Applied Energy 135 (2014) 142–157

Contents lists available at ScienceDirect

Applied Energy

journal homepage: www.elsevier .com/ locate/apenergy

Quantitative engineering systems modeling and analysisof the energy–water nexus

http://dx.doi.org/10.1016/j.apenergy.2014.07.1010306-2619/� 2014 Published by Elsevier Ltd.

⇑ Corresponding author at: Masdar Institute, Engineering Systems & Manage-ment, P.O. Box 54224, Abu Dhabi, United Arab Emirates. Tel.: +971 2 810 9333.

E-mail addresses: wlubega@masdar.ac.ae (W.N. Lubega), afarid@masdar.ac.ae,amfarid@mit.edu (A.M. Farid).

William N. Lubega a, Amro M. Farid a,b,⇑a Masdar Institute, Engineering Systems & Management, P.O. Box 54224, Abu Dhabi, United Arab Emiratesb MIT Mechanical Engineering Department, 77 Massachusetts Avenue, Cambridge, MA 02139, United States

h i g h l i g h t s

� A quantitative, engineering systems model of the energy–water nexus is developed.� System model built from bond graph representations of descriptive SysML functions.� Use of physics-based models and increased scope improves on existing approaches.� Model feasibility, advantages and application demonstrated in illustrative example.

a r t i c l e i n f o

Article history:Received 2 March 2014Received in revised form 25 July 2014Accepted 29 July 2014

Keywords:Power systemsWater resourcesWastewaterSustainable developmentBond graph modeling

a b s t r a c t

The energy–water nexus has been studied predominantly through discussions of policy options supportedby data surveys and technology considerations. At a technology level, there have been attempts tooptimize coupling points between the electricity and water systems to reduce the water-intensity oftechnologies in the former and the energy-intensity of technologies in the latter. To our knowledge, therehas been little discussion of the energy–water nexus from an engineering systems perspective. A previouswork presented a reference architecture of the energy–water nexus in the electricity supply, engineeredwater supply and wastewater management systems developed using the Systems Modeling Language(SysML). In this work, bond graphs are used to develop models that characterize the salient transmissionsof matter and energy in and between the electricity, water and wastewater systems as identified in thereference architecture. These models, when combined, make it possible to relate a region’s energy andmunicipal water consumption to the required water withdrawals in an input–output model.

� 2014 Published by Elsevier Ltd.

1. Introduction

1.1. Motivation

The energy–water nexus can be defined [1–4] as a system-of-systems composed of one infrastructure system with the artifactsnecessary to describe a full energy value chain and another infra-structure system with the artifacts necessary to describe a fullwater value chain. Large volumes of water are withdrawn andconsumed from water sources every day for electricity generationprocesses [5]. Simultaneously, extraction, treatment and convey-ance of municipal water and treatment of wastewater are depen-dent on significant amounts of electrical energy [5].

This energy–water nexus, which couples the critical systemsupon which human civilization depends, has long existed but isbecoming increasingly strained due to a number of global mega-trends [6]: (i) growth in total demand for both electricity andwater driven by population growth (ii) growth in per capitademand for both electricity and water driven by economic growth(iii) distortion of availability of fresh water due to climate change(iv) multiple drivers for more electricity-intensive water and morewater-intensive electricity such as enhanced water treatment stan-dards,water consuming flue gas management processes at thermalpower plants and aging infrastructure which incurs greater losses.

1.2. Literature gap

A number of discussions on the energy–water nexus have beenpublished in recent years. Overviews of the various challengesrelated to the nexus, as well as discussions of various policy optionsfor the amelioration of the risks can be found in [6–12]. Empirical

Nomenclature

DTW permissible temperature increases of water sourcesCG induced photocurrents at locations of solar photovoltaic

installationskG wind speed at locations of wind farmsH flue

G specific sensible heat content of generator flue gasH G lower heating value of fuel used at generatorspW osmotic pressure of water sources_MG process steam flow rate in thermal generators_MG fuel consumption of generators

IP RMS current drawn by pumps in water distributionnetwork

ID RMS current drawn by wastewater treatment facilitiesIF RMS current drawn by water treatment plantsIG RMS current supplied by generatorsIL RMS current drawn by all electrical load nodesIN RMS current drawn by pipes and pumps in non-potable

recycled wastewater distribution networkIL0 RMS current drawn by all electrical nodes excluding

current for water system purposesPD pressures imposed by wastewater treatment plants on

non-potable recycled wastewater distribution networkPF pressures imposed by water treatment plants on water

distribution networkPJ pressures at non-potable recycled wastewater demand

nodesPJ pressures at water demand nodesPW pressures of water sourcesQ P water flow rate through pipesQ D throughput of wastewater treatment facilities

Q dispD disposable effluent production rate at wastewater treat-

ment facilitiesQ rec

D non-potable recycled wastewater production rate atwastewater treatment facilities

Q E water demand rate at non-potable recycled wastewaterdemand nodes

Q F water supplied by water treatment and desalinationplants

Q brineF brine produced by water treatment and desalination

plantsQ evap

G water evaporation rate associated with electricalgeneration units

Q inG water withdrawal rate by generators

Q outG generator effluent flow rate

Q J water demand rate at demand nodesQ S water withdrawal rate from water storage unitsQ evap

S water evaporation rate from water storage unitsQ W water withdrawal rate from sourcesQ El water leakage rate at non-potable recycled wastewater

demand nodesQ Jl water leakage rate at demand nodesVP RMS voltages applied to pumps in water distribution

networkVF RMS voltages applied to water treatment plantsVG RMS voltage at generatorsVL RMS voltage at electrical loadsVN RMS voltages applied to pumps in non-potable recycled

wastewater distribution networkpatm atmospheric pressure

W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157 143

evaluations of the electricity-intensity of water technologies and thewater-intensity of electricity technologies have been reported andanalyzed in [13–18]. Efforts have been made towards physics-basedmodels in [19,20] in which formulations for estimating water use bythermal power plants based on the heat balance of the plant havebeen derived. An integrated operational view of the water andpower networks has also been presented as a simultaneous co-opti-mization for the economic dispatch of power and water [21–27].

Less literature is available on the development of tools forintegrated management of electricity and water supply systems.A decision support system for the United States based on an under-lying system dynamics model is described in [28]. The modelenables the exploration of various water and electricity policiesand relies on statistical relationships between the independentvariables of population and economic growth and the dependentvariables of electricity and water demand. Recent work [29] hasinterfaced the well known Regional Energy Deployment System(ReEDS) and Water Evaluation and Planning (WEAP) tools to createa platform for determining the water resource implications ofdifferent electricity sector development pathways. The platformuses empirical consumption and withdrawal coefficients reportedin [16] for the interface.

To the authors’ knowledge however, a transparent physics-based model that interfaces a model of the electricity system tomodels of the municipal water and wastewater systems enablingan input–output analysis of these three systems in unison hasnot been presented. This work attempts to present and apply thissystem-of-systems model.

1.3. Scope

This paper adopts, as it’s modeling scope, the engineered elec-tricity, water and wastewater infrastructure as well as critical

energy and matter flows across a system-of-systems boundaryencompassing these three interconnected systems.

1.4. Relevance

The holistic, integrated modeling approach presented in thispaper is of particular relevance to places with integrated electricityand water utilities (e.g. countries in the Gulf Cooperative Council(GCC)). The model enables the evaluation and comparison of differ-ent technology levers across the three systems of interest thusinforming management and government policy around water,environment and energy. The modeling approach may also beapplied to regions with separate electricity, water and wastewaterutilities to demonstrate potential areas of coordination.

1.5. Contribution

In this paper, a quantitative, physics-based, engineering sys-tems model of the energy–water nexus is developed as a first ofits kind. This is in contrast to the existing literature which eitherhas a smaller scope or uses empirical evaluations of water andenergy intensity. The model uses first-pass but often-cited engineer-ing models of various exchanges of mater and energy in andbetween the electricity, water and wastewater systems. Hence,the paper has a foundational nature in two regards. The first-passengineering models replace the various empirical data surveys onthe water intensity of energy technologies and energy intensityof water technologies [13,15]. Also, the first-pass models may berefined in the future as per the needs of the analytical application.The presented model builds upon a reference architecture previ-ously provided in [2] and is thus a reference model that can: (i)provide a foundation for qualitative discussions in the general case,

144 W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157

and (ii) be instantiated for quantitative analysis of a particular caseas in Section 7.

1.6. Paper outline

The remainder of this paper continues in four sections. The bulkof the paper is devoted to the development and presentation ofmodels. This is achieved sequentially. First, a high level systems-of-system view is given in Section 2. Then each of three componentinfrastructure systems: electricity, water, and wastewater areaddressed in Sections 3–5 respectively. Section 6 gives a recapitula-tion of the model in terms model outputs of interest. Section 7 thenpresents an illustrative example in which the developed models areused to analyze the exchanges of energy and water in a simple,hypothetical geographical region. Finally, Section 8 concludes.

2. Systems-of-systems modeling

This section delineates the system boundary and proceeds todescribe the modeling strategy that is employed for the followingthree modeling sections.

2.1. System boundary and context

Fig. 1 chooses the system boundary around the three engineer-ing systems of electricity, water and wastewater [1–4]. It alsodepicts the high level flows of matter and energy between themand the natural environment. The labels A through S representkey flows of interest across the system boundary and betweenthe three infrastructure systems. A subset of these are determinedfor the illustrative example in Section 7 and used to determinecertain measures of interest.

Potable Re

Delivered Water

Electrical energy for water supply

Disposable effl

Water withdrawal for electricity genera�on & storage

With

draw

al b

y w

ater

sup

ply

syst

em

Procesfuel

Fuel produ

Heat for cogenera�on

Boundary: Engineered W

Engineered water supply

systemLeaked w

ater

Evapora�velosses

Brine

S

M

L1

C

A

K

Electricsupply sy

D

N

Surface fresh water

Ground waterSea water

Fig. 1. System context diagram for combined e

Electricity, potable water, and wastewater are all primarily sta-tionary within a region’s infrastructure. In contrast, the traditionalfuels of natural gas, oil, and coal are open to trade. Consequently,the fuel processing function, though it has a significant water foot-print, is left outside of the system boundary. An advantage of thischoice of system boundary is that the three engineering systemsall fall under the purview of grid operators; and in some nations,such as the United Arab Emirates, all three grid operations areunited within a single semi-private organization. The systemcontext diagram shown in Fig. 1 makes it possible to relate aregion’s energy and municipal water use to the required waterand energy withdrawals in a complex input–output model.

2.2. Modeling strategy

The model presented here is built upon the foundation of a previ-ously developed graphical model described in SysML [1–4]. Thismodel showed that the energy–water nexus is not just composedof many component artifacts but also a large heterogeneity of func-tions; each with multiple input and output flows of matter andenergy. The model presented here is the quantification of theprevious work and thus develops a set of equations for each functionfound within the energy–water nexus. While the number ofequations and variables for each function is modest, the shear heter-ogeneity of functions within the energy–water nexus necessitatesthat the full model be quite large to cover the full scope of interest.

The modeling strategy for this work progresses in three steps asshown in Fig. 2. The system context diagram shown above allowsfor the mathematical modeling of each of the individual infrastruc-ture systems and their respective components. Each infrastructuresystem is taken in turn using the same modeling strategy. First, aSysML activity diagram [30] is presented for the full scope of the

Electricity

cycled Wastewater

Electrical energy for wastewater treatment

uent

sed

Raw fuel c�on

ater, Electricity, Wastewater systems

Wastewater management

system

Electrical losses

Solar irradia�on and wind

R

F E

B

ity stem

J

P

Evapora�ve Losses

Water

Altered Water

Other Energy

Collectable wastewater Q

Electrical energy for non-water use

G

H

Thermal losses

Power Plant Effluent I

Non-potable Recycled Wastewater

T

L2

Leaked water

lectricity, water and wastewater systems.

Presentation of SysMLActivity Diagram

Identification of Independentand Dependent Power Variables

Development of Function Model

Fig. 2. Modeling strategy.

W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157 145

engineered infrastructure system of concern. This allows for activ-ity blocks to be defined for each of the component functions withinthe engineered system. Second, in order to transition from thegraphical SysML language to a mathematical model each inputand output associated with an activity block is represented as aflow of power in its respective energy domain (e.g. electrical, flu-idic, thermal). Each power flow has two power associated variablesthat can be referred to as an effort and a flow; the product of whichis the power. Examples of the former are voltages and pressures,while examples of the latter are currents and volumetric flow rates.The power variables associated with each power flow on the SysMLdiagram can be distinguished as independent variables which areimposed on the function and dependent variables. Finally, mathe-matical models that express the dependent variables as functionsof the independent variables are developed. As mentioned in Sec-tion 1.5, the chosen models for each function are first-pass butoften-cited engineering models. In the future, because the modelingstrategy is very much modular, a practitioner may replace any ofthese functions with a more detailed physics-based model so longas it respects the high level functional interfaces. Alternatively, thismodular modeling strategy can incorporate functional modelsdeveloped through rigorous system identification.

The development of a transparent physics-based model for theenergy–water nexus naturally brings about considerations of itspractical implementation; both in physical modeling as well asnumerical computation. In regards to physical modeling, thereexist industrial grade multi-energy domain physical modeling soft-ware packages well-accepted by industry and academia alike.Dymola/Modelica [31], for example, has an active user community,lends itself to optimization problems [32] and has been demon-strated on models with over 100,000 equations [33]. Similarly,the General Algebraic Modeling System [34] provides an optimiza-tion-oriented modeling language. It’s in-built nonlinear solvershave also been demonstrated on systems with over 100,000 equa-tions [35].

Grid equations of the electricity, water and wastewater gridsare modeled by means of edge-node incidence matrices. Theremaining identified functions are developed using the bond graphmethodology. The interested reader is referred to [36,37] for anintroduction to bond graph modeling. The bond graph methodol-ogy presents a number of practical advantages in this work. First,this methodology readily facilitates the inter-energy-domain mod-eling necessitated by the heterogeneous nature of the energy–water nexus. Next, it clearly distinguishes between the directional-ity found in the SysML activity diagrams and the causality of theassociated quantitative models. As a result, it clarifies the depen-dence and independence of variables in the input–output modelat the level of each individual function and thus ultimately for

the full scope of the energy–water nexus model. This aspect isparticularly necessary to address the potential for closed loops ofmatter and energy. The bond graph methodology also allows thereuse of mathematical models within multiple functions as willbe seen repeatedly with components such as pumps. Finally, thebond graph methodology facilitates the development of morecomplex models of the system functions either to incorporatepreviously neglected functionality or introduce dynamic effects.

2.3. Conventions of modeling notation

A wide range of variables across the three systems of interestare modeled in this work. The reader is referred to the nomencla-ture section for a full description of variables used in the models inSections 3–5. As a general guide however, the convention used forthe power variables is that boldface upper case letters are used torepresent vector quantities and lower case letters are used to rep-resent scalar quantities. For uniformity, this convention is followedeven with variables for which it is common to represent scalarquantities with upper case letters; for example, lower case t is usedto represent scalar temperatures rather than the more commonupper case T. The letters used to represent the different powervariables are as follows:

� i 2 I – RMS current� v 2 V – RMS voltage� q 2 Q – Volumetric flow rate� p 2 P – Pressure� h 2 H – Specific enthalpy� t 2 T – Temperature� m 2M – Fuel consumption rate� m 2M – Steam mass flow rate

Additionally, the subscripts of these variables are denoted bythe set of facilities to which they belong. Lower case subscriptsare used for scalar quantities while boldface upper case subscriptsare used for vector quantities:

� f 2 F – Node in the water distribution network with knownpressure (water treatment plant)� j 2 J – Node in water distribution network with unknown

pressure� e 2 E – Node in non-potable recycled wastewater distribution

network with unknown pressure� p 2 P – Water distribution link (includes both pipes and

distribution pumps)� n 2 N – Non-potable recycled wastewater distribution link

(includes both pipes and distribution pumps)� g 2 G – Generator� l 2 L – Electrical load� t 2 T – Electrical transmission line� w 2W – Water source i.e. river, lake, aquifer or the sea� s 2 S – Water storage units, both tanks and artificial reservoirs� d 2 D – Wastewater treatment plant

3. Model of the electricity system

Fig. 3 shows the electricity system as a value chain. The fourmost prominent electricity generation technologies of thermal,hydroelectric, wind and solar PV are shown. These technologieshave varying water withdrawal and consumption footprints.Thermal generation requires large volumes of water for coolingpurposes and this requirement is one of the chief concernsassociated with the energy–water nexus. Hydroelectric powerrequires flowing water to drive generating turbines. Solar panelsand wind turbines have embedded water consumption due to their

Fig. 3. Activity diagram of electricity system functions.

146 W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157

manufacturing processes but do not have any significant waterrequirements in operation. Bond graph models for each of theseare developed below following the process in Fig. 2.

3.1. Generate electricity-hydro

Independent variables: pinw ; p

outw ; ig .

Dependent variables: qing ; q

outg ;vg .

Model function:

qing ¼ fhydro

qinð½pin

w ;poutw ; ig �Þ

qoutg ¼ fhydroqout

ð½pinw ;p

outw ; ig �Þ

vg ¼ fhydrovð½pin

w ; poutw ; ig �Þ

Fig. 4 shows a bond graph model of a hydroelectric power genera-tion station maintaining the input and output interfaces of the asso-ciated activity in Fig. 3. Water drawn from a reservoir at a pressurepin

w drives a turbine that converts the hydraulic energy of the fallingwater into mechanical energy and that is represented in Fig. 4 withan ideal gyrator of modulus k1 as in [38]. The turbine, in turn, drivesan AC generator for which a simple model presented in [36] is agyrator modulated by the displacement of the rotor. As the interestin this work is not in the dynamic behavior of this generator, thismodel is further simplified to an ideal gyrator with modulus k2 as

Se : pinw 1

qing

1 GYk1

Se : poutw

qoutg

Rf

1

Rm

GYk2

1

Re

Sf : igvg

Fig. 4. Bondgraph representation of hydroelectric power generation.

shown in the figure. Elements Re;Rm and Rf represent resistancesin the electrical, mechanical and fluidic domains.

Making the simplifying assumptions that the penstock resis-tance (Rf ) is linear, and that there is no spillage (qin

g ¼ qoutg ¼ qg),

the voltage vg generated by the power station and the water qg

withdrawn from the water source to drive generation can be deter-mined as:

vg ¼k1k2

k21 þ RmRf

" #pw �

k2Rf

k21 þ RmRf

þ Re

" #ig ð1Þ

Rm

k1 þ Rf Rm

� �pw þ

k2

k1 þ Rf Rm

� �ig ð2Þ

where pw ¼ pinw � pout

w . The model in Fig. 4 does not show the evap-orative losses associated with hydroelectric generation. There doesnot exist any causality between the power generation process andthese losses as the rate of evaporation from a particular reservoirwill not be altered if the dam is generating below or at its full capac-ity. Furthermore, the water stored in the reservoir is often used notonly for power generation but for water supply and thus the evap-oration cannot be fully ascribed to electricity generation. The evap-orative consumption is therefore best allocated to the reservoir as inSection 4.4 and not to the power generation process.

3.2. Generate electricity thermal

Independent variables: ig ; hg ; tinw ; t

outw .

Dependent variables: vg ; _mg ; qing ; q

outg .

Model function:

vg ¼ fthermalv ð½ig ;hg ; tinw ; t

outw �Þ

_mg ¼ fthermal _mð½ig ;hg ; tin

w ; toutw �Þ

qing ¼ fthermal

qinð½ig ; hg ; tin

w ; toutw �Þ:

qoutg ¼ fthermalqout

ð½ig ;hg ; tinw ; t

outw �Þ

Many types of thermal power generation facilities exist today.They may be classified by fuel type (e.g. natural gas, coal,heavy fuel oil), cooling system type (once-through, semi-closed,

Fig. 6. Bondgraph representation of boiler feed pump.

W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157 147

closed-circuit), number of thermodynamic cycles (single, double,triple), and finally number of integrated products (e.g. electricity,steam, syngas) [39] Naturally, it is beyond the scope of this paperto present a model for all of these possible combinations. Rather,for the purposes of demonstrating the holistic nature of this work,a thermal generation facility is taken to be a single rankine cyclepower plant with a once-through cooling system. Future workmay introduce models for other types of thermal generationfacilities as is deemed necessary.

Fig. 5 shows a word bond graph for such a plant. The bondsinternal to the cycle (shown in red) are convection bonds whichhave two effort variables and a single flow variable [37]. The flowvariable is the mass flow rate of the steam, while the effortvariables can be any two intensive properties – in this work thepressure and enthalpy are used as in [37].

T=G in Fig. 5 refers to the turbine and generator; i�g is the totalcurrent generated by the power plant, which is the sum of thecurrent required by the pump and the current ig drawn by theelectrical network identified as an independent variable.

In order to develop an input–output model for thermal genera-tion in the same manner as above for hydroelectric generation, it ishelpful to develop individual input–output models for the differentcomponents of the rankine cycle and subsequently combine them.These models are presented in the following subsections. Themodels lean significantly on the convection bond graph conceptsdeveloped in [37]. The subscripts 1 to 4 used in the followingsubsections to represent different states of the process steam/water as follows:

(i) State 1 is the state of the process water at the condensorexit.

(ii) State 2 is the state of the process water at the pump exit.(iii) State 3 is the state of the process steam at the boiler exit.(iv) State 4 is the state of the process steam at the turbine exit.

It is assumed that there is a control system that ensures that theprocess water exiting the condensor is always saturated at aknown temperature and pressure, that is to say that state 1 is a ref-erence point for the rankine cycle. All other states vary with elec-tric power demand. Each of the rankine cycle components is nowdiscussed in turn taking into consideration that the dependentvariables of an ‘‘upstream’’ component are the independentvariables of the ‘‘downstream’’ component.

3.2.1. Component 1: PumpIndependent sub-variables: Inlet pressure and specific

enthalpy, p1, h1; Voltage applied to pump vg .Dependent sub-variables: Outlet pressure and specific

enthalpy, p2, h2; Current drawn by pump ipump.Component model: A bond graph representation of the boiler

feed pump, assumed isentropic, is shown in Fig. 6. Typically acentrifugal pump is used [40], which is best represented by amodulated gyrator, (see Section 4.5), however a simple gyrator will

Fig. 5. Rankine cycle word bond graph.

be used here. The vertical bond from the 1-junction is a simplerather than convection bond with a single effort equal to theenthalpy difference between states 1 and 2; it allows connectionof convection bond graphs to other parts of the system [37].

From the model, h2 and the current ipump are readily determinedas:

h2 ¼ h1 þk1

k2vg

ipump ¼k1

k2m1 _mg

ð3Þ

where m1 is the specific volume of the water (assumed saturated)entering the pump. Assuming incompressibility ðdm ¼ 0Þ and anegligible change in water temperature in the pump, it follows thatdh ¼ mdP. The outlet pressure p2 is thus given by:

p2 ¼ p1 þðh2 � h1Þ

m1ð4Þ

3.2.2. Component 2: BoilerIndependent sub-variables: Inlet pressure and specific

enthalpy, p2;h2; Lower heating value of fuel hg .Dependent sub-variables: Outlet pressure and specific

enthalpy, p3;h3; Fuel consumption rate _mg .Component model: A bond graph of a boiler is presented in

Fig. 7. tg and tamb represent the boiler flame and ambient tempera-tures respectively. Heat input to the boiler ð _mghgÞ is lost throughthe flue gas and the boiler wall with the rest being usefully trans-ferred to the process steam. The heat lost in the flue gas consistssolely of sensible heat since the lower heating value of the fuel,hg is used and thus the latent heat of the water vapor content ofthe flue gas has already been excluded. Mass of both the processsteam and the fuel are conserved. The HS element represents anon-reversible heat exchanger [37] and is used here to representthe heat transfers from the fuel to the boiler wall, and from theboiler wall to the process steam. By conservation of energy, _mg isgiven by:

_mg ¼ _mgðh3 � h2Þ þ k4ðtg � tambÞ

hg � hflueg

ð5Þ

where k3 is the thermal conductivity of the boiler-ambient interfaceand hflue

g is the specific sensible heat content of the flue gas in J/kgwhich is a function of the specific heat capacity and temperature(approximately equal to tg) of the flue gas. The right hand side ofEq. (5) includes the unknown h3 which can be determined withthe aid of the turbine model below. There is no change in pressurein the boiler and thus the outlet pressure p3 ¼ p2.

3.2.3. Component 3: TurbineIndependent sub-variables: Pressure of the inlet steam p3;

Current i�g .Dependent sub-variables: Pressure and specific enthalpy at the

outlet p4;h4; Specific enthalpy at the inlet, h3; Voltage vg .

Fig. 7. Bondgraph representation of boiler.

Fig. 9. Bondgraph representation of once-through condensor.

148 W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157

Component model: Fig. 8 shows a bond graph of the turbineassumed to be isentropic. As described above, the simple bondallows connection of convection bond graphs to other parts ofthe system [37], in this case a turbine, modeled as a displacementmachine and thus represented by a transformer. The transformermodulus is a ratio m3=D where D is the volumetric displacementper radian of shaft rotation and m3 is the specific volume of thesteam entering the machine [37]. The turbine shaft in turn drivesa generator represented by a gyrator with modulus k4.

From the transformer and gyrator, it follows that:

h4 ¼ h3 �m3k4

Di�g ð6Þ

vg ¼m3k4

D_mg � Rei�g ð7Þ

Given that the expansion is assumed to be isentropic and thatp4 ¼ p1 (there is no pressure drop in the condensor), the unknownh3 in Eq. (6) can be determined by simultaneously solving with thefollowing equations:

s3 ¼ s4

s4 ¼ fs4ðp4; h4Þ ¼ fs4

ðp1;h4Þh3 ¼ fh3

ðp3; s3Þð8Þ

where fs4and fh3

are polynomial regressions of appropriate portionsof the superheated steam tables.

3.2.4. Component 4: CondensorIndependent sub-variables: Inlet pressure and specific

enthalpy p4;h4; Cooling water inlet temperature tinw; Cooling water

outlet temperature toutw .

Dependent sub-variables: Outlet pressure and specificenthalpy, p1;h1; Cooling water withdrawal rate qin

g ; Cooling watereffluent flow rate qout

g .Component model: Fig. 9 is a bond graph representation of a

once-through condensor using the H;HS, and two port resistanceR elements. The H element represents an ideal, reversible heat

Fig. 8. Bondgraph representation of isentropic turbine.

exchanger [37]. The R element represents the condensor wall, athermal resistance, across which heat is conducted to the coolingwater.

The true effort variable associated with the cooling water volu-metric flow rate is the specific enthalpy multiplied by the coolingwater density, however since the cooling water is not subjectedto a change of pressure or density in the condensor, changes inenthalpy are linearly related to changes in temperature. The useof temperature is particularly convenient because cooling wateroutlet temperature tout

w is typically limited by regulation for ecosys-tem protection purposes, and tin

w and toutw in Fig. 5 can be replaced

with a single effort variable Dtw associated with the flowqg ¼ qin

g ¼ qoutg . By conservation of energy:

qg ¼_mgðh4 � h1ÞqwcwDtw

ð9Þ

3.2.5. Combined thermal generation modelThe combined model for the thermal generation plant is thus

given by:

vg ¼m3k4

D� Re

k1m1

k2

� �_mg � Reig ð10Þ

qing ¼ qout

g ¼ qg ¼_mg

qwcwDtwh4 � h1ð Þ ð11Þ

_mg ¼ _mgðh3 � h2Þ þ k3ðtg � tambÞ

hg � hflueg

ð12Þ

where the specific enthalpies h2;h3;h4 are determined by simulta-neously solving Eqs. 3, 4, 6 and 8, and where Eq. (10) is obtainedby substituting for i�g ¼ ig þ ipump in Eq. (7).

3.3. Generate electricity – wind

Independent variables: kg ; ig .Dependent variables: vg . (The wind force corresponding to kg

is not of interest and is thus not modeled.)Model function:

vg ¼ fwindv ð½kg ; ig �Þ

Fig. 10 is a simplified bond graph representation of a wind turbine.A more detailed model can be found in [41]. The gyrator is modu-lated by the windspeed, that is to say, the torque s is a functionof k2

g [41]. The transformer is a combination of the generator andan inverter that effects change of causality from a current to a volt-age. The generator voltage vg is given by:

vg ¼k1

k2k2

g � igRe ð13Þ

Sf : λg MGY

k1

TF

k2

1

Re

Sf : igvg

Fig. 10. Bondgraph model of wind turbine.

W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157 149

3.4. Generate electricity – solar photovoltaic

Independent variables: Cg ; ig .Dependent variables: vg . (The voltage vg þ igRe imposed on the

photocell is not of interest.)Model function:

vg ¼ fsolarv ð½Cg ; ig �Þ

A simple bond graph model of solar photovoltaic installationpreviously presented in [4] is reproduced in Fig. 11. The solar PVinstallation is approximated as a current source, a linear resistance,and a gyrator. In this model, the gyrator represents both the inver-ter switching circuit and the DC-to-DC conversion circuit. Thefunction of the former is approximated by the change of causalityfrom flow source to effort source effected by the gyrator while thatof the latter is effected by the magnitude of the gyrator modulus.The resistance represents electrical losses in both processes.

From the model, the voltage vg imposed on the electricaldistribution network can be determined as:

vg ¼Cg

k� igRe ð14Þ

3.5. Transmit electricity

Independent variables: VG; IL .Dependent variables: IG;VL .Model function:

IG ¼ ftransmitIð½VG; IL�Þ

VL ¼ ftransmitVð½VG; IL�Þ

The transmission of electricity through an electricity distribu-tion network is typically modeled in the phasor domain. We definethe following phasor voltages and currents:

V G ¼ hVG VG

I G ¼ hIG IG

V L ¼ hVL VL

I L ¼ hIL IL

ð15Þ

where hVG ; hVL ; hIG ; hIL are diagonal ½G� G� matrices with entries ofthe form 1\hi; hi being the phase angle of the current or voltageat bus i. hVG ; hIL are assumed known. Two edge node-incidencematrices and one impedance matrix are also defined:

Sf : Γg GY

k

1

Re

Sf : igvg

Fig. 11. Bond graph representation of solar PV installation.

� Electrical load node incidence matrix AL with dimensions½T � L�.� Generator node incidence matrix AG with dimensions ½T � G�.� Transmission line impedance matrix RT with dimensions½T � T �.

Kirchoff’s Current Law and Ohm’s law respectively provide thefollowing equations:

AyLI T ¼ I L

AyGI T ¼ I G

AGV G þ ALV L ¼ RT I T

These can be rearranged to yield:

V L ¼ �A1I L þ A2V G

I G ¼ A3I L þ A4V Gð16Þ

where

A1 ¼ ðAyLR�1T ALÞ

�1

A2 ¼ A1AyLR�1T AG

A3 ¼ AyGR�1T ALA1

A4 ¼ AyGR�1T AG � AyGR�1

T ALA1AyLR�1T AG

The � denotes the transpose of a matrix. IG and VL can then bedetermined as the RMS values of I G and V L respectively. TheRMS load current IL can be decomposed into current drawn forwater purposes and current drawn for all other purposes as inFig. 3:

IL ¼ IL0 þ CLP IP þ CLFIF þ CLDID ð17Þ

where IL0 ; IP ; IF; ID are as defined in the nomenclature (expressionsto be derived in Sections 4 and 5) and are assumed to all be inphase; and CLP ;CLF;CLD are binary coupling matrices such thatCLP ðl;pÞ ¼ 1;CLFðl; f Þ ¼ 1 and CLDðl;dÞ ¼ 1 if pump p, water treat-ment plant f, and wastewater treatment plant d respectively areconnected to electrical load node l.

3.6. Summary of electricity system function models

Writing the model functions defined in Sections 3.1, 3.2, 3.3, 3.4and 3.5 in matrix form we have a combined model for the electric-ity system functions identified in Fig. 3:

VG ¼ fhydroVð½PW; IG�Þ þ fthermalV

½IG;H G;DTW� þ fwindVðkG; IG�Þ

þ fsolarVð½CG; IG�Þ

Q inG ¼ fhydro

Qinð½PW; IG�Þ þ fthermal

Q in½IG;H G;DTW�

Q outG ¼ Q in

G � Q evapG

_MG ¼ fthermal _M½IG;H G;DTW�

IG ¼ ftransmitIð½VG; IL�Þ

VL ¼ ftransmitVð½VG; IL�Þ

ð18Þ

For the particular case of the models developed in each of thepreceeding sections, Eq. (18) can be instantiated with Eqs. (1),(2), (10)–(13) and (14) written in matrix form:

VG ¼ K1 þ K2CGWPW þ K3kG þ K4CG � K5IG

Q inG ¼ F G1 ðIG;CGWDTWÞ

Q outG ¼ Q in

G � Q evapG

_MG ¼ F G2 ðIG;H GÞVL ¼ RMSðA1hIL IL � A2hVG VGÞIG ¼ RMSðA3hIL IL þ A4hVG VGÞ

ð19Þ

150 W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157

where CGW is a binary coupling matrix such that CGWðg;wÞ ¼ 1 ifgenerator g is connected to water source w. K1 is a ½G� 1� vectorwhose value K1ðgÞ is given by Eq. (10) for all thermal power gener-ators g. Similarly, K2;K3;K4 are ½G� G� diagonal matrices with diag-onal elements K2ðg; gÞ;K3ðg; gÞ;K4ðg; gÞ, given by Eqs. (1), (13) and(14) for all hydroelectric, wind, and solar power plants respectively.K5ðg; gÞ represents the electrical losses of each power plant g and isgiven by Eqs. (1), (10), (13) and (14) depending on the type of plant.F G1 and F G2 are vector functions with f g1

ðgÞ and f g2ðgÞ given by

Eqs. (11) and (12) respectively for all thermal generators g.

4. Model of the engineered water supply system

An activity diagram for the engineered water supply system isshown in Fig. 12. All water grid functions are dependent on electri-cal or thermal energy input. Pumping, either for extraction or dis-tribution, is responsible for the bulk of electrical energy consumedby the water system. Thermal desalination, typically in the form ofcogeneration plants that produce both water and electricity, is dri-ven by thermal energy input. Models of the four indicated watersupply options are presented below. Electricity consumption formunicipal water use is not modeled as it represents a plethora ofdifferent processes that are typically not under the purview of gridoperators.

4.1. Extract and treat ground water and surface water

Independent variables: qf ; pw;v f .Dependent variables: pf ; qw; if .Model function:

pf ¼ ftreatpð½qf ; pw; v f �Þ

qw ¼ ftreatqð½qf ;pw;v f �Þ

if ¼ ftreatið½qf ;pw;v f �Þ

The extraction and treatment of surface and ground water canbe effectively modeled as a pump as pumping consumes at least98% of the power at each treatment plant [15]. A bond graph

Fig. 12. Activity diagram of

representation of a treatment plant as a centrifugal pump drivenby an electric motor is shown in Fig. 13. The following input–out-put equations are derived from the model:

qw ¼ qf ð20Þ

if ¼k1k2

k21 þ RmRe

qf þRm

k21 þ RmRe

v f ð21Þ

pf ¼ �k2

1 þ ReRm þ k2Re

k21 þ ReRm

" #qf þ

k1k2

k21 þ ReRm

v f þ pw ð22Þ

4.2. Desalinate seawater (cogeneration)

Independent variables: Distillate demand from water distribu-tion network qf , Temperature of feed water tw, Heating steam flowrate _m.

Dependent variables: Enthalpy drop Dh of heating steam, brineoutput qbrine

f , Feed water withdrawal rate qw.Model function:

Dh ¼ fdesalChð½qf ; tw; _m�Þ

qbrinef ¼ fdesalC

qbrineð½qf ; tw; _m�Þ

qw ¼ fdesalCqð½qf ; tw; _m�Þ

Multistage flash (MSF) desalination, the dominant thermaldesalination process is typically integrated with thermal genera-tion in cogeneration plants as shown in Fig. 14 with the desalina-tion process deriving its requisite thermal energy from steamextracted from a back pressure turbine at the appropriate specificenthalpy to drive the desalination process [42–44].

Focusing on the flash desalination unit in Fig. 14 - the modelsfor the other units are unchanged from Section 3.2 as discussedbelow – a model of the desalination process can be developed fol-lowing the procedure in Fig. 2.

Fig. 15 is a bond graph representation of a flash desalination plant.Typically these plants consist of multiple stages of successively lower

water system functions.

Se : vf 1if

Re

GY

k1

1

Rm

GY

k2

1

Rf

Se : pw

qw

Sf : qf

pf

Fig. 13. Bondgraph representation of surfacewater/groundwater treatment plant.

Fig. 15. Bondgraph representation of a thermal (flash) desalination plant.

W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157 151

pressures and temperatures, however, as a simplification, a singlestage flash desalination plant is modeled in Fig. 15.

It is of interest to determine the enthalpy drop Dh of the heatingsteam. From the bond graph, assuming that there is no accumula-tion of vapor or liquid in the flashing stage, Dh can be determinedas:

Dh ¼ qf Z ð23Þ

where Z ¼ q hstgv ðtstg � twÞ

h i= _m Dtstg½ �. hstg

v is the vaporizationenthalpy at the stage temperature tstg ;q is the distillate density,and Dtstg is the temperature drop across the flashing stage.

In order to determine the brine output, an average recoveryratio [43] will be used:

qbrinef ¼ 1� RR

RRqf ð24Þ

The feedwater withdrawal rate qw is therefore:

qw ¼qf

RRð25Þ

As stated, thermal desalination is typically integrated with ther-mal power generation in a cogeneration plant. In this setup, theheating steam is taken from the exit of a back-pressure turbineand is superheated. It can be reasonably assumed that the enthalpyof the steam exiting the desalination unit is always greater than h1

in Section 3.2, that is, it is a saturated mixture. A condenser istherefore still required and the desalination unit can be thoughtof as utilizing ‘waste heat’. There is therefore no increase in fuelconsumption due to the addition of the desalination unit and fuelconsumption is still given by Eq. (12). The voltage vg generated bythe power generation cycle of the cogeneration plant as given byEq. (7) is also unchanged. The cooling water requirement for thecogeneration plant is reduced as some of the waste heat is usedfor the desalination process instead of being injected into thecooling water. This cooling requirement qg is readily determinedby comparison with Eq. (9) as:

qg ¼qw _mðh2 � Zqf � h3Þ

cwDtwð26Þ

Fig. 14. Word bond graph of MSF plant integrated with rankine cycle.

where Z is the coefficient identified in Eq. (23). Note that in the caseof a cogeneration plant, this cooling water is abundantly availablesea water. The primary concern from an energy water nexus per-spective of these withdrawals is therefore the long term effect ofthe returned water at an elevated temperature on marine ecosys-tems. Conversely, when the source is fresh water, there is an addedconcern of the vulnerability of power plant operations to freshwater scarcity. Ultimately, there is a trade-off between increasedwater temperatures Dtw and the freshwater requirement qg .

4.3. Desalinate seawater (membrane)

Independent variables: qf ; pw;v f .Dependent variables: pf ; qw; if ; qbrine

f .Model function:

pf ¼ fdesalMpð½qf ;pw;v f �Þ

qw ¼ fdesalMqð½qf ; pw;v f �Þ

if ¼ fdesalMið½qf ;pw;v f �Þ

qbrinef ¼ fdesalM

qbrineð½qf ; pw;v f �Þ

A reverse osmosis plant is similar to surface and ground watertreatment plants except that additional electrical energy is drawnto overcome the osmotic pressure pw, a parameter of the seawater.As a first approximation if can be given by:

if ¼k1k2

k21 þ RmRe

qf þRm

k21 þ RmRe

v f þ k5pw ð27Þ

Including brine output in a bond graph model of a reverseosmosis desalination plant would have necessitated the develop-ment of chemical bond graphs, a level of detail considered unnec-essary for this work. Instead, as in Section 4.2 a typical recoveryratio, RR can be used to determine the brine output:

qbrinef ¼ 1� RR

RRqf ð28Þ

The feedwater withdrawal rate qw is therefore:

qw ¼qf

RRð29Þ

4.4. Store water (tanks and artificial reservoirs)

Independent variables: Q S(artificial reservoirs), Q evapS .

Dependent variables: Q S(tanks), Q W.

Fig. 16. Bondgraph representation of centrifugal pump.

152 W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157

The two water storage functions in Fig. 12 are treated together.For tanks it is desired to have a model of the form Q S ¼ fstore1ðQ

evapS Þ

and for artificial reservoirs it is desired to have a model of the formQ W ¼ fstore2ðQ SÞ. Let lSðtÞ be an ½S� 1� vector of the quantities ofwater stored in a tank or artificial reservoir at a time t. Changesin this quantity of water are related to water injections into orout of the storage unit from the water distribution system in thecase of tanks and from upstream users in the case of artificial res-ervoirs Q S, evaporation losses Q evap

S and water withdrawals from awater source Q W as follows:

lSðt þ DtÞ ¼ lSðtÞ þ DtðQ S � Q evapS þ CSWQ WÞ ð30Þ

where CSW is a binary coupling matrix such that CSWðs;wÞ ¼ 1 if arti-ficial reservoir s withdraws water from water source w. Eq. (30) canbe recast for tanks as follows:

Q S ¼ Q evapS þ ðlSðt þ DtÞ � lSðtÞÞ=Dt ð31Þ

where Q S appears as a demand that can take on positive and nega-tive values in water distribution system as shown in Section 4.5.Similarly, Eq. (30) can be recast for artificial reservoirs as:

Q W ¼ CWSðQ S þ Q evapS þ ðlSðt þ DtÞ � lSðtÞÞ=DtÞ ð32Þ

In the case of the artificial reservoir, Q S is an independent var-iable determined by upstream uses such as water treatment plantsor hydroelectric power plants.

4.5. Distribute water

Independent variables: PF;Q J;VP ; patm;Q S;QevapS .

Dependent variables: Q F;PJ; IP ;Q Jl.Model function:

½Q F;PJ; IP ;Q Jl� ¼ fdistributeð½PF;Q J;VP ;Q S;QevapS �Þ

The distribution of water through a pipe network can be easilydescribed with the aid of two edge node-incidence matrices andone resistance matrix:

� Interior node incidence matrix BJ with dimensions ½P � J�.� Fixed pressure node incidence matrix BF with dimensions½P � F�.� Diagonal pipe and pump resistance matrix RP with dimensions½P � P �.

As the pressure loss due to the resistance of water pipes andpressure gain due to pumps are both non-linear functions of theflow rate, in the equations below, the dependent variable Q F isreplaced with the more computationally convenient pipe andpump flow rates Q P as is customary in water distribution networkanalysis [45]. Q F is then given by ByFQ P .

Leakages occur in pipes but are typically allocated to nodes aspressure-dependent demands for modeling purposes [45,46]. Asimple model, in which the node is treated as an orifice throughwhich the leakage Qjl is discharged [46] provides the relationship:

qjl ¼ bjpaj

j

where bj and aj are nodal parameters. Therefore, in matrix form, thenodal leakages are determined as follows:

Q Jl¼ bJðPJÞPJ ð33Þ

where bJ is a ½J � J� diagonal matrix with entries bjPaj�1j .

By continuity of mass, the previously discussed allocation ofnode leakages and the integration of storage as discussed inSection 4.4, the interior nodes ðJÞ are described by:

ByJ Q P þ Q J þ Q Jlþ CJSðQ S þ Q evap

S Þ ¼ 0 ð34Þ

The constitutive equations for the pipes and pumps provide:

RP ðQ P ÞQ P � BJPJ � BFPF ¼ 0 ð35Þ

Finally, in order to determine the current drawn by the pumpsin the hydraulic network, a model of a centrifugal pump is requiredsuch as the one in Fig. 16. A centrifugal pump can be modeled as amodulated gyrator with modulus w1 þ w2q [47] driven by anelectric motor. The current drawn therefore is given by:

ip ¼ kpðw1 þ w2qpÞqp ð36Þ

In matrix form, this can be written as:

IP ¼ KP WP ðQ P ÞQ P ð37Þ

where KP and WP are [P � P ] diagonal matrices with entries KP ðp;pÞand WP ðp; pÞ given by Eq. (36) for each pump p.

4.6. Summary of engineered water supply system equations

Writing the model functions defined in Sections 4.1, 4.2, 4.3 and4.5 in matrix form we have a combined model for the water systemfunctions identified in Fig. 12:

ðQ F;PJ;IP ;Q JlÞ¼ fdistributeð½PF;Q J;VP ;Q S;QevapS �Þ

IF¼ ftreatIð½Q F;PW;VF�Þþ fdesalMI

ð½Q F;PW;VF�Þþ fdesalCIð½Q F;PW;VF�Þ

PF¼ ftreatPð½Q F;PW;VF�Þþ fdesalMP

ð½Q F;PW;VF�Þþ fdesalCPð½Q F;PW;VF�Þ

DHF¼ fdesalCHð½Q F;TW; _M�Þ

Q brineF ¼ fdesalC

Q brineð½Q F;TW; _M�Þ

Q W¼ fdesalCqð½Q F;TW; _M�Þ

ð38Þ

For the particular case of the models developed in each of thepreceeding subsections, Eq. (38) can be instantiated with the scalarEqs. (20)–(23), (27) written in matrix form and combined with Eqs.(33)–(35) and (37), to yield the system in Eq. (39), where 1J and 1P

are ½J� J� and ½P � P � identity matrices respectively; CWF is a binarycoupling matrix such that CWFðw; f Þ ¼ 1 if water from source w iswithdrawn by treatment plant f ; K6;K7 and K8 are [F� F] diagonalmatrices with entries K6ðf ; f Þ;K7ðf ; f Þ and K8ðf ; f Þ given by the coef-ficients derived in Eq. (27) for all surfacewater, groundwater andreverse osmosis plants f ; K9 and K10 are are [F� F] diagonal matri-ces with entries K9ðf ; f Þ, and K10ðf ; f Þ given by the coefficientsderived in Eq. (22) for all surfacewater, groundwater and reverseosmosis plants f; and where ZF is an [F� F] diagonal matrix withentries ZFðf ; f Þ given by the coefficient derived in Eq. (23) for eachcogeneration plant f.

5. Model of the wastewater system

Fig. 17 is an activity diagram of wastewater system functions.As in the previous sections, the strategy in Fig. 2 will be followed.

5.1. Collect wastewater

Independent variables: Q J.Dependent variables: Q D.Model function:

Fig. 17. Activity diagram of wastewater functions.

W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157 153

Q D ¼ fcollectWWðQ JÞ

Wastewater collection, as shown in Fig. 17, typically does notrequire electric power input. Wastewater is typically conveyedby gravity-flow sewers [7] as wastewater treatment plants arebuilt at low elevations, close to the water bodies into which efflu-ent is to be discharged. The sewer hydraulics will thus be ignoredin this model and the collection represented by a simple massbalance:

Q D ¼ CDJQ J ð40Þ

where CDJ is a binary coupling matrix with CDJðd; jÞ ¼ 1 if wastewa-ter treatment plant d treats wastewater from water demand node j.

5.2. Treat wastewater

Independent variables: Q D;Q F;QrecD .

Dependent variables: ID;QdispD .

Model function:

ID ¼ ftreatWWIð½Q D;Q F;Q

recD �Þ

Q dispD ¼ ftreatWWQ

ð½Q D;Q F;QrecD �Þ

Electrical energy is required for the treatment of wastewaterand distribution of recycled wastewater. Bond graphs can be usedto model chemical reactions [48] such as biochemical processesfound in wastewater

bJðPJÞ 0 1J 0

0 0 1J ByJBJ 0 0 �RP ðQ P Þ0 1P 0 �KP WP ðQ P Þ

26664

37775

PJ

IP

Q Jl

Q P

26664

37775

þ

0Q J þ CJSðQ S þ Q evap

S ÞBFPF

0

26664

37775 ¼

0000

26664

37775

Q F ¼ ByFQ P

IF ¼ K6Q F þ K7VF þ K8CyWFPW

PF ¼ K9Q F þ K10VF þ CyWFPW

DHF ¼ ZFQ F

ð39Þ

treatment facilities [49]. However, as first-pass models are desirableto limit the complexity of this research, an approximation is madesuch that the current drawn by wastewater treatment linearlydepends on the volume flow rate of the water it treats. This isreasonable as power system voltages are typically maintained closeto 1.0 per unit and thus often reported empirical measures of

energy consumption per unit volume of treated water [49] can berelated to electric current demand:

ID ¼ KDQ D ð41Þ

where KD is a ½D� D� diagonal matrix consisting of the describedlinear factors. The treated wastewater may be recycled, as indicatedin Fig. 17, through a dedicated non-potable recycled wastewaterdistribution network (Section 5.3), or blended into the potablewater distribution network in which case the wastewater recyclingplant doubles as a water supply plant in Section 4. The water that isnot recycled and that is thus directly returned to a water sourceQ disp

D is given by:

Q dispD ¼ Q D � CDFQ F � Q rec

D ð42Þ

where CDF is a binary coupling matrix with CDFðd; f Þ ¼ 1 ifwastewater treatment plant d doubles as water supply plantf ;Q rec

D is the water recycled through a dedicated non-potablerecycled wastewater distribution network (Section 5.3) and Q F isas previously determined in Section 4.5.

5.3. Distribute non-potable recycled wastewater

Independent variables: PD;Q E;VN; patm.Dependent variables: Q rec

D ;PE; IN;Q El.Model function:

½Q recD ;PE; IN;Q El� ¼ fdistributeRWWð½PD;Q E�Þ

The distribution of recycled wastewater can be described usingthe same equations as the Distribute Water function (Eqs. (33)–(37)) replacing Q F;PF;Q J;PJ;VP ; IP , and Q Jl with Q rec

D ;PD;Q E;PE;

VN; IN and Q El respectively.

5.4. Summary of wastewater management system equations

Writing the model functions defined in Sections 5.1, 5.2 and 5.3in matrix form we have a combined model for the wastewatersystem functions identified in Fig. 17:

Q D ¼ fcollectWWðQ JÞID ¼ ftreatWWI

ð½Q D;Q F;QrecD �Þ

Q dispD ¼ ftreatWWQ

ð½Q D;Q F;QrecD �Þ

½Q recD ;PE; IN;Q El� ¼ fdistributeRWWð½PD;Q E�Þ

ð43Þ

For the particular case of the models developed in each of thepreceeding subsections, Eq. (43) can be instantiated by combiningthe equations in Sections 5.1, 5.2 and 5.3 providing the followingsystem of equations:

Table 1Energy–water nexus mode – outputs of interest.

Identifier Formulation Identifier Formulation

A 1GQ inG

K 1FQ F

B 1G_MG L 1JQ Jl þ 1EQ El

C VyP IP þ VyFIF M 1FQ brineF

D _myGCGFDHF N 1DCDFQ F

E VyLIL0P 1JQ J

F VyGIG � ðC þ Eþ JÞ Q 1DQ D

G _MyGH flueG

R 1DQ dispD

H 1GQ evapG

S 1SQ evapS

I 1GQ outG

T 1EQ E

J VyDID

154 W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157

bEðPEÞ 0 1E 00 0 1E ByE

BE 0 0 �RNðQ NÞ0 1N 0 �KNWNðQ NÞ

26664

37775

PE

IN

Q El

Q N

26664

37775þ

0Q E

BDPD

0

26664

37775 ¼

0000

26664

37775

Q recD ¼ ByDQ N

Q D ¼ CDJQ J

ID ¼ KDQ D

Q dispD ¼ Q D � CDFQ F � Q rec

D

ð44Þ

where BD;BE;KN and WN are the equivalents of BF;BJ;KP and WP inSection 4.5.

6. Model outputs of interest

Once the quantitative engineering systems model has beendeveloped, its independent and dependent variables may be usedto extract model outputs of interest. Referring back to Fig. 1 the

Fig. 18. Illustration in

quantities A through T are given by the energy and matter flowsmodeled in Sections 3–5 (Table 1).

7. Illustrative example

In this section, the developed model is applied to an illustrativeexample inspired by Egypt. The threefold purpose of the example isto (1) demonstrate the practical feasibility of the model, (2) discussits advantages over existing work, nd (3) discuss how it may bepractically applied by an industrial practitioner. Fig. 18 presentsa conceptual illustration of a geographical region served by anumber of different water and power sources. A complete descrip-tion of the hypothetical system is described in Section 7.1. Then,the models developed in Sections 3–5 are used to solve formeasures of interest described in Section 6. These results aresummarized and discussed in Section 7.2. Section 7.3 concludeswith a brief discussion of potential industrial application.

7.1. System description and model parameters

The water distribution system is modeled as consisting of threewater sources, as indicated, which are equidistant from an aggre-gated demand node of 6 m3/s or approximately 140 million gallonsa day. The required hydraulic pressure for the distribution is pro-vided entirely by pumping to the clearwell at the treatment plants.In other words, there are no additional pumps in the distributionnetwork. A pressure equivalent to 100 m head is assumed for allthe plants. The resistance-coefficients of the three pipes servingthe demand node (RP in Eq. (39)) are determined using theHazen–Williams formulation assuming a pipe diameter of 2 m, aC-factor of 100 and a length of 500 m. The leakage model parame-ters aj and bj in Eq. (33) are taken as 0.5 and 0.001 respectively.

The electricity distribution network for this example is modeledwith the standard IEEE14 bus system [50] with the followingmodifications:

spired by Egypt.

Table 3Results.

Quantity Value Quantity Value

A 81 m3 s�1 K 9.4 m3 s�1

B 11.9 kg s�1 L 0.9 m3 s�1

C 8.5 MWe M 2.437 m3 s�1

D 116 MWth N 0 m3 s�1

E 254.2 MWe P 6 m3 s�1

F 8 MWe Q 6 m3 s�1

G 9.5 MWth R 6 m3 s�1

H 0 m3 s�1 S 5 m3 s�1

I 81 m3 s�1 T 0 m3 s�1

J 0 MWe

W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157 155

1. The 40 MW generator at bus 2 is replaced with a 100 MW gen-erator representing a hydroelectric power plant.

2. The synchronous compensators at buses 3, 6 and 8 are replacedwith 10 MW generators representing a solar PV plant, a windpower power plant and a thermal power plant respectively.The generator at slack bus 1 is taken as a cogeneration powerplant.

3. The groundwater treatment plant, and surface water treatmentplant are attached to the network at buses 5 and 14.

Functions fs4and fh3

in Eq. (8) were regressed with quartic poly-nomials using data from ASME superheated steam and saturatedwater tables over the pressure range 5–50 bar with the specificenthalpies in MJ/kg and the specific entropies in kJ/(kg K):

s4 ¼ �15:7881� 0:1049p4 þ 19:3438h4 þ 0:0039p24

�6:1322h24 � 0:0001p3

4 þ 0:9316h34 � 0:0547h4

4

ð45Þ

h3 ¼ �61:3940þ 0:0841p3 þ 33:4630s3 � 0:0032p23

�6:7618s23 þ 0:0001p3

3 þ 0:6129s33 � 0:0206s4

3

ð46Þ

These model parameters and others necessary for the completeexample are provided in Table 2.

The model was implemented in the General AlgebraicModelling System (GAMS) [34] and solved with the built-in CON-OPT solver [35]. No special computational facilities were required.

7.2. Results

The goal of this work was to develop a transparent physics-based model of the energy–water nexus model capable calculatingthe flows of matter and energy across and between the systemboundaries of the engineered electricity, water, and wastewatersystems. To this effect, this illustrative example calculates thevalues of A� T in Fig. 1 and summarizes the results in Table 3.Electricity system evaporative losses H are zero because onlyhydroelectric and once-through cooling plants have been modeledin this work. The wastewater system has not been modeled inthis simple example and thus it can be assumed thatR ¼ Q ¼ P ¼ 6 m3 s�1 and that N ¼ 0 m3 s�1.

The strength of the developed energy–water nexus becomesapparent when considering the various ratios of values A� T as

Table 2Model parameters.

Parameter Equation Units Value

k2=k1 (2) m3/As 0.072p1 (4) Pa 2� 106

h1 3, 4, 11 MJ/kg 0.909m1 (3), (4) m3/kg 0.001_mg (10)–(12) kg/s 70

hg (12) MJ/kg 26

hflueg

(12) MJ/kg 0.8

qw (11) kg/m3 1000cw (11) kJ/kg K 4.1813Dtw (11) K 10k1=k2 (3), (10) A s/m3 100k4=D (6), (10) V s/m3 or J/(A m3) 1000k2=k1 (21) A s /m3 100Z (23) (kJ/kg)/(m3/s) 720RR (24) – 35 %qevap

s 8s (32) m3/s 5qj8j (39) m3/s 6aj8j (39) – 0.5bj8j (39) (m3/s)/Pa 0.001pf 8f (39) Pa 9:8� 105

rp8p (39) Pa/(m3/s) 1057

aggregate measures of interest. While the existing literature pro-vides empirical evaluations of the electricity-intensity of watertechnologies and water-intensity of electricity technologies[13–18], this model calculates these results a priori over a largersystem boundary. Consider, for example, the following ratios:

� A measure of the degree of coupling between the electricity andwater systems given by ðC þ JÞ=ðC þ Eþ F þ JÞ ¼ 3:2%.� Water supply required to sustain the electricity and water

supply systems given by Aþ K ¼ 91 m3=s.� Ratio of water displaced from its original source to total water

withdrawn for water and electricity systems given by ðLþ HÞ=ðAþ KÞ ¼ 1:1%.� Proportion of water withdrawn that is returned with signifi-

cantly altered quality (a measure of environmental impact)given by ðM þ RÞ=ðAþ KÞ ¼ 9:3%.

In a detailed analysis, the values of each these high-level aggre-gate measures can be functionally traced back to each of the powervariables associated with the functions connected to values A� T.Thus, a change in exogenous variables like water and electricitydemand growth will have subsequent and predictable changes inthe water intensity of energy technologies and the energy intensityof water technologies. Furthermore, the resulting flows across thesystem boundary such as leaked water, brine rejection, disposableeffluent, thermal and electrical and energy losses can all becalculated. In contrast, the empirically-based studies found in theliterature would assume that the water and energy intensity valueswere constant irrespective of measurement or operating condi-tions. Similarly, the flows across the system boundary would bebased upon these measurement assumptions. Additional measure-ments would have to be conducted to provide values other thanjust water and energy intensities. The model also well addresseschanges to endogenous variables. These include the prevalence,arrangement, and type of water and energy technologies. For exam-ple, the model can resolve the effect of a change in the penetrationlevel, and placement of a new (potentially undeployed) renewableenergy technology. In contrast, an empirical approach assumesexisting technology and does not consider the effects of technologyplacement. The developed model can be utilized to conduct suchdetailed analyses for large changes to the structure of an energy–water nexus in a particular region. Alternatively, a sensitivityanalysis approach based upon the jacobian of the energy–waternexus’ system of equation has already been reported [51].

7.3. Opportunities for industrial application

The advantages of a transparent physics-based energy–waternexus model described in the previous subsection can lead to anumber of practical industrial applications. At the smallest scale,the merits of competing water and energy technologies can be

156 W.N. Lubega, A.M. Farid / Applied Energy 135 (2014) 142–157

objectively assessed not just on cost, or their individual impactsbut also their impacts within the energy–water nexus as a system.A governmental regulator can use this information in permitting ofvarious types of facilities as it evaluates the net environmentalimpact. On a much larger scale and over a longer time horizon,governments can apply such a model to make planning decisionsacross the three networks. The derived aggregate measures canbe assigned an economic value to conduct integrated energy–water planning studies. Finally, because the model is physics-based it can be operationalized by integrated energy–waterutilities as is commonly found in the Gulf Cooperation Councilcountries. Such utilities can thus monitor energy and water inten-sities in real-time and perhaps make energy and water dispatchingdecisions accordingly. In regions with separate energy and waterutilities, such a model can be used to highlight areas of potentialcooperation. Further discussion of such opportunities for industrialapplication have been reported in [52,53].

8. Conclusion

This work has presented an integrated engineering systemsmodel for the electricity, water and wastewater systems basedon a previously presented reference architecture and developedwith the aid of bond graphs. The model provides a set of algebraicequations that relate electricity and water demands to: (i) therequisite system inputs and concomitant system outputs for a sys-tem boundary encompassing all three grids, and (ii) the salientexchanges of matter and energy between the three systems. Thismodeling approach allows the definition and determination ofaggregate system measures that can inform integrated planningprocesses. As an illustration, the presented model equations havebeen implemented in GAMS and used to determine a set of suchaggregate measures for a hypothetical geographical region.

Future work can apply the approach and models to both plan-ning and operations applications. The first-pass models used herecan be enhanced to the level of detail required for different appli-cations; critically however, in any such model refinements, theinputs and outputs of the individual functions identified in Figs. 3,12 and 17 remain the same.

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