Optimal design-for-control of self-cleaning water distribution networks using a convex multi-start algorithm

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Optimal design-for-control of self-cleaning water distribution networks

using a convex multi-start algorithm

Bradley Jenksa, Filippo Peccia, Ivan Stoianova

aDepartment of Civil and Environmental Engineering, Imperial College London, London SW7 2BB, United Kingdom

Abstract

The provision of self-cleaning velocities has been shown to reduce the risk of discolouration in water distri-

bution networks (WDNs). Despite these ﬁndings, control implementations continue to be focused primarily

on pressure and leakage management. This paper considers the control of diurnal ﬂow velocities to maximize

the self-cleaning capacity (SCC) of WDNs. We formulate a new optimal design-for-control problem where

locations and operational settings of pressure control and automatic ﬂushing valves are jointly optimized.

The problem formulation includes a nonconvex objective function, nonconvex hydraulic conservation law

constraints, and binary variables for modelling valve placement, resulting in a nonconvex mixed integer

nonlinear programming (MINLP) optimization problem. Considering the challenges with solving noncon-

vex MINLP problems, we propose a heuristic algorithm which combines convex relaxations (with domain

reduction), a randomization technique, and a multi-start strategy to compute feasible solutions. We evaluate

the proposed algorithm on case study networks with varying size and degrees of complexity, including a

large-scale operational network in the UK. The convex multi-start algorithm is shown to be a more robust

solution method compared to an off-the-shelf genetic algorithm, ﬁnding good-quality feasible solutions to all

design-for-control numerical experiments. Moreover, we demonstrate the implemented multi-start strategy

to be a fast and scalable method for computing feasible solutions to the nonlinear SCC control problem. The

proposed method extends the control capabilities and beneﬁts of dynamically adaptive networks to improve

water quality in WDNs.

Keywords: water quality, discolouration, self-cleaning capacity, design-for-control, mixed integer

nonlinear programming, convex optimization

1. Introduction

The management of water quality in water distribution networks (WDNs) presents a complex opera-

tional challenge. As a direct consequence of ageing and deteriorating infrastructure, the mitigation of dis-

colouration incidents is becoming one of the key operational challenges within water quality management

programmes. In addition to discolouration being the largest source of customer complaints (Vreeburg and

Boxall, 2007, Husband and Boxall, 2011, Armand et al., 2017), there is a growing body of evidence sug-

gesting its occurrence harbours increased microbial activity (Liu et al., 2013, 2014, van der Wielen and Lut,

2016). These conditions can accelerate bioﬁlm growth and drastically reduce the efﬁcacy of disinfectant

residuals in protecting against waterborne illness and contaminants intrusion. Moreover, WDNs in the UK

are highly sectorized and operated with ﬁxed topology for purposes of leakage management. This type of

network conﬁguration, referred to as district metered areas (DMAs), has been demonstrated to exacerbate

arXiv:2210.06388v2 [math.OC] 3 Feb 2023

water quality deterioration and increase the risk of discolouration incidents (Machell and Boxall, 2014, Ar-

mand et al., 2018). With progressively stringent water quality regulations, water companies are seeking

effective and cost-efﬁcient operational control strategies to reduce the risk of discolouration.

Discolouration is primarily a consequence of resuspended material accumulated within WDNs (Vree-

burg and Boxall, 2007). It can materialize from the cumulative impact of the following processes (Boxall and

Dewis, 2005): (i) the ingress and/or development of particulate matter; (ii) the accumulation of particulates

at the pipe invert and/or formation of cohesive layers at the pipe wall; and (iii) a hydraulic disturbance (i.e.

trigger event), which mobilizes loose particulates and generates sufﬁcient shear stress to overcome cohesive

forces at the pipe wall. Such hydraulic disturbances can be generated from different phenomena, including

pressure transients during unsteady hydraulic conditions (Aisopou et al., 2012). Apart from their origin,

the physical pathways of discolouration are intrinsically connected to network hydraulics. In a recent study

focusing on the impact of network sectorization on water quality, Armand et al. (2018) proposed a set of

surrogate hydraulic variables for discolouration risk assessment. Central to their ﬁndings was the role of

diurnal ﬂow velocities on particle transport and fate. This connection between discolouration and hydrody-

namic conditions has been supported by numerous experimental and theoretical studies; see van Summeren

and Blokker (2017) and Armand et al. (2018) for reviews on the topic. These studies have mainly focused

on the development of predictive tools for modelling particle transport and accumulation processes. Most

notably, Boxall et al. (2001) developed the Prediction of Discolouration in Distribution Systems (PODDS)

model, an empirically-based numerical tool which aims to characterize cohesive layer strength at the pipe

wall. The PODDS model was later updated to account for material regeneration in Furnass et al. (2014),

where both erosion and regeneration processes require calibration using continuous ﬂow and turbidity data.

Because such tools require extensive ﬁeld testing and are generally limited to pipe-level assessments, their

use in practice has not yet been widespread. Recognizing this limitation, van Summeren and Blokker (2017)

presented a theoretical particle transport model, combining the effects of gravitational settling, hydraulic

shear stresses, and bed-load transport. To complement this, several laboratory-based experimental studies

have emerged to better understand the complex interactions between particle properties and pipe hydraulics

(e.g. Sharpe et al., 2019; Braga et al., 2020).

In addition to predictive modelling, research has also focused on reducing the severity and frequency

of discolouration incidents through network design, maintenance, and control. Water companies in the

Netherlands have been conducting experimental research on the design and implementation of controls for

self-cleaning networks. The self-cleaning capacity (SCC) of a WDN is deﬁned as the ability for pipes to

experience peak daily ﬂow velocities above a threshold required to routinely re-suspend particles and thus

prevent accumulation (Vreeburg et al., 2009). Previous experimental programmes have suggested resuspen-

sion velocities on the order of 0.2 m/sto 0.25 m/sin distribution pipes (Ryan et al., 2008, Blokker et al.,

2010). This has been corroborated with a recent ﬁeld study monitoring turbidity under various ﬂow rates,

where an increase in turbidity levels were observed at ﬂow velocities greater than 0.2 m/s(Prest et al.,

2021). Water companies in the Netherlands have demonstrated successful self-cleaning implementations by

redesigning looped, oversized networks to branched layouts with smaller diameter pipes (Vreeburg et al.,

2009). A recent study has also investigated the trade-off between self-cleaning velocities and ﬁre ﬂow ca-

pacity in North American WDNs (Gibson et al., 2019). However, since the redesign of WDN infrastructure

becomes cost-prohibitive at scale, there have been recent forays in the reconﬁguration of existing network

topology to promote self-cleaning networks (Blokker et al., 2012, Abraham et al., 2016, 2018).

Combining UK and Dutch experience, Abraham et al. (2016, 2018) formulated an optimization prob-

lem for increasing SCC by redistributing ﬂow through changes in network topology. More speciﬁcally, the

optimization problem aimed to maximize the number of pipes with ﬂow velocities above a self-cleaning

threshold through two separate strategies: (i) optimal closure of isolation valves and (ii) optimal operational

settings of existing pressure control valves (Abraham et al., 2016, 2018). Abraham et al. (2018) solved the

problem of optimizing valve closures using a linear graph analysis tool. Following Schaub et al. (2014),

a line-outage distribution factor (LODF) matrix was computed to estimate the ﬂow redistribution resulting

from an outage (closure) of an (or multiple) edge-to-edge relation(s). For the optimal control problem,

Abraham et al. (2016) computed a local solution by approximating the nonsmooth objective function as

a continuous nonlinear function, followed by application of a tailored sequential convex programming al-

gorithm. While the beneﬁts of the LODF solution method for optimal valve closures were demonstrated

numerically using an operational network in the Netherlands, results from the control problem were limited

to a small-scale theoretical network. Moreover, decision variables were restricted to the control of existing

unidirectional pressure reducing valves (PRVs). Building on the SCC optimization problem posed in Abra-

ham et al. (2016, 2018), this manuscript considers both control and design-for-control problem formulations.

The latter involves the simultaneous optimization of valve placement and operational settings for both ex-

isting and new control valves. In addition to unidirectional PRVs, this work also considers bidirectional

dynamic boundary valves (DBVs) and automatic ﬂushing valves (AFVs) as dynamic hydraulic controls.

These hydraulic controls were developed to facilitate the novel operational framework of dynamically adap-

tive networks (Wright et al., 2014, Ulusoy et al., 2022). The resulting optimization problem is formulated as

a nonconvex mixed integer nonlinear program (MINLP).

Both mathematical optimization and heuristic methods have been used to solve design and control prob-

lems in WDNs (see literature review in Mala-Jetmarova et al., 2017). For mathematical optimization meth-

ods, scalability is recognized as a current limitation in solving MINLP problems to global optimality (Koch

et al., 2012, Sahinidis, 2019); that is, the implementation of global solvers become impractical for large

problem cases. Consequently, heuristic approaches are often employed to compute satisfactory feasible so-

lutions. A common heuristic method used for WDN optimization problems is the genetic algorithm (GA).

While GAs have been successfully applied to design problems, the computational effort required to ﬁnd

solutions sufﬁciently close to the global optimum grows rapidly with problem size (Maier et al., 2014). In

this manuscript, we develop a heuristic algorithm based on convex optimization and a multi-start scheme to

compute feasible solutions to the considered MINLP problem. To handle integer variables, we ﬁrst formulate

a convex subproblem through polyhedral relaxations of nonconvex terms and the continuous relaxation of

binary variables. We subsequently employ a randomization heuristic to sample Ncandidate valve conﬁgu-

rations from the set of fractional values generated from the convex subproblem. We then ﬁx binary variables

for each sampled valve conﬁguration and compute (locally) optimal operational settings from a nonlinear

programming (NLP) control problem. This follows the heuristic algorithm presented in Pecci et al. (2022),

extending its application to the SCC design-for-control problem and to the nonsmooth Hazen-Williams fric-

tion model. Since the degree of nonlinearity of the SCC problem is higher than the problem investigated in

Pecci et al. (2022), we include a multi-start strategy and a feasibility restoration problem for selecting start-

ing points. This step aims to minimize the risk of poor local optima as well as ensure hydraulic feasibility

of the NLP control problem. Finally, the best feasible solution is selected from the set of sampled valve

conﬁgurations. The proposed heuristic algorithm further increases the beneﬁts from the implementation of

dynamically adaptive networks as it expands their control capabilities to enhance water quality in WDNs.

This manuscript is organized as follows. In Section 2, we formulate the design-for-control problem of

maximizing the network SCC through dynamic hydraulic controls. We then present the proposed heuristic

algorithm in Section 3. Finally, in Section 4, we demonstrate the performance of the developed heuristic

algorithm using three case study networks with varying size and degrees of complexity. To facilitate a

broader discussion on heuristic approaches for the design and control of WDNs, we compare the results

with an off-the-shelf GA implementation, which is a common approach used in the literature.

2. Problem formulation

We investigate a design-for-control problem to maximize the length of network pipes experiencing ﬂow

velocities above a given self-cleaning capacity (SCC) threshold. This is achieved by installing new valves

and/or controlling their operational settings. For this purpose, our problem formulation considers three valve

types as pressure and connectivity control actuators. First, pressure reducing valves (PRVs), which are mod-

elled having unidirectional ﬂow. Second, bidirectional dynamic boundary valves (DBVs), for which ﬂow

is permitted in both directions across discrete model time steps. Here, DBVs represent the operation of

remote-controlled isolation valves, which modulate ﬂow and pressure between adjacent zones. Third, au-

tomatic ﬂushing valves (AFVs), whose ﬂushing rate is bounded by a set maximum value. Throughout this

manuscript, we refer to either PRVs or DBVs as control valves, as both have the capability of controlling

pressure and are modelled at network links. On the other hand, AFVs are simply referred to as ﬂushing

valves and are modelled at network nodes. We consider operational scenarios, for which PRV locations have

been ﬁxed to minimize average zone pressure (AZP), and thus decision variables include only their opera-

tional settings. In comparison, both locations and operational settings of DBVs and AFVs are considered as

decision variables. The operational settings of valves are modelled as continuous variables, whereas their

placement (location) are modelled through binary variables. All network links and nodes are considered as

potential locations of DBVs and AFVs, respectively. As the current stage of this work focuses on the self-

cleaning capacity of pipes at the DMA or distribution level, we do not consider storage tanks or pumping

activity as forms of hydraulic control. Therefore, we assume discrete and hydraulically independent model

time steps. Finally, it is noted that this work relies on the availability of a calibrated hydraulic model.

2.1. Hydraulic variables and constraints

The problem considers a water distribution network (WDN) with nplinks, nndemand nodes and n0

known head nodes (e.g. water sources, reservoirs). The network is modelled as a directed graph with

npedges (links) and nn+n0vertices (nodes). A demand-driven hydraulic analysis is used to simulate

steady-state network hydraulics over ntdiscrete time steps. For each time step t∈ {1, . . . , nt}, known

hydraulic conditions are given by vectors of nodal demands dt∈Rnnand source hydraulic heads h0

t∈Rn0.

Moreover, vectors ηt∈Rnpand αt∈Rnnare included to model local losses introduced by the action of

control valves and operational demands at ﬂushing valves, respectively. Unique vectors of hydraulic states

qt∈Rnpand ht∈Rnnare computed by solving the following steady-state energy (1a) and mass (1b)

conservation equations governing pipe ﬂow:

A12ht+A10h0

t+φ(qt) + ηt= 0 (1a)

12qt−dt−αt= 0,(1b)

where A12 ∈np×nnand A10 ∈np×n0are the link-node incidence matrices for demand and known

head nodes, respectively; and the vector φ(qt)=[φ1(q1,t). . . φnp(qnp,t)]Tmodels frictional head losses

associated with ﬂows qt. Omitting time index t,φj(qj)is deﬁned in general form for ﬂow conveyed across

link jas:

φj(qj) = rj|qj|nj−1qj,∀j∈ {1, . . . , np},(2)

where the resistance coefﬁcient rjand exponent nj, both independent of time t, take different values de-

pending on the link type (e.g. pipe or valve) and on the frictional head loss model. For valve links, nj= 2

and rj=8Kj

gπ2D4

, with Kjand Djrepresenting the valve loss coefﬁcient and diameter, respectively (Larock

et al., 1999). In this work, we apply the Hazen-Williams (HW) model to characterize frictional head losses

across pipe links. The HW model is an explicit and empirical relationship between pipe ﬂow and frictional

head loss, with nj= 1.852 and rjis deﬁned as follows for all j∈ {1, . . . , np}:

rj=10.67Lj

C1.852

jD4.871

,(3)

where Cis the HW coefﬁcient, a dimensionless number representing frictional characteristics; Lis pipe

length in meters; and Dis pipe diameter in meters (Larock et al., 1999). Similarly, explicit approximations

of the Darcy-Weisbach formula (e.g. Valiantzas, 2008) could be used to model frictional head losses. Finally,

it is convenient to isolate the nonlinear term φ(qt)in (1a). Here, we introduce a vector of auxiliary variables

θt∈Rnp, which separates the energy conservation constraint into its linear and nonlinear components, as

follows:

A12ht+A10h0

t+θt+ηt= 0 (4a)

θt−φ(qt)=0.(4b)

Valve placement and operation are modelled as follows. For each time step t∈ {1, . . . , nt}, the contin-

uous variable ηt∈Rnppresented in (1a) models the local losses introduced by the action of control valves

and the continuous variable αt∈Rnnpresented in (1b) models the ﬂow emitted at ﬂushing valves. More-

over, binary variables z∈ {0,1}npare included to model PRV and DBV placement, and v+

t∈ {0,1}np

and v−

t∈ {0,1}npto assign their control capabilities in the positive or negative ﬂow direction, respectively,

across each time step t. Thus, for all links j∈ {1, . . . , np}and time steps t∈ {1, . . . , nt}, binary variables

zj,v+

j,t, and v−

j,t are set as

zj=1control valve on link j

0no valve

j,t =1control valve on link jin positive direction

0no valve

v−

j,t =1control valve on link jin negative direction

0no valve

(5)

Analogously, the placement of AFVs at network nodes is modelled using binary variables y∈ {0,1}nn,

deﬁned as

yi=1ﬂushing valve placed at node i

0no valve (6)

These binary variables are subject to the following physical and economical constraints, which limit pressure

control capabilities in a single direction at each time step t(7a) and enforce a maximum number of control

valves nvand ﬂushing valves nfconsidered for installation (7b)-(7c):

j,t +v−

j,t ≤zj,∀j∈ {1, . . . , np},∀t∈ {1, . . . , nt}(7a)

j=1

zj=nv(7b)

i=1

yi=nf.(7c)

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Optimaldesign-for-controlofself-cleaningwaterdistributionnetworksusingaconvexmulti-startalgorithmBradleyJenksa,FilippoPeccia,IvanStoianovaaDepartmentofCivilandEnvironmentalEngineering,ImperialCollegeLondon,LondonSW72BB,UnitedKingdomAbstractTheprovisionofself-cleaningvelocitieshasbeenshowntoreducethe...

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Optimal design-for-control of self-cleaning water distribution networks using a convex multi-start algorithm

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