m
n
Q
m
n
Q
m
n
Q
m
, (1)
where
001
,
0
/
2724
.
1
Н
ДВ
P
Kt
m
;
2
2
11
2
2
2
WEAR
2
]
)
(
)
1
(
/[
)
1
(
c
m
c
P
Kt
K
A
m
N
П
N
N
;
2
2
11
3
2
c
a
m
m
;
)
2
(
2
2
2
2
11
4
c
b
a
m
m
;
2
2
11
5
2
b
a
m
m
;
)
/(
WEAR
2
6
N
P
Kt
K
c
m
;
)
/(
WEAR
2
7
N
P
Kt
K
a
m
;
2
2
11
8
b
m
m
;
)
/(
WEAR
2
9
N
P
Kt
K
b
m
;
)]
1
(
/[
)
1
(
.
.
10
П
Н
ДВ
Н
ДВ
P
A
A
m
;
2
2
1
/
N
d
Н
n
i
A
a
;
)
/(
1
d
N
Н
i
n
B
b
;
4
1
/
d
N
i
C
c
;
2
2
2
/
Н
d
p
n
i
A
a
;
)
/(
2
d
N
p
i
n
B
b
;
3
5
2
/
Н
d
p
n
i
C
c
;
pr;
00000833
,
0
cha
T
e
K
2
0
000196
,
0
24
,
1
t
Kt
;
CAT
F
d
D
D
i
/
.
МЕЖДУНАРОДНАЯ НАУЧНО-ТЕХНИЧЕСКАЯ КОНФЕРЕНЦИЯ
АКТУАЛЬНЫЕ ПРОБЛЕМЫ ЦИФРОВИЗАЦИИ ЭЛЕКТРОМЕХАНИЧЕСКИХ И
ЭЛЕКТРОТЕХНОЛОГИЧЕСКИХ СИСТЕМ
164
Here:
WEAR
K
- wear and tear coefficient of the pump working parts;
t
K
- coefficient that takes into account changes in the nominal power of the
pump unit's electric motor depending on the ambient temperature;
CAT
D
- catalog diameter of the pump impeller.
Let us denote the pump unit supply as
1
х
and the rotation frequency as
2
х
.
The optimization problem of the operating modes of the pumping station can be
formulated as the following nonlinear programming problem:
min
)
(
j
x
Э
; (2)
0
)
(
j
ij
x
g
,
m
i
,....,
1
; (3)
0
)
(
j
lj
x
g
,
p
m
l
,....,
1
; (4)
0
j
x
,
k
j
,....,
1
. (5)
Based on the generalized rule of Lagrange multipliers and the well-known Kuhn-
Tucker theorem, we minimize the objective function
)
(
x
Э
over the feasible set defined
by constraints (3)-(5). Then, the minimum point of the specified objective function can be
found as the solution to the following system of equations with additional variables
i
,
i
k
x
,
m
i
,....,
1
;
j
,
k
j
,....,
1
;
l
,
p
m
l
,....,
1
:
m
i
p
m
l
j
j
j
lj
l
j
j
ij
i
j
j
x
x
g
x
x
g
x
x
Э
1
1
0
)
(
)
(
)
(
0
)
(
2
1
k
j
ij
x
x
g
0
)
(
j
lj
x
g
( 6)
0
1
k
i
x
0
j
j
x
At the same time
0
j
x
,
0
j
,
0
i
,
0
l
.
Here:
)
(
j
ij
x
g
- limitations presented in the form of inequalities.;
)
(
j
lj
x
g
- restrictions presented in the form of equalities;
j
x
- dependent variables
1
х
and
2
х
)
2
(
k
.
Since the regulation of the rotational speed of the pump unit, which is structurally
combined to work together in the pumping unit, and therefore its delivery, is carried out
between the stages, then for the given range of changes in head
max
min
H
H
H
provided
during step (aggregate) regulation of the pump unit performance by means of automation,
the limitations will be recorded as follows:
0
1
x
N
Q
Q
N
Q
Q
Q
ГР
ГР
Т
ГР
(7)
0
max
2
max
n
x
n
n
(8)
0
2
min
min
x
n
n
n
(9)
2
2
2
2
3
3
2
3
Q
N
R
H
Q
N
c
n
N
Q
b
n
a
ТР
СТ
0
)
(
3
2
1
2
2
1
3
2
2
3
СТ
ТР
H
R
c
x
N
x
x
N
b
x
a
(10)
МЕЖДУНАРОДНАЯ НАУЧНО-ТЕХНИЧЕСКАЯ КОНФЕРЕНЦИЯ
АКТУАЛЬНЫЕ ПРОБЛЕМЫ ЦИФРОВИЗАЦИИ ЭЛЕКТРОМЕХАНИЧЕСКИХ И
ЭЛЕКТРОТЕХНОЛОГИЧЕСКИХ СИСТЕМ
165
Here:
Н
d
SH
n
i
A
a
/
2
3
;
)
/(
3
d
Н
SH
i
n
B
b
;
4
3
/
d
SH
i
C
c
, (11)
where
SH
A
,
SH
B
,
SH
C
- approximation coefficients of the total head-flow
characteristic N of the jointly operating pump units..
As a result, we obtain a system of nonlinear equations:
0
)]
(
2
[
)
,
(
1
3
1
2
2
3
4
1
1
2
1
ТР
R
c
x
N
x
N
b
N
x
x
x
Э
0
)
2
(
)
,
(
2
1
3
2
3
4
3
2
2
2
1
x
N
b
x
a
x
x
x
Э
0
2
3
1
x
x
N
Q
ГР
0
2
4
max
2
x
n
x
(12)
0
2
5
2
min
x
x
n
0
)
(
3
2
1
2
2
1
3
2
2
3
СТ
ТР
H
R
c
x
N
x
x
N
b
x
a
0
3
1
x
;
0
4
2
x
;
0
5
3
x
;
0
1
1
x
;
0
2
2
x
;
0
,
2
1
x
x
;
0
,
2
1
;
0
,
,
,
4
3
2
1
.
This system of equations can be solved by the Newton-Raphson method, which is
the most common and widely used method for solving a system of nonlinear equations [2-
12] described, as is known, by the recurrent formula:
)]
(
/
)
(
[
1
K
K
K
K
K
x
x
x
x
, ( 13 )
where
)
(
K
x
- scalar function of some argument
K
x
;
K
- iteration step length.
When solving (13) using the Newton-Raphson method, it is necessary to set the
relative error
, the number of equations of the system, the maximum number of
iterations, as well as the initial approximation for each
i
x
,
m
i
,....,
1
.
For
1
х
and
2
х
, the actual flow of the pumping unit and its current rotation speed
n
are taken as initial approximations:
N
Q
Q
x
Т
/
10
;
n
x
20
. (14)
Then:
ГР
Q
Q
N
x
30
;
n
n
x
max
40
;
min
50
n
n
x
(15)
Conclusion
Based on the expressions defining the mathematical model of the pumping unit, as
well as the system of nonlinear equations (12) and the initial conditions (14), (15), optimal
values of the pump unit rotation frequency and the pump supply for the machine water
lifting system can be determined, ensuring the minimum specific power consumption
while maintaining constant parameters
ТR
R
= const,
ТR
H
const,
d
i
= const.
References:
1. Allaev K.R., Khokhlov V.A., Sytdykov R.A. Transient processes of pumping
stations. Ed. prof. MM. Muhammadieva. – T.:, ―Science and Technology‖, 2012, 180 p.
МЕЖДУНАРОДНАЯ НАУЧНО-ТЕХНИЧЕСКАЯ КОНФЕРЕНЦИЯ
АКТУАЛЬНЫЕ ПРОБЛЕМЫ ЦИФРОВИЗАЦИИ ЭЛЕКТРОМЕХАНИЧЕСКИХ И
ЭЛЕКТРОТЕХНОЛОГИЧЕСКИХ СИСТЕМ
166
2. Allaev K.R., Khokhlov V.A., Titova Zh.O. Increasing the energy efficiency of
pumping stations with long pipelines // Problems of energy and resource saving. –
Tashkent, 2013 - No. 1-2 - P. 10-15.
3. Karelin V.Ya., Minaev A.V. Pumps and pumping stations. -M.: Stroyizdat, 1986. -
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machine irrigation systems // Bulletin of Tashkent State Technical University, 2011. No.
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5. Kamalov T.S., Khamudkhanov M.M. Electric drive system for machine irrigation
pumping units. -Tashkent: ―Fan‖, 1985. -96 p.
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coefficients of approximation of pump operating characteristics // Automated electric
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9. Shevelev F.A. Tables for hydraulic calculations of steel, cast iron, asbestos-cement
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