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5INTERNATIONAL SCIENTIFIC AND TECHNICAL CONFERENCE "DIGITAL TECHNOLOGIES: PROBLEMS AND SOLUTIONS OF PRACTICAL IMPLEMENTATION IN THE SPHERES" APRIL 27-28, 2023
ALGORITHM FOR SEASONALLY ADJUSTABLE RESERVOIR MODELING AND DEVELOPMENT OF A MATHEMATICAL MODEL 1Bobirjon Zayniddinov, 2Shokhrukh Erkinov, 3Doston Islomov
1Dean of faculty "Computer engineering", Doctor of Philosophy in Technical Sciences (PhD), Nurafshon branch of Tashkent University of Information Technologies named after Muhammad al-Khwarazmi, Uzbekistan, 2Head of sector of scientific researches, innovations and scientific-pedagogical personnel training, Doctor of Philosophy in Pedagogical Sciences (PhD), Nurafshon
branch of Tashkent University of Information Technologies named after Muhammad al-Khwarazmi, Uzbekistan, 3First degree MD student, Nurafshon branch of Tashkent University of Information Technologies named after Muhammad al-Khwarazmi, Uzbekistan https://doi.org/10.5281/zenodo.7857799
Abstract. Our article demonstrates creating the mathematical models and modelling algorithms for regulated watersourses.
Keywords: algorithm, seasonally, modeling.
The Hisorak reservoir is designed to provide water storage for the irrigation of Kashkadarya region. The reservoir has a rock dam with a maximum height of 140 m and a crest length of 528 m. The intake facility is located in the dam, and the deep operating water flow is located in the middle of the dam [1].
The dynamics of water in the reservoir in time can be described by the differential equation of the following form:
x(k +1) = f (x(k), u (k ),0, k) + g (0, x(k ))w(k), x(k0) = x0, (1)
Here x(k) e R2 - vector state, (x (k) -the volume of water in the reservoir, x2 (k) - water level in the reservoir); u(k) e R3 - control effect vector, (u1(k) - consumption of water entering the reservoir; u2 (k) - the amount of water released from the reservoir; u3 (k) - amount of water consumption); 0 e Rs - unknown vector of parameters; x0 - the initial volume of the reservoir at the present time k0; f and g - some functions; w(k) - a sequence of Gaussian random variables with zero mean and covariance
E{w(k)wT (j)} = Q(k)Sk].
Initial state x0- covariance P0 and x0 is also a random variable with a mean. Optimizing
reservoir operation modes consists in choosing such control measures in the water release structure, according to which the water supply schedule is provided, and at the same time, water losses for evaporation and filtration from the surface of the reservoir are minimal [2].
In the process of filling the water reservoir, the control functions are the consumption of water input u (t), water output u2 (t) and water consumption u3 (t) and - the working
process of the water reservoir.
Object state constraints [3] have the form:
Hmn < h(t) < Hmax,
dh(t)/ dt < UKp,
INTERNATIONAL SCIENTIFIC AND TECHNICAL CONFERENCE "DIGITAL TECHNOLOGIES: PROBLEMS AND SOLUTIONS OF PRACTICAL IMPLEMENTATION IN THE SPHERES" APRIL 27-28, 2023
here: h(t) - water level in the reservoir; #min, #max - respectively, the minimum and
maximum permissible value of the ordinate of the free water level in the reservoir; UKp critical
value of reservoir filling and operating speed.
(1) To linearize equation, we use the extended Kalman filter [3], as a result of which
we get
x(k +1) = A(0, k) x(k) + B(0, k )u(k) + G(k )w(k), (2)
y(k) = C(0, k)x(k) + q(k),
here A(0, k), B(0, k) and C (0, k) - linear functions of the unknown parameter vector 0 of dimension , Tj(k) - is a sequence of Gaussian random variables with zero mean and covariance.
x(k +1) = A(k, 0) x(k) + B(k, 0)u(k) + G(k )w(k) = D(k )u(k) + G(k )w(k)
(3)
here D(k) - from the function x(k) and u(k). Assuming 0(k / k) had a guess 0 for k at
the given time Xk, x and 0 from conventional estimation procedures to obtain one-step estimates of [4] can be used.
x(k +1/ k) = A (k / k, 0) x(k) + B (k / k, 0)u(k)
Recurrent parameter estimation and error covariance r(k +1/ k +1) defined by the following expressions:
0(k +1/k +1) = 0(k +1/k) + K (k +1) P-\k +1/k)[ x(k +1) - x(k +1/k)],
K(k +1) = F(k)r(k / k)DT (k), r(k +1/k +1) = r(k +1/k) - K(k + 1)P~\k +1/k)KT (k +1). Now we can write the control algorithm using the optimal system synthesis method [4] to minimize the average value of the quality criterion in the following form:
f 1 N-1 1
J (UN-1) = e\ 12 xT (i + 1)M (i)x(i +1) + uT (i)N(i)u(i)\Zk ,Uk}, (4)
12 i=k
here M(i) = MT (i) > 0 and N(i) = NT (i) > 0. To find the sequence UkN+-11, Considering (3), we use the method of minimizing (4), dynamic optimization [4]:
u (j/k) = -C(j/k)x(j), j = k +1,...,N -1, C(j/k) = [E{BT (j,0)S(j +1/k)B(j,0) + R(j)}]-1 x E{BT (j)S(j +1/k)A(j,0)},
(5)
S(j/k) = E{[A(j,0) - B(j,0)C(j/k)]T (M(j) + S(j +1/k)) x x [ A(j, k ) - B(j,0)C (j / k )] + C (j / k ) N (j )CT (j / k )} . The above formalization of the process allows choosing the structure of the mathematical model in the following form:
xk+1 = Akxk + Bkuk + wk , xk0 = x0 , yk = Ckxk +Vk, k e {ko, kT } ,
here
INTERNATIONAL SCIENTIFIC AND TECHNICAL CONFERENCE "DIGITAL TECHNOLOGIES: PROBLEMS AND SOLUTIONS OF PRACTICAL IMPLEMENTATION IN THE SPHERES" APRIL 27-28, 2023
A =
a11,k : ai2,k ________I________
a21,k ! a22,k
, Bk =
b11,k b12,k b13,k , Q = 1 0
b21,k b22,k b23,k 0 0
M[X0] = 0, M[X0xT0 ] = X0, M ] = 0, M[X0 rT] = 0, M[X0wTk ] = 0,
M
(w ^
w rT)
W G
GT V
8V.
Then the process and observation equations are written in the following form:
Xk+1 = AkXk + BkUk + wk , xk0 = X0 , (7)
Ok+1 =0k + w, (8)
A = CkXk +Vk , (9)
Advanced vector state input xp = [xT \OT ]T, (7)-(8) we write the equations in the following form
xp+i = fp [ xk, uk, k ] + wp, +i = xp, uk, k ] + w,
here: xk
x,,
a
fp =
f[ xk ,6k, uk, k ]
.........a..........
wp =
"W.
"W.
The required values of the registration and discretization time of the implementation of the observed random processes turned out to be equal, respectively: T = 30 hours, At =20 min. A total of 90 measurements were taken.
To estimate the extended state vector xpT = [xT | OT] simple algorithms of the object in
question are used. Analysis of state variables and parameter vector setting processes shows that the required variables have acceptable accuracy and convergence speed. The parameters of the model turned out to be equal:
Ak = 0.625 0.375 , Bk = 0.627 0.317 1 0
0.189 0.801 0.185 - 0.962 0 0
To determine the appropriateness of the developed models, we use a criterion based on the analysis of the properties of the residuals and related to the verification of assumptions about the model. The maximum value of the empirical correlation calculated for the residuals was equal to 0.174.
Since this value is less than the critical value, 2/(N)1/2 =2/(90)1/2 = 0.21, the hypothesis of independence of residuals can be considered consistent with the initial data, that is, the developed model adequately describes the process under consideration in the field of experience and can be used to develop a control system.
Summary. The developed mathematical model of the reservoir allows establishing quantitative relationships between the main input and output variables, predicting the state of the process with existing or selected control elements, and synthesizing optimal adaptive control laws for the considered process.
INTERNATIONAL SCIENTIFIC AND TECHNICAL CONFERENCE "DIGITAL TECHNOLOGIES: PROBLEMS AND SOLUTIONS OF PRACTICAL IMPLEMENTATION IN THE SPHERES" APRIL 27-28, 2023
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