# A Method for Modeling Urban Water Infrastructures Combining Geo-Referenced Data

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## Abstract

**:**

## 1. Introduction

## 2. State-of-the-Art

#### 2.1. Spatial Disaggregation of Water Demand

#### 2.2. Generation of Water Distribution Systems

#### 2.3. Resilience of Water Distribution Systems

## 3. Framework

## 4. Data and Methods

#### 4.1. Data

_{Total,blue}= 0.33 · Q

_{Total,yellow}= 31.4 Mio.m

^{3}/y.

#### 4.2. Disaggregation of Water Demand

#### 4.3. Identification of the Possible Water Network

#### 4.4. Water Distribution System Design Optimization Problem

#### 4.5. Optimization Instance

## 5. Results and Discussion

#### Limitations

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | Linear dichroism |

LCZ | Local Climate Zones |

WDS | water distribution system |

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**Figure 2.**(

**a**) Satellite image (Google Earth, 2021) and (

**b**) local climate zones of Cologne, Germany [15]. The yellow polygon marks the administrative boundary, and the blue one, the inner city, which is investigated in this paper. (

**c**) Streets from OpenStreetMap with the possible network topology are represented by blue edges. (

**d**) The possible network topology with sizes of the nodes (yellow points) represent the water demand disaggregated to the graph.

**Figure 3.**Cost composition of the total pipe installation cost per meter in dependence on the diameter.

**Figure 4.**Cost-optimal WDS (blue) with different line widths to represent the change in pipe diameter and maximum possible network topology (grey).

**Figure 5.**Quasi-stationary behavior of the generated WDS. (

**a**) Volume flow in the pipes, (

**b**) pressure at demand nodes.

**Table 1.**Local climate zones adapted from [13]. * The value for the local climate zone class 10 (heavy industry) is calculated by doubling the highest value occurring in the other classes (i.e., 12.5 m/A for class 1).

LCZ | Description | Built-Up Surface Fraction in % | Height in m | Average Surface Fraction in % | Average Height in m | Av. Volume per Area in m |
---|---|---|---|---|---|---|

1 | Compact high-rise | 40–60 | >25 | 50 | 25 | 12.5 |

2 | Compact midrise | 40–70 | 10–25 | 55 | 17.5 | 9.625 |

3 | Compact low-rise | 20–40 | 3–10 | 30 | 6.5 | 1.95 |

4 | Open high-rise | 20–40 | >25 | 30 | 25 | 7.5 |

5 | Open midrise | 20–40 | 10–25 | 30 | 17.5 | 5.25 |

6 | Open low-rise | 60–90 | 3–10 | 75 | 6.5 | 4.875 |

7 | Lightweight low-rise | 30–50 | 2–4 | 40 | 3 | 1.2 |

8 | Large low-rise | 10–20 | 3–10 | 15 | 6.5 | 0.975 |

9 | Sparsely built | 20–30 | 3–10 | 25 | 6.5 | 1.625 |

10 | Heavy industry | 20–30 | 5–15 | - | - | 25 * |

Set | Subset of | Description |
---|---|---|

$\mathcal{N}$ | Set of nodes. | |

${\mathcal{A}}_{n}$ | $\mathcal{N}$ | Set of adjacent nodes $\forall n\in \mathcal{N}$. |

$\mathcal{C}$ | $\mathcal{N}$ | Set of consumer nodes. |

$\mathcal{R}$ | $\mathcal{N}$ | Set of reservoir nodes. |

${\mathcal{R}}_{0}$ | $\mathcal{R}$ | Root reservoir node to constrain one connected network. |

$\mathcal{E}$ | $\mathcal{N}\times \mathcal{N}$ | Set of directed edges with start- and end-node ($i,\phantom{\rule{0.166667em}{0ex}}j$). |

$\mathcal{D}$ | Set of available pipe diameters. |

Variable | Domain | Description |
---|---|---|

${q}_{i,j}$ | ${\mathbb{R}}_{0}^{+}$ | Volume flow rate for each directed pipe. $(i,j)$. |

${s}_{n}$ | ${\mathbb{R}}_{0}^{+}$ | Source volume flow $\forall n\in \mathcal{R}$. |

${p}_{n}$ | ${\mathbb{R}}_{0}^{+}$ | Pressure at each node $\forall n\in \mathcal{N}$. |

$\Delta {p}_{i,j}$ | ${\mathbb{R}}_{0}^{+}$ | Pressure loss $\forall i,j\in \mathcal{E}$. |

$\Delta {p}_{i,j}^{\mathrm{pump}}$ | ${\mathbb{R}}_{0}^{+}$ | Pressure increase due to pumps $\forall i,j\in \mathcal{E}$. |

${d}_{i,j,d}$ | $\{0,1\}$ | Decision variable for choosing the edges on which to place pipes and their diameters. $\forall i,j\in \mathcal{E}$, $\forall d\in \mathcal{D}$ |

${c}_{i,j,r}$ | $\{0,1\}$ | Decision variable for choosing a path between all reservoirs to guarantee one connected network $\forall i,j\in \mathcal{E}$, $\forall r\in \mathcal{R}$. |

${b}_{i,j}$ | $\{0,1\}$ | Decision variable for choosing the pump locations $\forall i,j\in \mathcal{E}$. |

Parameter | Domain | Description |
---|---|---|

Scalar parameters | ||

${Q}^{\mathrm{max}}$ | ${\mathbb{R}}_{0}^{+}$ | Maximum volume flow rate |

${P}^{C,\mathrm{min}}$ | ${\mathbb{R}}_{0}^{+}$ | Minimum pressure at consumer nodes. |

$\Delta {P}^{\mathrm{pump},\mathrm{min}}$ | ${\mathbb{R}}_{0}^{+}$ | Minimum pressure increase for pumps. |

${P}^{\mathrm{max}}$ | ${\mathbb{R}}_{0}^{+}$ | Maximum pressure for big-M expression. |

${P}^{\mathrm{stat},\mathrm{max}}$ | ${\mathbb{R}}_{0}^{+}$ | Maximum static pressure to protect pipes. |

${U}^{\mathrm{mean}}$ | ${\mathbb{R}}_{0}^{+}$ | Mean velocity for pressure loss calculation. |

G | ${\mathbb{R}}_{0}^{+}$ | Gravitational constant. |

$\varrho $ | ${\mathbb{R}}_{0}^{+}$ | Density of water. |

${l}^{\mathrm{min}}$ | ${\mathbb{R}}_{0}^{+}$ | Minimum average link density [29]. |

$<O{>}^{\mathrm{min}}$ | ${\mathbb{R}}_{0}^{+}$ | Minimum average node degree [29]. |

${R}^{\mathrm{m},\mathrm{min}}$ | ${\mathbb{R}}_{0}^{+}$ | Minimum meshed-ness coefficient [29]. |

Indexed Parameters | ||

${Q}_{r}^{\mathrm{res},\mathrm{max}}$ | ${\mathbb{R}}_{0}^{+}$ | Maximum volume flow $\forall r\in \mathcal{R}$. |

${Q}_{n}^{\mathrm{demand}}$ | ${\mathbb{R}}_{0}^{+}$ | Water demand $\forall n\in \mathcal{N}$. |

${Q}_{d}^{\mathrm{max}}$ | ${\mathbb{R}}_{0}^{+}$ | Maximum volume flow to limit maximum velocity $\forall d\in \mathcal{D}$. |

${P}_{r}^{\mathrm{res}}$ | ${\mathbb{R}}_{0}^{+}$ | Static pressure $\forall r\in \mathcal{R}$. |

${H}_{n}$ | ${\mathbb{R}}_{0}^{+}$ | Elevation $\forall n\in \mathcal{N}$. |

${L}_{i,j}$ | ${\mathbb{R}}_{0}^{+}$ | Pipe length $\forall (i,j)\in \mathcal{E}$. |

${D}_{d}$ | ${\mathbb{R}}_{0}^{+}$ | Pipe diameter $\forall d\in \mathcal{D}$. |

${r}_{i,j}$ | ${\mathbb{R}}_{0}^{+}$ | Pressure loss coefficient $\forall (i,j)\in \mathcal{E}$. |

${C}_{d}^{\mathrm{pipe}}$ | ${\mathbb{R}}_{0}^{+}$ | Installation cost for pipes $\forall d\in \mathcal{D}$. |

Parameter | Instance | Description |
---|---|---|

Scalar parameters | Value | |

${Q}^{\mathrm{max}}$ | 2.71 m^{3} s^{−1} | Maximum volume flow rate |

${P}^{C,\mathrm{min}}$ | 4 bar | Minimum pressure at consumer nodes. |

$\Delta {P}^{\mathrm{pump},\mathrm{min}}$ | 1 bar | Minimum pressure increase for pumps. |

${P}^{\mathrm{max}}$ | 14.18 bar | Maximum pressure for big-M expression. |

${P}^{\mathrm{stat},\mathrm{max}}$ | 8 bar | Maximum static pressure to protect pipes. |

${U}^{\mathrm{mean}}$ | $1\mathrm{m}{\mathrm{s}}^{-1}$ | Mean velocity for pressure loss calculation. |

G | 9.81 m s^{−2} | Gravitational constant. |

$\varrho $ | 1000 kg m^{−3} | Density of water. |

${l}^{\mathrm{min}}$ | 0.0008 | Minimum average link density [29]. |

$<O{>}^{\mathrm{min}}$ | 2 | Minimum average node degree [29]. |

${R}^{\mathrm{m},\mathrm{min}}$ | 0.04 | Minimum meshed-ness coefficient [29]. |

Indexed Parameters | Range | |

${Q}_{r}^{\mathrm{res},\mathrm{max}}$ | $[0.189,1.12]\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}{\mathrm{s}}^{-1}$ | Maximum volume flow $\forall r\in \mathcal{R}$. |

${Q}_{n}^{\mathrm{demand}}$ | $[0,0.046]\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}{\mathrm{s}}^{-1}$ | Water demand $\forall n\in \mathcal{N}$. |

${Q}_{d}^{\mathrm{max}}$ | $[0.157,1.414]\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{3}{\mathrm{s}}^{-1}$ | Maximum volume flow to limit maximum velocity $\forall d\in \mathcal{D}$. |

${P}_{r}^{\mathrm{res}}$ | $[0,0]\phantom{\rule{0.166667em}{0ex}}bar$ | Static pressure $\forall r\in \mathcal{R}$. |

${H}_{n}$ | $[41,63]\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | Elevation $\forall n\in \mathcal{N}$. |

${L}_{i,j}$ | $[143,5086]\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | Pipe length $\forall (i,j)\in \mathcal{E}$. |

${D}_{d}$ | $[0.2,0.6]\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ | Pipe diameter $\forall d\in \mathcal{D}$. |

${r}_{i,j}$ | $[5.26,769]$ | Pressure loss coefficient $\forall (i,j)\in \mathcal{E}$. |

${C}_{d}^{\mathrm{pipe}}$ | $[219,634]\phantom{\rule{0.166667em}{0ex}}$€$/m$ | Installation cost for pipes $\forall d\in \mathcal{D}$. |

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## Share and Cite

**MDPI and ACS Style**

Rehm, I.-S.; Friesen, J.; Pouls, K.; Busch, C.; Taubenböck, H.; Pelz, P.F.
A Method for Modeling Urban Water Infrastructures Combining Geo-Referenced Data. *Water* **2021**, *13*, 2299.
https://doi.org/10.3390/w13162299

**AMA Style**

Rehm I-S, Friesen J, Pouls K, Busch C, Taubenböck H, Pelz PF.
A Method for Modeling Urban Water Infrastructures Combining Geo-Referenced Data. *Water*. 2021; 13(16):2299.
https://doi.org/10.3390/w13162299

**Chicago/Turabian Style**

Rehm, Imke-Sophie, John Friesen, Kevin Pouls, Christoph Busch, Hannes Taubenböck, and Peter F. Pelz.
2021. "A Method for Modeling Urban Water Infrastructures Combining Geo-Referenced Data" *Water* 13, no. 16: 2299.
https://doi.org/10.3390/w13162299