Hydraulic failure at a given node occurs when the supplied flow or pressure is less than

the required minimum flow or the required minimum pressure at that node. Considering we will use EPANET for simulation that the hydraulic simulator always satisfies water demand (because of applying the nodal mass balance equation) but not pressure head, this approach automatically assumes that water demand is satisfied. The nodal mass balance equation can be expressed as:

where D_i is demand at node i; H_i is head at node i; H_j is head at node j; N is total number of the demand nodes with unknown heads; is set of nodes connected directly to node i; is a nonlinear function relating the hydraulic loss and flow rate in the pipe connecting node i to node j. The nonlinear function used here is the Hazen- Williams or the Colebrook-White equation.

In the above equation, nodal pressure head is an unknown decision variable, and water demands are state variables. This equation can be expressed in a more compact form:

where F_i(*) is a vector of functions representing the mass balance at each node; H_i is a vector of supplied pressure heads at each node; is a vector of state variables and k is the total number of variables. In the study of the hydraulic performance failure, we define water demands D_i, pipe coefficients C_i, and tank/storage levels L_i as known state variables. Thus, the vector of state variables in this study can be expressed as; T represents the matrix transpose operation. Moreover, due to uncertainty, those known state variables will be treated as random values. Each random variable is expressed as a probability distribution, and then a random number generator is used to generate the values of D_i for each node, the values of C_i for each pipe, and the values of L_i for each tank/storage. Incorporation with EPANET, Monte Carlo simulation (MCS) technique can be applied to produce the random values of pressure head.

EPANET: EPANET is free public domain software developed by USEPA. It performs extended period simulation of hydraulic behavior within pressurized pipe networks [9] . A network consists of pipes, nodes, pumps, valves and storage tanks or reservoirs. EPANET tracks the flow water in each pipe, the pressure at each node, the level of water in each tank during a simulation period. Running under Windows, EPANET provides an integrated environment for editing network input data, running hydraulic and water quality simulations, and viewing the results in a variety of formats. In this study, water demands, pipe coefficients, tank/storage levels are inputting data during the simulation; pressure heads are outputs. We assume that pumps and valves are in good operation condition. The failure of pumps and valves is considered in the area of mechanical failure.

Monte Carlo Simulation: Monte Carlo simulation involves repeated generation of pseudovalues for the modeling inputs, drawn from known probability distributions within the ranges of possible values. In this study, this process will be completed by computer, where three random variables (water demand D_i, pipe coefficients C_i, and tank/storage level L_i) are produced synchronously at each time. The generated pseudovalues are then used as inputs for EPANET model to produce the supplied pressure head values of H_i. The Monte Carlo simulation implementation in a node i includes steps: (1) development of the representative probability distribution functions for selected model input parameters; (2) generation of pseudovalues for each of the selected input parameters from the distribution developed in the previous step; (3) implementation of EPANET model with the pseudovalues to generate a pressure head value of H_i; (4) comparison of the calculated value of H_i with the minimum allowable head; (5) repeated application of the previous two steps; (6) presentation of the results, and analysis of the results for decision making.

In the above step (4) of the MCS procedure, we defined as the minimum allowable head at a demand node. Thus, the probability of hydraulic performance failure at a demand node can be expressed as

where is the probability density function of the supplied pressure head; D_i^S is the supplied flow rate at the node i, and D_i^D is the water demand at the node i.

Mechanical failure involves pipe breakages and electric component failures (such as pump failure, power outages, control valve failure, etc). There are many factors that may cause pipe breaks, for instance, aging, physical bending stress, chemical and biological corrosion, etc. Among these factors, aging, material type, dimension, and bedding quality are important factors in predicting pipe breakage. Here we demonstrate two exponential break rate equations. Shamir and Howard’s exponential break rate model describes an exponential relationship between aging and breakage [17] .

where A_j is the growth rate coefficient of the pipe j; is the number of breaks per 1000 ft. length of pipe j in year t; t is time in years; t₀ is base year for the analysis (pipe installation year, or the first year for which data are available); M is total number of pipes.

In real situation, a primary cause of unreliability is the removal of water lines or pumps from service primarily due to breaks or power outages. A probability of an electromechanical component (k = 1, 2, …, U; where U is total number of electric components), such as a pump or a valve, being out of service can be included as another random variable that supported by historical data.

Different water demand has different importance. For example, a hospital water demand has the highest importance, and the water demand of a park area may have a lower importance. We weight failure probabilities to reflect their different importance

impacts. Then, the system failure probability can be defined as the maximum individual failure probability in the system

where are weighting coefficients; is the probability of hydraulic performance failure at the node i; is the failure probability of pipe j; is the failure probability of electromechanical component k; is the fire flow failure probability.

The system hydraulic reliability is given by

Although the pipe repair cost is much lower than the replacement fee, frequent breaks will result in the cumulative repair expense exceeding the replacement cost. Therefore, we need a methodology to find the critical (or threshold) break rate to determine the timing of repair or replacement [19] .

If we assume that the pipe will be replaced at the time of the n^th break (it also implies that for the previous (n − 1) breaks only repairs have been performed), we can write the present worth of the total cost of the pipe as

where is the discount rate; is the time of the i^th break measured from the installation year; is the repair cost of the i^th break; is the replacement cost at time; is the present worth.

For the total cost at time to be a minimum, it must satisfy the condition

However, the critical check is to determine the first instance when the condition holds true. Solving for, we have

Recognizing is the time between the nth and (n + 1)^th breaks or time interval for the occurrence of one break, we obtain the threshold break rate, Brk_th, as the inverse of (where). The threshold break rate is defined as the break rate between subsequent breaks.

Now, from the observed data for any given pipe we can derive a current break rate. Whenever the current break rate is equal to or more than Brk_th, the pipe should be replaced. It is worth noting that the optimal rehabilitation schedule information will be transferred to the upgrading optimization process (shown in Figure 4).

The hydraulic reliability can indicate the hydraulic performance information for each demand node. If the hydraulic failure probability on a node is lower than the allowable probability, the water provider should replace the inlet pipe with a larger diameter pipe or build a paralleling pipe. Then, the next few questions for the decision maker are: 1). Replace the existing pipeline by a bigger diameter pipeline or build a paralleling pipeline? 2) What size of pipe should be applied in this replacement or paralleling? 3) The duration of the new inlet pipe that can transfer sufficient water?

The design of water distribution systems is often viewed as a least-cost optimization problem in the previous literatures. The pipe diameters were considered as decision variables. Obviously other parameters should be considered as objectives in the optimization process, such as reliability, redundancy and water quality [20] . However, problems with quantifying these objectives for use within optimization design models kept researchers concentrating on the single, least-cost objective. Therefore, the overall objective in the upgrading optimization in this paper is to minimize the total cost. The following formulation describes this objective:

where is the cost of pipe j with diameter R_j and length L_j; m is the number of pipes that must be upgraded. The objective is subject to the following constraints:

where Equation (12) refers to conservation of flow constraints and energy equations (loop equations); Equation (13) is head boundary constraints; Equation (14) is designed water demand boundary constraints; Equation (15) represents general constraints including financial constraints.

The decision on either replacement or paralleling pipeline is depended on cost, geographical situation, and the existing pipe condition. Some pipeline geographical conditions are not suitable for the laying of parallel pipelines that the only choice is to replace the existing pipelines by larger diameter pipes. However, the geographical condition consideration of the pipe replacement planning is out of the scope of this study. In addition, we have to consider the duration of the upgraded pipeline. If one pipe needed upgrade because low hydraulic capability, simultaneously it was aging pipe that would be replaced in the next few years, it was better to directly be replaced by a new large diameter pipe rather than to build a parallel pipeline.

We assume the length of the term is l, the objective formulation is modified as follows

where l is the length of the planning period; and k (k = 1, 2, … l) is the year index of pipeline upgrading. The total cost includes the following components:

represents the pipeline costs including installation, paralleling, replacement and repair costs; is the cost of setting up the construction plant and machinery.

Pipe Roughness and Water Demand: The pipe roughness and the water demand are two model input parameters that vary over time. The pipe roughness increases over time at a rate that varies according to the pipe type, water quality, and operation and maintenance practices. To model the effect of aging on the carrying capacity of pipes, the equation of Sharp and Walsli [21] was used

where C(t) is the Hazen-Williams coefficient in year t; e_o is the roughness at the time of installation (mm); a is the roughness growth rate (mm/year); and R is the pipe diameter (mm).

The function developed by Dandy et al. [22] is used to predict nodal water demand.

where D_0j is the base year water demand for the node j and DGR is the annual percentage rate of increase in the base demand. DGR was assumed to follow demographic data patterns closely, and so it was taken as the population growth rate with typical values of about 2 to 5; t is the time in years; P_t and P₀ are the price per unit volume of water for year t and the base year, respectively; PREL is the price elasticity of demand. A typical range of PREL values is −0.2 to −0.5. Generally it is advisable to increase the price of water in the last 1 or 2 years of the design period for the best results in order to delay the need to upgrade the network. When there is adequate capacity after upgrading, prices can be reduced in order to utilize the system capacity as fully as possible and to maximize economic efficiency. Other demand management options could be used, and self-evidently, the joint effect of all the demand management techniques deployed should be considered.

Genetic Algorithms: We have chosen the genetic algorithms (GA) as the optimization method. GA is a search technique used in computing to find exact or approximate solutions to optimization and search problems. It is based on the mechanics of natural selection and natural genetics that using biological techniques including inheritance, mutation, selection, and crossover [23] . GA provides a robust and efficient way to search complex parameter spaces for ever better solutions to an optimization problem [23] . Although there are no guarantees that a GA will actually attain the optimum, it generally finds one or more extremely good solutions with relatively little computational effort.

The GA code can also be written by MATLAB. Moreover, EPANET simulation will be run during the GA optimization process. The following steps give the details of the optimization procedure for a repair schedule.

Step 1. Use Equation (4) to calculate each pipeline’s break rate B(i, j); and use Equation (10) to compute each pipeline’s Brk_th and then obtain the corresponding replacement year Yr(j), where i represents the time of year, and j is the pipe index.

Step 2. Set i = 0;

Step 3. i = i + 1;

Step 4. Run the hydraulic reliability model and calculate nodal hydraulic reliability R(i, j) for all studied nodes in a WDS in the i^th year;

Step 5. Compare the predicted hydraulic reliability with the minimum allowable hydraulic reliability Rm(i, j) at the i^th year, to find the pipes to be upgraded P(i, j);

Step 6. If the replacement year Yr(j) of the upgraded pipe is equal to or more than the planning term N, then, no paralleling is considered;

Step 7. Run Optimization Code (GA) to find optimal diameter pipe P(i, j, m) (where m is the index of pipe diameter, both replacement and paralleling use this diameter index), with the objective of minimizing total cost, and subject to the constraint that the R(i, j) should be greater than the maximum of Rm(i, j) during the whole planning period N. The computation of R(i, j), of course, needs to run the hydraulic reliability prediction model;

Step 8. If i < N, then go to step 3;

Step 9. Calculate the total cost and record the optimal rehabilitation and upgrading schedule.