## Abstract

The boundaries of existing cities are expanding rapidly due to the exponential growth in urban population. Therefore, the existing water distribution networks (WDNs) need to be expanded up to the newly developed areas to meet the additional water demand. The optimal design of a sub-network planned for network expansion requires multiple simulations under various constraints. Simulating the additional sub-network along with the existing network takes a lot of CPU time. In this study, a methodology is proposed to replace an existing large pipe network with its equivalent network consisting of a single source and a single pipe by applying the non-linear Thevenin theorem being used for electrical circuits. The equivalent network model parameters are extracted by fitting the driving-node head-loss characteristics at the connecting node. Unlike all other available methods except the traditionally used reservoir–pump model, the equivalent network presented in this study reduces to only two elements. The theoretical aspect of the reservoir–pump model can be explained by the proposed Thevenin reduction method. The advantage of the proposed method is put forward by deriving an analytical expression for the condition of maximum power transfer from the equivalent main network to the sub-network. The economic diameter value of the connecting pipe is subsequently determined. The proposed network reduction method is demonstrated on different WDNs for various demand patterns. The reduced networks yield accurate results and simulate faster when compared with those of the original networks. The proposed methodology is beneficial for a focused analysis of a sub-network and to transfer maximum power to the sub-network connected to a large existing hydraulic network.

## HIGHLIGHTS

Using the non-linear Thevenin theorem, a methodology is proposed for water distribution network simplification.

The equivalent network consists of only two elements.

The equivalent network is giving exact results as that of the original network.

This methodology gives computational advantage.

The relation between connecting pipe diameter and sub-network demand for maximum power transfer is derived using the equivalent network.

### Graphical Abstract

## INTRODUCTION

The urban population is expected to increase to 68% by 2050 (DESA 2018). The city outskirts of many countries are experiencing rapid expansion as the migration of people from rural to urban areas is increasing in manifolds. The reorganization of an existing water distribution network (WDN) to accommodate new areas is economical in developing economies (Swamee & Sharma 1990) because the working life of a pipeline (60–120 years) is more than its design period (20–40 years). An existing WDN can thus be expanded by attaching sub-networks for the newly developed areas to meet the additional water demands. Analysing and forecasting urban water demand is a complex task, as it depends on future climatic conditions and population growth. Nithila Devi *et al.* (2019) studied the expansion (i.e., urban sprawl) of Chennai city in India by 2030 using an urban growth model. Mukherjee & Singh (2020) studied the spatial and temporal changes in land use and pattern of vegetation and their impacts on land surface temperature. On the contrary, Wasko & Sharma (2017) concluded that rapid urbanization and global warming may be changing the hydro-meteorological condition, which in turn will change the rate of water consumption by the urban population. Global mean surface temperature is the most important indicator of global warming and climate change. Valipour *et al.* (2021) investigated the changes of 20-year (2000–2019) mean surface temperature, wind speed and albedo data from the Global Land Data Assimilation System (GLDAS) over the globe with respect to those in 1961–1990. They concluded that the rapid change of all these parameters happened only in the last few decades. All these parameters have an indirect effect on urbanization and consequently water demand in the changing climate scenario. The above studies are helpful to address the challenges of urbanization by assisting officials of the local governments, land management professionals and city planners to select the areas for the implementation of systematic urban planning practices. Hence, urban sprawl under the climate change scenario may significantly influence the forecast of water demand. There are two types of demand forecasts, short-term and long-term. Short-term forecasts are used for operation and management, and long-term forecasts are used for planning and infrastructure design. Peak water demand of WDN influences infrastructure expansion strategies, including the size and operation of reservoirs, pumping stations and pipe capacities (House-Peters & Chang 2011). The demands are modelled as a stochastic process (Piller & Brémond 2002; Alcocer-Yamanaka *et al.* 2012) with all possible failure conditions to ensure the reliability of a WDN (Yang *et al.* 1996). In addition, control, maintenance and pump scheduling of a WDN require optimization techniques to meet various constraints. Therefore, the optimal design of a WDN is complicated because it contains a large number of discrete elements (pumps, valves and pipe segments). The problem is non-linear and non-convex (Eiger *et al.* 1994). The size of an optimization problem thus becomes more extensive than an equivalent hydraulic simulation problem. Literature (Deuerlein 2008; Broad *et al.* 2010; Martinez Alzamora *et al.* 2014) emphasized that the requirement of extensive computation time was hindering the progress in developing optimization methods for large WDNs. Therefore, for expanding an existing large WDN, an efficient pipe network analysis tool that focuses primarily on the sub-network without investing much computation time for the already existing mainstream network can be a handy tool for hydraulic engineers. An efficient network reduction method can achieve such an objective.

Eggener & Polkowski (1976) first studied the skeletonization process by the systematic removal of pipes in Menomonie WDN, Wisconsin. They found that removing a large number of pipes under usual demands did not significantly affect pressures. Jeppson (1982) studied network reduction by replacing several series or parallel connected pipes with a single equivalent pipe. Jung *et al.* (2007) derived formulae for calculating equivalent diameters for series and parallel pipes based on the Hazen–Williams head-loss equation. The above two methods are not applicable if the pipes in a network are neither in series nor in parallel. To overcome the limitation, Anderson & Al-Jamal (1995) proposed a parameter-fitting approach using non-linear programming suitable only for small networks. Hamberg & Shamir (1988a, 1988b) proposed two models reducing the size of networks: (a) a stepwise combination of elements; and (b) a non-linear continuum representation of the system as a whole. In the stepwise combination method, pipes and water withdrawals that vary along the pipes and with time are replaced with their equivalents. An aerial distributed conductance, a function of the pipe properties and aerial density, is the link between a potential function related to the heads and the flow field. Using this, a reduced model can be obtained for the flow field of an existing system at a prescribed potential.

Later on, several network simplification methods such as skeletonization (Walski *et al.* 2003; Saldarriaga *et al.* 2008), Artificial Neural Network (ANN)-based metamodels (Rao & Alvarruiz 2007; Broad *et al.* 2010), variables elimination (Martinez Alzamora *et al.* 2014), decomposition **(**Ostfeld 2012; Zheng *et al.* 2013) and segmentation (Giustolisi & Ridolfi 2014) were proposed. The skeletonization of any network can be achieved either by removing the pipes with small diameters and nodes with lower demands or by reallocating removed demands to the remaining neighbouring junctions (Bahadur *et al.* 2008). Since this is not a single process, as the removal of many low-level elements in a series needs to be applied, the utilization of this technique thus becomes difficult and limited to looped networks. Saldarriaga *et al.* (2008) presented an automated skeletonization methodology to obtain a reduced model of a WDN that can accurately reproduce both the hydraulics and non-permanent water quality parameters (chlorine residual) of the original model by using the resilience index. As this method focuses only on pipe removal, it can be primarily applied to looped pipe networks. Hirrel (2010) proposed another method of automatic skeletonization by combining skeletonization with graph traversal algorithms. The graph traversal process is used to (i) obtain flow balance and head at each node, (ii) identify all the paths in the network and (iii) determine the head error and a total q-prime for each path. When skeletonization is performed manually, the accuracy of skeletonization depends on the experience and engineering judgement of the modeller. The Skelebrator module from the Bentleys Haestad Methods in WaterGEMS software (Hammer 2008) is one of the first commercial automatic skeletonization solutions. Skelebrator uses data scrubbing, branch trimming, series pipe removal and parallel pipe removal as a combination to automatically skeletonize water distribution systems. Due to the time-demanding training process, the ANN-based network models (Rao & Alvarruiz 2007; Broad *et al.* 2010) are not suitable for optimizing large networks. Martinez Alzamora *et al.* (2014) proposed a reduction method by linearizing the non-linear network at an operating point. The linear model is then reduced to the required size by eliminating unwanted elements. The non-linear model is again derived from the reduced linear model to preserve the non-linearity of the original system. While eliminating a node, the demand is arbitrarily distributed among the connected nodes. This may not be correct because the flow in a branch depends not only on the conductance value but also on the head difference between the branch nodes. Zheng *et al.* (2013) used graph theory to decompose a whole water network into different sub-networks based on the connectivity of the network components. The original network is simplified to a directed augmented tree, in which the sub-networks are substituted by augmented nodes and directed links to connect them. Then, each sub-network is optimized based on the sequence specified by the assigned, directed links in the augmented tree. A solution choice table is established for each sub-network (except for the sub-network that includes a supply node) and the optimal solution of the original network is finally obtained by the use of the solution choice tables. In order to simplify the analysis and the management tasks, the complex and large-size hydraulic systems are divided into modules based on the modularity index called segmentation **(**Giustolisi & Ridolfi 2014). Water distribution infrastructures also fall under complex network theory as these systems connect the nodes (vertices) by pipes (edges) and transfer water to customers **(**Giustolisi *et al.* 2019). Sitzenfrei *et al.* (2020) identified several graph characteristics based on a systematic complex network analysis of Pareto-optimal solutions of different WDNs. The optimized WDN is then obtained after a systematic investigation of graph weights combined with graph measures (e.g., analysis of the shortest paths and the customized edge betweenness centrality).

When there are not many changes in the existing network, especially demands and network parameters, the existing network can also be modelled as a reservoir element with a constant head. The reservoir can then be connected to the sub-network planned for expansion. It is a very simplified model, but it does not account for any fluctuation in the head due to the change in sub-network demand. To improve the reliability of the method, the existing network can be modelled as a reservoir with a fictitious pump at the connecting node. The three-point pump characteristics curve can be derived using static and residual pressures obtained from the two-hydrant flow tests at the connecting node. Here, the fictitious pump compensates for the head drop due to sub-network demand (Walski *et al.* 2003). Unlike the constant-head reservoir approach, this model (i.e., reservoir–pump model) considers the change in the head due to the variation of sub-network demand at the connecting node. However, the pump–reservoir model cannot be used to determine the diameter of the connecting pipe for maximum power transfer from the main to the sub-network.

Expansion of a WDN deals with the inclusion of additional demand nodes at predetermined locations (nodes) of the existing WDN. The maximum size of the newly connected area is suggested to be less than 20–25% of the existing WDN (Swamee & Sharma 2008). Because of the substantial economy of scale in pipe sizing and the long useful life, there may be beneﬁts in installing piping with some excess capacity in the short term depending on the expected long-term growth (Walski 2014). Agrawal *et al.* (2007) proposed a method for the reliability-based strengthening and expansion of WDNs. The model reservoir connected with the extended sub-network can be used in the optimization process. Todini (2000) developed a multi-objective approach for the expansion and rehabilitation of an existing WDN. Neelakantan *et al.* (2014) proposed a simulation optimization model for optimal up-gradation and expansion of an existing WDN using EPANET (Rossman 2000) as a hydraulic simulation tool and differential evolution algorithm for finding an optimal solution. In the analysis of all the above WDN expansion problems, hydraulic models were repeatedly simulated for the existing along with the sub-network for the expanded area to satisfy the optimization constraints. The hydraulic model for simulating a pipe network system solves the system of flow through each pipe and non-linear head balance equations at each junction. The solution schemes for the non-linear system of equations involve several matrix operations through iterations, which take a significant CPU time. The computation time increases nonlinearly when the same network needs to be solved repeatedly for variable demand patterns, control and operation of various hydraulic components and optimal design. Efficient network reduction methods can help reduce the size of the prototype network, thereby the computation time by preserving the nonlinearity of the original network and approximating various operating conditions.

All the network reduction methods described above consider the entire WDN even when an existing WDN is expanded by adding a sub-network of the size of 20–30% of the existing WDN. As a consequence, the size of the reduced hydraulic problem is still quite large and needs signiﬁcant computation time. The computation burden further increases when the optimal design of the sub-network is planned. Ideally, the existing WDN should not add an unnecessary computational burden while focusing primarily on the sub-network. However, it should be ensured that the operation of the existing WDN does not get affected in the process. Under this circumstance, a network reduction method may be necessary to reduce the existing WDN so that the analysis can focus on the sub-network for optimal design without compromising the operation of the existing WDN. However, the operational efficiency of a WDN has become more important in recent years. The WDN performance depends on the ability of the network to meet the demands with minimum pressure. Park *et al.* (1998) proposed the concept of hydraulic power and energy transmission to test the hydraulic reliability of a WDN. The hydraulic power capacity is defined as the probability that there exists a feasible flow of hydraulic power in the WDN. Todini (2000) also used the concept of hydraulic power in connection with the resilience index to characterize the surplus of energy capability to overcome sudden failures in looped networks. Using the concept of resilience, Prasad & Park (2004) introduced a new resilience measure called network resilience, which includes the effects of both surplus power and reliable loops. Blackmore & Plant (2008) used the resilience theory to study risk management in urban water systems. Vaabel *et al.* (2008) introduced the coefficient of the critical outlet power, *k*, and the surplus power factor, *s*, to evaluate the hydraulic power capacity of a WDN based on both flow within pipes and pressure head at the inlets of pipes. The value of *k* characterizes the potential of the hydraulic power used by the hydraulic system, which is also used to determine the reserve of hydraulic power. The factor *s* represents the reliability of the hydraulic system. If *s* = 0, the hydraulic system works at a maximum capacity. The increase in the value of *s* will improve the hydraulic reliability of the system. Calculating the *s* factor for a WDN depends on the network resistance coefficient, *C*, which characterizes overall head losses in water pipelines. Vaabel *et al.* (2014) proposed a theoretical approach for determining the value of *C* through matrix equations. So, hydraulic power capacity is important in maintaining the reliability of a WDN. Therefore, the hydraulic power capacity and its flow throughout the network (Park *et al.* 1998; Todini 2000) need to be ensured for a resilient WDN and its efficient operation.

The similarity between an electrical circuit and a pipe network is used for solving some of the WDN problems. Mcilroy (1950) developed a pipeline network analyser based on the pipe network and electrical circuit analogy. Stephenson & Eaton (1954) extended the concept to compressible fluid flow by adjusting the voltage and current relationship. Millar (1951) introduced the content and the co-content functions, also called energy and co-energy, to solve electrical networks. For a resistive network, the dissipation due to current flow is called content and the dissipation due to voltage is called co-content. These functions are dual to each other. Using the analogy, Collins *et al.* (1978) derived content and co-content functions for WDN to solve the optimization problem. Both the models represent the mathematical equivalent description of the pipe network analysis problem. So, any one of the models can be used for computational analysis. Deuerlein *et al.* (2009) used content and co-content theory to characterize the conditions that ensure the existence and uniqueness of solutions in demand-driven systems with flow control devices. Elhay *et al.* (2016) showed that the steady-state hydraulic equations are necessary and sufficient conditions for the optimality of the content model and its dual counterpart, the co-content model. In recent years, the content and co-content theory have been extended to solve pressure-driven models. Deuerlein *et al.* (2019) presented a content-based active set method (ASM) for pressure-driven systems without any link flow constraints. The ASM is reported to be fast and reliable for a wide range of pressure outflow relationships. The content and co-content-based theories are also used to derive globally convergent algorithms for dealing with flow control valve constraints (Piller *et al.* 2020). Oh *et al.* (2012) reviewed the application of electrical circuit methods to analyse pressure-driven microfluidic networks. The study shows that circuit analysis enables rapid predicting pressure-driven laminar flow in microchannels and is very useful for designing complex microfluidic networks. Liu & Hodges (2014) developed a software tool called Simulation Program for River Networks (SPRINT) to study river networks with many branches and nodes.

Thevenin published a series of studies on electrical circuits (Thévenin 1883a, 1883b, 1883c, 1883d, 1883e). The theory states that any linear two-terminal network can be replaced by its equivalent network having a single source and a single resistor. This concept became very popular because of its deep engineering insight and is now referred to as Thevenin's theorem (Johnson 2003). Later, the concept was extended to non-linear networks using the driving-point characteristics (Chua 1969). Today, it serves as a handy tool for circuit designers to perform a focused analysis on a particular portion of a large circuit by replacing the whole complicated circuit with a simple equivalent circuit consisting of only a single source and resistance (or impedance if reactive elements are present) (Alexander & Sadiku 2000). Hence, Thevenin's theorem can potentially be used to reduce a large, complicated WDN with a single source (i.e., a reservoir) and single resistor (i.e., an equivalent pipe friction) for faster analysis and optimal design. Balireddy *et al.* (2021) proposed a methodology for reducing a WDN using the single-port *linear* Thevenin theorem. The main network was first linearized at an operating point to be replaced with its equivalent network. The Thevenin equivalent parameters were calculated in the second step by implementing the linear network in a circuit simulator. Although the main network was reduced to a miniature form of an equivalent network, the methodology was not straightforward. The application of the linear Thevenin theorem requires several steps for reducing a large network. An electrical circuit simulator is also needed for implementing and solving the reduced hydraulic network as a linear electrical network. In addition, as the network is linearized at an operating point, the method may return erroneous results when the sub-network demand is significantly different from that of the operating point.

To overcome the above issues related to network reduction, in this work, a methodology is proposed to replace a large existing pipe network system using the *non-linear* Thevenin theorem to an equivalent network consisting of a single reservoir and an equivalent pipe. The proposed reduction method is derived by modifying the non-linear Thevenin theorem for electrical circuits. The model parameters of the equivalent network are extracted by fitting the driving-node head-loss characteristics (equivalent to driving-point characteristics of an electrical circuit) of the main network at the connecting node with a suitable head-loss formula. The Thevenin equivalent network is then attached to the sub-network that is planned for expansion. The two networks need to be connected with a pipe whose diameter can determine the amount of power that can be transferred from the main network to the sub-network and consequently the economic operation of the WDN. In this study, an analytical expression is derived for the maximum hydraulic power transfer from the equivalent main network to the sub-network so that the diameter of the connecting pipe can be determined. The proposed network reduction method can be considerably efficient when several simulations are required for the optimal design of an expanded network. The proposed methodology is demonstrated on two realistic WDNs by implementing them in a freely downloadable pipe flow simulator, EPANET V2.0 (Rossman 2000). The results obtained from the reduced networks are compared with the solutions of the original full networks. The Thevenin network reduction method is shown to be highly accurate with a significant advantage in computation time. The proposed concept is also compared with the industrial practice on network expansion using a reservoir and a fictitious pump (i.e., a reservoir–pump model) in place of the main network (Kampa *et al.* 2013). The Thevenin equivalent and reservoir–pump models are found to be very similar. However, the major advantage of the proposed method over the reservoir–pump model is that one can derive the condition for maximum power transfer from the main to the sub-network by extending the concept of the Maximum Power Transfer Theorem (MPTT) for electrical networks. Therefore, the proposed network reduction method using the non-linear Thevenin theorem for electrical circuits can be an efficient tool for hydraulic engineers to analyse a large pipe network expanded by attaching a sub-network.

## METHODOLOGY

The work being presented here is demonstrated for the analysis of the expansion of an existing WDN. When an existing network is expanded to serve the water demand of a particular area, the optimization problem is primarily focused on the sub-network since the main network is supposed to be already optimized or may not need a major modification. Due to the expansion of the network, there can be a reduction of the overall supply to meet the required demands in the main network itself. This can be addressed within the framework of the proposed scheme by changing the reservoir heights or pump capacity or connecting parallel pipes to meet the extra demands. The solution can be straightforward and requires as such no further optimization of the main network. Since prior to the network expansion, the main network is already optimized to meet a certain set of demands at various nodes and any of the straightforward solutions will not force the main network into a significant sub-optimal operation. To be more precise, the performance can deviate from the optimal operation at a minimum and that may be ignored. Therefore, in cases of such WDN expansion problems, we assume that the newly developed sub-network requires optimization while ensuring that the operation of the main network is not compromised. The expansion of an existing network to accommodate a new area is not uncommon in developing economies (Swamee & Sharma 2008). Under this circumstance, we present a network simplification methodology for the main network in order to speed up the overall analysis without sacrificing accuracy. Prior to simplification, the main network is monitored if its operation meets a certain set of criteria. Once the pre-set requirements are met, we simplify the main network to a bare minimum and connect with the sub-network. Therefore, the whole network optimization problem can be split into two separate problems, part of which involves performance monitoring and adoption of straightforward solutions (for the existing main network) and the remaining part (the new sub-network) may undergo rigorous optimization. In order to obtain reliable results under this framework, the accuracy of the network simplification scheme needs to be examined for various scenarios. We validate the results of the reduced network for static as well as extended period (i.e., dynamic) simulations.

### Non-linear Thevenin theorem for electrical networks

According to the Thevenin theorem, any resistive network can be replaced with an equivalent circuit consisting of a source in series with an equivalent resistor with respect to a node. Analogously, with respect to a node, a large WDN can be replaced with an equivalent reservoir connected to an equivalent pipe. The equivalent parameters for an electrical circuit are calculated using the driving-point characteristic plot (henceforth, referred to as DP-plot). Similarly, the equivalent parameters for a WDN can be derived using the driving-node head-loss characteristic curve (henceforth, referred to as DN-plot). The procedure for finding the Thevenin equivalent network for a WDN is explained in the subsequent section. DP-plot is the *v* (voltage) − *i* (current) characteristic curve measured across a pair of terminals of a network. The voltage source of the equivalent network is equal to the voltage intercept on the DP-plot. The *v* − *i* characteristic curve of the equivalent non-linear resistor is obtained by translating the DP-plot of the non-linear network along its voltage axis until it passes through the origin (Chua 1969).

*V*

_{Th}= Thevenin equivalent voltage) with a nonlinear resistor (

*R*

_{Th}= Thevenin equivalent resistance) using the DP-plot. To measure the

*v*−

*i*characteristics of this network, a variable source (i.e., voltage source,

*v*

_{in}) is connected across the two-terminal (A and B) dc-resistive network as shown in Figure 1(a). The input voltage source (

*v*

_{in}) is varied from its negative maximum to its positive maximum in steps and the input voltage and corresponding current (

*i*

_{in}) are noted. The DP-plot of the dc-resistive network is obtained by plotting

*i*

_{in}on the

*y*-axis for different values of

*v*

_{in}on the

*x*-axis as shown in Figure 1(b).

*V*

_{Th}of the network is equal to the voltage intercept (

*V*

_{oc}) on the DP-plot. Now, the equivalent non-linear resistance characteristics of the network are obtained by shifting the DP-plot to the origin along the voltage axis as shown in Figure 1(c). By fitting this curve with a power function (

*v*=

*R*

_{Th}

*i*), the

^{n}*v*−

*i*relationship for

*R*

_{Th}is obtained. Finally, the non-linear two-terminal dc network shown in Figure 1(a) is replaced with a single voltage source and single non-linear resistor as shown in Figure 1(d). It should be noted that the application of the non-linear Thevenin theorem is highly limited in electrical circuits. The present study, for the first time, explores the possibility of reducing a large pipe network (i.e., a non-linear hydraulic network system) using the non-linear Thevenin theorem.

### Implementation of the non-linear Thevenin theorem for hydraulic pipe networks

*et al.*2021) is relatively straightforward compared to the linear Thevenin theorem, which is highly popular in electrical circuits. Figure 2 illustrates the methodology to convert a large hydraulic network (Figure 2(a)) to a simple equivalent network with a single reservoir and single pipe (Figure 2(b)). Figure 2(a) shows a large hydraulic network, where N

_{t}is a terminal node (or, a connecting node where the sub-network is to be connected) having head

*h*and variable demand

_{i}*q*. When

_{i}*q*= 0, the head at terminal N

_{i}_{t}is called the open network head (analogous to open-circuit voltage),

*H*

_{oc}. The open network head can be imagined as the nodal head when the pipe at N

_{t}is opened to the atmosphere and the demand at the node is set to

*q*= 0. As the demand (

_{i}*q*) increases, the head (

_{i}*h*) at the node N

_{i}_{t}decreases in a non-linear manner. This is because, when demand

*q*increases, the flow through the network has to increase leading to an increase in head loss in the network. It is obvious that the head loss across the network increases nonlinearly as the demand increases. From the concept of network equivalence, the large existing network can be replaced with a simple equivalent network that follows a similar head-loss pattern. The use of the non-linear Thevenin theorem thus allows reducing a large hydraulic network with a small network having a single source (i.e., a reservoir) and a single pipe (i.e., an equivalent pipe) as shown in Figure 2(b). Therefore, head at N

_{i}_{t}is

*H*

_{oc}=

*H*

_{eq}, when the demand is zero, otherwise

*h*for different values of

_{i}*q*. That means the total head loss in the full network is compensated by a single equivalent pipe (

_{i}*P*

_{eq}) in the Thevenin equivalent hydraulic network. The detailed step-by-step implementation procedure for replacing a large hydraulic network with its non-linear Thevenin equivalent hydraulic network is explained in the following steps.

Identify the connecting node (N

_{t}) at which the new sub-network is planned to be connected to the main network.Obtain the open network head (i.e.,

*H*_{eq}) for zero demand (*q*= 0) at N_{i}_{t}.By varying the demand (

*q*) with small step size Δ_{i}*q*up to a possible maximum limit,*Q*_{max}(after which it may drop head below the allowable limit or lead to a negative pressure at some nodes or pumps in the main network exceeding their limits) and note down the corresponding head (*h*) at N_{i}_{t}. Any constraints, such as a minimum head (*H*_{min}), maximum flow rate, etc. need to be considered in this step for estimating the maximum capacity of the existing main network that can be supplied to the sub-network without compromising the operating condition of the main network.Calculate the head loss (

*h*_{Li}*=**H*_{oc}–*h*) at N_{i}_{t}for all demands.Plot the head loss versus demand (i.e.,

*h*_{Li}versus*q*) curve. The authors are calling this curve the_{i}*driving-node head-loss characteristics curve or DN-plot*.Fit this curve with an appropriate head-loss formula (e.g., for Hazen–Williams:

*h*_{Li}= [(10.67*L*_{eq})/(*C*_{eq}^{1.852}*D*_{eq}^{4.87})]*q*_{i}^{1.852}) and note the equivalent pipe parameters such as*L*_{eq}(pipe length),*D*_{eq}(pipe diameter) and*C*_{eq}(Hazen–Williams constant).Replace the large hydraulic network by its Thevenin equivalent network with the equivalent reservoir head

*H*_{eq}and an equivalent pipe*P*_{eq}.

### Similarities and differences between Thevenin and reservoir–pump models

In the case of network expansion, the industrial practice is to use the reservoir–pump model (Kampa *et al.* 2013). In this approach, the reservoir supplies water from the existing network system. The reservoir elevation should be equal to or slightly higher than the elevation at the connecting node. A fictitious pump is assumed to be connected with the reservoir. The characteristic curve is prepared based on the two-hydrant flow test data and the third one from a mathematical relationship based on the test data. The hydrant tests are carried out at the connecting node or as near to this node as possible. One test is conducted at zero flow through the connecting node that yields the static pressure (i.e., *P*_{static} for *Q* = 0) and the other at any demand point (i.e., *P*_{test} for *Q*_{test}). The third point on the characteristic curve is obtained for the residual pressure (*P*_{res}) for any flow (*Q*_{res}) and can be calculated using a mathematical formula as described subsequently.

*H*is the available head for the expanded zone,

*z*is the elevation of residual gage,

*a*is the conversion factor for head to pressure units and

*K*is the coefficient to calculate the effective head loss from the source to the residual hydrant for each test case.

*P*

_{pu}) is equal to the static head (

*P*

_{static}) minus the loss due to pump flow (

*Q*

_{pu}). It is assumed that the reservoir and the fictitious pump are connected by a smooth, short and wide pipe that does not contribute to head loss. The reservoir–pump system is now connected to the sub-network for the analysis. Since only three points are used to generate the pump characteristics, this is a practical but approximate method for a wide range of demand variation in the sub-network.

Theoretically, the reservoir–pump and Thevenin network reduction methods are equivalent. However, the Thevenin theorem provides us with a generalized view and reveals the underlying theoretical basis of the reservoir–pump model. On the contrary, the two reduction methods have some differences. The proposed Thevenin model consists of an equivalent reservoir and a pipe with no fictitious pump as in the reservoir–pump model. In the Thevenin model, the head loss due to a change in the sub-network demand is compensated by the equivalent pipe, whereas in the reservoir–pump model, such head loss is compensated by the fictitious pump. The fictitious pump characteristic curve may not accurately represent the head-loss characteristic if the demand in the sub-network varies in a wide range. Furthermore, unlike the traditional use of the fictitious pump, the underlying principle described by the Thevenin theorem allows us to derive a condition for the maximum power transfer from the main network to the sub-network and subsequently the diameter of the connecting pipe. The economical diameter of the connecting pipe is important, especially in the case when two networks need to be connected by a long pipe. However, the advantage of the proposed Thevenin method can be realized in a much better way in the case of multiple expansion zones by extending the single-port linear Thevenin theorem (Balireddy *et al.* 2021) to multi-ports using the dependent sources (Bandyopadhyay 2015).

### MPTT and its application in WDNs

In a water supply system, nearly 70 − 85% of the total cost is contributed toward water transmission and the WDN. The total cost of a water distribution system is broadly classified as capital cost and recurring cost. The cost of pumps and pumping stations, pipes, storage reservoirs and residential connections come under capital costs. Costs such as energy usage, operation and maintenance of the system components come under the recurring costs. As head loss in the system depends on pipe diameters, both the capital and running costs effectively depend on the pipe diameters (Swamee & Sharma 2008). As the connecting pipe carries the total sub-network demand, there is a significant head loss in the connecting pipe, especially in the situation where the pipe can be sufficiently long. Choosing a smaller connecting pipe diameter involves less capital expenditure but requires high pumping energy costs. Choosing a larger connecting pipe diameter leads to high capital expenditure but requires less pumping energy cost. Therefore, it is important to choose the optimal size of the connecting pipe for the economical operation of the entire WDN.

The energy required to transfer the water from the main network to the sub-network is directly proportional to the power. As the main network is fixed, the power transferred from the main network to the sub-network depends on the open-circuit head (i.e., static head), the equivalent resistance of the main network, connecting pipe diameter and the demand in the sub-network. In this work, the relation between sub-network demand and connecting pipe diameter is derived to ensure the transfer of maximum power from the main network to the sub-network through the connecting pipe. It should be noted that the term maximum power transfer indicates only the transfer of maximum power by choosing a suitable diameter of the connecting pipe using the concept described subsequently. By plotting the relation between connecting pipe diameter and sub-network demand for maximum power transfer, one can choose the diameter of the connecting pipe.

*Q*

_{Pmax}) transfer theorem, first, we need to replace both the main and sub-networks with their equivalents. In Figure 4, the main network is replaced with its Thevenin equivalent (

*H*

_{eq}

*, K*

_{eq}) and the sub-network with its total demand (

*Q*

_{s}). The two equivalent networks are then connected with a pipe (

*K*

_{c}).

We can derive an equation for the maximum power transfer as follows:

From the above expression, it is evident that the value of *Q*_{Pmax} depends on the *K*_{c} value. The length of the connecting pipe is fixed and the Hazen–Williams coefficients (*C*_{HW}) of the connecting pipe are also fixed based on the pipe material. Hence, the value of *K*_{c} depends directly on the connecting pipe diameter (*D*_{c}), which can now be calculated explicitly.

## RESULTS AND DISCUSSION

A variety of networks are considered to examine the performance of the non-linear Thevenin theorem on a WDN and the concept of maximum power transfer from the main network to a sub-network. This section is divided into two parts. In Part 1, the derivation of equivalent WDN using the non-linear Thevenin theorem and its accuracy are demonstrated on two networks. In Part 2, the equivalent network of a realistic WDN is tested for demand variation in the main network. The application of the non-linear Thevenin theorem for maximum power transfer from the main network to the sub-network is also demonstrated on the same realistic WDN.

### Equivalent WDNs using the non-linear Thevenin theorem

The proposed methodology for reducing a large hydraulic network to its equivalent simple network having a single reservoir and a single pipe using the driving-node head-loss characteristics is tested on two pipe network systems: (i) Network 1: an example problem taken from an instructional manual (Jeppson 1974) and (ii) Network 2: a large benchmark WDN (Richmond WDN) from the ASCE Library (Qiu *et al.* 2019). The proposed network reduction method is evaluated in terms of accuracy and computation time. The reference solutions for comparisons are generated by solving the entire networks (i.e., the main and sub-networks) using EPANET.

#### Evaluation of the proposed methodology on Network 1

*C*

_{HW}for all the pipes is given at the bottom left side of the main network. The pump characteristics are given at the bottom right side of the main network. The water levels in Reservoir 1 and Reservoir 2 are 277.4 and 329.2 m, respectively. Pump 1 and Pump 2 are installed in the pipes connected to the reservoirs to increase the potential head, whereas Pump 3 is used to maintain a sufficient head in the sub-network. The demand at nodes N4, N6, N9 and N11 in the main network is specified in m

^{3}/s.

*H*

_{oc}= 483.85 m. Now, the demand at N13 is varied from 0 to 0.22 m

^{3}/s and the corresponding variations in the head at the same node are measured. Pump 3 generates negative heads when demand at N13 is more than 0.22 m

^{3}/s as shown in Figure 6. Table 1 summarizes heads and head losses at N13 for different demands at the same node. The DN-plot of the network at N13 is obtained by plotting variable demands on the

*x-*axis and corresponding head losses on the

*y*-axis as shown in Figure 7. Now, these characteristics are fitted with the Hazen–Williams head-loss formula,

*h*

_{Li}= [(10.67

*L*

_{eq})/(

*C*

_{eq}

^{1.852}

*D*

_{eq}

^{4.87})]

*q*

_{i}

^{1.852}], as shown in Figure 7. The resulting parameters of the equivalent pipe are obtained as

*L*

_{eq}= 195.23 m,

*D*

_{eq}= 132 mm and

*C*

_{eq}=

*C*

_{HW}= 202.35, respectively. The main network can now be replaced by its Thevenin equivalent network, having a reservoir with a head of 483.85 m connected to an equivalent pipe with its equivalent length, diameter and Hazen–Williams coefficient.

Demand . | Head . | Head loss . | Demand . | Head . | Head loss . |
---|---|---|---|---|---|

q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} | q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} |

0.00 | 483.85 | 0.00 | 0.12 | 442.39 | 41.46 |

0.02 | 481.83 | 2.02 | 0.14 | 428.88 | 54.97 |

0.04 | 477.79 | 6.06 | 0.16 | 413.56 | 70.29 |

0.06 | 471.79 | 12.06 | 0.18 | 396.42 | 87.43 |

0.08 | 463.86 | 19.99 | 0.20 | 377.48 | 106.37 |

0.10 | 454.05 | 29.80 | 0.22 | 356.75 | 127.10 |

Demand . | Head . | Head loss . | Demand . | Head . | Head loss . |
---|---|---|---|---|---|

q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} | q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} |

0.00 | 483.85 | 0.00 | 0.12 | 442.39 | 41.46 |

0.02 | 481.83 | 2.02 | 0.14 | 428.88 | 54.97 |

0.04 | 477.79 | 6.06 | 0.16 | 413.56 | 70.29 |

0.06 | 471.79 | 12.06 | 0.18 | 396.42 | 87.43 |

0.08 | 463.86 | 19.99 | 0.20 | 377.48 | 106.37 |

0.10 | 454.05 | 29.80 | 0.22 | 356.75 | 127.10 |

*Q*| ≈ 0 m

^{3}/s) as those connected to the main network. It is clear from the comparison that the reduced network can be used to simulate the sub-network for different demand scenarios without sacrificing accuracy. It can, therefore, be concluded that the non-linear Thevenin theorem for electrical circuits can be used to reduce a hydraulic network for focussed analysis of a sub-network in the context of network expansion without compromising accuracy. Although different pipe layouts for the sub-network are not considered here, the same can be demonstrated since the DN-plot for the main network remains unchanged in the given range of sub-network demands. It is found that EPANET takes almost the comparable computational time to simulate both original (Figure 5) and reduced (Figure 8) networks with no noticeable advantage in computation time. This is because the network in Figure 5 is very small and the computation time is also very small.

Demand (m^{3}/s) at the nodes. | Simulated pressure head (m) at the nodes . | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

13 . | 14 . | 15 . | 16 . | 13 . | 14 . | 15 . | 16 . | ||||

. | . | . | . | Org. . | Eqv. . | Org. . | Eqv. . | Org. . | Eqv. . | Org. . | Eqv. . |

0.00 | 0.00 | 0.00 | 0.02 | 182 | 182 | 182 | 182 | 182 | 181 | 181 | 182 |

0.00 | 0.00 | 0.02 | 0.02 | 178 | 178 | 177 | 178 | 177 | 177 | 177 | 177 |

0.00 | 0.02 | 0.02 | 0.02 | 172 | 172 | 170 | 170 | 170 | 170 | 170 | 170 |

0.02 | 0.02 | 0.02 | 0.02 | 164 | 164 | 162 | 162 | 162 | 162 | 162 | 161 |

0.02 | 0.02 | 0.02 | 0.04 | 154 | 154 | 151 | 151 | 150 | 150 | 150 | 150 |

0.02 | 0.02 | 0.04 | 0.04 | 142 | 141 | 138 | 137 | 137 | 136 | 137 | 136 |

0.02 | 0.04 | 0.04 | 0.04 | 129 | 128 | 123 | 121 | 122 | 121 | 121 | 120 |

0.04 | 0.04 | 0.04 | 0.04 | 114 | 112 | 107 | 106 | 107 | 105 | 106 | 105 |

0.04 | 0.04 | 0.04 | 0.06 | 96 | 94 | 88 | 86 | 88 | 86 | 86 | 84 |

0.04 | 0.04 | 0.06 | 0.06 | 77 | 75 | 67 | 65 | 66 | 63 | 64 | 62 |

Demand (m^{3}/s) at the nodes. | Simulated pressure head (m) at the nodes . | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

13 . | 14 . | 15 . | 16 . | 13 . | 14 . | 15 . | 16 . | ||||

. | . | . | . | Org. . | Eqv. . | Org. . | Eqv. . | Org. . | Eqv. . | Org. . | Eqv. . |

0.00 | 0.00 | 0.00 | 0.02 | 182 | 182 | 182 | 182 | 182 | 181 | 181 | 182 |

0.00 | 0.00 | 0.02 | 0.02 | 178 | 178 | 177 | 178 | 177 | 177 | 177 | 177 |

0.00 | 0.02 | 0.02 | 0.02 | 172 | 172 | 170 | 170 | 170 | 170 | 170 | 170 |

0.02 | 0.02 | 0.02 | 0.02 | 164 | 164 | 162 | 162 | 162 | 162 | 162 | 161 |

0.02 | 0.02 | 0.02 | 0.04 | 154 | 154 | 151 | 151 | 150 | 150 | 150 | 150 |

0.02 | 0.02 | 0.04 | 0.04 | 142 | 141 | 138 | 137 | 137 | 136 | 137 | 136 |

0.02 | 0.04 | 0.04 | 0.04 | 129 | 128 | 123 | 121 | 122 | 121 | 121 | 120 |

0.04 | 0.04 | 0.04 | 0.04 | 114 | 112 | 107 | 106 | 107 | 105 | 106 | 105 |

0.04 | 0.04 | 0.04 | 0.06 | 96 | 94 | 88 | 86 | 88 | 86 | 86 | 84 |

0.04 | 0.04 | 0.06 | 0.06 | 77 | 75 | 67 | 65 | 66 | 63 | 64 | 62 |

#### Application of the proposed methodology on Network 2

*et al.*2019) as shown in Figure 9. This main network (dashed part) has 848 nodes, 8 reservoirs and 934 pipes. The pipe and reservoir data and other details of the network are obtained from the cited reference. A hypothetical sub-network (dotted part) with 40 nodes and 50 pipes is connected to the main network at N713 as shown in Figure 9. This network is tested for two head-loss formulae, namely, the Hazen–Williams and the generalized head-loss equation, , to understand the effect on accuracy and flexibility. In addition, the network is analysed for different sub-network demand scenarios. For the sub-network, length, diameter and

*C*

_{HW}of all the pipes are taken as 300 m, 100 mm and 120, respectively, for demonstration purposes.

The main network is simulated in EPANET for the open network head at Node N713 as described before and the equivalent reservoir head is found to be *H*_{oc} = 238.19 m. Now, the demand at the same Node N713 is varied from 0 to 0.013 m^{3}/s. The resulting heads at the same connecting node are noted and the differences with *H*_{oc} are also calculated as given in Table 3. The maximum demand here is fixed based on the constraint that the pressure head at any node cannot be negative. However, one can apply any constraints such as a minimum head and maximum flow velocity simultaneously while estimating the maximum supply capacity of the main network to the sub-network without compromising its operating condition.

Demand . | Head . | Head loss . | Demand . | Head . | Head loss . |
---|---|---|---|---|---|

q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} | q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} |

0.000 | 238.19 | 0.00 | 0.007 | 226.93 | 11.26 |

0.001 | 237.01 | 1.18 | 0.008 | 224.85 | 13.34 |

0.002 | 235.66 | 2.53 | 0.009 | 222.69 | 15.50 |

0.003 | 234.17 | 4.02 | 0.010 | 220.45 | 17.74 |

0.004 | 232.54 | 5.65 | 0.011 | 218.15 | 20.04 |

0.005 | 230.79 | 7.40 | 0.012 | 215.82 | 22.37 |

0.006 | 228.91 | 9.28 | 0.013 | 213.46 | 24.73 |

Demand . | Head . | Head loss . | Demand . | Head . | Head loss . |
---|---|---|---|---|---|

q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} | q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} |

0.000 | 238.19 | 0.00 | 0.007 | 226.93 | 11.26 |

0.001 | 237.01 | 1.18 | 0.008 | 224.85 | 13.34 |

0.002 | 235.66 | 2.53 | 0.009 | 222.69 | 15.50 |

0.003 | 234.17 | 4.02 | 0.010 | 220.45 | 17.74 |

0.004 | 232.54 | 5.65 | 0.011 | 218.15 | 20.04 |

0.005 | 230.79 | 7.40 | 0.012 | 215.82 | 22.37 |

0.006 | 228.91 | 9.28 | 0.013 | 213.46 | 24.73 |

*L*

_{eq}= 260.32 m,

*D*

_{eq}= 80 mm and

*C*

_{eq}

*=*

*C*

_{HW}= 120, respectively. Similarly, the resulting parameters of the equivalent pipe for the generalized head-loss formula are obtained as

*K*

_{eq}= 5,758 and

*n*

_{eq}= 1.26.

It is evident from Figure 10 that a better fit is achieved using the generalized head-loss formula. Here, the purpose is to analyse the sub-network for different demand scenarios, *q*_{i}. For the exponential formula, the equivalent reservoir head for each demand in the sub-network is different. One needs to change the equivalent reservoir head accordingly for every change in the sub-network demand. So, this model may not be useful when implemented in EPANET.

*P*

_{s}= 238.19 m and the other for any flow, for example,

*Q*

_{t}= 13 lps, which yields the static pressure head,

*P*

_{s}= 213.46 m are chosen as hydrant flow test data. The residual pressure head (

*P*

_{r}) for

*Q*

_{r}= 6 lps is calculated using Equation (7) as

*P*

_{r}= 232.28 m. The pump characteristics curve is now generated using the three data points. The reservoir–pump system is now directly connected to the sub-network as shown in Figure 11(b) for further analysis.

Now, the total demand in the sub-network is varied from 1 × 10^{−3} to 13 × 10^{−3} m^{3}/s for generating demand scenarios. The total sub-network demand is equally distributed among all the 40 nodes for demonstration purposes. The reduced networks shown in Figure 11 are now simulated in EPANET. For comparison, full network (Figure 9) is simulated in EPANET, and the results are considered as reference solutions. The ﬂow solutions of the reduced networks are found to be exactly in agreement (|ΔQ| ∼ 0 × 10^{−3} m^{3}/s) with those of the reference solutions. The Node N40, with an elevation of 160 m, shows the highest pressure variation in all the simulations. Hence, the pressure head at N40 is compared in Table 4. From Table 4, it is evident that the Thevenin equivalent network derived using the Hazen–Williams head-loss formula gives an error of less than 5%. In contrast, the reservoir–pump method gives an error of less than 7%. The reduced network shown in Figure 11(a) is also implemented in QucsStudio (Balireddy *et al.* 2022) using the values of *K*_{eq} and *n*_{eq}. The head values at N40 of the sub-network for variable demands are also compared in Table 4. The equivalent network derived by fitting the DN-plot with the generalized head-loss formula gives more accurate results with a maximum error less than 1.5%.

Total demand . | Node demand . | Pressure at Node N40 of the sub-network . | |||
---|---|---|---|---|---|

Original network . | Equivalent network . | ||||

(×10^{−3}m^{3}/s)
. | Thevenin generalized . | Hazen–Williams . | Reservoir–pump model . | ||

2 | 0.05 | 74.8 | 75.0 | 76.5 | 76.6 |

4 | 0.10 | 69.5 | 69.7 | 72.0 | 72.4 |

6 | 0.15 | 62.5 | 62.6 | 65.1 | 65.9 |

8 | 0.20 | 53.9 | 54.1 | 56.0 | 57.2 |

10 | 0.25 | 43.9 | 44.3 | 44.6 | 46.4 |

12 | 0.30 | 32.6 | 33.0 | 31.1 | 33.7 |

Total demand . | Node demand . | Pressure at Node N40 of the sub-network . | |||
---|---|---|---|---|---|

Original network . | Equivalent network . | ||||

(×10^{−3}m^{3}/s)
. | Thevenin generalized . | Hazen–Williams . | Reservoir–pump model . | ||

2 | 0.05 | 74.8 | 75.0 | 76.5 | 76.6 |

4 | 0.10 | 69.5 | 69.7 | 72.0 | 72.4 |

6 | 0.15 | 62.5 | 62.6 | 65.1 | 65.9 |

8 | 0.20 | 53.9 | 54.1 | 56.0 | 57.2 |

10 | 0.25 | 43.9 | 44.3 | 44.6 | 46.4 |

12 | 0.30 | 32.6 | 33.0 | 31.1 | 33.7 |

Both the networks (original and equivalent) are simulated in EPANET 1,000 times because one simulation is not good enough for comparison of simulation times. The equivalent networks in Figure 11 are observed to be four times faster than the original network in Figure 9. In Table 5, the proposed network reduction method is compared with different WDN reduction methods in terms of the number of hydraulic elements in the reduced network (Tao *et al.* 2009; Perelman & Ostfeld 2011; Jiang *et al.* 2013; Di Nardo *et al.* 2018). It should be noted that all the existing network reduction methods essentially bring down the number of hydraulic elements of the entire network. However, unlike other network reduction methods, the proposed methodology results in only two elements in the reduced network irrespective of its original size; hence, a reduction in computation time is obvious for a large WDN. In the case of a large sub-network, any efficient network reduction method can also be applied to the sub-network and then it can be connected to the Thevenin equivalent of the main network. Such an approach can further bring down the number of hydraulic elements drastically leading to a significant reduction in overall computation time.

Method . | Case study: number of elements in . | % Reduction in number of elements . | |
---|---|---|---|

main network . | reduced network . | ||

Skeletonization (Tao et al. 2009). | 7,378 | 1,247 | 83 |

Gaussian elimination (Martinez Alzamora et al. 2014). | 3,733 | 1,477 | 60 |

Branch collapsing, series and parallel pipe merging (Jiang et al. 2013) | 48,660 | 9,901 | 79 |

Identiﬁcation of a primary network (Di Nardo et al. 2018) | 284 | 195 | 31 |

Non-linear Thevenin theorem (present work) and reservoir–pump model | 942 (variable) | 2 (fixed) | 99 |

Method . | Case study: number of elements in . | % Reduction in number of elements . | |
---|---|---|---|

main network . | reduced network . | ||

Skeletonization (Tao et al. 2009). | 7,378 | 1,247 | 83 |

Gaussian elimination (Martinez Alzamora et al. 2014). | 3,733 | 1,477 | 60 |

Branch collapsing, series and parallel pipe merging (Jiang et al. 2013) | 48,660 | 9,901 | 79 |

Identiﬁcation of a primary network (Di Nardo et al. 2018) | 284 | 195 | 31 |

Non-linear Thevenin theorem (present work) and reservoir–pump model | 942 (variable) | 2 (fixed) | 99 |

From the investigated test cases presented previously, it is evident that the proposed network reduction method can provide accurate results along with the computational advantage.

### Evaluation of the Thevenin equivalent network for demand variation in the main network and maximum power transfer to the sub-network

*et al.*(2008). The network consists of 71 pipes, 46 nodes, 1 reservoir and 45 demand nodes. The details of the network are obtained from the cited reference and the reservoir head is considered as 20 m. However, in the present study, the pipe P45 (represented with a dashed line in Figure 12) between N36 and N34 is removed to demonstrate the application of the single-port non-linear Thevenin theorem. The network is divided into main (dashed part) and sub-networks (dotted part) as marked in Figure 12. The main upstream network has 63 pipes, 1 reservoir and 40 demand nodes, and the sub-network has 6 pipes and 5 demand nodes (Figure 12).

#### Thevenin equivalent network of Tiruppur WDN

*H*

_{oc}= 18.25 m. Now the demand at N15 is varied from 0 to 28 × 10

^{−3}m

^{3}/s with a step size of 4 × 10

^{−3}m

^{3}/s. For all these demands, the heads at all the nodes of the main network are compared with the minimum required head, for example, in this case,

*H*

_{min}is set at 13 m. In this step, the operating constraints of the main network are checked. From Figure 13(a), it is evident that N15 is first going below 13 m. So, this is the most sensitive node in the main network for variations in the sub-network demand. For finding the maximum possible sub-network demand (

*Q*

_{max}), the variation in the head at N15 for increasing connecting node demand and

*H*

_{min}line are plotted as shown in Figure 13(b). The demand at which the

*H*

_{min}line touches the head variation curve is marked as the

*Q*

_{max}and is found to be 26 × 10

^{−3}m

^{3}/s. Therefore, the main network can meet a maximum sub-network demand of 26 × 10

^{−3}m

^{3}/s through the connecting node without affecting its operation.

_{oc}are calculated for the corresponding head losses as shown in Table 6. The DN-plot of the main network at N15 is obtained as shown in Figure 14. It is attempted to fit this curve with the Hazen–Williams head-loss formula. The resulting values of the equivalent pipe length, diameter and the Hazen–Williams coefficient are obtained as

*L*

_{eq}= 70.53 m,

*D*

_{eq}= 106.75 mm and

*C*

_{eq}

*=*

*C*

_{HW}= 142, respectively. The DN-plot is also fitted using the generalized head-loss formula. The resulting values of the equivalent pipe resistance and exponent are obtained as

*K*

_{eq}= 209.22 and

*n*

_{eq}

*=*1.125, respectively.

Demand . | Head . | Head loss . | Demand . | Head . | Head loss . |
---|---|---|---|---|---|

q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} | q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} |

0.000 | 18.25 | 0 | 0.014 | 16.52 | 1.73 |

0.002 | 18.04 | 0.21 | 0.016 | 16.22 | 2.03 |

0.004 | 17.82 | 0.43 | 0.018 | 15.91 | 2.34 |

0.006 | 17.59 | 0.66 | 0.02 | 15.58 | 2.67 |

0.008 | 17.34 | 0.91 | 0.022 | 14.24 | 3.01 |

0.001 | 17.08 | 1.17 | 0.024 | 14.88 | 3.37 |

0.012 | 16.81 | 1.44 | 0.026 | 14.52 | 3.73 |

Demand . | Head . | Head loss . | Demand . | Head . | Head loss . |
---|---|---|---|---|---|

q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} | q (m_{i}^{3}/s)
. | h (m)
. _{i} | (H_{oc} − h) (m)
. _{i} |

0.000 | 18.25 | 0 | 0.014 | 16.52 | 1.73 |

0.002 | 18.04 | 0.21 | 0.016 | 16.22 | 2.03 |

0.004 | 17.82 | 0.43 | 0.018 | 15.91 | 2.34 |

0.006 | 17.59 | 0.66 | 0.02 | 15.58 | 2.67 |

0.008 | 17.34 | 0.91 | 0.022 | 14.24 | 3.01 |

0.001 | 17.08 | 1.17 | 0.024 | 14.88 | 3.37 |

0.012 | 16.81 | 1.44 | 0.026 | 14.52 | 3.73 |

*K*

_{eq}and

*n*

_{eq}. The head at N33 of the sub-network for variable demands is also compared in Table 7. The equivalent network derived by fitting the DN-plot using the generalized head-loss formula is giving exact results. The ﬂows in the sub-network pipes when it is connected to the Thevenin equivalent network are almost the same (|Δ

*Q*| ≈ 0 m

^{3}/s) as those when it is connected to the main network. To compare the simulation times, both the networks (the original full network and the reduced equivalent network) are simulated in EPANET 1,000 times. It is observed that the equivalent network derived using the proposed reduction method can be simulated two times faster than the original network.

Sub-network demand (×10^{−3}m^{3}/s)
. | Demand discharge at different nodes (×10^{−3}m^{3}/s). | Pressure at N33 . | ||||||
---|---|---|---|---|---|---|---|---|

. | Equivalent network using head loss formula . | |||||||

31 . | 32 . | 33 . | 34 . | 35 . | Original network . | Hazen–Williams . | Generalized . | |

0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 18.3 | 18.3 | 18.3 |

2.36 | 0.55 | 0.26 | 0.40 | 0.82 | 0.33 | 17.7 | 17.9 | 17.7 |

3.36 | 1.55 | 0.26 | 0.40 | 0.82 | 0.33 | 17.5 | 17.7 | 17.5 |

4.36 | 2.55 | 0.26 | 0.40 | 0.82 | 0.33 | 17.2 | 17.5 | 17.2 |

5.36 | 3.55 | 0.26 | 0.40 | 0.82 | 0.33 | 16.9 | 17.2 | 16.9 |

6.36 | 4.55 | 0.26 | 0.40 | 0.82 | 0.33 | 16.6 | 16.9 | 16.6 |

7.36 | 4.55 | 1.26 | 0.40 | 0.82 | 0.33 | 15.9 | 16.3 | 15.9 |

8.36 | 4.55 | 1.26 | 1.40 | 0.82 | 0.33 | 15.1 | 15.5 | 15.1 |

9.36 | 4.55 | 1.26 | 2.40 | 0.82 | 0.33 | 14.1 | 14.5 | 14.1 |

10.36 | 4.55 | 1.26 | 2.40 | 0.82 | 1.33 | 13.4 | 13.8 | 13.4 |

11.36 | 4.55 | 1.26 | 2.40 | 0.82 | 2.33 | 12.7 | 12.8 | 12.7 |

12.36 | 4.55 | 1.26 | 2.40 | 1.82 | 2.33 | 11.6 | 11.9 | 11.6 |

13.36 | 4.55 | 1.26 | 3.40 | 1.82 | 2.33 | 10.2 | 10.5 | 10.2 |

14.36 | 4.55 | 2.26 | 3.40 | 1.82 | 2.33 | 8.8 | 9.0 | 8.8 |

15.36 | 5.55 | 2.26 | 3.40 | 1.82 | 2.33 | 8.2 | 8.3 | 8.2 |

16.36 | 6.55 | 2.26 | 3.40 | 1.82 | 2.33 | 7.5 | 7.5 | 7.5 |

Sub-network demand (×10^{−3}m^{3}/s)
. | Demand discharge at different nodes (×10^{−3}m^{3}/s). | Pressure at N33 . | ||||||
---|---|---|---|---|---|---|---|---|

. | Equivalent network using head loss formula . | |||||||

31 . | 32 . | 33 . | 34 . | 35 . | Original network . | Hazen–Williams . | Generalized . | |

0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 18.3 | 18.3 | 18.3 |

2.36 | 0.55 | 0.26 | 0.40 | 0.82 | 0.33 | 17.7 | 17.9 | 17.7 |

3.36 | 1.55 | 0.26 | 0.40 | 0.82 | 0.33 | 17.5 | 17.7 | 17.5 |

4.36 | 2.55 | 0.26 | 0.40 | 0.82 | 0.33 | 17.2 | 17.5 | 17.2 |

5.36 | 3.55 | 0.26 | 0.40 | 0.82 | 0.33 | 16.9 | 17.2 | 16.9 |

6.36 | 4.55 | 0.26 | 0.40 | 0.82 | 0.33 | 16.6 | 16.9 | 16.6 |

7.36 | 4.55 | 1.26 | 0.40 | 0.82 | 0.33 | 15.9 | 16.3 | 15.9 |

8.36 | 4.55 | 1.26 | 1.40 | 0.82 | 0.33 | 15.1 | 15.5 | 15.1 |

9.36 | 4.55 | 1.26 | 2.40 | 0.82 | 0.33 | 14.1 | 14.5 | 14.1 |

10.36 | 4.55 | 1.26 | 2.40 | 0.82 | 1.33 | 13.4 | 13.8 | 13.4 |

11.36 | 4.55 | 1.26 | 2.40 | 0.82 | 2.33 | 12.7 | 12.8 | 12.7 |

12.36 | 4.55 | 1.26 | 2.40 | 1.82 | 2.33 | 11.6 | 11.9 | 11.6 |

13.36 | 4.55 | 1.26 | 3.40 | 1.82 | 2.33 | 10.2 | 10.5 | 10.2 |

14.36 | 4.55 | 2.26 | 3.40 | 1.82 | 2.33 | 8.8 | 9.0 | 8.8 |

15.36 | 5.55 | 2.26 | 3.40 | 1.82 | 2.33 | 8.2 | 8.3 | 8.2 |

16.36 | 6.55 | 2.26 | 3.40 | 1.82 | 2.33 | 7.5 | 7.5 | 7.5 |

#### Evaluation of the derived Thevenin equivalent network for main network demand variation

*H*

_{L}=

*KQ*. The Thevenin equivalent network parameters for different cases are also noted in Table 8.

^{n}Case . | Demand at each node . | Open-circuit head . | K_{eq}
. | n_{eq}
. |
---|---|---|---|---|

Case 1 | 80% of the base demand | 18.84 | 0.057 | 1.241 |

Case 2 | 90% of the base demand | 18.56 | 0.066 | 1.218 |

Case 3 | Base demand | 18.25 | 0.074 | 1.201 |

Case 4 | 110% of the base demand | 17.91 | 0.082 | 1.185 |

Case 5 | 120% of the base demand | 17.55 | 0.090 | 1.171 |

Case . | Demand at each node . | Open-circuit head . | K_{eq}
. | n_{eq}
. |
---|---|---|---|---|

Case 1 | 80% of the base demand | 18.84 | 0.057 | 1.241 |

Case 2 | 90% of the base demand | 18.56 | 0.066 | 1.218 |

Case 3 | Base demand | 18.25 | 0.074 | 1.201 |

Case 4 | 110% of the base demand | 17.91 | 0.082 | 1.185 |

Case 5 | 120% of the base demand | 17.55 | 0.090 | 1.171 |

*K*and

*n*values for different main network demands are plotted in Figure 17. It is evident from Figure 17 that

*K*and

*n*values are almost constant although there is a variation in the main network demands. The reason is that

*K*and

*n*values weakly depend on the demand variation (in the range of ±20% variation of the base demand as in Case 3) and strongly depend on the network characteristics such as topology, pipe length, diameter and roughness coefficient.

Again, to test the performance of the derived equivalent networks, the sub-network demand is also varied from 0 to 20 lps step-by-step and head values at the connecting node are measured. The obtained head values at the connecting node for all the cases are compared with the results obtained with that of Case 3 and it is observed that the maximum error is <7%. Since *K* and *n* values are almost invariant as shown in Figure 17, it can be stated that the error contributed by the equivalent pipe is negligible. Therefore, the primary source of the error lies in the equivalent reservoir head. For this, the test is repeated by keeping the equivalent pipe parameters to those of the base case (i.e., Case 3) and varying the equivalent reservoir head as in Table 8. It is observed that the maximum error is reduced to <2.5%.

From the abovementioned analysis, we can conclude that the derived equivalent network using the non-linear Thevenin theorem can handle variation in main network demand up to a significant level. Only, an equivalent reservoir head needs to be estimated for any change in the main network demand. However, when there is a (significant) change in the topology of the main network, it is needed to derive the equivalent network once again as the driving-node head-loss characteristics change.

#### Determination of the diameter of the connecting pipe for maximum power transfer

The Thevenin equivalent network of the main network is connected with a pipe that supplies demand from the main network to the sub-network as shown in Figure 4. The equivalent reservoir head is determined as *H*_{oc} = 18.25. The equivalent pipe resistance of the main network *K* = (10.67×*L*_{eq})/(*C*_{eq}^{1.852}×*D*_{eq}^{4.87}) = (10.67×70.53)/(142^{1.852}×0.10675^{4.87}) = 4,191.3 is calculated using the equivalent pipe parameters obtained in the previous section. Now, the diameter of the connecting pipe is varied from 20 to 300 mm and the corresponding *K*_{c} and *Q*_{Pmax} for maximum power transfer are calculated using Equation (11) derived for maximum power transfer are shown in Table 9.

Diameter (D_{c})
. | Length (L_{c})
. | C_{HW}
. | K_{c}
. | Q_{Pmax}
. |
---|---|---|---|---|

0.02 | 85 | 120 | 24,039,398.60 | 0.0030 |

0.04 | 85 | 120 | 822,067.58 | 0.0017 |

0.06 | 85 | 120 | 114,115.09 | 0.0050 |

0.08 | 85 | 120 | 28,111.98 | 0.0100 |

0.10 | 85 | 120 | 9,482.87 | 0.0158 |

0.12 | 85 | 120 | 3,902.36 | 0.0209 |

0.14 | 85 | 120 | 1,842.03 | 0.0245 |

0.16 | 85 | 120 | 961.34 | 0.0266 |

0.18 | 85 | 120 | 541.70 | 0.0278 |

0.20 | 85 | 120 | 324.28 | 0.0285 |

0.22 | 85 | 120 | 203.86 | 0.0289 |

0.24 | 85 | 120 | 133.86 | 0.0291 |

0.26 | 85 | 120 | 90.37 | 0.0293 |

0.28 | 85 | 120 | 62.99 | 0.0294 |

Diameter (D_{c})
. | Length (L_{c})
. | C_{HW}
. | K_{c}
. | Q_{Pmax}
. |
---|---|---|---|---|

0.02 | 85 | 120 | 24,039,398.60 | 0.0030 |

0.04 | 85 | 120 | 822,067.58 | 0.0017 |

0.06 | 85 | 120 | 114,115.09 | 0.0050 |

0.08 | 85 | 120 | 28,111.98 | 0.0100 |

0.10 | 85 | 120 | 9,482.87 | 0.0158 |

0.12 | 85 | 120 | 3,902.36 | 0.0209 |

0.14 | 85 | 120 | 1,842.03 | 0.0245 |

0.16 | 85 | 120 | 961.34 | 0.0266 |

0.18 | 85 | 120 | 541.70 | 0.0278 |

0.20 | 85 | 120 | 324.28 | 0.0285 |

0.22 | 85 | 120 | 203.86 | 0.0289 |

0.24 | 85 | 120 | 133.86 | 0.0291 |

0.26 | 85 | 120 | 90.37 | 0.0293 |

0.28 | 85 | 120 | 62.99 | 0.0294 |

As the diameter of the connecting pipe increases the demand for maximum power transfer increases but saturates at a certain value. This implies that after a certain diameter, the maximum power cannot be transferred beyond a certain value (e.g., after 0.2 m diameter in this case, there is not much variation in the value of the sub-network demand for maximum power transfer).

If the connecting pipe parameters are known, one can estimate the demand of the sub-network for maximum power transfer (e.g., if the diameter of the connecting pipe is 0.15 m in this case, the sub-network demand for the maximum power transfer is 0.025 m

^{3}/s).On the other hand, if the sub-network demand is known, one can find the economical diameter of the connecting pipe for maximum power transfer (e.g., if the sub-network demand is fixed to 0.0158 m

^{3}/s in this case, the economical diameter of the connecting pipe is 0.1 m).If we choose a diameter less than 0.1 m, losses in the connecting pipe will be higher and the transfer of maximum power to the sub-network may not be possible. So, the operational cost of the system will increase. If we choose a diameter greater than 0.1 m, we can decrease the head loss in the connecting pipe and we can transfer power more than the maximum power corresponding to 0.1 m. But the capital cost of the WDN will increase as the diameter of the connecting pipe and other corresponding costs increases. Therefore, the economical diameter of the sub-network demand is fixed to 0.1 m for

*Q*_{Pmax}= 0.0158 m^{3}/s.

*C*

_{HW}= 120 and diameter = 0.1 m. For these values, the value of sub-network demand for the maximum power transfer obtained from the proposed methodology is

*Q*

_{Pmax}= 0.0158 m

^{3}/s (Table 9). Now to verify this, the demand at the connecting node is varied and the corresponding head values at that node are noted. Power transferred from the main network to the sub-network through the connecting pipe is calculated using

*P*

*=*

*ρgQ*

_{s}

*H*

_{s}. A graph is plotted as shown in Figure 19. From Figure 19, it is evident that the maximum power is delivered at the calculated

*Q*

_{Pmax}= 0.0158 m

^{3}/s. So, the Thevenin equivalent network is useful in designing the interlinking pipe to ensure a maximum power transfer from the main network to a sub-network.

A sample cost analysis is presented here to explain the concept of economical connecting pipe diameter. Three pipes with internal diameter 90, 100 and 110 mm with a thickness of 6 mm are considered as shown in Table 10. The length of each connecting pipe is 85 m and sub-network demand is 16 lps. Cast iron with a density of 7,200 kg/m^{3} is chosen as the pipe material. The cost of cast iron is taken as ₹60/kg. By using these data, the cost of each individual pipe is calculated and noted in Table 10. When a pipe with a diameter of 100 mm is chosen as a connecting pipe, the minimum head value in the sub-network is 7.53 m at N33. Similarly, when 110 mm is chosen as connecting pipe diameter, the head value is 9.26 m at N33 of the sub-network. These two pipe diameters satisfy the minimum required head value condition which is 7 m at all nodes. Whereas, a 90 mm connecting pipe diameter gives only a 4.39 m head value at N33. To satisfy the minimum head condition, a pump that can deliver 2.61 m at 16.36 lps demand needs to be connected in series with the 90 mm pipe. The required pump capacity is calculated as *P**=**ρgQ*_{pump}*H*_{pump} = 1,000*9.81*0.01636*2.61 = 410.10 W. A pump with a capacity near this value is chosen; its cost and running cost (considering ₹5/unit as the electricity bill) per day are noted in Table 10. All the calculations are carried out with respect to the Indian currency, Rupee (₹) and prevailing market cost of pipes.

Pipe diameter | 90 mm | 100 mm | 110 mm |

Pipe cost | ₹32,169 | ₹35,628 | ₹39,087 |

Pump cost | ₹3,800 | – | – |

Running cost | ₹49/day | – | – |

Total cost | ₹35,969 + 49/day | ₹35,628 | ₹39,087 |

Pipe diameter | 90 mm | 100 mm | 110 mm |

Pipe cost | ₹32,169 | ₹35,628 | ₹39,087 |

Pump cost | ₹3,800 | – | – |

Running cost | ₹49/day | – | – |

Total cost | ₹35,969 + 49/day | ₹35,628 | ₹39,087 |

The total cost required for different connecting pipe diameters is compared in Table 10. It is evident that a 100 mm connecting pipe diameter is the most economical to supply the demand of 16 lps to the sub-network. It is also noted that 100 mm diameter satisfies the condition for maximum power transfer from Figure 18. Choosing a larger diameter (110 mm) than an economical diameter may give high head values at the sub-network node, but requires a high capital cost. On the other hand, choosing a smaller diameter (90 mm) requires an additional pump to compensate for the head loss causing an additional cost (capital and running costs of the pump). Therefore, the proposed methodology plays an important role in determining an appropriate connecting pipe diameter.

## SCOPE AND LIMITATIONS

The proposed Thevenin equivalent network method reduces the main network to only two hydraulic elements. The Thevenin equivalent and reservoir–pump methods are found to be very similar. So, it is computationally efficient irrespective of the size of the main network. It is demonstrated that the proposed reduction method gives negligible errors for large variations in the sub-network demand. One can estimate the connecting pipe diameter for transferring maximum power from the main network to the sub-network. The authors call this diameter an economical connecting pipe diameter. The derived equivalent network can handle the variation in the main network up to a certain level, but it cannot handle the topological variation in the main network. The proposed methodology is suitable only for a demand-driven approach (DDA). However, the methodology needs modification in order to make it applicable in the case of a pressure-driven demand approach (PDA). One possible solution could be the use of an external source (i.e., a reservoir) method to obtain the driving-node head-loss characteristics. In the case of PDA, an external reservoir with a variable head needs to be attached to the connecting node without any modification of the main network. By varying the external reservoir head, the flow (inward or outward depending on the external reservoir head) and the head at the connecting node need to be noted. Thus, the driving-node head-loss characteristics curve (i.e., DN-plot) can be obtained by plotting the head versus flow at the connecting node. The parameters of the equivalent network are then derived from the driving-node head-loss characteristics. The remaining procedure remains unaltered. However, this process is not experimented at our end yet. In our future work, we will try to implement the concept of the Thevenin equivalent network for PDA. The presented method is also not applicable when there are multiple connections between the main network and sub-networks. This problem may be addressed by using the multi-port linear Thevenin theorem.

## CONCLUSIONS

An accurate and computationally efficient network reduction method is presented in this study for a focussed analysis of a sub-network to be connected with a large existing pipe network. The non-linear Thevenin theorem for electrical circuits is applied to reduce the existing pipe network system. Unlike the other network reduction techniques, the proposed methodology allows us to replace a large hydraulic network with a simple Thevenin equivalent network consisting of a single reservoir connected to a single pipe. The parameters of the simplified equivalent network are extracted by fitting the driving-node head-loss characteristics (DN-plot) of the connecting node. The equivalent network is then connected to a sub-network planned for supplying water to a newly developed area. The reservoir–pump model which is used by practising hydraulic engineers for modelling existing systems is discussed and compared with the proposed Thevenin model. This network is shown to be simulated accurately with much less computational effort. The equivalent network is also found to maintain reasonable accuracy for limited demand variation in the main network. Such a network reduction scheme is useful when the sub-network needs to be simulated repeatedly for its optimal design. The proposed methodology is demonstrated using the freely downloadable pipe flow simulation package EPANET. For validation, different networks are considered for simulations with variable demands of the sub-network. The results of the reduced networks are found to be in agreement with those of the original networks and, additionally, the results are obtained with a computational advantage. Since the proposed network reduction method is highly accurate and computationally efficient, it can be instrumental for rapid analysis of a pipe network system, when an existing large network is expanded to new zones. The Thevenin theorem is also found to be useful for determining the economical diameter of the connecting pipe between main and sub-networks based on the concept of maximum power transfer. The proposed network reduction method and finding economical diameter can be useful for hydraulic engineers when a large network needs to be expanded by attaching a sub-network to serve a new area.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Communities.Bentley*. Available from: https://communities.bentley.com/products/hydraulics___hydrology/w/hydraulics_and_hydrology__wiki/9293/modeling-a-connection-to-an-existing-system.

ResearchGate, DOI,10