# Urban Drainage Network Rehabilitation Considering Storm Tank Installation and Pipe Substitution

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

- The computational models for the drainage networks are going to be tested with several rain scenarios, scenarios based on different climate change predictions [22].Potentially dangerous scenarios are the ones to be considered during the optimization process. The first selected rainfall is a synthetic design rainfall obtained from the IDF curves of 10-year return periods using the alternate block method. The second rainfall is obtained after processing the first through a climate change adaptive scenario. The rehabilitation will be carried out considering only the worst-case scenario.
- The rainfall-runoff transformation model used is the one included in the SWMM model. Specifically, the Curve Number model is used. For this, according to the characteristics of the terrain, the curve number has been specified for each of the sub catchments defined in the model.
- The drainage system models must go through a calibration process, since the analysis must be as accurate as possible. That is, the starting point of the process is a calibrated hydraulic model of the drainage network. The hydraulic model used would be SWMM. Traditionally, this type of simulation is performed considering uniform flow. However, in this case, each configuration is analyzed using the dynamic wave model, because it provides a better representation of floods than the kinetic wave model or uniform model.
- A simplification process is necessary for every model, yet the accuracy of the result must not be compromised. This simplification will highly reduce computational times for every hydraulic simulation.
- The optimization problem will be addressed in monetary units. Thus, the first step would be to find the cost functions that characterize the value of hydraulic variables in monetary units. So, the functions that together form the optimization total cost problem are: pipe replacement cost, ST installation cost and total flood damage cost as introduced by Cunha et al. [23].
- From all described mathematical approaches, it seems heuristics approaches can give the best advantages for the process. Therefore, based on previous experience [24], a PGA method was used.

#### 2.1. Decision Variables

_{C}is the number of network conduits; m is the number of feasible conduits selected to be replaced, varying between 1 and N

_{C}; and ND is the number of candidate diameters, between ND

_{0}and ND

_{max}.

_{S}, B

_{S}and C

_{S}are characteristic coefficients that adjust the tank’s section to different expressions, and z is the water level of the node. In the case of considering tanks of constant section, A represents the cross section while the coefficients B and C are null. However, considering tanks with variable section does not imply a major difficulty in the problem implementation beyond choosing the right DVs.

_{N}is the number of network nodes and n is the number of nodes selected to potentially install a ST, varying between 1 and N

_{N}. Each node in which an ST can be installed has the cross section (S) as DV. Since a heuristic optimization model is used, it is necessary to perform a discretization of S. For this reason, a maximum value of the tank cross section (S

_{max}) is defined for each node. In this way, N is the number of divisions in which S

_{max}is divided. Therefore, N determines the resolution of the section S, varying between N

_{0}and N

_{max}. A simulation performed with the ST cross section divided into N

_{0}parts is faster than a simulation performed N

_{max}divisions. So, to obtain better calculation times, the number of divisions of the ST cross section could be reduced.

#### 2.2. Objective Function

- The investment cost related to the substitution of each selected pipe of the network.
- The investment cost linked to required volumes of STs to be installed in each solution.
- The damage cost caused by the flooding level in various nodes of the network.

_{DR}(j) of the n STs installed in the drainage network. This cost concerns the existing STs whose volume will be expanded, and the network nodes where new STs will be installed. The third term represents the total flood damages costs [27] caused by the N

_{F}nodes in which a certain flooding volume V

_{I}(i) appears. All the terms of the objective function have a weight coefficient λ

_{i}, in order to prioritize one term versus another. If λ

_{i}is minimum (eventually null) the term is not considered, but if λ

_{i}is greater, the term is considered in the objective function.

#### 2.2.1. Pipe Replacement Cost Functions

_{nit}cost of the pipes with their diameter. Finally, the pipe substitution cost is in the form of a second-grade polynomial:

#### 2.2.2. Storm Tank Installation Cost Functions

_{DR}) is the cost associated with the installation of an ST of volume V

_{DR}and A, B and C are coefficients that must be adjusted in each case according to the specifics of the project.

#### 2.2.3. Flood Damage Cost functions

^{2}of area as a function of the flood level was obtained for 6 different social stratums, a commercial area and an industrial area [23]. There is also a curve representing the average value of the study area. They can be observed in Figure 2.

_{max}represents the maximum cost, when flood level y

_{max}is reached. y is the existing flood level in the specific node; λ = 4.89 and b = 2 are adjustment coefficients of the curve; and the parameter y

_{max}= 1.4 is the level from which the maximal economic damage is produced. In all the cases, Equation (6) depends totally on the value of C

_{max}, which is presented in Table 1, for different land uses and represents the maximum per area unit.

_{f}) in each node. In this way, the flood level is obtained as the relation between the flood volume V

_{f}and the area A

_{f}.

## 3. Methodology

_{0},…,N

_{max}). On the other hand, the number of candidate pipe diameters was ND (ND = ND

_{0},…,ND

_{max}).

- Reduce the number of nodes (n) in which STs could potentially be installed.
- Reduce the number of lines (m) in which there could potentially be a change in diameter.
- Reduce the discretization N that is made of the section of each of the STs.
- Reduce the number of candidate diameters ND in the pipes.

#### 3.1. Pre-locating Storm Tanks

_{it}) are performed with all the nodes of the network (n = N

_{N}), but without including the diameters in the optimization process (m = 0). Thus, N

_{it}optimizations are made, considering only the cross section (S) of the tanks as DVs and without modifying the diameters of the network. Since the objective is the reduction of the SS, the discretization of the cross sections of the tanks (S) is carried out with the smallest number of divisions (N = N

_{0}). This coarse discretization of each section is carried out since the objective is not to calculate its exact value, but to determine in which nodes the installation of an ST is adequate. That is, the objective of this step is selecting the nodes where STs could be installed in the rehabilitation of the network, i.e., a pre-location of STs.

_{n}of the best simulations is selected. The analysis of these simulations allows identifying the nodes where an ST could be installed and a list of n

_{s}possible locations is created. These nodes are selected because they are repeated as the location of an ST in all the p

_{n}selected solutions.

#### 3.2. Locating Lines of Possible Pipe Substitutions

_{it}optimizations is run. The DVs of each optimization are the cross section of the n

_{s}selected nodes in the previous process and the N

_{c}conduits of the network. For both types of variables the reduction of the SS is applied. Therefore, the discretization of the cross section of the tanks is the coarsest (N = N

_{0}) and the range of pipes available is the smallest (ND = ND

_{0}). In this step, the aim is to find a pre-location of pipes to be substituted.

_{m}of the best solutions are selected. The conduits selected are those that appear repeated in the solutions defined by the percentage p

_{m}. Analyzing these solutions, the list of m

_{s}pipes whose replacement is repeated in the p

_{m}best solutions is selected.

#### 3.3. Final Optimization, Location and Optimization of Storm Tanks and Pipe Diameters

_{s}selected nodes and the m

_{s}selected lines. Although the number of DVs is smaller, the exploration of each of these variables must now be greater. Therefore, the STs are discretized to the maximum (N = N

_{max}) and the list of candidate diameters for the conduits is also the largest (ND = ND

_{max}). This final optimization determines the location and size of the STs to be installed and the diameters of the pipes to be rehabilitated.

## 4. Case Study

^{3}, representing 18% of the total runoff of the network (21,233 m

^{3}). Figure 5 shows the nodes in which the main floods occur (flood level over 10 cm), indicating the flood volume V; the maximum level y reached by the water in the node and the cost associated with flood damage. Table 2 shows the detail of the flooded nodes: the flood volume, area and level, and the damage cost obtained from Equation (6). The nodes shown in Figure 5 are highlighted in bold in Table 2.

#### 4.1. Application of the Drainage Network Rehabilitation Methodology to E Chico

- Scenario 1: Rehabilitation of the network based only on the modification of conduits of the network and substituted them by another with different diameter. This scenario has 35 DVs, as all the conduits are potentially changed.
- Scenario 2: Rehabilitation of the network by installing only STs. This scenario also has 35 DVs, corresponding to the 35 potential nodes in which STs can be installed. The maximum section to be installed in the tank potentially installable is defined in each node. Subsequently, the optimization method selects the cross section according to the discretization (N) of this variable. It should be noted that the section minimum value corresponds to a diameter of 1.2 meters, corresponding to the cross-sectional value of a manhole.
- Scenario 3: Rehabilitation of the network combining the installation of conduits and STs. The number of DVs is 70.

_{max}= 25, since it must be combined with an additional state corresponding to the event of leaving the drainage line as it is in the model; that is, without rehabilitation. Table 4 also shows the installation cost of each diameter. These costs are obtained from the application of Equation (4) with the coefficients defined in Table 3.

#### 4.2. Application of the Solution Space Reduction Methodology to E-Chicó

- The number of simulations defined is one hundred (N
_{it}= 100). - The discretization of the ST area is reduced to its minimum value (N = N
_{0}= 10). - Only the sections of the tanks potentially to be installed in the nodes of the network are considered as DVs (n = N
_{N}= 35). - No conduit can be modified during the process (m = 0).
- The basic parameters of the PGA algorithm, population size (N
_{pop}) and the end criterion based on a number of generations (N_{gen}) without change, are fixed at 100.

_{n}of solutions with the best value in the objective function is selected. In short, the 10 best solutions are selected. In each one, it is analyzed in which nodes an ST is installed. This generates a list of pre-locations of STs in the network. In this case, this list contains a total of 15 possible locations of the STs. In Figure 6, the shaded cells represent the selected nodes.

_{max}= 40). The results obtained lead to the same list of selected nodes as indicated in Figure 6.

- The number of simulations is the same as in the previous stage (N
_{it}= 100). - The DVs are the areas of the ns nodes selected in the first stage and the diameters of all the conduits (n = N
_{C}= 35) of the network. - The discretization of the area of the STs is kept at the minimum value, as it happened with the previous stage of the process.
- The basic parameters of the PGA algorithm are the same as in the previous phase (N
_{pop}= 100, N_{gen}= 100). - Instead of using the full range of diameters (Table 4), a range of reduced diameters is used, the details of which can be seen in Table 5. This table shows the diameters D of the reduced range and the unit costs (C) obtained from the application of the cost function (4) with the parameters of the Table 3. Note that although Table 5 only has 9 values, the number of options for the DV is ND = 10. This additional value corresponds to the option of not taking any action on the pipe.

_{m}

_{1}= 10%) and another one considering only 5% of solutions with a lower value of the objective function (p

_{m}

_{2}= 5%). The preselected lines in each case are shown in Figure 7. In the left part of the figure, the values corresponding to 10% are collected, highlighting the preselected lines by gray color. On the right hand side, the pipes selected for 5% of the best solutions obtained in this process are represented in an analogous way.

_{max}= 25). The results obtained, in terms of the lines that would be pre-selected, are the same as those obtained in the case of using only the reduced range (ND = ND

_{0}= 10).

- Scenario 4: Rehabilitation of the network combining the possible installation of STs in the 15 selected nodes and the 15 conduits that can be substituted.
- Scenario 5: It is the same scenario as the previous one (scenario 4), with the difference that the number of potentially replaceable conduits is only 8.

#### 4.3. Results Analysis

^{3}) and flood levels are very low. Therefore, from a practical point of view it can be considered that the solutions are acceptable.

_{max}) according to Equation (2). Therefore, in Table 9 the size of the problem of each scenario is presented. The table also includes a measurement of the calculation time, expressed as the number of times it is necessary to evaluate the objective function.

#### 4.4. Sensitivity Analysis of the SS Reduction

_{pob}= 100 and a termination criterion N

_{gen}= 100. In order to validate the selection, a sensitivity analysis of the selection process of STs and conduits has been carried out. The process of reducing the SS was repeated with different values of N

_{pop}and N

_{gen}. The different values of the population size were 35, 50, 75, 100 and 300 elements, while the values used as finalization criteria were 50, 100 and 150 generations without a change in the value of the objective function. The results obtained are shown in the following figures. In Figure 10, the nodes that could potentially be a ST location are collected for each combination of values of N

_{pop}and N

_{gen}. On the other hand, Figure 11 shows the lines potentially replaceable. In both cases, the node or line selected has been indicated with an X in the corresponding cell.

_{pop}and N

_{gen}the preselected nodes would have been almost the same, and in any case those that appear in the final solutions of scenarios 3, 4 and 5 would have always been selected.

_{pop}and N

_{gen}are almost the same. That is, selected lines are those that are finally found in the final solutions of scenarios 3, 4 and 5. Moreover, N

_{pop}and N

_{gen}parameters have no influence on the pre-location of STs or on the pre-selection of pipes. Therefore, the SS reduction methodology can be considered reliable.

## 5. Conclusions

- The increase in rainfall intensities caused by climate change causes originally well-designed networks to present flooding problems. The use of STs has been shown as an effective technique to solve this problem. However, the effectiveness of this method is even greater when it is combined with the rehabilitation of some pipes of the drainage network. The results shown by scenarios 1, 2 and 3 presented above show how the combined action of STs and pipe renewal (scenario 3) is much more effective than the isolated action of any of the other two actions (scenarios 1 and 2).
- The number of DVs required for the rehabilitation of a drainage network can be very high. This may cause (Figure 9) the optimization model to give solutions that are quite dispersed and sometimes far from the optimal values required. It is well known that the importance of meta-heuristic algorithms does not lie in the optimality of their solutions, but in the ability to obtain large sets of good solutions. That is why the disparity of solutions observed with large numbers of DVs can become a problem.
- An alternative methodology has been proposed for the reduction of the SS that permit us to locate solutions that are better and less dispersed than those obtained with the initial model. This methodology is based on the pre-location of STs and the pre-selection of possible conduits. The methodology uses the PGA developed, reducing the SS by using a smaller discretization of the size of the STs, a smaller range of diameters of pipes, and a smaller number of elements (tanks and pipes). The final result has proven to be effective for the cases analyzed, not only because it reduces SS and computing times, but also because it obtains better solutions (scenario 4 and 5). Likewise, the solutions obtained with the scenarios that use a reduced SS (scenarios 4 and 5) are much more concentrated and less dispersed (Figure 9) than those obtained initially.
- The SS reduction methodology in the case study has been shown to be reliable and stable since variations in the N
_{pop}and N_{gen}parameters of the PGA model hardly modify the preselected lines and nodes.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DV | Decision variable |

GA | Genetic Algorithm |

IDF | Intensity, Duration, Frequency |

NSGA | Non-dominated Sorting Genetic Algorithm |

PGA | Pseudo-Genetic Algorithm |

SS | Search Space, Solution Space |

ST | Storm Tanks |

SWMM | Storm Water Management Model |

WTP | Wastewater Treatment Plant |

## Notation

α, β | characteristic coefficients for the cost function related to the installation of new conduits |

A, B | characteristic coefficients for the cost function related to the installation of a new ST of volumen V_{DR} |

A_{f} | ponded area of a node to represent the flood level |

A_{S}, B_{S,} C_{S} | characteristic coefficients of tank cross section equation |

b | adjusts coefficients of the damage cost curve |

C(DN(i)) | ccost related to the installation of the diameter i |

C(V_{DR}(j)) | cost related to the construction or expansion of the volumen V_{DR}(j) |

C(y) | damage cost related to a flood level y |

C_{max} | maximum economic damage cost, when flood level y_{max} is reached |

G | number of generations of the PGA |

L_{i} | length of the conduit i |

λ | adjusts coefficients of the damage cost curve |

λ_{i} | Lagrange multipliers of the objective function |

m | number of feasible conduits selected to be replaced, varying between 1 and N_{C} |

m_{s} | number of possible pipe locations created after the pre-selecting pipes location step |

n | number of nodes selected to potentially install a ST, varying between 1 and N_{N}. |

N | number of divisions for the cross section S, varying between N_{0} and N_{max} |

N_{C} | number of conduits of the network |

N_{D} | number of candidate diameters, between ND_{0} and ND_{max} |

N_{gen} | stop criteria: maximal number of generations G without change in the objective function value |

N_{it} | number of optimizations (runs) of the PGA algorithm to develop one of the steps of the SS reduction method |

N_{N} | number of nodes of the network |

N_{pop} | population size of the PGA |

n_{s} | number of possible ST locations after the pre-locating STs step |

p_{m} | percentage of the best solutions selected in the pre-selecting pipes location step |

p_{n} | percentage of the best solutions selected in the pre-locating STs step |

PS_{max} | maximum size of the optimization problem |

S | tank’s cross section |

S_{max} | maximum value of the tank cross section |

V_{DR}(j) | volume of the j-th ST |

V_{f} | volume of water flooded in a node |

y | existing flood level on the specific node |

y_{max} | level from which the maximal economic damage is produced |

z | water level in a storm tank |

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**Figure 1.**Flow chart of the optimization methodology based on the PGA algorithm and the toolkit of the SWMM model.

**Figure 11.**Sensitivity analysis results for pre-selecting conduits with different N

_{pop}and N

_{Gen}.

Land Use | Str. 1 | Str. 2 | Str. 3 | Str. 4 | Str. 5 | Str. 6 | Commercial | Industrial | Average |
---|---|---|---|---|---|---|---|---|---|

C_{max} (€/m^{2}) | 142 | 245 | 257 | 584 | 732 | 1168 | 3975 | 3041 | 1268 |

Node. | Flood Volume (m^{3}) | Flood Area (m^{2}) | y_{max} (m) | C(€) |
---|---|---|---|---|

N02 | 123.56 | 1240 | 0.100 | 135,857 |

N04 | 132.56 | 930 | 0.143 | 181,375 |

N06 | 501.79 | 1890 | 0.265 | 875,502 |

N07 | 23.95 | 1250 | 0.019 | 6644 |

N09 | 1.82 | 1130 | 0.002 | 45 |

N10 | 385.12 | 700 | 0.550 | 646,838 |

N11 | 25.83 | 820 | 0.032 | 11,288 |

N23 | 949.54 | 450 | 2.110 | 569,922 |

N32 | 36.65 | 1500 | 0.024 | 12,727 |

N33 | 469.82 | 3030 | 0.155 | 671,908 |

N34 | 1181.87 | 3270 | 0.361 | 2,131,929 |

TOTAL | 3832.51 | 5,244,034 |

α | Β | A | B | C |
---|---|---|---|---|

40.69 | 208.06 | 16923 | 318.4 | 0.65 |

D (mm) | 300 | 350 | 400 | 450 | 500 | 600 | 700 | 800 | 900 | 1000 | 1100 | 1200 |

C (€/m) | 30.93 | 39.73 | 49.56 | 60.44 | 72.36 | 99.31 | 130.43 | 165.71 | 205.15 | 248.75 | 296.51 | 348.43 |

D (mm) | 1300 | 1400 | 1500 | 1600 | 1800 | 1900 | 2000 | 2200 | 2400 | 2600 | 2800 | 3000 |

C (€/m) | 404.51 | 464.76 | 529.16 | 597.73 | 747.35 | 828.4 | 913.61 | 1096.52 | 1296.07 | 1512.27 | 1745.11 | 1994.6 |

D (mm) | 300 | 400 | 600 | 800 | 1000 | 1200 | 1500 | 1800 | 2000 |

C (€/m) | 30.93 | 49.56 | 99.31 | 165.7 | 248.74 | 348.43 | 529.16 | 747.35 | 913.61 |

Scenario | No. DVs | Objective Function | Terms in the Objective Function | No. Elements in the Solution | ||||
---|---|---|---|---|---|---|---|---|

Nodes | Lines | Floods | STs | Pipes | STs | Pipes | ||

1 | 0 | 35 | 791,214 | 24,753 | 0 | 766,461 | 0 | 21 |

2 | 35 | 0 | 273,455 | 5392 | 268,063 | 0 | 6 | 0 |

3 | 35 | 35 | 268,292 | 20,238 | 230,087 | 17,968 | 4 | 4 |

4 | 15 | 15 | 245,547 | 8353 | 213,133 | 24,061 | 4 | 5 |

5 | 15 | 8 | 213,981 | 12,701 | 186,353 | 14,927 | 3 | 3 |

Node | Flood Volume (m^{3}) | Flood Area (m^{2}) | y_{max} (m) | C (€) |
---|---|---|---|---|

N10 | 3.77 | 700 | 0.013 | 307.94 |

N32 | 12.96 | 1500 | 0.010 | 1680.29 |

N33 | 36.04 | 3030 | 0.009 | 6364.57 |

TOTAL | 52.77 | 8352.80 |

Node | Flood Volume (m^{3}) | Flood Area (m^{2}) | y_{max} (m) | C (€) |
---|---|---|---|---|

N02 | 12.82 | 1240 | 0.013 | 3024.66 |

N03 | 10.82 | 1080 | 0.010 | 1619.42 |

N32 | 12.96 | 1500 | 0.009 | 1681.56 |

N33 | 36.08 | 3030 | 0.012 | 6775.01 |

TOTAL | 75.78 | 12,700.64 |

Scenario | No. DVs | Problem Size | No. Evaluations of the Objective Function |
---|---|---|---|

1 | 35 | 4.0 × 10^{38} | 51,200 |

2 | 35 | 5.8 × 10^{61} | 65,485 |

3 | 70 | 2.3 × 10^{100} | 709,000 |

4 | 30 | 2.8 × 10^{76} | 205,300 |

5 | 23 | 4.1 × 10^{69} | 151,300 |

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**MDPI and ACS Style**

Ngamalieu-Nengoue, U.A.; Iglesias-Rey, P.L.; Martínez-Solano, F.J.; Mora-Meliá, D.; Saldarriaga Valderrama, J.G.
Urban Drainage Network Rehabilitation Considering Storm Tank Installation and Pipe Substitution. *Water* **2019**, *11*, 515.
https://doi.org/10.3390/w11030515

**AMA Style**

Ngamalieu-Nengoue UA, Iglesias-Rey PL, Martínez-Solano FJ, Mora-Meliá D, Saldarriaga Valderrama JG.
Urban Drainage Network Rehabilitation Considering Storm Tank Installation and Pipe Substitution. *Water*. 2019; 11(3):515.
https://doi.org/10.3390/w11030515

**Chicago/Turabian Style**

Ngamalieu-Nengoue, Ulrich A., Pedro L. Iglesias-Rey, F. Javier Martínez-Solano, Daniel Mora-Meliá, and Juan G. Saldarriaga Valderrama.
2019. "Urban Drainage Network Rehabilitation Considering Storm Tank Installation and Pipe Substitution" *Water* 11, no. 3: 515.
https://doi.org/10.3390/w11030515