standard deviation respectively of the head at node i (which depends on vector of uncertain parameters X). The formulas for expected value and standard deviation are:

2

−αij ) ηj (Xj )dXj αij =

+∞ Z (Hi (Xj ) − Hi (ξX )) ηj (Xj )dXj

(9) (10)

−∞

Integrals in (9) and (10) are 1D and can be calculated using conventional numerical formulas.

To estimate the standard deviation for all nodes in a network with M uncertain parameters using formulas (8)–(10) one needs to perform (n−1)M + 1 model runs, where n is the odd number of points in the formula for numerical integration (for odd n the point in the center (corresponding to ξX ) is common for all dimensions and the value of the function at this point can be computed only once). It is clear from (10) that amendments αij in (8) and (9) account for non-symmetry of heads around mean value of uncertain variables. In case of non-independent uncertain variables expected value still can be computed with formula (8), but the formula for standard deviation for correlated parameters is:

Nn X

σcorr,Hi ≈ σnoncorr,Hi + +∞ Z +∞ Z X X (Hi (Xj ) − H(ξX ) − αij ) ∗

j=1 k=1,k6=j−∞ −∞

(Hi (Xk ) − H(ξX ) − αik ) ∗

1) + 1 where Λ is the number of significant variables and n is the number of quadrature formula points. Finally, note that as the GA run progresses, the sets of critical nodes and significant variables may have to be updated periodically. This can be done using the procedure described in the previous step. In comparison with first-order second-moment reliability method the methodology described here can provide better accuracy for the same computational overheads (3-point numerical integration scheme has 4th order of accuracy and requires the same number of model run as First Order Second Moment reliability method). Another commonly used method, FORM, although providing higher accuracy, is much more computationally demanding, with complexity increasing rapidly as the number of uncertain parameters increases (for each network configuration it requires conversion of matrix of order M (operation with complexity ∼ M 3 ) and solving of the system of non-linear equations of the same order).

ηj (Xj , Xk )dXj dXk The numerical computation of the above integral does not require more model runs, however one has to provide the joint PDF, which is not always readily available. Formula (9) allows us to estimate the relative contribution of uncertainty in each variable to the nodal heads uncertainty. This information can be used to build up the list of ’significant’ variables and model the rest of the network as certain, leading to significant computational time savings. To summarize, the following algorithm is proposed to solve the least cost design problem under uncertainty using GA: (1) Identify the sets of critical nodes and significant variables. Take some initial configuration of the network (e.g. the existing network configuration). Then, compute the head mean and deviation at each node using formulae (8)-(10). Nodes at which (ξHi − Hi,min )/σHi < 3 are added to the set of ”critical nodes” (Ω). Uncertain variables whose relative contribution to standard deviation in critical nodes is more then some prescribed level (say, 5%) form the set of ”significant variables” (Λ). Note that this step requires M ∗ n state estimate calculations, where M is the number of uncertain variables and n is the number of points in the quadrature formula used. (2) Run GA to find the optimal robust design. Use the GA to obtain solution of problem (1),(6).Note that the total number of state estimate calculations (i.e. hydraulic solver runs) necessary to obtain the value of fitness function for the second objective is |Λ|(n −

4. CASE STUDY 4.1 Problem description The robust least cost design methodologies presented here is tested and compared on the wellknown literature problem of New York Tunnels reinforcement (Schaake and Lai (1969)) (see Figure 1). The original objective of the problem was to identify the least cost network rehabilitation solution which satisfies minimum head requirements at all nodes for a set of fixed nodal demands. The only means of rehabilitation allowed is to duplicate existing pipes with new ones. A total of 16 solutions are possible for each pipe in the network: do not duplicate that pipe or duplicate the pipe with a new pipe having one of 15 available diameters (range 91 − 518 cm). Therefore, even though New York Tunnels network is fairly small, the optimization problem is quite large (total number of possible solutions, i.e. network configurations is 1621 ≈ 1.9 ∗ 1024 ). So far, a number of authors have solved the problem as a deterministic one. A review of these approaches can be found in (Savic and Walters (1997)). The two deterministic solutions identified previously on opposite ends of the cost domain are presented here in Table 1. Unlike in the deterministic approaches, it is assumed here that some parameters in problem formulation are uncertain. To demonstrate that the methodology presented here is able to handle uncertainties in different types of parameters and with various probability distribution functions, the following case is considered: (1)nodal demands are assumed to be

relevant optimization process was finished. As one would expect the cost of solution raises exponentially with the robustness level. Non-smoothness and non-convexity of the parts of the curve can be explained (1) by discrete nature of the problem, (2) by using indirect index for robustness measurement and (3) by errors caused by using simplified methodology. Table 1. Selected Deterministic and Optimal Robust Solutions. Pipe

Fig. 1. Layout for New York City water supply system uncertain variables following Gaussian PDF with mean equal to the deterministic demand value and standard deviation equal to 10% of the mean value; (2) the friction coefficients in the old pipes are assumed to be uniformly distributed stochastic variables on the interval [90, 110]. The deterministic demands, friction coefficients and other network data used here are taken from Murphy et al. (1993). The EPANET hydraulic solver (Rossman (2000)) was used to calculate unknown heads and flows for each demand sample. During the first step of the algorithm the set of significant uncertain variables, consisting of demands in 7 nodes (9, 11, 16, 17, 18, 19 and 20) and pipe roughness coefficients in two pipes (13 and 14) was formed. Therefore each fitness function evaluation required 19 runs of the EPANET solver.

4.2 Results and discussion The results obtained for the problem formulated in the previous section are shown in Figure 4.2. The curve represents part of the the Pareto front obtained by solving the optimization problem (1), (6)using the Non-dominating sorting GA (NSGAII) (Deb et al. (2000)) with population size 400 and 1000 generations. Note that the level of robustness of each solution shown in Figure 2. was re-calculated using 100,000 MC samples once the

1–6 7 8–10 11 12 13 14 15 16 17 18 19 20 21 Cost (mln.$) Robust.

Det. Solutions Quindry Murphy et al. et al. (1981) (1993) — — — — — — 302 — 341 — 337 — 337 — 334 305 49 213 233 244 185 213 185 183 — — 140 183 63.58 38.80 58.5%

48.0%

90%

Stoch. Solutions 95% 99%

— — — — — — — 488 274 243 243 243 — 183 49.03

— 305 — — — — — 518 243 243 243 243 — 183 52.64

— — — — — 366 — 427 213 274 213 182 — 213 57.45

91.6%

95.0%

99.1%

Details of the two optimal robust solutions for robustness levels 90% and 95% are shown in Table 1. The following can be noted from this table: (1) Both stochastic (i.e. robust) solutions have higher cost than the optimal deterministic solution identified by Murphy et al.(Murphy et al. (1993)). This is the price that has to be paid for increased robustness; (2) Robustness of the two deterministic solutions is quite low. In the case of the Murphy et al. (Murphy et al. (1993)) solution this is the consequence of the deterministic optimization which left no or very little redundancy in the system to cope with demand fluctuations. In the case of the Quindry et al. (Quindry et al. (1981)) solution, even though the rehabilitation cost is very high, the robustness is low because of the inappropriate selection of duplication diameters. This solution also demonstrates that spending a lot of money on rehabilitation does not guarantee that a high level of system robustness will be achieved.

5. CONCLUSIONS In this paper a methodology for the multiobjective least cost robust design of water distribution networks under uncertain demand is presented. The problem is solved using GAs after converting the original problem formulation to an

Fig. 2. Trade-off curve cost vs. robustness equivalent, simplified deterministic optimization problem. This way computational efficiency of the methodology is significantly increased. The methodology is verified on a case study where, among other things, optimal robust solutions obtained are compared to well known deterministic solutions from the literature. The results clearly demonstrate that neglecting uncertainty in the design process may lead to serious underdesign of water distribution networks. The methodology proposed here is of a general type in terms that different uncertain parameters with different PDF types can be considered. The disadvantage of the methodology shown here is that the target level of the design robustness cannot be specified explicitly in the problem formulation phase, i.e. it has to be specified indirectly (by specifying the target value of parameter ). As a consequence, the actual level of robustness can be calculated only once the optimization process has converged and the final solution is obtained.

REFERENCES A.V. Babayan, D.A. Savic, and G.A. Walters. Least cost design of water distribution network under demand uncertainty. Journal of Water Resources Planning and Management, ASCE, page accepted for publication, 2004. Y. Bao and L.W. Mays. Model for water distribution system reliability. Journal of Hydraulic Engineering, ASCE, 116(9):1119–1137, 1990. G.C Dandy, A.R. Simpson, and L.J. Murphy. A review of pipe network optimisation techniques. In Proc. WATERCOMP, Melbourne, Australia, pages 373–383. 1993.

K. Deb, S. Agrawal, A. Pratap, and T. Meyarivan. A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: Nsgaii. In Proceedings of the Parallel Problem Solving from Nature VI Conference , 16-20 September. Paris, France, pages 849–858. 2000. I.C. Goulter. Systems analysis in waterdistribution network design: From theory to practice. Journal of Water Resources Planning and Management, ASCE, 118(3):238–248, 1992. A. Haldar and S. Mahadevan. Probability, Reliability and Statistical Methods in Engineering Design. John Wiley & Sons, 2000. Z. Kapelan, D.A. Savic, and G.A. Walters. Robust least cost design of water distribution systems using gas. In C. Maksimovic, D. Butler, and F. A. Memon, editors, Proc. Computer Control for Water Industry (CCWI), London (UK), pages 147–155. 2003. K.E. Lansey. Optimal design of water distribution systems. In L.W Mays, editor, Water Distribution System Handbook. McGraw-Hill, 2000. K.E. Lansey, N. Duan, L.W. Mays, and Y.K. Tung. Water distribution system design under uncertainties. Journal of Water Resources Planning and Management, ASCE, 115(5):630– 645, 1989. L.J. Murphy, A.R. Simpson, and G.C. Dandy. Pipe network optimization using an improved genetic algorithm. In Report No. R109, Department of Civil and Environmental Engineering. University of Adelaide, 1993. D. Obradovic and P. Lonsdale. Public Water Supply Models, Data and Operational Management. E & FN Spon, 1998. G.E. Quindry, E.D. Brill, and J.C. Liebman. Optimisation of looped water distribution sys-

tems. Journal of the Environmental Engineering, ASCE, 107(4):665–679, 1981. L.A. Rossman. Epanet2 Users Manual. US EPA, 2000. D.A. Savic and G.A. Walters. Genetic algorithms for the least-cost design of water distribution networks. Journal of Water Resources Planning and Management, ASCE, 123(2):67–77, 1997. J. Schaake and D. Lai. Linear programming and dynamic programming applications to water distribution network design. In Report No. 116, Dept. of Civil Engineering. MIT, 1969. B.A. Tolson, H.R. Maier, A.R. Simpson, and B.J. Lence. Genetic algorithms for reliabilitybased optimisation of water distribution systems. Journal of Water Resources Planning and Management, ASCE, 130(1):63–72, 2004. C. Xu and I.C. Goulter. Probabilistic model for water distribution reliability. Journal of Water Resources Planning and Management, ASCE, 124(4):218–228, 1998. C. Xu and I.C. Goulter. Reliability-based optimal design of water distribution networks. Journal of Water Resources Planning and Management, ASCE, 125(6):352–362, 1999. Y.-G. Zhao and T. Ono. Moment methods for structural reliability. Structural Safety, 23:47– 75, 2001.