Modeling of Two Sub-reach Water Systems: Application to Navigation

Canals in the North of France

Pau Segovia

1,2

, Klaudia Horv

ath

, Lala Rajaoarisoa

, Fatiha Nejjari

, Vicenc¸ Puig

and Eric Duviella

Unit

e de Recherche en Informatique et Automatique, IMT Lille Douai, Lille, France

Automatic Control Department, Universitat Polit

ecnica de Catalunya, Terrassa, Spain

Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

Keywords:

Large-scale Systems, Water Systems, Saint-Venant Equations, Modeling, IDZ Models.

Abstract:

Inland navigation networks are large-scale systems that can be described by using the nonlinear Saint-Venant

partial differential equations. However, as there is no analytical solution for them, simpliﬁed models are used

instead for modeling purposes. This work addresses the modeling of two sub-reach systems by means of the

well-known Integrator Delay Zero model. Two main scenarios are considered: in the ﬁrst one, the two partial

models are independently computed one from each other; the second one uses previous knowledge of the

whole two sub-reach system in order to ensure the ﬂow consistency along the system. The application of these

two methodologies to a part of the navigation network in the north of France serves as the case study for this

work.

1 INTRODUCTION

Inland navigation networks cover more than 38000

km in Europe and are used principally for trans-

port. The navigation transport takes part in the

Trans-European network program (TEN-T

), which

promotes the development of transport infrastructure

policies to close the gaps between Member States’

transport networks and to guarantee seamless trans-

port chains for passengers and freight. In France, an

intensiﬁcation in the use of inland waterways is ex-

pected in the near future. The gauge of the allowed

boats and the navigation schedule will be risen up.

Hence, constraints on the inland waterway manage-

ment will be more severe.

The accommodation of navigation requires the

control of the water levels in each part of the navi-

gation network, which is composed of several inter-

connected reaches that represent portions of a water

stream between at least two hydraulic structures such

as locks. These reaches are large-scale, free-surface

systems that exhibit large delays between the genera-

tion of an upstream input and its measurement along

the water course and at the downstream end of the

reach. Some perturbations can travel back and forth,

http://ec.europa.eu/transport/themes/infrastructure/ten-t-

guidelines/index en.htm

resulting in resonance phenomena. The dynamics of

the reach can be accurately described by the Saint-

Venant’s partial differential equations (Chow, 1959).

However, as there is no known analytical solution for

these equations, simpliﬁed models such as transfer

functions (Litrico and Georges, 1999) or the Integra-

tor Delay (ID) model (Schuurmans et al., 1999) are

used instead. These models are designed by consider-

ing some assumptions on the linearity of the reach dy-

namics. The ID model was improved by considering

an additional zero that allows taking into account high

frequency phenomena. The obtained Integrator De-

lay Zero model (IDZ) was tested and compared with

the ID model on several canals (Litrico and Fromion,

2004). More recently, an Integrator Resonance model

(IR) has been proposed (van Overloop et al., 2010),

(Horv

ath et al., 2014b) and (van Overloop et al.,

2014) to reproduce the resonance phenomena. Gray-

box models can be used when lacking prior knowl-

edge of the physical characteristics of canals, such as

dimensions or the Manning-Strickler coefﬁcient (Du-

viella et al., 2013) and (Horv

ath et al., 2014a). Fi-

nally, approaches based on linear parameter-varying

models or multi-models that enable to take into ac-

count the nonlinearities due to the consideration of

large operating ranges were proposed (Duviella et al.,

2007), (Duviella et al., 2010), (Bolea et al., 2014) and

(Bolea and Puig, 2016).

Segovia, P., Horváth, K., Rajaoarisoa, L., Nejjari, F., Puig, V. and Duviella, E.

Modeling of Two Sub-reach Water Systems: Application to Navigation Canals in the North of France.

DOI: 10.5220/0006418604590467

In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 1, pages 459-467

ISBN: 978-989-758-263-9

459

Many man-made, hydraulic structures such as

gates can be located along the navigation reaches,

as well as secondary inputs. Therefore, they can-

not be considered as a single reach with only two

controlled points, the upstream and the downstream

ends. This particularity demands a decomposition of

the reaches in several sub-reaches according to the

hydraulic structures and secondary inputs that affect

the water stream. The models that represent the in-

ﬂuence of the discharges on the water levels for each

sub-reach have to be computed. The next step is to

interconnect these partial models to reproduce the be-

havior of the whole canal. In this work, this modeling

step is based on IDZ models. However, this model-

ing procedure might cause the ﬂow proﬁles for each

of these partial models to be not consistent with the

ﬂow proﬁles of the adjacent sub-reaches. This means

that the interconnected model will most probably not

reproduce adequately the real dynamics of the whole

reach. This work aims to develop a modeling ap-

proach in which the prior knowledge of the whole

system dynamics are taken into account when each

of these sub-reaches is modeled.

The content of this paper is structured as follows:

Section 2 is dedicated to the description of the IDZ

model. Section 3 addresses the modeling step of two

sub-reach systems introducing some considerations

as well as the interconnection method. In Section

4, the proposed approach is used by considering a

real navigation reach located in the north of France,

the Cuinchy-Fontinettes reach. The comparison of

the interconnection approaches with and without prior

knowledge of the whole system with a reference

model is performed. This reference model is provided

by an hydraulic simulation software that solves nu-

merically the Saint-Venant equations. Finally, conclu-

sions about the performed work are drawn in Section

2 SELECTING A MODEL TO

DESCRIBE A REACH

As already mentioned, the Saint-Venant differential

equations accurately describe the real dynamics of

the system. However, since no analytical solution is

known for these equations, as well as being extremely

sensitive to errors in the geometry and other unmod-

eled dynamics, simpliﬁed models are needed. Among

all the existing possibilities, the IDZ model (Litrico

and Fromion, 2004) is used in this paper. It results

from the linearization of the Saint-Venant’s differen-

tial equations around an operating point q

, and con-

stitutes a simple yet efﬁcient option to accurately de-

scribe a canal in high and low regimes.

The IDZ model, as its name implies, consists of an

integrator, a delay and a zero: the two ﬁrst terms cap-

ture the low frequencies tank-like behavior, whereas

the zero accounts for the high frequencies. In Laplace

form, its structure is as follows:

i j

(s) =

i j

s + 1

i j

−τ

i j

, (1)

where α

i j

represents the inverse of the zero, A

i j

the in-

tegrator gain and τ

i j

the propagation time delay. The

exact values of these parameters cannot be computed,

but an accurate approximation can be used instead

(Litrico and Fromion, 2004). Since the used parame-

ters are an approximation of the theoretical ones, the

notation ˆp

i j

(s) replaces p

i j

(s) hereinafter.

The integrator gain illustrates how the volume

changes according to the variation of the water level.

This parameter is also known as the equivalent back-

water area due to its dimensions. The time delay rep-

resents the minimum required time for a perturbation

to travel from its origin to the measurement points.

The zero approximates through a constant gain the os-

cillatory phenomena that occurs in high frequencies.

The model representing the inﬂuence of the dis-

charges on the water levels at the boundaries is given

by:



y(0,s)

y(L,s)





ˆp

(s) ˆp

(s)

ˆp

(s) ˆp

(s)



| {z }

P(s)



q(0,s)

q(L,s)



, (2)

where 0 and L are the abscissas for the initial and ﬁnal

ends of the reach; ˆp

i j

(s), the IDZ model that links the

water level and the j

discharge and follows the

structure given in (1); y(0,s) and y(L,s), the upstream

and downstream water levels, respectively; q(0,s) and

q(L,s), the upstream and downstream discharges, re-

spectively.

In order to compute α

i j

, A

i j

and τ

i j

(with i, j =

1,2), it is necessary to know where the transition be-

tween the upstream uniform and downstream backwa-

ter ﬂows occurs. The value of this abscissa is named

and can be obtained as follows:

(

max

L −

−y

i f s

6= 0

L i f s

= 0

, (3)

with y

[m] the downstream boundary condition, y

[m] the normal depth and s

(dimensionless) the de-

viation from bed slope of the line tangent to the water

curve at the downstream end of the pool. The reader

is referred to (Litrico and Fromion, 2004) for further

details about the computation of these magnitudes.

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

460

According to (3), x

can either be 0, L or take

an intermediate value between 0 and L. The reach

is completely under backwater ﬂow if x

= 0, com-

pletely under uniform ﬂow if x

= L or present both

kinds of ﬂow if 0 < x

< L. In particular, the interval

(0,x

) is under uniform ﬂow whereas the interval (x

L) is under backwater ﬂow. This is an important fact

for the computation of the parameters α

i j

, A

i j

and τ

i j

as they have to be computed for each kind of ﬂow that

is present in the reach. The same formulas are applied

for the uniform and backwater parts but are evaluated

according to the length of each of these parts. The

partial uniform and backwater parameters are merged

into the so-called equivalent parameters, which rep-

resent the whole pool. In the event that x

= 0 or

= L, they will only have to be computed once, for

the whole length of the reach.

Finally, the following consideration serves to in-

troduce the problem that is studied in this work, which

is no other than the presence of secondary inputs in a

canal; in particular, the relative position of x

and the

abscissa in which the inﬂow takes place (x

in f low

here-

inafter). The two possibilities are depicted in Fig. 1.

(a)

inflow

(b)

inflow

Figure 1: (a) Water proﬁle for x

> x

in f low

. (b) Water proﬁle

for x

< x

in f low

The same water proﬁle is obtained for both cases

as the upstream and downstream boundary conditions

are the same. The difference, however, appears when

the whole reach is regarded as the interconnection of

two sub-reaches due to the presence of the inﬂow. Fig.

1(a) is composed of a ﬁrst sub-reach which is only un-

der uniform ﬂow and a second sub-reach under uni-

form ﬂow in the interval (x

in f low

, x

) and backwater

ﬂow in the interval (x

, L). On the other hand, Fig.

1(b) shows that the ﬁrst sub-reach is under uniform

ﬂow in the interval (0, x

) and under backwater ﬂow

in the interval (x

, x

in f low

) whereas the second sub-

reach is only under backwater ﬂow.

The objective, therefore, is to compute the IDZ

model as an interconnection of two sub-reaches di-

vided by a secondary inﬂow within the ends of the

reach.

3 MODELING A TWO

SUB-REACH SYSTEM

In this section, the general formulation of intercon-

nected sub-reaches (Litrico and Fromion, 2004) is

given. It is then extended by considering different

criteria regarding the use of the previous knowledge

of the global dynamics. Two different interconnected

models are proposed and discussed.

3.1 Formulation of the Interconnected

IDZ Model

One possible approach to model the systems that were

presented in Fig. 1 is to consider two different sub-

reaches. This division of a reach into sub-reaches is

always done when there is an hydraulic structure that

physically divides the reach, but also under other cir-

cumstances, i.e. the presence of a secondary input or

a change in the cross section of the the water stream.

Moreover, since there is no such hydraulic structure,

this situation is hereinafter referred to as simple in-

terconnection. Fig. 2 (where big-sized markers at

sections 0 and L denote the presence of an hydraulic

structure) illustrates this situation. For the sake of

convenience, 0, x

in f low

and L denote the initial, inter-

mediate (where the inﬂow takes place) and ﬁnal ab-

scissas of the reach, respectively.

From relation (2), and according to Fig. 2, the

following set of equations can be considered. SR

and

denote sub-reaches 1 and 2, respectively.



y(0,s)

y(x

in f low

,s)



ˆp

(1)

(s) ˆp

(1)

(s)

ˆp

(1)

(s) ˆp

(1)

(s)



q(0,s)

q(x

in f low

,s)



(4)

Modeling of Two Sub-reach Water Systems: Application to Navigation Canals in the North of France

461

Figure 2: Water inﬂow ﬂowing into a canal, leading to a

simple interconnection structure.



y(x

in f low

,s)

y(L,s)



ˆp

(2)

(s) ˆp

(2)

(s)

ˆp

(2)

(s) ˆp

(2)

(s)



q(x

in f low

,s)

q(L,s)



(5)

This paper studies the particular case in which

q(x

in f low

) = 0, which can occur, for instance, when

a lateral hydraulic device such as a controlled valve is

closed and does not let the water ﬂow into the main

stream. The more general case q(x

in f low

) 6= 0, i.e.

when the controlled valve is open, will be addressed

in the future.

For the case q(x

in f low

) = 0, the ﬁnal model that

represents the interconnection is obtained by impos-

ing the following conditions at the interconnection

node (Litrico and Fromion, 2004):

(1)

in f low

,s) = y

(2)

in f low

,s) (6a)

(1)

in f low

,s) = q

(2)

in f low

,s) (6b)

These two equations express the continuity con-

ditions that hold true at the interconnection abscissa

in f low

. In particular, Eq. (6a) ensures that no sudden

change in the water depth occurs, whereas Eq. (6b)

guarantees the ﬂow consistency in the abscissa x

in f low

and is a direct consequence of q(x

in f low

) = 0.

Therefore, the ﬁnal model (2) that represents the

two interconnected sub-reaches is given by:



y(0,s)

y(L,s)



ˆp

(G)

(s) ˆp

(G)

(s)

ˆp

(G)

(s) ˆp

(G)

(s)



q(0,s)

q(L,s)



(7)

where ˆp

(G)

i j

represent the interconnecting transfer

functions. Their expressions are:

ˆp

(G)

= ˆp

(1)

ˆp

(1)

ˆp

(1)

ˆp

(2)

− ˆp

(1)

(8a)

ˆp

(G)

= −

ˆp

(1)

ˆp

(2)

ˆp

(2)

− ˆp

(1)

(8b)

ˆp

(G)

ˆp

(1)

ˆp

(2)

ˆp

(2)

− ˆp

(1)

(8c)

ˆp

(G)

= ˆp

(2)

−

ˆp

(2)

ˆp

(2)

ˆp

(2)

− ˆp

(1)

(8d)

Note: the Laplace variable s has been omitted for

readability in all ˆp

(k)

i j

terms.

In order to obtain the global model for the two

sub-reach system, it seems it might be enough to com-

pute a separate model for each sub-reach and then use

(8a)–(8d) to obtain the interconnected model. The

model obtained by means of this procedure is here-

inafter referred to as a two sub-reach system with in-

dependent ﬂow proﬁles.

However, as the two sub-reaches are not divided

by any hydraulic structure, it does not seem correct to

treat them independently. Instead, since q(x

in f low

) =

0 in this paper, it is possible to obtain the value of

for the whole canal and use this information when

modeling each of the two sub-reaches. The model that

is obtained by using this information about the global

dynamics is hereinafter referred to as a two sub-reach

system with consistent ﬂow proﬁle.

3.2 Two Sub-reach Systems with

Independent Flow Proﬁles

The ﬁrst and simplest option is to compute the model

for each sub-reach separately according to the pro-

cedure described in Section 2 and then interconnect

them. However, since there is no hydraulic struc-

ture dividing the two sub-reaches, a consistent ﬂow

proﬁle is not obtained, which does not seem logical.

This is caused by the need to compute x

for each

sub-reach, which will probably result in obtaining a

different ﬂow proﬁle with a completely different dy-

namic response. This idea is depicted in Fig. 3(a) and

4(a): a value of x

is computed for each sub-reach

(blue dash-dot lines), which results in a completely

different proﬁle than the real one, shown in Fig. 1.

3.3 Two Sub-reach Systems with

Consistent Flow Proﬁles

In this approach, the value of x

computed for the

reach is used to build the interconnected two sub-

reach system model afterward. The ﬁrst sub-reach is

forced to change from uniform to backwater ﬂow in

when x

< x

in f low

, whereas in the situation x

in f low

the second sub-reach is forced from uniform

to backwater ﬂow in x

. This behavior is shown in

Fig. 3(b) and 4(b), which results in the same curves

as those depicted in Fig. 1.

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

462

(a)

inflow

(b)

x x

inflow inflow

Figure 3: Flow proﬁles for a two-reach system when x

in f low

. (a) Independent ﬂow proﬁle. (b) Consistent ﬂow

proﬁle.

4 APPLICATION

The aim is to verify which of the two approaches pre-

sented in Section 3 leads to a better model, even if one

has the intuition that taking into account the previous

knowledge of the system to ensure the ﬂow continu-

ity should yield better results than treating them as

two sub-reaches with independent ﬂow proﬁles. In

order to compare them, both models are computed

for a real system and then compared with the results

provided by SICˆ2

(Malaterre et al., 2014), an hy-

draulic simulation software. Since it solves numer-

ically the Saint-Venant equations without simpliﬁca-

tions, the obtained results with this software will be

used as the basis to compare the accuracy of both ap-

proaches.

The chosen two sub-reach system is the Cuinchy-

Fontinettes reach (CFr), which belongs to the inland

navigation network in the north of France and is il-

lustrated in Fig. 5. This reach is bounded both up-

stream and downstream by the locks of Cuinchy and

Fontinettes, respectively. The gate Port de Garde,

which is located within both ends and placed outside

of the CFr, can be used to regulate the amount of water

that is sent to Aire. According to the current manage-

http://sic.g-eau.net/

(a)

inflow

(b)

inflow

Figure 4: Flow proﬁles for a two-reach system when x

in f low

. (a) Independent ﬂow proﬁle. (b) Consistent ﬂow

proﬁle.

ment strategy, this gate is generally closed, which is

the reason why this value is considered to be equal to

0 in this work. The discharges in the locks of Cuinchy

and Fontinettes are considered as the inputs of the

IDZ model given by (2). On the other hand, they are

modeled as discharge boundary conditions in SICˆ2.

Figure 5: Schematic view of the CFr.

Table 1 sums up the physical and geometrical

data used to model the CFr: n

[s/m

1/3

] is the Man-

ning’s roughness coefﬁcient, m (dimensionless) is the

side slope of the cross section (m = 0 for rectangular

shape), B

[m] is the bottom width of the reach, q

Modeling of Two Sub-reach Water Systems: Application to Navigation Canals in the North of France

463

/s] is the considered ﬂow for the linearization of

the Saint-Venant equations, L [m] is the length and y

[m] is the downstream water depth. Besides, the bot-

tom slope s

is equal to 10

−4

for both reaches. On

the other side, C, A and F stand for Cuinchy, Aire and

Fontinettes, respectively.

Table 1: Physical data for CFr.

m B

L y

C-A 0.035 0 52 0.6 28700 2.44

A-F 0.035 0 52 0.6 13600 3.8

C-F 0.035 0 52 0.6 42300 3.8

Two real scenarios are considered: the ﬁrst one

presents the operation of the Cuinchy lock; the sec-

ond one, the operation of the Fontinettes lock. Table

2 contains the details of these lock operations. A pos-

itive sign for the dispatched water volume means that

the water volume is ﬂowing into the system, whereas

a negative sign means that it is leaving the system.

Table 2: Considered scenarios.

Scenario Volume [m

] Duration [min]

1 3000 15

2 -23000 20

The following ﬁt coefﬁcients are computed (be-

tween the reference and each of the proposed models)

for a quantitative comparison of the accuracy of the

results:

• Pearson product-moment correlation coefﬁcient,

which measures the linear dependance between

two variables. It is deﬁned in the following way:

r =

∑

t=1



(t) −Y



(t) −Y



∑

t=1



(t) −Y



∑

t=1



(t) −Y



(9)

with T the horizon for which the data have been

acquired, Y

(t) the observed water depth at time

t, Y

(t) the predicted water depth at time t and Y

and Y

the mean value of observed and modeled

water depths, respectively.

This coefﬁcient is bounded between +1 (total pos-

itive linear correlation) and -1 (total negative lin-

ear correlation), and 0 means that there is no linear

correlation.

• Nash-Sutcliffe model efﬁciency coefﬁcient, which

is used to assess the predictive power of hydrolog-

ical models as follows (Nash and Sutcliffe, 1970):

E = 1 −

∑

t=1

(t) −Y

(t))

∑

t=1



(t) −Y



(10)

E can range from 1 to −∞, where 1 indicates a

perfect match of modeled and observed values, 0

corresponds to the case in which the model pre-

dictions are as accurate as the mean of observed

data and E < 0 means that the model predictions

are less accurate than the mean of observed data.

• Maximum difference between the modeled and ob-

served data as a measure of the magnitude of the

maximum error. It is computed as:

∆ = max

1≤t≤T

(t) −Y

(t)| (11)

The results for both scenarios are presented below.

For each of them, the previous ﬁt coefﬁcients between

the reference (SICˆ2) and the three different modeling

approaches (model of the reach, two sub-reaches with

independent and consistent ﬂow proﬁles) are summa-

rized in Tables 3 and 4. In addition, the simulation re-

sults for Y

and Y

are presented for the four models.

Each ﬁgure is zoomed in the area of interest; how-

ever, the same simulation time has been used for both

scenarios.

Remark: the computation of the IDZ model for the

reach results in x

= 5760 m, which indicates that the

present case study falls under the situation described

by Fig. 3 as x

in f low

= 28700 m. This means that the

C-A reach is under both uniform and backwater ﬂow

(with ﬂow transition at x = x

) and that A-F is under

backwater ﬂow only. On the other hand, the obtained

proﬁle for the two-reach model with independent ﬂow

proﬁle is only backwater for both of them: when the

IDZ model is computed for each sub-reach, a value of

= 0 is obtained for both sub-reaches, which means

that they are both under backwater.

4.1 Scenario 1

Figures 6 and 7 and Table 3 summarize the results

obtained for the Cuinchy lock operation.

Table 3: Fit coefﬁcients for an upstream lock operation.

Reach Indep. ﬂow Cont. ﬂow

0.2831 -0.0193 0.2831

0.7311 0.4328 0.7312

∆

[m] 0.1225 0.1703 0.1272

-1.2637 -5.3193 -1.0506

0.5110 0.0641 0.5613

∆

[m] 0.0016 0.0128 0.0015

None of the three models is capable of represent-

ing the observed peak in Y

with satisfying accuracy,

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

464

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

Time [h]

SIC

Reach

Indep. flow

Cont. flow

Figure 6: Upstream water levels for an upstream lock oper-

ation.

0 2 4 6 8 10 12 14 16 18 20 22 24

3.798

3.8

3.802

3.804

3.806

3.808

3.81

3.812

3.814

3.816

Time [h]

SIC

Reach

Indep. flow

Cont. flow

Figure 7: Downstream water levels for an upstream lock

operation.

even if the equilibrium value is well ﬁtted. Never-

theless, the consistent ﬂow proﬁle model, which is

overlapped with the reach model, offers a better peak

response than the two sub-reach model with indepen-

dent ﬂow proﬁle.

Moreover, the two sub-reach model with indepen-

dent ﬂow proﬁle clearly predicts a much worse re-

sponse for Y

than the other two models, represented

by the nonexistent predicted peak according to the

reference. The appearance of this peak is physically

justiﬁed by the fact that the computation of the ﬂow

proﬁle for the two sub-reach system with independent

ﬂow yields backwater ﬂow only. The systems that ex-

hibit this kind of ﬂow proﬁle are more sensitive to res-

onance phenomena. An example of this behavior was

obtained in a work with ﬂat systems (Segovia et al.,

2017).

It is also worth noting that both the reach model

and the consistent ﬂow proﬁle model predict a much

faster response than it is actually observed, which

causes the Nash-Sutcliffe indicators to be negative for

. Nevertheless, when the consistent ﬂow model is

considered, the maximum difference is lower and the

dynamics are better reproduced, as shown by the com-

puted correlation coefﬁcients.

4.2 Scenario 2

In this case, Figures 8 and 9 and Table 4 summarize

the results for the Fontinettes lock operation.

0 1 2 3 4 5 6 7

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

Time [h]

SIC

Reach

Indep. flow

Cont. flow

Figure 8: Upstream water levels for a downstream lock op-

eration.

0 0.5 1 1.5 2

3.66

3.68

3.7

3.72

3.74

3.76

3.78

3.8

3.82

3.84

Time [h]

SIC

Reach

Indep. flow

Cont. flow

Figure 9: Downstream water levels for a downstream lock

operation.

For the upstream level Y

, the ﬂat signal cor-

responds to the overlapping of the observed water

depths and the predicted responses with the reach

model and the consistent ﬂow model. This fact is

shown by the maximum difference indicator in Ta-

ble 4, which prevents the computation of the Nash-

Sutcliffe indicators for these two models. The two

sub-reach model with independent ﬂow proﬁle pre-

dicts a very different response from the observed data.

Indeed, a peak is predicted in the response of the

two sub-reach system with independent ﬂow proﬁle,

whose justiﬁcation is the same as given for Y

in Sec-

tion 4.1. The Nash-Sutcliffe indicator takes an ex-

Modeling of Two Sub-reach Water Systems: Application to Navigation Canals in the North of France

465

Table 4: Fit coefﬁcients for a downstream lock operation.

Reach Indep. ﬂow Cont. ﬂow

— -3.38 ·10

—

0.3894 0.2388 0.3785

∆

[m] 0 0.0764 0

0.2394 0.0444 0.5671

0.9549 0.4370 0.8607

∆

[m] 0.0471 0.0630 0.0383

tremely large, negative value, showing the deviation

between this model and the observed values.

With regard to Y

, the predicted dynamics for the

three models are similar, albeit the reach model and

the consistent ﬂow proﬁle models offer a better pre-

diction than the other model.

A conclusion for both scenarios is that an oper-

ation performed at one end of the system does not

have a major impact at the other end, which is due

to the large dimension of the system. Another fac-

tor that might play a role in this behavior is the fact

that a bed slope equal to 10

−4

is considered, which

results in different values for the upstream and down-

stream bottom elevation. For the second scenario, a

large volume of water is dispatched outside the sys-

tem, and the observed data show that the water level

remains (almost) constant upstream. The ﬁnal water

level variation is due to the mass balance along the

system.

Another conclusion that can be drawn for both

scenarios is that the two sub-reach model with inde-

pendent ﬂow proﬁle predicts signiﬁcant peaks in Y

for the upstream action and in Y

for the downstream

action results from considering two backwater ﬂow

dynamics. If a system has to be modeled as the inter-

connection of two sub-reaches, the best choice seems

to consider the consistent ﬂow proﬁle model.

5 CONCLUSIONS AND FUTURE

WORK

This work presented the study of a two sub-reach sys-

tem based on IDZ models. Some considerations that

need to be taken into account in order to ensure the

ﬂow consistency of the system were addressed, and

those steps were illustrated by means of a case study

based on a real system in the north of France. Ac-

cording to the obtained results, it is possible to state,

as one could previously anticipate, that the accuracy

with respect to the reference is greater if the previous

knowledge of the system (namely x

) is considered to

ensure the continuity of the ﬂow.

In the light of the outcome, although the IDZ

model yields acceptable results, other aspects may

need to be considered to possibly come up with some

rules about its applicability. Further work includes ad-

dressing the general case q(x

in f low

) 6= 0. In this case,

the structure of the global model (2) will be different,

but the modeling approach will be similar. In addi-

tion, canals characterized by a different topography

such as tributaries and distributaries will be consid-

ered. The obtained models are expected to be used in

fault detection and isolation (FDI) and fault-tolerant

control (FTC).

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