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Vestnik ^AUNC. Fiz.-Mat. nauki. 2023. vol. 44. no. 3. P. 121-129. ISSN 2079-6641
MATHEMATICAL MODELING
" https://doi.org/10.26117/2079-6641-2023-44-3-121-129 Research Article Full text in English MSC 49Q22
Algorithms for Clarification of the Operating Modes of Channel Sections in the Management of Water Resources in Channel
Irrigation Systems
M. N. Esonturdiyev*
Chirchik state pedagogical university, 111700, Tashkent region, Chirchik city, A. Temur street, 104, Uzbekistan
Abstract. The optimal operating modes are determined for channel sections based on the condition that all lateral water intakes are guaranteed to receive the planned flow rates of water resources with minimal filtration and evaporation water losses. Lateral outlets are guaranteed to receive water flows if they have appropriate heads of water in front of the facility. These necessary heads determine the values of water levels in the channel sections, which are determined in the process of water distribution. In the process of operational management of water resources in mechanical water-lifting systems, established water distribution limits are implemented, taking into account the current actual situation with the availability of water resources and the technical characteristics of pumping stations and hydrotechnical structures of the irrigation system, as well as the technical characteristics of hydrotechnical structures and canal sections of the entire irrigation system. Currently, optimal management of water resources is carried out with the help of a dispatch service, and the operating modes of canal sections are determined by dispatchers. Therefore, significant deviations of actual regimes from planned values, unevenness and instability of water supply to consumers are constantly observed. Based on the above, the article developed mathematical models of the unsteady flow of water resources in the section of the main canal and seasonal reservoir based on the nonlinear differential equations of Saint-Venant. An algorithmic sequence for determining the operating modes of a hydraulic structure is also given, with the help of which it is possible to manage the water resources of the main canal, which will satisfy the needs of water users with minimal losses of water and energy resources.
Key words: optimal control, information systems, numerical methods, channel, necessary conditions for the optimality of water resources, discrete water supply.
Received: 18.07.2023; Revised: 17.09.2023; Accepted: 30.10.2023; First online: 02.11.2023
For citation. Esonturdiyev M.N. Algorithms for clarification of the operating modes of channel sections in the management of water resources in channel irrigation systems. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 44: 3,121-129. EDN: XWOLYU. https://doi.org/10.26117/2079-6641-2023-44-3-121-129. Funding. Not applicable.
Competing interests. There are no conflicts of interest regarding authorship and publication.
Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for submitting the final version of the article to the press.
* Correspondence: A E-mail: esonturdiyev80@mail.ru ^
The content is published under the terms of the Creative Commons Attribution 4.0 International License © Esonturdiyev M.N., 2023
© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)
Вестник КРАУНЦ. Физ.-мат. науки. 2023. Т. 44. №3. C. 121-129. ISSN 2079-6641
МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ " https://doi.org/10.26117/2079-6641-2023-44-3-121-129 Научная статья
Полный текст на английском языке УДК 517.977.5
Алгоритмы уточнения режимов работы участков каналов при управлении водными ресурсами в канальных ирригационных
системах
М. Н. Эсонтурдиев*
Чирчикский государственный педагогический университет, 111700, г. Чирчик, ул. А. Темура, 104, Республика Узбекистан.
Аннотация. Определены оптимальные режимы работы участков канала осуществляется из условия, что все боковые водозаборы гарантированно получают плановые расходы водных ресурсов при минимальных потерях воды на фильтрацию и испарение. Боковые отводы гарантированно получают расходы воды в том случае, если у них имеются соответствующие напоры воды перед сооружением. Эти необходимые напоры определяют значения уровней воды на участках канала, которые определяются в процессе водораспределения. В процессе оперативного управления водными ресурсами в механических водоподъемных системах реализуются установленные лимиты водораспределения с учетом сложившейся фактической ситуации с обеспеченностью водными ресурсами и технических характеристик насосных станциях и гидротехнических сооружений оросительной системы, а также технических характеристик гидротехнических сооружений и участков каналов целое оросительной системы. В настоящее время оптимальное управление водных ресурсов осуществляется с помощью диспетчерской службы, а режимы работы участков канала определяются диспетчеров. Поэтому постоянно наблюдаются значительные отклонения фактических режимов от плановых значений, неравномерность и нестабильность подачи воды потребителям. Исходя из выше сказанного, в статье разработаны математические модели неустановившегося течения водны ресурсов участке магистральном канале и водохранилище сезонного регулирования на основе в нелинейных дифференциальных уравнений Сен-Венана. Также приведена алгоритмическая последовательность определения режимов работы гидротехнического сооружения, с помощью которой можно управлять водными ресурсами магистрального канала, что позволит удовлетворить потребности водопользователей с минимальными потерями водных и энергетических ресурсов.
Ключевые слова: оптимальное управление, информационные системы, численные методы, канал, необходимые условия оптимальности водных ресурсов, дискретная водоподача.
Получение: 18.07.2023; Исправление: 17.09.2023; Принятие: 30.10.2023; Публикация онлайн: 02.11.2023
Для цитирования. Esonturdiyev M. N. Algorithms for clarification of the operating modes of channel sections in the management of water resources in channel irrigation systems // Вестник КРАУНЦ. Физ.-мат. науки. 2023. Т. 44. №3. C. 121-129. EDN: XWOLYU. https://doi.org/10.26117/2079-6641-2023-44-3-121-129. Финансирование. Исследование выполнялось без финансовой поддержки фондов. Конкурирующие интересы. Конфликтов интересов в отношении авторства и публикации нет. Авторский вклад и ответственность. Автор участвовал в написании статьи и полностью несет ответственность за предоставление окончательной версии статьи в печать.
* Корреспонденция: А E-mail: esonturdiyev80@mail.ru ф
Контент публикуется на условиях Creative Commons Attribution 4.0 International License © Esonturdiyev M. N., 2023
© ИКИР ДВО РАН, 2023 (оригинал-макет, дизайн, составление)
Introduction
Let us consider the statement of the problem of calculating the operating modes of the channel sections for water distribution control.
The calculated planned (limited) costs at the beginning of sections, water outlets and the end of the channel must be implemented at each section of the channel.
The operating modes of the channel sections are determined on the basis of the given water flow rates of the lateral outlets and the water level in the end cross sections of the channel sections, i.e., water levels of the up-stream side of the control structures, and these regimes are considered constant for ten days.
Lateral outflows and inflows can be concentrated or distributed. As concentrated inflows and outflows, lateral water outlets or concentrated inflows are considered, and as distributed outflows, losses due to filtration and evaporation are considered.
At present, there are various methods for calculating the surface curve of the nonuniform motion of the water flow, based on the integration of the differential equation for the non-uniform motion of water in free channels without lateral outflows and inflows.
These techniques are based on the use of graphical dependencies or table functions and are not suitable for use in modern computers.
This article presents a numerical algorithm for calculating the surface curve of nonuniform water flow in free channels with lateral outflows and inflows, based on the integration of the differential equation for non-uniform water motion using the finite difference method and the quasi-linearization method for approximation of nonlinear dependencies [1].
Theoretical Computations
Lateral inflows and outflows are defined as follows [1]
N
q(x,H) = qf(x,h) + qt(x,H) + ^ qn(HJ6(x - aj, (1)
n=1
where qf(x,H),qi(x,H)are the intensities of filtration and evaporation losses, qn(Ha)is the water flow rate of n-lateral outlet, 6(x — an) is the delta function characterizing the location of the outlet of water consumers along the length of the channel, an is the distances to n-lateral outlet [3].
The flow rate and water level at the end of the channel sections are given as the initial conditions.
Q(l) = Qk, H(l) = Hk, I = vn, k = vn, (2)
where Qk is the water flow in the k-hydrotechnical structure, Hk is the water level in the k- hydrotechnical structure. In the channel cross section, where the lateral water outlets are located, the corresponding restrictions on water levels are set, which provide the given flow rates as follows [2]
h(an) > H*an, n = 1,...,N, (3)
where h* an is the level value required to supply the water flow to the water outlet.
The task of determining the operating modes of the channel section in the presence of backwater from the lower control structure is reduced to determining that value of the water level at the end of the hkchannel section, which would minimize filtration and evaporation losses in the channel section. At the same time, the water levels in the channel sections, where the lateral outlets are located, satisfy the restrictions (1) on the water head in front of the outlet structure and lateral outlets.
To solve the formulated problem, the main point is the calculation of the free water surface in the channel section with lateral water intakes.
Given that the P(x,h)and w(x, h) functions are the functions of the x and h variables, the second equation can be written as follows [2]
9P 9Pdh 2Qw dQ - Q2dw (dz0 Q |Q|\
dX + dhdh + Q wV + +F' (4)
Having made simple algebraic transformations and taking into account
dw 0 Q
Ft + a7 = q' (5)
i^Q + A (p+?) = -w (izo + SM) + F, (6)
91 9 x V w ) \dx K2 J w
^(x, h) = JhB(x, a)da; P(x,h) = ^^(H — a)B(x, a)da;
F(x,h) = gJ0H(h — a) ^ da;
(7)
where Q = Q(x,t) is the water discharge; h = h(x,t) is water depth; B = B(x,h) is the width of the bed or channel at the h depth; w = w(x, h) is the cross-sectional area at the h depth; P = P(x, h) is the force of hydrostatic water pressure; F = F(x, h) is the wall reaction force caused by the non-prismatic nature of the channel or bed; K = K(x,h) = wCR1/2 is the flow rate module (set according to empirical formulas); z0 = z0(x) is the bottom level; q = q(x, 1, h) is the lateral inflow per unit length; g is the acceleration of gravity, C is the Chezy factor, R is the hydraulic radius. Further on, the following values should be taken into account: z = z0 + h is the free water surface level; v = Q/w is water velocity.
we obtain the following equation [3]
dP dPdh 2Qq Q2 (9w dw dh\ _ (dz0 Q|Q|\
9X + ^dX + ^ + w^Ux + 9hd^J = -gw Ux + ^T) + F' (8)
Having made simple transformations, we finally obtain
9P + Q29w\ dh = -gw (dzc + QIQI\ + F - 9P - 2Qq - Qjaw (g) 9h + w2 9hy dx dx + K2 J + F 9x w w2 9x ' (9)
Let us assume that the channel bed, flow rate Q, water depth hn, for instance, at the channel end in (N - N) cross sectionand the hyd raulic parameters of the section are given. Let us divide the channel section, which has the length L into separate sections of relatively small length equal to lm. In this case, we consider each selected channel
section with a length lm separately, going upstream: first, let us calculate section I, then section II, etc. The calculation of each section consists in determining the depth hm and the flow rate Qm at the beginning of this section, using known values lm and Hm+i.
Applying finite difference methods for equations and , we obtain the following difference equations [3], [5]
dP i q2 M)
9h + w2 ah ;m+1
Qm+1 Qm
T-m+1 — hm _
In
= —gwm+1
dzo + Q1QI , +
dx + K ' m+1 +
= qT
F — 9P — 2Qq — q2 3w\
9x w w2 9x J m+i
(10) (11)
Here (.)m+i means that the corresponding expression is calculated from the known values Qm+i and Hm+i and corresponds to small sections with number m + 1.
The calculation is carried out from the end section of the channel to the beginning, i.e., the unknown values are Qm and hm, which are calculated by formulas (12) and (1), i.e., the flow rate Qm and depth hm are determined recursively at the boundary sections (N - 1), (N -2), ..., (2), (1) [6].
Qm = Qm+1 + qmlm
hm = h
m+1
+
9w i dzo + qjQA
9Wm+1 ( dx + 172 I
_ + ( F — dP — 2Qq — QidwA \
K2 ) m+1 V 9x w w2 9Vm+1
V
9P + Q2 9w) 9h + w2 m+1
(12)
lm, (13)
/
where (.)mand (.)m+i are the parameters for the section m and m + 1 lm is the length increment.
For the prismatic bed of the channel without lateral inflow, the equation has the following form [7]
/ ,
( (\ — qiQA \
hm = h
m+1
+
m+1
V
1 — Q^A,
9 wV m+1/
lm.
(14)
The main empirical variable in dependences (13) and (14) is the flow rate modulus of the channel section. In numerical calculations, the approximate formula [8] is used to calculate the free surface of the water flow:
K = 2 (K(xm+1 > hm+1 ) + K(xm+1 > hm+1 + lmK(xm+1, hm+1 ))) .
(15)
We performed computer calculations using expressions (13), (14) in the form of programming modules for calculating the surface curve of the water flow.
Having an algorithm for determining modes with different values of the water level at the end of the channel section with known values of the water flow rates at the end and side water consumers, we calculate the surface curves of the water flow for the corresponding level values. Further, according to the surface curve, the conditions for fulfilling the restriction on the pressure in front of the water outlets are checked and such a value of the water level at the channel end is selected, at which all restrictions for discharge heads would be met and the values of the total evaporation and filtration loss in the channel section would be minimal [9].
l
m
Research Results
We have determined the operating modes of the first section of the channel. Table 1 shows the hydraulic parameters of the channel section, in which the calculations of the operating modes were performed.
Table 1
Hydraulic parameters of the channel section
b m n y i g 1 Kf
25 3 0.02 0.2 0.00007 9.8 30000 1
Table 2 shows the results of calculation of the operating modes of the channel section, which shows the main calculated operating modes of the channel for three values of the water level at the channel end.
Table 2
Calculation of operating modes of the channel section
Mode 1 Mode 2 Mode 3
Distance to outlet Name of outlet Water discharge Water level Water discharge Water level Water discharge Water level
m3/s m m3/s m m3/s m
0 74.52 2.67 74.52 2.37 74.58 2.82
1200 1 72.03 2.70 74.41 2.38 72.09 2.85
1800 2 68.80 2.71 71.93 2.40 68.85 2.86
2700 3 65.66 2.74 68.69 2.41 65.71 2.89
5100 4 65.33 2.82 65.56 2.43 65.38 2.98
6900 5 63.03 2.88 65.23 2.51 65.08 3.04
9900 6 61.35 3.00 62.94 2.57 61.39 3.17
12000 7 60.60 3.09 61.27 2.67 60.63 3.26
14100 8 59.16 3.18 60.52 2.75 59.19 3.36
16200 9 58.40 3.28 59.09 2.83 58.43 3.46
18600 10 57.04 3.39 58.35 2.92 57.06 3.57
21000 11 55.47 3.51 56.99 3.03 55.48 3.70
26400 12 51.67 3.80 51.66 3.40 51.68 3.99
28800 13 50.09 3.93 50.09 3.53 50.09 4.13
30000 50.00 4.00 50.00 3.60 50.00 4.20
Fig. 1. Free surface curves at different level regimes in the channel section
Figure 1 shows the free water surface curves in the channel section for mode 1 and mode 2. It is evident from Fig. 1 that the free water surface curve of mode 2 does not cover all the restrictions (3) for the side water consumers of the section, i.e., four water intakes in this section cannot receive their limited waters in this regime. The water level in front of this outlet is less than their allowable values and they are shown in the table in dark colour.
Fig. 2. Calculation of volumes and losses of water in the channel section
Figure 2 shows the calculation of the volumes and losses of water in the channel section at various level operating modes of the channel and shows the area that does not satisfy the conditions for limiting the water levels of the lateral water outlets.
In modes 1 and 3, the free water curve data table covers all allowable water levels, i.e., satisfies the constraints (3), but in mode 3 the losses are greater than in mode
1, so mode 1 is an acceptable mode of operation. The obtained channel level modes are supported by automatic water level control systems and centralized control and management [4].
The developed algorithm makes it possible to calculate the surface curve of the water flow on non-prismatic channels and has very good convergence, since it is based on the significance of the flow movement. The calculation accuracy depends on the length increment lm. On the basis of a numerical experiment using the program for calculating the free surface of the flow according to the above formulas at various increments lm and, by examining the accuracy of the results obtained at different steps of calculation, for the channel section considered on the example with the step length lm = 300 m, the accuracy of the results was obtained, which satisfies the practice of calculating hydraulic dependencies.
Based on the solution of the problem of determining the operating modes of the channel sections in the operational planning of water distribution on the main channel, the following sequence is calculated for all sections of the main channel
^M =1 [m, Qmn, Qírm, Qwmn, Qfmn, Qf\Vmn, qOCmn, h!un, Wiuu, Qímn] ,
1 - (16)
Vm e M, Vn e Nw j.
here Qmn, Q^ix is the water consumption at the beginning and end of the section implementing the requests of consumers of the section, QWmn, QRmn the total flow rate under requests for water intakes and inflows, hmn , Wmn and QRmn is the water level set at the end of the channel section, the volume of water in the channel section and losses of water resources in the channel section, QRWmn>qOCmn are the total flow rates under requests for irrigation and other consumers in the section m for ten days n of the growing season.
Conclusion
The developed algorithm for calculating water flow rates for sections of the main canal makes it possible to determine the operating modes of key structures and water intakes based on the conditions for ensuring water intake limits, while organizational losses of water resources will be reduced and the uniform distribution of water resources across all water intakes will be ensured.
References
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Information about the author
Esonturdiyev Mamatkobil Nurmamatovichfa - Senior teacher of Chircik State Pedagogical University, Chirchik, Uzbekistan, ORCID 0000-0002-3925-5129.
Информация об авторе
Эсонтурдиев Маматцобил Нурмаматович Ä - Старший преподаватель Чирчикского государственного педагогического университета, Чирчик, Узбекистан, © ORCID 0000-0002-3925-5129.
