Mathematical Problems in Engineering

Volume 2018, Article ID 6938483, 11 pages

https://doi.org/10.1155/2018/6938483

## The Sensitivity of a Water Distribution System to Regional State Parameter Variations

Built Environment, Council for Scientific and Industrial Research (CSIR), Pretoria 0184, South Africa

Correspondence should be addressed to Philip R. Page; moc.liamg@7rpegap

Received 4 August 2017; Revised 6 April 2018; Accepted 23 April 2018; Published 29 May 2018

Academic Editor: Suzanne M. Shontz

Copyright © 2018 Philip R. Page. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The sensitivity of a pressurised water distribution system (WDS) to state parameter variations is studied. A novel* local regional sensitivity analysis* (LRSA) approach is introduced which applies the same change to a collection of parameters, called a* region*. For example, sensitivity to suburbs can be studied. General analytical (using algebraic methods) results are derived. They show how sensible conclusions arise from LRSA and state this dependence of the WDS on regions for the first time. For most cases, the WDS is 1.852–2 times more sensitive to pipe roughness coefficients than to pipe lengths. In most cases, when certain pipes do not have minor losses, the WDS is 4.871–5.333 times more sensitive to pipe diameters than to pipe lengths. Hence, the WDS is the most sensitive to pipe diameters, medium sensitive to pipe roughness coefficients, and least sensitive to pipe lengths. For most cases, when all reservoir and tank elevations (and heads) remain the same, changes of other elevations do not change flow rates and change the pressures in a simple additive way. In most cases, when all the reservoir water surface elevations are changed together, the flow rates remain unchanged, and the pressures change in a simple additive way.

#### 1. Introduction

The study of sensitivity analysis [1, 2], employed here, is concerned with the following:(1)Forward use: how much the outputs depend on each or some of the inputs.(2)Inverse use: how uncertainty in the outputs can be apportioned to different sources of uncertainty in the inputs.

Sensitivity analysis may be performed for a variety of reasons, including exploration of model response to specific inputs (a forward use) or identification of key model inputs (an inverse use). The former can be useful for verification or validation of the model [3] and the latter for the calibration of the model [4, Section ].

A pressurised water distribution system (WDS) depends on various* state parameters* (i.e., “inputs”, e.g., pipe lengths) that are characteristics of the infrastructure and water status [5–7]. Numerical models then predict the water hydraulic* variables* (i.e., “outputs”, e.g., flow rates and pressures) from the state parameters (water quality properties are not considered here). For the objective of a particular sensitivity study, there is typically interest in a particular hydraulic water property, or a combination of such properties, called the indicator (or index) function [8–13]. For example, the minimum pressure in the system [14, p. 131], or the average pressure in the entire system, might be of interest. The sensitivity analysis answers the question: What is the sensitivity of an indicator function to change in the parameters that it depends on?

Various sensitivity analysis methods exist: some are more mathematical and others more statistical, in nature [1, 3].* Statistical* methods typically approach the subject by assuming that parameters are distributed according to a probability density function. There are numerous applications to the WDS, as recently reviewed [15]. Common methods used in WDS applications include Monte Carlo [16, 17] and an accompanying approximation called first-order second moment (FOSM) [8, 16], which uses the sensitivity-matrix evaluated at the mean of the distribution. Fuzzy sets, as recently reviewed [18], have also been used for WDS applications.

The following* mathematical* methods exist:* nominal range sensitivity analysis* (NRSA) [3] (also known as “local sensitivity analysis”, “threshold analysis”, “one-at-a-time”, or “one-factor-at-a-time”) and sensitivity-matrix methods [1, 17]. NRSA and sensitivity-matrix methods are applicable to deterministic models [2], such as the hydraulic pipe network models used for a WDS, considered here. These methods, respectively, apply to changes of single and multiple state parameters. NRSA evaluates the effect on model outputs exerted by individually varying* only one* of the model inputs across its entire range of plausible values, while holding* all other* inputs at their nominal or base-case values (i.e., fix them locally) [2]. NRSA can be repeated for any number of individual model inputs. An extension to multiple parameters, with a linearization approximation, is the sensitivity-matrix, which consists of first-order partial derivatives of outputs with respect to (w.r.t.) inputs (Jacobian matrices) [1, 3].* Local* sensitivity analysis methods perturb some inputs at a time, while the remaining inputs are fixed to base-case values. NRSA and sensitivity-matrix methods are* local* methods [3].

Recent applications of mathematical sensitivity analysis to the hydraulics of a WDS include a GIS-based sensitivity analysis method based on NRSA used in several analyses of an Austrian-based group. For the effect of continuous change of parameters on hydraulic variables, of interest here, this method was applied to calibration, pipe diameter design, and input uncertainty assessment [10], and vulnerability identification [11]. Other recent applications include the sensitivity-matrix method for ranking the relative importance of pipes [9], leakage localisation [19], and studying demands [20].

Since NRSA is the easiest sensitivity analysis method to implement and understand [3], a novel sensitivity analysis formalism that extends this method is developed, called* local regional sensitivity analysis* (LRSA). Like the NRSA and sensitivity-matrix methods, it is* local* and should hence preferably be applied to problems where this is an advantage, or not a disadvantage. Examples of such problems are model verification or validation [3].

In NRSA and sensitivity-matrix methods different changes are applied to each parameter* individually*. LRSA extends these methods to allow the same change to apply to a collection of parameters, called a* region*. The importance of the study of parameter regions has been emphasized [16]. For example, by breaking the entire system into suburbs, a complicated calibration problem can be studied incrementally [4, Section ].

The novelty of LRSA is that it allows, in an easy way, consideration of the sensitivity of the WDS to state parameter* regions*, not just individual state parameters. The unique contributions that are made to the study of a WDS are as follows:(i)Ease of understanding: a complicated WDS is easier to understand if the number of parameters it is sensitive to is reduced. This is done by lumping parameters together in regions.(ii)Simplicity of analysis: because the parameters in the WDS can be grouped into regions, the process of sensitivity analysis is simplified.(iii)Dependence on the collective (the region): because a change of the collective is made, the sensitivity of the WDS to the collective can be studied.

Numerical sensitivity calculations have been performed for two networks [21], using the LRSA techniques in Section 3.2. In this paper, several results are derived, which are of a general nature. The results explicitly state the dependence of a WDS on state parameter* regions* for the first time. The results also serve as consistency checks with the current body of knowledge to show how sensible results arise from LRSA. One motivation for using these results as consistency checks is that they do not depend on a linearization approximation.

The results apply to the standard NRSA and sensitivity-matrix methods as special cases. Hence the results do not require adoption of the LRSA method and stand in their own right. For the NRSA and sensitivity-matrix methods, the results have not been found to be explicitly stated elsewhere.

The paper is organized as follows. LRSA is introduced in Section 2 and then developed in Section 3 for a single state parameter region. The* single* region results are introduced in Section 4; the pipe parameter scaling laws are stated in Section 5 and the node elevation uncertainty results in Section 6. The* multiple* state parameter region results are introduced in Section 7, with the generalized pipe parameter scaling laws and node elevation uncertainty results in Sections 8 and 9, respectively. The types of sensitivity analysis covered by LRSA, and how these relate to conventional methods, are described in Section 10; and Section 11 summarizes the conclusions. The Appendix contains single region proofs of results and illustrative examples.

#### 2. Introduction to Local Regional Sensitivity Analysis

A central characteristic of the sensitivity analysis developed below is that the* same change* is made for* all* the state parameters in a* single* set of parameters (either or ), as is partially implemented in some current network models for demand [4, Section ], [5]. The effect of the change on a* single* indicator function (composed of* all* the water flow rates and pressures in a single set of links and nodes ) is then determined mathematically (generalization beyond a “single” set of parameters will be made later in the paper but only serves to complicate matters at this stage). There is hence an associationThe two regions are in general unrelated, and a region does not have to be spatially connected. However, in practical problems or often represents a connected region in space and similarly for (, ). For example, or can represent a suburb inside a WDS, while (, ) can represent the end-nodes of the entire system, where pressure is often the lowest.

The sensitivity analysis considers two types of continuous state parameter change.

*Proportional Change (** for all **)*. These are appropriate for parameters which can only have one sign, e.g., pipe length. There are three important characteristics of proportional change:(1) is dimensionless, so change made to different types of parameters can be compared.(2)The change emphasizes the larger parameter values. For example, if a set of pipe lengths is changed by the same amount , the pipes with the larger lengths are changed more. This is sensible, since the larger length pipes should be more “important” when change due to pipe lengths is considered.(3)The change to the individual parameters can naturally be interpreted as a single property of the collective. For example, if a set of pipe lengths in a connected region is changed by the same amount , the total pipe length of the region is changed by . This means that the collective pipe length of the region is changed in a sensible way.

*Additive Change (** for all **)*. These are appropriate for parameters which can have different signs, e.g., elevation. There are two important characteristics of additive change:(1)The change to the parameters is independent of the parameter values. For example, elevations are changed by the same amount . This is sensible, since the absolute size of an elevation is arbitrary, because all elevations refer to a reference elevation (e.g., mean sea level [5, 14]).(2)The change to the individual parameters can naturally be interpreted as a single property of the collective. For example, if a set of elevations in a connected region is changed by the same amount , the elevation of the entire region is changed by . This means that the collective elevation of the region is changed in a sensible way.

*All *the parameters in a state parameter region are changed together, so that the changes of the parameters are* correlated* in a specific way. It is proposed that it is sufficient for a parameter region to be changed according to only one of two types of changes: proportional or additive.

#### 3. Sensitivity Analysis Formulation: Single Region

##### 3.1. Parameter Variation: Proportional Change

Standard modelling of a WDS requires the construction of a network with internal nodes, and links (or lines) joining the external and internal nodes [5–7] (Figure 1). Water flows at rate through a link, and every internal node has pressure head at the position of the node. Pressure is defined in such a way that atmospheric pressure is zero. The variablesare obtained as a solution of a numerical model. Here denotes* all* links and * all* internal nodes. By convention, flow rates are positive for the base-case solution, and a pressure is usually positive (when a flow rate is negative, reversing the direction of the corresponding link will make it positive). Reservoirs or tanks are by definition external nodes.