Optimizing the Design of Water Distribution Networks Using Mathematical Optimization

Abstract

Decaying infrastructure in municipalities is becoming a problem of increasing importance as growing populations put increasing stress on all service systems. In tough economic times, renewing and maintaining infrastructure has become increasingly difficult. As an example, many municipal water networks were installed several decades ago and were designed to handle much smaller demand and additionally have decayed due to age. This chapter discusses an efficient approach for the problem of replacing all the pipes using the same network topology, at minimum cost, to achieve current pressure demands at junctions of the network.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Decaying infrastructure in municipalities is becoming a problem of increasing importance as growing populations put increasing stress on all service systems. In tough economic times, renewing and maintaining infrastructure has become increasingly difficult. As an example, many municipal water networks were installed several decades ago and were designed to handle much smaller demand and additionally have decayed due to age.

For example, consider the case of Modena, a city northwest of Bologna in the Emilia-Romagna region of Italy. We can see in Fig. 9.1 how the population has increased by more than a factor of 2.5 over the last century. The Modena water distribution network comprises 4 reservoirs, 272 junctions, and 317 pipes. Its complexity can be gleaned from Fig. 9.2.

Fig. 9.1

Population of Modena

Fig. 9.2

Water distribution network of Modena

The aim is to replace all the pipes using the same network topology at minimum cost to achieve pressure demands at junctions of the network. Pipes are only available from commercial suppliers that produce pipes in a limited number of diameters. Reservoirs pressurize the network while most pressure is lost due to friction in pipes (some pressure is also lost at the junctions). Our goal is to model the problem using continuous variables as flow rates in the pipes, and pressures at the junctions; and discrete variables for the diameters of the pipes. Noting that pressure loss due to friction behaves nonlinearly, this puts us in the domain of mixed integer nonlinear programming (MINLP). Thus, we will try to obtain a model which is tractable by standard MINLP solvers.

In fluid dynamics, hydraulic head equates the energy in an incompressible fluid with the height of an equivalent static column of that fluid. The hydraulic head of a fluid is composed of pressure head and elevation head. The pressure head is the internal energy of a fluid due to the pressure exerted on its container and the elevation head is the energy given by the gravitational force acting on a column of fluid, in this case water. As a convention, the units of hydraulic, pressure, and elevation head are meters, i.e., each unit represents the energy provided by a column of water of the height of 1 m. The water in its way from the reservoir to the rest of the junctions loses energy. This loss is called head loss. Head loss is divided into two main categories, “major losses” associated with energy loss per length of pipe, and “minor losses” associated with bends and relatively small obstructions. For our purposes, considering that we are interested in networks covering relatively large geographic areas, it suffices to ignore minor losses.

2 Nomenclature

In this section we introduce the sets and parameters, i.e., the data input of our problem.

2.1 Sets

The water network will be represented as a directed graph $G=(N,E)$ where, N is the set of nodes that represent the junctions and E is the set of pipes. Moreover, we define the set of reservoirs S as a subset of N. Finally, for ease of notation, we define $\delta_-(i)$ ($\delta_+(i)$) as the sets of pipes with an head (tail) at junction i.

2.2 Parameters

In the following, we introduce the notation for the data elements that we call parameters.

For each junction but the reservoirs $i \in N \setminus S$ we have:

• elev(i) = physical elevation of junction i, i.e., the height of junction i ($[m]$).

• dem(i) = water demand at junction i ($[m^3/s]$).

• ph$_{min}(i) =$ lower bound on pressure head at junction i ($[m]$).

• ph$_{max}(i) =$ upper bound on pressure head at junction i ($[m]$).

Moreover, for each reservoir $i \in S$, we have h _s (i), i.e., the fixed hydraulic head at junction i.

Finally, for each pipe $e=(i,j) \in E$ the following parameters are needed:

• l$(e) =$ length of pipe e ([m]).

• $v_{max}(e) =$ upper bound on the velocity of water in pipe e ([$m/s$]).

• $k(e) =$ physical constant depending on the roughness of pipe e.

• $\mathcal{D} (e,r) =$ r-th diameter that can be chosen for pipe e ([m]).

• $\mathcal{C} (e,r) =$ cost of the r-th diameter that can be chosen for pipe e ([€/$m])$.

Note that all the parameters described above are data input and are used to define constraints and objective function of the model.

3 The Mathematical Model for the Problem

We proceed further in describing the problem by specifying a formulation of it.

3.1 Decision Variables of the Problem

First, we specify the variables which are also the expected output of our problem:

• $Q(e)=$ flow in pipe e, for all $\ e\ in E$ $[m^3/s]$. This is the volume of water which passes through pipe per unit time.

• $D(e)=$ diameter of pipe e, for all $\ e\in E$ $[m]$.

• $H(i)=$ hydraulic head of junction i, for all $\ i\in N$ $[m]$.

For modeling purposes, each pipe e has a nominal orientation, and negative flow corresponds to water flow in the direction opposite to the nominal orientation of the pipe.

The goals of the problem are to choose, for each edge, the diameter of the pipe that has to be installed in order to implement the drinkable water distribution network by minimizing the installation costs and satisfying physical and operational constraints.

3.2 Constraints in the Problem

Each reservoir has a fixed hydraulic head that is specified:

$$H(i) = h_s(i), \mbox{for all}\ i\in S.$$

(9.1)

Typically, there are inequalities imposed that bound the velocity of the water in each pipe. Because the water flow is given by the product between the cross-sectional area of the pipe $\pi(D(e)/2)^2$ and the velocity, for simplicity the velocity bounds can be written as bounds on the flow:

$$- \frac{\pi}{4} v_{max}(e) D^2(e) \le Q(e) \le \frac{\pi}{4} v_{max}(e) D^2(e), \mbox{for all}\ e\in E.$$

(9.2)

Notice how these equations are nonlinear in the diameter variables D(e), but we can think of them as linear in the cross-sectional area $\pi(D(e)/2)^2$.

Next, we have the usual flow-conservation equations:

$$\sum_{e\in \delta_-(i)} Q(e) - \sum_{e\in \delta_+(i)} Q(e) = \mbox{dem}(i), \mbox{for all}\ i\in N\setminus S,$$

(9.3)

where, $\mbox{dem}(i)$ is the demand for water at junction i, $\delta_-(i)$ denotes the set of pipes oriented into junction i, and $\delta_+(i)$ denotes the set of pipes oriented out of junction i. We only have these equations for junctions that are not reservoirs.

Next, we impose upper and lower limits on the pressure head at each junction:

$$\mbox{ph}_{\mbox{min}}(i) \le H(i)-\mbox{elev(i)} \le \mbox{ph}_{\mbox{max}}(i), \mbox{for all}\ i\in N \setminus S.$$

(9.4)

Head loss along each pipe is modeled via a nonlinear function of the diameter of the pipe and the flow in the pipe. Typically, one uses the so-called Hazen–Williams equation, which is an empirical formula relating the head loss caused by frictions to the physical properties of the pipe:

$$\begin{aligned} H(i)-H(j) = \frac{\text{sgn}(Q(e)) |Q(e)|^{1.852} \cdot 10.7 \cdot \mbox{l}(e)\cdot \mbox{k}(e)^{-1.852}} {D(e)^{4.87}}, \nonumber\\ \text{for all}\ e=(i,j)\in E,\end{aligned}$$

(9.5)

where, we consider the pipe e to be oriented from junction i to junction j. Notice how following intuition head loss is proportional to the length $\mbox{l}(e)$ of the pipe e. The dependence on the flow Q(e) and the diameter D(e) is nonlinear.

For each pipe e, the available diameters belong to a discrete set of r _e elements. For $e\in E$:

$$d_{min}(e):=\mathcal{D}(e,1)<\mathcal{D}(e,2)<\cdots<\mathcal{D}(e,r_e)=:d_{max}(e)~, $$

and so, we have the equations:

$$D(e) \in \left\{\mathcal{D}(e,1),\mathcal{D}(e,2),\ldots,\mathcal{D}(e,r_e) \right\}, \mbox{for all}\ e\in E.$$

(9.6)

For each pipe $e\in E$, there is a cost function $C_e()$ having a discrete specification as a (typically rapidly) increasing function of diameter. That is, $\mathcal{C}(e,r):=C_e(\mathcal{D}(e,r))$, $r=1,\ldots,r_e$ , where:

$$\mathcal{C}(e,1)<\mathcal{C}(e,2)<\cdots<\mathcal{C}(e,r_e)~. $$

Fig. 9.3

Diameter set for Fossolo instance: diameter (m) and cost per meter (€/m)

Our objective to be minimized is:

$$\sum_{e\in E} C_e(D(e))\ \mbox{l}(e).$$

(9.7)

Our water-network optimization problem is now completely specified as the minimization of (9.7), subject to (9.1)–(9.6).

The mathematical model is reported below:

$$\begin{aligned} \hspace{74pt}\min \sum_{e\in E} C_e(D(e))\ \mbox{l}(e)\\ \hspace{165pt} H(i) = h_s(i), \mbox{for all}\ i\in S\\ \hspace{45pt}- \frac{\pi}{4} v_{max}(e) D^2(e) \le Q(e) \le \frac{\pi}{4} v_{max}(e) D^2(e), \mbox{for all}\ e\in E\\ \hspace{33pt}\sum_{e\in \delta_-(i)} Q(e) - \sum_{e\in \delta_+(i)} Q(e) = \mbox{dem}(i), \quad\\ \hspace{97pt}\mbox{for all}\ i\in N\setminus S\\ \hspace{18pt}\mbox{ph}_{\mbox{min}}(i) \le H(i)-\mbox{elev(i)} \le \mbox{ph}_{\mbox{max}}(i), \\ \hspace{94pt} \mbox{for all}\ i\in N \setminus S\\ \hspace{-78pt}H(i)-H(j) = \frac{\text{sgn}(Q(e)) |Q(e)|^{1.852} \cdot 10.7 \cdot \mbox{l}(e)\cdot \mbox{k}(e)^{-1.852}} {D(e)^{4.87}},\\ \hspace{70pt} \mbox{for all}\ e=(i,j)\in E\\ \hspace{70pt}D(e)\in \left\{\mathcal{D}(e,1),\mathcal{D}(e,2),\ldots,\mathcal{D}(e,r_e)\right\}, \mbox{for all}\ e\in E.\end{aligned}$$

4 Solver

Because of the presence of discrete choices (the pipe diameters) and intrinsic nonlinearity (the dependence of head loss on flow and on pipe diameter), we are in the realm of MINLP. This is a very active area of current research (see [8], for example). We chose to use the open-source code Bonmin (Basic Open-source Nonlinear Mixed INteger programming) as our MINLP solver (see [3]). This choice was guided by the facts that (i) it is accessible from the modeling language AMPL (see [5]), (ii) it is freely distributed under an open-source license that has fairly liberal terms, (iii) we are able to modify the source code in case that is needed, and (iv) the primary developer, Pierre Bonami, is a very good friend (so he can modify the source code for us, in case that would be helpful).

Among other algorithms, Bonmin includes an NLP-based branch-and-bound implementation (see [2]). Bonmin is primarily aimed at searching for a global optimum of a formulation having a convex relaxation. Our formulation is not convex, in particular because of the nonlinear Eqs. 9.5, so our use of Bonmin is as a heuristic. We utilized several features of Bonmin that are designed to facilitate such a use. In fact, though these features have rather general value for nonconvex formulations, these features were implemented in Bonmin due to our needs for this project.

4.1 Decreasing Nonconvexity

Because Bonmin is really aimed at MINLP formulations having a convex relaxation, it is natural to expect that it will perform better as a heuristic on nonconvex formulations for which the nonconvexity is not too severe. We can soften nonconvexities by making a substitution of variables. Specifically, we can define the cross-sectional area variables:

$$A(e):= \pi(D(e)/2)^2, \mbox{for all}\ e\in E. $$

Then, the nonlinear flow bounds (9.2) become linear flow bounds:

$$- v_{max}(e) A(e) \le Q(e) \le v_{max} A(e), \mbox{for all}\ e\in E.$$

(9.8)

Moreover, the nonlinear head loss Eqs. (9.5) become:

$$\begin{aligned} H(i)-H(j) = \frac{\text{sgn}(Q(e)) |Q(e)|^{1.852} \cdot 10.7 \cdot \mbox{l}(e) \cdot k(e)^{-1.852} \left(\frac{\pi}{4}\right)^{2.435}} {A(e)^{2.435}}, \nonumber\\ \mbox{for all}\ e=(i,j)\in E.\end{aligned}$$

(9.9)

So, we consider the cross-sectional area variables A(e) to satisfy

$$A(e) \in \left\{\mathcal{A}(e,1),\mathcal{A}(e,2),\ldots,\mathcal{A}(e,r_e) \right\}, \mbox{for all}\ e\in E,$$

(9.10)

where,

$$\mathcal{A}(e,r):=\pi (\mathcal{D}(e,r)/2)^2, \mbox{for all}\ e\in E,~ r=1,\ldots,r_e. $$

In (9.9), sgn$(y)$ for any number y denotes the sign (+ or -) of y. Denoting $x = Q(e)$, the term sgn$(Q(e))|Q(e)|^{1.852}$ in (9.9) can be represented as the function f(x), where $f(x) = x^p$ $(p = 1.852)$ when x is nonnegative; and $f(x) = -(f(-x))$ when x is negative. In the next section, we will describe how we approximate f(x) by a smooth function.

4.2 Smoothing

The MINLP code Bonmin makes use of the NLP code Ipopt (see [7]) to solve continuous relaxations. Like many codes for NLP, it is aimed at smooth formulations. Unfortunately, the absolute value term in the head loss constraints (9.5) is nondifferentiable (at 0) but not so bad. Thus, to accommodate Ipopt, we smoothed the nondifferentiability. In order to do that effectively, our main goal is not to provide a fully accurate approximation near 0 because it is well known that Hazen–Williams Eq. (9.5) is in itself a poor approximation of the real pressure loss for small values of the flow. Instead, we smooth away the mild nondifferentiability by defining the head loss equation in a piecewise manner, so as to have accurate evaluations of the function. We insure the smoothness by matching function values as well as first and second derivatives at the breakpoints. Fortunately, piecewise constraints can be modeled in AMPL (see § 18.3 of [5]).

More precisely, let $f(x)=x^p$ (p = 1.852) when x is nonnegative, and $f(x)=-f(-x)$ when x is negative (x is standing in for Q(e)). This function misbehaves at 0 (the second derivative does not exist there). Choose a small positive δ, and replace f with a function g on $[-\delta,+\delta]$. Outside of the interval, we leave f alone. We will choose g to be of the following form: $g(x) = ax + bx^3 + cx^5$. In this way, we can choose $a,b,c$ (uniquely) so that f and g agree in value, derivative, and second derivative at $x=|\delta|$. So we end up with a smooth-enough antisymmetric function. It agrees in value with f at 0 and outside $[-\delta,+\delta]$. It agrees with f in the first two derivatives outside of $[-\delta,+\delta]$.

Formally, it is easy to prove that:

Proposition 1

 The unique polynomial $g(x)=ax+bx^3+cx^5$ having $f(x)=g(x)$ , $f'(x)=g'(x)$ and $f''(x)=g''(x)$ at $x=|\delta|$ is:

$$\begin{aligned} g(x) =& \left(\frac{3 \delta^{p-5}}{8}+\frac{1}{8} (p-\text{1}) p \delta^{p-\text{5}}-\frac{3}{8} p \delta^{p-\text{5}}\right) x^5 \\ &+\left(-\frac{5 \delta^{p-3}}{4}-\frac{1}{4} (p-\text{1}) p \delta^{p-\text{3}}+\frac{5}{4} p \delta^{p-\text{3}}\right) x^3 \\ &+\left(\frac{15 \delta^{p-1}}{8}+\frac{1}{8} (p-\text{1}) p \delta^{p-\text{1}}-\frac{7}{8} p \delta^{p-\text{1}}\right) x~.\end{aligned}$$

Proof

Via simple calculation one simply has to equate: (i) $g(\delta)=a\delta+b\delta^3+c\delta^5 = \delta^p = f(\delta)$, (ii) $g'(\delta)= a + 3b\delta^2 + 5c\delta^4 = p\delta^{p-1} = f'(\delta)$, and (iii) $g''(\delta) = 6b\delta + 20c\delta^3 = p(p-1)\delta^{p-2} = f''(\delta)$. This is now a square linear system in the $a,b,c$ variables. We solve it (symbolically), using Mathematica (see [9]). Finally, we just observe that f and g are antisymmetric, so we have the same $a,b,c$ for $x=-\delta$.

Figure 9.4, drawn for $\delta=0.1$, shows that g provides a good approximation of f. Indeed the quintic curve fits very well on $(-\delta,+\delta)$, and of course it matches up to second order with the true function f at $\pm\delta$. This is all no surprise because we are operating in a small interval of 0, and the function that we approximate is not pathological. The NLP solver Ipopt responds well to this technique, as does our MINLP solver Bonmin.

Our experience is that the inaccuracy in using this smoothed function is minimal compared to the other inaccuracies (e.g., numerical and modeling inaccuracies).

Fig. 9.4

Smoothing f near x = 0.

4.3 Continuous Cost Function

The NLP solver passes useful branching information to the MINLP solver by working with a good fully-continuous model. Because the cost data is truly discrete, we need to fit a continuous function to the data points. We use and advocate weighted fits to minimize relative error. For example, our least-squares fit for pipe e minimizes

$$\sum_{r=1}^{r_e} \left[1 - \left(\frac{\sum_{j=0}^{t_e} \beta(j,e) \left(\mathcal{A}(e,r)^2\right)^j}{\mathcal{C}(e,r)}\right) \right]^2~, $$

where, t _e is the desired degree and $\beta(j,e)$ are the coefficients of the polynomial¹ approximating C _e . Note that, for our purposes, a feature was added to Bonmin to accommodate one objective function for calculating lower bounds to guide the branch-and-bound search (for us, that is the fitted objective function) and another to calculate actual objective values to improve the incumbent solution (for us this just amounts to using the real objective function determined by the discrete cost data).

4.4 SOS

It would be nice if Bonmin could directly handle branching for imposing constraints like (9.10), but at the present time, this is not the case. So, we simply define additional binary variables $X(e,r)$, where $X(e,r)=1$ indicates that cross-sectional area $\mathcal{A}(e,r)$ is selected for pipe e ($r=1,\ldots,r_e$ , $e\in E$). Then, we use the “SOS type-1” branching (see, [1]) that is available in Bonmin. As standard, we use AMPL suffixes to pass along the SOS information needed by the solver:.sosno (“SOS number”) is used to group variables into separate SOS constraints and.ref (“reference value”) is used to indicate the value symbolized by a variable. In this way, for $e\in E$, in AMPL we naturally set:

$$X(e,r).\mbox{sosno}:= e~, \mbox{for}\ r=1,\ldots,r_e~, $$

and

$$X(e,r).\mbox{ref}:= \mathcal{A}(e,r)~, \mbox{for}\ r=1,\ldots,r_e~. $$

4.5 Heuristic Features of Bonmin

NLP-based branch-and-bound is only a heuristic for a formulation having a nonconvex continuous relaxation, because relaxations are not solved to global optimality. But Bonmin includes several options tailored to improve the quality of the solutions it provides in such a context.

First, the NLP-solver Ipopt may end up in different local optima when started from different starting points. The two options num_resolve_at_root and num_resolve_at_node allow for solving the root node or each node of the tree, respectively, with a user-specified number of different randomly-chosen starting points, saving the best solution found. Note that the function to generate a random starting point is very naive: it chooses a random point (uniformly) between the bounds provided for the variable.

Secondly, because the solution given by Ipopt does not truly give a lower bound, the user can adjust the fathoming rule to continue branching even if the solution value to the current node is worse than the best-known solution value. This is achieved by setting allowable_gap, allowable_fraction_gap and cutoff_decr to negative values.

5 Example

In this section, we describe a real-world instance as an example. The data are taken from a neighborhood of Bologna called Fossolo. We have 37 junctions (of which 1 reservoir, identified by the index 37), and 58 pipes. The topology of the network is represented in Fig. 9.5 and provided in detail in Table 9.3.

Fig. 9.5

Fossolo network

The hydraulic head at the reservoir is always fixed. In our example it is fixed to 121.0 m. The value of $v_{max}(e)$ is set to 1 $m^3/s$ and the roughness coefficient is set to 100 for all $e \in E$. The lower bound on the pressure head $ph_{min}(i)$ is equal to 40 m and the upper bound $ph_{max}(i)$ is equal to $121 - eval(i)$ meters for all $i \in N$. The values of the elevation and the demand for each junction besides the reservoir are reported in Table 9.1. The network topology and the length of the pipes are reported in Table 9.3.

Table 9.1 Elevation (meters) and water demand (m ³/s) for instance Fossolo

Table 9.2 Diameter set (meters) and relative cost per meter (€/m) for instance Fossolo for each pipe

Table 9.3 Network topology and length of the pipes (meters) for instance Fossolo

Concerning the available diameters for each pipe, we have 13 different possibilities. Table 9.2 reports for each $e \in E$ the value of $(\mathcal{D}(e,r),\mathcal{C}(e,r))$ for all $r=1,\dots,r_e$.

6 Solution Found by the MINLP Model

The solution found by Bonmin is depicted in Fig. 9.6 where the size of each diameter is proportional to the thickness of the arc. The diameters are expressed in meters, and the diameter is equal to 0.06 for the pipes without explicit number, i.e., the minimum diameter permissible for this data set.

The analysis of this solution shows a configuration in which the size of the selected diameters decreases from the reservoir toward the parts of the network farther away from the inlet point. This characteristic of the allocation of diameters to pipes plays in favor of a correct hydraulic operation of the network and has a beneficial effect on water quality, see, e.g., the discussion in [10].

Fig. 9.6

Solution for Fossolo network: the size of each diameter is proportional to the thickness of the arc. The diameter (expressed in meters) is equal to 0.06 m for the pipes without explicit number

This characteristic could be noticed in the solution of MINLP for different instances, see Bragalli et al. [4] for details.

Note that the proposed solution is not guaranteed to be a global optimum of the problem. However, because of the intrinsic difficulty of the problem at hand, the proposed solution is, from a practical viewpoint, a very good quality feasible solution. Other approaches based on heuristic algorithms, mixed integer linear programming, or nonlinear programming models are not effective for medium/large instances. In this application, modeling the problem in the most natural way seems to be the most successful approach.

7 Conclusions

More details concerning our methodology and results can be found in [4].

8 A Practical Exercise

In this section, we describe a small but interesting example taken from the literature, in particular it was introduced by Fujiwara and Khang [6]. The network is a simplified version of the water network of the city of Hanoi, Vietnam.

We have 32 junctions (of which 1 reservoir, identified by the index 1), and 34 pipes. The topology of the network is represented in Fig. 9.7.

Fig. 9.7

Hanoi network

The hydraulic head at the reservoir is always fixed. In our example it is fixed to 100.0 m. The roughness coefficient is set to 130 and the elevation of each junction is 0. The lower bound on the pressure head $ph_{min}(i)$ is equal to 30 m and the upper bound $ph_{max}(i)$ is equal to 100 m for all $i \in N$. The values of the demand for each junction besides the reservoir, the network topology, the upper bound on the speed of the water within the pipes, and the length of the pipes are reported in Table 9.5.

Table 9.4 Diameter set (meters) and relative cost per meter (€/m) for instance Hanoi for each pipe

Table 9.5 Water demand (m ³/s), topology, pipe length (m), and maximum speed of the water within the pipes ($m^3/s$) for instance Hanoi

Concerning the available diameters for each pipe, we have 6 different possibilities. Table 9.4 reports for each $e \in E$ the value of $(\mathcal{D}(e,r),\mathcal{C}(e,r))$ for all $r=1,\dots,r_e$.