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Research Papers: Energy Systems Analysis

# Topology Optimization of Robust District Heating NetworksOPEN ACCESS

[+] Author and Article Information
Alberto Pizzolato

Department of Energy,
Politecnico di Torino,
Torino 10129, Italy
e-mail: alberto.pizzolato@polito.it

Birmingham Centre for Energy Storage (BCES),
School of Chemical Engineering,
University of Birmingham,
Birmingham B15 2TT, UK
e-mail: a.sciacovelli@bham.ac.uk

Vittorio Verda

Department of Energy,
Politecnico di Torino,
Torino 10129, Italy
e-mail: vittorio.verda@polito.it

1Corresponding author.

Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received February 1, 2017; final manuscript received September 18, 2017; published online November 14, 2017. Assoc. Editor: George Tsatsaronis.

J. Energy Resour. Technol 140(2), 020905 (Nov 14, 2017) (9 pages) Paper No: JERT-17-1054; doi: 10.1115/1.4038312 History: Received February 01, 2017; Revised September 18, 2017

## Abstract

Large district heating networks greatly benefit from topological changes brought by the construction of loops. The overall effects of malfunctions are smoothed, making existing networks intrinsically robust. In this paper, we demonstrate the use of topology optimization to find the network layout that maximizes robustness under an investment constraint. The optimized design stems from a large ground structure that includes all the possible looping elements. The objective is an original robustness measure, that neither requires any probabilistic analysis of the input uncertainty nor the identification of bounds on stochastic variables. Our case study on the Turin district heating network confirms that robustness and cost are antagonist objectives: the optimized designs obtained by systematically relaxing the investment constraint lay on a smooth Pareto front. A sudden steepness variation divides the front in two different regions. For small investments topological modifications are observed, i.e., new branches appear continuously in the optimized layout as the investment increases. Here, large robustness improvements are possible. However, at high investments no topological modifications are visible and only limited robustness gains are obtained.

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## Introduction

During the last decades, district heating has distinguished among other space heating technologies for its limited primary energy cost [1]. This is possible thanks to the easy integration with high efficiency cogeneration plants [2,3], industrial waste-heat [4,5], or distributed renewable energy sources [69]. Despite these advantages, security of supply of distribution networks is often questioned. The effect of malfunctions can rapidly propagate in the network with dramatic effects for the users. Adding loops can increase the resilience of the system but raises the question on which is the optimal looping strategy with a limited investment constraint. A great body of literature has been devoted to this topic in the civil engineering community in the context of water distribution networks (WDNs). Several authors have proposed different ways to maximize either the reliability or the robustness of the network.

Reliability is defined as the ability of the system to provide adequate performance for customers during abnormal operating conditions [10]. Adequate performance includes both the demand satisfaction in terms of flows to be supplied (total volume and flow rate) and the range of pressures at which those flows must be provided [11]. A direct quantification of reliability involves the probabilistic mapping and integration of the failure region, which can be achieved with a brute-force sampling approach through Monte Carlo simulations [12] or through a more efficient first-order approximation of the failure surface [13]. In order to limit central processing unit time, an indirect estimation of reliability trough performance indicators has been proposed by several authors. For instance, Todini [14] performed an heuristic optimization with a systematic diameter reduction from a superstructure using an original resilience index. This quantity was calculated summing up the surplus hydraulic power as a proportion of the available hydraulic power at each node of the network. A similar “energetic” indicator was used by Farmani et al. [15], which used the minimum surplus pressure head across the network as meaningful reliability indicator. Another popular approach is the entropy concept proposed by Tanyimboh and Templeman [16], which aims at assessing the “disorder” of the flow in the network providing a surrogate measure of connectivity. In reliability-based optimization studies, uncertainty is introduced into the problem through stochastic variables with given probability density function. The accuracy of each input probability density function is of crucial importance and requires a detailed probabilistic analysis to yield accurate results [17,18].

Robustness is defined as the ability of the system to complete its task under a defined set of disturbances, e.g., fluctuations of design parameters or noise [19]. It is generally quantified in terms of dispersion of the system performance from its mean [20]. Different dispersion measures, sometimes combined with average quantities, have been adopted for robust design of WDNs. For instance, Babayan et al. [21] investigated the effect of demand uncertainty proposing a robust design methodology that minimizes the standard deviation of the pressure head. Similarly, Giustolisi et al. [19] used a robustness indicator that keeps into account both the mean and the standard deviation of nodal pressure. Another interesting approach is the one of Jung et al. [20], which used a coefficient of variation computed as the ratio of the nodal pressure standard deviation to its average value. Robust approaches can cope with little probabilistic information on the design and/or noise variables. The uncertain parameters can be described as uncertain but bounded (UBB). In this case, the performance dispersion is estimated using first-order sensitivities multiplied by the amplitudes of the variables uncertainty [22]. In many cases, the utilization of UBB-type parameters reduces the risk of optimization results affected by incorrect statistics. In the case of uncertain design variables, bounds can be easily identified by engineering experience. However, when dealing with uncertain noise factors as in fluid network robustness maximization, this step is not straightforward. Hence, robustness formulations requiring no characterization of the noise factors are needed.

Most of the reviewed literature in reliability and robustness maximization of WDNs adopted heuristic or meta-heuristic global search methods to conduct the optimization. The genetic algorithm is the most popular approach; it has been used either in its original form or in slightly modified versions by Tolson et al. [13], Babayan et al. [21], Farmani et al. [15], Creaco et al. [23], Jung et al. [20], and Giustolisi et al. [19]. Evolutionary methods guarantee a thorough exploration of the design space toward global optimum but require a large number of system evaluations per design variable. Using sampling approaches to compute statistics on the results of the analysis further increases the computational burden. Furthermore, even if sensitivities-based methods are used to calculate the robustness of the system with respect to UBB-type parameters, evolutionary approaches greatly suffer the curse of dimensionality. Design optimization of large and complex networks requires computationally affordable procedures.

In this paper, we aim at introducing a fully deterministic approach for the robust design of fluid-networks that does not require any form of a priori characterization of the input uncertainty. Taking inspiration from the work of Gong et al. [24], Han and Kwak [25], and Kim et al. [26], we formulate robustness as a weighted sum of first-order minimum pressure sensitivities to a fluid-dynamic resistance noise. The network design optimization is performed with a gradient-based optimizer. The objective and constraint gradients are obtained by the discrete adjoint approach, which is computationally cheap and nearly insensitive to the problem dimensions. This makes the presented procedure well suited to the design optimization of large and complex networks. The remainder of this paper is organized as follows: Section 2 presents the numerical model used to predict the fluid-dynamic performance of the network. Section 3 discusses the optimization problem formulation including the design variables, the objective, and the constraints definition. Section 4 reports the most relevant algorithmic details of the numerical implementation. Section 5 presents and discusses the optimized layouts of the Turin district heating network along with their performance. Finally, a summary and some concluding remarks are presented in Sec. 6.

## Physical Model of the Network

This section describes the discretized governing equations solved to predict the thermal and fluid-dynamic performance of the network. In this paper, we use a one-dimensional finite volume model [27] to calculate the mass flow rate vector G, the pressure vector P, and the temperature vector T.

The network is represented through a graph approach [28], where each pipe is considered as a branch delimited by two nodes, which physically correspond to either a bifurcation or a conjunction. A schematic of the network model is given in Fig. 1. The degrees-of-freedom corresponding to the mass flow rates are defined at each branch while those corresponding to pressures are defined at each node in a staggered grid fashion. The network topology is uniquely represented by the incidence matrix $A:ℝnb→ℝnn$ [27], where nb is the number of branches, and nn is the number of nodes. Let N be the graph representation of the network with branches $Ξ(N)={ξ1,ξ2,…,ξnb}$ and nodes $Ψ(N)={ψ1,ψ2,…,ψnn}$. The incidence matrix A is defined such as: Display Formula

(1)$Aij={+1 if ψi is the inlet node of branch ξj−1if ψi is the outlet node of branch ξj0otherwise$

We assume a steady-state and incompressible flow with constant fluid properties and negligible viscous dissipation. The continuity equation is readily written as Display Formula

(2)$AG+Gextr=0$

where Gextr is a nodal mass flow rate extraction vector. The discrete version of the momentum equation is expressed through a nonlinear system of the form Display Formula

(3)$BG−ATP−t=0$

where $B$ is a diagonal matrix which contains the fluid-dynamic resistance of each branch in the network as following: Display Formula

(4)$Bjj(Gj)=Gj(fjDjLj+βj)2ρSj2$

where fj is the Fanning friction factor, βj is the sum of local friction sources of the pipe, Dj is the pipe diameter, Lj is the pipe length, Sj is the pipe cross section, and ρ is the fluid density. The column vector $t=[Δp1,Δp2,…,Δpnn]T$ contains the pumping pressure heads $Δpi$ applied in each branch of the network. The vector of fluid-dynamic residuals can be written in a unique coupled system of the form Display Formula

(5)$Rfd=Ku−f=0$

where the pseudo-stiffness matrix K is defined as Display Formula

(6)$K=[A0B−AT]$
the total state variable vector u is defined as Display Formula
(7)$u={GP}$

and the pseudo-load vector f is defined as Display Formula

(8)$f={−Gextt}$

To obtain a well-conditioned system, we prescribe the pressure Pin at the inlet of the network and the mass flow rates $Goutj$ at the outlet. Mathematically Display Formula

(9)$Pi=Pin if ψi ∈ Γ1$
Display Formula
(10)$Gj n=−Goutj if ψi ∈Γ2,ψi is the outlet node of ξj$

where n is the inward-pointing normal depicted in Fig. 1.

## Optimization Problem Formulation

In this section, we discuss the optimization problem formulation including the design variables, the objective, the constraints, and the sensitivities calculation. Furthermore, we check the accuracy of discrete adjoint sensitivities against finite difference derivatives.

###### Design Variables.

In this paper, we search for the topology of the district heating network that maximizes robustness adopting the so-called ground structure [29] or superstructure [30] method. A large set of connections between fixed nodes is created as potential or vanishing branches. The network layout is then controlled by acting on the pipe diameters as following: Display Formula

(11)$Di(si)=Dmaxsi$

where s is a vector of abstract design variables and Dmax = 0.2 m is the maximum allowable branch diameter. Equation (11) converts the topology optimization problem into a sizing problem of a reference ground structure. The latter is identified by enumeration of all the possible looping paths on the available road-ways.

###### Objectives and Constraints.

In robust district heating networks, the minimum supply pressure head is affected by random fluctuations of the local fluid-dynamic resistance. Those fluctuations can be the result of fouling, chemicals deposition or valve actions needed to handle malfunctions. The objective function z is thus formulated as Display Formula

(12)$z(s)=∑i=1nbγi(si)=∑i=1nbd(minP)dβiωi$

where $ω$ is a vector of weights which quantifies the importance of the noise factors, and $γ$ is the weighted sensitivity field. We will refer to this latter quantity as local robustness hereafter. The weights correspond to the branch lengths normalized as following: Display Formula

(13)$ωi=Limax(L)$

To obtain a differentiable objective, we approximate the minimum function in Eq. (12) using the generalized p-mean Display Formula

(14)$Pmin=(1nn∑i=1nnPip)1p$

where the factor p is set to $−10$. Since Eq. (14) is not explicitly dependent on the design variables s, differentiating yields Display Formula

(15)$dPmindβ=−λ1T∂Rfd∂β$

where the column vector, $λ1$, is the set of adjoint variables. It can be calculated solving the following adjoint problem: Display Formula

(16)$W=(∂Rfd∂u)Tλ1−∂Pmin∂u=0$

The partial derivatives appearing in Eqs. (15) and (16) are reported in Appendix A of this paper. Note that our robustness metric (12) is fully deterministic and does not rely on statistical moments of the outputs. We expect agreement with conventional robust or reliability-based design approaches dealing with uniformly distributed and tightly bounded stochastic variables with a linear objective–noise relation. In water distribution network, the latter assumption is verified in the limit of infinitesimal β variations. However, when large variability is expected, Pmin scales sublinearly with β and our method potentially yields under-estimation of the failure region.

The increased robustness of the network comes at the price of additional investment cost to build the looping branches. To account for it, we augment the optimization problem with the following investment cost constraint: Display Formula

(17)$h(s)=∑i=1Nsci(si)Li−C*≤0$

where C* is the maximum allowable investment cost, and ci is the design-dependent specific investment cost per unit length. The latter is obtained as a quadratic fit to the construction cost data of the Turin district heating network Display Formula

(18)$ci=a0+a1Di+a2Di2$

where $a0=63€/m$, $a1=614$$€$/m2, and $a2=8505 €/ m3$. To properly account for topological changes, the original cost function in Eq. (18) is modified in a such a way that $ci*→0$, when $Di→0$. Hence, we smooth the fixed term of the original cost relationship through the logistic distribution to obtain Display Formula

(19)$a0̃i=a0(11+exp (−kDi)−1)$

where k = 212 is the statistical quasi-variance set in order to recover 99% of the fixed term at D = 25 mm. This value is considered as a lower limit due to manufacturing considerations. The smoothed cost function along with the original version is represented in Fig. 2. To sum up, the optimization problem to be solved is Display Formula

(20)$minsz(u(s),λ1(u(s),s),s)s.t.h(s)≤0s∈S={ℝNs|smin

where the box constraint sets a lower bound smin = 1e–4 and un upper bound smax = 1 to each design variable si in the Ns-dimensional control space. Problem (20) is solved through the nested analysis and design approach. For this reason, the fluid-dynamic response and the adjoint states used in the objective function should be computed at each optimization iteration by solving Eqs. (5) and (16).

###### Sensitivities Calculation.

The gradients of the objective and constraints with respect to the design variables vector s are needed to drive a gradient-based optimizer. Since the objective is formulated as a linear combination of sensitivities (Eq. (12)) differentiating with respect to s leads to second-order sensitivities. Those are obtained using again the adjoint approach, which requires solving one additional system per objective and constraint. The discrete sensitivity field is thus calculated as Display Formula

(21)$dzds=∂z∂s−λ2T∂Rfd∂s−λ3T∂W∂s$

where $λ2$ and $λ3$ are the adjoint vectors enforcing fluid-dynamic equilibrium (Eq. (3)) and the first-order adjoint solution (Eq. (16)). They are calculated solving in sequence Display Formula

(22)$(∂W∂λ1)Tλ3=∂z∂λ1$
Display Formula
(23)$(∂Rfd∂u)Tλ2=∂z∂u+λ3T∂W∂u$

For the sake of completeness, we report all the partial derivatives involved in Eqs. (21)(23) in Appendix B of this paper.

###### Finite Difference Validation.

To check the accuracy of the presented framework, we have tested both first-order and second-order sensitivities against central finite difference derivatives. The relative absolute error of the first-order gradients on four selected branches is reported in Fig. 3(a) for different values of the perturbation size $δβ$. A minimum plateau is visible in the region $1e−6≤δβ≤1e−2$. For $δβ<1e−6$, a high oscillatory trend is visible due to machine error. For $δβ>1e−2$, inaccuracies are due to truncation error. A similar behavior can be observed for the trend of the second-order sensitivities presented in Fig. 3(b). In this case, the plateu is shrinked and moved to the region $1e−9≤δs≤1e−7$.

## Numerical Implementation

The discretized version of the Navier–Stokes equations is solved with the SIMPLE algorithm of Patankar [31]. The nonlinearity of the momentum equation is dealt through an under-relaxed fixed-point iteration method. Numerical experiments have shown that setting the under-relaxation parameter to 0.8 yields an acceptable trade-off between stability and computational cost. Outer iterations are stopped when L2 norm of the relative residuals drops below 1e – 9. The linear systems arising at each fixed-point, at each SIMPLE iteration and in the sensitivity analysis are solved through a direct method in matlab. The optimization problems are solved using the method of moving asymptotes (MMA) of Svanberg [32]. The relevant MMA parameters are provided in Table 1. Convergence is considered satisfactory when the relative change in the objective is less than 1e – 5 for five consecutive iterations, and all constraints are satisfied. The presented modeling framework has been validated the with experimental results obtained from the Turin district heating network, see for instance Ref. [33].

## Results and Discussion

The system considered in this work is a portion of the Turin district heating network that delivers thermal power through 231 branches to 110 closely located buildings. The total thermal capacity that is able to deliver in design conditions is 17.6 MW. The original network along with the ground structure considered for the optimization are shown on the city map of Fig. 4. The performance of the system in design conditions is represented in Fig. 5, where the mass flow rate vector field and the scalar pressure field are shown. A total mass flow rate of 82.7 kg/s at 6.5 bar is injected in the network from the inlet node shown on the eastern side. The mass flow rate is extracted at the user nodes, which are identifiable on the map as outlet nodes of dead-end branches. We have identified seven local minima in the pressure field, hereafter referred as pressure hotspots (PHs). These nodes largely contribute to the objective function value obtained through the smooth minimum of Eq. (14). The strict global minimum is located in correspondence of PH 1 and has a value of 5.889 bar. This means that the maximum pressure drop along the supply line amounts to 0.611 bar.

We first investigate how the allocated investment C* affects the robustness of optimized layouts. To this aim, we systematically relax the cost constraint of Eq. (20). For each value, an optimization run is performed. Since robustness and cost are antagonist and independent objectives, the optimized values lay on the smooth Pareto front depicted in Fig. 6(a). A sudden steepness variation divides the front in two different regions:

1. (1)The region of topological modifications (for $C*≤200$ k$€$), where new branches appear continuously in the optimized layout as the allocated investment increases. This region corresponds to the steepest portion of the Pareto Front. Here, remarkable robustness improvements are possible: a 200 k$€$ investment yields a 29.3% gain in our robustness index. The minimum investment required for each branch to exist (i.e., $sj>smin$) is shown in Fig. 7. The design ground field is nearly completely saturated for a 200 k$€$ investment. The black branches are never found to be economically convenient in the range of investments considered. The optimized topology with ${20,40,60,80,100,200}$ k$€$ of investment counts ${2,5,11,13,17,19}$ looping branches, respectively.
2. (2)The region of nontopological modifications (for $C*>200$ k$€$), where the design ground field is nearly saturated. Only modifications to the branch diameters are observed. This region corresponds to the flat portion of the Pareto front. Here, only marginal robustness improvements are possible: a 800 k$€$ investment yields only a 2.5% robustness increase.

Differentiation of the Pareto trend yields the marginal cost of robustness reported in Fig. 6(b). The function ranges from a minimum value of 694 $€$/Pa to a maximum of 50,682 $€$/Pa. Substantial differences are visible between the two regions identified previously: in region (i) the marginal cost is nearly flat, and in region (ii) the trend is exponential. For subsequent analysis, we select three points on the Pareto frontier presented in Fig. 6(a): a small investment point (20 k$€$), an intermediate investment point (200 k$€$) and a large investment point (1000 k$€$). The corresponding optimized layouts are depicted in Fig. 8. If the allocated investment is minimal, only two looping branches are visible. Their position is very close to the inlet branch of the network. A rationale behind this choice can be gathered from the examination of the local robustness field $γ$ in Fig. 9. Here, it is easy to identify the most critical branch (depicted in blue in zone (a)), with a local robustness of −358 Pa. We observe that the optimized layout in Fig. 8(a) is selected in order to bypass the most critical branch with the maximum number of alternative routes. The optimized diameters calculated for those two branches are 28 mm and 31 mm, respectively, which is slightly above the manufacturing limit. For all the other elements of the ground structure, the abstract design variable hits the lower bound. For an intermediate allocated investment cost (200 k$€$), nearly all the branches of the ground structure are present. The diameters range from a minimum of 25 mm to a maximum of 323 mm. The 2 looping branches that populated the 20 k$€$ layout are the biggest pipes in the network. An additional examination of Fig. 9 reveals that the pipe with the third biggest diameter, i.e., 23.8 mm (indicated in yellow), is an alternative path to an other group of critical branches (zone (b)) with local robustness between −142 Pa and −124 Pa. Finally, the optimized design obtainable with an allocated investment of 1000 k$€$ (Fig. 8(c)), appears as a simple rescaling of the 200 k$€$ one: no topological modifications nor significant trends are reported. The only visible difference is the diameters scale. The pressure fields obtained for the three Pareto points are reported in Fig. 10. In all the cases considered, the pressure field is morphed around the most critical branches in zone (a) and (b) and considerably flattened in the area close to PH 6 and PH 7. Again, most of the performance differences are noticeable in the transition from a 20 k$€$ investment to a 200 k$€$ investment. Over this threshold, the pressure field looks unchanged. Topological changes allow increasing dramatically the supply pressure in PH 6 and PH 7. The total drop from the inlet node is reduced by roughly 40% and 33%, respectively (Fig. 11(a)). In the other PHs, no substantial pressure drop reduction is achievable. This is because most of the fluid-dynamic irreversibilities are due to a badly designed end-user branch, which cannot be looped. As a summary, Fig. 11(b) reports the PHs pressures registered for the cases considered. Topological modifications of the network can raise the minimum supply pressure from 5.889 bar to 5.964 bar, while an additional 0.014 bar can be gained by sizing correctly the looping branches.

## Conclusions

In this paper, we demonstrated the use of topology optimization to obtain an optimized looping strategy that maximizes robustness in district heating networks under a limited investment constraint. The optimized layout was obtained modifying the branch diameters of a large ground structure, which includes all the possible looping paths. We also introduced a novel robustness measure that is calculated as a weighted sum of the minimum supply pressure sensitivity to the fluid-dynamic resistance. A key advantage of this figure of merit is that it does not require any detailed probabilistic characterization of the input uncertainty nor the identification of bounds of stochastic variables.

The framework was tested on the Turin district heating network. The results highlighted the importance of considering the full design space in robustness maximization of fluid-networks. The benefits connected with the optimal sizing of the network were not comparable with those brought by the identification of the optimal topology. In fact, large performance improvements were observed only in the region of topological modifications, where new branches appears continuously in the optimized layout as the allocated investment increases. Here, a 30% increase of our robustness measure was achieved at the cost of 200 k$€$. When the ground structure saturated, limited robustness gains were possible: an additional 800 k$€$ investment improved our robustness measure by only 2.5%.

## Appendices

###### Appendix A: Partial Derivatives for First-Order Sensitivities Calculation

Here, we report all the partial derivatives involved in the calculation of the first-order sensitivities. The partial derivatives appearing in Eq. (16) are calculated as Display Formula

(A1)$(∂Pmin∂u)i={1nnPi(p−1)(1nn∑j=1nnPjp)(1p−1)if 1≤i≤nb0otherwise$
Display Formula
(A2)$∂Rfd∂u=K+LK$

where $LK$ is the strictly lower triangular portion of K in block form (Eq. (6)) that includes only B. The remaining term in Eq. (15) is obtained as Display Formula

(A3)$(∂Rfd∂β)i,j={Gj22ρSj2 for i={nn+1,…,nn+nb},j={1,…,nb}0otherwise$

###### Appendix B: Partial Derivatives for Second-Order Sensitivities Calculation

Here, we report all the partial derivatives involved in the calculation of the second-order sensitivities. The first term on the right-hand side of Eq. (23) is obtained as Display Formula

(B1)$∂z∂u=λ1T(∂2Rfd∂β∂uω)$

where the term in brackets reads Display Formula

(B2)$(∂2Rfd∂β∂uω)i,j={3Gjωj2ρSj2 for i={nn+1,…,nn+nb},j={1,…,nb}0otherwise$

The second term on the right-hand side of Eq. (23) can be written as Display Formula

(B3)$(∂W∂u)i,j={2λj(fjDjLj+βj)2ρSj2for i={nn+1,…,nn+nb},j={1,…,nb}lifor i={nb+1,…,nn+nb},j={nb+1,…,nn+nb}0otherwise$

where Display Formula

(B4)$li=1nn(1nn∑j=1nnPjp)(1p−1)(p−1)Pi(p−2)+1nn2(1nn∑j=1nnPjp)(1p−2)(1p−1)pPi2(p−1)$

The term on the left-hand side of Eq. (22) is readily obtainable again from Eq. (A2)Display Formula

(B5)$∂W∂λ1=∂Rfd∂u$

The partial derivative on the right-hand side of Eq. (22) is calculated as Display Formula

(B6)$∂z∂λ1=∂Rfd∂βω$
which reduces to exploiting the result of Eq. (A3).

Moving to the explicit dependencies appearing in Eq. (21) we have Display Formula

(B7)$∂z∂s=λ1T(∂2R∂β∂sω)$

where the term in brackets reads Display Formula

(B8)$(∂2R∂β∂sω)i,j={−Gj2ωjρSj2DmaxDj for i={nn+1,…,nn+nb},j={1,…,nb}0otherwise$

The second explicit dependency in Eq. (21) has the form Display Formula

(B9)$(∂Rfd∂s)i,j={−Gj2ρSj2DmaxDj(3fLj2Dj+βj) for i={nn+1,…,nn+nb},j={1,…,nb}0otherwise$

The third explicit dependency in Eq. (21) can be calculated as Display Formula

(B10)$(∂W∂s)i,j={−3GjλjρSj2DmaxDj(3fLj2Dj+βj) for i={nn+1,…,nn+nb},j={1,…,nb}0otherwise$

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Babayan, A. , Kapelan, Z. , Savic, D. , and Walters, G. , 2005, “ Least-Cost Design of Water Distribution Networks Under Demand Uncertainty,” J. Water Resour. Plann. Manage., 131(5), pp. 375–382.
Chakraborty, S. , Bhattacharjya, S. , and Haldar, A. , 2012, “ Sensitivity Importance-Based Robust Optimization of Structures With Incomplete Probabilistic Information,” Int. J. Numer. Methods Eng., 90(10), pp. 1261–1277.
Creaco, E. , Fortunato, A. , Franchini, M. , and Mazzola, M. , 2014, “ Comparison Between Entropy and Resilience as Indirect Measures of Reliability in the Framework of Water Distribution Network Design,” Procedia Eng., 70, pp. 379–388.
Gong, W. , Khoshnevisan, S. , and Juang, C. H. , 2014, “ Gradient-Based Design Robustness Measure for Robust Geotechnical Design,” Can. Geotech. J., 51(11), pp. 1331–1342.
Han, J. , and Kwak, B. , 2004, “ Robust Optimization Using a Gradient Index: Mems Applications,” Struct. Multidiscip. Optim., 27(6), pp. 469–478.
Kim, N.-K. , Kim, D.-H. , Kim, D.-W. , Kim, H.-G. , Lowther, D. A. , and Sykulski, J. K. , 2010, “ Robust Optimization Utilizing the Second-Order Design Sensitivity Information,” IEEE Trans. Magnetics, 46(8), pp. 3117–3120.
Sciacovelli, A. , Verda, V. , and Borchiellini, R. , 2013, Numerical Design of Thermal Systems, CLUT, Torino, Italy.
Bondy, J. A. , and Murty, U. S. R. , 1976, Graph Theory With Applications, Elsevier, Amsterdam, The Netherlands.
Bendsoe, M. P. , and Sigmund, O. , 2013, Topology Optimization: Theory, Methods, and Applications, Springer Science & Business Media, Berlin.
Grossmann, I. E. , 1996, “ Mixed-Integer Optimization Techniques for Algorithmic Process Synthesis,” Adv. Chem. Eng., 23, pp. 171–246.
Patankar, S. V. , 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Washington, DC.
Svanberg, K. , 1987, “ The Method of Moving Asymptotes a New Method for Structural Optimization,” Int. J. Numer. Methods Eng., 24(2), pp. 359–373.
Guelpa, E. , Toro, C. , Sciacovelli, A. , Melli, R. , Sciubba, E. , and Verda, V. , 2016, “ Optimal Operation of Large District Heating Networks Through Fast Fluid-Dynamic Simulation,” Energy, 102, pp. 586–595.
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Jung, D. , Kang, D. , Kim, J. H. , and Lansey, K. , 2013, “ Robustness-Based Design of Water Distribution Systems,” J. Water Resour. Plann. Manage., 140(11), pp. 13–28.
Babayan, A. , Kapelan, Z. , Savic, D. , and Walters, G. , 2005, “ Least-Cost Design of Water Distribution Networks Under Demand Uncertainty,” J. Water Resour. Plann. Manage., 131(5), pp. 375–382.
Chakraborty, S. , Bhattacharjya, S. , and Haldar, A. , 2012, “ Sensitivity Importance-Based Robust Optimization of Structures With Incomplete Probabilistic Information,” Int. J. Numer. Methods Eng., 90(10), pp. 1261–1277.
Creaco, E. , Fortunato, A. , Franchini, M. , and Mazzola, M. , 2014, “ Comparison Between Entropy and Resilience as Indirect Measures of Reliability in the Framework of Water Distribution Network Design,” Procedia Eng., 70, pp. 379–388.
Gong, W. , Khoshnevisan, S. , and Juang, C. H. , 2014, “ Gradient-Based Design Robustness Measure for Robust Geotechnical Design,” Can. Geotech. J., 51(11), pp. 1331–1342.
Han, J. , and Kwak, B. , 2004, “ Robust Optimization Using a Gradient Index: Mems Applications,” Struct. Multidiscip. Optim., 27(6), pp. 469–478.
Kim, N.-K. , Kim, D.-H. , Kim, D.-W. , Kim, H.-G. , Lowther, D. A. , and Sykulski, J. K. , 2010, “ Robust Optimization Utilizing the Second-Order Design Sensitivity Information,” IEEE Trans. Magnetics, 46(8), pp. 3117–3120.
Sciacovelli, A. , Verda, V. , and Borchiellini, R. , 2013, Numerical Design of Thermal Systems, CLUT, Torino, Italy.
Bondy, J. A. , and Murty, U. S. R. , 1976, Graph Theory With Applications, Elsevier, Amsterdam, The Netherlands.
Bendsoe, M. P. , and Sigmund, O. , 2013, Topology Optimization: Theory, Methods, and Applications, Springer Science & Business Media, Berlin.
Grossmann, I. E. , 1996, “ Mixed-Integer Optimization Techniques for Algorithmic Process Synthesis,” Adv. Chem. Eng., 23, pp. 171–246.
Patankar, S. V. , 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, Washington, DC.
Svanberg, K. , 1987, “ The Method of Moving Asymptotes a New Method for Structural Optimization,” Int. J. Numer. Methods Eng., 24(2), pp. 359–373.
Guelpa, E. , Toro, C. , Sciacovelli, A. , Melli, R. , Sciubba, E. , and Verda, V. , 2016, “ Optimal Operation of Large District Heating Networks Through Fast Fluid-Dynamic Simulation,” Energy, 102, pp. 586–595.

## Figures

Fig. 1

Schematic of the one-dimensional model of the network

Fig. 2

Original and smoothed cost function

Fig. 3

Finite difference check on the accuracy of the adjoint approach: (a) first-order sensitivities and (b) second-order sensitivities

Fig. 4

Original network and ground structure

Fig. 5

Performance of the current network in design conditions: (a) mass flow rate and directions in each branch of the network and (b) pressure field

Fig. 6

(a) Pareto front expressing the trade-off between robustness and cost and (b) marginal cost of robustness

Fig. 7

Minimum investment required for each branch to exist. Black branches are never found to be economically convenient.

Fig. 8

Optimized network layout at selected Pareto points. (a) 20 k€ investment, (b) 200 k€ investment, (c) 1000 k€ investment.

Fig. 9

Local robustness field for the Pareto point at C*=200k€

Fig. 10

Supply pressure field at selected Pareto points: (a) 20 k€ investment, (b) 200 k€ investment, (c) 1000 k€ investment

Fig. 11

(a) Pressure drop reduction obtained in the PHs at selected Pareto points and (b) supply pressure in the PHs at selected Pareto points

## Tables

Table 1 MMA parameters utilized

## Discussions

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