Sediment minimization in canals: An optimal control approach · dynamics and sediment transport and deposition equations, for the cases of cohesive and non-cohesive sediments. As

Noname manuscript No.(will be inserted by the editor)

Sediment minimization in canals:An optimal control approach

L.J. Alvarez-Vazquez · A. Martınez ·C. Rodrıguez · M.E. Vazquez-Mendez

Received: 27 Oct 2017 / Accepted: date

Abstract This work deals with the computational modelling and control ofthe sedimentation of suspended particles in canals. To analyze this environ-mental problem, two alternative mathematical models are proposed (1D and2D, respectively), coupling the system for shallow water hydrodynamics withthe Exner equations for sediment transport. The main goal here is relatedto establishing the optimal management of a canal to avoid the settling ofsuspended particles and their unwanted effects. The problem is formulatedas an optimal control problem of partial differential equations, where a setof design variables (the shape of the channel section and the water inflow)is considered, in order to control flow velocity and, therefore, sedimentationof suspended particles. In this novel simulation-based optimization approachto the problem, in addition to a well-posed mathematical formulation of theproblem, both theoretical and numerical results are presented for a realisticcase (interfacing MIKE21 package with the authors’ own MATLAB code forNelder-Mead optimization algorithm).

Keywords Optimal control · Computational modelling · Sedimentation ·Canal

L.J. Alvarez-Vazquez, A. MartınezE.I. Telecomunicacion. Universidade de Vigo. 36310 Vigo. SpainE-mail: [email protected], [email protected]

C. RodrıguezF. Matematicas. Universidade de Santiago de Compostela. 15782 Santiago. SpainE-mail: [email protected]

M.E. Vazquez-MendezE.P.S. Universidade de Santiago de Compostela. 27002 Lugo. SpainE-mail: [email protected]

2 L.J. Alvarez-Vazquez et al.

1 Introduction

Channel networks have been used throughout the years to convey, distributeand apply water to land. Channels may be constructed on different topogra-phies and soil conditions with different cross sections and longitudinal profiles.Nowadays, the most widely used channel sections are parabolic, rectangular,trapezoidal and circular sections. The section variables of these channels (sideslope, bottom width, depth, hydraulic radius, etc.) are designed according tothe laws of open channel flow, combined with the particular aims and con-straints (which include not only agricultural needs, but also industrial, munic-ipal and power needs).

There exist various studies on the values of section variables for differentchannel geometries. For instance, in his classical text Chow [1] analyzed theproperties of optimal sections, giving the relations between the section vari-ables of the most hydraulically efficient sections for different channel types (inhis study the optimality is understood in the sense of considering the con-veyance of a given discharge with minimum flow area). Some of the resultsof this study are still in use for several channel types in order to determinehydraulically and/or economically efficient sections. The relations obtained forthese optimal section variables were modified by considering other different pa-rameters. So, for instance, including the effect of freeboard, Guo and Hughes [2]made a study on the optimal values of section variables which either minimizesfrictional resistance or minimizes construction cost, presenting their solutionsfor the trapezoidal channel sections. Consideration of freeboard in channel de-sign had not been limited to trapezoidal sections. For example, Loganathan[3] derived optimality conditions for the parabolic-canal design accounting forfreeboard and limitations on velocity and canal dimensions. Monadjemi [4]has performed a detailed study of the relationships derived for various channeltypes, and proved that considering the minimization of flow area as an objec-tive will result in the same optimal values of section variables as of consideringthe minimization of wetted perimeter. Later, Froehlich [5] also proposed simpleexpressions for the optimal section variables of width and depth constrainedtrapezoidal channels. Swamee [6] proposed explicit equations of section vari-ables for the minimal flow area problem considering resistance equation of uni-form flow as a function of roughness, height of channel bottom and kinematicviscosity of water. After this study, Swamee et al. [7] proposed also equationsfor section variables in the case of triangular, rectangular, trapezoidal and cir-cular channel geometries, considering the minimization of channel cost as theobjective for optimization purposes.

However, it is not until more recent times that among the standard objec-tives to be optimized (earthwork costs, manteinance, flow area, cost of waterloss...) sediment minimization is addressed. Among the latest works on thisdirection, the authors can mention those of Kentli et al. [8], Ebtehah et al.[9], Bonakdari et al. [10], and the references therein, where evolutionary algo-rithms (neural networks, data mining, genetic algorithms...) are used in orderto optimize different canal features. Nevertheless, in all these works, minimiza-

Sediment minimization in canals: An optimal control approach 3

tion is applied in order to obtain optimal parameters for well-known formulaetaken from classical literature, usually related to quantifying shear stresses onchannel bed and/or sides and stresses required for incipient motion of sedimentparticles. In this paper, the authors present a novel and more realistic alter-native where they address the problem by a simulation-based optimizationalgorithm, that takes into account the whole process, including both hydro-dynamics and sediment transport and deposition equations, for the cases ofcohesive and non-cohesive sediments. As far as they know, this approach hasnot been previously addressed in the scientific literature.

A large number of the most important quality problems in canals are re-lated to the behaviour and characteristics of cohesive or non-cohesive sedi-ments, which are responsible, among others, for the malfunction of irrigationchannels, the instability in canals for surface drainage, or the loss of capacity inchannels. So, in section 2 the authors analyze a well-posed mathematical for-mulation of this sedimentation problem within a one-dimensional framework.Section 3 is devoted to present an alternative formulation of the problem in thetwo-dimensional case, the solution to which can be obtained by the numericalalgorithm introduced in section 4. Final sections of the paper are devoted topresent several numerical results and conclusions for a real-world example ofecological relevance.

2 Mathematical formulation and analysis of the problem: a 1Dapproach

To fix ideas, here will be presented the case for a channel collecting wastewaterfrom the cooling system of a thermoelectric plant, but this approach can beapplied to any type of irrigation canals, storm sewers, drainage channels andso on.

A wastewater treatment plant aims to achieve from wastewater an effluentwater with better quality features, based on certain standard parameters. In-side a treatment plant (consider, for example, one receiving wastewater froma cooling system corresponding to a thermoelectric plant) water transfers areoriginated from different containers through channels. In these canals typicallyoccurs natural deposition of the numerous particles in suspension, which causesa change in the geometry of the canal bed (sludge accumulation, vegetationgrowth...). All this will trigger a malfunction of the treatment processes, whichresults in the electric plant. The goal is then aimed to the study of sedimen-tation in the bed of a channel to optimally design the geometry of the canalsection and to control water inflow, so as to avoid above mentioned problems.

Because of its high computational load, solving a full 3D model is extremelytime-consuming. Thus, the development of 1D and 2D models has been an im-portant task since many decades ago, mainly via section- and depth-averagingof the 3D model equations.


L

EH(x,t)

z(x,t)

wD

aA(x,t)

A (x,t)S

0 x

b(x)

W

Fig. 1 Schematic section of the channel for the 1D model.

2.1 Setting of the problem

The 1D mathematical model presented in this section in order to study sed-imentation in a channel couples partial differential equations for the one-dimensional version of the hydrodynamics of shallow waters (formulated interms of the wet area and the averaged flow) with the convection-reaction-diffusion equation for sediment transport (formulated in terms of the con-centration of suspended particles) and with the Exner equation for sedimentdeposition and erosion (formulated in terms of the settled area). Specifically,the following coupled system is considered for a channel of length L over aperiod of time T :

∂A

∂t+∂Q

∂x=∑j

qj δ(x− pj) in (0, L)× (0, T ), (1)

∂Q

∂t+

∂

∂x

(Q2

A

)+ gA

∂

∂x(b+ z +H) (2)

= − gP

C2h

Q

A2|Q|+

∑j

qj Vj cos(εj) δ(x− pj) in (0, L)× (0, T ),

∂

∂t(Ac) +

∂

∂x(Qc)− ∂

∂x

(kA

∂c

∂x

)− φ =

∑j

mj δ(x− pj), (3)

ρs(1− η)∂As∂t

+ φ = 0 in (0, L)× (0, T ), (4)

with boundary conditions:

A(L, t) = AL(t), Q(0, t) = q(t) in (0, T ), (5)

c(0, t) = c0(t), k∂c

∂x(L, t) = cL(t), As(0, t) = As0(t) in (0, T ), (6)


and initial conditions:

A(x, 0) = A0(x), Q(x, 0) = Q0(x) in (0, L), (7)

c(x, 0) = c0(x), As(x, 0) = A0s(x) in (0, L), (8)

where A(x, t) is the wet area, that is, the cross sectional area of flow; Q(x, t) isthe flux of water (Q = Au, with u(x, t) the averaged velocity of water); c(x, t)is the section-averaged concentration of sediment transported in suspension;As(x, t) is the settled area; pj are the points of water inflow to the canal withwater flux qj(t), velocity Vj(t), angle εj between the discharge and the mainchannel, and sediment flux mj(t); δ(x − pj) denotes the Dirac delta functionconcentrated at point pj ; b(x) represents the geometry of the (fixed) bottomof canal; z(x, t) is the height of settled sediment on the channel bottom (ifthe shape of the section is assumed known: rectangular, circular, trapezoidal...then there exists a bijective mapping between height and corresponding area[11], that is: z = B(As) or, equivalently, As = B−1(z) = S(z)); H(x, t) is thedepth of water (in a similar way to previous case, H = B(A + As) − B(As)or, equivalently, A = S(z + H) − S(z)); P represents the wetted perimeter;g is the gravity acceleration; Ch is the Chezy friction coefficient; k is thediffusion coefficient; ρs is the density of sediment; η ∈ [0, 1] corresponds to thebed porosity; and φ measures the interchange of sediment with the bottom(balancing eroded vs. deposited matter).

In the case of cohesive sediments (that is, taking into account flocculationprocesses) the expression employed for φ will be:

φ = W (Se − Sd), (9)

for W the channel width at water surface. In this formula, deposition Sd isdetermined as:

Sd = pwsc, (10)

with ws the floc settling velocity given by expression [12]:

ws =

k1c

s1 if c ≤ cp,ws0 (1− k2c)s2 if c > cp,

(11)

for ws0 a reference velocity, k1, k2, s1 and s2 parameters, and cp the critical sed-iment concentration at which the floc settling velocity changes its trend fromincreasing to decreasing [13]; and p representing the probability of deposition,determined as:

p =

1 if τb ≤ τb,min,τb,max − τb

τb,max − τb,minif τb,min < τb ≤ τb,max,

0 if τb > τb,max,

(12)

for τb the bed shear stress given by expression τb = ρgQ |Q| /(ChA)2, ρ thefluid density, and τb,min and τb,max the critical lower and upper bed shearstresses for deposition, respectively.


Remark 1 Dynamic behaviour of cohesive sediment is governed by a lot ofother factors, such as, sediment size, shape, composition, organic matter con-tent, etc. In this work a simplified approach has been used, but other authors[14,15] have considered using other settling velocity equations that may cap-ture other particularities in the cohesive sediment dynamics. Unfortunately,these expressions cannot be used here due to their excessively heuristic (non-analytical) character.

Finally, erosion term Se is defined by the expression [16]:

Se =

0 if τb ≤ τb,e,

E(τb − τb,eτb,e

)mif τb > τb,e,

(13)

for E the erodability of bed, m an erosion power coefficient, and τb,e the criticalbed shear stress for erosion.

Alternatively, in the case of non-cohesive sediments, a possible expressionfor φ may be:

φ =Q

LA(c∗ − c), (14)

with c∗ the transport capacity of sediment, and LA the adaptation length, rep-resenting the distance required by clear water entering a uniform flow streamflowing over a uniform grain size bottom to reach the uniform sediment trans-port conditions. This adaptation length depends on the particle grain size andon the characteristics of the water flow, i.e., more precisely on the ratio be-tween friction velocity and particle settling velocity. It can be computed, forinstance, by Han’s formula [17]: LA = λu∗/νf , with λ a calibration parameter,u∗ the friction (or shear) velocity, and νf the settling velocity of sediment [18].

Remark 2 The mathematical analysis of the state system (1) − (8) is veryfar from being a closed issue. Existence and uniqueness of solution for thegeneral system is still an open question, although for very particular cases ofshallow water-Exner models (for instance, the simple Grass law [19]) severalresults have been achieved in last years. So, for the 1D shallow water (SaintVenant) equations partial existence results can be found, among others, in thework of Bermudez et al. [11]. On the other hand, Cordier et al. [20] provedthe hyperbolicity of the Saint Venant-Exner model, (hence, the possibility ofderiving existence and uniqueness results for the solution of the Cauchy initial-value problem governed by this model with appropriate initial and boundaryconditions) for simple expressions of the bedload transport rate. However, formore general morphodynamic models, (for instance, those including multiplesediment sizes using an active layer approach), hyperbolicity cannot be ingeneral demonstrated, and instead the model has been proven to be ellipticunder some circumstances [21]. Further technical details, both for the cohesiveand the non-cohesive cases, can be found, for instance, in the recent papers[22], [23] or [24], and the references therein.


Now, focusing the attention on the shape of the channel, the authors startfrom a rectangular channel whose original dimensions are a width D and adepth E. A modification of the shape of its section will be obtained by fillinga lateral side (for example, with concrete) so that the new modified width of thebasis of the channel is w, and that the modified side wall presents an inclinationangle α with relation to the vertical (see Fig. 1). By technical reasons, severalbound constraints will be imposed for this type of design variables (controlsentering the state system through the domain on which the problem is posed):

w ≤ w ≤ w, α ≤ α ≤ α, (15)

or, in reduced form, (w,α) ∈ [w,w] × [α, α]. (A geometrically acceptable setof bounds for the channel section depicted in Fig. 1 could be, for instance,w = D/4, w = D − E, α = 0, α = π/4.)

Then, for the first set of design variables (w,α), geometric functions B andS = B−1 relating the cross-sectional area a and the height h of the trapezoidalsection of the channel, as introduced at the beginning of in Subsection 2.1 inorder to compute wet area A and settled area As, take the particular form:

a = S(h) = wh+tan(α)

2h2, h = B(a) =

√w2 + 2 tan(α)a− w

tan(α). (16)

Thus, once replaced the previous expressions and transformed the concen-tration equation to a non-conservative formulation, the state system for thenon-cohesive case can be rewritten as:

∂A

∂t+∂Q

∂x=∑j

qj δ(x− pj) in (0, L)× (0, T ), (17)

∂Q

∂t+

∂

∂x

(Q2

A

)+ g

1√w2 + 2 tan(α)(A+As)

∂

∂x

(A2

2

)

+g

(∂b

∂x+

1√w2 + 2 tan(α)(A+As)

∂As∂x

)A (18)

= − gP

C2h

Q

A2|Q|+

∑j

qj Vj cos(εj) δ(x− pj) in (0, L)× (0, T ),

∂c

∂t+

∂

∂x

(Qc

A

)− ∂

∂x

(k∂c

∂x

)− 1

A

Q

LA(c∗ − c) =

1

A

∑j

mj δ(x− pj), (19)

∂As∂t

= − 1

ρs(1− η)

Q

LA(c∗ − c) in (0, L)× (0, T ). (20)

Remark 3 On the other hand, for the cohesive case, equations (17) and (18) ofthe state system remain unchanged, but equations (19) and (20) must be re-placed, taking into account the expression (9) for the interchange of sediment,


by:

∂c

∂t+

∂

∂x

(Qc

A

)− ∂

∂x

(k∂c

∂x

)=

1

A

∑j

mj δ(x− pj) +W

A(Se − Sd), (21)

∂As∂t

= − 1

ρs(1− η)W (Se − Sd) in (0, L)× (0, T ). (22)

In addition, as above commented, the authors will also control a seconddesign variable, water inflow q(t) entering the canal by the end x = 0, in orderto avoid the sedimentation problems mentioned in previous section by meansof a suitable velocity of water in the canal (this second type of control entersthe state system through the boundary condition (5) for the system of partialdifferential equations). Since water is thought to enter (that is, q(t) cannotbe negative), and due to technological requirements (related to the maximalcapacity of the canal), the authors are led to consider only the feasible fluxesinside the space of admissible controls given by:

Qad = q ∈ L2(0, T ) : 0 ≤ q ≤ β (23)

with β > 0 a suitable bound for water inflow.

Finally, as cost function to be optimized it will be taken a combinationof several factors including, necessarily, the minimization of the settled sludgelayer in the bottom of the channel. One of the simplest examples of costfunctional to be minimized could be of the form:

J(w,α, q) =1

2

∫ T

0

∫ L

0

A2s dx dt (24)

Remark 4 Taking into account the relations between As, z, φ, Se and Sd, asset in the formulation of the state system, another simple cost functions could

be taken under alternative expressions such as, for example, J =∫ T0

∫ L0|As| dx dt,

J = 12

∫ T0

∫ L0z2 dx dt, J =

∫ T0

∫ L0|z| dx dt, J =

∫ L0|φ(T )| dx, or J =

∫ L0

(|Se(T )|+|Sd(T )|) dx.

Then, the optimal control problem to be solved consists of finding theoptimal shape parameters w and α, and the optimal inflow of water q(t) insuch a way that, verifying the corresponding state system, minimize the costfunction J (given, for instance, by (24)), and satisfy the control constraints(15) and (23). Thus, the problem can be written is a short form as:

min(w,α,q)∈Uad

J(w,α, q) (25)

where the admissible set Uad is given by Uad = [w,w]× [α, α]×Qad.


2.2 Optimality conditions

It is worthwhile remarking here that above optimal control problem will benon-convex because of the strong nonlinearities of the state system, so unique-ness of solution is not expected. In the following the authors will center theirattention in obtaining a formal first-order optimality condition satisfied by thesolutions of this optimal control problem, in order to characterize them. Forthe sake of simplicity, and also in order to avoid too large and cumbersomeexpressions, it will be analyzed here the case where only one of the three con-trols needs to be optimized (the other two remaining fixed and known), butthe general case can be studied using analogous techniques.

For instance, for the particular case of a rectangular channel (i.e., for α = 0fixed) with a given water inflow q(t), only the channel width w needs to beoptimized in the space of admissible controls given by the real interval [w,w],that is, the authors look for the value w, satisfying the corresponding boundconstraints w ≤ w ≤ w, that minimizes the functional J(w) given, for example,by expression (24).

For this particular case, for instance within a non-cohesive framework, andin order to express this necessary optimality condition in the simplest way, theadjoint state (r, p, s, v) is introduced as the solution of the following coupledsystem:

− ∂r

∂t+

(Q2

A2− g 1

wA

)∂p

∂x+ g

(∂b

∂x+

1

w

∂As∂x− 2P

C2h

Q

A3|Q|)p+

Qc

A2

∂s

∂x

+

1

LA

Q

A2(c∗ − c) +

1

A2

∑j

mj δ(x− pj)

s = 0 in (0, L)× (0, T ), (26)

− ∂p

∂t− ∂r

∂x− 2

Q

A

∂p

∂x− c

A

∂s

∂x+

2gP

C2h

|Q|A2

p

+1

ρs(1− η)

(c∗ − c)LA

v − 1

LA

(c∗ − c)A

s = 0 in (0, L)× (0, T ), (27)

− ∂s

∂t− Q

A

∂s

∂x− ∂

∂x

(k∂s

∂x

)+

1

LA

Q

As− 1

ρS(1− η)

Q

LAv = 0, (28)

− ∂v

∂t− g

wA∂p

∂x− g

w

∂A

∂xp = As in (0, L)× (0, T ), (29)

with final conditions:

r(x, T ) = p(x, T ) = s(x, T ) = v(x, T ) = 0 in (0, L), (30)

and boundary conditions:

p(0, t) = p(L, t) = s(0, t) = 0 in (0, T ), (31)(r +

c

As)

(L, t) =

(k∂s

∂x+Q

As

)(L, t) = 0 in (0, T ). (32)


Thus, it can be derived the following formal first order optimality conditioncharacterizing the optimal solutions of the control problem:

Theorem 1 Let w ∈ [w,w] be an optimal solution of the control problemcorresponding to minimizing the functional J(w) in the admissible interval[w,w]. Then, there exist (A,Q, c,As), solutions of the state system (17) −(20) with boundary/initial conditions (5) − (8), and (r, p, s, v), solutions ofthe adjoint system (26)− (32), such that the following optimality condition isverified:

(w − w)

∫ T

0

∫ L

0

∂

∂x(A+As)Apdx dt ≥ 0, ∀w ∈ [w,w]. (33)

Proof Since w is a solution of the constrained minimization problem (25),the following inequality holds:

DJ(w) · (w − w) ≥ 0, ∀w ∈ [w,w]. (34)

Let (A,Q, c,As) be the state corresponding to the optimal control w, then:

DJ(w) · w =

∫ T

0

∫ L

0

As As dx dt

where (A, Q, c, As) = D(A,Q, c,As)(w) · w is given by the linearized system:

∂A

∂t+∂Q

∂x= 0 in (0, L)× (0, T ),

∂Q

∂t+

∂

∂x

(2QQ

A

)− ∂

∂x

(Q2A

A2

)− g w

w2

∂

∂x

(A2

2

)+ g

1

w

∂

∂x(AA)

+g

(∂b

∂x+

1

w

∂As∂x

)A− g w

w2

∂As∂x

A+ g1

w

∂As∂x

A

= − 2gP

C2h

Q

A2|Q|+ 2gP

C2h

Q

A3|Q| A in (0, L)× (0, T ),

∂c

∂t+

∂

∂x

(Qc

A

)+

∂

∂x

(Qc

A

)− ∂

∂x

(QcA

A2

)− ∂

∂x

(k∂c

∂x

)− 1

A

Q

LA(c∗ − c)

+1

A

Qc

LA+

A

A2

Q

LA(c∗ − c) = − A

A2

∑j

mj δ(x− pj),

∂As∂t

= − 1

ρs(1− η)

Q

LA(c∗ − c) +

1

ρs(1− η)

Q

LAc in (0, L)× (0, T ),


A(L, t) = 0, Q(0, t) = 0 in (0, T ),

c(0, t) = 0, k∂c

∂x(L, t) = 0, As(0, t) = 0 in (0, T ),



A(x, 0) = 0, Q(x, 0) = 0 in (0, L),

c(x, 0) = 0, As(x, 0) = 0 in (0, L),

Then, taking into account above definition of the adjoint system, it can bedemonstrated that the derivative of the cost functional J can be expressed as:

DJ(w) · w =

∫ T

0

∫ L

0

As As dx dt =

∫ T

0

∫ L

0

− ∂v

∂t− g

w

∂

∂x(Ap)As dx dt

=

∫ T

0

∫ L

0

v ∂As∂t

+g

wAp

∂As∂x dx dt

=

∫ T

0

∫ L

0

v(− 1

ρs(1− η)

Q

LA(c∗ − c) +

1

ρs(1− η)

Q

LAc) +

g

wAp

∂As∂x dx dt

=

∫ T

0

∫ L

0

(− ∂p

∂t− ∂r

∂x− 2

Q

A

∂p

∂x− c

A

∂s

∂x+

2gP

C2h

|Q|A2

p− 1

LA

(c∗ − c)A

s)Q

+1

ρs(1− η)

Q

LAvc+

g

wAp

∂As∂x dx dt

=

∫ T

0

∫ L

0

p∂Q∂t

+ r∂Q

∂x+ p

∂

∂x

(2QQ

A

)+ s

∂

∂x

(cQ

A

)

+(2gP

C2h

|Q|A2

p− 1

LA

(c∗ − c)A

s)Q+1

ρs(1− η)

Q

LAvc+

g

wAp

∂As∂x dx dt

=

∫ T

0

∫ L

0

( ∂∂x

(Q2A

A2

)+ g

w

w2

∂

∂x

(A2

2

)− g 1

w

∂

∂x(AA)

−g(∂b

∂x+

1

w

∂As∂x

)A+ g

w

w2

∂As∂x

A+2gP

C2h

Q

A3|Q| A)p

+r∂Q

∂x+ s

∂

∂x

(cQ

A

)− 1

LA

(c∗ − c)A

sQ+1

ρs(1− η)

Q

LAvc dx dt

=

∫ T

0

∫ L

0

( ∂∂x

(Q2A

A2

)+ g

w

w2

∂

∂x

(A2

2

)− g 1

w

∂

∂x(AA)

−g(∂b

∂x+

1

w

∂As∂x

)A+ g

w

w2

∂As∂x

A+2gP

C2h

Q

A3|Q| A)p

+r∂Q

∂x+

1

ρs(1− η)

Q

LAvc+ (−∂c

∂t− ∂

∂x

(Qc

A

)+

∂

∂x

(QcA

A2

)

+∂

∂x

(k∂c

∂x

)− 1

A

Qc

LA− A

A2

Q

LA(c∗ − c)− A

A2

∑j

mj δ(x− pj))s dx dt

=

∫ T

0

∫ L

0

( ∂∂x

(Q2A

A2

)+ g

w

w2

∂

∂x

(A2

2

)− g 1

w

∂

∂x(AA)


−g(∂b

∂x+

1

w

∂As∂x

)A+ g

w

w2

∂As∂x

A+2gP

C2h

Q

A3|Q| A)p

+r∂Q

∂x+

1

ρs(1− η)

Q

LAvc+ c

∂s

∂t+Qc

A

∂s

∂x− QcA

A2

∂s

∂x

+c∂

∂x

(k∂s

∂x

)− 1

A

Qcs

LA− A

A2

Q

LA(c∗ − c)s− A

A2

∑j

mj δ(x− pj)s dx dt

=

∫ T

0

∫ L

0

−Q2A

A2

∂p

∂x+ g

w

w2

∂

∂x

(A2

2

)p+ g

1

wAA

∂p

∂x

−g(∂b

∂x+

1

w

∂As∂x

)Ap+ g

w

w2

∂As∂x

Ap+2gP

C2h

Q

A3|Q| Ap

+∂r

∂tA− QcA

A2

∂s

∂x− A

A2

Q

LA(c∗ − c)s− A

A2

∑j

mj δ(x− pj)s dx dt

=

∫ T

0

∫ L

0

g ww2

∂

∂x

(A2

2

)p+ g

w

w2

∂As∂x

Ap dx dt

=gw

w2

∫ T

0

∫ L

0

∂

∂x(A+As)Apdx dt

So,

DJ(w) · (w − w) = (w − w)g

w2

∫ T

0

∫ L

0

∂

∂x(A+As)Apdx dt. (35)

Now, taking expression (35) to (34), and bearing in mind the positivity of

the termg

w2, the optimality condition (33) can be derived, which concludes

the proof. utRemark 5 In the objective functional to be minimized other aspects can bealso included, such as the economic cost of the filling material added to modifythe shape of the channel. In this case, the functional to minimize could takethe form:

J(w,α, q) =1

2

∫ T

0

∫ L

0

A2s dx dt+ µE(D − w)− E2

2tan(α) (36)

where the last term measures the area of the modified section and µ is aweight parameter (related to the length of the canal and the economic costof the material). For the above particular case of a rectangular canal (α = 0)with a fixed water inflow q(t), this functional would read:

J(w) =1

2

∫ T

0

∫ L

0

A2s dx dt+ µE(D − w) (37)

Thus, a new optimality condition - similar to that one achieved in Theorem 1- could be obtained, only bearing in mind that now:

DJ(w) · w =

(g

w2

∫ T

0

∫ L

0

∂

∂x(A+As)Apdx dt− µE

)w. (38)


3 An alternative formulation of the environmental problem: a 2Dapproach

In this section the authors introduce an alternative formulation for the opti-mization problem, simulating the whole processes by means of a 2D model forsedimentation in a channel, which is obtained by coupling the partial differ-ential equations for the two-dimensional version of the shallow waters hydro-dynamics (formulated in terms of the depth of water and the depth-averagedflow [25]) with the equations for sediment transport (formulated in terms ofthe concentration of suspended particles and the height of sediment). Specifi-cally, the following system is proposed for a channel with ground surface areaΩ ⊂ R2 over a time period of length T :

∂h

∂t+∇ ·Q = 0 in Ω × (0, T ), (39)

∂Q

∂t+∇ ·

(Q

h⊗Q

)+ gh∇(b+ z + h) = −g Q |Q|

C2hh

2+ f in Ω × (0, T ), (40)

∂

∂t(hc) +∇ · (Qc)−∇ · (kh∇c)− φ = l in Ω × (0, T ), (41)

ρs(1− η)∂z

∂t+ φ = 0 in Ω × (0, T ), (42)


Q · n = 0, k∂c

∂n= 0 on γ0 × (0, T ), (43)

Q = q n, c = c1, z = z1 on γ1 × (0, T ), (44)

h = h2, k∂c

∂n= c2 on γ2 × (0, T ), (45)


h(0) = h0, Q(0) = Q0, c(0) = c0, z(0) = z0 in Ω, (46)

where h(x, y, t) is the depth of the water column at point (x, y) ∈ Ω andat time t ∈ (0, T ); Q(x, y, t) = (Q1, Q2) is the flux of water (Q = hU , forU(x, y, t) = (U1, U2) the depth-averaged velocity of water); c(x, y, t) is thedepth-averaged concentration of sediment transported in suspension; z(x, y, t)is the height of sediment layer; b(x, y) represents the (steady) geometry of thecanal bottom; g is the gravity acceleration; Ch is the Chezy friction coefficient;k is the diffusion coefficient; ρs is the density of sediment; η ∈ [0, 1] correspondsto the porosity of bottom; and φ measures the interchange of sediment with thebottom, and f(x, y, t) and l(x, y, t) collect the source terms. In the definitionof the boundary conditions, three different parts in the boundary ∂Ω of thecanal plant Ω are considered: the boundary of the canal corresponding tolateral walls, denoted by γ0, the inflow boundary, denoted by γ1, and theoutflow boundary, denoted by γ2, such that ∂Ω = γ0 ∪γ1 ∪γ2. For these threesubsets, n represents the unit inner normal vector to the boundary.


In the case of cohesive sediments (that is, taking into account flocculationprocesses) the expression employed for φ will be:

φ = Se − Sd. (47)

In this formula, the erosion term Se and the deposition term Sd are defined,as in the 1D case, by the expressions (13) and (10) − (12), respectively. Theonly difference to the 1D case is that the bed shear stress τb in (13) and (12)

is now computed as τb = ρg |Q|2 /(Chh)2.Since the main aim is focused in minimizing the sedimentation effects by

controlling the shape of the section (given by w and α) and the water inflowq(t) through the boundary γ1, the cost function to be optimized should includethe minimization of deposition and erosion on the bottom of the channel. Then,another one of the simplest examples of cost functional to be minimized couldtake the form:

J(w,α, q) =

∫Ω

(|Se(T )|+ |Sd(T )|) dx (48)

where erosion Se and deposition Sd are computed from the solution of thestate system (39)− (46).

Within this two-dimensional approach, the design variables w and α (re-spectively, the width of the channel bottom and the inclination angle of thelateral wall) enter the formulation of the problem via the function characteriz-ing the canal bottom, that is, b = b(w,α), where (w,α) must satisfy the boundconstraints (15).

Again, in a similar way to the one-dimensional case, the control also in-cludes the normal flux q(t), which enters the problem via the condition onthe inflow boundary γ1. By already commented technological reasons, waterinflow q(t) must lie in the space of admissible controls Qad defined by (23).

Then, the optimal control problem, denoted by (P), consists of finding theoptimal shape parameters w and α, and the optimal flux q(t) on the inflowboundary in such a way that, verifying the state system (39)− (46), minimizethe cost function J given by (48), and satisfy the control constraints (15) and(23). Thus, the problem can be written as:

(P) min(w,α,q)∈Uad

J(w,α, q)

for the admissible set Uad = [w,w]× [α, α]×Qad.A mathematical analysis of this 2D problem could be developed following

the same steps as in the 1D case. However, not to extend this paper too much,it will not be presented here.

4 The numerical problem

In order to minimize the objective function J in above optimal control problem(P) it can be used a wide range of numerical algorithms (both with and without


derivatives) [26]. The simplest alternative (avoiding the high computationalefforts arising from the numerical calculation of the gradients, that can beobtained - as shown for the 1D case - via the numerical resolution of thecorresponding adjoint systems) is the utilization of any derivative-free method.So, in this paper the authors propose the use of a direct search algorithm forsolving the discretized optimization problem (this algorithm has already showna very effective performance in several related environmental control problems[27]). However, in order to do this, and as a previous step, it is needed tochange the bound constrained problem into an unconstrained optimizationproblem by introducing, for instance, a penalty function involving the controlconstraints w − w ≤ 0, w − w ≤ 0, α − α ≤ 0, α − α ≤ 0, −q ≤ 0 andq − β ≤ 0.

Moreover, due to technological reasons (mainly related to the fact thatflow control mechanisms cannot act upon water flow in a continuous way, butdiscontinuously at short time periods) the authors are led to seek the optimalboundary control among the piecewise-constant L2(0, T ) functions. So, forthe time interval [0, T ] they choose a number M ∈ N, consider the time step∆τ = T/M > 0, and define the discrete times τm = m∆τ , for m = 0, 1, . . . ,M .Thus, any function q ∈ L2(0, T ), constant at each subinterval defined by thegrid τ0, τ1, . . . , τM, is completely determined by the discrete set of valuesq∆τ = (q1, q2, . . . , qM ) ∈ RM , where qm represents the constant value of qat each time subinterval [τm−1, τm), for m = 1, . . . ,M . Taking this discretecontrol q∆τ ≡ q to the first boundary condition in (44), the state system(39)− (46) is solved, yielding a discretized solution (h∆τ , Q∆τ , c∆τ , z∆τ ) thatallows us to compute the objective function J(w,α, q∆τ ), given by expression(48). Then, the penalty function Φ can be defined in the following way:

Φ(w,α, q∆τ ) = γ J(w,α, q∆τ ) +

M∑m=1

max−qm, qm − β, 0

+ maxw − w,w − w,α− α, α− α, 0 (49)

where the parameter γ > 0 determines the relative contribution of the objec-tive function and the penalty term. It is worthwhile remarking here that thisfunction Φ is an exact penalty function in the sense that, for any sufficientlysmall γ, the solutions of the original constrained problem (P) correspond tothe minimizers of function Φ (see, for instance, Han [28]).

In order to compute a minimal value of this non-differentiable function Φ inthis work it is proposed the use of a direct search algorithm: the Nelder-Meadsimplex method [29], given the essentially geometric nature of the problem.This algorithm is a gradient-free method, based on the mere comparison of val-ues of the minimizing function, that constructs a sequence of simplices (sets ofsample points) as an approximation to the optimal point, and that has beensuccessfully used by the authors in other related environmental problems [30,31,27]. A short description of the above algorithm can be found, for instance,in [32]. Although the Nelder-Mead algorithm is not guaranteed to convergein the general case, it has good convergence properties in low dimensions (cf.


[33] for a detailed analysis of its convergence under convexity requirements).Moreover, to prevent stagnation at a non-optimal point, it is employed a mod-ification proposed by Kelley (cf. [34] for details): when stagnation is detected,the simplex is modified by an oriented restart, replacing it by a new smallersimplex.

For the numerical resolution of the sedimentation system (39) − (46) hasbeen used the two-dimensional model MIKE21, which is a standard softwarefor advanced numerical modelling of hydrodynamics, sediment transport andecological modelling. MIKE21 model, a software originally developed by Dan-ish Hydraulic Institute (DHI) [35], has been widely used in the simulation ofwater quality processes in rivers, estuaries, bays and coastal areas, and, in ourcase, can solve precisely the state system presented in above sections, thanksto the versatile adaptability of the software.

Fig. 2 Flow diagram illustrating the full computational algorithm.

Thus, in order to solve the optimal control problem the Mud Transportmodule MIKE21-MT has been interfaced with the authors’ own MATLABcode for the optimization Nelder-Mead algorithm, by using the MATLAB DFSInterface Library provided by DHI. It should be also noted that this processis not straightforward since, for each evaluation of the penalty cost function


Φ(w,α, q∆τ ) in the optimization procedure, thev algorithm previously need togenerate a new mesh for the updated domain defined from geometric param-eters w and α, and to solve the state system (39)− (46) for a new set of datab(w,α) and q∆τ , which increases in a very significant way the computationalcost of the algorithm. A detailed outline of the full computational process canbe found in the flowchart shown in Fig. 2.

5 Numerical results

Although a large number of numerical tests have been developed, only one ofthem will be shown in this final part of the paper. To be precise, the authorspresent here the numerical results obtained by using above method in orderto determine the optimal shape and the optimal inflow flux for a wastewatercanal of L = 50m length with a slope of 0.6%, and a time interval of T =2500 s. The original rectangular section to be optimized presents dimensionsD = 1.732m and E = 1.5m. In this particular example, for the optimalshape design of the section have been considered a fixed width of the channelbottom w = D/2 = 0.866m, and an inclination angle α varying betweenα = 0 and α = π/6 = 0.523. For the optimal inflow have been chosen a boundβ = 1.1m2/s, and a discretization in M = 3 time subintervals. Finally, thefollowing initial/boundary conditions have been chosen: for the depth of waterh0 = h2 = 1.5m, for the height of sediment z0 = z1 = 0.052m, and for theconcentration of sediment c0 = c1 = 0.075 kg/m3.

Then, starting from five random feasible configurations (necessary for theinitialization of the Nelder-Mead algorithm) with cost function values rang-ing from Φ = 0.2217 to Φ = 5.7182, the algorithm arrives - after 174 functionevaluations - to the optimal section given by α = 0.046 (that is, an almost rect-angular section), and an optimal water inflow q∆τ given by q1 = 0.033m2/s,q2 = 1.089m2/s and q3 = 0.272m2/s, yielding an optimal cost function valueof Φ = 0.0449.

Fig. 4 shows the optimized deposition and erosion at final time T = 2500 sobtained for this optimal solution. On the other hand, in Fig. 3 it can beobserved the non-optimized deposition and erosion at final time T = 2500 scorresponding to the best of the five initial guesses employed in the iterativeoptimization algorithm.

By a simple inspection of both figures, it can be confirmed the improvementon the behaviour of the sedimentation, with evident decreases in depositionand erosion.

6 Conclusions

In this paper the authors have formulated and solved an ecological controlproblem related to the management of a canal in order to avoid the settlingof suspended particles and their unwanted effects.


Fig. 3 Non-optimized deposition (up) and erosion (down) at final time T = 2500 s cor-responding to one of the initial guesses in the optimization algorithm: α = 0.1, andq∆τ = (0.099, 0.095, 0.050). In this uncontrolled case, deposition values vary widely between0 and 1.2 × 10−4, and erosion ranges from 0 to 4.0 × 10−4.

Fig. 4 Optimized deposition (up) and erosion (down) at final time T = 2500 s correspondingto the optimal solution: α = 0.046, and q∆τ = (0.033, 1.089, 0.272). In this optimal case,deposition remains almost null for the entire channel, and erosion only reaches (at veryreduced regions) small peaks of at most 9.0 × 10−5.

As a first achievement, the environmental problem is mathematically well-posed in terms of an optimal control problem of partial differential equations(where the design variables are related to the shape of the canal section andthe water inflow entering the channel), an optimality condition (characterizingthe optimal solutions) is derived, and a numerical algorithm is also proposedfor solving the related discretized optimization problem.

Furthermore, the efficiency of the suggested algorithm is confirmed by thenumerical experiments developed for a realistic case, where the authors com-bine the utilization of the MIKE21-MT model for the simulation of sedimenta-tion with their own software for the resolution of the optimization problems.Attained results show that, by a suitable choice of channel section and ofwater inflow, the sediment accumulation at the bottom of the canal can be


reduced in a significant way, avoiding the harmful consequences derived fromthis phenomenon.

Acknowledgements This work was supported by funding from project MTM2015-65570-P of Ministerio de Economıa, Industria y Competitividad (Spain) and FEDER. The helpand support provided by DHI with the MIKE21 modelling system is deeply appreciated.The authors also thank the two anonymous referees for their interesting comments andsuggestions that have greatly improved the readability of the paper.

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Documents

Sediment minimization in canals: An optimal control approach · dynamics and sediment transport and deposition equations, for the cases of cohesive and non-cohesive sediments. As