Simplification of water distribution networks using non-linear Thevenin theorem and its application for maximum power transfer

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.

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.

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).

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.

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).

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.

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.

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.

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.

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 m³/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.

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.

Now, the total demand in the sub-network is varied from 1 × 10⁻³ to 13 × 10⁻³ m³/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 m³/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%.

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.

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.

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.

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.

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/m³ 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_pumpH_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.

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.

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.

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.

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

The authors declare there is no conflict.