Fractal-Based Approach to Simultaneous Layout Routing and Pipe Sizing of Water Supply Networks

Abstract

The process of designing water distribution networks is divided into two main stages: network layout routing and pipe sizing. However, routing and sizing are not separate tasks—the shape of the network affects the diameters of the pipes, and vice versa. This paper presents an innovative fractal-based method, which enables the simultaneous layout routing and pipe sizing of water supply networks. The developed pipe routes and diameters selected according to the method are mathematically justified; the selection considers the total length of the pipes, the number of rotation angles of the base section, the cost of the water supply system construction and the priority of water supply to individual customers. The novelty of the method lies in the possibility of carrying out the processes of routing and sizing of the network in a recursive manner by the adoption of the principles of fractal geometry and Murray’s law. The method was tested under the conditions of a synthetic settlement. The obtained results enable us to conclude that the method is universal and suitable for shaping water supply networks, while determining the pipes’ diameters, both under the conditions of a single- and multi-sided water supply source.

1. Introduction

One of the first steps in designing a water distribution network (WDN) is pipeline routing, defined as giving the water supply network a geometric shape [1]. The shape of the geometric structure depends on many factors, including the spatial arrangement of the settlement, topography, the presence of natural and artificial obstacles in pipe routing, the location of the water demand centers or the layout of communication routes [2]. Therefore, the routing of the water supply network is subject to various rules and restrictions, depending on both external factors and the type of water supply pipelines being designed. The routing process has an impact on further design steps, such as hydraulic calculations, investment costs, and subsequent operation of the water supply system. That is why developing an effective and profitable way of designing water supply networks, including the search for the optimal method of routing and sizing, is still a current issue.

Water supply network structures are generally classified in terms of topology, distinguishing networks into branched, looped, or mixed structures [1,3,4]. To find the correlations between geometrical properties and design or operational aspects of WDN, the complex network theory (CNT), graph theory (GT), and fractal geometry (FG) are applied [5,6,7,8,9]. All the above-mentioned approaches provide the tools (indicators) for a precise description of network structures, comparison of the networks [10], and application to analyses of WDNs [11], including analyses of their optimization [12] and reliability [13,14]. Advanced methods of describing and classifying WDNs are still relevant due to the fact that accurate network management also involves socioeconomic and environmental issues [15]. In the context of the routing of water supply networks, presenting networks in the form of graphs makes it possible to use least cost path algorithms (LCPA) or path of least resistance (PLR) algorithms, combined with geographic information systems (GIS) [16], to determine the connection route between water nodes. The CNT optimization approach has been used by researchers to formulate different design methodologies [5,17,18,19]. In accordance with the application of FG, routing of a WDN can be achieved by reproducing dendric (tree) structures by copying and rotating the initial section (axiom) [20,21,22,23]. However, despite many approaches observed, none of the routing methods are fully universal, mainly with regard to extensive looped networks with multiple sources. The search for new methods is still ongoing; as the benefits of the optimal geometric layout of the network may be greater than the savings obtained from the pipe sizing [24].

Sizing of water networks is a complex combinatorial problem (NP-hard problem NPH) [25] and should be treated as an optimization task. It includes the selection of diameters from a predefined pipe type series in order to find the minimum total cost while meeting hydraulic requirements [26] as well as assuming constant geometric shape and nodal demands [27]. As the shape of the network influences the dimensioning process, both tasks should be solved simultaneously. Numerous different approaches have been implemented, including mathematical programming [28,29,30,31], heuristic [32,33], metaheuristic [34,35,36,37,38,39,40,41,42,43], and multi-objective optimization approaches [44,45,46,47,48,49]. However, there is still no automated network design method under the conditions of a complex, real settlement.

Considering the premises above, it is reasonable to search for and develop new methods supporting the process of designing water supply networks, including the routing and sizing processes. The aim of this paper is to present and evaluate a novel fractal-based approach of simultaneous routing and sizing (SRS) of water network structures. The main objectives of the paper are testing the developed method in different initial conditions, the verification of the possibility of obtaining repeatable results and evaluation of the hydraulic operation parameters of the designed water supply networks.

2. Materials and Methods

The analyses and results presented in this paper are a part of a larger scope of research, the aim of which is the development and implementation of a new method of designing WDNs. The full scope of research is presented in Figure 1. After developing the rules of the method, the solution was tested in various conditions, including artificial settlements plans, quasi-real networks (virtual cities) and real condition networks. Finally, the proposed sizing method was evaluated in comparison to the genetic algorithm method. In this section of the manuscript we present the scientific background of the original method, a description and novelty of the proposed SRS method, and exemplary calculations as well as the method and results evaluation methodology under the conditions of artificial settlements created randomly for the purpose of this research.

2.1. Scientific Background of the Original Method

The SRS method [50] is a modification to the original approach proposed by Kowalski [20]. In his original method, Kowalski proved that after considering certain assumptions, branched, looped, and mixed water supply networks can be reflected and described using dendritic (tree-shaped) structures. Such structures are common in nature; for example, in the architecture of trees, river basins, water transport in plants, or the circulatory system of living organisms [51,52,53,54]. Moreover, Kowalski verified that tree-shaped WDNs can be treated as fractal sets which are difficult to describe using classical geometry because of their high irregularity [55]. Fractals are characterized by such main features as: fine structure at arbitrarily small scales, recursive procedures of construction, intricate detailed structures, self-similarity, and size not quantified by the usual measures, e.g., length [56]. In fractal geometry, a basic property of a fractal is its dimension, which describes how fractals fill space and how irregular that arrangement is [7]. Fractal dimension often refers to the so-called box counting method (Formula (1)) [56]. Its application in WDNs is wide, including reliability, connectivity, redundancy, and operational aspects [7,8,57].

$$D_b\left(F\right)=\underset{\delta \to 0}{\mathit{lim}}{\displaystyle \frac{\mathit{log}{N}_{\delta }\left(F\right)}{-\mathit{log}\delta }},$$

(1)

where F—a non-empty bounded subset of a finite-dimensional Euclidean space, $D_b\left(F\right)$—the box-counting dimension of the set F, $N_{\delta }\left(F\right)$—the smallest number of sets of diameters δ covering F.

Tree-like structures found in nature (e.g., blood vessels) are “optimized” in some way through evolutionary processes. Researchers often observe the solutions found in nature and then, by analogy, apply them to technical solutions. In the 1920s, Murray postulated one of the fundamental principles that the organization of vessels in the circulatory system of living organisms is based on minimizing the energy required to ensure adequate blood flow [58,59]. According to Murray’s theory, the process of increasing vascular composition is based on the relationship between the volumetric flow rate Q and the vessel diameter d (Formula (2)). In his works, Murray claimed that the total energy expenditure of the flow is minimized when x = 3, while the relationship between the diameters of the vessels of subsequent generations of the optimal vascular tree are given by Formula (3).

$$Q=b·d^x,$$

(2)

$$d_0^x=d_1^x+d_2^x,$$

(3)

where Q—volumetric flow, b—a certain constant, the same before and after the division of vessels, d—vessel diameter, x—positive exponent, d₀—parent vessel diameter, d₁, d₂—child vessel diameter.

Using the self-similarity properties of a geometric set representing a water supply network, based on tree structures, allows for the application of Murray’s law, which states that each branching of a base pipe leads to a reduction in the diameters of the child pipes. In the case of water supply networks, due to technical constraints, this law can be modified [20]. The diameters of the child pipes should be reduced by one value in the pipe series, relative to the base pipe. This represents a divergence from Formula (3) while retaining the very idea of Murray’s law.

2.2. Description and Novelty of the Proposed SRS Method

The main novelty of the proposed SRS method refers to the simultaneous performance of the routing and sizing process, while in the method [20] these processes were conducted independently. In accordance with the fractal-based nature of the method, the routing process involves copying and rotating (affine transformations) the base section of the water pipeline (the first link of the network directly connected to the water source) and conducting a mathematical analysis of the potential connection routes. The sizing process, depending on the iteration, involves determination of diameters in accordance with the modified Murray’s law [58,59] or selecting diameters from an adopted pipe type series. The application of the proposed method is performed in a recursive manner—the processes of routing and sizing are carried out cyclically within subsequent groups (classes) of demand nodes. Each subsequent iteration leads to a more detailed structure of the designed water supply network. Successive transformations of the base section are called bifurcations of a given iteration. The routing process is carried out by copying and rotating the base segment L_i. Creating a new segment L_i+1 is performed by copying the base segment L_i and rotating it by an angle α. The length of the segment L_i+1 is the product of the length of the base segment L_i and the length parameter a of the newly constructed section (Formula (4)). The example of routing symmetrical and asymmetrical water supply structure through successive transformations of the base section L₀ is presented in Figure 2a,b [20].

$$L_{i+1}\to \left\{\begin{array}{c}a·L_i,{\alpha }^{\prime }\\ b·L_i,{\alpha }^{″}\\ c·L_i,{\alpha }^{‴}\end{array}\right.$$

(4)

where L_i—length of the segment in i-th bifurcation, L_i+1—length of the next section, a, b, c—parameters of the length of the newly constructed sections, α′, α″, α‴—angles describing the position of the newly created sections in relation to the preceding section.

Using the Formula (4), it is also possible to create looped structures. Figure 3 shows a single-loop network. Additionally, a diagram of the hierarchical dependence of the segments forming this network is presented. Considering Figure 3a, it can be noticed that the loop cannot be closed, regardless of whether segment 5 is created from segment 2 or 4. As a solution to this problem, an additional connecting node (Figure 3b) was proposed, which allows for loop closure. This means that when creating looped structures using the tree structures procedure, it may be necessary to introduce additional connection nodes [20].

The SRS method introduces additional modifications [50] to the original approach:

• Water demand nodes are classified in a ranking considering the quantity of water demand and priority of water supply delivery.
• In the first two iterations (to the two first classes of demand nodes), the network is routed from the water source to the nodes of a given class. In subsequent iterations, the direction of routing is reversed.
• If a lower-class node is located on the water pipeline route developed as a result of previous iterations, it is assumed that the consumers represented by this node have already been supplied with water, which excludes it from further routing process.
• The application of the proposed method requires the following: (i) determining the plan of a settlement, including street grids, number, capacity, and location of water sources; (ii) determining the spatial distribution of water demand nodes; (iii) estimating the unit costs of constructing a water supply pipeline.

The connection routes created in the routing process between two given water nodes should have the characteristics of minimum paths in terms of two basic criteria: the sum of the lengths (ΣL) and the sum of the rotation angles (Σα) of the base section. While the ΣL criterion refers to finding the shortest route, the Σα criterion allows determining which of the identical minimal paths should be selected during the routing process. Additionally, a third criterion for selecting the most favorable (economically) route is proposed: the cost (ΣK) of making a given connection, in case where criteria ΣL and Σα are insufficient. This approach makes it possible to assess which path is more financially profitable. Detailed definitions of minimum routing paths are presented through Formulas (5)–(7).

$$\sum L=\mathrm{m}\mathrm{i}\mathrm{n}\left(L_1=\sum _{j=1}^p\sum _{i=1}^r{L}_{ji},\dots ,L_k=\sum _{j=1}^p\sum _{i=1}^t{L}_{ji}\right)$$

(5)

$$\sum \propto =min\left({\alpha }_1=\sum _{j=1}^p\sum _{i=1}^r\left|{\alpha }_{ji}\right|,\dots ,{\alpha }_z=\sum _{j=1}^p\sum _{i=1}^t\left|{\alpha }_{ji}\right|\right)$$

(6)

$$\sum K=min\left(K_1=K_{L1}+K_{\alpha 1},\dots ,K_w=K_{Lw}+K_{\alpha w}\right)$$

(7)

where ΣL—total length of water supply sections, k—number of alternative water supply paths to nodes of a given class, p—number of nodes of a given class, r,t—the number of bifurcations necessary to connect the base section with a given node, z—number of alternative paths satisfying Formula (2), Σα—total sum of the rotation angles of the base sections in subsequent bifurcations, ΣK—total cost of construction of a given section, w—number of alternative paths satisfying Formulas (2) and (3), $K_{Lw}$—cost of constructing a straight pipe in the w-th section, $K_{\alpha w}$—cost of constructing the pipe elbows in the w-th section.

To assess the connection route in terms of construction costs, knowledge of the diameters of individual pipelines is required. This means that the routing and sizing processes should be carried out in parallel because they determine each other. This requires a recursive procedure within the developed method. The dimensioning process is based on the application of modified Murray’s law [58,59] but is also dependent on the available range of pipe type series and the required minimum diameter approved by water utility. Initially, a smaller diameter value from the pipe type series is assigned to each subsequent branch of the network. Next, the pipe diameter values are assigned to specific dimensioning stages (last iteration—the smallest permissible value from the pipe type series). A detailed description of the routing and sizing processes in individual iterations is presented in

Table 1. The routing and sizing process divided into iterations.

Iteration	Routing	Sizing
1st and 2nd	Direction: from supply points to class I and II nodes End: when connected to all class I and II nodes If there are 2 or more water sources—establish connection between sources Analysis of minimal efficiency paths—ΣL, Σα When alternative paths exist—analysis of ΣK When equivalent routes exist—all are maintained	Continuity equation and assumed flow velocity (e.g., 1 m/s) for determining the main water pipe diameter If 2 or more water sources—establish percentage share between sources Diameter sizing—via modified Murphy’s law When identical routes—larger diameter is used
3rd and further	Direction: from class III nodes and further to the network routed in 1st and 2nd iteration End: when connected to all nodes of a given class No minimal efficiency paths analysis—all pipes from the bifurcation that provide a connection to the demand node are maintained	Each iteration is assigned with specific diam-eter: Last iteration—smallest diameter (e.g., DN80) Penultimate iteration—second smallest di-ameter in the type series (e.g., DN100) The third last iteration—next smallest diam-eter in the type series (e.g., DN125)

, while the block diagram of the consecutive steps in developed method is shown in Figure 4. In the presented diagram, the occurrence of double loops can be noticed—these are the interactions between the routing and sizing processes in subsequent iterations. The application of the method for networks with single and multiple water sources differs in one step: establishing a connection between the sources.

The individual steps of routing and dimensioning water supply networks according to the developed SRS method are listed below:

• Preparing the settlement grid, potential connection, and demand nodes.
• Determining the location, demand, and priority of water supply to individual demand nodes. Classification of demand nodes in the ranking.
• Determining the number and location of water sources. Possible definition of the percentage share of individual water sources.
• 1st iteration of pipe routing to class I nodes. Analysis of connection paths in terms of ΣL, Σα, and ΣK. In the case of networks with two or more water sources, the routing process should be carried out between the connecting individual sources.
• Determining the distribution of flows in the routed network based on the water consumption of class I demand nodes.
• Conducting the pipe sizing process. The dimensioning process is linked back to step 4—routing of water pipes.
• Carrying out the 2nd iteration routing and sizing processes to class II nodes.
• Routing and sizing processes of subsequent iterations to the remaining node classes.
• The process of routing and sizing of the water supply network is complete when demand nodes of all classes are connected with the water source.

2.3. Calculation Example

The process of implementing the SRS method is presented in an example calculation for a simple settlement (Figure 5a), containing 9 water demand nodes located in a development area plan (in the form of a grid of 5 × 5 squares with a side length of 150 m) with streets (30 m wide) between them. The location of the demand nodes is random, specifying the so called “centers of water consumption”. The water source is marked as a Z node and the total water demand of the settlement is assumed to be 75 dm³/s. Demand nodes are divided into 3 classes: one node (A) is assigned to class I, three nodes are classified as class II (B, C, D), and 5 nodes are placed in class III (E–I).

The subsequent stages (bifurcations) of the routing process in the 1st iteration are presented in Figure 5b. The process of the rotation of the base section started from the water source Z. From the source node, potential connections are routed in all possible directions to the nearest intersections (continuous line). Next, from all the ends of the continuous lines, further potential connections are made to further intersections (dashed lines). In this way, through subsequent bifurcations, routing is carried out until a connection with a node of a given category is reached. The assumed goal was to connect the source to the class I node (A) which was achieved in the 3rd bifurcation. The connecting route was 540 m long and had a rotation angle of the base section of 0° (marked in light blue in Figure 5b). After the routing process, the diameter of the Z–A pipe had to be determined. For this purpose, the continuity equation was used, assuming total water consumption to be 75 dm³/s and the water flow velocity to be v = 1 m/s. The calculated diameter of the Z–A pipe was 0.309 m, so the DN300 pipe was selected from the DN/ID type series line of ductile iron pipes. This completed the 1st iteration of the SRS process. The assumed velocity of 1 m/s refers only to the selection of the pipe diameter routed in the first iteration. This may correspond to the pipes originating from the water supply stations. Due to the water flow in further directions and the minimum diameters that can be used in the WDNs, it is expected that the velocities in the further parts of the network (farther from the source) will be lower. Due to the dimensioning process being carried out in accordance with the modified Murray’s law, no pressure constraints were introduced—according to the applied law, the appropriate selection of diameters should guarantee the minimization of flow energy.

The routing process of the 2nd iteration was carried out from the water pipeline route obtained in the previous iteration to class II nodes (nodes B, C, D). As a result of the routing, a connection with nodes B and C in the 3rd bifurcation was achieved. The resulting routes to nodes B and C provided a simultaneous connection to the nodes and were equivalent—they had an identical length (450 m) and an identical sum of the rotation angles of the base section required to obtain the connection (90°). A visualization of the routing process of the 3rd bifurcation of the 2nd iteration is shown in Figure 5c. In subsequent bifurcations, attempts were made to obtain a connection with node D. As a result of the routing, three potential paths were distinguished (Figure 5d).

● 1st alternative route: ∑L = 630 m	∑α = 90°
● 2nd alternative route: ∑L = 540 m	∑α = 180°
● 3rd alternative route: ∑L = 630 m	∑α = 270°

To determine the final connection route, it was necessary to analyze the sum of the lengths of the sections (∑L), the sum of the rotation angles (∑α) and the total cost (∑K) of making a given connection. To determine the cost of implementing individual routes, it was necessary to know the pipe diameters. For this purpose, sizing was carried out according to the modified Murray’s law. The sections created in the 3rd bifurcation of the 2nd iteration (the connections with nodes B and C) were assigned a diameter of DN250, and the connection with node D was made with a diameter of DN200 mm. For the purposes of this example, the unit costs of constructing 1 m of DN200 water supply pipe (600.00 PLN/m) and a 90° pipe elbow (4000.00 PLN) were calculated, which were used to analyze the costs of constructing three potential connection routes to node D. The calculation results of the total cost of implementing the given connection routes are presented in

Table 2. Total cost of alternative routes to node D.

Route	Length	Unit Cost of DN200 Pipe	Pipeline Cost	Number of 90° Elbow	Unit Cost of 90° Elbows	∑K
	m	PLN/m	PLN	-	PLN	PLN
1	630	600	378,000	1	4000	382,000
2	540		324,000	2	8000	332,000
3	630		378,000	3	12,000	390,000

. The lowest cost was estimated to be route 2 (Figure 5d), with a length of 540 m and a sum of the rotation angles of the base section equal to 180° (double rotation of the base section required to connect to node D). The resulting network, ensuring the connection to all class II nodes with the existing water supply, is presented in Figure 5e.

In the 3rd iteration, in accordance with the assumptions of the SRS method, there was a change in the routing direction: from class III nodes towards the existing water supply pipelines. Additionally, the method of sizing the pipes was also changed—specific values from the DN/ID type series were assigned to specific iterations. Due to the fact that in the 2nd iteration, the smallest specified diameter was DN200, specific diameter values were also used for individual bifurcations (1st bifurcation of the 3rd iteration—DN150, 2nd bifurcation of the 3rd iteration—DN125, etc.). Connections to all class III nodes were achieved in the 2nd bifurcation of the 3rd iteration. Therefore, the selected pipe diameters were DN150 and DN125. Because the minimum diameter DN80 was not achieved, in accordance with the principles of the SRS method, all selected diameters were reduced by 2 positions from the DN/ID type series. The final diameter range of the routed network is DN80-DN200. During the routing process, it turned out that one of the class III nodes (F) was located in a place where the WDN had already been routed, and it was excluded from the further routing process. The final routing results are shown in Figure 5f.

During the routing and sizing of the exemplary water supply network, it can be observed that the assumptions of the SRS method were met; the pipes were sized with descending diameter gradation and consideration of the permissible minimum diameter; thus, there was no need to perform tedious hydraulic calculations of the looped system.

2.4. Methodology of the SRS Method Evaluation

The application of the presented SRS method was carried out as a multi-stage study and numerical research. After the implementation of preliminary tasks (assessment of unit costs of constructing water pipes and defining general principles for creating a node ranking), the method was tested under the conditions of a virtual settlement.

2.4.1. Description of the Synthetic Settlement Plan

The goal of the model tests was to evaluate the applicability of the SRS method in terms of (i) the hydraulic conditions of the designed networks, (ii) the absence of any blockages when using the presented method (iii) repeatability of the obtained results. The synthetic settlement plan was represented by a grid map consisting of development areas and streets. The structure of the model grid is presented in Figure 6. The area of the synthetic settlement plan was mapped as a 10-row and 10-column grid of squares (green color—development areas), each with a side of 150 m. It was assumed that there were 30 m wide streets (white color) between the development areas and a flat terrain. The total area was 380.25 ha, and the estimated number of inhabitants of the model unit was 85,556 people. The average daily water demand was 21,769.26 m³/d (≈252 dm³/s). During the model test studies, the irregular location of consumers as well as single- and multi-sided water supply cases were taken into account. Potential connection points for the water supply network located at street intersections (with a total number of 100) are marked with red points in Figure 6. Potential water demand nodes (total: 220 nodes) are marked with gray points. The model studies included the routing and sizing of 7 different WDNs:

• 3 single-side supplied water networks (A, B, C)—one water source,
• 3 multi-side supplied water networks (D, E, F)—two water sources,
• 1 multi-side supplied water network, G, with a top-down water source capacity (source no. 1—capacity equal to 70% of total demand; source no. 2–30%).

In each test, different locations of the water sources and demand nodes were assigned randomly for each of the A–G networks. Each of the water demand nodes in the analyzed networks was classified into one of the subtypes in the water consumption volume category (KQ1-KQ3) and the water delivery priority category (KP1-KP3). Subsequently, the demand nodes were ranked based on the sum of points obtained based on the assigned weights. In this way, 5 classes of nodes were distinguished, listed in

Table 3. Ranking of demand nodes divided into classes.

Class	Total Weight	Water Demand KQ		Delivery Priority KP
		Subtype	Weight	Subtype	Weight
I	6	KQ1	3	KP1	3
II	5	KQ1	3	KP2	2
		KQ2	2	KP1	3
III	4	KQ1	3	KP3	1
		KQ2	2	KP2	2
		KQ3	1	KP1	3
IV	3	KQ2	2	KP3	1
		KQ3	1	KP2	2
V	2	KQ3	1	KP3	1

. The nodes with the largest demand were classified as class I (KQ1, weight: 3) with high priority for water delivery (KP1, weight: 3). In turn, class V included the nodes with the smallest demand (KQ3, weight: 1) with low priority for water delivery (KP3, weight: 1). The number of classes of water demand nodes directly determines the number of iterations during the process of network routing and sizing according to the SRS method. With the proposed division of nodes into 5 classes, there were 5 iterations. Each iteration corresponds to the connection of nodes from a given class with a water source or existing water network.

According to the classification of the nodal demands carried out in [60], it was assumed that out of the 220 potential demand nodes, water was demanded by 42% (92 consumer nodes) and the largest consumer had a base water demand equal to 5% of the total demand (13 dm³/s). The structure of the demand nodes was determined as follows:

• the largest consumers: 23 nodes (25% of the 92) with base demand 4–13 dm³/s,
• the medium consumers: 9 nodes (10% of the 92) with base demand 2–3 dm³/s,
• the smallest consumers: 60 nodes (65% of the 92) with base demand equal to 1 dm³/s.

On average, there were 7 nodes in the A–G networks in class I and 20 nodes in class V, respectively. The largest group was class III, in which, on average, 32 out of 92 nodes were qualified. A maximum of 38 nodes were qualified in each class (in networks C and E). A detailed summary of the number of nodes in networks A–G is presented in

Table 4. Number of nodes in individual classes in networks A–G.

Class	Network							Average
	A	B	C	D	E	F	G
I	9	7	5	10	4	7	5	7	7.30%
II	11	9	7	7	5	9	12	9	9.32%
III	24	29	38	31	38	35	31	32	35.09%
IV	30	22	23	23	28	22	25	25	26.86%
V	18	25	19	21	17	19	19	20	21.43%
Total	92	92	92	92	92	92	92	92	100%

. The synthetic settlement plans with the locations of water sources and the division of demand nodes into individual classes in networks A–G are shown in Figure 7.

2.4.2. Results, Analysis, and Methodology

Networks A–G, routed and sized via the SRS method, were further compared to each other in order to determine the repeatability of the method. The assessment of the results was performed with regard to two major aspects: the geometric characteristics of the networks and the operating hydraulic conditions. The assessment of the geometric structures was carried out by determining the sum of the lengths (ΣL) of water pipes constituting the routed network, by specifying the number of iterations and bifurcations needed to route the network, and by determining the sum of the rotation angles (Σα) of the base section in the first two iterations. Additionally, the geometric structure was characterized by a fractal dimension (Formula (1)), calculated via Fractalyze 3.0 software [61]. The developed networks were also analyzed in terms of the diameter range and the existence of branched endings.

The operating hydraulic conditions were analyzed in accordance with numerical steady state models of each of the designed networks. Simulations were performed in the EPANET 2.2 software [62]. The basic settings of the model were as follows: LPS flow units, Darcy–Wiesbach head loss formula, and 1.5 mm pipe roughness. When analyzing the flow velocity in the model networks, a unified scale was adopted. The pressure and the water age in the networks were analyzed with regard to the occurrence of minimum, maximum, and average values. In order to compare the A–G networks, it was assumed that the minimum required pressure in the network was 40 m H₂O and the characteristics of individual pumping stations were developed with this in mind. The single-point pump curves were defined by the head (H) and flow (Q). The pump heads were determined using a trial and error approach, so that the minimum pressure in the network was reached. Additionally, for each of the networks, pump energy consumption was evaluated assuming 75% pump efficiency. To determine the water age, it was necessary to change the type of simulation to an Extended Period Simulation (EPS) and define the simulation time, which had to be longer than the time of water inflow from the source to its most distant nodes.

3. Results

3.1. Networks A, B, and C

As the result of applying the SRS method on the model grid, the following networks with a single water source were created:

• network A: total pipe length ΣL = 23,670 m, diameter range DN80-DN600, total number of iterations: 5, total number of bifurcations in all iterations: 46, total rotation of the base sections in the first two iterations: Σα = 1800°,
• network B: ΣL = 23,940 m, DN80-DN500, 5 iterations, 21 bifurcations, Σα = 1260°,
• network C: ΣL = 25,110 m, DN80-DN350, 5 iterations, 20 bifurcations, Σα = 1080°.

The geometric structures of the designed water supply networks A, B, and C with the pipe diameters are shown in Figure 8. In the designed networks, source nodes are located in different places: for network A it is the upper right corner of the area, for network B it is the right side of the area, and for network C it is the left side of the synthetic settlement plan area. The location of water sources has a direct impact on the number of bifurcations needed to route the network to class I nodes. When analyzing the geometric structures of the designed water supply networks A, B and C, it can be noticed that the routed structures cover the area of the entire synthetic settlement. In all three networks, the smallest selected diameter is DN80, which is achieved during the dimensioning process in the fifth iteration (up to class V nodes). However, the maximum diameters differ between networks, from DN600 in network A, to DN500 in network B, to DN350 in network C. This is a consequence of the randomly selected locations of water sources and demand nodes. In each of the analyzed networks A, B and C, there were visible “end branches” of the networks with significant diameters. Such situations should be treated as unfavorable, because they would require structural protection of the water pipe ends against the resulting stress forces.

A detailed list of water velocities in the analyzed networks A, B, and C is presented in

Table 5. Velocities in water pipes in networks A, B, and C.

Velocity	Network A		Network B		Network C
	Length
m/s	m	%	m	%	m	%
≥1.5	0	0	0	0	0	0
1.0–1.49	0	0	990	4.14	720	2.87
0.7–0.99	540	2.28	1260	5.26	1530	6.09
0.3–0.69	6300	26.62	7830	32.71	8280	32.98
0.0–0.29	16,830	71.10	13,860	57.89	14,580	58.06
ΣL	23,670	100	23,940	100	25,110	100
Average Velocity (m/s)	0.26		0.32		0.31

. When analyzing the distribution of water flow velocities in the pipes, it can be noticed that the pipes in which the water flow velocities range from 0 to 0.29 m/s predominate. They constitute 71.10% of all sections in network A, 57.89% in B and 58.06% in C. Although this velocity is unsatisfactory in terms of the hydraulic operating conditions of the network, it is also typical for real water supply networks operating under standard water flow conditions due to the minimum required diameter DN80. The average value of water flow velocity in network A was 0.26 m/s, in network B 0.32 m/s, and in network C 0.31 m/s.

The contour maps of pressures in networks A, B, and C are shown in Figure 9. In all A, B, and C networks, there are very small differences in pressure within the designed networks. The maximum pressure can be observed in network B (44.62 m H₂O), while the lowest average pressure occurs in network A (40.94 m H₂O). The water age in networks A, B and C is presented in the form of contour maps in Figure 10. In network A, the maximum value of water age is 3 h 39 min, with an average of 70.2 min. In network B, the maximum and average values are 4 h 37 min and 67.8 min, respectively. In turn, in network C, one can notice the relatively highest maximum value of water age (4 h 55 min) with the lowest average value (54.6 min). The estimated electricity consumption by pumps in networks A, B, and C was 138.33, 148.51, and 142.88 kW, respectively.

3.2. Networks D, E, and F

As the result of applying the SRS method to the model grid, the following networks with a double water source were created:

• Network D: ΣL = 23,400 m, DN80-DN450, 5 iterations, 16 bifurcations, Σα = 1440°,
• Network E: ΣL = 22,500 m, DN80-DN400, 5 iterations, 21 bifurcations, Σα = 720°,
• Network F: ΣL = 22,770 m, DN80-DN400, 5 iterations, 19 bifurcations, Σα = 1170°.

The geometric structures of the designed water supply networks D, E, and F, along with the pipe diameters, are shown in Figure 11.

Similarly to the case of the networks supplied from one side, networks D–F also cover the entire area of the synthetic settlement. In all three cases, the minimum diameter in the networks is DN80, while the maximum diameter is DN400 or DN450. As in the case of networks supplied from one side, in each of them there are “end branches” of the water pipeline with significant diameters.

When analyzing the distribution of water velocities in networks D, E, and F, it can be observed that the pipes with the lowest flow velocities dominate (0–0.29 m/s). In the case of the D network, they constitute 63.85% of the pipe length of the entire network, in the E network, 58.40%, and in the F network, 68.38%. The average value of water flow velocity in network D was 0.28 m/s, in network E, 0.34 m/s, and in network F, 0.25 m/s (

Table 6. Velocities in water pipes in networks D, E, and F.

Velocity	Network D		Network E		Network F
	Length
m/s	m	%	m	%	m	%
≥1.5	0	0	0	0	0	0
1.0–1.49	90	0.38	720	3.20	0	0
0.7–0.99	900	3.85	1620	7.20	1080	4.74
0.3–0.69	7470	31.92	7020	31.20	6120	26.88
0.0–0.29	14,940	63.85	13,140	58.40	15,570	68.38
ΣL	23,400	100	22,500	100	22,700	100
Average Velocity (m/s)	0.28		0.34		0.25

).

The contour maps of pressure in networks D, E, and F are shown in Figure 12. The maximum pressure can be observed in the E network (46.87 m H₂O), while the lowest average pressure occurs in the D network (40.94 m H₂O). The designed networks are characterized by an overall low water age—a maximum time of approximately 3 h (Figure 13). The total electricity consumption necessary for the operation of the pumps was 138.06 kW in the D network, 153.97 kW in the E network, and 139.30 kW in the F network.

3.3. Network G

As the final part of the model studies, the SRS method was applied under the conditions of a top-down assumed water source capacity. In the case of the G network, the capacity of the first water source was 70% of the total demand. Application of the proposed method resulted in the designing of a network ∑L equal to 23,130 m and the diameter range of DN80-DN450. As in the previous cases, the network was routed in five iterations with a total number of bifurcations equal to twenty. The sum of the rotation angles of the base section ∑α in the first two iterations was 1170°. The geometric structure of the G network along with the specification of the selected diameters of water pipes is shown in Figure 14. Red markers, numbered 1 and 2, indicate the location of water sources. The G network covered the entire settlement. As in all previously considered cases, the resulting network includes “end branches” of the network with significant diameters (DN350-DN400).

Network G is characterized by the share of pipes with the lowest water velocities equal to 46.3%. Additionally, this network includes sections with velocities ranging from 1.0 to 1.49 m/s (8.95% of all), as well as single sections with velocities above 1.5 m/s and even 2 m/s. Velocities above 1 m/s are the consequence of the capacities of the top-down water sources and occur in the pipes located far from the pumping station. The maximum pressure in the network was 55.28 m H₂O, while the average was 49.07 m H₂O. The maximum water age was 4 h 5 min, with the average equal to 45.6 min. Electricity consumption amounted to a total of 180.05 kW.

4. Discussion

A detailed list of the parameters of A–G networks designed using the SRS method is presented in

Table 7. Characteristics of networks A–G.

Network	No. of Water Sources	ΣL	Σα	Db	DN	Velocity (Average)	Pressure (Average)	Water Age		Energy
								Max	Average
		m	°	-	mm	m/s	m H2O	h:min	min	kW
A	1	23,670	1800	1.109	80–600	0.26	40.94	3:39	70.2	138.33
B	1	23,940	1260	1.108	80–500	0.32	43.54	4:37	67.8	148.51
C	1	25,110	1080	1.162	80–350	0.31	41.62	4:55	54.6	142.88
D	2	23,400	1440	1.105	80–450	0.28	40.94	2:43	57.6	138.06
E	2	22,500	720	1.101	80–400	0.34	45.12	2:43	43.8	153.97
F	2	22,770	1170	1.144	80–400	0.25	41.49	3:16	60.0	139.30
G	2	23,130	1170	1.111	80–450	0.49	49.07	4:05	45.6	180.05

. When comparing the networks with a single water source (A, B, and C) with the networks with two water sources (D, E, and F), one can notice an increased value of the total length of water pipes (on average 24,240 m in networks A, B, and C, and 22,890 m in networks D, E, and F—an increase of 5.5%). Double-side supplied networks are additionally characterized by lower sums of rotation angles of the base section, calculated in the first two iterations. In all analyzed cases, the minimum diameter in the networks is DN80, while the maximum diameter is DN350, DN500 and DN600 in networks A, B, and C, respectively, and DN400 and DN450 in networks D, E, and F. In each network, the existence of “branch ends” with significant diameters (DN300–DN400) can be noticed. In all analyzed networks, the fractal dimension is defined as a non-integer number. This confirms that the geometric structures created by the SRS method possess the properties of fractal figures.

Average flow velocities and pressure heads in the networks did not show significant differences depending on the number of water sources. In networks A, B, and C, the average velocity in the pipes was 0.30 m/s, while in networks D, E, and F it was 0.29 m/s. The average water pressure was 42.03 m H₂O in networks A, B, and C and 42.51 m H₂O in networks D, E, and F. The water age in all analyzed networks was relatively small. Additionally, there was a tendency for the water age to decrease as the number of water sources increases. The maximum water age was 2 h 43 min (network B), up to 4 h 55 min in network C. In networks D, E, and F the average water age was 53.8 min, while in networks A, B, C it was 62.4 min. Regardless of the number of water sources, all networks showed very similar electricity consumption necessary to supply water at the appropriate pressure. There was an average of 143.24 kW in networks A–C and 143.77 kW in D–F.

Compared to the networks in which the source capacities were defined in accordance with the SRS Method (A–F), the G network clearly stands out, in which the source capacity ratio was assigned top-down. Comparing the percentage of pipe length with specific flow velocities, a clear decrease in the share of the pipes with the lowest velocities can be noticed: 46.3% in the G network to an average of 62.95% in the A-F networks. In network G, the highest average water flow velocity can also be observed (0.49 m/s), as well as water flow velocities above 1.5 m/s. Such high velocities under standard flow conditions may indicate the limited capacity of the selected pipes, as well as high water losses during water flow under fire flow conditions. The pressure amplitude in the G network was over 15 m H₂O. Additionally, in the G network there were nodes with water age more than 4 h. High pressure losses were also confirmed by the high electrical power consumption—the highest value, 180.05 kW, was approx. 25% higher compared to the A-F networks.

5. Conclusions and Future Directions of Research

Considering the results obtained from these model tests, it can be concluded that the main research objectives set in this article have been met. The proposed method of simultaneous routing and sizing allows us to obtain repeatable results with various numbers as well as locations of water sources and recipients. Similar features characterizing the geometric systems were obtained, as well as similar hydraulic parameters of the operation of individual networks. The SRS method is characterized by the following: (1) the method allows for simultaneous routing and sizing processes, (2) the method includes a benchmarking mechanism for classifying demand nodes in terms of water consumption and water supply priority, (3) a new criterion for selecting water path is proposed—the cost of constructing the water pipeline, (4) the selection of pipe diameters is carried out according to the modified Murray’s law, (5) the method is applicable to systems with single and multiple water sources, (6) the SRS method enables routing a network in which the capacities of water sources are pre-defined.

However, the developed method has certain limitations, which opens the possibility of further research. The proposed research directions include: testing the method under the conditions of a quasi-real and real settlement, verifying the sizing process by comparison with an existing recognized method (e.g., the genetic algorithm approach), checking the impact of different demand node weights and priorities on the shape of the developed water supply networks, determining other ranking categories, modification of the SRS method so that network ends with significant diameters are not created during the routing process, and development of a computer software for network routing and sizing with the implemented SRS method. To comprehensively assess the potential application of the developed method, a comparison with other design methods is also planned, both traditional (e.g., the Hardy–Cross method) as well as those using advanced heuristic and metaheuristic methods.