Tiku T. Tanyimboh

Redundancy model for water distribution systems

Abstract This paper presents a model, based on head driven simulation, for assessing the redundancy of water distribution systems (WDS). The formulation recognises the pressure dependency of water consumption in the solution procedure. A new algorithm for pressure dependent modelling of WDS has, therefore, been developed. Notable features of the proposed network analysis technique include the introduction of a new subcategory of nodes called key partial-flow nodes and the use of a joint head-flow system of equations. The algorithm is reliable, quick and easy-to-implement. The redundancy assessment methodology addresses the randomness of component failure or unavailability. Results are presented which demonstrate the suitability and meaning of the redundancy measure. In particular, it is recommended that redundancy be evaluated along with reliability when assessing system performance. The computer program developed can seamlessly calculate several performance indicators including reliability.

OPTIMUM DESIGN OF FLEXIBLE WATER DISTRIBUTION NETWORKS

Abstract A method for designing flexible water distribution networks is presented. Flexibility is the extent to and ease with which a distribution network can cope with eventualities for which it was not specifically designed. This paper shows that some flexibility can be achieved by maximizing the entropy of the flows. A sample network is considered and designs for various levels of entropy are examined. Several indices including energy and head loss are used to compare the designs. The results suggest that an entropy constraint can reduce the tendency towards implicitly branched configurations characteristic of cost minimization models. A striking feature of the proposed methodology is its apparent ability to produce resilient designs without a substantial increase in cost. The results further highlight some implications for connectivity-based reliability measures and core tree approaches to layout optimization.

Modelling errors, entropy and the hydraulic reliability of water distribution systems

This paper reports on an investigation of the possible influence of modelling errors on the relationship between the entropy and hydraulic reliability of water distribution systems. The errors are due to minor differences between the design optimisation and subsequent simulation models, which lead to small discrepancies between the capacity of the network and the required supply. Pressure-dependent analysis was used for the hydraulic simulations. It is shown that any correlation between the redundancy or undercapacity due to the modelling errors and the hydraulic reliability is insignificant. The results, therefore, provide yet more evidence that the entropy-reliability relationship is strong.

Pressure-Dependent EPANET Extension

In water distribution systems (WDSs), the available flow at a demand node is dependent on the pressure at that node. When a network is lacking in pressure, not all consumer demands will be met in full. In this context, the assumption that all demands are fully satisfied regardless of the pressure in the system becomes unreasonable and represents the main limitation of the conventional demand driven analysis (DDA) approach to WDS modelling. A realistic depiction of the network performance can only be attained by considering demands to be pressure dependent. This paper presents an extension of the renowned DDA based hydraulic simulator EPANET 2 to incorporate pressure-dependent demands. This extension is termed “EPANET-PDX” (pressure-dependent extension) herein. The utilization of a continuous nodal pressure-flow function coupled with a line search and backtracking procedure greatly enhance the algorithm’s convergence rate and robustness. Simulations of real life networks consisting of multiple sources, pipes, valves and pumps were successfully executed and results are presented herein. Excellent modelling performance was achieved for analysing both normal and pressure deficient conditions of the WDSs. Detailed computational efficiency results of EPANET-PDX with reference to EPANET 2 are included as well.

Maximum entropy flows for single-source networks

This paper was prompted by growing evidence that Shannons measure of uncertainty can be used as a surrogate reliability measure for water distribution networks. This applies to both reliability assessment and reliability-governed design. Shannons measure, however, is a non-linear function of the network flows. Therefore, the calculation of maximum entropy flows requires non-linear programming. Hence, a simpler, more accessible method would be most useful. This paper presents an alternative and rigorous method for calculating maximum entropy flows for single-source networks. The proposed method does not involve linear or non-linear programming. Also, it is not iterative. Consequently, the method is very efficient. In this paper, the methodology is described, several examples are presented and an algorithm is suggested.

CALCULATING MAXIMUM ENTROPY FLOWS IN MULTI-SOURCE, MULTI-DEMAND NETWORKS

Abstract Previous work has shown how maximum entropy flows in a water distribution network can be calculated by maximizing a nodal entropy function. This requires the use of numerical nonlinear optimization. In the case of single-source networks a much simpler path entropy formulation exists which permits solutions to be obtained quickly by manual calculations not requiring numerical optimization. This paper extends the path entropy formulation to general multi-source, multi-demand networks and develops a simple manual calculation method for maximum entropy flows in these general networks. The method is quick, non-iterative and does not directly involve the use of numerical optimization. Examples are presented and discussed.

A MAXIMUM ENTROPY BASED APPROACH TO THE LAYOUT OPTIMIZATION OF WATER DISTRIBUTION SYSTEMS

This paper investigates the idea that minimum cost maximum entropy designs of water distribution systems can be used to identify good layouts of water distribution systems in the sense that designs based on these layouts could achieve a reasonable compromise between reliability and cost. The proposed approach presupposes that there is a sufficiently strong relationship between entropy and reliability to achieve this purpose. Evidence is provided to support this view. A fundamental issue relating to the invariance of the entropy function with respect to permutations of its arguments is also addressed. The proposed technique is assessed using an example, which shows that the method yields good solutions to the joint layout, reliability and pipe-size optimization problem. It is also shown that if two or more layouts of a water distribution system have the same maximum entropy value, then minimum cost maximum entropy designs based on these layouts can be expected to be similar in terms of their overall performance.

Peaking demand factor-based reliability analysis of water distribution systems

Water demands vary and consideration of the probabilistic nature of the variations should lead to more instructive assessments of the performance of water distribution systems. Water consumption data for several households were analysed using the chi-square technique and it was found that distributions worth considering under certain circumstances include the normal and lognormal.Reliability values were calculated for a range of critical demand values and the corresponding confidence levels determined from the probability distributions. Water consumption was assumed to be pressure dependent and the modelling of the water distribution system was carried out accordingly. This peaking factor approach coupled with the statistical modelling of demands provides a more realistic way of incorporating variations in demands in the evaluation and reporting of system performance than the traditional single demand value approach in that the extent to which a network can satisfy any demand and the probability that the demand will occur can be recognized explicitly. The method is illustrated by an example.

Penalty-Free Feasibility Boundary Convergent Multi-Objective Evolutionary Algorithm for the Optimization of Water Distribution Systems

This paper presents a new penalty-free multi-objective evolutionary approach (PFMOEA) for the optimization of water distribution systems (WDSs). The proposed approach utilizes pressure dependent analysis (PDA) to develop a multi-objective evolutionary search. PDA is able to simulate both normal and pressure deficient networks and provides the means to accurately and rapidly identify the feasible region of the solution space, effectively locating global or near global optimal solutions along its active constraint boundary. The significant advantage of this method over previous methods is that it eliminates the need for ad-hoc penalty functions, additional “boundary search” parameters, or special constraint handling procedures. Conceptually, the approach is downright straightforward and probably the simplest hitherto. The PFMOEA has been applied to several WDS benchmarks and its performance examined. It is demonstrated that the approach is highly robust and efficient in locating optimal solutions. Superior results in terms of the initial network construction cost and number of hydraulic simulations required were obtained. The improvements are demonstrated through comparisons with previously published solutions from the literature.

Calculating the reliability of single-source networks by the source head method

Abstract This paper proposes two new reliability measures that, for small networks in particular, are easy to understand and interpret and straightforward to calculate. The measures fully recognise the pressure dependency of demand and are fully capable of quantifying partial failure. The methodology is based on a recent parameter referred to as the notional net source head to fully satisfy all nodal demands. An important feature of the present formulation consists of a subtle reversal of the roles of normal and subnormal service, wherein lies the key to the proper quantification of partial failure. The final component of the methodology is a probabilistic analysis which addresses the random nature of component failures or unavailability. The final result is a sufficiently meaningful and accurate reliability measure. Damage tolerance is also characterised and quantified. The efficacy of the reliability and damage tolerance measures are highlighted using a numerical demonstration. Finally, despite the overall simplicity of the proposed approach, it is computationally efficient.