## Abstract

Most studies about the automatic control of open canal irrigation systems only focus on the distant downstream water level, which ignores the fact that the offtakes may be located anywhere along the canal. Such a simplified control strategy is likely to result in uncontrollable and inefficient water delivery. Therefore, a multi-point hydraulic control method is proposed, in which a simplified Saint-Venant model is formulated to describe the hydraulic states of multiple controlled points. Then, it is underlined that the controlled points with and without the offtakes may have different control objectives. It is suggested to implement soft constraints to the downstream end when there is no offtake, meaning that moderate water level fluctuations are acceptable. By comparing with the common model predictive control (MPC) controller, where the Integrator Delay model and hard constraint are used for distant downstream water level control, the proposed MPC controller successfully improved the water level control stability before the offtakes and the water supply reliability by 91 and 69.5% under the conventional condition and by 54.9 and 27.1% under the water-deficient condition. Accordingly, the proposed multi-point hydraulic control method shows great potential for the precision irrigation of large irrigation districts.

## HIGHLIGHTS

Simplified Saint-Venant equations are formulated for the offtakes control along the irrigation canal.

The differences in control objectives between various controlled points are underlined.

A soft constraint is applied to the canal downstream end for operation safety.

The control performance of irrigation water is remarkably enhanced with the proposed control method.

### Graphical Abstract

## ACRONYMS

- ASCE
American Society of Civil Engineers

- ID
Integrator Delay

- IDZ
Integrator Delay Zero

- IR
Integrator Resonance

- MPC
Model predictive control

*NISE*Non-dimensional Integrated Square of Error

- OCIS
Open canal irrigation systems

*PD*Dependability of surface water distribution

- PID
Proportion–integral–differential

*RMSE*Root mean square error, m

- SCCS
Simulation and Control of Canal System

- SV
Saint-Venant

- SVLD
Proposed Saint-Venant simplified model

## NOTATIONS

*A*Wetted area, m

^{2}**A**System matrix

*A*_{s}Average storage area of the ID model, m

^{2}*a*_{1}–*a*_{3}Parameters of Equation (4)

*B*Water surface width, m

**B**_{d}Known disturbance matrix

**B**_{u}Control input matrix

**C**Output matrix

*CV*_{T}Coefficient of variation

*c*Average celerity of the flow, m/s

*c*_{1}–*c*_{2}Parameters of Equation (4)

*d*Offtake flow deviation, m

^{3}/s**d**Disturbance vector

*d*_{1}–*d*_{3}Parameters of Equation (5)

*E*Gate opening, m

*e*Water level deviation, m

*e*_{1}–*e*_{3}Parameters of Equation (5)

*e**Water level deviation outside of the target band, m

*g*Gravity constant, m/s

^{2}*J*Objection function of MPC

*k*Control step

*L*Canal length, m

*m*Control horizon

*n*Manning friction coefficient

*p*Prediction horizon

*Q*and*q*Flow and flow deviation, m

^{3}/s**Q**Weight matrix to the water level deviations

*Q*and_{d}*Q*_{r}Discharge delivered to the offtake and the water demand discharge, m

^{3}/s**R**Weight matrix to the flow changes

*s*and_{0}*s*_{f}Bottom slope and friction slope

*T*Operation duration, h

*T*_{s}Time step, min

*t*Simulation time, s

**u**Input vector calculated by the controller

*v*Average velocity of the flow, m/s

*x*Space coordinate of the computed cross-section or the offtake, m

**x**Controlled water system states

- Δ
*x* Lengths of divided segments, m

*y*Water depth, m

**y**Output vector

*τ*Delay time of the ID model, s

*φ*_{1}and*φ*_{2}Improvements in

*NISE*and_{offtake}*PD**χ*Wetted perimeter, m

## INTRODUCTION

With climate change and population growth, the demands for agricultural productivity and irrigation water are under increasing pressure (Xu *et al.* 2020). Taking China's agricultural irrigation development in the past 40 years as an example, the number of irrigation districts and the irrigated area have increased by 50.8 and 65.7%, respectively (Ministry of Water Resources 2020). These figures reveal that traditional rain-fed cultivating activities have been replaced by irrigation-based agriculture. However, the backward manual operational management of irrigation water is hindering the development of sustainable agriculture. The irrigation water use efficiency is not more than 51% (Khiabani *et al.* 2020), implying that a significant portion of surface water resources is wasted and the farmers’ demands cannot be fully satisfied.

Open canal irrigation systems (OCIS) are the main way of water conveyance and distribution. Owing to the disturbances and unexpected hydraulic coupling within the canal networks, there are always unreliable and unfair water deliveries by manual operation. Many intractable issues arose. For example, inadequate water delivery somewhere would lead to undesired water stress on plants and the dependency of the farmers on the groundwater (Askari Fard *et al.* 2022), while over-irrigation somewhere would result in a large amount of water waste, waterlogging, or soil salinization (Hassani & Hashemy Shahdany 2021). Therefore, upgrading the OCIS operation through modernization and automation is among the essential needs.

The OCIS are complex systems with considerable delays, and its automatic control is a challenging issue of interest (Litrico 2009). In the last decades, there has been a growing number of studies about the canal automation mechanisms (Conde *et al.* 2021) from the earliest classical control characterized by proportion–integral–differential (PID) control, to the later modern control dominated by model predictive control (MPC), and then to the recently booming intelligent control powered by reinforcement learning. Before controller design, selecting an appropriate control-oriented model for the OCIS is crucial (Mao *et al.* 2019). The control-oriented model is a mathematical representation used for hydrodynamic prediction, which helps to reach a desirable control performance. In 1871, the Saint-Venant (SV) equations (Saint-Venant 1971) were derived for investigating the hydraulic behavior of the flow in open channels. Based on this, many related studies arose for the development of hydraulic engineering. For example, Lacasta *et al.* (2017) calibrated the discharge coefficients of lateral gates by a classical Monte Carlo optimization method and an adjoint method-based optimization method. The study provides an affordable way to model the outflow from a lateral hydraulic structure. Nonetheless, the direct use of the SV equations for controller design is impractical for its complexities. Accordingly, many kinds of simplified models come into being. For instance, the Integrator Delay (ID) model (Schuurmans *et al.* 1995) is one of the most reported and classic modeling strategies for the OCIS according to the latest review (Conde *et al.* 2021). The canal is assumed to be divided into a normal depth section and a backwater section. And then, the fluctuation of downstream water level with various flow changes is estimated. Though such an approximated model does not have rigorous physical fundament, the reported works have shown its broad applicability for its conciseness and precision (Wahlin & Clemmens 2006; Mao *et al.* 2019; Askari Fard *et al.* 2022). Other commonly used control-oriented models include the Integrator Delay Zero (IDZ) model (Litrico & Fromion 2004), the Integrator Resonance (IR) model (van Overloop *et al.* 2010), and reduced SV models (Xu *et al.* 2011).

However, most researchers only focus on the downstream water level deviation, indicating that all offtakes are assumed to be aggregated into one at the downstream end of each pool (Wahlin & Clemmens 2006; Hassani & Hashemy Shahdany 2021). In this way, the smooth control of the downstream water level suggests that the lateral canals or the farmers can draw water successfully as planned. Undeniably, such a control strategy is reasonable if the check gate is located downstream of the offtakes closely for drawing water or a suitable flow velocity (Shah *et al.* 2016) like the laboratory canals (Horváth *et al.* 2014). Unfortunately, in many real-world irrigation projects, the number of check gates is limited due to the cost or the topography. And the offtakes may be anywhere along the OCIS (Shah *et al.* 2016) for many reasons, including the plantation structure, administrative division, water rights management mode, new water demands, and others. For example, on the Canale Emiliano Romagnolo, one of the most important irrigation canals in Northern Italy, more than half of the offtakes are positioned upstream and middle of the canal pools (Luppi *et al.* 2018). The Zhanghe Irrigation System (Wang *et al.* 2017) is taken as another example, which is one of the nine major irrigation districts in China and is characterized by its ‘melon-on-the-vine’ irrigation network. The number of offtakes located downstream of the main canal only accounts for 15%, which is recruited from field visits. Other cases are widespread (Georges 1994; Wahlin & Clemmens 2006; Munir *et al.* 2012; Hong *et al.* 2014; Shah *et al.* 2016). Consequently, the conventional control method for a single point, like the distant downstream water level control, may be inappropriate for many real irrigation canals. The uncontrolled water level before the offtake gates could inflict considerable water mismatch and water resource waste.

The hydraulic fluctuations before the offtakes have a great impact on the water supply reliability during the key growth stage. Hence, it is vital to consider every offtake section, no matter where it is. Furthermore, when there is no offtake at the downstream end of the canal pool, it is still important to control this point for the safety of canal system operation and continuous water supply downstream. Nonetheless, maintaining this water level at the predefined target level could be regarded as an abuse of limited control capacity and computing resources. Keeping the downstream water level in a deadband around the target level (target band) (van Overloop *et al.* 2008; Hashemy Shahdany *et al.* 2013) is believed to be enough. Accordingly, the control strategy for multi-points with different control targets has strong practical implications for irrigation water management and is worth studying. But, to the best of the authors’ knowledge, it is still an unattended subject. Cen *et al.* (2017) have succeeded in controlling multi-point water levels from the initial values to the target values on a short irrigation canal by nonlinear optimal control with discretized SV equations. Nevertheless, both the offtakes and the difference in control targets of different controlled points were not considered in her study.

The more specific objectives of the study are (i) developing an applicable control-oriented model for multi-point hydraulic control, with a focus on the offtake sections; (ii) highlighting the control target difference between the offtake sections and the downstream end of the canal pool; (iii) designing advanced MPC controllers based on the ID model and the proposed model, respectively; (iv) investigating the prediction accuracy and control performance by the proposed model and comparing the results with the popular ID model; and (v) evaluating the advantages and potential of the proposed control method.

This paper is organized as follows. Section 2 describes the details of the proposed control method and the MPC controller design. The study case and test scenarios are also introduced. Section 3 shows the simulation results, demonstrating the superiorities of the proposed control method. Discussion and conclusions follow in Sections 4 and 5, respectively.

## MATERIALS AND METHODS

### The Integrator Delay model

*et al.*(1995) proposed the ID model to take inflow delay into account, see Equation (1). Delay time (

*τ*in s) and average storage area (

*A*in m

_{s}^{2}) are two main model properties, which can be estimated by the parameter identification method (Zhong 2016). In Equation (1), the computation of

*e*(

*t*) is independent of the offtake location with an assumption that all offtakes are aggregated into one at the most downstream end. But the reality is not. Furthermore, the previous study has proven that this coarse assumption has a great impact on the model prediction accuracy (Guan

*et al.*2022). The system construction of a real-time controller based on the ID model and distant downstream control is shown in Figure 1(a).where

*e*(

*t*) is the water level deviation at the downstream end of the canal pool, m;

*q*(

_{u}*t*−

*τ*) is the inflow deviation to the backwater with delay time

*τ*, m

^{3}/s;

*q*(

_{d}*t*) is the downstream outflow deviation, m

^{3}/s;

*d*(

*t*) is the offtake flow deviation according to the irrigation schedule, m

^{3}/s. The deviation means the difference between the momentary value and the initial steady-state value.

### The simplified Saint-Venant model

*A*is the wetted area, m

^{2};

*Q*is the flow, m

^{3}/s;

*t*is the simulation time, s; is the space coordinate with

*L*being the canal length, m;

*y*is the water depth, m;

*B*is the water surface width, m;

*g*is the gravity constant, m/s

^{2};

*s*

_{0}is the bottom slope;

*s*is the friction slope and given by:where

_{f}*χ*is the wetted perimeter, m;

*n*is the Manning friction coefficient.

*t*

_{0};

*a*

_{1}–

*a*

_{3}and

*c*

_{1}–

*c*

_{2}are parameters that can be calculated by: and .

*x*

_{1}to

*x*

_{2}, and then the ordinary differential equations are derived (Wang 2004):where is the space distance from

*x*

_{1}to

*x*

_{2};

*d*

_{1}–

*d*

_{3}and

*e*

_{1}–

*e*

_{3}are parameters computed by:

*M*− 1 segments with

*M*sample points according to the offtake locations. For each segment, Equation (5) is applicable. To calculate in

*c*

_{1}and

*c*

_{2}, the direct difference method is adopted. More specifically, the first-order forward difference method is used for the first point, the central difference method is applied for the intermediate points, and the backward difference is employed for the last point (Cen

*et al.*2017):

*v*and

*c*are the average velocity and average celerity, respectively, m/s.

### Model predictive control

*J*with penalties on water level deviations and flow changes is minimized to obtain optimal control actions, as shown in Equation (9).where

**x**(

*k*) represents the controlled water system states at control step

*k*;

**A**is the system matrix;

**B**is the control input matrix;

_{u}**B**is the known disturbance matrix;

_{d}**C**is the output matrix;

**u**(

*k*) is the input vector calculated by the controller, i.e. the control commands for check gates;

**d**(

*k*) is the disturbance vector, i.e. the watering schedule;

**y**(

*k*) is the output vector, i.e. the water level deviation of each controlled point.where

*J*represents the objection function constrained by water level limitations and flow capacities;

*p*and

*m*represent the prediction horizon and control horizon, respectively;

**Q**and

**R**are the weight matrixes for a trade-off between the water level deviations and flow changes.

*e*to

_{min}*e*. But in case a violation occurs, a considerable higher penalty is given to the state outside of its limited range, i.e. . And the optimization module must consequently try to avoid this violation. The calculation of is shown in Equation (10). Soft constraints are introduced here to safely weaken the downstream water level control, so that more powerful control could be taken for the offtake sections (for more details about soft constraints, refer van Overloop

_{max}*et al.*(2008) and Hashemy Shahdany

*et al.*(2013)).where is the water level deviation outside of the target band at control step

*k*;

*e*and

_{max}*e*are the security limitations to the downstream water level deviation.

_{min}### Study area and test scenarios

*et al.*1998), as shown in Figure 4. The basic design and modeling parameters are listed in Table 1, where the delay time and average storage area of the ID model are calculated based on the design flow condition. It is assumed that there is only one offtake in the test canal reach, located at

*x*

_{2}(,

*x*

_{1}= 0,

*x*

_{3}= 7,000 m).

Length (m) . | Bed slope . | Manning coefficient . | Slide slope . | Bottom width (m) . | Design flow (m^{3}·s^{−1})
. | Target depth (m) . | Delay time (min) . | Storage area (m^{2})
. |
---|---|---|---|---|---|---|---|---|

7,000 | 1/10,000 | 0.02 | 1.5 | 7 | 14 | 2.1 | 21 | 53,311 |

Length (m) . | Bed slope . | Manning coefficient . | Slide slope . | Bottom width (m) . | Design flow (m^{3}·s^{−1})
. | Target depth (m) . | Delay time (min) . | Storage area (m^{2})
. |
---|---|---|---|---|---|---|---|---|

7,000 | 1/10,000 | 0.02 | 1.5 | 7 | 14 | 2.1 | 21 | 53,311 |

*The first test scenario*(see Figure 5(a)): There is an inflow oscillation from*T*= 2 h to*T*= 10 h to examine the prediction accuracy of the proposed SVLD model under the severe disturbance condition around the equilibrium state. Moreover, the prediction results of the most downstream end, i.e.*x*_{3}, are compared with the ID model later to prove the feasibility of canal automation by the SVLD model. Afterward, the offtake starts to withdraw water at*T*= 13 h and keeps a steady offtake flow of 1 m^{3}/s after*T*= 15 h. In consequence, the influences of the offtake demand change and offtake location on model prediction performance are calculated. During the simulation time, the downstream check gate keeps a stable outflow.*The second test scenario*(see Figure 5(b)): This test is used to assess the MPC controllers based on the ID model and the proposed SVLD model under the conventional operating condition. Noteworthy is there is a constant limitation on the canal inflow. In other words, only the downstream check gate can be regulated by the MPC controllers for the water level stability of every controlled point. In general, the canal head draws water from the upstream reservoir, which is managed by the regional water board. Therefore, the intake flow cannot always be changed timely on demand. In this test scenario, the offtake flow has an initial demand of 2 m^{3}/s, but it is increased to 4 m^{3}/s between*T*= 5 h and*T*= 10 h. Different from the first test scenario, the actual offtake flow change is governed by the sluice free flow formula during unsteady flow simulation.*The third test scenario*(see Figure 5(c)): This test is employed to evaluate the ability of the proposed control method to handle the severe mismatch between supply and demand. This situation happens in arid and semiarid regions, in which the canal inflow is less than the total demands between*T*= 5 h and*T*= 10 h. The only difference between this and the previous test scenario is the inflow fluctuation.

### Simulation configurations and controller alternatives

The unsteady flow simulation is carried out with the computation time step *D _{t}* = 1 min on the Simulation and Control of Canal System (SCCS) platform (Wang & Guan 2011), on which the SV equations are resolved with the Preissmann four-point implicit difference scheme. This control simulation platform has been applied and verified by previous studies (Liu

*et al.*2013; Guan

*et al.*2018).

To construct a more accurate state-space model, the strategy of constant control action (Horváth *et al.* 2015) is adopted for the second and third test scenarios. In this strategy, *T _{s}* = 1 min and

*m*= 1. As for the prediction horizon, it is always bigger than the sum of the delay time of all canal pools (van Overloop 2006), and then

*p*= 90 is appropriate by trial-and-error, i.e. the prediction horizon is 1.5 h long. According to the operational experience of the OCIS, the gate control time interval is set as 15 min, which means the MPC controller is recalled every 15 min, and then the control commands are sent to the downstream check gate.

Based on the above controller configurations, five controller alternatives are developed in Table 2. Hard constraint or soft constraint is applied to the downstream water level, while only hard constraint is imposed on the offtake section. Alternative B3 represents the proposed control method. , , , and are the penalty coefficients on the water level deviations of the upstream end, the offtake section, the downstream end, and the violated part of the downstream end, respectively. and are the penalty coefficients on the flow changes of upstream inflow and downstream outflow, respectively.

Alternatives . | Internal model . | Constraints . | . | . | . | . | . | . |
---|---|---|---|---|---|---|---|---|

A1 | ID | Hard | \ | \ | 1.0 × 10^{+5} | 0 | 1 | 1 |

A2 | Soft | \ | \ | 1 | 1.0 × 10^{+5} | 1 | 1 | |

B1 | SVLD | Hard 1 | 1.0 × 10^{−5} | 1.0 × 10^{+5} | 1.0 × 10^{+5} | 0 | 1 | 1 |

B2 | Hard 2 | 1.0 × 10^{−5} | 1.0 × 10^{+6} | 1.0 × 10^{+5} | 0 | 1 | 1 | |

B3 | Soft | 1.0 × 10^{−5} | 1.0 × 10^{+6} | 1 | 1.0 × 10^{+5} | 1 | 1 |

Alternatives . | Internal model . | Constraints . | . | . | . | . | . | . |
---|---|---|---|---|---|---|---|---|

A1 | ID | Hard | \ | \ | 1.0 × 10^{+5} | 0 | 1 | 1 |

A2 | Soft | \ | \ | 1 | 1.0 × 10^{+5} | 1 | 1 | |

B1 | SVLD | Hard 1 | 1.0 × 10^{−5} | 1.0 × 10^{+5} | 1.0 × 10^{+5} | 0 | 1 | 1 |

B2 | Hard 2 | 1.0 × 10^{−5} | 1.0 × 10^{+6} | 1.0 × 10^{+5} | 0 | 1 | 1 | |

B3 | Soft | 1.0 × 10^{−5} | 1.0 × 10^{+6} | 1 | 1.0 × 10^{+5} | 1 | 1 |

### Operational performance indicators

*RMSE*) is used to evaluate the model prediction accuracy, see Equation (11). The smaller the

*RMSE*, the smaller the prediction error, and the higher the prediction accuracy.where

*n*is the number of measurements;

*X*is the measured result from the SCCS platform, m;

_{i}*Y*is the model prediction result, m.

_{i}*NISE*) is used to investigate the water level control performance (Guan

*et al.*2018). The smaller the

*NISE*is, the less the water level fluctuation at the controlled point is.

*PD*demonstrates the dependability of surface water distribution (Molden & Gates 1990). As

*PD*approaches zero, the relative water delivery is becoming more uniform over time, indicating a more dependable delivery.where

*T*is the operation duration, set as 20 h;

*y*is the water depth at time

_{t}*t*, m;

*y*is the target water depth, m;

_{target}*Q*is the discharge delivered to the offtake, m

_{d}^{3}/s;

*Q*is the water demand discharge, m

_{r}^{3}/s;

*CV*is the coefficient of variation of the

_{T}*Q*/

_{d}*Q*time series.

_{r}## RESULTS

### Comparative analysis of model prediction accuracy

*x*

_{2}= 4,000. Since the ID model can only predict the downstream water level fluctuation, so it is compared with the SVLD model in Figure 6(c). It is shown that the prediction performance of both models is good, and the ID model even performs slightly better than the SVLD model in the inflow oscillation stage. On the one hand, this result confirms the reliability of the ID model, though this model is much simpler in form. On the other hand, it proves that the proposed model could be as applicable as the ID model for advanced controller design. The predictions for the upstream end and the offtake section also turned out well (see Figure 6(a) and 6(b)).

It is also interesting that the prediction accuracy of the ID model is greatly affected by the actual offtake location, especially in Figure 7(c). This result is imputed to the unreasonable assumption of the ID model about offtake location, and it has been discussed in detail in the previous study (Guan *et al.* 2022).

### Control performance evaluation

*x*

_{2}= 4,000. Take the second test scenario as an example, the control performances with each controller alternative are displayed in Figure 8. The simulation results in the third test scenario are similar but with more hydraulic fluctuation due to the severe water shortage.

Figure 8(a) and 8(b) presents the control results of the downstream end by each controller alternative. The controllers began to work before the demand change and restored the water level to the setpoint within 10 h, showing the superiority of the MPC algorithm. For alternatives A2 and B3, a target band based on the existing freeboard of the downstream end is employed for storing more water than hard constraints (Hashemy Shahdany *et al.* 2013). The stored water was used to compensate for the delay time of water traveling and cover upcoming demand changes, which contributes to a bigger but safe water level fluctuation and faster stabilization.

The hydraulic responses of the offtake section are provided in Figure 8(c) and 8(d). The results for alternatives A1 and A2 show the improvement in the water level control stabilities by the ID model-based MPC with soft constraints. It should be noted that this improvement is not generated by the control ability of the ID model but by the water level rise in the backwater area (see Figure 8(a)). The controllers based on the ID model can only take the downstream end into account, so it is a coincidental result in the conventional condition. When it comes to the water-deficient condition, the improvement in *NISE _{offtake}* changes from 57.9 to −2.3% (see Table 3).

Scenarios . | Alternatives . | Internal model . | Constraints . | NISE (m)
. _{offtake} | φ_{1}
. | PD
. | φ_{2}
. |
---|---|---|---|---|---|---|---|

2 | A1 | ID | Hard | 1.40 × 10^{−4} | \ | 7.29 × 10^{−3} | \ |

A2 | Soft | 5.90 × 10^{−5} | 57.9% | 4.15 × 10^{−3} | 43.0% | ||

B1 | SVLD | Hard 1 | 1.02 × 10^{−4} | 27.4% | 6.51 × 10^{−3} | 10.7% | |

B2 | Hard 2 | 2.14 × 10^{−5} | 84.8% | 3.35 × 10^{−3} | 54.0% | ||

B3 | Soft | 1.27 × 10^{−5} | 91.0% | 2.22 × 10^{−3} | 69.5% | ||

3 | A1 | ID | Hard | 4.30 × 10^{−3} | \ | 3.80 × 10^{−2} | \ |

A2 | Soft | 4.40 × 10^{−3} | − 2.3% | 3.42 × 10^{−2} | 10.2% | ||

B1 | SVLD | Hard 1 | 5.63 × 10^{−3} | − 31.0% | 4.54 × 10^{−2} | − 19.5% | |

B2 | Hard 2 | 2.20 × 10^{−3} | 48.9% | 2.91 × 10^{−2} | 23.4% | ||

B3 | Soft | 1.94 × 10^{−3} | 54.9% | 2.77 × 10^{−2} | 27.1% |

Scenarios . | Alternatives . | Internal model . | Constraints . | NISE (m)
. _{offtake} | φ_{1}
. | PD
. | φ_{2}
. |
---|---|---|---|---|---|---|---|

2 | A1 | ID | Hard | 1.40 × 10^{−4} | \ | 7.29 × 10^{−3} | \ |

A2 | Soft | 5.90 × 10^{−5} | 57.9% | 4.15 × 10^{−3} | 43.0% | ||

B1 | SVLD | Hard 1 | 1.02 × 10^{−4} | 27.4% | 6.51 × 10^{−3} | 10.7% | |

B2 | Hard 2 | 2.14 × 10^{−5} | 84.8% | 3.35 × 10^{−3} | 54.0% | ||

B3 | Soft | 1.27 × 10^{−5} | 91.0% | 2.22 × 10^{−3} | 69.5% | ||

3 | A1 | ID | Hard | 4.30 × 10^{−3} | \ | 3.80 × 10^{−2} | \ |

A2 | Soft | 4.40 × 10^{−3} | − 2.3% | 3.42 × 10^{−2} | 10.2% | ||

B1 | SVLD | Hard 1 | 5.63 × 10^{−3} | − 31.0% | 4.54 × 10^{−2} | − 19.5% | |

B2 | Hard 2 | 2.20 × 10^{−3} | 48.9% | 2.91 × 10^{−2} | 23.4% | ||

B3 | Soft | 1.94 × 10^{−3} | 54.9% | 2.77 × 10^{−2} | 27.1% |

Compared with alternative A1, *φ*_{1} and *φ*_{2} represent the improvements in *NISE _{offtake}* and

*PD*, respectively.

The control results by the SVLD model-based MPC are presented in Figure 8(d). In alternative B1, the penalty coefficients on both the downstream end and the offtake section are 1.0 × 10^{+5}, which is the same as that of alternative A1 on the downstream end. In this way, the results of the two control alternatives are similar. Further, the penalty coefficient on the offtake section is increased to 1.0 × 10^{+6} in alternative B2, and then the control performance is significantly better. This improvement greatly proves the ability of the proposed SVLD model to control the offtake section, which is not controllable with the ID model. The selection of the penalty coefficients determines the control performance. Though the control objectives of alternatives B1 and B2 are both to maintain the water level of the two controlled points at the target values, the penalty coefficient on the offtake section needs to be bigger than that on the downstream end. The main reason is that the downstream controlled point is near the downstream check gate, which is the only controllable check gate in this test scenario. Hence, the controller has a stronger ability to control the downstream water level, so that a smaller penalty coefficient can lead to acceptable control results. The farther the controlled point is from the control devices, the larger the penalty coefficient is supposed to be.

The control alternative B3, which is the suggested multi-point hydraulic control method in the paper, shows the best water level control performance at the offtake section (see Figure 8(d)). In this control scheme, a strong hard constraint is applied to the offtake section for water level stability, while a soft constraint is employed to the downstream end for operation safety when there is no offtake. With the relaxation of the control constraints on the downstream water level deviation, the feasibility area for solving the objection function of the MPC (i.e. Equation (9)) is extended, and the controller can search for a more efficient solution for restraining the hydraulic fluctuation at the offtake section. Furthermore, Table 3 shows that the suggested control method can also greatly improve the dependability of surface water distribution. It means that the water supply could be more reliably guaranteed during the key growth stage of crops.

## DISCUSSION

Given that there always are offtakes along the irrigation open canal, rather than concentrating on the downstream end of each canal pool, the SVLD model is proposed. Although compared with the popular ID model, there are slightly bigger modeling workloads and more complex state space without greatly improving the model prediction accuracy, the SVLD model is successful in controlling the hydraulic fluctuation of the upstream offtake section. It is hard for the ID model or similar simplified models. The multi-point hydraulic control method is suggested by designing the advanced controllers with the SVLD model and imposing different constraints on different controlled points. In consequence, much better water level control at the offtake section and a more dependable water supply during the key growth stage of crops could be guaranteed.

A simple case is modeled and controlled in this paper for highlighting the superiority of the proposed control method. However, real-world OCIS are much more complex. For guidance, the general operating rules are set out for practical applications. Firstly, each canal pool needs to be simplified and modeled using the control-oriented model. The proposed SVLD model is suggested if there are many controlled points or the offtakes are located somewhere upstream. The canal pool is divided into several segments according to the locations of the controlled points or offtakes. Accordingly, Equation (5) can be applied and the state-space model of the OCIS can be preliminarily structured for advanced controller design. Then the most important thing is clarifying the control objectives of different controlled points. If the controlled point works for the final water users, like the offtake section in this study, the hard constraint is recommended for the stable water level and reliable water intake. On the other hand, if the controlled point works for other purposes, like the connection with lateral canals or downstream water level security, it is suggested to use the soft constraint to better allocate the limited control resources. Ultimately, the advanced controller is designed for the OCIS.

From the simulation results, it can be seen that only when the control objective difference between controlled points is fully considered, can the controller function be brought into full play, and can the best control effect be achieved. In the test case of this paper, there is presumed to be only one offtake for the simplicity of control modeling. The offtake section is naturally the control focus, which is directly related to the success of irrigation water distribution. But in addition, the downstream end of the canal pool should also be taken into consideration for operation safety. The control requirements of these two controlled points are different. The former needs to keep the water level at the target value as much as possible, while the latter only needs to be within the target band without being subject to tough restrictions. By this control method, the advanced controller can find operation instructions that are more credible for farmers to draw water. If the same control goals are set, the irrigation performance would decrease due to the abuse of control resources, and the time for optimization solutions may be longer because of the tighter constraints. Furthermore, the water use units behind the offtake gates are also an important consideration. It means that the direct water supply to the farmers for irrigation, eco-environmental water use, or water delivery to lateral canals may be managed with different control attitudes. To maximize water delivery benefits, the economic values of the irrigated crops and water applications are also significant reference factors for setting control objectives, which can be studied in the future.

According to Figure 7, the proposed SVLD model could adequately capture the hydraulic dynamics of the open canal, not worse than the popular ID model. The maximum segment length in the presented test case is 7,000 m when the offtake is located at the downstream end. In this condition, the overall prediction error of the SVLD model for the downstream water level fluctuation is nearly half as much as that of the ID model. Kong *et al.* (2019) realized the real-time automatic control of the Middle Route Project of the South-to-North Water Transfer Project by the ID model-based MPC controller. In his study, the length of canal pools ranged from 9.2 to 26.6 km, which is much longer than 7,000 m. Therefore, the confidence in applying the SVLD model to the OCIS controller design is greatly enhanced. When higher model prediction accuracy is needed, setting up state observers (Bonet *et al.* 2016; Rodriguez *et al.* 2020) or more sensor monitoring points is feasible to increase the number of divided segments. Nonetheless, a higher computing burden or greater control cost is the concomitant problem. It is adverse for real-time online control, especially for large-scale water systems or long prediction horizons. It is noteworthy that only offtake demands and the open canal with prismatic geometry are concerned in the study. However, the practical engineering characteristics, like the abrupt changes of cross-sections, crossing structures, and others, may affect the prediction accuracy and control performance of the proposed SVLD model. Accounting for the time-varying head loss in the SVLD model or incorporating it with other control-oriented models may be the solution, but this is beyond the scope of this paper.

Keeping the water level stable at the offtake sections is indispensable for precision irrigation. The suggested multi-point hydraulic control method is promising in improving the transmission efficiency of irrigation water and ensuring the sustainable development of agricultural water resources. Moreover, the SVLD model is also a user-friendly model that can help managers make more accurate decisions for manual operations in developing countries and backward areas. For these regions, there is still a long way to irrigation canal automation. Manual operations may still be the main method for water delivery management for quite a long time. However, the numerical simulation and prediction of the open canal hydrodynamics before the manual operations can help the managers to make more accurate decisions, much better than relying on experience alone. Compared with the hydraulic simulation packages governed by the SV equations, the developed SVLD model is simple enough for irrigation district managers to use, and it is more powerful than the popular ID model. Any action toward reducing the operational team burdens and upgrading the accuracy of the water delivery process would be a long-lasting alternative in rehabilitating the operational activities in the irrigation districts. Last but not least, the proposed multi-point hydraulic control method creatively studies the water level controllability before the offtake gates along the open canal. It is an important basis for further research about the automatic control of complex branching canal networks.

## CONCLUSIONS

For the actuality that the offtakes are widely located along the OCIS, a multi-point hydraulic control method for advanced controller design is proposed in the paper. Two innovation points could be paid attention to (i) a simplified SV model, i.e. the SVLD model, is formulated by linearizing and integrating the SV equations and then dividing the canal pool only according to the locations of the offtakes. The model prediction results are evaluated by comparing them with the popular ID model. (ii) The differences in control objectives between controlled points are concerned. More precisely, when there is no offtake at the most downstream end of the canal pool, the soft constraint is suggested for this controlled point. It applies a target band based on the existing freeboard for operation safety, rather than the usual target values. The test canal 2 proposed by the ASCE is taken as the controlled object, and then the MPC algorithm is applied. The following conclusions can be drawn:

- 1.
The proposed SVLD model has good prediction accuracy for the hydraulic responses of the downstream end and upstream offtake sections. Taking the test case as an example, without adding monitoring points or state observers, the canal pool is divided into segments only by the offtake location, and the prediction error is less than 0.02 m. The prediction performance is comparable to the popular ID model, showing that the proposed SVLD model is qualified for advanced controller design.

- 2.
The ID model cannot actively dominate the hydraulic fluctuation of upstream offtake sections. Under the conventional condition, the soft constraint on the downstream end helps the water level control before upstream offtake gates, which is benefited from the water level increase in the downstream backwater area. However, the opposite result is obtained in the water-deficient condition.

- 3.
The proposed multi-point hydraulic control method for the OCIS could purposefully control the hydraulic responses of each controlled point along the open canal. The soft constraint on the downstream end contributes to the rational allocation of control resources and results in more outstanding control performance. Taking the ID model-based MPC with hard constraint as the comparison object, which is a common control alternative, the water level control stability before the offtake gate and the water supply reliability are increased by 91 and 69.5% under the conventional condition and by 54.9 and 27.1% under the water-deficient condition.

## ACKNOWLEDGEMENTS

This work was financially supported by the National Natural Science Foundation of China (Nos.51979202 and Nos. 51009108). We are grateful to the editors and the anonymous reviewers.

## DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

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