An Optimisation-based Approach for Integrated Water Resources Management

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An Optimisation-based Approach for Integrated

Water Resources Management

Songsong Liu,a Petros Gikas,b,c Lazaros G. Papageorgioua a

Centre for Process Systems Engineering, Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK, E-mail: l.papageorgiou@ucl.ac.uk

b

Department of Environmental Engineering, Technical University of Crete, Chania, 73100, Greece

c

Hellenic Ministry of Environmental Planning and Public Works, (a) General Secretariat of Public Works,Special Service of Public Works for Greater Athens Sewerage and Sewage Treatment, Varvaki 12, Athens, 11474, and (b) Central Water Agency, Patission 147, Athens, 11251, Greece

Abstract

This paper considers an integrated water management problem for a region lacking fresh and ground water resources, which comprises (a) the optimal placement of desalination, water reclamation and wastewater treatment plants, (b) the calculation of the optimal capacities of the above facilities, and (c) the calculation of the optimal conveyance system for desalinated, reclaimed water and wastewater. This problem is formulated as a mixed-integer linear programming (MILP) model, with an objective to minimise the annualised total cost including capital and operating costs. Finally, the proposed model is applied to a real case for the Greek island of Syros in the Aegean Sea. Keywords: integrated water resources management, MILP, desalination, wastewater treatment, water reclamation

1.Introduction

In the last decade, optimisation techniques have widely been used in the field of integrated water resources management. Medellín-Azuara et al. [1] applied an economic-engineering optimisation model to explore and integrate water management alternatives such as water markets, reuse and seawater desalination in Ensenada, Mexico. Han et al. [2] presented a multi-objective linear programming model to allocate various water resources among multiusers and applied it to obtain reasonable allocation of water supply and demand in Dalian, China. Cunha et al. [3] presented an optimisation model for regional wastewater systems planning, together with a simulated annealing (SA) algorithm to optimise layout of sewer networks, location of treatment plants, etc. With the increasing water consumption worldwide, study of using various water sources available to fulfill the water demand has become an important issue recently. In this work, we aim to develop an optimisation-based approach using mixed-integer linear programming (MILP) techniques for the integrated water resources management in a water deficient insular area, where fresh water importation is a particularly expensive and non-sustainable option [4]. The alternative water recourses, which can meet the demands for water, are seawater desalination and water reclamation from wastewater.

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2.Problem Statement

In this work, we consider an insular area which is water deficient. The water demand is exclusively satisfied by desalinated seawater and reclaimed water from wastewater. Desalination yields potable quality water at a relatively high cost, while reclaimed water can be used for non-potable urban, industrial and agricultural applications at production cost significantly lower to that of desalinated water.

The area is divided into several sub-regions based on the population distribution and land terrain. The optimal allocation in each region of desalination, wastewater treatment and water reclamation plants are to be determined. Wastewater is collected from all sources and is conveyed to a wastewater treatment plant, where it is treated to meet the specific discharge limits. Then, part of the treated wastewater may undergo further treatment (at an extra cost) in order to meet the reclaimed water quality criteria, while the remaining is discharged into the sea. For simplicity, it is assumed that there is no water loss in the process of wastewater treatment and water reclamation. The desalinated water can be used as potable water; but it may also be augmented with reclaimed water for non-potable uses. The schematic graph of the water/wastewater flows in a region is given in Fig. 1. It is assumed that both qualities of water and wastewater are allowed to be freely distributed among most of the regions. Thus, the infrastructure needs for water distribution and storage, including the main pipeline network, pumping stations, and storage tanks, are also optimised.

Wrw

Qdw

Sww _{Primary & Secondary} Treatment Tertiary Treatment Www Wsw Www _Wrw _Dnp Qww Qww Qrw Qrw Wastewater Treatment Plant

Desalination Pdw D p Qdw Q dwnp Desalination Plant

Water Reclamation Plant

Wrw

Qdw

Sww _{Primary & Secondary} Treatment Tertiary Treatment Www Wsw Www _Wrw _Dnp Qww Qww Qrw Qrw Wastewater Treatment Plant

Desalination Pdw D p Qdw Q dwnp Desalination Plant

Water Reclamation Plant

Fig. 1. Schematic graph of the flows of various types of water and wastewater

In this problem, given are the pairwise distances, pumping distances and elevations between the relative population centres of the regions, daily potable/non-potable water demand, wastewater production, capital costs of the relative plants with different sizes, production costs of desalinated and reclaimed water, treated wastewater, diameters and unit costs of pipelines (installed), unit costs, maximum flow rates and pumping elevations of pumps, unit storage cost and storage retention time, cost of electricity, and water/wastewater velocity to determine the locations and capacities of the plants, pipeline main network characteristics, daily production volumes of the plants, daily mains flows of desalinated water, reclaimed water and wastewater, and pumps (number and capacities) at each established link, so as to minimise the annualised total cost, including capital costs for plants, pipelines, pumping stations, and storage tanks, and operating/energy costs for water production, wastewater treatment and pumping. 3.Mathematical Formulation

The integrated water resources management problem is formulated as an MILP model, as described next:

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3.1.Mass Flow Balance

From Fig. 1, we can see that before the wastewater treatment in region i, collected wastewater flows, ww

jit

Q , plus wastewater supply, ww it

S , is equal to distributed wastewater flows, ww

ijt

Q , plus total wastewater for treatment, ww it W . t i W Q S Q ww it i ww ijt ww it j ww jit + =

∑

+ , ∀,

∑

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The treated wastewater production, ww it

W , is equal to the summation of discharged wastewater amount, sw

it

W , and reclaimed water production, rw it W . t i W W W_itww= _itsw+ _itrw, ∀, (2)

The reclaimed water production, rw it

W , plus incoming reclaimed water flows, rw jit

Q , and interactive desalinated water flow, dwnp

it

Q , is equal to outgoing reclaimed water, rw ijt

Q , and non-potable water demand, np

it D . t i D Q W Q Q np it j rw ijt rw it dwnp it j rw jit + + =

∑

+ , ∀,

∑

(3)

In the desalination plant, the summation of desalinated water production, dw it

P , and incoming desalinated water, dw

jit

Q , is equal to the summation of outgoing desalinated water, dw

ijt

Q , potable water demand, p it

D , and desalinated water flow, dwnp it Q . t i D Q Q P Q p it dwnp it i dw ijt dw it j dw jit + =

∑

+ + , ∀,

∑

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3.2.Plant Capital Cost

The capital costs of the desalination plants (DPCC), wastewater treatment plants (WTPCC) and water reclamation plants (WRPCC), which are functions of plant capacities, are piecewise linearised in the model.

, dw ik i k dw k CC DPCC=

∑∑

⋅λ (5) , ~ dw ik k dw k dw i A A =

∑

⋅λ and Edw i i k dw ik = ∀

∑

λ , (6) where dw k CC and dw k

A~ are the cost and capacity at the break point k in the desalination plant capital cost function in region i, and dw

i

A are positive variables for the capacity and dw

ik

λ are SOS2 variables, while binary variable dw i

E indicates whether a desalination plant is allocated in region i. Similar constraints and variables are also defined for the capital cost of wastewater treatment plants and water reclamation plants.

3.3.Pipeline Network Capital Cost

There are potentially three different pipeline networks, each used for desalinated water, wastewater or reclaimed water. The capital cost (PipeCC) is calculated by the unit cost of pipe type p multiplied by the length of pipes (Lij). We also

introduce binary variables Yijp to indicate if pipe type p is selected between regions i and j for water and wastewater transportation.

The flow rate of water/wastewater, Qp

~

, is determined by the pipe diameter (dp) and the

flow velocity (v). The daily flow and flow rate of desalinated water are related by the operation time proportion variable, dw

ijt γ . We introduce dw ijp dw ijt dw ijpt Y YG ≡γ ⋅ to linearise the nonlinear term, as given below:

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t j i YG Q Y Q Q P p dw ijpt dw p P p dw ijp dw ijt dw p dw ijt , , , ~ ~ _⋅ _⋅ ₌ _⋅ _∀ =

∑

∈ ∈ γ (7)

Similar constraints for the daily flows and flow rates of wastewater and reclaimed water are also developed.

3.4.Pumping Station Capital Cost

The cost of a pumping station includes the cost of a pair of pumps (operating and standby) and a shell cost. By determining the number of each type of pumps selected, we can get the pumping station capital cost (PStaCC). For the selected pumps, the maximum allowable flow rate should be no less than the real flow rate, and the summation of the pump’s maximum pumping elevations should be no less the elevation between regions plus the head loss, which is given by the Hazen-William Equation. Incorporating the pipe selection, the head loss from region i to j can be written as:

j i d Y C Q b H p _p ijp p ij ij ) , , ~ ( 1.852⋅ ₄_.₈₇ ∀ ⋅ ⋅ = Δ α

∑

(8)

where b is conversion constant, αij is pumping distance between regions, and C is roughness coefficient.

3.5.Storage Capital Cost

The storage capital cost (StorageCC) is given by the unit storage cost, daily water demand (potable and non-potable), and water retention days.

3.6.Production Operating Cost

Similar to the plant capital costs in Section 3.2, the production operating costs are functions of the water/wastewater amounts. Piecewise linear approximations are used to calculate the above production operating costs of desalinated water, treated wastewater and reclaimed water. For desalination, dw

k

P~ and dw k

PEC are the production amount and unit energy usage at break point k, EPis the cost of electricity and Nt is the duration of season t. dw

it

P are the daily production variables during season t and dw ikt

ξ are SOS2 variables, while dw

it

X is a binary variable, which is equal to 1 if the desalination plant in region i has production during season t. The annual desalination production cost (DC) is given by:

∑∑∑

⋅ ⋅ ⋅ ⋅ = i k t dw ikt dw k dw k t EP PEC P N DC ~ ξ (9) , ~

∑

⋅ = k dw ikt dw k dw it P P ξ and Xdw i t it k dw ikt = , ∀,

∑

ξ (10)

Similar constraints for annual production costs of treated wastewater (WTC) and reclaimed water (WRC) are also used here.

3.7.Pumping Cost

The pumping cost (PumpingC) is equal to the pumping energy multiplied by the cost of electricity. The daily required pumping energy is given by the daily water flow, elevation, head loss and pump efficiency. By using Eqs. (7) and (8), we have the following equations for the pumping energy for desalinated water:

t YG C Q d L b H Q g PumpE dw ijpt i j p dw p p ij ij dw p dw dw t ⋅ ⋅ ∀ ⋅ + ⋅ ⋅ ⋅ ⋅ =

∑∑∑

) ] , ~ ( [ ~ 1 1.852 87 . 4 ρ β (11) where βdw

is the desalinated water pump efficiency, ρ is the water density, and g is the gravity. The required pumping energy for wastewater and reclaimed water is derived similarly.

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3.8.Objective Function

The objective is to minimise the annualised total cost, including the operating costs and capital costs, which are annualised by the recovery capital factor (CRF):

1 ) 1 ( ) 1 ( ) ( − + + ⋅ ⋅ + + + + + + + + + = n n r r r StorageCC PStaCC PipeCC WRPCC WTPCC DPCC PumpingC WRC WTC DC OBJ (12) where r is the annual interest rate and n is the length of the project in years.

4.Case Study

Now, we apply the proposed model to Syros, an island in Aegean Sea, Greece. The existing infrastructure on the island is not considered in the problem. In the problem, the project is for 20 years, and an annual interest rate of 5% is used. The island is divided into 6 regions (R1-R6), and the population centre for every region is at sea level, apart from R1 which is at elevation of 250 m. The water demands and wastewater supply vary between summer time (high, 4 months) and the rest of a year (low, 8 months): Table 1. Estimated daily water demands and wastewater supply in Syros (summer/winter) (m3/day)

R1 R2 R3 R4 R5 R6 potable water demand 150/50 4000/2800 500/200 650/350 500/200 500/300 non-potable water demand 250/0 900/100 600/50 880/30 580/30 380/30

wastewater supply 150/50 3700/2600 200/100 300/150 300/150 450/250 The capital cost and operating production cost of the plants are piecewise linearised by 4 break points, where the capacity/production is 100, 1000, 2500 and 5000 m3/day, respectively. There are 8 types of available pipes (4 for desalinated and reclaimed water and 4 for wastewater), and 8 types of pumps (4 for desalinated and reclaimed water and 4 for wastewater). The water storage should satisfy two days’ demand in summer time. The proposed MILP model has been implemented in GAMS 22.8 using solver CPLEX 11.1 on a Intel Pentium 4 3.40 GHz, 1.00 GB RAM machine. The optimality gap was set to 5%. There are 1636 constraints, 1009 continuous variables, and 756 integer/binary variables in the model. After a computational time of 164 s, an objective of $2,954,339 is obtained. The breakdown of the optimal annualised total cost is given in Fig. 2, while the optimal allocations of plants in each region and pipeline links are presented in Fig. 3.

Treated Wastewater Production Cost, $43,754 Reclaimed Water Production Cost, $6,774 Desalinated Water Production Cost, $1,048,695 Pumping Cost, $44,128 Desalination Plant Cost, $274,144 Wastewater Treatment Plant Cost, $461,876 Water Reclamation Plant Cost, $94,204 Pipeline Cost, $166,624 Pumping Station Cost, $20,542 Storage Cost, $793,599 1 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2 1 6 5 4 3 2

Fig. 2. Breakdown of the optimal objective Fig. 3. Optimal plant allocations/pipeline links The optimal details of the optimal solution are shown in Table 2, including information for each established link (flow/direction, type of pipes, type and number of pumps). The optimal daily production of each plant is given in Fig. 4.

Desalination plant Wastewater treatment plant Water reclamation plant Desalinated water pipeline Wastewater pipeline Reclaimed water pipeline Desalination plant Wastewater treatment plant Water reclamation plant Desalinated water pipeline Wastewater pipeline Reclaimed water pipeline

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Table 2. Solution details for each established link link water type* pipe diameter

(in) flow rate (m3/day) direction pump type (m3/day) operating pump no. 1---6 dw 4 560.4 1Æ6 - 0 2---3 dw 6 1260.9 2Æ3 2400 1 3---4 dw 6 1260.9 3Æ4 2400 1 4Æ5 1200 1 4---5 dw 4 560.4 5Æ4 1200 1 2---3 rw 6 1260.9 2Æ3 2400 1 3---4 rw 4 560.4 3Æ4 720 1 1---6 ww 4 700.5 1Æ6 - 0

* dw: desalinated water, rw: reclaimed water, ww: wastewater

0 950 750 380 5000 3630 0 150 3700 2600 200 ₁₀₀ 300 ₁₅₀ 300 ₁₅₀ 450 300 0 0 0 0 2060 150 180 300 300 380 0 1000 2000 3000 4000 5000 D T R D T R D T R D T R D T R D T R D T R D T R D T R D T R D T R D T R Summer Winter Summer Winter Summer Winter Summer Winter Summer Winter Summer Winter

R1 R2 R3 R4 R5 R6 Region/Season D a il y V o lu m e (m 3/d ay)

Desalination Treatment Reclamation

Fig. 4. Optimal daily production/treatment volumes of all plants in summer time and winter time 5.Concluding Remarks

The problem of integrated water and wastewater resources management of a water deficient island has been addressed. A mixed integer linear programming model has been proposed for the problem, by minimising the annualised total cost. The optimisation-based approach has successfully been applied to the Greek island of Syros. 6.Acknowledgements

The authors gratefully acknowledge Mr. George Vakongios for his assistance on the estimation of water demand and wastewater production, Mr. Nektarios Katsiris for his assistance in water and wastewater pumping prising and calculations and Mr. Christos Lioumis for the estimation of the treatment facilities costs. S.L. acknowledges the financial support from ORSAS, KC Wong Education Foundation, UK FCO, and CPSE. References

[1] J. Medellín-Azuara, L.G. Mendoza-Espinosa, J.R. Lund, and R.J. Ramírez-Acosta, Water Sci. Technol., 55 (2007) 339.

[2] Y. Han, S. Xu, and X. Xu, Water Resour. Manage., 22 (2008) 911.

[3] M.C. Cunha, L. Pinheiro, J. Zeferino, A. Antunes, and P. Afonso, J. Water Resour. Plan. Manage.-ASCE, 135 (2009) 23.