CC BY
2MATHEMATICAL MODEL OF OPTIMAL WATER DISTRIBUTION IN HYDROELECTRIC POWER PLANTS
RIZOEV SAMARIDDIN SUNATULLOEVICH
senior teacher of informatics department of the Tajik National University. Republic of
Tajikistan, Dushanbe
Annotation. In work it is constructed and investigated the best models of system such as «water reservoirs from hydroelectric power station» and improvement of model is proved due to reasonable distribution of water between turbines of hydroelectric power station in water reservoirs. The equation of optimum distribution is received.
Key words: model, systems of a water reservoirs, distribution of water, computing experiments.
Let's consider the problem of the best distribution of water between turbines in reservoirs. Suppose a system of reservoirs with given hydraulic and technical characteristics is given, and it satisfies the conditions of the optimal control problem in the formulation proposed in the works (Fig 1.) [9-14]:
Figure 1 - Conceptual model of water flow in a hydroelectric power station
-Г = Q - U1> ~t = U1 - U2,-->—T = Um-1 - Um , t 40,tk J I1)
dt dt dt
i ' _
U = max Va.u„, u. e U, Van-s = 1, 0 < a, < 1,
1 ¿—l J j ' У ' ¿—l J ' J '
J j=1
J=1
(2)
V (0) = V0, V (tk )=V -
(3)
m
i (u )=Va1 г (u )- max (4)
1=1
In problem (1)-(4) the following notations are adopted [1-4]: Vi = Vi(t) - according to the value of the volume of the i-th reservoir, - i = 1,2,3,.. .,m; w = w (x, t) flow rate; Q(x, t) - the amount of water passing through the section co = co(x, t) - at time t at point x and enters the first reservoir, Uj
is taken from the set of permissible controls U and is piecewise continuous controls, meaning the amount of water flowing from the turbines of i the -th reservoir to the i +1 reservoirs per unit of time [3-5; 7].
Assertion. Equation (2) and equation
ri
n X 1 n
h Un
U-
Uij G U
(5)
j=i
Equivalent.
Equation (5) is the best model for the distribution of water from the turbines i of a reservoir to i +1 the reservoirs per unit time. Thus, to solve the optimal control problem (1)-(4), we must be able to solve equation (5). Equation (5) is an algebraic representation relating the amount of water passing through the turbines and the amount of water entering the next reservoir [1, 7, 9-14].
Proof of Statement: Necessity. Let's enter the notation Z = ui
Then we will get
Xj
uu , j = 1>k
Zw = £ X
j=1
(6)
Let it take place (6) prove justice (4)
Z
max
coeM
V j=1
Let (X1, X2Xk, Z) equation (6) be the solution, then introducing the notation
(7)
- 2 :
n-s
Z'
=
V J
From (6) we get the following system [1, 3-5]:
eoxXs + ... + wkXs - Zs = 0, X
a
s n—s
Z' = 0,
(8)
Since (Xj, X2,..., Xk, Z ) (6) is the solution, that is, (5), it is easy to see
к j=1
n n—s
IX
_
Zn
i.e. ^
Let us show that the determinant of system (8) is equal to zero, i.e.
j=1
Indeed, using the assumption of equality
n n—s
— 1 = 0.
k
s
1
n _
A2 = D]-s + D2n-s -1 and Aк = X
к n
С
1, к = 2,3
,... zero, on the basis of the
j=i
method of mathematical induction, we show that Ak+1 X
j=1
we divide the determinants A k+1 and we get [9-14]:
к+1 П
n-s
С
1 = 0.
A
к+1
- 1
- D
CD'
С d2
1 0
0 1
Ск-1 Ск
00 00
- D'
- D'
0 0...1 0
0 0...0 1
For this purpose,
=(-1)
r
к + 2
л
С
V
У
D
s
n - s
D'
D СУ2 ••
1 0 •• 0 1 .•
D-1 Ск
00 00
d'
- D
s
n-s
0 0...1 0 0 0...0 1
+
+ (- 1)2 "+2 A „ =(- 1Г+3( - s (- 1)"+(„ =
к-1 s
n к-1 n
1 0 ... 0
0 1 ... 0
0...0 1
к-1 n
+
+ XC -s - 1 = (-1)2к+С-s + Xc-s - 1 = XC -s - 1
j=1
j=1
j=1
n
s
s
s
s
1
s
s
s
s
к-1 n
This shows that both CO e M, i.e. CO e M, ZCj S 1- that is, it means
j=1
A k+i = 0.
In virtue of the 1st equation (6) we have:
zs =
г к ^
Z*jxJ
V J=
and C =
v zn У
n - s n
■ Therefore, since,
^ к Л к Xn
Z"jX' < ZCX i.e. zS •zn-S = ZX.
j=i V J=i у J=iz j=1
Г \
hence for any
с e
m, zm = Z x; ,
j=1
n ^-.0
C =
X;
z x;
V j=1 У
Consequently in equality(7) the maximum is
reached [3-5; 9-14]. Thus, Z =
L ' J ' co<=M
( k
V j=i J
Sufficiency: Let equation(7) take place. Let us prove that (6) is true. Let it
f k
Z = ju{w) X
V j=1
с e M.
be known what
y ---7
condition of the extremum, and then we get a system of equations of type s s
= o
Qçq is a necessary
XS - с
n—s
j
Л1 n-S s-.^n—S
Ck •C
• CO
n - s
j
• x; = o, j = 1, к
and hence
XI
xs.
or
Г.Л n-S r.^n-S
Ск • C
XI
x;
Summarizing the Last
z X,
equality on j from 1 to k, have, Ск
n-S
j=1
с
Xn
n-S _ _J
к
к
X
n then
к
Z X
j=1
- is the maximum point of the function u(fà), ^ M as (ua® ^ 0 )• Now let's calculate the value ju(o)0 ) • of Easy to see, this is
к
к
S
S
n
n
n
r
Z=Z®x X;=Z
j=i j=i j=i
k Í xn ^
ryn
VZ У
• x; =Z
j=i
XnXn-;
Z'
V
У
Z
j=1
X jn-;
Z
V
Z
¿—i 7n-; That is
j=1 Z
Z; • Zn-; =
Z XJ
j=1
k
Therefore, Z" = Z X>J ' ^ G M • from here
j=i
Z = ) =
Л
Z
j=1
/ л
k
ZX
x;
n
j
V j=1 У
V У V
j—; \
Z
=1
n;
xJ • xp
k
Zx
V =1 У
^ ^ Xn ^
Z
,=T Zn—; V =1 У
And
1 k
Z* = И* (™o ) = Zn—; ZXJ that is
j=1
Г k
Z = )= Z X
n
j
V j=1 У
Thus
У
Z = jU(^o ) =
л
Í
k
Z
j=1
л
XJ
Z XJ
V j=1 У
V
Xs
У
k
Zk X V j =1 У
The statement has been proven.
n—s
;n
n
k
k
n—s
n—;
n
n
n — s
n
REFERENCES
1. Комилиён, Ф. С. Амсиласозии математикии марх,илах,ои хдёти популятсияи оилаи занбури асал [Матн] / Ф.С. Комилиён, И.М. Саидзода // Паёми Донишгох,и давлатии Бохтар ба номи Носири Хусрав. - 2022. - № 2-1 (96). - С. 5-14. - EDN: ILDIZS.
2. Комилов, Ф. С. Трансформация математической модели в инструмент по прогнозированию динамики рыбной популяции экосистемы мелководного водохранилища / Ф. С. Комилов, И. М. Саидов, М. Р. Еров, И. Л. Косимов // Вестник Таджикского национального университета. Серия естественных наук. - 2017. - № 1-5. - С. 27-32.
3. Саидзода, И. М. Барномасозии компютерии фаъолияти занбури асал вобаста ба таъсири беморих,о ва зараррасонх,о [Матн] / И.М. Саидзода, Ф.С. Комилиён // Паёми донишгох,и давлатии Хоруг. - 2023. - № 1. - С. 45-56.
4. Саидзода, И. М. Хусусиятх,ои экосистемаи обанборх,о ва амсиласозии компютерии онх,о / И. М. Саидзода, М. Р. Еров. - Душанбе : Донишварон, 2021. - 118 с.
5. Саидов, И. М. Асосх,ои амсиласозии риёзй [Матн] / И.М. Саидов. - Душанбе: Мех,роч-граф, 2020. - 152 с.
6. Юнуси М. Модель возмущенного водохранилища и некоторые вопросы его эксплуатации. Докл. АИ РТ, том XLVI, №11-12, 2003, - С. 58-61.
7. ЮнусиМ. Об одной модели возмущенной системы водохранилищ. Вестник национального университета. Душанбе: Сино, 2003, с. 16-20.
8. Saidzoda, I. M. МаШетайса1 model of a bee colony by gender in a stationary case / I. M. Saidzoda // Scientific research of the SCO countries: synergy and integration : Proceedings of the International Conference, Beijing, 14 октября 2023 года. Vol. Part 1. - Beijing: Инфинити, 2023. - P. 143-148. - DOI 10.34660/INF.2023.24.14.431.
9. Yunus M.K. Optimization methods /M.K. Jonah, R.N. Одинаев. - Dushande. -2014. - 179 p.
10. Yunusi M. Movement of perturbed wave in water reservoir. The book abstracts of Second Inter. Seminar on the applied mathematics. Zanjon, Iran, October 3-4, 2000, - 78 с.
11. Yunusi, M. A model of a perturbed reservoir and some issues of its operation /M. Yunusi // Report of the AI RT. - 2003. - No. 11-12, Volume XLVI. -Pp.58-61.
12. Yunusi, M. Movement of perturbed wave in water reservoir /M. Yunusi //The book abstracts of Second Inter. Seminar on the applied mathematics. Zanjon, Iran. -October 3-4, 2000. -Pp.78-83.
13. Yunusi, M. On one model of a perturbed system of reservoirs /M. Yunusi, T. Karimov // Bulletin of the National University. -2003. -No. 2. - Pp.16-20.
14. Yunusi, M.M. Numerical solution of one problem of a multisectoral economy on matlab / M.M. Yunusi, S.S. Rizoev, M.K. Yunusi // Bulletin of the Tajik National University. Natural Sciences Series. - 2016. - No. 1/2 (196). - Pp.52-56.
