# Economic Analysis of Flood Risk Applied to the Rehabilitation of Drainage Networks

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## Abstract

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## 1. Introduction

- The hydrologic study and the runoff model are beyond the scope of this work;
- No changes in the network topology are considered. The actions allowed to improve the network are the replacement of pipes, the installation of storm tanks and the inclusion of hydraulic control elements in the network;
- The networks in which this methodology can be applied must be gravity-fed. Networks with pumping systems are not considered in this study;
- The hydraulic model is considered as a datum; its parameters and initial conditions are not questioned.

## 2. Materials and Methods

#### 2.1. Optimization Model

_{max}). This criterion considers that when the value of the objective function does not change during a certain number of generations, the objective function has reached its minimum and the process ends. G

_{max}is explained in depth in the work presented by Bayas-Jiménez et al. [14].

#### 2.1.1. Decision Variables

_{max}. The previously defined list of options for optimization, called ΔND, was composed of a small number of diameters immediately larger than the analyzed pipe diameter. This decision was taken because pipes in the optimization process would only take values larger than the current diameter, so defining lists using the whole range of pipes would only increase the computational effort unnecessarily; with this action, the size of the problem was considerably decreased.

_{0}was used and for the second stage, a refined option list NS

_{max}was used. Finally, the third type of DVs considered the head-loss values that the hydraulic control element installed in the pipes could generate. In the optimization process, the selected pipes v

_{s}could have hydraulic controls with a degree of openness from a previously defined option list Nθ.

#### 2.1.2. Cost Functions

_{p}(D

_{i}) represents the pipe replacement cost in euros, D

_{i}is the pipe diameter and L

_{i}represents the pipe length in meters. The coefficients α, β and γ are adjustment coefficients corresponding to each project.

_{T}(V

_{i}) represents the cost in euros of the construction of a storm tank. C

_{min}represents the minimum cost of building a storm tank. V

_{i}is the flood volume that the tank must store in cubic meters. C

_{var}and ω are coefficients corresponding to the analyzed project. The third structural investment cost function determined for this work was the cost of hydraulic control. To define this function, the cost of purchasing and installing valves of different diameters was analyzed. This analysis determined a second-degree polynomial function, shown in Equation (3).

_{v}(D

_{i}) is the cost of the hydraulic control in euros, D

_{i}is the diameter of the pipe where the hydraulic control would be installed, in meters, and σ, μ and φ are adjustment coefficients of the analyzed project.

_{y}(y

_{i}) represents the damage cost in euros and y

_{max}is the maximum depth at which the flooding reaches the maximum damage in meters. C

_{max}is the maximum flood damage cost obtained when y

_{max}is reached, Ai is the flood area of the analyzed subcatchment in square meters, y

_{i}is the reached depth of the analyzed node in meters and λ and υ are adjustment coefficients based on historical flood damage data.

_{y}(D

_{i})) as a function of its annual exceedance rate (p). In a more recent work, Olsen et al. [21] mention that a good approximation of the curve can be obtained with a log-linear relationship and that, when integrated, it can show the average annual cost of flood risk; Figure 3 shows this relationship.

_{0}representing the annual exceedance rate for which flood damage begins to occur (Equation (7)). Equation (8) shows how to calculate the annual flood risk cost C

_{F}(p).

#### 2.2. Optimization Process

#### 2.2.1. Search Space Reduction

_{0}for nodes. In the case of pipes, and in order to optimize the process, a ΔND list of diameters immediately larger than analyzed pipe was used.

_{max}was used because the objective was to eliminate DVs quickly and not to optimize the network. This process is applied iteratively until DVs could not be reduced further. Once the best regions of each sector were defined, a new search was made in the complete network, with the n

_{s}and m

_{s}selected in the sectors to finally delimit the optimal search region. It should be noted that the clustering method required significant computational use, so the method required the use of a cluster server to take advantage of its characteristics. Thus, despite the fact that several analyses were carried out, when they were carried out at the same time, and when working with sectors with fewer DVs, results were achieved in less time.

#### 2.2.2. Final Optimization

_{max}was used for STs. In the case of pipes, the same ΔND range of diameters immediately larger than the analyzed pipes was used. Finally, for the HCs, a list of options Nθ was used that defined the degrees of opening that the element could adopt. At this stage, a more demanding G

_{max}was also used to find the closest solution to the optimum. This is the process that was followed to obtain the solution to the problem analyzed with the proposed methodology.

#### 2.3. Case Studies

#### 2.3.1. Balloon Network

#### 2.3.2. ES-N Network

#### 2.3.3. Investment Costs

#### 2.3.4. Flood Costs

## 3. Results

#### 3.1. Balloon Network

#### 3.2. ES-N Network

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DV | Decision Variable |

EAD | Estimated Annual Damage |

IDF | Intensity–Duration–Frequency |

LID | Low-Impact Development |

OF | Objective Function |

PGA | Pseudo-Genetic Algorithm |

SS | Search Space |

SSR | Search Space Reduction |

SWMM | Storm Water Management Model |

Nomenclature | |

a | coefficient of the line of the flood cost |

A_{i} | flood area |

b | coefficient of the line of the flood cost |

C_{max} | maximum flood damage cost |

C_{min} | minimum cost of building a storm tank |

C_{P} (D_{i}) | cost of pipe replacement |

C_{T} (V_{i}) | cost of building a storm tank |

C_{v} (D_{i}) | cost of installing hydraulic controls |

C_{var} | adjustment coefficient for calculating the cost of storm tank |

C_{y} (y_{i}) | flood damage cost |

D_{i} | pipe diameter |

G_{max} | convergence criterion |

L_{i} | pipe length |

m_{s} | pipes selected to be optimized |

ND_{max} | diameter range available |

n_{s} | nodes selected to be optimized |

NS | list of options used for nodes |

NS_{max} | refined option list for nodes |

NS_{0} | coarse option list for nodes |

Nθ | option list for hydraulic controls |

p | annual exceedance rate |

p_{0} | annual exceedance rate for which flood damage begins to occur |

r | annual interest |

t | years to recover the investment |

T | return period |

V_{i} | flood volume at the node |

v_{s} | pipes selected to install hydraulic controls in the optimization process |

y_{i} | flood depth at node |

y_{max} | maximum depth at which the maximum cost of flood damage is reached |

α | adjustment coefficient for calculating the cost of replacing pipes |

β | adjustment coefficient for calculating the cost of replacing pipes |

γ | adjustment coefficient for calculating the cost of replacing pipes |

ΔND | range of diameters immediately larger than the analyzed pipe |

Λ | annual amortization factor |

λ | adjustment coefficient for calculation of flood damage |

μ | adjustment coefficient for calculating the cost of installing hydraulic controls |

σ | adjustment coefficient for calculating the cost of installing hydraulic controls |

υ | adjustment coefficient for calculation of flood damage |

φ | adjustment coefficient for calculating the cost of installing hydraulic controls |

ω | adjustment constant for the calculation of the cost of the construction of storm tanks |

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**Figure 1.**Damage–Depth curve (

**a**), Damage—annual exceedance rate curve (

**b**) and Flood risk density curve (

**c**).

**Figure 15.**Results of the Objective Function in each iteration analyzed. (

**a**) Balloon network. (

**b**) ES-N network.

**Table 1.**Elements, magnitude of SS and percentage of reduction of the sectors that make up the Balloon network.

Sector | Number of Pipes | Number of Nodes | DV | SS | Reduction of SS in Clustering Process |
---|---|---|---|---|---|

Sector 1 | 11 | 11 | 33 | 36 | 100% |

Sector 2 | 12 | 12 | 36 | 40 | 96% |

Sector 3 | 4 | 4 | 12 | 13 | 100% |

Main network | 44 | 43 | 130 | 144 | |

Total | 71 | 70 | 211 | 233 |

Terms in Objective Function | Pipes | Storm Tank | Hydraulic Control | Flood | Total | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Cost per year | 2739 € | 19,727 € | 153 € | 5608 € | 28,227 € | ||||||

Elements | C5 | C84 | C85 | C86 | N8 | N15 | N16 | N63 | C75 | ||

Present diameter (m) | 0.70 | 0.70 | 0.70 | 0.70 | |||||||

Optimized diameter (m) | 1.00 | 1.10 | 1.00 | 1.00 | |||||||

Volume (m^{3}) | 2496 | 4284 | 5031 | 594 | |||||||

Head-loss (m) | 72.55 |

**Table 3.**Elements, magnitude of SS and percentage of reduction in the clustering stage of the sectors that make up the ES-N network.

Sector | Number of Pipes | Number of Nodes | DVs | SS | Reduction of SS in Clustering Process |
---|---|---|---|---|---|

Sector 1 | 5 | 5 | 15 | 17 | 100% |

Sector 2 | 2 | 2 | 6 | 7 | 100% |

Sector 3 | 13 | 13 | 39 | 43 | 100% |

Sector 4 | 23 | 23 | 69 | 76 | 98% |

Sector 5 | 24 | 24 | 72 | 79 | 91% |

Sector 6 | 8 | 8 | 24 | 26 | 100% |

Sector 7 | 55 | 55 | 165 | 182 | 99% |

Sector 8 | 8 | 8 | 24 | 26 | 100% |

Sector 9 | 15 | 15 | 45 | 50 | 100% |

Sector 10 | 23 | 23 | 69 | 76 | 100% |

Sector 11 | 25 | 25 | 75 | 83 | 100% |

Sector 12 | 16 | 16 | 48 | 53 | 97% |

Sector 13 | 9 | 9 | 27 | 30 | 100% |

Sector 14 | 5 | 5 | 15 | 17 | 100% |

Sector 15 | 39 | 39 | 117 | 129 | 97% |

Sector 16 | 45 | 45 | 135 | 149 | 97% |

Sector 17 | 4 | 4 | 12 | 13 | 100% |

Sector 18 | 12 | 12 | 36 | 40 | 75% |

Sector 19 | 5 | 5 | 15 | 17 | 100% |

Main network | 49 | 49 | 147 | 162 | |

Total | 385 | 385 | 1155 | 1271 |

Terms in Objective Function | Pipes | Storm Tank | Hydraulic Control | Flood | Total | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Cost per year | 2672 € | 41,930 € | 125 € | 27,722 € | 72,449 € | |||||||

Elements | P266 | P293 | P294 | P335 | N71 | N126 | N131 | N161 | N216 | P25 | ||

N252 | N276 | N308 | N343 | |||||||||

Present diameter (m) | 0.40 | 0.40 | 0.25 | 0.30 | ||||||||

Optimized diameter (m) | 0.70 | 0.70 | 0.45 | 0.50 | ||||||||

Volume (m^{3}) | 1700 | 500 | 750 | 1950 | 1950 | |||||||

1100 | 1250 | 1050 | 800 | |||||||||

Head-loss (m) | 72.55 |

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**MDPI and ACS Style**

Bayas-Jiménez, L.; Martínez-Solano, F.J.; Iglesias-Rey, P.L.; Boano, F. Economic Analysis of Flood Risk Applied to the Rehabilitation of Drainage Networks. *Water* **2022**, *14*, 2901.
https://doi.org/10.3390/w14182901

**AMA Style**

Bayas-Jiménez L, Martínez-Solano FJ, Iglesias-Rey PL, Boano F. Economic Analysis of Flood Risk Applied to the Rehabilitation of Drainage Networks. *Water*. 2022; 14(18):2901.
https://doi.org/10.3390/w14182901

**Chicago/Turabian Style**

Bayas-Jiménez, Leonardo, F. Javier Martínez-Solano, Pedro L. Iglesias-Rey, and Fulvio Boano. 2022. "Economic Analysis of Flood Risk Applied to the Rehabilitation of Drainage Networks" *Water* 14, no. 18: 2901.
https://doi.org/10.3390/w14182901