Download The Study of Optimal Allocation Model for Water Deliveries to... Reaches of Tarim River

The Study of Optimal Allocation Model for Water Deliveries to Lower
Reaches of Tarim River
LU Haishu
School of Economics and Management,Jiangsu Teachers University of Technology, 213001, China
[email protected]
Abstract The main purpose of this paper is to establish and solve the optimal allocation model for the
water deliveries to the Lower Reaches of Tarim River. Based on nonlinear optimization theory with
constrained conditions, a large-scale optimal allocation model for the water deliveries to the Lower
Reaches of Tarim River is established. By employing the famous Kuhn-Tucker Condition method, this
paper solves the optimal allocation model for the water deliveries to the Lower Reaches of Tarim River
and obtains reasonable allocation of water quantity for various sub-regions of the Lower Reaches of
Tarim River. The reasonable allocation of water quantity will play an important role in water deliveries
to the Lower Reaches of Tarim River in emergency in the future.
Key words Water deliveries, Water delivery allocation, Optimal solution, Kuhn-Tucker condition, Tarim
River.
1 Introduction
Lying in northwest China, Tarim River is the longest landlocked river which covered an extensive area
(1020000km2) with insufficient water resources. The irrational increasing utilization of the water
resources has led to outstanding deterioration of ecological environment of the whole Tarim River Basin,
especially for lower reaches of Tarim River, since water flow is lack, the riverbed of lower reaches of
Tarim River becomes dried, and ecological environment deteriorated. The dried lower reaches of Tarim
River have resulted in the dropping of underground water levels, the worsening of water quality, and the
abandoning of farmland. For instance, the related data indicate that the underground water level in lower
reaches of Tarim River has been decreased from 4 to 16 m. Poplar forest area shranks notably. Taitema
Lake, at which Tarim River ended, dried up in 1974. The lower reaches of Tarim River were left with
little water for survival. The Daxihaizi reservoir dried up in 1993 for first time. The degradation of
natural vegetation has contributed to the enlarging of Taklimakan Desert.
How to improve environmental benefits from the limited water resources of Tarim River Basin has
become an urgent problem. Having complete a lot of investigations of Tarim River Basin, this paper
divides Tarim River Basin into five major regions, that is, I region (from Daxihaizi Reservoir to Yingsu),
II region (from Yingsu to Alagan ), III region (Laota River), IV region (from Alagan to Yiganbujima), V
region (from Yiganbujima to Taitema Lake). For the convenience of study, every region is divided into
two sub-regions, as shown in the above Figure 1.
S01
S1
S1
S5
S02
S3
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V region
Yiganbujima
IV region
Alagan
S4
eto t
III region
S6
S5
Taitema Lake
II region
Yingsu
Daxihaizi Reservoir
I region
S2
Fig1. The skeleton diagram of the water resources system
The meanings of symbols of the above diagram are given as follows:
S01 ------- volume of water drawn from Daxihaizi Reservoir into I region.
S02 ------- volume of water drawn from Daxihaizi Reservoir into III region.
Si-1 ------- run-off volume of the upper part of the river section i, i=I, II, III, IV, V.
Si -------- -run-off volume of the lower part of the river section i, i=I, II, III, IV, V.
Wi -------- volume of water consumed by i sub-region i, i =1,…, 10.
The above cable network illustrates water quantity transition laws between large regions:
I region: S1=S01-(W1+W2); II region: S2=S1-(W3+W4); III region: S3=S02-(W5+W6); IV region: S4=S2+S3,
S5=S4-(W7+W8);V region: S6=S5-(W9+W10). But it is need to point out that the water quantity transition
laws also exist in the inner of large regions although they don’t show in Figure 1, for example, the water
quantity transition laws for the inner of II region is given as follows:
The water consumption II region: W3+ W4
Water consumption W4 of sub-region 4
---------------------------------------------Water consumption W3 of sub-region 3
S1
S2
Fig2. Water quantity transition of the ith river section (large region)
2 Mathematical Model
We know that the supplied water S0 (108m3) of Daxihaizi Reservoir can meet the demands for water
in sub-region i, i=1,…, 10, where sub-region 1 and sub-region 2 are belong to I region, sub-region 3 and
sub-region 4 are belong to II region, sub-region 5 and sub-region 6 are belong to III region, sub-region 7
and sub-region 8 are belong to IV region, sub-region 9 and sub-region 10 are belong to V region.
The problem to be solved is stated as follows: when S0, the volume of water mentioned above is
fixed and at last, there is S6, the volume of water which will flow into Taitema Lake, we try to find the
optimum water allocation between ten sub-regions in order to achieve the maximum improved ecology
area. According to the actual conditions of Tarim River Basin, we know that S0=3.5(108m3),
S6=0.1(108m3), which is considered bound constrained conditions. The connection styles between
regions are complex. Up till now, this type of connection has not been discussed in other documents;
however, this type of connection really exists and should be considered.
2.1 Sub-regions models
To make the best rational use of the limited water resources of a sub-region, the maximum sum of
the improved ecology area of the various vegetation is adopted as the objective function. Suppose that
there is vegetation 1, vegetation 2 and vegetation 3 in i sub-region. Let Wi1, Wi2 and Wi3 denote the water
consumption of vegetation 1, vegetation 2 and vegetation 3 in sub-region i, respectively. Hence, the
objective function of sub-region i is:
Bi (Wi) = max {Bi1 (Wi1) + Bi2 (Wi2) + Bi3 (Wi3)}
i =1,…, 10.
(1)
In order to obtain Bi (Wi ), it is necessary to determine the functional relationship between Wik and
Bik (i =1,..,10, k=1,2,3). For instance, B12 (W12) indicates the maximum improved ecology area obtained
from the water volume W12 allocated in the vegetation 2 on the land of sub-region 1 among three kinds
of vegetation. The balances of the amounts of water are as follows:
Wi = Wi1 + Wi2 + Wi3
i =1,…, 10.
(2)
The upper and lower limits are as follows:
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≦ ≦
LWik Wik UWik
i =1,...,10, k=1,2,3
(3)
By synthesizing the objective function and the constrained conditions as introduced in the above
model, the optimal allocation mathematical model for sub-regions can be obtained. Now we give some
theoretical analysis about the above optimal allocation model for sub-regions. Obviously, the feasible set
of the above optimum problem is bounded closed set in R3, thus, according to the fact that bounded
closed set is equivalent to compact set in finite dimensional spaces, we know that the feasible set is
compact. Since for all i, Bi1 (Wi1) + Bi2 (Wi2) + Bi3 (Wi3) is continuous on the feasible set, then by the
famous Weierstrass Theorem, we know that the solution of the above optimal allocation model for
sub-regions must exists.
Next step is to solve the above optimal allocation mathematical model for sub-regions. There are
many methods about how to achieve the solution for this model. Among these methods, we think
Dynamic Programming (in short, D.P.) introduced by R. Bellman (1957) is practicable to solve the
above optimal allocation model for sub-regions. If the functional relationships between Wik and Bik (Wik),
can be expressed by analytical functions, this problem can be solved by using derivative, when there is
only a corresponding relationship between discrete points, dynamic programming can be employed. By
using the above methods, we can obtain the following functional relationships between the water
consumption Wi and the improved ecology area Bi (Wi ), for all i as follows.
Bi (Wi ) =fi (ai ln (bi Wi)+ci )
i =1,...,10.
(4)
where, ai, bi, ci are constant, i =1,..,10.
2.2 The basin model
By synthesizing the optimal water allocation models of the sub-regions, the optimal allocation
mathematical model for the basin can be obtained.
2.2.1 The basin model
Suppose that the ecological objective is the maximum sum of nets from water utilizations, the
objective function of ecology is:
10
(5)
B(W ) = max{ f ( a ln(b W ) + c )}
∑
i
i
i
i
i
i =1
2.2.2 The constraint conditions
The constrain conditions of the optimal allocation mathematical model of ecology water use of the
whole basin should also include the followings:
W1 + W2 +…W10 = 3.4 (108m3).
(6)
LWi Wi UWi
i =1,..., 10
(7)
, where LWi, UWi denote the lower limits and the upper limits of Wi , respectively.
2.2.3 Solution to the Mathematical Model
In solving the optimal allocation mathematical model, we adopt the method of “breaking up a
major part into minor parts and then putting minor parts together into a major part”. By “breaking up a
major part into minor parts”, it is meant that the original complicated large-system optimization problem
is broken up into several minor subsystem optimization problems so as to obtain the local optimal
solutions for the optimization problems of the subsystems, which are relatively easy to be solved. By
“putting minor parts together into a major part”, it is meant that the obtained local optimum solutions are
put together through their mutual connections so as to obtain the overall optimum solution by adopting
the methods of large-system optimization.
Through repeated calculation and theoretical analysis, and learning from foreign and domestic
theories concerning large-system optimization, this paper adopts the advanced approach of large-system
and Kuhn-Tucker condition method which is a flexible aid to the solution of optimization problem
concerning non-linearity and multi-variables. The above basin model is equivalent to the following
standard model:
10
(8)
B(W ) = min{ − f ( a ln(b W ) + c )}
≦ ≦
∑
i
i
i =1
The constraint conditions are as follows:
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i
i
i
W1 + W2 +…W10 = 3.4 (108m3).
(9)
Wi -UWi 0
(10)
LWi -Wi 0
(11)
Based on the above analysis, we know that the above model must have optimal solution. It is easy to
prove that the function f (W1,…,W10) = f1 (a1 ln (b1W1) +c1) +…+ f10 (a10 ln (b10W10) +c10) is concave. In
fact, for any two points (W1,…,W10), (W’1,…,W’10) X, and t [0,1], where X denotes the feasible set of
the optimal problem. Combining the fact that for each i =1,..., 10, Bi (Wi ) =fi (ailn (bi Wi )+ci) is concave
function, we have the following:
f (t(W1,…,W10) +(1-t)( W’1,…,W’10)) t f (W1,…,W10) +(1-t) f (W’1,…,W’10)
(12)
Hence, the function f (W1,…,W10) = f1 (a1 ln (b1 W1) +c1) +…+ f10 (a10 ln (b10 W10)+c10) is concave. Since
the constraint conditions are linear. So the above nonlinear optimum problem may be solved by
Kuhn-Tucker condition method. Furthermore, the local solutions of the above nonlinear optimum
problem must be global solutions.
In general, it is not easy to find these points which satisfy all the constrain conditions. In order to
obtain the solutions of the optimal allocation mathematical model of ecology water use in Tarim River
Basin, we discuss the complementary slackness conditions.
By using the method mentioned above, we can obtain the solutions of the optimal allocation
mathematical model of ecology water use in Tarim River Basin. See Table I.
Table I. The water consumption, the improved ecology area in different sub-regions and regions
Water consumption
Ecology area
Divided regions
(108m3)
(km2)
Sub-region 1
0.1326
62
I region
≦
≦
∈
∈
≧
From
Daxihaizi to
Yingsu
Sub-region 2
0.4491
198
total
0.5817
260
I region
From Yingsu
to Alagan
Sub-region 3
Sub-region 4
total
Sub-region 5
Sub-region 6
total
Sub-region 7
Sub-region 8
0.3003
0.4456
0.7459
0.3283
0.5213
0.8496
0.2164
0.4082
136
252
388
142
262
404
94
228
total
0.6246
322
Sub-region 9
Sub-region 10
0.1825
0.4157
38
84
0.5982
122
3.4
1496
III region
Laotalimu
River
IV region
From Alagan
to
Yiganbujuma
IV region
From
Yiganbujuma
to Taitema
Lake
total
total
3 Analysis of results
3.1 Analysis of results of I region
The allocated water quantity of Sub-region 1 and Sub-region 2 is 0.1326 (108m3) and 0.4491
8 3
(10 m ), respectively. The ecological improvement area of Sub-region 1 and sub-region 2 is about 62
(km2) and about 198 (km2), respectively. The total water consumption from Daxihaizi reservoir to
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Yingsu section is 0.5817 (108m3). The ratio of water quantity allocation and the ecological improvement
area of region I is percent 17.1% and 17.4%, respectively. The above data show that the water quantity
delivered to this area has a marked influence on improving the ecological environment.
3.2 Analysis of results of II region
From Table I, we know that the allocated water quantity of Sub-region 3 and Sub-region 4 is
0.3003 (108m3) and 0.4456 (108m3), respectively. The ecological improvement area of Sub-region 3 and
sub-region 4 is about 136 (km2) and about 252 (km2), respectively. The total water consumption from
Yingsu to Alagan is 0.7459 (108m3). The ratio of water quantity allocation and the ecological
improvement area of region II is percent 22% and 26%, respectively. The above data show that the water
quantity delivered to this area has a marked influence on improving the ecological environment.
3.3 Analysis of results of III region
The allocated water quantity of Sub-region 5 and Sub-region 6 is 0.3283 (108m3) and 0.5213
8 3
(10 m ), respectively. The ecological improvement area of Sub-region 5 and sub-region 6 is about 142
(km2) and about 262 (km2), respectively. The total water consumption of Lao Tarim River is 0.8496
(108m3). The ratio of water quantity allocation and the ecological improvement area of region III is
percent 25% and 27%, respectively. The above data show that the water quantity delivered to this area
has a marked influence on improving the ecological environment.
3.4 Analysis of results of IV region
The water quantity consumption of Sub-region 7 and Sub-region 8 is 0.2164 (108m3) and 0.4082
8 3
(10 m ), respectively. The ecological improvement area of Sub-region 7 and sub-region 8 is about 94
(km2) and about 228 (km2), respectively. The total water consumption of region IV is 0.4157 (m3). The
ratio of water quantity allocation and the ecological improvement area of region IV is percent 17.6% and
8.2%, respectively. The above data show that the water quantity delivered to this area has a marked
influence on improving the ecological environment.
3.5 Analysis of results of V region
The water quantity consumption of Sub-region 9 and Sub-region 10 is 0.1825 (108m3) and 0.4157
8 3
(10 m ), respectively. The ecological improvement area of Sub-region 9 and sub-region 10 is about 38
(km2) and about 84 (km2), respectively. The total water consumption of region IV is 0.6246 (m3). The
ratio of water quantity allocation and the ecological improvement area of region IV is percent 18.4% and
21.5%, respectively. From the above data, it can be seen that the living vegetation in this area is
becoming fewer. The consequences of water delivery are ineffective. But in order to protect the 218
National Road, a certain quantity of water must be supplied to region IV to ensure that the 218 National
Road covered by a certain width of the vegetation on both sides.
4 Non--engineering measures
Emergent water delivery to lower reaches of Tarim River Basin is an important engineering
measure for protecting and improving ecological environment. Water delivery can coincide with the
expected result of model or not depends on the continuous and effective investment management. To
ensure and increase the input of emergent water delivery and strengthen the management of the Tarim
River through the establishment of special rules and regulations will be the preconditions for ensuring
the outcome of the emergency water supply, it is also entirely possible to expand ecological
improvement results.
Water management practice of Tarim River Basin shows that since the organization of Tarim River
Basin was founded in 1991, it can not coordinate the relations between local regions, between army and
local government, between upstream and downstream, and between tributary and mainstream effectively.
Water management agencies in both horizontal and vertical, basin unified management with the local
regional management are obvious disharmony.
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These existing management problems shows that in order to manage water resources of basin in the
future effectively, government must strengthen its social management capacity in water resource
management. At the same time, government should play an active role in strengthening the unified
management of basin and stimulating the initiative of region management. Governmental functions
should be transformed at a higher speed with the sense of administration according to law built up and
administrative efficiency raised.
The organization of Tarim River Basin should allocate the initial water right and water pollution
right rationally, thus lay a good foundation for the subsequent establishment of water rights transaction
markets and pollution rights market so as to achieve the purpose to rationally allocate water right and
pollution right of Tarim River Basin by using market mechanisms and enhance the efficiency of the
using of water resources.
5 Conclusions
Based on the methods of system engineering and nonlinear optimization, this paper established the
optimal allocation model of ecology water in Tarim River Basin. By using Kuhn-Tucker condition
method, we obtained the solutions of the optimal allocation mathematical model of ecology water use in
Tarim River Basin. Since the system of Tarim River Basin water resources is rather complicated, it is
felt that some topics need to be further discussed, involving many sectors and factors, for example, time
factor etc. The preparatory works of collecting and working up fundamental data need to be conducted
simultaneously among various units. This is indeed quite a research topic, which calls for large
quantities of data. Kinds of facts show that the techniques suggested in this paper are effective and
widely applicable and the allocation schemes of the lower reaches of Tarim River Basin will play a key
role in water deliveries in emergency in the future.
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