CHAPTER 3 WATER DISTRIBUTION SYSTEM MAINTENANCE OPTIMIZATION PROBLEM

Transcription

1 41 CHAPTER 3 WATER DISTRIBUTION SYSTEM MAINTENANCE OPTIMIZATION PROBLEM 3.1 INTRODUCTION Water distribution systems are complex interconnected networks that require extensive planning and maintenance to ensure that water is delivered to the consumers without service interruptions. All water distribution systems deteriorate over time. These systems are more susceptible to leaks and breaks causing disruption of water supply to consumers, and requiring time and money for repair (Bhave 2003). The reliability of such deteriorating systems is continually decreasing and therefore the maintenance of existing water distribution systems is essential to maximize the infrastructure availability and thereby to enhance the service level. Maintenance decisions are usually made based on intuition and experience, without giving due consideration to the maintenance cost. Hence, there is a need to develop a framework to assist water utility managers to evaluate various maintenance strategies for water distribution systems (Luong and Nagarur 2005). The objective of this study is to propose a maintenance model to obtain a near-optimal maintenance strategy that minimizes the total discounted maintenance cost over a defined planning horizon. The various alternatives considered in the water distribution system maintenance problem are, different scheduled maintenance time periods, type of various

2 42 maintenance actions such as (i) replacing pipes with different materials and of different diameters; or (ii) rehabilitating a pipe (by either cleaning and leaving the pipe as it is, or cleaning and lining it with cement mortar). All these options have costs associated with them. Given the nonlinear nature of the equations required to describe the cost of the water distribution system maintenance, the proposed model combines the use of Stochastic Search Algorithms with a Monte Carlo simulation model. 3.2 FORMULATION OF THE WDS MAINTENANCE PROBLEM The study considers a Water Distribution System (WDS) which comprises of a pipe network, junction joints and components in the pumping system. Several maintenance strategies for the pipes in the water distribution system are proposed in the study to improve the infrastructure availability of the system. The maintenance alternatives considered in the WDS optimization problem are different scheduled maintenance time periods, (viz., 10 years, 15 years and 20 years) and the type of maintenance action, (viz., pipe rehabilitation and pipe replacement). The pipe rehabilitation effort includes cleaning and leaving the pipe, and cleaning and lining it with cement mortar. The pipe replacement actions include, using the same or different pipe material, with the same diameter or commercially available next higher pipe diameter. The diameters of the commercially available pipes that are found suitable for the WDS under study are considered. The pipe materials considered in the study are Cast Iron (CI), Ductile Iron (DI), and Prestressed Concrete (PSC). Therefore the combination of the distinct maintenance decision variables viz. scheduled maintenance time period (p m ), type of maintenance action (a m ), rehabilitation effort (r s ), pipe material (m n ) and the pipe diameter (D), creates many possible maintenance alternatives, each associated with different levels of maintenance effort.

3 43 The WDS maintenance problem is concerned with the minimization of the total discounted maintenance cost of the WDS over a twenty-five year planning horizon, subject to the infrastructure availability constraint. The total discounted maintenance cost is the sum of the discounted unscheduled maintenance cost and the discounted scheduled maintenance cost. The discounted unscheduled maintenance cost includes the discounted unscheduled repair cost of the pipes in the network, the discounted unscheduled repair cost of the pumping system and the discounted unscheduled repair cost of the junction joints. The discounted scheduled maintenance cost includes the discounted cost of rehabilitation and replacement actions carried out at specified maintenance time periods on the pipes in the water distribution network. formulated as Thus the WDS maintenance optimization problem can be Minimize C = DPI + DPU + DJN + DSM (3.1) subject to where A A mi C DPI DPU = total discounted maintenance cost of WDS over a defined maintenance planning horizon = Discounted unscheduled repair cost of pipes in the network over the planning horizon = Discounted unscheduled repair cost of the pumping system over the planning horizon

4 44 DJN = Discounted unscheduled repair cost of junction joints in the network over the planning horizon DSM = Discounted cost of scheduled maintenance (rehabilitation and replacement) action on pipes in the network over the planning horizon A A mi = availability of WDS infrastructure = specified target availability of WDS infrastructure given by, The discounted unscheduled repair cost of pipes in the system is DPI = y n m x1 i 1 j 1 1 k R x ijx (3.2) where R ijx n m y k = unscheduled repair cost of pipe j due to i th repair action in x th year = number of unscheduled pipe repairs in x th year = number of pipes in the WDS = maintenance planning horizon = discount rate given by, The discounted unscheduled repair cost of the pumping system is

5 45 DPU = q y px p1 x (3.3) x1 B 1 k. P where B px P q = ineffective utilization time of WDS due to p th pumping system failure (in hours) in x th year = average repair cost of pumping system per hour = number of pumping system failures given by, The discounted unscheduled repair cost of the junction joint is DJN = s y ux u1 x (3.4) x1 O 1 k. Q where O ux Q s = ineffective utilization time of WDS due to u th junction joint failure (in hours) in x th year = average repair cost of a junction joint per hour = number of junction joint failures The discounted cost of the scheduled maintenance action on pipe j in time step t sj is given by,

6 46 DSM(t sj ) = m j1 H j 1 k T tsj. j if t sj 2, 3, 4 (3.5) 0 otherwise where H j = rehabilitation cost of pipe j T j = replacement cost of pipe j t sj = time step when a scheduled maintenance action (rehabilitation or replacement) is initiated on pipe j (2,3,4) = length of the time step (5 years) The WDS infrastructure availability is computed as, A = yt ie y (3.6) where y = a twenty five year maintenance planning horizon T ie = Total ineffective utilization time of WDS and is computed as T ie = y n m y q y s Eijx Bpx Oux F sm (3.7) x1 i1 j1 x1 p1 x 1u 1 where E ijx = ineffective utilization time of WDS due to i th repair action on pipe j (in hours) in x th year F sm = ineffective utilization time of WDS due to scheduled maintenance action on pipes over the planning horizon = w max F rj 1 j m r 1

7 47 w F rj = number of scheduled maintenance time periods = ineffective utilization time of WDS due to scheduled maintenance during r th maintenance time period on pipe j (in hours) Penalized Objective Function As the WDS maintenance optimization problem under study is a constrained optimization one, it is necessary to transform it into an unconstrained problem to solve it using stochastic search methods. Using the penalty approach, the constrained maintenance optimization problem is converted into an unconstrained problem (Deb 2002). If C is the total discounted maintenance cost of WDS, the penalty objective function is defined as Minimize Z = C (1 + V c ) (3.8) where is the penalty constant which is judiciously selected in order to steer the search towards the feasible region. The value of is assumed as 10,000 in this study. The value of the violation coefficient, V c is computed based on whether the constraint equation is violated or not. The pseudo code used to compute the value of V c is given below. if A < A mi A V c = 1 Ami else V c = 0

8 48 For convenient presentation and comparison of the results, the best objective function value, Z * is divided by THE METHODOLOGY The methodology used to solve the Water Distribution System maintenance problem under study is presented in this section. The study comprises of two phases: Phase I Failure study (to identify the potential failure modes) of the Water Distribution System and the performance evaluation of Water Distribution System (to assess the performance level of an existing Water Distribution System). Phase II Generation of various maintenance alternatives in order to strengthen and extend the useful life of the WDS, and evaluation of the proposed maintenance strategies to determine the near- optimal maintenance strategy which would minimize the total discounted maintenance cost (and maximize WDS availability) over the specified planning horizon, using Stochastic Search Algorithms. given in Figure 3.1. The methodology proposed for the WDS maintenance problem is

9 49 Failure study of Water Distribution System (to identify potential failure modes) Assessment of WDS availability using Monte Carlo Simulation Model Generation of maintenance alternatives (to improve the WDS availability) Formulation of WDS maintenance optimization model Application of proposed Stochastic Search Algorithms for a real-life WDS Figure 3.1 Methodology Proposed to Solve WDS Maintenance Problem 3.4 WDS MAINTENANCE OPTIMIZATION SOLUTION APPROACH Many feasible WDS maintenance alternatives in order to minimize the total discounted maintenance cost over a fixed planning horizon are considered in the maintenance optimization problem under study. The combination of these maintenance alternatives creates many possible maintenance strategies each associated with different degrees of maintenance effort. Therefore, the problem of determining the cost-effective maintenance

10 50 strategy is computationally hard. This combinatorial optimization problem is very complex in nature and quite hard to solve by conventional optimization techniques. Even with the use of advanced computing techniques, it is indeed a difficult task to evaluate all possible maintenance strategies and determine the optimal maintenance strategy that maximizes WDS availability at least-cost. Search technique is one of the methods for solving such problems where one cannot determine a priori the sequence of steps leading to a solution. There are two important issues in search strategies: exploiting the best solution and exploring the search space. Hill-climbing is an example of a strategy which exploits the best solution for possible improvement while ignoring the exploration of the search space. Random search is an example of a strategy, which explores the search space while ignoring the exploitation of the promising regions of the search space. Simulated Annealing (SA), Tabu Search (TS) and Genetic Algorithms (GA) are a class of general-purpose search techniques combining the elements of directed and stochastic search which can make a remarkable balance between exploration and exploitation of the search space (Pirlott 1996). In this study, these stochastic search algorithms have been employed to solve the WDS maintenance optimization problem under study and the Monte Carlo simulation model is used to compute the objective function value. The link between the maintenance optimization model and the simulation model is shown in Figure 3.2. The simulation model is used to compute the objective function value Z, which is the total discounted maintenance cost over the planning horizon. The decision variables viz., the scheduled maintenance time period (p m ), the type of maintenance action (a m ), rehabilitation effort (r s ), pipe material (m n ) and the pipe diameter (D) are the input parameters for the Monte Carlo simulation model. Whenever a new solution (maintenance strategy) is generated, the simulation model computes

11 51 the objective function value, Z. The optimization model (which uses one of the stochastic search methods - Simulated Annealing technique, or Tabu Search algorithm or Genetic Algorithms) is used to search for the nearoptimal maintenance strategy. OPTIMIZATION MODEL (Stochastic Search Algorithm) Decision variables Total discounted (p m, a m, r s, m n, D) maintenance cost (C) SIMULATION MODEL (Monte Carlo) Figure 3.2 Optimization-Simulation Model Link In order to demonstrate the application of the proposed approach to solve the WDS maintenance optimization problem, Velachery water distribution system, in Chennai, India is considered in the study. A preamble on the Chennai water distribution system is presented below. 3.5 APPLICATION Till about 1870, the people of Chennai (Madras), India were dependent on shallow wells situated in their own houses or on public wells and tanks in the neighborhood for their water supply needs. There was no protected water supply at that time and these sources were not satisfactory. Organized water supply to Chennai commenced in 1872 which is the nucleus of the protected surface water supply system now existing in Chennai city. The Chennai Metropolitan Water Supply and Sewerage Board (CMWSSB) was established in 1978 as a statutory Body for exclusively attending to the growing needs and for planned development and appropriate regulation of

12 52 water supply and sewerage services in the Chennai Metropolitan Area. At present the quantity of water supply in Chennai is 645 MLD per day out of which 570 MLD of water per day is supplied through pipelines and only 75 MLD of water is delivered through mobile service. The total length of water distribution network in Chennai is 2924 km. The water distribution station at Velachery, which is one of the 16 water distribution stations in Chennai city is considered in the maintenance optimization study. The Velachery water distribution network was commissioned for water supply in The Velachery water distribution system is considered as a representative system to demonstrate the application of the proposed maintenance decision model. The modified Velachery water distribution system in Chennai, India, is shown in Figure 3.3. The major components considered in the maintenance study of the water distribution system are the pipes and junction joints in the water distribution network and the critical components in the water supply pumping system. All the pipes in the existing system are made of cast iron. Figure 3.3 Schematic Representation of Water Distribution System under study

13 53 The pipe characteristics of the water distribution system under study is given in Table 3.1. Table 3.1 Water Distribution Pipe Characteristics Pipe ID Length (m) Diameter (mm) P P P P P P P P P P P The failure time and repair time data considered in the study are taken from the published data (CPHEEO 1999, Bhave 2003) and from the inhouse records of the Chennai Metropolitan Water Supply and Sewerage Board (CMWSSB) for their own use. The various costs involved in the maintenance optimization study are the repair cost of a pipe in the unexpected event of pipe failure, the replacement cost and the rehabilitation cost of a pipe during the scheduled maintenance time period, the repair cost of the junction joint and the repair cost of the pumping system. It is found from the in-house records of the CMWSSB that the average repair cost of the junction joint is Rs.800 per hour and the average pumping system repair cost is Rs.300 per hour. The cost equations used in the WDS maintenance optimization study are given below:

14 54 The cost of a pipe repair depends on the type of break, size of the pipe, ease of access to the break, time of repair and cost of transportation to site. The repair cost of a pipe can be expressed as a function of pipe diameter D j, (Bhave 2003) i.e. Pipe repair cost = gd. (3.9) h j where D j = diameter of pipe j (in mm) which undergoes an unscheduled repair. The regression coefficients g and h are obtained using regression analysis. The replacement cost of pipe j is calculated as Replacement cost, T j = Lj. C r (3.10) where L j is the length of pipe j (in m) and C r is the unit cost of the pipe being replaced, which is a function of the pipe diameter, and is estimated from the formula, 1 C r = a D for cast iron pipe (3.11). b 1 j 2 C r = a D for ductile iron pipe and 3.12). b 2 j 3 C r = a D for prestressed concrete pipe (3.13). b 3 j where D j is the pipe diameter in mm, The regression coefficients, a 1, b 1, a 2, b 2, a 3 and b 3 are obtained by carrying out regression analysis for a set of data corresponding to different pipe diameters and pipe materials. The pipe rehabilitation cost depends on the length of the pipe to be rehabilitated and the unit cost of rehabilitation. Unit cost of rehabilitation depends on several factors like pipe diameter, alignment and depth of cover;

15 55 location and access to the site; and length of pipe which can be cleaned and lined at a time (Bhave 2003). The rehabilitation cost of a pipe of length L j is H j = L. C (3.14) j c where C c is the unit cost of rehabilitation. computed as The unit cost of rehabilitation through cleaning and lining can be 1 C c = e (3.15). k 1 D j computed as The unit cost of rehabilitation through cleaning and leaving is 2 C c = e (3.16). k 2 D j The values of regression coefficients, e 1, k 1, e 2, and k 2 are obtained using regression analysis. 3.6 SUMMARY In this chapter, the need for maintaining a water distribution system and the objectives of the proposed maintenance optimization study are presented. The maintenance optimization problem of the water distribution system under study is formulated. Several maintenance alternatives for the pipes in the water distribution network are proposed in the study. The details of the unscheduled maintenance/repair costs of pipes, junction joints and pumping components and the scheduled maintenance costs due to the rehabilitation and replacement actions on the water distribution pipe network are presented. The total discounted WDS maintenance cost over the specified planning horizon subject to a target infrastructure availability is the measure

16 56 of performance considered in the study. The methodology and the solution approach used in the study are also presented. The test problem, viz., a reallife Velachery water distribution system in Chennai, India, used to demonstrate the application of the proposed maintenance decision model is described.