CHAPTER 5 MAINTENANCE OPTIMIZATION OF WATER DISTRIBUTION SYSTEM: SIMULATED ANNEALING APPROACH

Transcription

1 79 CHAPTER 5 MAINTENANCE OPTIMIZATION OF WATER DISTRIBUTION SYSTEM: SIMULATED ANNEALING APPROACH 5.1 INTRODUCTION Water distribution systems are complex interconnected networks that require extensive planning and maintenance to ensure that the drinking water is delivered to the consumers. Lack of adequate water supply in urban areas is due to aging water distribution systems and inadequate maintenance of this critical infrastructure (Bhave 2003). The reliability of such deteriorating systems is continually decreasing and therefore good maintenance planning is required to improve the reliability of these aging water distribution systems. The various maintenance activities of pipe network that are buried underground involve cost. Hence it is necessary to consider maintenance of existing water distribution systems as economically as possible. Maintenance optimization problem of existing water distribution systems considering various maintenance alternatives such as different scheduled maintenance time periods, employing different maintenance actions such as replacing or rehabilitating a pipe, cleaning and leaving the pipe or cleaning and lining with cement mortar, replacing the pipe with different pipe materials as well as with different pipe diameters has received less attention in the literature than it deserves. The problem of determining the near-optimal

2 80 maintenance strategy to minimize the total discounted maintenance cost is computationally hard. The stochastic search methods such as Simulated Annealing, Tabu Search, and Genetic Algorithms have created an immense interest among researchers to solve this type of combinatorial optimization problems. Simulated Annealing (SA) is a randomized local search method that has been used to derive near-optimal solutions for computationally complex optimization problems (Johnson et al 1991) Applications of SA algorithm to solve optimization problems in the area of telecommunication network design (Fetterolf and Anandalingam 1992), production scheduling (Palmer 1996), water distribution network design (Cunha and Sousa 1999), and reservoir system operation (Ramesh et al 2002), have been reported in the literature. But the use of Simulated Annealing algorithm for maintenance optimization problem in water supply distribution systems has not received much attention in the literature. In this chapter, the application of Simulated Annealing technique to solve the maintenance optimization problem with large computational complexity, in a real-life water distribution system is presented. The objective is to determine the near-optimal maintenance strategy that maximizes system availability at minimum total discounted maintenance cost over the planning horizon. The Velachery water distribution system considered in the study is described in chapter 3 (Section 3.5). The organization of this chapter is as follows. The need for the application of the stochastic search algorithms for the maintenance optimization problem is presented in Section 5.2. In Section 5.3, a description of Simulated Annealing technique is presented. The implementation of the SA algorithm for the WDS maintenance optimization

3 81 study is detailed in Section 5.4. The results along with discussions are presented in Section 5.5. A summary is given in Section NEED FOR STOCHASTIC SEARCH ALGORITHMS The maintenance optimization problem of WDS under study involves various maintenance actions on the pipes in the water distribution system, in order to maximize the system availability at minimum cost. The various maintenance alternatives considered in the study are: different maintenance time periods 10 years, 15 years, or 20 years; the type of maintenance activity - pipe replacement or pipe rehabilitation; the pipe material used in replacement Cast Iron (CI), Ductile Iron (DI) or Prestressed Concrete (PSC); Cleaning and lining with cement mortar or cleaning and leaving without lining while rehabilitation and the pipe diameter using a pipe with same diameter or with commercially available next higher diameter, while replacing, for one or more of the eleven pipes in the water distribution system. Therefore, the number of possible combinations of maintenance strategies (solutions) for the pipes in the water distribution system is very large. It is not easy to search for the global optimal solution from such a large solution space. It requires a large computational time for local traditional optimization methods to search for quality solutions. Moreover, the cost equations used in the maintenance problem under study are non linear. The literature reveals that the deterministic optimization methods are unable to cope up with the non linear water distribution network problems and that the stochastic optimization algorithms are quite successful in solving such problems, though requiring large number of evaluations (Mohan and Babu 2010). Hence, the use of search algorithms such as Simulated Annealing the tool used to solve combinatorial optimization problems, is inevitable.

4 SIMULATED ANNEALING TECHNIQUE Simulated annealing is a randomized improvement algorithm that has been successfully applied to yield good results for combinatorial intractable problems (Kirkpatrick et al 1983). Simulated Annealing is based on the analogy between the annealing of a solid and the optimisation of a system with many independent variables. Solids are annealed by raising the temperature to a (maximal) value for which the particles randomly arrange in the liquid phase, followed by cooling to force the particles into the lowenergy states of a regular lattice. At high temperatures all possible states can be reached, although low-energy states are occupied with a larger probability; lowering the temperature decreases the number of accessible states and the system finally will be frozen into its ground state, provided that the maximum temperature is high enough and the cooling is sufficiently low. In combinatorial optimization a similar situation occurs; the system may occur in many different configurations. Any configuration has a cost that is given by the value of the cost function for that particular configuration. Similar to the simulation of the annealing of solids, one can statistically model the evolution of the system that has to be optimized into a state that corresponds with the minimum value of the cost function (Aarts and van Laarhoven 1985). 5.4 OPTIMAL MAINTENANCE STRATEGY SA APPROACH WDS Simulation Model to Compute Z The objective function value Z is calculated using simulation of the water distribution system for a period of 25 years, since a twenty five year planning horizon is considered. The assumptions made in the WDS simulation model are given in Section

5 Notations and Terminology Z Objective function value p m Scheduled maintenance time period a m Type of maintenance action (rehabilitation or replacement) m n Pipe material to be used during replacement D Diameter of a pipe r s Rehabilitation action (clean & leave or clean & line) R j T j H j Unscheduled repair cost of pipe j Replacement cost of pipe j Rehabilitation cost of pipe j P Average repair cost of pumping system per hour Q Average repair cost of a junction joint per hour T ie Total WDS ineffective utilization time T ie ' WDS ineffective utilization time due to a repair action on a pipe or pump component T ie " WDS ineffective utilization time due to a rehabilitation or replacement action on a pipe Failure rate of a pipe or a pump component MTTR Mean time to repair of a pipe or a pump component T f Failure time of a pipe or a pump component T c T a Current clock time Time of occurrence of a failure

6 84 E s T r T rc End simulation time Repair time of a pipe or a pump component Repair completion time T sc Scheduled maintenance action completion time NMIE Next most imminent event C Total discounted maintenance cost of WDS The Simulation Logic The flowchart of the WDS simulation model is shown in Figure 5.1. The logic of the WDS simulation model in order to compute the objective function value, Z is briefly described below. Initially, the life characteristics viz., the failure rate and the mean time to repair (MTTR) of the pipes in the network and the pumping components are determined. When a pipe in the water distribution network or a component in the pumping system fails, then repair action is initiated on that pipe/component immediately. The WDS ineffective utilization time is updated after each unscheduled repair action whenever a pipe/component in the system fails, and also after each scheduled replacement/rehabilitation action carried out on a pipe. The failure rate of the pipe or pump component is updated after each unscheduled repair. Simulation is terminated after the simulation clock time reaches twenty five years, as the maintenance planning horizon considered is twenty five years.

7 85 Start Set T c = T ie = 0 Input E s, P and Q Input maintenance strategy: p m, a m, r s, m n and D of all pipes Input R j, T j and H j of all pipes Input and MTTR of all pipes and pump components Determine T f of all pipes and pump components T a = min (T f ) T c = T a Remove T a from the set of failure times, (T f ) Make crew busy; Determine T r Compute T rc p m NMIE Repair completion E s Carry out Scheduled maintenance T c = T sc ; Update T ie " and Calculate total discounted maintenance cost, C T c = T rc Update T ie ' and T ie =T ie ' + T ie " Calculate Availability, A Compute Z Stop Figure 5.1 Flowchart of WDS Simulation Model

8 Setting the Run Length of the WDS Simulation Experiment The run length of the WDS simulation experiment used to compute Z has been varied for a given combination of maintenance actions on the WDS pipes and the results are given in Table 5.1. Table 5.1 Objective Function values for Various Run Lengths Objective function value, Z Replication 1 Replication 2 Replication Run length Replication H o H 1 : No significant difference in objective function value Z due to changes in run length. : Significant difference in objective function value Z due to changes in run length. ANOVA is conducted with the Z values and the ANOVA for run length is given in Table 5.2. Table 5.2 ANOVA for Run Length Source of Sum of Degrees of Mean sum Variation squares freedom of squares F cal Run length Error Total

9 87 Since the value obtained for F cal does not exceed 2.57, the value of F 0.05, with 6 and 21 degrees of freedom, the null hypothesis H o is accepted at the 0.05 level of significance. Therefore the run length of the WDS simulation experiment is fixed at 40. An initial ANOVA conducted with the Z values for less than 40 runs showed that there is a significant difference in the objective function value due to changes in run length Simulated Annealing Methodology Simulated Annealing (SA) is a combinatorial optimization approach that uses the Metropolis algorithm to evaluate the acceptability of alternate strategies and slowly converge to an optimum solution (Kirkpatrick et al 1983). The SA approach is based on ideas from statistical mechanics and motivated by an analogy to the behaviour of physical systems in the presence of heat bath. The non-physicist, however, can view it simply as an enhanced version of the familiar technique of local optimization or iterative improvement, in which an initial solution is repeatedly improved by making small local alterations until no such alteration yields a better solution. Simulated Annealing randomises this procedure in a way that allows for occasional uphill moves (changes that worsen solution), in an attempt to reduce the probability of becoming stuck in a poor but locally optimal solution. Because of its ability to avoid poor local optima, it offers hope of obtaining significantly better results (Johnson et al 1989). The temperature T 0 is a parameter, which acts like an iteration/time counter for the algorithm. The temperature is successively reduced by means of a temperature reduction factor (cooling rate). When the temperature reaches a pre-specified value (called the freezing temperature) the procedure is said to be frozen and is terminated. At every temperature, iterations are carried out (L p is the iteration limit) in the search for better solutions. The strength of the method lies in the fact that inferior solutions are accepted (with

10 88 a certain acceptance probability) with the hope that the procedure can clear local optimal troughs and find a global optimum. The Simulated Annealing algorithm used for the problem of determining the near-optimal maintenance strategy of the water distribution system under study is given below: Notations and Terminology T 0 T f L p Z * Z c P a Initial Temperature Final Temperature Cooling Rate Length of Plateau (i.e. number of iterations at every stage of refinement) The best objective function value Computed objective function value Probability of accepting a bad solution R no Uniform random number from [0,1] SA Algorithm Initialization Determine T 0, T f,, L p. Generate randomly an initial solution S. Call WDS simulation model and compute Z c. Z * Z c Phase transformation Do the following L p times: Pick a random neighbor S 1 from S. Call WDS simulation model and compute Z c. If (Z c Z * ) then {

11 Z * Z c S * S 1 89 else Z * Z c S * S 1 } { determine P a = / 0 e m T where m = Z c Z * ; and generate a random number R no U[0,1]. then if (R no P a ) { else } leave Z c as bad solution } Reducing T 0 T 0 x T 0 Stopping Criterion If (T 0 T f ) then goto Phase transformation else report Z * and the corresponding best solution S *. S * gives the best maintenance strategy yielded by SA algorithm.

12 SA Parameter Setting using Orthogonal Experiments Simulated Annealing (SA) parameters, namely the initial temperature T 0, final temperature T f, cooling rate and the length of plateau L p play a vital role in finding the solution to the given problem. Orthogonal experiments are used to determine suitable values for these SA parameters. Initially these parameters viz., T 0, T f, and L p are assigned the values 200, 5, 0.80 and 60 respectively. These values are assumed to be the mid-values to be considered in the 3-level orthogonal experiments. A set of values for each parameter with a certain percentage deviation from the mid-values is then considered. Based on the observations made on the quality of solutions obtained with this set of values during the pilot runs conducted on the SA, a lower limit and upper limit values for each parameter are selected. The SA parameters and the levels considered in the study are given in Table 5.3. Table 5.3 Simulated Annealing Parameters and Level Settings SA parameter Level 1 Level 2 Level 3 Initial temperature T 0 (A) Final temperature T f (B) Cooling rate (C) Length of plateau L p (D) The L Orthogonal Array (OA) is used in the study (Ross 2005). 9 The experiment settings and the objective function value, Z obtained from the orthogonal experiments for a given maintenance strategy, are shown in Table 5.4.

13 91 Table 5.4 SA-Orthogonal Experiment Settings and Objective Function values Parameter settings Objective function value, Z Sl. No A B C D Replication 1 Replication H 0 : No significant difference in the objective function value Z due to changes in SA parameter values. H 1 : Significant difference in the objective function value Z due to changes in SA parameter values. The ANOVA of L 9 OA for SA parameters is given in Table 5.5. Table 5.5 ANOVA of L 9 OA for SA Parameters Source Sum of squares Degrees of freedom Mean sum of squares F cal F table at =0.05, 1 =2 and 2 =9 A B C D Error Total

14 92 Based on the ANOVA of the orthogonal experiments conducted with the objective function values, the following values are fixed for the SA parameters. Initial temperature, T 0 = 100 Final temperature, T f = 10 Cooling rate, = 0.70 Length of plateau, L p = Neighborhood Structure In order to implement Simulated Annealing, it is necessary to define the neighborhood structure for the specific problem under consideration. The neighborhood structure has a significant effect on the efficiency of the search and the quality of solutions produced. The generation procedure of this neighborhood structure considered in this water distribution maintenance optimization study is described below. The maintenance alternative for each pipe in the water distribution network viz., the maintenance time period, type of maintenance action, the pipe material and the pipe diameter, is represented using a four digit code respectively. The first digit with a value 1 or 2 or 3 represents maintenance time period of 10 years or 15 years or 20 years respectively. The second digit with a value 1 or 2 represents pipe rehabilitation or pipe replacement respectively. When the second digit assumed the value 1, then the third digit with a value 1 or 2 would represent cleaning and lining with cement mortar or cleaning and leaving respectively. If the second digit assumed the value 2, then the third digit with a value 1 or 2 or 3 would represent, using a Cast Iron pipe or a DI pipe or a PSC pipe respectively. And the fourth digit with a value 1 or 2 indicates replacing the pipe with the same diameter or replacing with

15 93 the next higher commercially available diameter. Hence a 44 digit code is used to represent maintenance alternative on each of the eleven pipes in the water distribution system. An example of a code representing a solution is illustrated below The description detail of the above solution code is given in Table 5.6. Table 5.6 Description of Solution Code Pipe ID P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 P 10 P 11 Maintenance period (yrs) Maintenance action H R R R H H H H R H R Maintenance effort CLe CI CI DI CLi CLi CLe CLi PSC CLe PSC Pipe diameter SD SD SD ND SD SD SD SD SD SD ND (R- replace, H- rehabilitate, SD- same diameter, and ND- next higher available diameter, CI-use cast iron pipe, PSC-use PSC pipe, DI-use DI pipe, CLe-Clean and leave, CLi-Clean and line with cement mortar) Neighborhood Generation Procedure The perturbation on a given solution is performed by generating a random number for each of the four digits representing the maintenance alternative on a pipe. Hence, totally 44 random numbers are considered in order to generate several maintenance strategies for the eleven pipes in the water distribution network. This procedure is described below: For j = 1 to 11 do the following: /To fix the maintenance time period Generate a random number R no U[0,1] If R no < then

16 94 else S(i) 1 If R no < then else /To fix the maintenance action S(i) 2 S(i) 3 Generate a random number R no U[0,1] If R no <0.50 then else S(i+1) 1 S(i+1) 2 /To fix the maintenance effort If S(i+1) = 1 then { Generate a random number R no U[0,1] If R no < 0.50 then else } S(i+2) 1 S(i+3) 1 S(i+2) 2 S(i+3) 1 If S(i+1) = 2 then { Generate a random number R no U[0,1] If R no < then else S(i+2) 1

17 95 If R no < then else S(i+2) 2 S(i+2) 3 /To decide diameter Generate a random number R no U[0,1] ` } Next j If R no < 0.50 then S(i+3) 1 else S(i+3) RESULTS AND DISCUSSIONS In this Chapter, the application of Simulated Annealing algorithm to determine the near-optimal maintenance strategy with the objective of minimizing the total discounted maintenance cost over the specified planning horizon and maximizing system availability in a real-life water distribution system is addressed. The measure of performance considered in the study is the total discounted maintenance cost of WDS over the planning horizon subject to satisfying the target availability. The Simulated Annealing algorithm is proposed to solve this combinatorial maintenance optimization problem. The optimal SA parameter values are obtained from the Taguchi orthogonal experiments. The SA algorithm and the Monte Carlo simulation procedure to compute the objective function value Z are coded in C++ and executed on a personal computer with Pentium GHz processor and 760 MB RAM. The convergence graph of the SA algorithm is shown in Figure 5.2. It is found from Figure 5.2 that the near-optimal solution is obtained at 177 th

18 96 iteration and there is no improvement in the quality of the solution after this iteration number. From Figures 5.2, it is observed that the SA heuristic (under the perturbation scheme used in the study) has successfully searched for solutions to the maintenance optimization problem Z Iteration Number Figure 5.2 Convergence Graph SA Algorithm The results of Simulated Annealing algorithm are shown in Table 5.7. Table 5.7 Simulated Annealing Algorithm Results Search Algorithm Simulated Annealing Total Cost, Z * (Rs. X10 6 ) Maximum Availability achieved Ineffective utilization time per year (days) Iteration Number CPU Time (seconds) It is found from Table 5.7 that the maximum availability is achieved with the minimum total discounted cost (Rs x 10 6 ) when the maintenance strategy proposed by the SA algorithm is adopted. The total

19 97 ineffective utilization time of WDS when the maintenance strategy proposed by Simulated Annealing algorithm is implemented is 25.5 days. The proposed maintenance strategy (yielded by SA algorithm) to be implemented on the water distribution system in order to minimize the total discounted WDS maintenance cost over the planning horizon is shown in Table 5.8. Table 5.8 Proposed WDS Maintenance Strategy (SA Algorithm) Pipe ID Maintenance period (years) Maintenance action Pipe material/ Rehabilitation strategy Pipe diameter (mm) P 1 15 Rehabilitation CLe 200 P 2 10 Replacement DI 100 P 3 10 Rehabilitation CLe 80 P 4 10 Rehabilitation CLi 100 P 5 10 Rehabilitation CLe 80 P 6 20 Rehabilitation CLe 125 P 7 15 Rehabilitation CLi 100 P 8 10 Replacement PSC 100 P 9 20 Replacement CI 100 P Rehabilitation CLi 125 P Replacement CI 80 The water management expert can make an appropriate decision to adopt the maintenance strategy depending on the maintenance budgetconstraint and the availability of resources needed to carry out maintenance. The results obtained in the study provide insights into the working of the Simulated Annealing search technique and its application to water distribution system maintenance problems.

20 SUMMARY Despite a lot of earlier work in the field of water distribution system design optimization, there is scarcity of techniques to tackle maintenance optimization problems of urban water distribution system. The evaluation of large number of maintenance strategies to arrive at a nearoptimal maintenance strategy for maximum water distribution system performance is a complex combinatorial optimization problem. The Stochastic search technique, Simulated Annealing, is effective in tackling such problems which are computationally hard. In this chapter, the application of Simulated Annealing technique to evaluate various maintenance strategies, and thereby to assist the management select the best maintenance strategy which maximizes the water distribution system infrastructure availability at least cost is demonstrated. The total discounted maintenance cost of the water distribution system over the planning horizon is considered as the measure of performance. Monte Carlo simulation technique is employed taking into account the failure characteristics of pipe network and the pumping components. A Simulated Annealing algorithm is implemented on the maintenance optimization problem for a real-life urban water distribution system. Results obtained from this application suggest that Simulated Annealing can be used for obtaining near-optimal solutions for water distribution system maintenance problems that are computationally intractable.