Fuzzy Analysis of Pressure-Deficient Water Distribution Networks

Water distribution network analysis is traditionally done under the assumption that the demand requirements are met at all the nodes. In reality, a network could be met with various abnormal and uncertain conditions, resulting in variations at the demand nodes. Softwares such as EPANET 2 analyze a network using demand dependent analysis (DDA). Under pressure deficient conditions, DDA will not give satisfactory results. In such cases pressure-dependent or pressure driven analysis called as node flow analysis (NFA) is required, where outflows at the demand nodes are treated as functions of pressure using node head-flow relationships. Fuzzy analysis helps in understanding how the uncertainty in various independent parameters of water distribution network such as, nodal demands, pipe roughness values, reservoir heads, pipe diameters and so forth will affect the dependent parameters such as pipe velocities, discharges and nodal pressures. The membership functions of dependent parameters are obtained by considering membership functions of uncertain independent parameters. The Impact Table method from literature suggests a repetitive analysis by considering the monotonous relationship between dependent and independent parameters. The Impact Table method was also employed for carrying out fuzzy analysis under pressure deficient condition to obtain fuzzy membership function of nodal outflows. Optimization based methods of fuzzy analysis are more useful when relationship between dependent and independent parameters are non-monotonous.

Optimization based methods of fuzzy analysis are more useful when relationship between dependent and independent parameters are non-monotonous. Genetic algorithm is used in this study for performing fuzzy analysis on three networks by pressure dependent approach. The objective functions which are evaluated are (1) nodal pressures and (2) the actual demand at the nodes under the uncertain conditions. The membership function considered is triangular. The analysis is performed by setting up hydraulic model in the software EPANET and linking it with the MATLAB for performing optimization through an EPANET-MATLAB toolkit. Fuzzy NFA incorporates nodal outflows as additional dependent parameters. The outflows at pressure-deficient nodes are also observed to change monotonically as a result of changing the nodal demands and pipe roughness coefficients. Using this observation of monotonic change, the fuzzy analysis methodology of Gupta and Bhave (2007) is extended to obtain a membership function of available nodal flows under pressure-deficient conditions.The fuzzy NFA procedure has the following steps:Step 1. Carry out a NFA of the network considering normal values of uncertain parameters.Step 2. Change an uncertain parameter to its maximum value keeping other uncertain parameters at their normal value; and obtain the values of dependent parameter. Note the impact of changing an uncertain parameter on the dependent parameter, i.e. whether increasing, decreasing or none. Step 2 is to be repeated for all uncertain parameters.Step 3. To determine the maximum (or minimum) value of the dependent parameters, take an appropriate extreme value of an uncertain parameter based on its impact and carry out the analysis. Step 3 can be repeated for all dependent parameters and for different α-cuts.For validation, two networks were selected from previous researches, and the networks were analyzed with the developed model and WaterNetGen software, and finally the results were compared. Then paid to hydraulic analysis of the water distribution network of Jangal city under uncertain conditions and lack of pressure as a case study.

Fuzzy analysis is the approach adopted in this study to quantify thevagueness of a water distribution network. An optimization model is employed to find out theextremities in the responses of the system, as the relationship between independent and dependentvariable are not necessarily monotonous. The hydraulic system is analyzed by node flow analysis totake care of the head-demand relationships under pressure deficient conditions. As most of the hydraulic simulation software perform analysis considering that demands are met at all the nodes, the results will vary when there is a pressure deficiency in the network. This assessment of network can be incorporated into the better designing of a WDN. The obtained results showed that the proposed model can perform hydraulic simulation well in the condition of lack of pressure. This study has considered only the uncertainties in the nodal demands, but it can be further extended to any input parameter just by considering them as variables in the algorithm and extreme responses can be obtained.