Keywords

1 Introduction

Sewer networks are an essential part of human society, which collect wastewater from residential, commercial and industrial areas and transports to wastewater treatment plant. Construction and maintenance of this large-scale sewer networks required a huge investment. A relatively small change in the component and construction cost of these networks, therefore leads to a substantial reduction in project cost. The design of a sewer network problem includes two sequential sub problems: (1) generation of the network layout and (2) optimal sizing of sewer network components. The component size optimization of sewer network problem consists of many hydraulic and technical constraints which are generally nonlinear, discrete and sequential. Satisfying such constraints to give an optimal design is often challenging even to the modern heuristic search methods. Many optimization techniques have been applied and developed for the optimal design of sewer networks, such as linear programming [1, 2], nonlinear programming [3, 4] and dynamic programming [5–7]. Evolutionary strategies, such as genetic algorithms [8, 9], ant colony optimization algorithms [10, 11], cellular automata [12] and particle swarm optimization algorithms [13], have received significant consideration in sewer network design problems. Recently, Ostadrahimi et al. [14] used multi-swarm particle swarm optimization (MSPSO) approach to present a set of operation rules for a multi-reservoir system. Haghighi and Bakhshipour [15] developed an adaptive genetic algorithm. Therefore, every chromosome, consisting of sewer slopes, diameters, and pump indicators, is a feasible design. The adaptive decoding scheme is set up based on the sewer design criteria and open channel hydraulics. Using the adaptive GA, all the sewer system’s constraints are systematically satisfied, and there is no need to discard or repair infeasible chromosomes or even apply penalty factors to the cost function. Moeini and Afshar [16] used tree growing algorithm (TGA) for efficiently solving the sewer network layouts out of the base network while the ACOA is used for optimally determining the cover depths of the constructed layout.

In this paper, PSO algorithm with new modification is applied to get optimal sewer network component sizes of a predetermined layout.

2 Formulation of Sewer System Design

2.1 Sewer Hydraulics

In circular sewer steady-state flow is described by the continuity principle and Manning’s equation which is

$$Q = VA$$
(1)
$$V = \frac{1}{n}R^{2/3} S^{1/2}$$
(2)

where Q = sewage flow rate, V = velocity of sewage flow, A = cross-sectional flow area, R = hydraulic mean depth, n = Manning’s coefficient and S = slope of the sewer. Common, partially full specifications for circular sewer sections are also determined from the following equations:

$$K = QnD^{ - 8/3} S^{ - 1/2}$$
(3)
$$\theta = \frac{3\pi }{2}\sqrt {1 - \sqrt {1 - \sqrt {\pi K} } }$$
(4)
$$\left( {\frac{d}{D}} \right) = \frac{1}{2} \times \left( {1 - \cos \frac{\theta }{2}} \right)$$
(5)
$$R = \frac{D}{4}\left( {\frac{\theta - \sin \theta }{\theta }} \right)$$
(6)

where K = constant, D = sewer diameter, θ = the central angle in radian and (d/D) = proportional water depth. Equation (4) is applicable for K values less than (1/π) = 0.318 Saatci [17].

2.2 Sewer Design Constraints

For a given network, the optimal sewer design is defined as a set of pipe diameters, slopes and excavation depths which satisfies all the constraints. Typical constraints of sewer networks design are:

  1. 1.

    Pipe flow velocity: each pipe flow velocity must be greater than minimum permissible velocity to prevent the deposit of solids in the sewers and less than maximum permissible velocity to prevent sewer scouring. The minimum permissible velocity of 0.6 m/s and maximum velocity of 3.0 m/s have been adopted in the present paper.

  2. 2.

    Flow depth ratio: wastewater depth ratio of the pipe should be less than 0.8.

  3. 3.

    Choosing pipe diameters from the commercial list.

  4. 4.

    Pipe cover depths: maintaining the minimum cover depth to avoid damage to the sewer line and adequate fall for house connections. The minimum cover depth of 0.9 m and maximum cover depth of 5.0 m have been adopted.

  5. 5.

    Progressive pipe diameters: The diameter of ith sewer should not be less than the diameter of immediately preceding sewer.

The optimal design of a sewer system for a given layout is to determine the sewer diameters, cover depths and sewer slopes of the network in order to minimize the total cost of the sewer system. The objective function can be stated as

$${\text{Minimize}}\left( C \right) = \sum\limits_{i = 1}^{N} {({\text{TCOST}}_{i} + {\text{PC}}_{i} )}$$
(7)

where I = 1,…, N (total number of sewers), TCOST i (total cost) = (Cost of sewer i  + Cost of manhole i  + Cost of earth work i ) and PC i  = penalty cost (it is assigned if the design constraint is not satisfied).

3 Particle Swarm Optimization (PSO)

Kennedy and Eberhart [18] were first to introduce particle swarm optimization technique in 1995. In PSO techniques, every problem solution is a flock of birds and denoted to the particle. In this technique, birds develop personal and social behaviour and reciprocally manage their movement towards a destination [13, 19].

Each particle is affected by these components: (i) its own velocity, (ii) the best location or position it has attained so far called particle best position and (iii) the overall best position attained by all particles called global best position. Initially, the group of particles starts their movement in the first iteration randomly, and then they try to search the optimum solution. The procedure can be described mathematically, as below [14, 20–22].

The current location of the ith particle with D-dimensions at tth iteration is indicated as

$$X_{i} \left( t \right) = \left\{ {x_{i1} ,x_{i2} ,x_{i3} , \ldots ,x_{\text{id}} } \right\}^{t}$$
(8)

Earlier best position or location,

$$P_{i} \left( t \right) = \left\{ {p_{i1} ,p_{i2} ,p_{i3} , \ldots ,p_{\text{id}} } \right\}^{t}$$
(9)

and velocity

$$V_{i} \left( t \right) = \left\{ {v_{i1} ,v_{i2} ,v_{i3} , \ldots ,v_{\text{id}} } \right\}^{t}$$
(10)

Every particle’s location in the search space is updated by

$$X_{i} \left( t \right) = X_{i} (t - 1) + V_{i} \left( t \right)$$
(11)

where the new velocity

$$V_{i} \left( t \right) = \omega \cdot V_{i} \left( {t - 1} \right) + c_{1} \cdot R_{1} \left\{ {P_{i} \left( t \right){-}X_{i} \left( {t - 1} \right)} \right\} + c_{2} \cdot R_{2} \left\{ {P_{g} \left( t \right){-}X_{i} \left( {t - 1} \right)} \right\}$$
(12)

where i = 1, 2,…, N (N denotes population size); t = 1, 2,…,T (T denotes a total number of iterations); ω = factor of inertia; R 1 and R 2 are the random values (which between 0 and 1); c 1 and c 2 are the learning or acceleration coefficients. X i (t) (location of every particle) is calculated by its earlier location X i (t − 1) and its current velocity. V i (t) (particle’s velocity) changes the location of the particles towards a better solution, at every iteration. V i (t − 1) is the velocity from the earlier iteration, P i is the best location of every particle and P g is the best position or location ever found by any particle.

The inertia weights of each time interval (or iteration) ω(t) and acceleration coefficient (c 1 and c 2) are updated with these equations:

$$\omega (t) = \omega_{\hbox{max} } - \frac{{\omega_{\hbox{max} } - \omega_{\hbox{min} } }}{T} \times t$$
(13)
$$c_{1} = c_{{1,{ \hbox{max} }}} - \frac{{c_{1,\hbox{max} } - c_{1,\hbox{min} } }}{T} \times t$$
(14)
$$c_{2} = c_{{2,{ \hbox{max} }}} - \frac{{c_{2,\hbox{max} } - c_{2,\hbox{min} } }}{T} \times t$$
(15)

where T = total number of iterations; ω min and ω max are the minimum and maximum inertia weights, and their values have been taken as 0 and 0.8, respectively, in the present problem; c 1,max and c 2max = maximum accelerations factors, their values have been taken as 2; c 1,min and c 2,min = minimum accelerations factors, their values have been taken as 0.5.

Particle velocities on every dimension are limited to minimum and maximum velocities.

$$V_{\hbox{min} } \le V_{i} \le V_{\hbox{max} }$$
(16)

The particle velocities are an important factor. V max and V min must be limited. Otherwise, the solution space may not be discovered precisely. V max is generally considered about 10–20 % of the range of the variable on every dimension [19].

According to the above-mentioned Eqs. (11) and (12), a possible structure of the PSO algorithm is shown below.

  1. 1.

    Initialise a population of particles by randomly assigning initial velocity and location of every particle.

  2. 2.

    Calculate the optimal fitness function for every particle.

  3. 3.

    For every particle, compare the fitness value with the best particle (P i ) fitness value. If the current value is better than P i , then update the position with the current position.

  4. 4.

    Calculate the best particle of the swarm with the best fitness value, if the best particle value is better than global best (P g ), then update the P g and its fitness value with the location.

  5. 5.

    Determine new velocities for all the particles using Eq. (12).

  6. 6.

    Update new position of each particle using Eq. (11).

  7. 7.

    Repeat steps 2–6 until the stopping criterion is met.

  8. 8.

    Show the result given by the best particle.

Above-mentioned PSO algorithm deal with both discrete and continuous variables. PSO algorithm with discrete variables is requisite for the design of sewer networks.

4 Optimization of Sewer System

The live example (Sudarshanpura, Jaipur, India sewer network) is considered to check the above-proposed approach. The Sudarshanpura sewer network (Fig. 1) consists of 105 manholes and 104 pipes.

Fig. 1
figure 1

Sudarshanpura sewer network

Full size image

The following steps were used to optimize the component sizing of sewer system using PSO algorithm:

  1. 1.

    Start with first link (i = 1) of the first iteration.

  2. 2.

    Calculate constant value K,

    • If K > 0.305, then increase diameter.

  3. 3.

    If K < 0.305, then calculate sewer hydraulics.

  4. 4.

    Calculate invert levels of upstream and downstream nodes of a particular link.

  5. 5.

    Calculate cost of pipe, cost of manhole and cost of earthwork.

  6. 6.

    Calculate total cost of sewer network (TCOST).

  7. 7.

    Add the respective penalty cost (PC) in TCOST where constraints are violated.

  8. 8.

    Calculate feasible solution using PSO.

  9. 9.

    Check solutions obtained are feasible or not.

  10. 10.

    If feasible solution is not obtained increase iteration by 1 and go to step 1.

  11. 11.

    If feasible solution is obtained, then take output.

  12. 12.

    End.

The cost of pipe (RCC NP4 class), manhole and earth work were taken from Integrated Schedule of Rates, RUIDP [23].

5 Results

The performance of the proposed PSO procedure for optimization of the sewer system is now tested against Sudarshanpura sewer network. The optimal results are obtained using 60 iterations and population size of 1000, respectively. The total cost of the sewer system using PSO approach was found to be Rs. 8.505 × 106 and 9.232 × 106 for the traditional design approach. Thus, there is 7.87 % reduction in total cost by applying PSO approach to the present problem. Table 1 shows the solution obtained by PSO approach.

Table 1 Results of the Sudarshanpura sewer network obtained by PSO
Full size table

6 Conclusion

A particle swarm optimization with new modification was applied in this paper to the optimal solution of sewer system design problems. Using the PSO approach, the total cost of the sewer system was reduced by 7.87 % compared to the traditional design approach. The results indicated that the proposed approach is very promising and reliable, that must be taken as the key alternative to solve the problem of optimal design of sewer system.