Network Structure Optimization Method for Urban Drainage Systems Considering Pipeline Redundancies

This section introduces the methods used including layout selection, hydraulic design, and complex network analysis, and then applies them to a study area in Dongying City of Shandong Province in China.

2.1

Overview

The research framework is shown in Fig. 1. The proposed framework primarily consists of three steps: (1) determine the initial layout of the urban drainage system by applying graph theory algorithm; (2) obtain the optimal hydraulic design according to the adaptive genetic algorithm; and (3) determine the crucial nodes for increasing loop and redundancy based on the complex network analysis, and evaluate the resilience performance. The evaluation mainly focuses on the performance of urban drainage systems when subjected to functional failure. Thus, two indicators were selected: total overflow volume (TOV) and mean flood duration (MFD) (Mugume, Gomez, et al. 2015). The TOV is the volume of stormwater that flows out of the drainage channel when the runoff discharge is more than the drainage capacity.

Overview of the proposed research framework. Termination criteria (1): get all the subgraphs; termination criteria (2): all layouts in Ω are hydraulically designed. SWMM storm water management model; TOV total overflow volume; MFD mean flood duration

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2.2

Methodology

The urban drainage system network is created using layout selection and hydraulic design as a foundation, and complex network analysis is used to further optimize the network structure.

2.2.1

Layout Selection

Layout selection mostly involves determining the position and quantity of manholes, choosing the pipes involved, and determining the water flow direction. In general, graph theory algorithms are used to choose layouts (Navin and Mathur 2016; Turan et al. 2019). A graph is a collection of vertices and edges. Manholes and pipelines are analogous to vertices and edges, respectively. As the foundational graph, all potential pipes are connected. The loop-by-loop cutting algorithm forms a feasible tree layout by cutting the undirected base graph step-by-step (Haghighi 2013). Figure 2 is a summary of the layout selection methodology used in this study: (1) to construct the base graph G (V, E), where V is the collection of vertices and E is the collection of edges, and identify all the loops in the base graph; (2) to take any loop, remove an edge in the loop, and update the remaining edges (maintain connectivity during the edge removal process); (3) to check if there are still loops in the graph, and if so, go back to (2), if not, go to (4); and (4) to get all subgraphs that do not have loops (Fig. 2).

2.2.2

Hydraulic Design

The hydraulic design is mainly used to determine the diameter and slope, with the criteria of avoiding the use of pumping stations and pressurized pipes.

1. (1)

   
   

   Objective function

   The problem of optimization of an urban drainage system may be expressed as:

   $${\text{Minimize}}\,{\text{F}} = \mathop \sum \limits_{i = 1}^{N} C_{i} \left( {D_{i} ,H_{i} } \right) \times L_{i}$$

   (1)

   where F is the objective function; N is the total number of pipes; Ci is the construction cost of pipe i; Di is the diameter; Hi is the buried depth; and Li is the length of the pipe.

2. (2)

   
   

   Design constraints

   The hydraulic design of the urban drainage system needs to meet the corresponding pipe diameter constraints, flow velocity constraints, and buried depth constraints:

   $$D_{\min } \le D \le D_{\max }$$

   (2)

   $$D_{{{\text{down}}}} \ge D_{{{\text{up}}}}$$

   (3)

   $$D \in D_{s} = \left\{ {D_{1} ,D_{2} ,D_{3} , \ldots ,D_{Z} } \right\}$$

   (4)

   $$v_{\min } \le v \le v_{\max }$$

   (5)

   $$dh_{\min } \le H \le dh_{\max }$$

   (6)

   where Dmin is the minimum pipe diameter; D is the pipe diameter; Dmax is the maximum pipe diameter; Ddown is the downstream pipe diameter; Dup is the upstream pipe diameter; Ds is an optional set of pipe diameters; v is the flow velocity; vmin is the minimum flow velocity; vmax is the maximum flow velocity; dhmin is the minimum buried depth; H is the buried depth; and dhmax is the maximum buried depth.

3. (3)

   
   

   Adaptive genetic algorithm for optimization

   Genetic algorithm was originally proposed by Holland (1975), and it is often used to solve combinatorial optimization problems with particularly large solution space after development. Genetic algorithm originated from Darwin’s theory of biological evolution. It searches for optimal solutions by simulating the process of natural selection and biological evolution. Genetic algorithms are used for optimal hydraulic design optimization (Palumbo et al. 2013; Hassan et al. 2018). Figure 3 illustrates the steps of adaptive genetic algorithm for hydraulic optimization.

   • Integer coding. Integer encoding can improve computational efficiency.

   • Generate initial population. The initial population is the initial solution generated according to the coding rules. The individuals in the initial population are the parameters of the pipes.

   • Decoding. The related parameters are decoded for the hydraulic calculation of the pipes.

   • Fitness evaluation. The selection of the fitness function directly affects the convergence speed of the genetic algorithm and whether the optimal solution can be found. Every time fitness is calculated, it will be sorted from largest to smallest. In this study, the reciprocal of the cost of the network is selected as the objective function. The fitness function is expressed as:

     $$f = \frac{a}{F\left( x \right) + G}$$

     (7)

     $$G = P_{i} \times \, N \, \times \, C_{\max } \times \, L_{\max }$$

     (8)

     where f is the fitness; G is the penalty function, when the constraints are not met, Pi = 1, and the penalty function is executed on the fitness; a is the coefficient (mainly because the construction cost of the drainage pipe network is relatively high, to avoid the reduction of the optimization potential due to inadequate adaptation to obtain a local optimum).

   • Crossover. Adaptively adjust the crossover probability. The crossover probability calculation formula is:

     $$P_{c} = \left\{ \begin{gathered} P_{c\max } - \frac{{\left( {P_{c\max } - P_{c\min } } \right)\left( {f - f_{\min } } \right)}}{{f_{\max } - f_{\min } }},\quad f \ge f_{{{\text{avg}}}} \hfill \\ P_{c\max } , \quad f < f_{{{\text{avg}}}} \hfill \\ \end{gathered} \right.$$

     (9)

     where Pc is the crossover probability; Pcmax is the maximum crossover rate; Pcmin is the minimum crossover rate; fmax is the maximum fitness; fmin is the minimum fitness; and favg is the average fitness.

   • Mutation. Adaptively adjust the mutation probability. The probability of mutation is calculated as:

     $$P_{m} = \left\{ \begin{gathered} P_{m\max } - \frac{{\left( {P_{m\max } - P_{m\min } } \right)\left( {f - f_{\min } } \right)}}{{f_{\max } - f_{\min } }},\quad f \ge f_{{{\text{avg}}}} \hfill \\ P_{m\max } ,\quad f < f_{{{\text{avg}}}} \hfill \\ \end{gathered} \right.$$

     (10)

     where Pm is the probability of mutation; Pmmax is the maximum rate of mutation; and Pmmin is the minimum rate of mutation.

   Fig. 3

   Flowchart of adaptive genetic algorithm

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The termination criterion of the algorithm is to achieve the preset number of iterations.

2.2.3

Complex Network Analysis

Two-layer complex network analysis is developed, consisting of a global network analysis for all nodes and a local network analysis applied individually for each node (Fig. 4).

Relationship between the global network analysis and the local network analysis

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(1) Global network analysis

Global network analysis is applied to find the crucial nodes in the urban drainage system. The role of particular nodes in the graph and their effects on the network may be determined using centrality, which can aid in the identification of significant nodes. Betweenness centrality and closeness centrality are essential in network analysis (Freeman 1977; Brandes 2001). In a big water distribution system, demand was indicated by the betweenness centrality (Sitzenfrei 2021). The edge betweenness centrality for sewage systems was adjusted by Hesarkazzazi et al. (2020) to reflect how frequently an edge is included in the shortest path from the source vertices to the outlet. In this study, customized modifications are introduced: betweenness centrality refers to the frequency at which a node appears on the shortest path in the network; closeness centrality represents the average distance from a node to the outlet:

$$C_{B} (v) = \sum\limits_{s \ne t \ne v \in V} {\frac{{\sigma_{st} (v)}}{{\sigma_{st} }}}$$

(11)

$$C_{C} (v) = \frac{1}{{\sum\limits_{t \in G} {d_{G} (v,t)} }}$$

(12)

$$I = w_{1} \times C_{B} \left( v \right) + w_{2} \times C_{C} \left( v \right)$$

(13)

where CB(v) is the betweenness centrality; s, v, t are nodes; V is node set; \({\sigma }_{st}(v)\) is the number of shortest paths from s to t through v; \({\sigma }_{st}\) is the number of shortest paths from s to t; Cc(v) is the closeness centrality; G is the graph; \({d}_{G}(v,t)\) is the minimum length of any path connecting nodes v and t in G; I is node value; and w1 and w2 are weights determined by the Analytic Hierarchy Process. In this study w1 is 0.2, w2 is 0.8. The Analytic Hierarchy Process analysis was done according to Zhang et al. (2022).

(2) Local network analysis

Local network analysis, which includes degree (d), in-degree (din), out-degree (dout), and maximum degree (dm), is a focused investigation of nodes with higher values derived from global network analysis. Degree describes how many edges are connected to a node; in-degree describes how many edges enter the node; out-degree describes how many edges leave the node; and maximum degree describes how many edges can connect to the node:

$$d = d_{{{\text{in}}}} + d_{{{\text{out}}}}$$

(14)

If d < dm, the redundancy can be increased, and if d = dm, the redundancy cannot be increased. Take node c in Fig. 4 as an example, where its maximum degree is 3; the current in-degree is 1; the out-degree is 1; there is room to increase redundancy. There is no room to add redundancy at node b because it can only have a maximum degree of 4, while the in-degree and out-degree are already 3 and 1, respectively.

2.3

Study Area and Datasets

The study area is located in the eastern part of the Dongying City center, Shandong Province, China, with an area of 8.916 km2 (Fig. 5). The Storm Water Management Model (SWMM) was used for hydraulic simulation (Rossman 2015). The characteristics of the subcatchment and original stormwater engineering were obtained through a digital elevation model dataset and current status. The elevation of the study area is high in the south and low in the north. The impervious coverage rate is 90.9%. We used elevation data with a resolution of 30 m × 30 m, and land use data with a resolution of 10 m × 10 m (Fig. 6). The total length of the original drainage pipe is 25,142 m, and the pipe density is 2.82 km/km2. The pipe diameter ranges from 300 to 2000 mm. The research area has monitoring equipment for ponding points as well as a rain gauge station nearby. The rainfall data were recorded by the rain-gauge station from 18 to 20 August 2018, and the data of the three waterlogging points’ flooding depth are used to calibrate the parameters (Fig. 7a). The model is acceptable because all errors are within 10% (Table 1). The final calibration parameter values of SWMM are shown in Table 2.

Location of the study area in Dongying City, Shandong Province, China. a China; b Dongying City;  c Base graph of the study area

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Overview of the study area. a Ground elevation; b land use

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Rainfall data used in the study. a Rainfall event used for calibration; b design storm under different return periods (5-year, 10-year, and 20-year): 2-h design hyetograph

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Full size table

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The verification results show that the coefficient Re is greater than 0.9, indicating that the validity and accuracy of the model are acceptable (Table 1). The calibrated values of the parameters for SWMM are shown in Table 2.

There is a strong correlation between street networks and urban water infrastructures—around 80% of total sewer networks correlate with 50% of the street networks (Mair et al. 2017). Subcatchments are divided according to the distribution of buildings and streets in the study area. The digital elevation model is subjected to a spatial analysis to ascertain the flow direction. The design return period of drainage pipes in important areas is 3–5 years. The study area belongs to the northern residential area of Dongying City, which is a relatively densely populated residential area. The return period for design pipes is fixed at 5 years for safety concerns. For the hydraulic assessment of the urban drainage system, rainfall with 10-year and 20-year return periods is used (see Fig. 7b). The rainfall of 10-year and 20-year return periods is 80.7 mm and 92.3 mm, respectively. According to the rainstorm intensity formula of Dongying City (Di et al. 2017), the rainfall design is shown in Fig. 7b.