Building theories from case studies is a common research approach that can also be applied to the analysis of networks. Although case studies of real systems bridge the gap between theory and practice, they are nevertheless investigations of a subset of cases, each with specific characteristics. To tackle this problem, a multitude of networks with diverging characteristics are generated using the graph-theory-based Modular Design System (MDS). In this paper the application of this MDS is demonstrated by generating a set of 2280 virtual Water Supply Systems. The layout and the properties of these systems are representative of typical examples encountered in practice. Scatterplots, density, and cumulative distribution functions are used to characterize several network parameters. A comparison of the virtual set with three real-world case studies shows similar characteristics. Finally, the potential of the methodology is demonstrated by analyzing the impact of increased water demand on hydraulic performance. It can be shown that the allowed maximum and minimum diameters envelop a range of impacts on mean values for nodal pressure.
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 Even if the use of case studies is common for building theories and the analysis of systems [Eisenhardt, 1989], they are nevertheless investigations of only specific pieces of our world, a subset of cases that can differ greatly from one another. Generalizing conclusions from individual investigations of networks is questionable and sometimes inappropriate. The number of available case studies for analysis of water supply networks is restricted due to tedious data collection, difficulties in model setup, and sensitive network data. In order to overcome these obstacles, this paper will present a novel method for the generation of network systems with different characteristics that will assist case study research. The example featured in this work involves the generation of water supply systems (WSSs), but the methodology can likewise be applied to other (natural or man-made) network systems (e.g., rivers, sewers, railroads, electricity).
 The field of graph theory defines a network as a directed graph with edges that have capacities. In hydraulic networks the term capacity denotes transport ability and is represented by the pipe diameter and the flow that is delivered from sources to sinks. Simple, complete, and planar graphs; trees; or fractals are examples of possible network shapes. A simple graph has no loops and not more than one edge between any two different vertices. A complete graph is a simple graph in which every pair of distinct vertices is connected by a unique edge. A planar graph is a graph that can be drawn on a plane so that the edges intersect only at the vertices (i.e., two edges do not cross each other at any other point). A tree [Aldous, 1993] is a graph in which any two vertices are connected by exactly one path, (i.e., any connected graph without loops is a tree).
 A fractal specifies a geometric shape that is composed of smaller components, each of which is a replica of its original shape. Researchers have reported on the fractal nature of rivers [Tarboton et al., 1988; Rinaldo et al., 1992; Rinaldo et al., 1993]. Tarboton et al.  presented fractal dimensions by means of Tokunaga cyclicity parameters. As Tokunaga fractals are tree shaped, rivers are assumed to be branched. Building upon this theory, Cui et al.  introduced a stochastic Tokunaga model for stream networks.
 Urban drainage systems, like rivers, have been found to resemble tree graphs. Ghosh et al.  used dendritic and space-filling Tokunaga fractal tree geometry to generate artificial urban drainage systems. Möderl et al.  developed a case study generator based on the Galton–Watson branching process to generate virtual urban drainage systems. Similarly, Urich et al.  investigated the generation of virtual sewer systems using an agent-based modeling approach. Therein, sewer systems are designed based on a stochastically generated virtual urban fabric detailed by Sitzenfrei et al. .
 Even though water supply systems consist of loops, their degree of meshing does not allow them to be represented as planar or tree graphs. The aforementioned methods are therefore not appropriate to generating WSSs and other looped systems.
 The Modular Design System (MDS) approach was introduced as a consequence of the idea of identifying simple building blocks in complex networks [Milo et al., 2002], Möderl et al. . This method forms the basis for the systematic generation of virtual WSSs. In this paper, the generation of a set of virtual systems is demonstrated first. Second, the characteristics of the systems are analyzed and compared with three real-world networks. Third, as an example for an application of such a set, we analyze one for the effect of an increase in water demand. Other applications that employ sets of case studies are numerous. Their most common use involves the (model-based) test of a range of measures for overcoming various design or management scenarios (e.g., calibration algorithms, sensor placements, rehabilitation strategies). As the sets of case studies encompass a fairly broad range of characteristics, results from their use in research will provide more credibility to the corresponding output. In sensitivity analysis, virtual case studies provide the gradual variation of parameter values required to assess their influence on model output. A final aspect worth mentioning is that sets of virtual case studies are essential in software testing and fixing bugs.
2. Materials and Methods
 For the systematic generation of virtual WSSs using the MDS approach [Möderl et al., 2007], a MATLAB toolbox was developed. In what follows, this approach is explained using MATLAB syntax (to enhance the reader's understanding), but the method can also be applied outside the MATLAB environment. To clarify issues of notation, brackets [ ] will denote either a matrix or a concatenation operator (introduced later). Indexing into a matrix is a means of referencing a subset of elements from the matrix and is denoted by parentheses ( ).
 The core of the MDS consists of a topological generation algorithm, which creates water networks, starting from basic building blocks according to the following boundary conditions: (1) nodal demands (2) number of water resources, and (3) network connectivity.
 Boundary conditions and/or parameter values are varied for each generated WSS (see section 2.1). Once layout and topology of the water distribution network have been designed, diameters of newly generated pipes are determined according to state-of-the-art design rules. Here, a simple algorithm based on the principle of economic flow velocity was used to estimate realistic pipe diameters (see section 2.2). Finally, the network can be exported for external simulations (in this case with the hydraulic solver EPANET 2 [Rossman, 2000]).
2.1. Generation of WSSs Using the MDS
 The algorithmic generation of WSSs using the MDS involves the concatenation of different WSS modules to form a complex network. The MDS junctions of WSSs are represented as points of a grid. For reasons of simplicity a rectangular grid was chosen with a constant spatial resolution of 500 m (but polar, elliptic, or even irregular coordinates can also be used). Each grid point corresponds to an element of a graph matrix where the value of such an element represents the connections in the four cardinal directions in binary coded form (e.g., east, 20 = 1; north, 21 = 2; west, 22 = 4; south, 23 = 8; as illustrated in Figure 1). For each grid point in the graph matrix, additional parameters such as types (e.g., reservoir, junction, pipe) and attributes (e.g., demand, head) can be set. Values of attributes can be changed without affecting the layout.
 Using the MDS, an empty grid with r rows and c columns is created with the initialization function G=mds(r,c), which results in an r * c zero matrix G. Additionally, an r1 * c1 graph M, representing an arbitrary module, can be inserted in the greater matrix G(r1 < r, c1 < c) by G(i,j) = M where (i,j) is the position of the upper left corner of the module. The feature responsible for automatically connecting nodes (connecter) is used for linking modules over longer distances. Several graph matrices of arbitrary size and layout can be assembled to build more complex WSSs using an adapted concatenation operator to merge these graph matrices. For repetitive concatenation of modules, it is also possible to use loops with changing module positions and/or modules based on random or deterministic algorithms. Here, only deterministic algorithms are used. In Figure 2, different layouts from a looped basic demand raster (BDR) are shown and described as graphs, symbols, and matrices. In addition, the initialization of a module and the insertion procedure are presented.
 The WSS layout is influenced by water demand of the spatially distributed population, i.e., demand distribution. Beginning with a basic demand raster of 25 junctions in 4,000,000 m2 (denoted BDR; see Figure 2, middle row), different sizes of demand area and varying population densities can be represented.
 The variation of demand is here expressed by different basic junction demands. Values of 0.1 L/s (de1), 1 L/s (de2), and 5 L/s (de3) represent daily demand peaks of villages, cities, and metropolises, respectively (see Figures 3a–c). To avoid using identical demand values across all junctions of the WSS, basic junction demands were randomly varied by a uniform distributed random multiplying factor ranging from 0.75 to 1.25 (see Figure 3d).
 A variation in the size of the demand area (spatial distribution) is achieved either by expansion of the BDR in one or two dimensions, or by using complex expansion (see Figure 4 and Figure 5). One-dimensional expansion would result in a longitudinal WSS and represent urban areas located in valleys (di1). A two dimensionally expanded BDR represents WSSs of plain urban areas with equally long diagonals (di2). A complex expansion is used where these di2-BDRs are assembled with intermediate distances. These building blocks could, for example, represent water supply systems of several municipalities along a river (dic).
 The number of sources property represents the allocation of water sources within the network, which is accomplished in the same manner as the definition of demand boundary conditions. Every WSS is supplied by a minimum of one (so1) and a maximum of four (so4) sources (see Figure 6 and Figure 7).
 Numerous publications [e.g., Kalungi and Tanyimboh, 2003; Tung et al., 1989] have analyzed differences in network connectivity and have found that alternative pathways from water source to demand junctions increase network redundancy. Therefore, both standardized branched (ne1) and looped (ne2) networks are generated and compared to quantify differences in performance.
 The standardized branched network can be assembled with the elements of Figure 1. Alternatively, the selected binary-coded connections of the looped BDR (ne2) in the graph matrix (Figure 8a) can be manually overwritten.
 If a complex spatial distribution variation (dec) is applied, two modes for linking complexes that mimic the natural evolution of such networks are provided. The first option mimics two or more networks gradually growing toward each other (a mostly unplanned evolution). These networks are frequently linked together by one or several connections at their outer branches. This is modeled by the mode of connecting pipes (li2), thus linking complexes with one or several pipes of small diameter. The second mode of linking is given by connecting two or more networks with a strong pipe, which acts as the backbone for the supply. This is modeled by means of an integrated pipe (li1), chosen here with a diameter of 0.5 m (see Figure 9). As with the graph matrices depicting variation in structure (Figure 4), examples of variations in linking between complexes (li) are created using intermediate distances between building blocks and an auto connecting function (described by Möderl et al. ).
 In Figure 10 the variation in number of complexes (Figure 10a, 2 complexes; Figure 10b, 3 complexes) and the variation of intermediate distance of complexes (Figure 10a, in2; Figure 10b, in1) between complexes are shown. These parameters can be used to construct scenarios of municipalities in close proximity (e.g. in an alpine valley as shown in Figure 11).
 Compiling all variations made to the network resulted in a collection of 2280 different WSS layouts. This set of conceptual layouts covers the majority of the network types found in reality. An extensive description of the layout that can be acquired by means of this simple algorithm supports the use of the methodology described in this paper.
2.2. Realistic Design of Model Parameters
 All model parameters are stored within the MDS as attributes of each node (e.g., demand, elevation) or link (e.g., length, diameter, roughness). Attributes can be easily altered without affecting the setup of the layout.
 Pipe lengths are determined from spatial resolution of the grid. In this study, a uniform grid resolution (and thus equal pipe length) is applied for convenience. The methodology allows grid resolution to be easily scaled, which results in variable pipe lengths.
 Optimum design of a WSS with respect to pipe diameter has been extensively studied in the past [Maier et al., 2003; Savic and Walters, 1997; Loganathan et al., 1995; Eusuff and Lansey, 2003]. The goal of this work, however, is to create WSSs that are as realistic as possible and take into account the evolving planning process of such systems that leads to suboptimal design. A simple flow velocity driven pipe-sizing method is thus used by assuming an economic flow velocity of 1 m/s (according to Trifunovic [2006, p. 132]). There is no restriction for either low flow velocity or pressure. The algorithm will augment pipe diameters incrementally (i.e., 0.08, 0.1, 0.125, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5, and 0.600 m) to satisfy the condition that velocities do not exceed 1 m/s. Pipe roughness is kept constant for this study, but randomization can be implemented into the algorithm. Invert elevations of junctions and reservoirs are also kept constant at 0 m and 100 m, respectively. Applying a spatial joint between nodal elevations and digital terrain models (real or virtual) would enhance the realism of the case studies. To maintain simplicity, this option was neglected. Standard values are chosen for all other model parameters.
2.3. Description of Real-World Case Studies
 The three real-world systems used in section 3 for a comparison are located in alpine valleys on ground with elevation of 500 up to 700 m. Water flows from hillside reservoirs and tanks, driven by gravity. The number of nodes in the systems range between 133 (real 1) and 804 (real 3). A similar range is seen for the number of links. The total demand served for real 1 and real 3 are approximately 6 L/s and 320 L/s, respectively. Pipes in the systems have a diameter distribution between 80 mm and 800 mm. The slightly modified layouts of the real-world systems are shown in Figure 12. To comply with legal policies set by the water utility (which provided the data), 5% of the pipes and junctions in the network were erased and the layouts were rotated, scaled, and mirrored.
3. Results and Discussion
 This paper outlines the systematic generation of virtual network systems based on the MDS methodology resulting in 2280 typical network layouts. This set (denoted as WSS-Set-2280) has been generated for further development or applications and can be downloaded from the homepage of the Unit of Environmental Engineering, University of Innsbruck (available at www.hydro-it.com/extern/IUT/mds_app/). The generation process, as described above, creates virtual water supply networks with different properties (listed in Table 1). These properties are used to characterize the network systems.
Table 1. Definition of Properties of WSSs
One-dimensional expansion (di1)
Two-dimensional expansion (di2)
complex expansion (dic)
Number of sources
branched network (ne1)
looped network (ne2)
Linking of complexes
integrated pipe (li1)
connecting pipe—pipes growing over the years (li2)
Number of complexes
Intermediate distance of complexes
1 km (in1)
5 km (in2)
10 km (in3)
3.1. Network Statistics
 To better understand WSS-Set-2280, statistical parameters were determined, the results of which are shown in Figures 13–16. The aim is to demonstrate that the characteristics of the generated set of networks resemble those of the real-world ones (even at a range of different system sizes, e.g., 2000–20,000 inhabitants supplied). Figure 13 specifies the number of nodes and links in each network of the set by density function (Figure 13, left) and cumulative distribution function (CDF) (Figure 13, right) of number of nodes and number of links. It can be seen that 10% and 75% of the systems consist of not more than 100 and 1000 nodes, respectively (similar observations are seen in the number of links). The properties of the real-world systems are shown to correspond with those of the virtual systems. Hence, the ranges of link and node counts of the three real-world systems are within 10 and 70% of the CDF of the virtual systems. The ratio of links per nodes ranges from 1 to 1.5 for looped and branched systems, respectively.
 The definition of water supply network logically implies that sources and sinks are also part of the system. Figure 14 shows density function (Figure 14, left) and CDF (Figure 14, right) of total demand and demand per source. Approximately 66% of virtual systems deliver a total demand lower than 1000 L/s and 29% of the virtual systems have total demand lower than 100 L/s. On the basis of these distributions, groups for low, medium, and high demands can be defined to allow comparison of different demand scales. Demand thresholds are chosen as 100 L/s and 1000 L/s. Once again, the three real-world case studies correspond with the characteristics of the virtual systems.
 For WSSs, the capacities of the edges in a directed graph are represented by the pipe diameters. Figures 15 and 16 highlight the influence of network properties on pipe sizing. Median diameters are plotted against total demand of each system, taking into account the properties of network connectivity, linking of complexes, and demand distribution (Figure 15). Looped systems exhibit lower median diameter than branched layouts due to redundant flow paths. Head loss (which is a function of diameter) is thus smaller in looped systems despite delivering the same amount of water to satisfy the same demand area. In spatially large branched systems, median diameters equal the maximum possible diameter, which indicates that more than half of the pipes have diameters of 0.6 m. While this is not realistic in practice, it is still of interest to the investigation for comparison purposes. Integrated pipes with large diameters cause a shift in the pipe size distribution (Figure 15, middle). The influence of 0.5 m diameter integrated pipes is greater than the effect of smaller pipes, due to lateral inflow. An increase in demand density results in larger diameters being required to satisfy economic flow velocity. As shown above, the number of components and the size of real-world studies are largely similar to the characteristics of the virtual counterparts.
Figure 16 shows CDFs of pipe diameters in WSSs, taking into account the network connectivity and total demand properties. Black and gray lines show CDFs of 5th and 95th percentiles (P5 and P95) of pipe diameters, respectively. The solid and dashed lines indicate looped and branched systems. For 50% of the looped systems, the 95th percentile of diameters is less than 0.15 m. In contrast, the P95 value of branched systems is less than 0.25 m. The differences are more pronounced when results are distinguished based on total demand.
 The WSS-Set-2280 collection encompasses a rich diversity of WSS layouts and can be used as a surrogate for case study research. If there is a concern about the lack of representation of specific circumstances in the set, previously outlined concepts can be used to simply generate an alternative collection. A MATLAB toolbox has made available at the authors' home page for this purpose (available at www.hydro-it.com/extern/IUT/mds/). For convenience, the MDS toolbox also provides an export interface to the widely used water supply software EPANET 2 [Rossman, 2000], which allows direct evaluation of the hydraulic performance of the generated systems.
3.2. Application Example
 As an example of application, the effect of an increase in water demand is analyzed using sets of virtual WSS networks. The original WSS-Set-2280 is used as a reference set for the status quo. For simplicity, we will base our analysis on a modified set with the assumption of a 50% increase in demand. To allow comparison, each virtual system is evaluated by EPANET 2 for hydraulic performance in its current status and after applying the 50% increase in demand. The CDFs of mean pressures at the nodes are plotted for looped and branched systems in Figure 17. Due to the demand increase, 4.2% of the branched systems and 2.3% of the looped systems experienced a significant drop in minimum pressure (i.e., below 25 m). The higher percentage of failures in looped systems can be attributed to smaller pipe diameters. Mean pressure values under normal conditions plotted against the hydraulic performance under increased demand are shown in Figure 17 (right). Although there are differences in their hydraulic behavior, the performance of both looped and branched systems is affected. The EPANET 2 hydraulic computation shows good agreement for pressure in both the real-world and the virtual systems. A clear benefit of the methodology is that the impacts of the tested scenario are reflected across the whole range of feasible WSS layouts, which thus allows for generic performance assessment and more informed decision support.
 This paper investigated the possibility of generating virtual networks based on the Modular Design System approach described in past research. The method assembles complex conceptual networks from simple building blocks, giving rise to a robust mathematical description of water supply system network layout. This symbolic description allows itself to be easily programmed. An open source toolbox in MATLAB has been developed by the authors and is used within this investigation.
 The design of virtual water supply networks involves (1) the layout of the network and (2) the pipe diameter design process. While the first aspect is a scientific innovation, pipe design is a common engineering problem and has been implemented here following state-of-the-art guidelines. The layout design is based on system properties such as spatial distribution of demands, number of sources, type of connections as well as number, linking, and intermediate distance of complexes, among others. Any variation in these properties leads to changes in the network layout. The systematic generation of a set of virtual WSSs is subsequently performed by varying these properties within realistic ranges.
 A set of 2280 virtual networks was generated covering a broad range of layouts to offer a suitable surrogate for case study research. Statistical analyses of the network properties revealed that the generated networks are sufficiently diverse for this purpose. The characteristics of the virtual set are in close accordance with three real-world water supply systems, which further supports the credibility of the methodology. In the research there are substantial assumptions and simplifications and more research is needed to create enhanced virtual systems that are closer to reality. Currently, only junctions, reservoirs, and pipes are represented within the MDS. Pumps, valves, and other components can be added later in performance assessment tools such as EPANET 2. In a later version it will be possible to insert these objects directly into modules of the MDS.
 Applications of the methodology are manifold and cover different aspects such as analysis and optimization of measures, sensitivity analysis, test of calibration algorithms, software testing, and more. Here, the value of a multitude of virtual networks is demonstrated by analyzing the impact of increased water demand on hydraulic performance. It can be shown that the allowed maximum and minimum diameters envelops a range of impacts on pressure.
 The two major benefits of the methodology illustrated in this paper are the following: (1) the ability to stringently describe the network layout by means of a simple algorithm and (2) the capacity to perform scenario testing and receive a generic output (where impacts are reflected across the entire range of virtual case studies). It is believed that the methodology provides effective decision support in adapting our water supply systems to the impending challenges posed by the environment. The method has been developed in the context of water supply networks, but can in principle also be applied to any other kind of natural or man-made network structure (e.g., rivers, sewers, roads).
 The work reported was funded by project KIRAS PL 3: Achilles, project 824,682 under the Sicherheitsforschungs—Förderprogramm KIRAS of the Austrian Federal Ministry for Transport, Innovation, and Technology (BMVIT) and the Austrian Research Promotion Agency (FFG).