# Exploring Explicit Delay Time for Volume Compensation in Feedforward Control of Canal Systems

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## Abstract

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^{3}/s, and the upstream flow of the canal pool changes from 6 m

^{3}/s to 7.2 m

^{3}/s, among the three existing algorithms, the volume step compensation algorithm has better performance in terms of time to achieve stability, i.e., 1.25 h. The actual adjusted storage accounts for 99.6% of the target adjusted storage, which can basically meet the requirement of compensated storage of the canal pool. The delay time explicit algorithm only needs 1.47 h to stabilize the regulation system. The fluctuation of water level and discharge in the regulation process is small. The actual adjusted storage accounts for 99.6% of the target adjusted storage, which can basically meet the requirement of compensated storage for the canal pool. The delay time calculated by explicit algorithm can provide references for the determination of delay time in feedforward control.

## 1. Introduction

^{11}m

^{3}, accounting for 61% of the total water consumption for industry, agriculture, life and ecology [1]. However, the effective utilization coefficient of irrigation water is only 0.5 [2], and the potential of increasing agricultural production and saving water is still huge. One of the main ways to improve the water utilization efficiency of water conveyance is the automatic operation of the channel system. In the channel control algorithm, feedforward control algorithms adjust the input of the system in advance by predicting the possible state deviation in the future operation of the control system. This open-loop algorithm does not need to compare with the actual monitoring data of the channel operation. In order to solve the feedforwardcontrol problem of canals, gate stroking was proposed in 1969 [3]. According to the offtake discharges’ schedules, this method could determine the discharge variations of check structures by inverse solutions of unsteady open-channel flow equations. However, it was difficult to get a reasonable solution sometimes because it needs some extreme or unrealistic inflow variation, or no solution at all when calculating [4]. Nowadays, the open-loop algorithm of canals mainly considers the difference of stable storage in different flow states, and actively compensate the difference of storage through the opening and closing lag between gates [5]. Compared with gate stroking, this algorithm has an advantage in calculation, i.e., there is basically no extreme or unrealistic solution in the solution process. However, if the opening and closing lag time between gates is too small and the surge wave has not arrived when the intake gate is opened, the water supply will be insufficient, otherwise the water will be wasted and the excess water will be discharged through the downstream canal pool. Combined with the actual project, Wei simulated the influence of different feedforward control time on the water level fluctuation at the intake, and proposed a feedforward control time calculation method to effectively reduce the water level fluctuation at the intake [6]. In order to solve the nonlinear optimization problem with constraints on the gate movements in feedforward control, the sequential quadratic problem (SQP) method is used [7]. In order to shorten the time necessary to stabilize the new flow rate at the buffer reservoir in a traditional automated upstream controlled canal, the method is proposed which requires calculated, remote manual adjustments to all the canal check structure gate positions in addition to two flow rate changes made at the head of the canal, followed by a return to automated upstream control [8].

## 2. Delay Time Algorithm

#### 2.1. Volume Step Compensation Algorithm

_{d}at t

_{d}. From the assumed initial state, it can be deduced that the storage of the canal pool needs to be adjusted by ΔV, i.e., the initial flow Q

_{0}is adjusted Q

_{0}+ΔQ

_{s}at t

_{s}. At t

_{d}, the storage of the canal pool can be adjusted completely. The meaning of the volume step compensation is to adjust the discharge of the canal pool once in advance. When the intake begins to take water, the adjusted storage can satisfy the storage required by the current intake of the canal pool, and the flow need not be adjusted again. At this time, ΔQ

_{s}is equal to Δq

_{d}. The delay time Δτ is the difference between t

_{s}and t

_{d}. Its calculation is shown in Equation (1).

#### 2.2. Dynamic Wave Principle Algorithm

_{DW}is the delay time of dynamic wave principle calculation; L is the length of canal pool, (m); v

_{0}is the initial average velocity of canal, (m/s); c

_{0}is the initial average velocity, (m/s).

_{0}of the canal pool is adjusted to Q(t

_{s}) at t

_{s}, and from ΔQ

_{d}to the target value Q(t

_{d}) at t

_{d}. The purpose of secondary regulation is to avoid the waste of water when the intake opens and the inflow of the canal pool exceeds the flow required by the intake.

#### 2.3. Water Balance Model Algorithm

^{3}/s); τ(x) is the delay time at canal pool x, (s); K(x) is the time parameter at canal pool x, (s). In order to determine the time of water intake, the process of water intake and supply is simplified to a linear equation and superimposed [10], as shown in Equations (4)–(6).

_{w},

_{w},

_{p}is the time parameter introduced by the change of downstream intake flow; q

_{d}

^{(t)}is the change of downstream flow at t time, (m

^{3}/s); δQ

_{u}is the upstream water supply, (m

^{3}/s); k

_{d}is the flow coefficient, indicating the sensitivity of time parameter K to the influence of downstream flow and water level boundary, (m

^{2}/s); q

_{w,0}is the downstream intake flow, (m

^{3}/s); T

_{w}is the optimal intake time; a is the sudden drop of water level caused by intake, (s/m

^{2}), τ is the delay time of parameter identification of water balance model. In order to minimize the downstream discarded water, the upstream water supply quantity δQ

_{u}is equal to the downstream water intake quantity q

_{w}

_{,0}. At this time, the canal pool does not produce discarded water, and the calculation formula of the optimal water intake time is demonstrated in Equations (7) and (8). In Equation (7), t

_{w}is the difference between the optimal water intake time and the delay time of the canal pool. In the actual simulation process, K, τ, K

_{p}, a and k

_{d}parameters can be obtained by the parameter identification method.

## 3. Delay Time Explicit Algorithm

^{3}/s); K′ is the flow modulus, (m

^{3}/s); v is the average velocity of the canal, (m/s); C is the Chezy coefficient; R is the hydraulic radius, (m); n is the roughness; A is the area of the control point, (m

^{2}). When the bottom slope of the canal pool is gentle and the water demand changes a little, the difference of A, R and B upstream before and after the volume compensation of the canal pool is small and the average values of the two points are calculated.

_{d}and J

_{s}before and after the volume compensation. The estimation of the required compensation value ΔV of the canal pool is shown in Equations (11)–(13).

^{2}; $\overline{B}$ is the average water surface width upstream of the canal pool before and after the volume compensation, m; $\overline{R}$ is the average hydraulic radius upstream of the canal pool before and after volume compensation, m. According to Equation (1), the relationship between the delay time Δτ and the flow of the change of water demand Δq

_{d}can be attained. However, in practical engineering, due to the influence of length L, initial discharge Q

_{0}and different water demand variation, the delay time calculated by $\overline{A}$, $\overline{B}$, $\overline{R}$ is smaller than actual delay time. That is to say, there exists amplification coefficient α

_{am}to correct the calculation results. In Equation (13), when the initial and final discharge of the canal pool is determined, the main factors affecting the required volume are the geometric parameters of the canal pool, including the length, width of the canal bottom and so on. Combining with the geometric parameters of the canal pool, this paper introduce an empirical formula of α

_{am}on the length and width of the canal bottom to correct the delay time (Equation (14)). The relationship between the delay time Δτ and the flow Δq

_{d}of the change of water demand is shown in Equation (15). The coefficients a and c are calculated in Equation (16).

## 4. Simulation Settings

^{3}/s. As shown in Figure 4, in the second hour, the intake begins to take 1.2 m

^{3}/s, and the upstream flow of the canal pool changes from 6 m

^{3}/s to 7.2 m

^{3}/s (Figure 4).

_{am}of the canal pool is 1.75.

## 5. Results and Discussion

^{3}/s, and the IAQ value is small. However, the dynamic wave principle algorithm uses ideal delay time to compensate the storage by increasing the inflow of the canal pool. At this time, the maximum overshot flow exceeds the target flow by 0.644 m

^{3}/s, which is reflected in the IAQ value. The IAQ value of the dynamic wave algorithm is about 43.67 times that of the volume step compensation algorithm, and the flow regulation fluctuates significantly.

^{3}/s for compensation of the storage. After the water intake is opened, in order to avoid the waste of water, the flow of the canal pool is adjusted to the target flow for the second time. According to the compensated storage volume and the optimal intake time of the canal pool, the water balance model can calculate the flow value of the canal pool adjusted in advance. Affected by parameter identification, the optimal intake time of the calculation is slightly large. In order to meet the demand of the canal pool storage, the canal pool can adopt a small flow to compensate the storage capacity, i.e., the flow of the canal pool adjusted in advance is 6.843 m

^{3}/s, which is less than the target flow (7.2 m

^{3}/s). Then the flow of the canal pool increases to the target state, so as to avoid the phenomenon of insufficient water supply after opening the intake.

^{3}. The difference between the actual storage and the target storage of four algorithms are shown in Figure 7. Compared with the volume step compensation algorithm and delay time explicit algorithm, the actual compensation storage of the other two algorithms exceeds the target storage. The actual storage of the dynamic wave algorithm after water intake exceeds the target storage 121.79 m

^{3}, the actual storage of the water balance model algorithm after water intake exceeds the target storage 114.64 m

^{3}. In order to maintain the stability of the control system, the excess storage should be drained from the downstream. The dynamic wave principle algorithm is based on the velocity and velocity of the flow in the initial state of the canal pool, without considering the energy attenuation in the process of water flow propagation. The delay time calculated by this algorithm is small. In order to fully compensate the storage required by the canal pool, the flow of the storage compensation will be larger than the target flow. When the water intake is started, the flow of the canal pool will be adjusted to be equal to the target flow, so as to avoid the inflow of the canal pool being larger than the flow required by the canal pool, resulting in the waste of water resources. However, in the process of regulating the canal pool, the discharge has been adjusted twice, which results in the fluctuation of the water level and discharge of the canal pool, and the operation needed to stabilize the canal pool is more, which leads to a greater waste of water. Based on the principle of water balance in the canal pool, the water balance algorithm assumes that the canal pool does not produce discarded water, and derives the equation for calculating the optimal intake time. In practice, the optimal water intake time is obtained by identifying the K, τ, K

_{p}, a and k

_{d}parameters of the canal pool under water intake and supply processes. The identification accuracy of the five parameters will affect the optimal intake time. In the current example, the optimal intake time obtained is slightly large, and the actual adjusted storage exceeds the target.

^{3}less than the target storage, and the actual storage of the delay time explicit algorithm is 9.82 m

^{3}less than the target storage. The volume step compensation algorithm calculates the delay time by assuming the steady state at the beginning and the end. In order to calculate the storage of the canal pool, it is necessary to discretize the canal pool in space and obtain the water surface profile of the constant non-uniform flow. In order to ensure the continuity of the calculation of the water surface line, the water surface line will be smoothed at discrete points. At the same time, the algorithm ignores the intermediate state of the actual adjustment, so the actual storage adjustment value of the algorithm is slightly smaller than the target storage. The delay time explicit algorithm assumes that the change of the A, B and R of the canal pool before and after intake is small. Therefore, the delay time obtained by using the average value $\overline{A}$, $\overline{B}$, $\overline{R}$ is smaller than actual delay time and needs to be corrected by amplification coefficient. However, only considering the influence of the canal pool length and bottom width, this paper calculates the amplification factor by using a simple empirical formula (Equation (14)). The actual compensation storage is slightly smaller than the target storage. In practice, the measured data (flow or water level) and more geometric parameters of the canal pool should be taken into account to correct this empirical formula. The percentage of the actual compensation storage of the four algorithms to the target compensation storage is shown in Table 4.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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K/min | τ/min | K_{p}/min | a/(s∙m^{−2}) | k_{d}/(m^{2}∙s^{−1}) | T_{w}/min |
---|---|---|---|---|---|

67.6 | 29 | 63.8 | 0.026 | 15.9 | 51.46 |

Algorithm Name | Volume Step Compensation | Dynamic Wave Principle | Water Balance Model | Delay Time Explicit |
---|---|---|---|---|

Delay time/min | 39 | 24 | 51.46 | 37 |

Algorithm Name | Transition Time/h | Maximum Overshot Flow/(m^{3}∙s^{−1}) | IAE/% | IAQ/(m^{3}∙s^{−1}) |
---|---|---|---|---|

Volume step compensation | 1.25 | 7.201 | 3.65 × 10^{−6} | 0.030 |

Dynamic wave principle | 2.55 | 7.844 | 1.49 × 10^{−5} | 1.313 |

Water balance model | 1.77 | 7.201 | 1.53 × 10^{−5} | 0.033 |

Delay time explicit | 1.47 | 7.201 | 3.62 × 10^{−6} | 0.097 |

Algorithm Name | Volume Step Compensation | Dynamic Wave Principle | Water Balance Model | Delay Time Explicit |
---|---|---|---|---|

Percentage/% | 99.6 | 104.6 | 104.4 | 99.6 |

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**MDPI and ACS Style**

Liao, W.; Guan, G.; Tian, X.
Exploring Explicit Delay Time for Volume Compensation in Feedforward Control of Canal Systems. *Water* **2019**, *11*, 1080.
https://doi.org/10.3390/w11051080

**AMA Style**

Liao W, Guan G, Tian X.
Exploring Explicit Delay Time for Volume Compensation in Feedforward Control of Canal Systems. *Water*. 2019; 11(5):1080.
https://doi.org/10.3390/w11051080

**Chicago/Turabian Style**

Liao, Wenjun, Guanghua Guan, and Xin Tian.
2019. "Exploring Explicit Delay Time for Volume Compensation in Feedforward Control of Canal Systems" *Water* 11, no. 5: 1080.
https://doi.org/10.3390/w11051080