One of the main problems in water management of irrigation systems is the control of the equitable distribution of water among different orifice offtakes. The difficulty of managing a canal is partly caused by the lack of knowledge of the canal state because the scheduled demand is often not fulfilled, since farmers extract more water than is scheduled and it is impossible for the watermaster to determine the canal state. However, an innovative developed algorithm called CSE is proposed in this paper. This algorithm is able to estimate the real extracted flow and the hydrodynamic canal state (that is, the water level and velocity along the irrigation canal). The algorithm solves an inverse problem implemented as a nonlinear optimization problem using the Levenberg–Marquardt method. The algorithm is tested, taking into account several numerical examples, and a practical implementation is made for a real case study in the PAC-UPC canal, a 220 m laboratory canal especially designed for research into irrigation canal control area and irrigation canal modelling. This useful algorithm evaluates the real extraction flow and the canal state and could be a useful tool for a feedback controller.

## INTRODUCTION

Frequently, irrigation managers are worried about the equitable distribution of water among different orifice offtakes. Some farmers may demand much more water than was scheduled, and this normally involves several problems such as important disadjustment on canal management. In such cases, we propose solving this problem with the CSE algorithm, which is able to estimate the real extracted flow and the hydrodynamic canal state (that is, the water level and velocity along the irrigation canal). The water saved may then be used to increase the irrigated area or save money. Alternatively, reduction of water consumption and peak requirement may allow a reduction in the water withdrawal from rivers or reservoirs, reduction in the capacity, and hence cost, of storage works, and of conveyance and distribution works such as canals and pumping stations. The potential sophistication of on-farm water management is highly dependent upon the level of water delivery service provided to individual farms, which in turn, depends upon the conveyance manageability within the complete water distribution system (Plusquellec 1988). The Food and Agriculture Organization of the United Nations indicates that for the year 2030 agricultural production will have to be increased by +80% to fulfil food demand, but it will have to be done without the possibility of increasing water withdrawals by more than +12% (Faurès *et al.* 2010). This can be done by reducing spillages along canals, which is technically possible since studies indicate that in the Mediterranean region, by the year 2025, water savings can be 65% for irrigation, 22% for industrial use and 13% for domestic use (Institut Méditerranéen de l'Eau 2007). These savings can be obtained through the implementation of several measures allowing better management of water demand at several institutional and technical levels. The survey carried out in the south of France in 1997 confirmed the figures found in other sources (ASCE 1993; Mareels *et al.* 2005), showing that water losses at the distribution level were around 50% on average and could be reduced to less than 10% by the modernization of canal real-time operation including automation (Malaterre & Rogers 2008). In addition, the investment in automation allows the ‘simplification of operational control, and makes possible to reduce and/or optimize the number and qualifications of the operating staff’ (Goussard 1993).

In canal operations, the scheduling of gate operations to satisfy known changes in water demands is a necessity established by some authors. All our effort is focused on controlling and managing the flow deliveries of the system because in that way the profitability of the system can be increased. One of the obstacles to scheduling the gate operations for an irrigation period is providing real input data to the control algorithms, because it is possible to predict the initial conditions in a canal from the water level measures and introduce the backwater profile in the algorithm, but the problem is due to unknown flows extracted by farmers during the canal operation. In that sense, the input data introduced in the control algorithm is far from the reality and the challenge is to calculate a scheduling of gate operations to satisfy the water demands.

The management of water resources in a canal is sometimes a difficult task, since the delivery scheduled the day before could be modified by the users during the irrigation cycle without notice. For instance, the demand deliveries could be increased because a farmer extracted a greater flow than the scheduled delivery, so the watermaster would not know the reason why the water level decreased at the target points. In that sense, the proposed algorithm (CSE) would be a useful tool for the watermaster. CSE is able to obtain first the unknown flows withdrawn by farmers and, additionally, the hydrodynamic canal state, and thus the water level and velocity at each cross-section of the canal. On the other hand, if a farmer pumps water from an unknown point in the canal, CSE can obtain the approximate location of this point and the volume of water withdrawn by the farmer, so this can be a useful tool in the management of any canal.

There are a few authors who have developed algorithms involved in calculating the offtake discharges in irrigation canals, including Delgoda *et al.* (2013) and Van Overloop *et al.* (2008).

The performance of this algorithm is quite simple. In a case where we introduce a flow change in a particular section of a canal, the perturbations associated with this flow change can be estimated from a computer model based on the Saint-Venant equations. CSE solves the inverse problem from the registered perturbations (water level measurements at checkpoints), and it calculates the flow changes associated with them.

The input variables for the CSE algorithm are the water level measurements at certain points of the canal and the gate trajectories during a past time horizon. The output variables of CSE are the extracted flow during this time horizon and the hydrodynamic canal state at the current time obtained by implementing an inverse problem.

To achieve this objective, the CSE algorithm uses the hydraulic influence matrix (HIM), which establishes a relationship between the real extraction flow and the water depth and velocity at any point in the canal during a past time horizon. This matrix was defined by several authors (Soler 2003; Soler *et al.* 2013; Bonet 2015). In order to make the meaning of the HIM more understandable, the next example is given.

*t*=

*K*+ 1, we modify the water level and velocity in cross-sections close to the pump location (cross-section

*i*), but we do not modify the hydrodynamic variables in a far cross-section (

*i*+ 1). Instead, at

*t*=

*K*+ 2, we modify the hydrodynamic variables in all sections between

*i*to

*i*+ 1. In this sense, the HIM considers the range of influence of a flow change at every section of the canal during a time interval (the theoretical definition of the HIM matrix is introduced in the next section).

First, the algorithm was tested in a canal with two pools introducing an unknown extraction flow. In a second example, the CSE was tested with the test cases (Clemmens *et al.* 1998) introduced by the ASCE Task Committee on canal automation algorithms. The last example was done in a laboratory canal, PAC-UPC, especially designed to develop basic and applied research in irrigation canal control area located in Barcelona's School of Civil Engineering.

## METHODS

*Q*represents the flow disturbance,

_{b}*HIM′*(

*Q*) is the simplified hydraulic influence matrix that represents the influence of an extraction flow on the water level at different points of the canal, and

_{b}*HIM*(

*Q*) is the hydraulic influence matrix that represents the influence of an extraction flow on the water level and velocity at the canal.

_{b}The HIM matrix is a square matrix and positive definite matrix (see Bonet 2015). The method used to solve the nonlinear optimization problem is the Levenberg–Marquardt method, which is a robust method with easy implementation and is a special method to solve an ill-conditioned matrix such as the HIM matrix. Thus, the algorithm solves a nonlinear optimization problem using the Levenberg–Marquardt method to evaluate the last expression (1).

### The HIM matrix

The HIM matrix is based on the full Saint-Venant equations, which describe the free surface flow in canals. In partial derivatives this system is of the hyperbolic, quasi-linear and second-order type. The two equations are based on mass and momentum conservation.

*y*is the water level measured from the bottom of the canal,

*v*is the weighted average velocity in a cross-section,

*x*is the space,

*t*is the time,

*S*

_{0}is the canal bottom slope,

*S*(

_{f}*y*,

*v*) is the friction slope and

*c*is the celerity of a gravity wave, where

*A*(

*y*) is the area of the wetted surface or a cross-section, x

^{+}is the position of the upstream characteristic curve, x

^{−}is the position of the downstream characteristic curve and

*T*(

*y*) is the top width of the free surface.

*P-R*and

*Q-R*(Figure 2), respectively, are taken into account, we obtain the following equations: where

*S*

_{fR}*=*

*S*(

_{f}*y*,

_{R}*v*),

_{R}*S*

_{fP}*=*

*S*(

_{f}*y*,

_{P}*v*),

_{P}*S*

_{fQ}*=*

*Sf*(

*y*,

_{Q}*v*) and 0 ≤

_{Q}*θ*≤ 1 is the weighting coefficient that indicates the type of numerical scheme used. When

*θ*= 1, the numerical scheme is implicit, if

*θ*= 0 it is explicit, and when

*θ*= 1/2 the numerical scheme is in central differences, the method of the characteristic curves.

If the flow conditions at points *P′* and *Q′* are known, *x _{P′}*,

*t*,

_{P′}*y*,

_{P′}*v*and

_{P′}*x*,

_{Q′}*t*,

_{Q′}*y*,

_{Q′}*v*are also known, so

_{Q′}*x*,

_{R}*t*,

_{R}*y*,

_{R}*v*remain as unknowns (Figure 2), which can be found by calculating the last four Equation (3), using any of the methods to solve nonlinear equations, such as the Newton–Raphson method. The way of calculating the influences shown in this section is closely linked to the numerical scheme of characteristic curves. However, usually this scheme is not exactly used because it gives the solution at a point R whose coordinates (

_{R}*x*,

_{R}*t*) are unknown

_{R}*a priori*. These coordinates are part of the solution, and normally it is more important to know the solution of the flow conditions at a specific point and at specific time instants. To solve this problem, first interpolate and then solve.

A structured grid like this one (Figure 2) creates a new nomenclature. Indeed, every variable will have a double index, where *k* refers to time and *i* to space. Thus, *y _{ik}* and

*v*represent the values for water level and average velocity at the coordinates

_{ik}*x*

_{i}*=*

*i*Δ

*x*and

*t*

_{k}*=*

*k*Δ

*t*where Δ

*x*and Δ

*t*are selected by the user.

*x*,

_{P}*y*,

_{R}*v*and

_{R}*x*, where the values of

_{Q}*y*and

_{p}*y*are dependent on the value of

_{Q}*x*and

_{P}*x*evaluated using an interpolation function of second order too, (to be coherent with the numerical procedure used) we have used the Lagrange factors (a way of representing quadratic splines). For a dummy variable

_{Q}*z*, the result is: In this way the variables

*y*,

_{P}*v*,

_{P}*y*and

_{Q}*v*become functions of

_{Q}*x*and

_{P}*x*, as follows: On the other hand, there are many control structures in canals such as gates, orifice offtakes, lateral weirs, etc. which allow flow control according to the specification of the watermaster. The individual study of each one is impossible in this work, so for this reason the most common structures are introduced. A common one found is a checkpoint, a target point where the water level is measured with a depth gauge, and it includes a sluice-gate, a lateral weir outlet, offtake orifice or a pump, as can be seen in Figure 3. The interaction of this control structure with the flow can be described according to the mass and energy conservation Equation (6): where:

_{Q}*S*(*y _{e}*) is the horizontal surface of the reception area in the checkpoint position

*A*(*y _{e}*)

**v*is the incoming flow to checkpoint, defined in terms of water level and velocity

_{e}*A*(*y _{s}*)

**v*is the outgoing flow to checkpoint which continues along the canal, described in terms of water level and velocity

_{s}*Cd* is the discharge coefficient of the sluice-gate and *a*_{c} is the sluice-gate width

*d* is the checkpoint drop and *u* is the gate opening

*q _{b}* is the pumping offtake

*q _{s}*(

*y*) is the outgoing lateral flow through the weir where

_{e}*C*

_{s}is the discharge coefficient, a

_{s}is the weir width and y

*is the weir height measured from the bottom, called weir equation*

_{0}*q _{offtake}*(

*y*) is the outgoing offtake orifice flow where

_{e}*C*

_{d}is the discharge coefficient,

*A*

_{0}is the area of the offtake orifice, called orifice offtake equation.

*k*

*+*

*1*, that is, the incoming water level

*y*. In the same way, is defined as the existing water level at the first node of the downstream pool from the checkpoint at the same time

_{e}*k*

*+*

*1*, and

*y*the outgoing water level at the control structure (Figure 4). The same can be said for the velocities and : where: and

_{s}*x*

_{Q}are the unknown variables.

*q*

_{b}as our particular variable. Thus, applying the first derivative of the functions (7) with respect the extraction flow, we obtain:

In Equation (8), for the first time, the extraction flow *q*_{b} explicitly appears in the description. Despite the fact that the specific form of this function is still unknown, Equation (8) shows that the influence of the parameter *q*_{b} on flow conditions at time *k**+**1* is the sum of the indirect influence of the conditions at instant *k* and the direct influence at instant *k**+**1* through the term ‘*L*’, which represents the variation in the extraction flow.

As a result, the method of characteristics is applied to the Saint-Venant equations in order to obtain algebraic equations to establish a relation between the influence parameter *q*_{b} and the hydrodynamic canal state, and all the influences are lumped together in a global matrix, which is referred to as HIM(Qb). Based on this system of equations, and using the first derivative (∂y/∂qb, ∂v/∂qb) on an analytical process, we can establish the changes in flow behaviour (water level and velocity) due to a flow change at a point at a certain time instant.

### The optimization problem

The inverse problem (1) is formulated as an unconstrained optimization problem. The optimization problem applied is the classical nonlinear problem without constraints and the method used to solve it is the Levenberg–Marquardt method (Press *et al.* 1992). Some authors have used optimization problems to solve inverse problems, e.g., gate stroking (Wylie 1969) and CLIS (Liu *et al.* 1998), but all of them implement inverse problems in control algorithms.

*n*

_{c}, where

*n*

_{c}is the number of checkpoints: Finally, every vector (9) defined for each time instant of the past time horizon is combined to define the ‘measured water level vector’, whose dimension is

*n*

_{y}, where

*n*

_{y}=

*k*

_{F}×

*n*

_{c}, where

*k*

_{F}is the final instant of the past time horizon. We define this vector as: We can check the measured water level vector values in a computational grid in Figure 5.

*x*(k), which is defined as the vector containing the numerical solution at the time instant k of all the discretization points: where

*y*(

_{i}*k*) and

*v*

_{i}(

*k*) = water depth and mean velocity at point

*i*; and

*n*

_{s}= number of cross-sections in which the canal is discretized. In this way, the vector

*x*(1) is the known initial condition.

The state vector at the current time defines the current hydrodynamic state. The state vector is shown in a computational grid in Figure 5 (triangles).

*k*-instant during a past time horizon into a single vector that is called ‘prediction vector’ (12). The dimension of this vector is

*n*

_{x}= (2 ×

*n*

_{s}) ×

*k*

_{F}: We are only interested in the water level at target points where we also obtain the water level measurements. We define a new vector that contains the water depth values given at a prescribed number of points (

*n*

_{c}) at the time instant

*k*: This vector is constituted by a subset of values of the state vector (11).

*n*

_{Y}=

*k*

_{F}×

*n*

_{c}. The vector (14) contains all water depth values and is shown in Figure 5. If you look closely at this figure, the position of the elements of the vector (14) in the grid domain coincides with the elements of the measured water level vector.

*C*is a matrix, called a discrete observer matrix by Malaterre (1994), a matrix of dimension

*n*

_{Y}×

*n*

_{X}and whose components are only ‘zeros’ or ‘ones’. This matrix defines the direction of the control logics along a canal pool: downstream level control, upstream level control or control of intermediate water levels.

*K*. Then, the extracted flow trajectories can be approached with piecewise functions. The extracted flow vector is defined by lumping together all the extracted flows during the past time horizon, as follows: where the dimension of this vector is

*n*

_{Q}=

*n*

_{P}×

*K*

_{F},

*n*

_{P}is the number of pump stations and

*K*

_{F}is the final operation period of the past time horizon.

In this way, only *Q*_{b} determines canal behaviour along the past time horizon. When the extracted flow trajectories are implemented in the canal, the flow response given by the model will be unique. Inversely, one flow behaviour is caused by only one set of extracted flow vectors, as a flow change is also responsible for water level disturbances.

We can check the extracted flow vector values in a computational grid in Figure 5.

Once CSE has estimated the extracted flow vector, the algorithm can also estimate the state vector at the current time, considering all variables that control the flow in the canal such as the scheduled demands, the initial conditions, and the gate trajectories during the past time horizon. Thus, the CSE algorithm calculates the flow disturbance (Δ*Q _{b}*), which better explains the changes between the measured and predicted water levels (Δ

*Y*) (1).

*et al.*1982; Fletcher 1987). In mathematical terms, the objective is to obtain the extracted flow vector that minimizes the following performance criterion: where

*Q′*matrix is a weighing matrix and the dimension of the matrix is

*n*

_{Y}*×*

*n*. This matrix could be used to weight the water level error at a particular checkpoint. This matrix is defined as the identity matrix in CSE.

_{Y}*Q*contains the extracted flow trajectories (16).

_{b}## RESULTS AND DISCUSSION

### Numerical example: a canal introducing a single disturbance

The canal, with a trapezoidal section, is represented in Figure 7, and the general data are shown in Table 1. The characteristics of the checkpoints, sluice-gates, pump stations and orifice offtakes are shown in Tables 2 and 3.

Pool number | Pool length (km) | Bottom slope (%) | Side slopes (H:V) | Manning's coefficient (n) | Bottom width (m) | Canal depth (m) |
---|---|---|---|---|---|---|

I | 2.5 | 0.1 | 1.5:1 | 0.025 | 1 | 2.5 |

II | 2.5 | 0.1 | 1.5:1 | 0.025 | 1 | 2.5 |

Pool number | Pool length (km) | Bottom slope (%) | Side slopes (H:V) | Manning's coefficient (n) | Bottom width (m) | Canal depth (m) |
---|---|---|---|---|---|---|

I | 2.5 | 0.1 | 1.5:1 | 0.025 | 1 | 2.5 |

II | 2.5 | 0.1 | 1.5:1 | 0.025 | 1 | 2.5 |

Number of control structure or checkpoint | Gate discharge coefficient | Gate width (m) | Gate height (m) | Step (m) |
---|---|---|---|---|

0 | 0.61 | 5.0 | 2.5 | 0.6 |

1 | 0.61 | 5.0 | 2.5 | 0.6 |

Number of control structure or checkpoint | Gate discharge coefficient | Gate width (m) | Gate height (m) | Step (m) |
---|---|---|---|---|

0 | 0.61 | 5.0 | 2.5 | 0.6 |

1 | 0.61 | 5.0 | 2.5 | 0.6 |

Number of control structure or checkpoint | Discharge coef./diameter orifice offtake (m) | Orifice offtake height (m) | Lateral spillway height (m) | Lateral spillway width (m)/discharge coefficient |
---|---|---|---|---|

0 | – | – | – | – |

1 | 0.6/0.77 | 1.0 | 2.3 | 10/1.99 |

Number of control structure or checkpoint | Discharge coef./diameter orifice offtake (m) | Orifice offtake height (m) | Lateral spillway height (m) | Lateral spillway width (m)/discharge coefficient |
---|---|---|---|---|

0 | – | – | – | – |

1 | 0.6/0.77 | 1.0 | 2.3 | 10/1.99 |

In these examples, an upstream large reservoir is considered, whose water level H_{reservoir} is 3 m constantly throughout the test. This is the upstream boundary condition. At the end of the last pool, there is a control structure with orifice offtake and a pump station. The flow through the orifice offtake depends on the upstream water level of the orifice, and the disturbance is introduced by the pump station. This is the downstream boundary condition. This example starts from an initial steady state (Tables 4 and 5), with a specific and constant demand delivery at the end of the pools (5 m^{3}/s through the orifice offtake), and the disturbance is not introduced initially.

Control structure | Initial flow rate (m3/s) | Control structure | Initial water level (m) |
---|---|---|---|

Gate 1 | 10.0 | Checkpoint 1 | 2.0 |

Gate 2 | 5.0 | Checkpoint 2 | 2.0 |

Control structure | Initial flow rate (m3/s) | Control structure | Initial water level (m) |
---|---|---|---|

Gate 1 | 10.0 | Checkpoint 1 | 2.0 |

Gate 2 | 5.0 | Checkpoint 2 | 2.0 |

Control structure | Flow delivered through an orifice offtake (m^{3}/s) |
---|---|

Gravity outlet 1 | 5.0 |

Gravity outlet 2 | 5.0 |

Control structure | Flow delivered through an orifice offtake (m^{3}/s) |
---|---|

Gravity outlet 1 | 5.0 |

Gravity outlet 2 | 5.0 |

### The disturbance

^{3}/s (pump station 1 for 15 minutes, from minute 40 to 55). This disturbance is introduced to the numerical model, based on the model of characteristics introduced before, as a flow change, and in this way we get the water level values. Once the water levels are introduced in the CSE algorithm, it will propose the pump flow trajectories that describe with the best accuracy the variation of water level at the checkpoints during the past time horizon.

Flow disturbance reduces the water level at checkpoint 1 from 2 m to 1.60 m, and at checkpoint 2 from 2 m to 1.92 m (see Figure 8). In that sense, a flow change of 2 m^{3}/s at pump station 1 has a significant impact on the canal profile, although the water level at checkpoint 1 and 2 recovered to the desired water level (2 m) in just 160 minutes and 150 minutes, respectively, due to the short period of time that the disturbance was applied. As soon as the water level at the checkpoints recovers to the desired water level at these points, the flow through the orifice offtakes returns to 5 m^{3}/s.

#### Results

^{3}/s, so considering that the real pump flow is 2 m

^{3}/s, the percentage maximum error between real and estimated flow is 0.25%.

CSE computes the unscheduled offtake changes introduced to the canal and writes these offtake changes as a pump flow trajectory (temporal function) associated with a certain pump station. The algorithm assigns to the pump stations all flow changes between the initial steady state to the current state (Bonet 2015).

Both water profiles are similar, as the accuracy of CSE in calculating the extracted flow vector is so high. An error of 5 L/s in the extracted flow is equivalent to a water level error of 1 mm and a velocity error of 0.001 m/s in cross-sections close to the extracted point.

### Numerical example: a canal with multiple disturbances at the same time

#### Test cases

In this numerical example, we want to demonstrate that CSE is able to obtain the real extraction flow in a canal with several pools with multiple flow extractions at the same time. In that case, a test case proposed by the ASCE Task Committee to evaluate control algorithms, which we had already used to test CSE, is introduced.

*et al.*1998) in several scenarios, but here only one is considered, the Maricopa Stanfield canal. The canal has eight pools separated by undershot sluice-gates. All the canal pools have been discretized and numbered in the direction of flow from upstream to downstream. Geometric characteristics of canal 1 (Maricopa Stanfield canal) are shown in Table 6 and Figure 12, and the control structures in Table 7. In the Maricopa Stanfield canal, there are gravity outlet orifices at the downstream end of each canal pool.

Pool number | Pool length (km) | Bottom slope | Side slopes (H:V) | Manning's coefficient (n) | Bottom width (m) | Canal depth (m) |
---|---|---|---|---|---|---|

I | 0.1 | 2 × 10^{−3} | 1.5:1 | 0.014 | 1 | 1.1 |

II | 1.2 | 2 × 10^{−3} | 1.5:1 | 0.014 | 1 | 1.1 |

III | 0.4 | 2 × 10^{−3} | 1.5:1 | 0.014 | 1 | 1.0 |

IV | 0.8 | 2 × 10^{−3} | 1.5:1 | 0.014 | 0.8 | 1.1 |

V | 2 | 2 × 10^{−3} | 1.5:1 | 0.014 | 0.8 | 1.1 |

VI | 1.7 | 2 × 10^{−3} | 1.5:1 | 0.014 | 0.8 | 1.0 |

VII | 1.6 | 2 × 10^{−3} | 1.5:1 | 0.014 | 0.6 | 1.0 |

VIII | 1.7 | 2 × 10^{−3} | 1.5:1 | 0.014 | 0.6 | 1.0 |

Pool number | Pool length (km) | Bottom slope | Side slopes (H:V) | Manning's coefficient (n) | Bottom width (m) | Canal depth (m) |
---|---|---|---|---|---|---|

I | 0.1 | 2 × 10^{−3} | 1.5:1 | 0.014 | 1 | 1.1 |

II | 1.2 | 2 × 10^{−3} | 1.5:1 | 0.014 | 1 | 1.1 |

III | 0.4 | 2 × 10^{−3} | 1.5:1 | 0.014 | 1 | 1.0 |

IV | 0.8 | 2 × 10^{−3} | 1.5:1 | 0.014 | 0.8 | 1.1 |

V | 2 | 2 × 10^{−3} | 1.5:1 | 0.014 | 0.8 | 1.1 |

VI | 1.7 | 2 × 10^{−3} | 1.5:1 | 0.014 | 0.8 | 1.0 |

VII | 1.6 | 2 × 10^{−3} | 1.5:1 | 0.014 | 0.6 | 1.0 |

VIII | 1.7 | 2 × 10^{−3} | 1.5:1 | 0.014 | 0.6 | 1.0 |

Target points | Gate discharge coefficient | Gate width (m) | Gate height (m) | Step (m) | Length from gate 1 (km) | Orifice offtake height (m) |
---|---|---|---|---|---|---|

0 | 0.61 | 1.5 | 1.0 | 1.0 | 0 | – |

1 | 0.61 | 1.5 | 1.1 | 1.0 | 0.1 | 0.45 |

2 | 0.61 | 1.5 | 1.1 | 1.0 | 1.3 | 0.45 |

3 | 0.61 | 1.5 | 1.0 | 1.0 | 1.7 | 0.40 |

4 | 0.61 | 1.2 | 1.1 | 1.0 | 2.5 | 0.45 |

5 | 0.61 | 1.2 | 1.1 | 1.0 | 4.5 | 0.45 |

6 | 0.61 | 1.2 | 1.0 | 1.0 | 6.2 | 0.40 |

7 | 0.61 | 1.0 | 1.0 | 1.0 | 7.8 | 0.40 |

8 | – | – | – | – | 9.5 | 0.40 |

Target points | Gate discharge coefficient | Gate width (m) | Gate height (m) | Step (m) | Length from gate 1 (km) | Orifice offtake height (m) |
---|---|---|---|---|---|---|

0 | 0.61 | 1.5 | 1.0 | 1.0 | 0 | – |

1 | 0.61 | 1.5 | 1.1 | 1.0 | 0.1 | 0.45 |

2 | 0.61 | 1.5 | 1.1 | 1.0 | 1.3 | 0.45 |

3 | 0.61 | 1.5 | 1.0 | 1.0 | 1.7 | 0.40 |

4 | 0.61 | 1.2 | 1.1 | 1.0 | 2.5 | 0.45 |

5 | 0.61 | 1.2 | 1.1 | 1.0 | 4.5 | 0.45 |

6 | 0.61 | 1.2 | 1.0 | 1.0 | 6.2 | 0.40 |

7 | 0.61 | 1.0 | 1.0 | 1.0 | 7.8 | 0.40 |

8 | – | – | – | – | 9.5 | 0.40 |

#### Scenario: test case 1-2 (Maricopa Stanfield)

After the first 2 hours, the initial backwater profile is changing, so an unscheduled flow change is introduced in the canal modifying some offtake flow, as shown in Table 8.

Pool number | Offtake initial flow (m^{3}/s) | Check initial flow (m^{3}/s) | Unscheduled offtake changes at 2 hours (m^{3}/s) | Check final flow (m^{3}/s) |
---|---|---|---|---|

Heading | – | 2.0 | – | 2.0 |

1 | 0.2 | 1.8 | – | 1.8 |

2 | 0.0 | 1.8 | 0.2 | 1.6 |

3 | 0.4 | 1.4 | −0.2 | 1.4 |

4 | 0.0 | 1.4 | 0.2 | 1.2 |

5 | 0.0 | 1.4 | 0.2 | 1.0 |

6 | 0.3 | 1.1 | −0.1 | 0.8 |

7 | 0.2 | 0.9 | – | 0.6 |

8 | 0.9 | 0.0 | −0.3 | 0.0 |

Pool number | Offtake initial flow (m^{3}/s) | Check initial flow (m^{3}/s) | Unscheduled offtake changes at 2 hours (m^{3}/s) | Check final flow (m^{3}/s) |
---|---|---|---|---|

Heading | – | 2.0 | – | 2.0 |

1 | 0.2 | 1.8 | – | 1.8 |

2 | 0.0 | 1.8 | 0.2 | 1.6 |

3 | 0.4 | 1.4 | −0.2 | 1.4 |

4 | 0.0 | 1.4 | 0.2 | 1.2 |

5 | 0.0 | 1.4 | 0.2 | 1.0 |

6 | 0.3 | 1.1 | −0.1 | 0.8 |

7 | 0.2 | 0.9 | – | 0.6 |

8 | 0.9 | 0.0 | −0.3 | 0.0 |

The unscheduled deliveries are more significant at target 8 in test case 1-2, where the flow rate changes from 0.9 m^{3}/s to 0.6 m^{3}/s (33%).

#### Results

We show the results obtained by CSE divided by eight graphs, one for every pool.

Every graph shows the scheduled and unscheduled offtake changes (demanded extracted flow) by the farmers, the real value delivered by the offtake and the real extracted flow calculated by CSE.

In this test 1-2, the canal is in steady state during the first 2 hours. After the first 2 hours, unscheduled water deliveries are introduced to the system at 7,200 s, although the algorithm takes no notice until the next regulated period, once the water level is measured at the checkpoints. It is important to note that the unscheduled water deliveries are relevant in all targets but especially at target 8, because in just one regulation period, the water level at checkpoint 8 increases from 0.8 m to 1.08 m, due to the introduction of a water delivery change of 0.3 m^{3}/s at checkpoint 8.

### Numerical example in a laboratory canal

In this example, the CSE algorithm has been tested in a real canal. We want to verify that CSE is able to assess with accuracy a disturbance introduced in a laboratory canal, and in this way, some tests were done to check the good results obtained by CSE in the last examples.

The PAC-UPC is a laboratory canal specially designed to develop basic and applied research in irrigation canal control area and in all subjacent areas like irrigation canal instrumentation, irrigation canal modelling, water measurements, etc. The canal is located in the Laboratory of Physical Models inside the North Campus of the UPC (Barcelona's School of Civil Engineering).

#### General description

Pool | Pool length (m) | Canal depth (m) | Manning's coefficient (n) | Width (m) |
---|---|---|---|---|

1 (from G1 to G2) | 42 | 1.00 | 0.016 | 0.44 |

2 (from G2 to G3) | 45 | 1.00 | 0.016 | 0.44 |

3 (from G3 to G4) | 45 | 1.00 | 0.016 | 0.44 |

4 (from G4 to G5) | 45 | 1.00 | 0.015 | 0.44 |

5 (from G5 to W4) | 43 | 1.00 | 0.016 | 0.44 |

Pool | Pool length (m) | Canal depth (m) | Manning's coefficient (n) | Width (m) |
---|---|---|---|---|

1 (from G1 to G2) | 42 | 1.00 | 0.016 | 0.44 |

2 (from G2 to G3) | 45 | 1.00 | 0.016 | 0.44 |

3 (from G3 to G4) | 45 | 1.00 | 0.016 | 0.44 |

4 (from G4 to G5) | 45 | 1.00 | 0.015 | 0.44 |

5 (from G5 to W4) | 43 | 1.00 | 0.016 | 0.44 |

The canal is supplied by a small reservoir at the upstream end. The objective of this element is to provide the canal with enough water to feed the canal. Water comes from the reservoir through gate 1 (*G1*), which is normally in submerged conditions. This gate regulates the inflow by adjusting the gate opening.

The water that is not used is recirculated to the laboratory pumping system. It is possible for the user to arrange the canal with several pool configurations, i.e., a canal with only one very long pool, a canal with one long pool and one short pool, etc.

### Test geometry canal

In this example, the geometrical configuration of the canal is as follows: *G1* is the only sluice-gate operating because gates 3 and 5 are out of the water. Only the weirs *W2* and *W4* are operating, and the algorithm only uses the water level measurements at sensors *L1*, *L6*, *L10* and *L11*, although the data of sensor *L7* were used to check some results.

Gate 1 is made of methacrylate reinforced with a metal skeleton in order to provide enough stiffness and a low weight. The vertical movement of the gate is guided by metal frameworks embedded in the canal and is executed by three-phase servomotors. The gate features in this example are shown in Table 10.

Gate | Gate discharge coefficient | Contraction coefficient | L1 water level reservoir | Gate width (m) | Height of the gate opening (m) | Steep (m) |
---|---|---|---|---|---|---|

1 | 0.68 | 0.60 | 1.257 | 0.434 | 0.122 | 0.0 |

Gate | Gate discharge coefficient | Contraction coefficient | L1 water level reservoir | Gate width (m) | Height of the gate opening (m) | Steep (m) |
---|---|---|---|---|---|---|

1 | 0.68 | 0.60 | 1.257 | 0.434 | 0.122 | 0.0 |

Rectangular weirs are used to extract water laterally in order to emulate the effect of offtake discharges in real irrigation canals. These weirs have a width of 39 cm and were constructed starting from a 35 cm canal height, except for the end weir (*W4*) that starts from the canal invert (Sepúlveda 2007). From this minimum height of 35 cm, it is possible to increase the height of the weir by placing PVC pieces in metal rails, one on top of the other. These pieces are 5 cm, 10 cm, 20 cm and 35 cm. With different combinations, it is possible to cover a broad range of weir heights, and thereby to achieve a broad range of output flow through the weir. This structure allows easy measurement of the extracting flow rate measuring the water level upstream of the weir. The features of the sharp-crested weir are shown in Table 11.

Weir | Weir discharge coefficient (C_{W}) | Weir height (m) | Weir width (m) |
---|---|---|---|

4 | 0.5776 | 0.35 | 0.39 |

Weir | Weir discharge coefficient (C_{W}) | Weir height (m) | Weir width (m) |
---|---|---|---|

4 | 0.5776 | 0.35 | 0.39 |

#### Initial conditions for the example

A particular steady state is the initial condition for the canal. The total flow is 110 L/s through *G1*, and weirs 1, 2 and 3 are not operative, only *W4* works. Table 12 shows the water level measurements, which were measured manually, and the flow rate at particular points at initial time. The water level upstream from *G1* and the height of the gate opening were shown in Table 10.

Checkpoint | Initial water level (m) | Flow (m^{3}/s) |
---|---|---|

1 (L4) | 0.758 | 0.110 |

2 (L6) | 0.730 | 0.110 |

3 (L8) | 0.689 | 0.110 |

4 (L10) | 0.644 | 0.110 |

5 (L11) | 0.604 | 0.110 |

Checkpoint | Initial water level (m) | Flow (m^{3}/s) |
---|---|---|

1 (L4) | 0.758 | 0.110 |

2 (L6) | 0.730 | 0.110 |

3 (L8) | 0.689 | 0.110 |

4 (L10) | 0.644 | 0.110 |

5 (L11) | 0.604 | 0.110 |

#### Scenario

At the beginning of the test, the flow rate in the canal is 110 L/s. At a particular time (250 s after starting the test), some of the pieces that make up the lateral weir (*W2*) were removed, so the weir height changed to 55 cm. Later, at time 1950 s, the weir was closed again (Table 13).

Time (s) | G1 opening (m) | W1 height (m) | W2 height (m) | W3 height (m) | W4 height (m) |
---|---|---|---|---|---|

0 | 0.122 | 0.90 | 0.90 | 0.90 | 0.35 |

250 | 0.122 | 0.90 | 0.55 | 0.90 | 0.35 |

1,950 | 0.122 | 0.90 | 0.90 | 0.90 | 0.35 |

3,410 | 0.122 | 0.90 | 0.90 | 0.90 | 0.35 |

Time (s) | G1 opening (m) | W1 height (m) | W2 height (m) | W3 height (m) | W4 height (m) |
---|---|---|---|---|---|

0 | 0.122 | 0.90 | 0.90 | 0.90 | 0.35 |

250 | 0.122 | 0.90 | 0.55 | 0.90 | 0.35 |

1,950 | 0.122 | 0.90 | 0.90 | 0.90 | 0.35 |

3,410 | 0.122 | 0.90 | 0.90 | 0.90 | 0.35 |

*W2*was not measured directly. However, it was possible to estimate the flow through

*W2*because we obtained the water level measurements of the

*L7*sensor (Figure 15), which is the closest sensor to

*W2*, and the discharge coefficient of

*W2*was calibrated in previous works (see Horváth 2013).

Table 13 shows the changes in the weir height to introduce disturbance into the canal.

*L7*decreases quickly by 3 cm. Obviously, the flow extracted through the weir modified the water level surface and the flow along the canal. From sensors

*L1*,

*L6*,

*L10*and

*L11*were obtained the water level measurements used by CSE (see Figures 16 and 17).

The water level measurements at the reservoir collected by sensor *L1* were used by CSE for the establishment of the upstream boundary condition. Although the water level at the reservoir should be constant in time, due to limitations of laboratory installations the water level at the reservoir is variable in time as it depends on the water level downstream of the gate.

#### Results

The disturbances are introduced into the system by modifying the weir height. Sensors *L6*, *L10*, *L11* get the water level measurements along the canal and these values are introduced into the CSE algorithm, which calculates a discharge through the weir that generates a variation in the water levels at the checkpoints equal to the water level measured at sensors *L6*, *L10* and *L11*. The extracted hydrograph explains the evolution of the water level measurements at the sensors during the past time horizon.

*W2*) calculated using the equation of a sharp-crested weir (eq. weir) from the water level measurements at sensor

*L7*and the discharge coefficient calibrated by several authors (Galvis

*et al.*2011; Horváth

*et al.*2012; Horváth 2013).

After analyzing Figure 18, the following can be added:

The algorithm obtains an extracted hydrograph similar to the real flow extracted through the weir (

*W2*), especially at the initial moment of introducing the disturbance.Although the real extracted flow was obtained using the weir equation with the water level measurements of sensor

*L7*and the discharge coefficient calibrated by Hórvarth, the calculated flow extraction was quite accurate due to the noise error of the measurements not being significant.The flow rate difference between the hydrograph obtained by CSE and the real hydrograph was around 2.5 L/s.

Probably, the differences between both hydrographs may be the result of deviations in water level measurements at sensors

*L6*,*L7*,*L10*and*L11*and/or little deviation between the real Manning or hydraulic loss coefficients and those simulated, although several tests were done to adjust the coefficients as well as possible in the current conditions of the canal.

## CONCLUSIONS

The CSE algorithm is very stable numerically because it calculates the extracted hydrograph more similarly to the real hydrograph that has been extracted by the weir. The result is physically feasible, and CSE does not look for incoherent solutions that could also define unreal extraction flows and canal states. The results obtained in the numerical examples verify the good expectations for the algorithm.

The water levels measurements must be as accurate as possible, since the result (extracted hydrograph) obtained by CSE will be as accurate as the input data (water level measurements) are.

The CSE algorithm is sensitive to some physical parameters such as the Manning roughness coefficient and, by extension, other parameters such as the local energy losses in the canal bend. For this reason, the algorithm should not be used in canals where the physical parameters are not well known. Although it is beyond the scope of this paper (it is difficult to always determine the true conditions of any canal, as opposed to simulation models where they are presumed to be known exactly), we did some tests introducing changes in the Manning coefficient, measurement errors in depth gauges or by missing water level measurements of gauges (see Bonet 2015) to check the robustness of CSE.

The CSE is able to calculate the hydrodynamic canal state as shown in the first example. The accuracy of CSE in calculating the extracted flow vector is essential for calculating with accuracy the hydrodynamic canal state with CSE. The CSE calculates the hydrodynamic canal state from the scheduled demands, gate trajectories, initial conditions and the extracted flow vector. The hydrodynamic canal states obtained by a computer model using the real disturbances or from the results obtained by CSE are very similar.