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ABSTRACT
Water distribution networks are typically designed based on certain values of uncertain parameters such as reservoir head, nodal demand and friction coefficient to furnish acceptable performance by limiting nodal pressure head and pipes’ velocities. In this study, a fuzzy uncertainty analysis approach is developed to handle uncertainty of reservoir head and nodal demand over an extended period and long-term simulation, based on type-2 fuzzy logic. Furthermore, the long-term uncertainty of Hazen-Williams (HW) coefficient is also considered. A parallel genetic algorithm is introduced to solve a multi-objective optimization problem for fuzzy uncertainty analysis. The method is powered by a pressure dependent hydraulic simulator which is based on EPANET and an iterative procedure. It is implemented for a benchmark network from literature and a case study. Results show that network velocity is highly affected by uncertainties and the accumulation of different uncertainties may change the network performance significantly.
Notation list
The following symbols are used in this paper:
| ai = | = | physical defined parameter for pipe i (Equation 4a) |
| Ci°= | = | HW coefficient of pipe i at the time of pipe installation in the network |
| Ci(t) = | = | HW coefficient of pipe i at the end of the year t |
| Cicrisp(t) = | = | crisp value of Ci at the end of the year t |
| Cimax(t) = | = | maximum value of Ci at the end of the year t |
| Cimin(t) = | = | minimum value of Ci at the end of the year t |
| Di = | = | diameter of pipe i |
| Eh = | = | the head of reservoir h |
| ei = | = | physical defined parameter for pipe i (Equation 4b) |
| f = | = | network hydraulic simulator function |
| M = | = | number of decision variables |
| N = | = | number of objective functions |
| nn = | = | number of junction nodes in network |
| np = | = | number of pipes in network |
| nr = | = | number of reservoir |
| ns = | = | number of simulation periods |
| ny = | = | number of years |
| pj = | = | pressure head at node j |
| pjmin = | = | minimum pressure head at node j |
| pjreq = | = | required pressure head at node j |
| qjavl = | = | available demand at node j |
| qjreq = | = | required demand at node j |
| qj°= | = | demand at the beginning of network operation at node j |
| qj(t,s) = | = | demand at node j in hour s and at the end of the year t |
| qj crisp(t,s) = | = | crisp value of qj at the hour s and the end of the year t |
| qj max(t,s) = | = | maximum values of qj at the hour s and the end of the year t |
| qjmin(t,s) = | = | minimum values of qj at the hour s and the end of the year t |
| r = | = | geometric growth coefficient for demand |
| s = | = | time of day for extended period simulation |
| t = | = | year index for long-term simulation |
| vk = | = | velocity at pipe k |
| W = | = | vector of decision variables |
| Z = | = | vector of objective functions |
| α = | = | level of uncertainty |
| μ(x) = | = | fuzzy membership function of x |
| τ(s) = | = | function of demand coefficient at each time step in a day (demand pattern) |
Disclosure statement
No potential conflict of interest was reported by the authors.