Optimal Allocation Model of Reclaimed Water Reuse

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Procedia Engineering

ProcediaProcedia Engineering 00 (2011) 000–000 Engineering 28 (2012) 763 – 766

www.elsevier.com/locate/procedia

2012 International Conference on Modern Hydraulic Engineering

Optimal Allocation Model of Reclaimed Water Reuse Peng Bina, Zhang Yun-xina, Guo Weib, Gao Sujuanb, a* b

a Hebei University of Engineering，Handan 056021, China Handan Design and Research Institute of Water Conservancy and Hydropower，Handan 056021, China

Abstract In order to utilize the reclaimed water resources reasonably and maximize energy-consuming green benefit, the optimal allocation model is established for reclaimed water resources, which is based on the marginal benefit equilibrium theory of water-consuming and marginal utility theory of the economics, and there are two levels including industries and consumers. It promotes consumers to improve the water-consuming green marginal benefit, accelerate the process of generalizing reclaimed water resource recovery.

© 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Society for Resources, © 2011 Published by Elsevier Ltd. Environment and Engineering Keywords: Reclaimed water resources; the Optimal Allocation Model; Marginal Benefit Equilibrium

1. Introduction Sewage treatment will be one of the most potential measures of the all exploitation methods for the water resource. Now, in the world, the Israel’s degree of utilization can be the first in the world [1].In China, until the 1980s, city sewage recycling technology was gradually developed and certain achievements were achieved. For example, Li Mei established the water supply price model from the angles of the market, cost and value theory[2]; Liu Zhiqiang improved the model of the network construction cost and operation cost[3]; Li Jiaguo simulated and analyzed the water utilization system according to the process of reclaimed water utilization status and actual condition, then the from macroscopic angle of the cost, distance and obtained quantity [4], they built the model of using optimally water with 5 sets linear programming. But they neglected or did not consider thoroughly some objective limitation reasons about sewage resource recovery which led to the application of study lagging behind

* Corresponding author. Peng Bin. Tel.: 013831032882; fax: 0310-3123573. E-mail address: [email protected].

1877-7058 © 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of Society for Resources, Environment and Engineering doi:10.1016/j.proeng.2012.01.805

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the research level seriously. Therefore, the optimal utilization of the sewage after moderate treatment with lest investment which is in tolerable range of the current is the problem to be solved in the field of recycling reclaimed water in our country. 2. Theory analysis 2.1. Optimal allocation model of reclaimed water resources among industries 2.1.1 The optimal allocation model of reclaimed water resources among industries At the assumption that at t moment, the amount of reclaimed water resources provided by a certain area under the existing conditions is R, and there are N industries which accept reclaimed water resources, the water resources ( short for the conventional water ) quantity provided for economic development of the industries is Qi 0 , i = 1, 2, …N. The conventional water is assumed to be used firstly, and then the reclaimed water resources. Under the hypothesis that the marginal benefit of the industry consuming water resource is f i  f i Qi 0  , a quadratic function is used to fit the actual data of every industry. Based on the marginal benefit of the industries utilizing the water resources, the benefit is scheduled from big to

f i > f j ,  i < j.

small. Without loss of generality, we assume that

Now, The amount of reclaimed water resources obtained by each industries is Ri . The target is to get the maximum green benefit of reclaimed water resources after allocation. Assume that after each industry get relevant resource, the green benefit of total water resources is: Ui 

Qi

Ri

 f Q dQ   f Q i

i

i

Qi 0

Then the optimization problem is: N

N Qi

i 1

i 1 Qi 0

Max U  U i  

St

i

i0

 Ri dRi

(1)

0



N Ri

f i Qi dQi    f i Qi  Ri dRi i 1 0

df i Qi  df i Qi  Ri   0 dQi dRi

Qi  Qi 0  Ri

N

R   Ri i 1

(2) (3)

where is the consumption of the ith industry, the Eq.(3) is the decline constraints of marginal benefit of water consumption. The above optimization problem is a conditional extremum problem. Therefore, the Lagrange multiplier function is constructed as follows: (4)   N

Then

L  U    R   Ri  i 1  

(5) f i Qi 0  Ri    ,  i = 1, 2, ⋯,N It means that when the each marginal benefit of the industries consumption equals to  , the total economic benefits is the biggest. From Eq.(5), here is: (6) Ri  g i    Qi 0

In above formula, g i  represents inverse function of f  . From the above analysis,  can be interpreted as the marginal balance benefit of water consumption of each industry. The above result can be interpreted as the reference [5]. Therefore, the marginal benefit equilibrium principle of reclaimed resources allocation can be described as: when the industries of the water consumption marginal benefit is balanced, the total green benefit of water resources consumption of

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N area is maximum. And Qi 0 + Ri j can be expressed as Qi j0  Qi 0  Ri j . If y j 0  f j Q j 0  , there is: Qi 0  f i 1 y j 0   g i y j 0  j

(7)

2.1.2. Advanced discussion about optimal allocation of reclaimed water resources among industries 2.1.2.1 Scarcity of reclaimed water resources for allocation Assuming there are N industries which are likely to get the reclaimed water resources, and the supply quantity of reclaimed water resources is R. Due to the insufficient of the water supply capacity, namely, N*

N

*

R   Qi  Qi 0 

,that is to say, the resources can be allocated to N industries only,

i 1

y *  f * QN * 0 

*

relevant equilibrium marginal benefit is y , then there is is:

i 1

,

N*  N .

The

. Therefore the optimization problem

* N* R i

N * Qi

N*

R   Qi  Qi 0 

Max U  U i    f i Qi dQi    f i Qi  Ri dRi

i 1 i 1 Q i 1 0 (8) It’s constraint conditions are the same as the Eq (2) and (3). Under the circumstances, in order to get * * the maximum benefits, limited water resources are sufficiently used in the N industries, and N is i0

j

requested. And for  j,

R0j   Ri j i 1

.

2.1.2.2 Sufficient of reclaimed water resources for allocation If the reclaimed water resource is sufficient, the quantity is R, thus all of N industries can receive the resources. That is to say, then

y  f N QN

0

N

N

i 1

i 1

R   RiN  Qi  Qi 0 

 is the result, and there is N

. Now, the corresponding equilibrium marginal benefit is y,

N

(9)

 R   g  y   Q   R i 1

i

i 1

i

i0

Comby with Eq (9) and (6), the quantity received by industries can be obtained. Ri  g i  y   Qi 0 ，i= 1, 2, ⋯,N Then introducing g i  y  into Eq. (9), and there is

(10)

2

y R  

N N  N   N  N    bi    bi   4  ai   ci  Qi 0  R  i 1 i 1   i 1   i 1  i 1

(11)

N

2 a i i 1

According to the marginal benefit diminishing principle, the sign of Eq. (11) must assure that y  yR  is a decreasing function of R. If y  0 , when y=0, the maximum consumption of reclaimed water c  Q  0 c resources is when it is saturated. So, , and the maximum quantity of reclaimed allocation N

N

i 1

i

water resources should satisfy

i 1

N

N

i 1

i 1

R   ci   Qi 0

N

i

i 1

i0

N

. For short,

A   ai B  i 1

,

N

b i 1

i

,

N

N

i 1

i 1

C   ci   Qi 0  E

, A and B are constant, y

 B  B 2  4 AC

2A . C>0 and C is also a diminishing function of E. then the formula (11) can be also described From the above discussion, when A and B are constant, the sign should make y be an increasing function of C, and y  0 . Then the selection principle of selecting the sign is as follows: 1 ) when A〉0, B〉0, take positive; 2) when A〉0, B〈0, take negative; 3) when A〈0, B〉0, take negative; 4) when A 〈0, B〈0, take positive.

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2.2. Allocation model of reclaimed water resources among users Compared with the model among industries, the model among users discussed the cost of pollution w treatment furthermore. Assuming the pollution generated by different consumer is defined as ij ; the average governance charge of the user per unit pollution is k ij ; the quantity of reclaimed water resources received by the user is Rij , the cost of the users spending on pollution control is Pij  wij kij Rij . Based on the above assumptions, it’s similar to the model between industries, and the optimization problem is as following: M Qij  Rij

(12)

The same to the above solution, constructs the Lagrange multiplier function ,Then there is  j = 1, 2, ⋯, M f ij Qij  Rij   wij k ij  1 ,

(13)

j 1

 f Q

M

 Rij dRij   wij k ij Rij

Max Yi  U i  Pi  

ij

Qij

ij

j 1

Namely, when the difference between the marginal benefit of each user’s consumption and the coefficient of charge cost of controlling the pollution is equilibrium, the total green benefit of the industry is maximum. From Eq. (13), it is obtained (14) Rij  g ij 1  wij k ij   Qij Eq. (14) ascertains the quantity of reclaimed water resources received by each user. According the above analysis, based on the maximum green benefit, the marginal benefit equilibrium principle of the optimization allocation among users of reclaimed water resources can be described as: when the difference between the marginal benefit of each user’s consumption and the coefficient of pollution charge cost is equilibrium, the total social benefit of the industry is maximum. 3. Conclusion In the theory, the allocation model of the reclaimed water resources is established, the goal of which is to get the maximum green benefit of the water consumption. The model has practical reference value for the reclaimed water resources. And it is conductive to guide users to improve the marginal benefit of the utilization efficiency of water resources and water consumption, and promote the construction of watersaving society. Acknowledgements Supported by China Institute of Water Resources and Hydropower Research Open Research Fund. References [1] Guo xiao-yu. The ecological effect and its evaluation of reclaimed water irrigation on lawn. Beijing: Capital Normal University, 2006 [2] Li Mei. Reclaimed water reuse system analysis and simulation of city waste-water, Xi'an: Xi'an University Of Architecture And Technology, 2003 [3] Liu Zhi-qiang. Optimal design of water distribution system research, Tian jin: Tianjin University,2004 [4] LI Jia-guo; GONG Hui-li; Optimized Decision-Making Models of Reclaimed Water Reuse, China Science, 2001,31(5):421427 [5] Wang Jin-feng, Liu Chang-ming, Wang Zhi-yong; The marginal benefit equilibrium model of water resources special allocation, Resource Science, 2007,(1): 84-91