DEVELOPMENT OF TWO-STAGE FRACTIONAL PROGRAMMING METHODS...

DEVELOPMENT OF TWO-STAGE FRACTIONAL PROGRAMMING

METHODS FOR ENVIRONMENTAL MANAGEMENT

UNDER UNCERTAINTY

A Thesis

Submitted to the Faculty of Graduate Studies and Research

in Partial Fulfillment of the Requirements

for the Degree of

Master of Applied Science

in Environmental Systems Engineering

University of Regina

By

Xiong Zhou

Regina, Saskatchewan

December, 2014

Copyright 2014: X. Zhou

UNIVERSITY OF REGINA

FACULTY OF GRADUATE STUDIES AND RESEARCH

SUPERVISORY AND EXAMINING COMMITTEE

Xiong Zhou, candidate for the degree of Master of Applied Science in Environmental Systems Engineering, has presented a thesis titled, Development of Two-Stage Fractional Programming Methods for Environmental Management Under Uncertainty, in an oral examination held on December 19, 2014. The following committee members have found the thesis acceptable in form and content, and that the candidate demonstrated satisfactory knowledge of the subject material. External Examiner: *Dr. Mehran Mehrandezh, Industrial Systems Engineering

Supervisor: Dr. Guo H. Huang, Environmental Systems Engineering

Committee Member: Dr. Stephanie Young, Environmental Systems Engineering

Committee Member: Dr. Chunjiang An, Adjunct

Chair of Defense: Dr. Guoxiang Chi, Department of Geology *via Teleconference

i

ABSTRACT

Due to the increasing contamination and resource-scarcity issues, environmental

systems management is essential to socio-economic development. However, formulating

relevant policies and strategies is often associated with a variety of complexities. It is

necessary for decision makers to identify desired management plans to reflect

multiobjective features that involve a trade-off between environmental protection and

economic development. Moreover, these complexities will be further intensified by

multiple formats of uncertainties existent in the related factors and parameters, as well as

their interrelationships. Therefore, efficient system analysis techniques for supporting

multiobjective environmental systems management under such complexities are required.

In this dissertation research, a set of two-stage fractional programming methods were

developed for managing environmental systems under uncertainty, including (a) a two-

stage fractional programming method for managing multiobjective waste management

systems, (b) a two-stage chance-constrained fractional programming method for

sustainable water quality management under uncertainty, and (c) a dynamic chance-

constrained two-stage fractional programming method for planning regional energy

systems in the province of British Columbia, Canada.

The proposed multiobjective optimization methods could address the conflicts

between two objectives (e.g. economic and environmental effects) without the demand of

subjectively setting a weight for each objective. Economic penalties were taken into

consideration as corrective measures against any arising infeasibility caused by a particular

realization of uncertainty, such that a linkage to pre-regulated policy targets was

ii

established. Furthermore, the methods facilitated an in-depth analysis of the interactions

between economic cost and system efficiency. The developed methods could provide

desired decision alternatives for managing environmental systems under various conditions.

iii

ACKNOWLEDGEMENT

I would like to express my sincerest gratitude to my supervisor, Dr. Gordon Huang,

for his constant and patient guidance in my graduate study. Without his extreme

encouragement and support during my research, I would not have successfully completed

this research.

I sincerely and gratefully acknowledge the support of the Faculty of Graduate Studies

and Research and the Faculty of Engineering during my graduate study at University of

Regina.

My grateful appreciation also extends to Dr. Hua Zhu, Dr. Wei Sun, and Dr.

Chunjiang An for their constructive advice with respect to my research, as well as Dr. Cong

Dong for her insightful suggestions. My further gratitude goes to Mr. Renfei Liao, Ms.

Yuanyuan Zhai, Mr. Yao Yao, Mr. Guanhui Cheng, Mr. Yurui Fan, Ms. Zhong Li, Ms.

Jiapei Chen, Ms. Shan Zhao, Mr. Shuo Wang, Mr Yang Zhou, Ms. Xiujuan Chen, and

many others in the Institute for Energy, Environment and Sustainable Communities, for

their kind support, assistance, and friendship.

Finally, I would like to thank my parents for their unconditional support of my

research endeavors. I am indebted to them for everything they do for me.

iv

TABLE OF CONTENTS

ABSTRACT ........................................................................................................................ i

ACKNOWLEDGEMENT ............................................................................................... iii

LIST OF TABLES ........................................................................................................... vi

LIST OF FIGURES ....................................................................................................... viii

CHAPTER 1. INTRODUCTION .................................................................................... 1

CHAPTER 2. LITERATURE REVIEW ........................................................................ 5

2.1. Deterministic optimization modelling of environmental management

systems ............................................................................................................................ 5

2.1.1. Linear programming .......................................................................................... 5

2.1.2. Mixed-integer programming.............................................................................. 6

2.1.3. Multiobjective programming ............................................................................. 8

2.1.4. Linear fractional programming .......................................................................... 9

2.2. Stochastic optimization modelling of environmental management systems ... 10

2.2.1. Two-stage stochastic programming ................................................................. 10

2.2.2. Chance-constrained programming ................................................................... 11

2.3. Summary ............................................................................................................... 12

CHAPTER 3. A TWO-STAGE FRACTIONAL PROGRAMMING METHOD FOR

MANAGING MULTIOBJECTIVE WASTE MANAGEMENT SYSTEMS ........... 14

3.1. Background ........................................................................................................... 14

3.2. Methodology ......................................................................................................... 17

3.3. Case study ............................................................................................................. 22

3.3.1. Overview of study system ............................................................................... 22

3.3.2. TSFP model for municipal solid waste management ...................................... 30

3.3.3. Results and discussion ..................................................................................... 34

3.4. Summary ............................................................................................................... 51

CHAPTER 4. TWO-STAGE CHANCE-CONSTRAINED FRACTIONAL

PROGRAMMING FOR SUSTAINABLE WATER QUALITY MANAGEMENT

UNDER UNCERTAINTY.............................................................................................. 53

4.1. Background ........................................................................................................... 53

v

4.2. Methodology ......................................................................................................... 56

4.2.1. Development of TCFP model .......................................................................... 56

4.2.2. Solution methods ............................................................................................. 61

4.3. Case study ............................................................................................................. 64

4.3.1. Overview of the study system ......................................................................... 64

4.3.2. Water quality simulation model ...................................................................... 73

4.3.3. TCFP model for water quality management ................................................... 74

4.3.4. Results and discussion ..................................................................................... 81

4.4. Summary ............................................................................................................... 96

CHAPTER 5. DYNAMIC CHANCE-CONSTRAINED TWO-STAGE

FRACTIONAL PROGRAMMING FOR PLANNING REGIONAL ENERGY

SYSTEMS IN THE PROVINCE OF BRITISH COLUMBIA, CANADA ................ 98

5.1. Background ........................................................................................................... 98

5.2. Overview of the British Columbia energy system ........................................... 101

5.2.1. The province of British Columbia ................................................................. 101

5.2.2. British Columbia energy system ................................................................... 105

5.2.3. Statement of problems ................................................................................... 106

5.3. Development of DCTFP-REM model ............................................................... 108

5.3.1. Dynamic chance-constrained two-stage fractional programming (DCTFP)

method ..................................................................................................................... 108

5.3.2. Development of the DCTFP-REM model ..................................................... 114

5.4. Result analysis .................................................................................................... 125

5.5. Summary ............................................................................................................. 171

CHAPTER 6. CONCLUSIONS ................................................................................... 174

6.1. Summary ............................................................................................................. 174

6.2. Research achievements ...................................................................................... 176

6.3. Recommendations for future research ............................................................. 177

References ...................................................................................................................... 178

vi

LIST OF TABLES

Table 3.1 Transportation and operation costs for target waste flows .............................. 25

Table 3.2 Transportation and operation costs for excess waste flows ............................. 26

Table 3.3 Different waste-generation rates and probability levels .................................. 27

Table 3.4 Target waste-flow levels from the city to the landfill and the composting and

recycling facilities ............................................................................................................ 28

Table 3.5 Capacity-expansion options and costs for the landfill and the composting and


Table 3.6 Solutions of the TSFP model for binary variables ........................................... 35

Table 3.7 Solutions of the TSFP model ........................................................................... 36

Table 3.8 Solutions of the TMILP model ........................................................................ 44

Table 4.1 Water consumption and wastewater discharge rates with the associated

probabilities...................................................................................................................... 67

Table 4.2 BOD concentrations of wastewater discharged and treatment efficiencies ..... 69

Table 4.3 Allowable BOD loading for each source ......................................................... 70

Table 4.4 Pre-regulated targets, product demands, benefit, and costs analysis for the

sectors .............................................................................................................................. 71

Table 4.5 Solutions obtained from the TCFP model ....................................................... 78

Table 4.6 Solutions obtained from the TCLP model ....................................................... 89

Table 5.1 Population, labour force, employment, and households in the province of

British Columbia ............................................................................................................ 103

Table 5.2 GDP, goods GDP, and services GDP in the province of British Columbia .. 104

Table 5.3 Solutions of primary energy suppliers for power generation

under qs = 0.01 ............................................................................................................... 126

Table 5.4 Solutions of primary energy suppliers for heat generation under qs = 0.01 .. 128

Table 5.5 Solutions of primary energy suppliers for cogeneration under qs = 0.01 ...... 129

Table 5.6 Solutions of primary energy suppliers for end-users under qs = 0.01 ............ 130

Table 5.7 Binary solutions for capacity expansions of power generation

under qs = 0.01 ............................................................................................................... 143

vii

Table 5.8 Binary solutions for capacity expansions of heat generation

under qs = 0.01 ............................................................................................................... 145

Table 5.9 Binary solutions for capacity expansions of cogeneration under qs = 0.01 .. 146

Table 5.10 Solutions of primary energy suppliers for power generation from TCMIP

under qs = 0.01 ............................................................................................................... 152

Table 5.11 Solutions of primary energy suppliers for heat generation from TCMIP

under qs = 0.01 ............................................................................................................... 154

Table 5.12 Solutions of primary energy suppliers for cogeneration from TCMIP

under qs = 0.01 ............................................................................................................... 155

Table 5.13 Solutions of primary energy suppliers for end-users from TCMIP

under qs = 0.01 ............................................................................................................... 156

Table 5.14 Binary solutions from TCMIP for capacity expansions of power generation

under qs = 0.01 ............................................................................................................... 160

Table 5.15 Binary solutions from TCMIP for capacity expansions of heat generation

under qs = 0.01 ............................................................................................................... 163

Table 5.16 Binary solutions from TCMIP for capacity expansions of cogeneration

under qs = 0.01 ............................................................................................................... 164

viii

LIST OF FIGURES

Figure 3.1 Overview of the study system ........................................................................ 24

Figure 3.2 The target waste flow levels and optimized waste flows to the landfill facility

.......................................................................................................................................... 41

Figure 3.3 The target waste flow levels and optimized waste flows to the composting and


Figure 3.4 Comparison of capacity expansion schemes from optimal-ratio and least-cost

models .............................................................................................................................. 47

Figure 3.5 Comparison of optimized waste flows to the landfill facility from optimal-ratio

and least-cost models ....................................................................................................... 49

Figure 3.6 Comparison of optimized waste flows to the composting and recycling

facilities from optimal-ratio and least-cost models .......................................................... 50

Figure 4.1 Schematic diagram of the study system ......................................................... 65

Figure 4.2 Target and planning production level for the wastewater treatment plant ...... 83

Figure 4.3 Target and planning production level for the paper plant ............................... 84

Figure 4.4 Target and planning production level for the leather plant ............................. 85

Figure 4.5 Target and planning production level for the tobacco plant ........................... 86

Figure 4.6 Target and planning area for the recreational sector ....................................... 87

Figure 4.7 The comparison of net benefits between optimal-ratio and TCLP models .... 92

Figure 4.8 The comparison of water consumption between optimal-ratio and TCLP

models .............................................................................................................................. 93

Figure 4.9 The comparison of system efficiency between optimal-ratio and TCLP models

.......................................................................................................................................... 94

Figure 5.1 Primary energy suppliers for power generation technologies

under qs = 0.01 ............................................................................................................... 133

Figure 5.2 Primary energy suppliers for heat generation technologies

under qs = 0.01 ............................................................................................................... 134

Figure 5.3 Primary energy suppliers for cogeneration technologies under qs = 0.01 .... 135

Figure 5.4 Primary energy suppliers for end-users under qs = 0.01 .............................. 136

ix

Figure 5.5 Electricity productions from different non-renewable power generation

technologies under qs = 0.01 .......................................................................................... 138

Figure 5.6 Electricity productions from different renewable power generation


Figure 5.7 Heat generation from different generation technologies under qs = 0.01 ..... 140

Figure 5.8 Electricity generation from different cogeneration technologies

under qs = 0.01 ............................................................................................................... 141

Figure 5.9 Capacity expansion schemes for different non-renewable power generation


Figure 5.10 Capacity expansion schemes for different renewable power generation


Figure 5.11 Capacity expansion schemes for heat generation facilities

under qs = 0.01 ............................................................................................................... 149

Figure 5.12 Capacity expansion schemes for cogeneration facilities under qs = 0.01 ... 150

Figure 5.13 Electricity productions from hydropower under qs = 0.01 ......................... 158

Figure 5.14 Electricity productions from wave/tide power under qs = 0.01 .................. 159

Figure 5.15 Capacity expansion schemes of non-renewable power generation

technologies from TCMIP under qs = 0.01 .................................................................... 165

Figure 5.16 Capacity expansion schemes of renewable power generation technologies

from TCMIP under qs = 0.01 ......................................................................................... 166

Figure 5.17 Capacity expansion schemes of heat generation facilities from TCMIP

under qs = 0.01 ............................................................................................................... 167

Figure 5.18 Capacity expansion schemes of cogeneration facilities from TCMIP

under qs = 0.01 ............................................................................................................... 168

Figure 5.19 The comparison of system costs between DCTFP-REM and TCMIP models

........................................................................................................................................ 169

Figure 5.20 The comparison of system efficiencies between DCTFP-REM and TCMIP

models ............................................................................................................................ 170

1

CHAPTER 1

INTRODUCTION

Environmental systems management is crucial to socio-economic development due

to the increasing contamination and resource-scarcity issues (Maqsood and Huang, 2003).

There are many concerns that must be taken into account in planning environmental

systems, such as economic, environmental, social, technical, and political factors, leading

to a variety of complexities in formulating relevant policies and strategies (Wilson, 1985).

In addition, it is necessary for decision makers to identify preferred management plans to

reflect multiobjective features that involve a trade-off between environmental protection

and economic development. Moreover, these complexities will be further intensified by

multiple forms of uncertainties existing in the related factors and parameters, as well as

their interrelationships (Li and Huang, 2009; Zhu and Huang, 2011). Therefore, efficient

system analysis techniques for supporting multiobjective environmental management

under such complexities are highly sought after.

Over the past decades, broad spectrums of optimization methods were developed for

planning environmental systems. Among them, multiobjective optimization methods were

widely used to provide desired management schemes under various system conditions.

Although these methods were helpful in tackling multiobjective environmental

management problems, most of them transformed the multiple conflicting objectives into

a single monetary measure based on unrealistic or subjective assumptions (Zhu et al.,

2014). However, it has been found that environmental concerns in environmental

2

management systems often involve moral and ethical principles that may not be related to

any economic use or value (Kiker et al., 2005; Zhu, 2014).

Linear Fractional programming (LFP), which could balance objectives between two

functions, e.g., cost/volume, output/input, or cost/time, is effective for dealing with the

multiobjective optimization (Charnes et al., 1978; Mehra et al., 2007; Stancu-Minasian,

1997a, 1999). It could not only intensify the comparative analysis regarding the objectives

of different aspects through using their original magnitudes, but also provide an

unprejudiced measure of system efficiency (Zhu et al., 2014). Moreover, it is especially

suitable for situations where solutions with better achievements per unit of inputs (e.g.

time, resource, and cost) are desired. In the past, LFP has been widely employed in various

fields, such as resource management, finance, production and transportation (Mehra et al.,

2007; Schaible and Ibaraki, 1983; Stancu-Minasian, 1997a, 1999). Although LFP was

widely applied in various areas ranging from engineering to economics (Mehra et al.,

2007), there were few studies on LFP for environmental management under uncertainty.

In environmental management systems, however, various kinds of uncertainties exist

in numerous system components as well as their interrelationships (Babaeyan-Koopaei et

al., 2003; Ghosh and Mujumdar, 2006; Miao et al., 2014; Sabouni and Mardani, 2013). In

the past, a large number of optimization techniques were employed for dealing with

uncertainties, such as stochastic mathematical programming (SMP), interval mathematical

programming (IMP), and fuzzy mathematical programming (FMP) (Lv et al., 2010; Zhu

et al., 2009). Chance-constrained programming (CCP), as one of the major branch of SMP,

is effective in dealing with optimization problems where the right-hand-side coefficients

are expressed as probability distributions (Huang et al., 2001; Li et al., 2007c). It is

3

necessary to provide a linkage to economic implications due to the violation of pre-

regulated environmental policies. Two-stage stochastic programming (TSP) is an

appealing method to handle the recourse problems, where an analysis of multi-stage

decisions is desired and uncertainties are presented as random variables in the objective

(Huang and Loucks, 2000; Li et al., 2007b; Luo et al., 2003; Maqsood and Huang, 2003).

The motivation for TSP is to take recourse or corrective action when uncertain future

events have occurred. In the TSP method, a first-stage decision is undertaken based on

random short-term events. After the random events are later resolved, a second-stage

decision will be correspondingly taken in order to minimize the expected costs (Birge and

Louveaux, 1988; Birge and Louveaux, 1997; Datta and Burges, 1984; Liu et al., 2003). As

well, facility expansion is a crucial issue in planning environmental management systems,

where integer variables are typically employed to indicate whether specific facility

expansion options are to be taken (Li et al., 2006b). Mixed-integer linear programming

(MILP) is a remarkable mathematical programming method for this purpose (Baetz, 1990a;

Huang et al., 1995; Huang et al., 1997; Huang et al., 2013; Li et al., 2008c). However,

CCP, TSP, and MILP were incapable of effectively analyzing multiobjective

environmental management problems.

Therefore, as an extension of the previous works, the objective of this research is to

develop a set of two-stage fractional programing methods for multiobjective

environmental management under uncertainty. The developed methods will have

advantages in reflecting trade-offs among conflicting objectives, complexities of multi-

stage decisions, system reliability under constraint-violation conditions, and dynamic

features of system behaviors, as well as their interactions. The tasks of this research are as

4

follows:

i) development of a two-stage fractional programming method for managing

multiobjective waste management systems;

ii) development of a two-stage chance-constrained fractional programming method

for sustainable water quality management under uncertainty; and

iii) development of a dynamic chance-constrained two-stage fractional programming

method for planning regional energy systems in the province of British Columbia, Canada.

This dissertation is divided into six chapters. Chapter 2 presents a comprehensive

literature review of the previous studies in environmental management. Chapter 3

introduces a two-stage fractional programming method for managing multiobjective waste

management systems. Chapter 4 describes a two-stage chance-constrained fractional

programming method for sustainable water quality management under uncertainty.

Chapter 5 outlines the development of a dynamic chance-constrained two-stage fractional

programming method for planning regional energy systems in the province of British

Columbia, Canada. Chapter 6 presents the conclusions of this dissertation research.

5

CHAPTER 2

LITERATURE REVIEW

2.1. Deterministic optimization modelling of environmental management systems

In the past, broad spectrums of deterministic mathematical programming approaches

have been developed for supporting environmental systems management, such as linear

programming, nonlinear programming, dynamic programming, integer/mixed-integer

programming, and multiobjective programming.

2.1.1. Linear programming

Linear programming (LP) was the most commonly used mathematical programming

method in environmental systems management and planning. Peirce and Davidson (1982)

formerly investigated the relative costs of regional and statewide hazardous waste

management schemes through employing linear programming techniques. Najm et al.

(2002) developed a linear programming model within the framework of dynamic

optimisation, which was employed to support a MSW management system taking into

account both environmental and socio-economic considerations. Everett and Modak (1996)

presented a deterministic linear programming model to aid decision makers in the long-

term scheduling of disposal and diversion options in a regional integrated solid waste

management system. Kondo and Nakamura (2005) provided a waste input-output model,

which was based on the method of linear programming. Fishbone and Abilock (1981)

described a linear-programming model of national energy systems, which was driven by

6

useful energy demands, optimized over several time periods collectively, and allowing

multiobjective analyses to be carried out quite easily. Suganthi and Williams (2000)

developed a linear programming model for providing the optimal allocation of the

renewable energy in the rural sector in India.

2.1.2. Mixed-integer programming

Mixed-integer linear programming (MILP) is a useful tool for dealing with capacity

expansion issues (Baetz, 1990a; Huang et al., 1995; Huang et al., 1997; Huang et al., 2013;

Li et al., 2008c). In fact, capacity expansion for environmental management facilities is a

crucial issue in the planning of environmental systems, where integer variables are

typically employed to indicate whether particular facility expansion options are to be

undertaken (Li et al., 2006b). Previously, the MILP method had been broadly employed

for this purpose (Cerda et al., 1997; Cheng et al., 2003; Croxton et al., 2003; Ku and Karimi,

1988; Little, 1966; Morais et al., 2010; Niemann and Marwedel, 1997; Raman and

Grossmann, 1993; Richards et al., 2002). Pinto and Grossmann (1995) proposed a

continuous-time mixed-integer linear-programming method for short-term planning of

multistage batch plants. Chakrabarty (2000) tested scheduling for core-based systems

through using mixed-integer linear programming. Costa and Oliveira (2001) introduced

an evolutionary algorithms approach to the solution of mixed integer non-linear

programming problems. Chang et al. (2001) experimented with mixed integer linear

programming based approaches on short-term hydro scheduling. Moore and Bard (1990)

described the mixed integer linear bi-level programming problem. Dua and Pistikopoulos

(2000) proposed an algorithm for the solution of multiparametric mixed integer linear

7

programming problems. Recently, Fazlollahi and Marechal (2013) developed

multiobjective, multi-period optimization of biomass conversion technologies using

evolutionary algorithms and mixed integer linear programming (MILP). Shabani and

Sowlati (2013) presented a mixed integer non-linear programming model for tactical value

chain optimization of a wood biomass power plant. Rueda-Medina et al. (2013) provided

a mixed-integer linear programming approach for optimal type, size and allocation of

distributed generation in radial distribution systems. Baetz (1990b) presented a dynamic

programming model for determining the optimal capacity expansion patterns for waste-to-

energy and landfill facilities over time. Revelle et al. (1968) applied linear programming

to the management of water quality in a river basin, where the objective function was

structured in terms of the costs of the treatment plants and the principal constraints

prevented violation of the dissolved oxygen standards. Vieira and Lijklema (1989)

developed a dynamic programming model for determining the optimal extent of regional

water and wastewater treatment, as well as wastewater diversion schemes in a river basin.

Liebman and Lynn (1966) presented a dynamic programming model that minimized the

cost of providing waste treatment to meet specified dissolved oxygen concentration

standards in a stream. Malik et al. (1994) outlined an integrated energy system planning

method for Wardha District in Maharashtra State, India and presented an optimal mix of

new/conventional energy technologies through using a computer-based mixed integer

linear programming model.

8

2.1.3. Multiobjective programming

Chang and Wang (1996) applied multiobjective mixed integer programming

techniques for addressing the potential conflict between economic and environmental

objectives and for evaluating sustainable strategies of waste management in a metropolitan

region. Su et al. (2008) proposed an inexact multiobjective dynamic programming

(IMODP) model for supporting MSW management under uncertainty, where two major

objectives were to minimize both the system cost and the environmental impact.

Santibanez-Aguilar et al. (2013) developed a multiobjective mixed-integer linear

programming model for planning a distributed system of processing facilities to treat

MSW while simultaneously considering economic and environmental aspects. Lohani and

Adulbhan (1979) applied the goal programming to a regional water quality management

problem where the following two objectives were considered: (1) to minimize total waste

treatment cost, and (2) to maintain the water quality objective (dissolved oxygen) close to

the minimum level stated in the stream standards.

Ramanathan and Ganesh (1993) developed a multiobjective programming model for

the allocation of energy resources to various energy end uses. Ren et al. (2010) proposed

a multiobjective goal programming approach to analyze the optimization operating

strategy of a distributed energy resource system while simultaneously minimizing energy

cost and environmental impact which is assessed in terms of CO2 emissions. Zhang et al.

(2012) developed a short-term multiobjective economic environmental hydrothermal

scheduling model, where the objective was to simultaneously minimize energy cost as well

as the pollutant emission effects. However, multiobjective optimization methods could not

9

effectively tackle practical energy management problems due to its need to transform

multiobjectives into a single measure based on unrealistic or subjective assumptions.

2.1.4. Linear fractional programming

Linear Fractional programming (LFP), which can balance objectives between two

functions, e.g., cost/volume, output/input, or cost/time, is effective for dealing with

multiobjective optimization (Charnes et al., 1978; Mehra et al., 2007; Stancu-Minasian,

1997a, 1999). It could not only intensify the comparative analysis regarding the objectives

of different aspects through using their original magnitudes, but also provide an

unprejudiced measure of system efficiency (Zhu et al., 2014). In the past, LFP has been

widely employed in various fields, such as resource management, finance, production and

transportation (Mehra et al., 2007; Schaible and Ibaraki, 1983; Stancu-Minasian, 1997a,

1999). For example, Gómez et al. (2006) described a timber harvest scheduling problem

in order to obtain a balanced age class distribution of a forest plantation through presenting

a linear fractional goal programming model. Lara and Stancu-Minasian (1999) proposed

a multiple objective linear fractional programming (MLFP) model for an agricultural

system, where solutions were obtained through maximizing the gross margin and

employment levels per unit of water consumption. Although LFP was widely applied in

various areas ranging from engineering to economics (Mehra et al., 2007), there were few

studies on LFP for environmental management under uncertainty. Recently, Zhu and

Huang (2011) introduced a stochastic linear fractional programming (SLFP) approach in

order to support sustainable MSW management under uncertainty.

10

2.2. Stochastic optimization modelling of environmental management systems

Deterministic mathematical programming approaches were developed for dealing

with various environmental problems. However, in practical environmental management

systems, various kinds of uncertainties exist in numerous system components as well as

their interrelationships. Multi-stage decisions and multiobjective features that involve

balancing a trade-off between environmental protection and economic development will

further intensify such uncertainties. Advanced mathematical programming approaches are

desired for environmental management under such complexities.

2.2.1. Two-stage stochastic programming

Two-stage stochastic programming (TSP) is considered as an efficient method for

addressing this type of problems, where an analysis of multi-stage decisions is desired and

uncertainties are expressed as random variables in the objective (Huang and Loucks, 2000;

Li et al., 2007b; Luo et al., 2003; Maqsood and Huang, 2003). The motivation for TSP is

the desire to take recourse or corrective action when uncertain future events have occurred.

In the TSP method, a first-stage decision is undertaken based on random short-term events.

After the random events are later resolved, a second-stage decision will be correspondingly

taken in order to minimize the expected costs (Birge and Louveaux, 1988; Birge and

Louveaux, 1997; Datta and Burges, 1984; Liu et al., 2003). As a consequence, TSP can

present an effective linkage between policies and associated economic penalties caused by

unsuitable policies (Li et al., 2014; Li and Huang, 2007; Seifi and Hipel, 2001). In the past,

the TSP method has been widely explored. For example, Kall and Mayer (1976) introduced

a two stage stochastic linear programming with a fixed recourse. Pereira and Pinto (1985)

11

presented a dual dynamic programming algorithm for two-stage problems. Wang and

Adams (1986) developed a two-stage optimization method for the planning of reservoir

operations. Birge and Louveaux (1988) introduced a multi-cut algorithm for two-stage

stochastic linear programming. Higle and Sen (1991) described a cutting plane algorithm

for TSP programs. Eiger and Shamir (1991) proposed a model for an optimal multi-period

operation within a multi-reservoir system. Cheung and Chen (1998) addressed a dynamic

empty-container allocation problem through the use of a two-stage stochastic networking

approach. Darby-Dowman et al. (2000) presented a TSP method to identify robust plans

for horticultural management. Albornoz et al. (2004) described a thermal power system

expansion planning with an integer TSP model.

2.2.2. Chance-constrained programming

Chance-constrained programming (CCP) is a remarkable mathematical programming

method for effectively tackling optimization problems, where the reliability of satisfying

system constraints under uncertainty needs to be reflected. In fact, the CCP methods do

not require all the constraints to be fully satisfied; instead, they can be satisfied in a

proportion of cases under given probabilities (Loucks et al., 1981). In addition, the CCP

method is attractive in dealing with uncertainties in the model’s right-hand side values

when they are expressed as probability density functions (Morgan et al., 1993). Over the

past decades, a large number of CCP methods were proposed and applied to environmental

management problems (Charnes and Cooper, 1983; Charnes et al., 1971; Morgan et al.,

1993). For example, Charnes et al. (1970) developed incorporated an acceptance region

theory within a CCP framework. Fortin and McBean (1983) considered the management

12

of acid-rain abatement through using chance constraints to represent the uncertainty in the

transfer coefficients of a linear programming model. Hugh Ellis et al. (1991) incorporated

estimates from different long-range transport models into a multiobjective stochastic

framework. A linear CCP model to support decisions for acid rain abatement was

developed by Ellis et al. (1985, 1986). More recently, Li et al. (2007b) proposed an inexact

two-stage chance-constrained linear programming (ITCLP) method for planning waste

management systems. Tan et al. (2011) proposed a radial interval chance-constrained

programming (RICCP) approach for supporting source-oriented non-point source

pollution control under uncertainty. Zhang and Li (2011) introduced the chance-

constrained programming to optimal power flow under uncertainty. Bilsel and Ravindran

(2011) developed a multiobjective chance-constrained programming method for supplier

selection under uncertainty. Wang et al. (2012) presented a chance-constrained two-stage

(CCTS) stochastic program for a unit commitment problem with uncertain wind power

output. Tian et al. (2013) proposed a chance-constrained programming approach to

identify the optimal disassembly sequence.

2.3. Summary

Many previous research efforts have been made in the development of optimization

methods for supporting environmental management under uncertainty. However, the

majority of them are unable to address conflicts between environmental and economic

objectives under uncertainties. These existing mathematical programming methodologies

have difficulties in establishing a linkage between predefined policies and the implied

economic penalties within a multiobjective context. Therefore, as an extension of previous

13

studies, several two-stage fractional programming methods will be developed to

effectively support the environmental management under the complexities. The developed

methods will then be applied to hypothetical case studies of solid waste management and

water quality management, as well as a real-world case study in the province of British

Columbia, Canada.

14

CHAPTER 3

A TWO-STAGE FRACTIONAL PROGRAMMING METHOD FOR MANAGING

MULTIOBJECTIVE WASTE MANAGEMENT SYSTEMS

3.1. Background

Municipal solid waste (MSW) management is one of the most important issues for

urban communities throughout the world (Huang and Chang, 2003). Waste managers may

often encounter challenges of balancing a trade-off between economic development and

environmental protection (Cheng et al., 2002; Li et al., 2007c; Minciardi et al., 2008). Due

to the insufficiency in available facility capacities to meet future waste disposal demands,

identification of desirable expansion schemes is an important aspect in planning long-term

solid waste management systems (Maqsood et al., 2004; Simoes and Catapreta, 2013;

Tonjes and Mallikarjun, 2013). Furthermore, uncertainties presented in probability

distributions may exist in many related system parameters and their interrelationships,

leading to difficulties in providing a linkage to economic consequences of violated policies

pre-regulated by local authorities (Huang et al., 1993; Yeomans and Huang, 2003).

Therefore, developing effective approaches for reflecting system sustainability, dynamic

complexities, and policy effects would be preferred to support MSW management.

In the past, many optimization techniques were developed for planning

multiobjective MSW management systems (Adamides et al., 2009; Ahluwalia and Nema,

2006; Cheng et al., 2003; Galante et al., 2010; Kim et al., 2013; Li et al., 2013a; Mavrotas

et al., 2013; Minciardi et al., 2008; Rentizelas et al., 2014; Rerat et al., 2013; Srivastava

15

and Nema, 2012; Su et al., 2008; Suo et al., 2013). For example, Su et al. (2008) proposed

an inexact multiobjective dynamic programming (IMODP) model for supporting MSW

management under uncertainty, where two major objectives were to minimize both the

system cost and the environmental impact. Santibanez-Aguilar et al. (2013) developed a

multiobjective mixed-integer linear programming model for planning a distributed system

of processing facilities to treat MSW while simultaneously considering economic and

environmental aspects. However, when dealing with issues of multiple conflicting

objectives, most of the previous studies tended to combine multiple objectives into a single

one through identifying weighting factors or economic indicators on the basis of subjective

assumptions (Zhu et al., 2014).

Linear Fractional programming (LFP), which can potentially balance objectives

between two functions, e.g., cost/volume, output/input, or cost/time, is effective for

dealing with multiobjective optimization problems (Charnes et al., 1978; Mehra et al.,

2007; Stancu-Minasian, 1997a, 1999). It could not only intensify the comparative analysis

regarding the objectives of different aspects through using their original magnitudes, but

also provide an unprejudiced measure of system efficiency (Zhu et al., 2014). In the past,

LFP has been widely employed in various fields, such as resource management, finance,

production and transportation (Mehra et al., 2007; Schaible and Ibaraki, 1983; Stancu-

Minasian, 1997a, 1999). For example, Gómez et al. (2006) described a timber harvest

scheduling problem in order to obtain a balanced age class distribution of a forest

plantation through presenting a linear fractional goal programming model. Recently, Zhu

and Huang (2011) introduced a stochastic linear fractional programming (SLFP) approach

in order to support sustainable MSW management under uncertainty. Nevertheless, these

16

methods would encounter difficulties in providing desired allocation and expansion plans

under different policy scenarios, since the existing LFP approaches were not able to

simultaneously address both complexities of multi-stage decisions and dynamic variations

of system behaviors.

In fact, waste management facility expansion is a crucial issue in planning

environmental management systems, where integer variables are typically employed to

indicate whether specific facility expansion options are to be taken (Li et al., 2006b).

Mixed-integer linear programming (MILP) is a remarkable mathematical programming

method for this purpose (Baetz, 1990a; Huang et al., 1995; Huang et al., 1997; Huang et

al., 2013; Li et al., 2008c). Furthermore, in many real-world problems, it is necessary to

provide a linkage to economic implications due to the violation of pre-regulated

environmental policies. Two-stage stochastic programming (TSP) is considered as an

efficient method for dealing with this type of problems, where an analysis of multi-stage

decisions is desired and uncertainties are expressed as random variables in the objective

(Huang and Loucks, 2000; Li et al., 2007b; Luo et al., 2003; Maqsood and Huang, 2003).

In the TSP method, a first-stage decision is undertaken based on random short-term events.

After the random events are later resolved, a second-stage decision will be correspondingly

taken in order to minimize the expected costs (Birge and Louveaux, 2011; Birge and

Louveaux, 1988). However, a remarkable limitation of the MILP and TSP methods is their

incapability of effectively handling multiobjective optimization problems.

Therefore, as an extension of the previous efforts, the objective of this study is to

develop a two-stage fractional programming (TSFP) method for municipal solid waste

management. Techniques of two-stage stochastic programming (TSP) and mixed-integer

17

linear programming (MILP) will be integrated within a fractional programming (FP)

framework to tackle multiobjective optimization problems that involve issues of capacity

expansions for waste management facilities and uncertainties that exist in a number of

modeling parameters. With the capability of multiobjective optimization, the developed

TSFP will be able to help address conflicts between two objectives (e.g. economic and

environmental effects) within a MSW management system, without the demand of

subjectively setting a weight for each objective. Such a feature will help facilitate effective

exploration and reflection of trade-offs between two conflicting objectives, which implies

a significant improvement in terms of multiobjective environmental systems planning.

Moreover, TSFP can help establish a linkage between pre-formulated polices and the

implied economic penalties, and facilitate dynamic analysis for decisions of capacity-

expansion planning.

3.2. Methodology

Linear fractional programming (LFP) involves the optimization of two conflicting

objective functions subject to a decision space delimited by a set of constraints. A general

LFP problem can be defined as follows (Zhu and Huang, 2011):

Max CX

f xDX

(3.1a)

subject to:

AX B (3.1b)

0X (3.1c)

18

where X and B are column vectors with n and m components respectively; A is a real m ×

n matrix; C and D are row vectors with n components; α and β are constants.

The above model can be efficiently used to deal with deterministic multiobjective

optimization problems. However, it is incapable of effectively reflecting uncertainties

expressed as probabilistic distributions in practical planning problems. Moreover, it has

difficulties in establishing a connection to economic consequences when the previously

regulated policies are violated. When decisions need to be made periodically over time

and uncertainties in the model’s right-hand sides are presented as probability density

functions, the study problem can be formulated as a two-stage stochastic programming

(TSP) (Li et al., 2006b). The TSP method is effective for tackling optimization problems

where an analysis of multi-stage decisions is desired and the relevant data are mostly

uncertain (Li et al., 2008a). Such a method could be introduced into the above LFP

framework as a new possible approach for better analyzing various policy scenarios that

are associated with different levels of economic penalties when the previously regulated

policy targets are violated. This leads to a two-stage fractional programming (TSFP) model.

In the TSFP model, two subsets of decision variables are included: initial variables

that must be determined before the random short-term events are resolved, and recourse

variables that will be determined when the events are later disclosed. Generally, the TSFP

model can be formulated as follows:

Max 1 1

2 2

[ ]

[ ]

C X E D Yf

C X E D Y

(3.2a)

subject to:

19

AX A Y B (3.2b)

, 1, 2,...,i i iA X A Y i m (3.2c)

, 0X Y (3.2d)

where X and Y are first-stage and second-stage variables, respectively; C1, C2, D1, and D2

are coefficients in the ratio objective; A, A′, Ai, and A′i are coefficients in the constraints;

and i is a random right-hand parameter of the constraint i. By letting the random

variables (i.e. i ) take discrete values ih with probability levels hp ( 1,2,...,h v and

1hp ), the above TSFP can be equivalently transformed into a linear programming

model as follows (Ahmed et al., 2004; Li et al., 2007a):

Max 1 1

1

2 21

v

h hhv

h hh

C X p D Yf

C X p D Y

(3.3a)

subject to:

hAX A Y B (3.3b)

, 1, 2,..., ; 1, 2,...i i h ihA X A Y i m h v (3.3c)

, 0, 1, 2,...hX Y h v (3.3d)

where v is the number of possible realizations for the random parameter i .

Obviously, the developed model (3.3) can effectively tackle uncertainties in right-

hand sides expressed as probability distributions when coefficients in the objective

20

function and in the left-hand sides are deterministic (Li et al., 2008c). However, in long-

term planning problems, it is more useful to identify desirable capacity expansion schemes

for the systems during different periods with conflicting optimization objectives. Thus, the

introduction of mixed-integer linear programming (MILP) into the TSFP model is

considered to be feasible for this type of capacity expansion planning problem. Therefore,

when some decision variables are defined as integers to indicate whether or not specified

expansion options should be undertaken, the TSFP model can be reformulated as:

Max 1 1

1

2 21

v

h hhv

h hh

C X p D Yf

C X p D Y

(3.4a)

subject to:

hAX A Y B (3.4b)

, 1, 2,..., ; 1, 2,...i i h ihA X AY i m h v (3.4c)

10, , 1, 2,...j jx x X j k (3.4d)

20, , 1, 2,... ; 1, 2,...jh jh hy y Y h v j k (3.4e)

1 10, , and integer variables, 1,...,j j jx x X x j k n (3.4f)

2 20, , and integer variables, 1, 2,... ; 1,...,jh jh h jhy y Y y h v j k n (3.4g)

According to Charnes and Cooper (1962), if (i) the objective function is continuously

differentiable, (ii) the feasible region is non-empty and bounded, and (iii) the denominator

is strictly positive on the feasible region, the TSFP model can be equivalently transformed

into the following linear programming problems:

21

Max * *1 1

1

v

h hh

g C X p D Y r

(3.5a)

subject to:

* *hAX A Y r B (3.5b)

* * , 1, 2,..., ; 1,2,...i i h ihA X AY r i m h v (3.5c)

* *2 2

1

1v

h hh

C X p D Y r

(3.5d)

* *, , 1, 2,...h hX r X Y r Y h v (3.5e)

* * *10, , 1,2,...j jx x X j k (3.5f)

* * *20, , 1,2,... ; 1,2,...jh jh hy y Y h v j k (3.5g)

1 10, , and integer variables, 1,...,j j jx x X x j k n (3.5h)

2 20, , and integer variables, 1, 2,... ; 1,...,jh jh h jhy y Y y h v j k n (3.5i)

0r (3.5j)

Through using the algorithm of branch and bound, model (3.5) can be solved. The

optimization solutions of xj (j = 1, 2, …, k1) and yjh (j = 1, 2, …, k2) can be obtained through

the transformations of *j jx x r (j = 1, 2, …, k1) and * /jh jhy y r (j = 1, 2, …, k2, and h

= 1, 2, …, v), while the solutions for integer variables of xj (j = k1 + 1, k1 + 2, …, n1) and

yjh (j = k2 + 1, k2 + 2, …, n2, and h = 1, 2, …, v) can be obtained directly.

The developed TSFP method improves upon the conventional optimization methods

through introducing TSP and MILP methods into a general LFP optimization framework.

The TSFP method has three major advantages: (i) it can effectively balance two conflicting

22

objectives; (ii) it can efficiently be used for analyzing various scenarios that are associated

with different levels of economic penalties when the previously regulated policies are

violated; and (iii) it can support in-depth dynamic analysis in terms of capacity-expansion

planning.

3.3. Case study

3.3.1. Overview of study system

In this study, the developed TSFP model is applied to support the planning of a

municipal solid waste management system (Li et al., 2006b; Zhu and Huang, 2011). The

system manager is responsible for allocating waste flows generated from three districts to

two disposal facilities, including one landfill and one set of recycling/composting facilities.

A 15-year planning horizon, which is divided into three 5-years periods, is considered.

Figure 1 shows a schematic of the MSW management system. Landfilling, the most

economical approach, has been the most prevalent method for the disposal of MSW (Zhu

and Huang, 2011). However, landfilling is highly discouraged due to the negative impacts

on the environment and the associated threats to public health, as well as the limitation of

suitable land resources (Zhu and Huang, 2011). In contrast, composting and recycling can

reduce the waste, and the revenue can be obtained as a result of material recovery from

organic waste and recyclables. Thus, diverting waste flows to the composting and

recycling facilities will reduce the environmental impacts and extend the landfill lifetime.

A projected target of waste-flow levels from each district to each facility is pre-regulated

based on the local waste-management policies. Exceeding such a level will lead to system

23

penalties, which are presented in terms of raised operating and transportation costs. The

municipal waste-generation rates may continuously increase because of population

increase and economic development. When the capacities of the existing landfill and the

composting and recycling facilities are insufficient to meet the increasing waste-disposal

demands, capacity expansions are desired. The system manager is to identify desired

solutions of waste-flow allocation and facility-expansion schemes with the maximized net

diverted waste per unit of system cost.

The MSW generation rates of different cities vary temporally, and the costs of waste

transportation and operation also vary among different periods (Maqsood et al., 2004).

Tables 3.1 and 3.2 present transportation costs for the target and the excess waste flows

from districts to two facilities, operating costs of the two facilities, and revenues from

composting and recycling facilities in three time periods (Li et al., 2006b; Zhu and Huang,

2011). The MSW generation rates and the associated probabilities are listed in Table 3.3

(Li et al., 2008b). Table 3.4 contains target waste flows from the districts to landfill and

composting and recycling facilities. Excess waste flows will be produced when allowable

waste-flow levels pre-regulated by authorities are exceeded; the total waste flows will be

the sum of both fixed allowable and probabilistic excess flows (Li et al., 2006a). The

landfill and the composting and recycling facilities can only be expanded once during the

planning horizon according to the region’s environmental policy. Table 3.5 shows

capacity-expansion options and costs for the landfill and composting and recycling

facilities (Li et al., 2006b).

24

Figure 3.1 Overview of the study system

District 1 District 2 District 3

Landfill Recycling and

composting facilities

MSW MSW MSW

25

Table 3.1 Transportation and operation costs for target waste flows

Time period

k = 1 k = 2 k = 3

Facility operating cost ($/tonne):

OP1k (Landfill) 40 50 60

OP2k (Composting and recycling facilities)

70 80 90

Transportation cost to landfill (for waste flows) ($/tonne)

TR11k (District 1) 12.1 13.3 14.6

TR12k (District 2) 10.5 11.6 12.8

TR13k (District 3) 12.7 14.0 15.4

Transportation cost to composting and recycling facilities ($/tonne):

TR21k (District 1) 9.6 10.6 11.7

TR22k (District 2) 10.1 11.1 12.2

TR23k (District 3) 8.0 9.7 10.6

Cost for shipping residue to landfill ($/tonne):

FTk 4.7 5.2 5.7

Revenue ($/t) from composting and recycling facilities ($/tonne)

REk 28 30 32

26

Table 3.2 Transportation and operation costs for excess waste flows

Time period

k = 1 k = 2 k = 3

Facility operating cost ($/tonne):

DP1k (Landfill) 45 55 65

DP2k (Composting and recycling facilities)

75 85 95

Transportation cost to landfill (for waste flows) ($/tonne)

DR11k (District 1) 18.2 20 21.9

DR12k (District 2) 15.8 17.4 19.2

DR13k (District 3) 19.1 21 23.1

Transportation cost to composting and recycling facilities ($/tonne):

DR21k (District 1) 14.4 15.9 17.6

DR22k (District 2) 15.2 16.7 18.3

DR23k (District 3) 13.2 14.6 15.9

Cost for shipping residue to landfill ($/tonne):

DTk 5.5 6 6.5

Revenue ($/t) from composting and recycling facilities ($/tonne)

RMk 28 30 32

27

Table 3.3 Different waste-generation rates and probability levels

Level of waste-generation

Probabilities Waste-generation rate, WGjkh (tonne/day)

k = 1 k = 2 k = 3

WG1kh (District 1) (tonne/day)

h = 1 (low) 0.2 200 310 345

h = 2 (medium) 0.6 250 345 400

h = 3 (high) 0.2 290 380 425


h = 1 (low) 0.2 135 215 295

h = 2 (medium) 0.6 170 230 320

h = 3 (high) 0.2 205 245 355


h = 1 (low) 0.2 160 235 280

h = 2 (medium) 0.6 195 265 315

h = 3 (high) 0.2 230 275 350

28

Table 3.4 Target waste flows from the city to the landfill and the composting and

recycling facilities

Time period

k = 1 k = 2 k = 3

Target waste-flow level to landfill (tonne/day):

T11k (District 1) 115 120 125

T12k (District 2) 65 75 90

T13k (District 3) 90 95 100

Target waste-flow level to composting and recycling facilities (tonne/day):

T21k (District 1) 165 185 200

T22k (District 2) 95 115 140

T23k (District 3) 120 145 165

Maximum waste-flow level to landfill (tonne/day):

T11k max (District 1) 380 395 450

T12k max (District 2) 270 305 340

T13k max (District 3) 325 360 395

Maximum waste-flow level to composting and recycling facilities (tonne/day):

T21k max (District 1) 380 395 450

T22k max (District 2) 270 305 340

T23k max (District 3) 325 360 395

29

Table 3.5 Capacity-expansion options and costs for the landfill and the composting and


Time period

k=1 k=2 k=3

Capacity-expansion option and costs for landfill:

LCk (Expansion capacity)

(106 tonne) 1.55 1.55 1.55

FLCk (Capital cost)

($106 present value) 1.3 1.3 1.3

Capacity-expansion option for composting and recycling facilities (tonne /day):

TE1k (option 1) 50 50 50

TE2k (option 2) 52 52 52

TE3k (option 3) 55 55 55

Capital cost for composting and recycling facilities ($106 present value):

FTC1k (option 1) 1.05 0.83 0.65

FTC2k (option 2) 1.52 1.19 0.93

FTC3k (option 3) 1.98 1.55 1.22

30

3.3.2. TSFP model for municipal solid waste management

In the study system, the waste managers would face multiobjective issues, where the

conflict between environmental protection and economic development need to be

addressed. Moreover, this complexity would be further intensified due to the concerns of

capacity expansions for two waste disposal facilities and the uncertainties in numerous

modeling parameters for multi-stage decisions. The problems under consideration include:

(a) how to effectively allocate waste flows in the MSW management system; (b) how to

identify desired capacity expansions schemes for facilities under uncertainty; (c) how to

maximize net diverted waste with potential low system costs and environmental impacts;

and (d) how to reflect economic penalties of corrective measures due to the violation of

environmental policies. Consequently, the proposed TSFP method is considered to be a

feasible approach for dealing with this MSW management problem.

The objective is to maximize net diverted waste per unit of system cost, while a set

of constraints define the relationships between the decision variables and system

conditions. In detail, a two-stage mixed-integer fractional programming (TSFP) model can

be formulated as follows:

31

(1) Ratio objective:

3 3 3 3 3

2 21 1 1 1 1

2 3 3 3 3

2 11 1 1 1 1

3 3 3

1 1 1 1

net diverted wasteMax

system cost

1 1k jk k jh jkhj k j k h

k ijk ijk ik k jk k k ki j k j k

k jh ijk ijk iki j k h

f

L T FE L P X FE

L T TR OP L T FE FT OP RE

L p X DR DP

2 3 3 3

2 11 1 1

3 3 3

1 1 1

k jh jk k k kj k h

k k mk mkk m k

L p X FE DT DP RM

FLC Y FTC Z

(3.7a)

(2) Landfill capacity constraints:

3

1 1 2 21 1 1

, and 1,2,3k k

k jk jkh jk jkh k kj k k

L T X T X FE LC LC Y h k

(3.7b)

(3) Capacity constraints for composting and recycling facilities:

3 3

2 21 1 1

Z , 1,2,3k

jk jkh mk mkj m k

T X TE TE h k

(3.7c)

(4) Waste management demand constraints:

2

1

, ,ijk ijkh jkhi

T X WG j k h

(3.7d)

(5) Expansion option constraints:

32

1,

0,

integer, k

k

Y k

k

(3.7e)

1, ,

0, ,

integer, ,mk

m k

Z m k

m k

(3.7f)

3

1

1kk

Y

(3.7g)

3 3

1 1

1mkm k

Z

(3.7h)

(6) Non-negativity constraints:

max 0 , , ,ijk ijk ijkhT T X i j k h (3.7i)

where:

i = index for the MSW management facilities (i= 1 for the landfill, i = 2 for the composting

and recycling facilities);

j = index for the three districts (j =1, 2, 3);

k = index for the time periods (k = 1, 2, 3);

h = index of waste-generation rate in district j (h = 1, 2, 3);

m = name of expansion option for the composting and recycling facilities (m = 1, 2, 3);

pjh = waste-generation rate with probability h in district j;

Lk = length of time period k (day);

Tijk = target waste flow from district j to facility i during period k (tonne/day) (the first-

stage decision variable);

33

Tijk max = maximum target waste flow level from district j to facility i during period k

(tonne/day);

Xijkh = amount by which the target waste flow level Tijk is exceeded when the waste-

generation rate is WGjkh with probability pjh (tonne/day);

Yk = integer variable (=1 or 0) representing landfill expansion at the start of period k;

Zmk = integer variable (=1 or 0) representing composting and recycling facilities with

expansion option m at the start of period k;

DPik = operating cost of facility i for excess waste flow during period k ($/tonne), where

DPik ≥ OPik (the second-stage cost parameter);

DRijk = transportation cost for excess waste flow from district j to facility i during period

k ($/tonne), where DRijk ≥ TRijk (the second-stage cost parameter);

DTk = transportation costs of excess waste residue from the composting and recycling

facilities to the landfill in period k ($/tonne) where DTk ≥ FTk (the second-stage cost

parameter);

FE = residue flow rate from the composting and recycling facilities to the landfill (% of

incoming mass);

FTk = transportation costs from the composting and recycling facilities to the landfill in

period k ($/tonne);

LC = existing landfill capacity (tonne);

kLC = amount of capacity expansion for the landfill (tonne);

OPik = operating costs of facility i in period k ($/tonne);

REk = revenue from the composting and recycling facilities in period k ($/tonne);

34

RMk = revenue from facility i because of excess waste flow during period k ($/tonne) (the

second-stage cost parameter);

TE = existing capacity of composting and recycling facilities (tonne);

mkTE = amount of capacity expansion option m for composting and recycling facilities at

the start of period k (tonne);

TRijk = transportation costs from district j to facility i in period k ($/tonne);

WGjkh = waste generation rate in district j in period k with level h (tonne/day);

FLCk = capital cost of landfill expansion in period k ($/tonne);

FTCmk = capital cost of expanding composting and recycling facilities by option m in

period k ($/tonne).

3.3.3. Results and discussion

The solutions of binary decision variables obtained through the TSFP model (3.7) are

presented in Table 3.6. It is indicated that the landfill would be expanded once at the start

of period 2 with an incremental capacity of 1.55 × 106 tonnes (i.e. Y2 = 1). However, since

sufficient capacities have been developed in period 2, no expansion would be undertaken

in period 3. Similarly, the composting and recycling facilities would be expanded once at

start of period 1 with an incremental capacity of 55 tonne/day (i.e. Z31 = 1). However, a

further expansion for the composting and recycling facilities would not be needed in

periods 2 and 3.

Table 3.7 presents the solutions of continuous decision variables, which represent the

excess waste flows from the districts to the facilities with minimized system costs and

maximized net diverted waste under different waste generation levels. It is revealed that,

35

Table 3.6 Solutions of the TSFP model for binary variables

Symbol Facility Expansion plan

Period Solution

Y2 Landfill 2 1

Z31 Composting and recycling facilities

3 1 1

36

Table 3.7 Solutions of the TSFP model

Waste flow (tonne/

day)

i j k

Level of

waste

generation

Probability (%)

Target waste flow

Excess waste flow

Optimized waste flow

X1111 1 1 1 Low 20 115 0 115

X1112 1 1 1 Medium 60 0 115

X1113 1 1 1 High 20 40.0 155.0

X1121 1 1 2 Low 20 120 0 120

X1122 1 1 2 Medium 60 0 120

X1123 1 1 2 High 20 95.0 215.0

X1131 1 1 3 Low 20 125 0 125

X1132 1 1 3 Medium 60 25.0 150.0

X1133 1 1 3 High 20 100.0 225.0

X1211 1 2 1 Low 20 65 0.0 65

X1212 1 2 1 Medium 60 0 65

X1213 1 2 1 High 20 65.0 130.0

X1221 1 2 2 Low 20 75 0 75

X1222 1 2 2 Medium 60 5.0 80.0

X1223 1 2 2 High 20 75.0 150.0

X1231 1 2 3 Low 20 90 0 90

X1232 1 2 3 Medium 60 15.00 105.0

X1233 1 2 3 High 20 90.0 180.0

X1311 1 3 1 Low 20 90 0 90

37

Table 3.7 Continued.

Waste flow (tonne/

day)

i j k

Level of

waste

generation

Probability (%)

Target waste flow

Excess waste flow


X1312 1 3 1 Medium 60 0 90

X1313 1 3 1 High 20 0 90

X1321 1 3 2 Low 20 95 0 95

X1322 1 3 2 Medium 60 0 95

X1323 1 3 2 High 20 30.0 125.0

X1331 1 3 3 Low 20 100 0 100

X1332 1 3 3 Medium 60 0 100

X1333 1 3 3 High 20 80.0 180.0

X2111 2 1 1 Low 20 165 45.0 210.0

X2112 2 1 1 Medium 60 30.0 195.0

X2113 2 1 1 High 20 25.0 190.0

X2121 2 1 2 Low 20 185 0 185

X2122 2 1 2 Medium 60 40.0 225.0

X2123 2 1 2 High 20 0 185

X2131 2 1 3 Low 20 200 0 200

X2132 2 1 3 Medium 60 30.0 230.0

X2133 2 1 3 High 20 0.0 200.0

X2211 2 2 1 Low 20 95 0 95

X2212 2 2 1 Medium 60 55.0 150.0

38


Waste flow (tonne/

day)

i j k

Level of

waste

generation

Probability (%)

Target waste flow

Excess waste flow


X2213 2 2 1 High 20 70.0 165.0

X2221 2 2 2 Low 20 115 0 115

X2222 2 2 2 Medium 60 35.0 150.0

X2223 2 2 2 High 20 55.0 170.0

X2231 2 2 3 Low 20 140 0 140

X2232 2 2 3 Medium 60 0 140

X2233 2 2 3 High 20 35.0 175.0

X2311 2 3 1 Low 20 120 120.0 240.0

X2312 2 3 1 Medium 60 80.0 200.0

X2313 2 3 1 High 20 70.0 190.0

X2321 2 3 2 Low 20 145 100.0 245.0

X2322 2 3 2 Medium 60 25.0 170.0

X2323 2 3 2 High 20 45.0 190.0

X2331 2 3 3 Low 20 165 40.0 205.0

X2332 2 3 3 Medium 60 10.0 175.0

X2333 2 3 3 High 20 5.0 170.0

Waste diversion (103tonne) 2088.71

Cost ($106) 361.73

Waste diversion/cost (103tonne per $106) 5.77

39

over the planning horizon, the patterns of excess waste flow allocation vary among

different time periods as a result of the temporal and spatial variations of waste

management conditions under uncertain inputs. The results indicate that, when the waste

generation rates are low, medium, and high with probabilities of 20%, 60%, and 20%

respectively, the excess wastes from district 1 in period 1 should mainly be shipped to the

composting and recycling facilities; those from district 2 should be transported to either

the landfill or the composting and recycling facilities; and those from district 3 should all

be delivered to the composting and recycling facilities. There are similar characteristics in

the solutions for other periods. This is because both the regular and penalty transportation

costs from district 3 to the composting and recycling facilities are lower than the landfill.

Thus, it can be seen that the regular and penalty transportation costs have significant

effects on the solutions of the TSFP model compared with the regular and penalty

operation costs.

The results also show the optimized waste allocation pattern including information

of target and excess flows to the landfill and the composting and recycling facilities. For

example, for excess waste flows from district 1 to the landfill during period 1, the solutions

of X1111 = X1112 = 0 indicate that there would be no excess in reference to the target waste

flow level when the waste generation rates are low and medium associated with

probabilities of 20% and 60%; therefore, the optimized waste flows would equal to the

target one (i.e. A1111 = A1112 = 115 tonne/day). However, the solution of X1113 = 40

tonne/day indicates that there would be an excess of 40 tonne/day to the landfill under the

high waste generation rate with the probability of 20%, and the optimized flow is the sum

of target and excess waste flows (i.e. A1113 = T1113 + X1113 = 115 +40 = 155 tonne/day). In

40

period 2, the excess flows would be 0, 0, and 95 tonne/day under low, medium, and high

waste generation rates (i.e. X1121 = 0, X1122 = 0, and X1123 = 95 tonne/day), the optimized

flows are 120, 120, and 215 tonne/day, respectively. In period 3, the results of X1131 = 0,

X1132 = 25, and X1133 = 100 tonne/day indicate that, under low, medium, and high waste

generation rates, the excess flows to the landfill would be modified to 0, 25, and 100

tonne/day. Similarly, the excess flows from other districts to the landfill and the

composting and recycling facilities in period 1, 2, and 3 can be interpreted based on the

results in Table 3.7. Figures 3.2 and 3.3 show the optimized waste allocation patterns

where details are presented for the target and excess waste flows to the landfill and the

composting and recycling facilities under different waste generation levels.

The results in Table 3.7 also provide the resulting system cost ($361.73 × 106) which

covers expenses for dealing with target waste flows and probabilistic excess flows, and for

expanding the landfill and the composting and recycling facilities. It is indicated that

variations in the values of target waste flow levels could reflect different policies for

managing waste allocation, which are associated with different levels of waste

management costs. The regular costs for disposing/diverting target waste flows would be

$291.43 × 106; the penalty for handing the excess flow would be $67.01 × 106; the cost for

facility expansions would be $3.28 × 106.

When waste managers are more concerned with the economic aspect to minimize the

system cost, a conventional two-stage mixed-integer linear programming (TMILP) model

is compared to further illustrate the effects of the TSFP model in waste management.

Therefore, the optimal-ratio objective in model (3.6) can be converted into a least-cost

problem with the following objective:

41

Figure 3.2 The target waste flow levels and optimized waste flows to the landfill facility

0

50

100

150

200

250W

aste

flo

w (

t/d)

Target waste flow Optimized waste flow allocation

District 1 District 3 District 2 Period 1 Period 3 Period 2 Period 1 Period 3 Period 2 Period 1 Period 3 Period 2

42

Figure 3.3 The target waste flow levels and optimized waste flows to the composting and


0

50

100

150

200

250

300W

ast

e flo

w (

t/d)

Target waste flow Optimized waste flow allocationDistrict 1 District 3 District 2

Period 1 Period 3 Period 2 Period 1 Period 3 Period 2 Period 1 Period 3 Period 2

43

2 3 3 3 3

2 11 1 1 1 1

2 3 3 3 3 3 3

2 11 1 1 1 1 1 1

3 3 3

1 1 1

Min system cost

k ijk ijk ik k jk k k ki j k j k

k jh ijk ijk ik k jh jk k k ki j k h j k h

k k mk mkk m k

f

L T TR OP L T FE FT OP RE

L p X DR DP L p X FE DT DP RM

FLC Y FTC Z

(3.6j)

Thus, the generated TMILP model subject to the constraints 3.6(b) to 3.6(g) can be

solved through using the TMILP method (Maqsood et al., 2004). Table 3.8 presents the

results of the least-cost model. The optimized waste allocation patterns obtained from the

TSFP and conventional TMILP method are significantly varied. The comparisons of the

system performance between the optimal-ratio model and least-cost model are presented

in Table 3.8. It is indicated that, with the same parameters, the system cost obtained from

the least-cost model is $342.18 × 106, which is slightly lower than $361.73 × 106 from the

optimal-ratio model. However, the waste diversion per unit of cost obtained from the least-

cost model is 5.08 × 106 tonne per $106, which is significantly lower than 5.77 × 106 tonne

per $106 from the optimal-ratio model. As shown in Tables 3.7 and 3.8, the TSFP model

leads to a higher system efficiency, which can be expressed as the ratio of waste diversion

to system cost.

Comparisons of the detailed expansion schemes for the waste management facilities

under different expansion options are illustrated in Figure 3.4. It is indicated that the

landfill would be expanded once at the start of period 1 from the least-cost model, which

is earlier than the result from the optimal-ratio model. In comparison, even with the same

conditions, a higher capacity of the composting and recycling facilities from the TSFP

44

Table 3.8 Solutions of the TMILP model

Waste flow (tonne/

day)

i j k

Level of

waste

generation

Probability (%)

Target waste flow

Excess waste flow


X1111 1 1 1 Low 20 115 0 115

X1112 1 1 1 Medium 60 30 145

X1113 1 1 1 High 20 65 180

X1121 1 1 2 Low 20 120 0 120

X1122 1 1 2 Medium 60 40 160

X1123 1 1 2 High 20 95 215

X1131 1 1 3 Low 20 125 0 125

X1132 1 1 3 Medium 60 55 180

X1133 1 1 3 High 20 100 225

X1211 1 2 1 Low 20 65 0 65

X1212 1 2 1 Medium 60 55 120

X1213 1 2 1 High 20 65 130

X1221 1 2 2 Low 20 75 0 75

X1222 1 2 2 Medium 60 40 115

X1223 1 2 2 High 20 75 150

X1231 1 2 3 Low 20 90 0 90

X1232 1 2 3 Medium 60 15 105

X1233 1 2 3 High 20 90 180

X1311 1 3 1 Low 20 90 0 90

45


Waste flow (tonne/

day)

i j k

Level of

waste

generation

Probability (%)

Target waste flow

Excess waste flow


X1312 1 3 1 Medium 60 25 115

X1313 1 3 1 High 20 70 160

X1321 1 3 2 Low 20 95 0 95

X1322 1 3 2 Medium 60 25 120

X1323 1 3 2 High 20 75 170

X1331 1 3 3 Low 20 100 0 100

X1332 1 3 3 Medium 60 10 110

X1333 1 3 3 High 20 85 185

X2111 2 1 1 Low 20 165 0 165

X2112 2 1 1 Medium 60 0 165

X2113 2 1 1 High 20 0 165

X2121 2 1 2 Low 20 185 0 185

X2122 2 1 2 Medium 60 0 185

X2123 2 1 2 High 20 0 185

X2131 2 1 3 Low 20 200 0 200

X2132 2 1 3 Medium 60 0 200

X2133 2 1 3 High 20 0 200

X2211 2 2 1 Low 20 95 0 95

X2212 2 2 1 Medium 60 55.0 150.0

46


Waste flow (tonne/

day)

i j k

Level of

waste

generation

Probability (%)

Target waste flow

Excess waste flow


X2213 2 2 1 High 20 70.0 165.0

X2221 2 2 2 Low 20 115 0 115

X2222 2 2 2 Medium 60 35.0 150.0

X2223 2 2 2 High 20 55.0 170.0

X2231 2 2 3 Low 20 140 0 140

X2232 2 2 3 Medium 60 0 140

X2233 2 2 3 High 20 35.0 175.0

X2311 2 3 1 Low 20 120 120.0 240.0

X2312 2 3 1 Medium 60 80.0 200.0

X2313 2 3 1 High 20 70.0 190.0

X2321 2 3 2 Low 20 145 100.0 245.0

X2322 2 3 2 Medium 60 25.0 170.0

X2323 2 3 2 High 20 0 145

X2331 2 3 3 Low 20 165 0 165

X2332 2 3 3 Medium 60 0 165

X2333 2 3 3 High 20 0 165

Waste diversion (103 tonne) 1739.96

Cost ($106) 357.95

Waste diversion/cost (103 tonne per $106) 4.86

47

Figure 3.4 Comparison of capacity expansion schemes from optimal-ratio and least-cost

models

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Period 1 Period 2 Period 3

Cap

acity

exp

ansi

on

(106

t)

Landfill Facility

0

10

20

30

40

50

60

Period 1 Period 2 Period 3C

apa

city

exp

ans

ion

(t/d

)

Composting and recycling facilities

Optimal-ratio model Least-cost model

48

model would be achieved. The composting and recycling facilities would also be expanded

earlier from the optimal-ratio model. The excess flows are related to the capacity

expansion activities, as planning for more capacity expansion would correspond to a raised

capital cost and a decreased excess flow. Moreover, further comparisons between the two

models are provided in Figures 3.5 and 3.6. The results indicate that the least-cost model

leads to a lower daily waste flow to the composting and recycling facilities due to its higher

operational cost. In contrast, the landfill facility from the optimal-ratio model will be

operated at a lower capacity, which indicates that the waste diversion from districts to the

landfill would be minimized, and thus the environmental impacts would be reduced and

the landfill lifetime would be extended. Therefore, compared with the least-cost model,

the TSFP model could more effectively tackle the MSW management problem.

The TSFP improves upon the conventional TSP methods through integrating MILP

and FP within its modeling framework such that the developed method can identify

schemes with optimal system efficiency under different policy scenarios. As an effective

tool for supporting MSW management, the solutions obtained through the TSFP model

could not only balance the conflicts among multiple objectives without modifying their

original magnitudes, but also provide a linkage between pre-regulated policies and

economic implications expressed as penalties. Moreover, the TSFP model can account for

the dynamic variations of system capacity due to the capacity expansions of waste-

management facilities and support an in-depth analysis of the interactions between system

efficiency and economic cost. Consequently, the technique of the TSFP model could also

be applied to other fields of systems planning with comprehensive consideration of two

conflicting objectives, capacity expansion issues, and multiple policy scenarios.

49

Figure 3.5 Comparison of optimized waste flows to the landfill facility from optimal-

ratio and least-cost models

0

50

100

150

200

250W

aste

flow

(t/d

)

Least-cost model Optimal-ratio model

District 1 District 3 District 2 Period 1 Period 3 Period 2 Period 1 Period 3 Period 2 Period 1 Period 3 Period 2

50

Figure 3.6 Comparison of optimized waste flows to the composting and recycling

facilities from optimal-ratio and least-cost models

0

50

100

150

200

250

300W

aste

flo

w (

t/d)

Least-cost model Optimal-ratio modelDistrict 1 District 3 District 2

Period 1 Period 3 Period 2 Period 1 Period 3 Period 2 Period 1 Period 3 Period 2

51

3.4. Summary

In this study, a two-stage fractional programming method has been developed for

planning environmental management systems under uncertainty. The method is based on

an integration of the existing two-stage programming and mixed-integer linear

programming techniques within a fractional programming framework. It can solve ratio

optimization problems involving capacity expansion issues and pre-regulated policy

analysis with random information. The merits of the proposed approach include (1)

balancing objectives of two aspects, (2) reflecting different policy scenarios, (3) generating

capacity expansion schemes, and (4) optimizing system efficiency.

The TSFP method has been applied to a case study of a solid waste management

system associated with decisions of waste allocation and planning of facility expansion. It

is revealed that the conflicts between economic development objective of minimizing

system cost and environmental protection objective of maximizing net diverted waste can

be effectively addressed without setting a weight factor for each objective. Moreover, the

results also reveal that the regular and penalty transportation costs are more sensitive to

the solutions of TSFP compared with operation costs. The TSFP model will be able to help

address conflicts between two objectives (e.g. economic and environmental effects) within

a MSW management system, without the demand of subjectively setting a weight for each

objective. Such a capability will help facilitate effective exploration and reflection of

trade-offs between conflicting objectives, which implies a significant improvement in

terms of multiobjective environmental systems planning.

52

Solutions of the TSFP model provide desired MSW management schemes and

capacity expansion plans with maximized system efficiency under different policy

scenarios. The results indicate that decisions at a lower level of allowable waste-loading

would lead to a low system cost but a decreased reliability in system requirements; in

contrast, a desire for increasing the system reliability could run into a higher system cost.

Moreover, the model can facilitate an in-depth analysis of the interactions between system

efficiency and economic cost.

This method is an attempt for planning MSW management systems through

proposing a new modeling framework, which can solve ratio optimization problems

involving policy scenario analyses and capacity expansion issues. As a new method of

mathematical programming under uncertainty, the results suggest that the TSFP technique

is applicable and can be potentially extended to other problems, such as water resource

management and air pollution control planning. Extensions of the TSFP method in

considering three and more objective problems and integrating other methods of fuzzy set

and interval analysis within its framework would be an interesting topic that deserves

future research efforts.

53

CHAPTER 4

TWO-STAGE CHANCE-CONSTRAINED FRACTIONAL PROGRAMMING

FOR SUSTAINABLE WATER QUALITY MANAGEMENT

UNDER UNCERTAINTY

4.1. Background

The challenge of sustainable water quality management, which reflects both

environmental and socio-economic aspects in watershed systems, has been of concern to

many researchers and managers in the recent decades (Huang and Xia, 2001). Thus it is a

necessity to propose integrated strategies for effective water quality management among

water users from different sectors. Although a number of the optimization techniques have

been developed, there are still many difficulties in planning sustainable water quality

management systems. A fundamental difficulty is the need to simultaneously account for

multiple conflicting objectives (Minciardi et al., 2008), and this may be further intensified

due to the associated economic penalties when the promised targets are violated under

random uncertainties (Li and Huang, 2009; Zhu and Huang, 2011). It is thus desired to

develop effective mathematical programming approaches for supporting water quality

management under such complexities.

In the past decades, a broad spectrum of optimization methods was developed for

multiobjective water quality management (Afshar et al., 2013; Chang et al., 1997; Dandy

and Engelhardt, 2006; Das and Haimes, 1979; Han et al., 2011; Kanta et al., 2011; Liu et

al., 2012; Rosenberg and Madani, 2014; Singh et al., 2007; Xu et al., 2014; Xu and Qin,

54

2013). For instance, Qin et al. (2007) developed an interval-parameter fuzzy nonlinear

programming model for water quality management under uncertainty, where the objective

was to minimize the operating cost under the environmental criteria. Liu et al. (2012)

developed an interval-parameter chance-constrained fuzzy multiobjective programming

(ICFMOP) model for the water pollution control in a wetland management system, which

involved multiple interactive objectives, such as environmental protection, economic

development, and resources conservation. Although these methods were helpful in

tackling practical water quality management problems, most of them transformed the

multiple conflicting objectives into a single monetary measure based on unrealistic or

subjective assumptions. Actually, environmental concerns in water quality management

often involve ethical and moral principles that may not be related to any economic use or

value (Kiker et al., 2005; Zhu, 2014).

As an effective measure to balance conflicting objectives, linear fractional

programming (LFP) was employed in various management problems (Gómez et al., 2006;

Stancu-Minasian, 1997b, 1999) for facilitating the analysis of system efficiency and

optimizing the ratio between two quantities (e.g. cost/volume, cost/time, output/input).

Moreover, it is especially suitable for situations where solutions with better achievements

per unit of inputs (e.g. cost, resource, time) are desired. For example, Lara and Stancu-

Minasian (1999) proposed a multiple objective linear fractional programming (MLFP)

model for an agricultural system, where solutions were obtained through maximizing the

gross margin and employment levels per unit of water consumption. Although LFP was

widely applied in various areas ranging from engineering to economics (Mehra et al.,

2007), there were few studies on LFP for environmental management under uncertainty

55

(Chang, 2009; Hladík, 2010).

However, in water quality management systems, various kinds of uncertainties exist

in numerous system components as well as their interrelationships (Babaeyan-Koopaei et

al., 2003; Ghosh and Mujumdar, 2006; Miao et al., 2014; Sabouni and Mardani, 2013).

For example, random characteristics of stream conditions (e.g. point/nonpoint source

pollution, stream flow, and water supply), and natural processes (e.g. precipitation and

climate change) can be possible sources of uncertainties (Li and Huang, 2009). The only

exceptions were the works of Zhu and Huang (2011, 2013), where two chance-constrained

fractional programming methods [stochastic linear fractional programming (SLFP) and

dynamic stochastic fractional programming (DSFP)] were proposed for supporting

sustainable waste management and energy system planning (Zhu and Huang, 2011, 2013).

In their methods, chance-constrained programming (CCP) is integrated into a linear

fractional programming (LFP) framework in order to solve ratio optimization problems

associated with random information. However, the SLFP and DSFP methods could only

handle uncertainties in the constraints’ right-hand sides; they were unable to deal with

more complicated problems, where an analysis of multi-stage decisions is desired and

uncertainties are presented as random variables in the objectives.

Two-stage stochastic programming (TSP) an effective method to solve the recourse

problems associated with randomness. The motivation for TSP is the desire to take

recourse or corrective action when uncertain future events have occurred. In the TSP

method, a first-stage decision is undertaken based on random short-term events. After the

random events are later resolved, a second-stage decision will be correspondingly taken in

order to minimize the expected costs (Birge and Louveaux, 1988; Birge and Louveaux,

56

1997; Datta and Burges, 1984; Liu et al., 2003). As a consequence, TSP can present an

effective linkage between policies and associated economic penalties caused by unsuitable

policies (Li et al., 2014; Li and Huang, 2007; Seifi and Hipel, 2001). However, TSP was

not able to deal with ratio optimization problems associated with random information.

Consequently, as an extension of the previous works, this study aims to develop a

two-stage chance-constrained fractional programming (TCFP) method for water quality

management. Techniques of chance-constrained programming (CCP) and two-stage

stochastic programming (TSP) will be incorporated into a linear fractional programming

(LFP) framework. Thus, the developed method can not only address conflicts between

environmental and economic objectives under uncertainties, but also establish a linkage

between predefined policies and the implied economic penalties. TCFP will be applied to

a case study of water quality management for demonstrating its applicability.

4.2. Methodology

4.2.1. Development of TCFP model

Consider a water quality management system in a region, where multiple sectors

utilize water and discharge wastewater into a stream (Du et al., 2013; Xu and Qin, 2014).

The water quality managers are responsible for making production plans for multiple water

users. In order to obtain higher system benefits, the economic objective of such a problem

can be simply formulated as maximizing the net benefits under the environmental

requirements constraints, which are derived from water quality simulation models. The

environmental requirements include allowable BOD (biochemical oxygen demand)

57

discharge constraints for each water user as well as allowable BOD and DO (dissolved

oxygen) constraints in each stream segment. When uncertainties in the model’s objective

are expressed by random variables and decisions need to be made periodically over time,

the study system can be formulated through a two-stage stochastic programming (TSP)

approach with the recourse model (Li and Huang, 2009). In the TSP model, there are two

subsets of decision variables: first-stage decision variables that must be determined before

random variables are disclosed, and second-stage decision variables (recourse variables)

that will be determined after the uncertainties are disclosed. Therefore, in a TSP model for

the water quality management system, the objective is to maximize water-related

economic benefits, while a set of constraints define the interrelationships between the

decision variables and environmental criteria requirements. Specifically, the economic

objective can be formulated as follows:

1 1

Max [ ( )]I K I K

ik ik ik ik iki k i k

f NB T E U X T

(4.1a)

where Tik is the first-stage decision variable, and ikX is the second-stage decision variable.

For each wastewater source, there is a maximum limit for the BOD discharge:

1 , ,ik ik ik ik ikX W C BA i k (4.1b)

For each steam segment, the BOD concentration and DO deficit should be restricted

by the water quality requirements for supporting aquatic life and maintaining an aerobic

condition:

58

, , ,jk ik ik jk BODL X W R j k (4.1c)

, , ,jk ik ik jk DOD X W R j k (4.1d)

The product demand for each water user should be limited by minimum and

maximum levels:

min max , ,ik ik ikT T T i k (4.1e)

The non-negativity constraints are needed for decision variables in practical problem:

0, ,ikX i k (4.1f)

where:

i = index for the wastewater dischargers (i = 1, 2, ..., I);

j = index for the stream segments (j = 1, 2, ..., n);

k = index for the planning periods (k = 1, 2, ..., K);

BAik = the BOD discharge allowance for source i during period k (tonne/day);

Cik = the BOD concentration of raw wastewater generated at source i in period k (kg/m3);

,jk ik ikD X W = a simulation function for DO deficit at the end of reach j (mg/L) which

can be derived from water quality simulation models;

E[ ] = the expected value of a random variable;

f = the value of objective function;

,jk ik ikL X W = a simulation function for BOD concentration at the beginning of reach j

(mg/L) which can be derived from water quality simulation models;

59

NBik = net benefits per unit product for source i in period k, i.e. the first-stage economic

parameter ($/unit product);

ηik = the BOD treatment efficiency at source i during period k (%);

Rjk BOD = designated BOD concentration at the beginning of reach j in period k (mg/L);

Rjk DO = allowable DO deficit at the end of reach j in period k (mg/L);

Tik = the product target (unit/day or ha/yr) pre-regulated by source i during period k, i.e.

the first-stage decision variable;

Tik min = minimum demands for product i during period k (unit/day);

Tik max= maximum demands for product i during period k (unit/day);

ikW = the random wastewater discharge rate at source i in period k (m3/unit product);

ikX = the production level by which the pre-regulated target Tik is violated under the

random wastewater discharge rate ikW , i.e. the second-stage decision variable (unit/day or

ha/yr).

Obviously, model (4.1) can effectively tackle uncertainties in the objective expressed

as random variables (with known distributions). However, in real-world water quality

management problems, it is necessary to address the environmental objectives related to

water conservation, as well as uncertainties existed in constraints’ right-hand sides.

Furthermore, tradeoffs in conflicting objectives between water-related economic benefits

and water consumption need to be reflected. Such complexities cannot be reflected in

model (4.1). The stochastic linear fractional programming (SLFP) method is useful for

balancing two conflicting objectives and addressing randomness existed in the right-hand

parameters (Zhu and Huang, 2011). A SLFP model can be formulated as follows:

60

Max ( )CX

f xDX

(4.2a)

subject to:

Pr{ ( )} 1 , 1,2,...,s s sA X b q s S (4.2b)

0X (4.2c)

where X is a column vector of decision variables; C and D are row vectors; α and β are

constants; As(τ) is a vector of coefficients in constraints; bs(τ) is the random right-hand

parameter in constraint s; qs (qs ∈[0,1]) is a given level of probability for constraint s (i.e.

significance level), indicating that the constraint should be satisfied with at least a

probability of 1 sq ; S is the number of constraints.

Although SLFP can handle the ratio objective and uncertainties in right-hand

parameters, it has difficulties in investigating economic consequences of violating some

overriding policies. Consequently, one potential method for addressing such complexities

is to introduce the SLFP technique into the TSP framework. This leads to a two-stage

chance-constrained fractional linear programming (TCFP) model as follows:

1 1

Water co

Net

nsu

benef

mption

itsMax

[ ( )]

[ ]

I K I K

ik ik ik ik iki k i k

I K

ik iki k

f

NB T E U X T

E V X

(4.3a)

subject to:

61

Pr{(1 ) ( )} 1 , 1, 2,..., , ,ik ik ik ik ik sX W C BA q s S i k (4.3b)

, , ,jk ik jk BODL X W R j k (4.3c)

, , ,jk ik jk DOD X W R j k (4.3d)

min max , ,ik ik ikT T T i k (4.3e)

0, ,ikX i k (4.3f)

where:

BAik (τ) = the random BOD discharge allowance for source i during period k (tonne/day);

qs = the given level of probability for constraint s (i.e. significance level);

Uik = the net benefits of violation target for source i during period k, i.e. the second-stage

economic parameter;

Vik = the water consumption rate for source i during period k.

Hence, the proposed TCFP model consists of a ratio objective and a set of water

quality constraints derived from a water quality simulation model, where randomness in

both the objective and constraints with known probability distributions can be addressed.

The developed TCFP method is effective to deal with recourse problems, where an analysis

of multi-stage decisions is desired and the relevant data are mostly uncertain.

4.2.2. Solution methods

Generally, the TCFP model (4.3) can be formulated as follows:

Max 1 1

2 2

[ ]

[ ]

C X E D Yf

C X E D Y

(4.4a)

62

subject to:

Pr{ ( )} 1 , 1,2,...,s s s sA X A Y b q s S (4.4b)

, 1, 2,...,r r rA X A Y r m (4.4c)

, 0X Y (4.4d)

where X and Y are respectively first-stage and second-stage variables; C1, C2, D1, and D2

are row vectors of coefficients in the ratio objective; ( ) ( ), sA A ( ) ( ),sA A

( ) ( ),sb B ; As(τ) and ( )sA are row vectors of random coefficients in the

constraint s; A(τ), ( )A and B(τ) are sets with random elements defined on a probability

space Г; Ar and A′r are row vectors of coefficients in the constraint r; r is random right-

hand parameters of the constraint r.

For a given set of first-stage variables X, the second-stage problem decomposes into

independent linear subproblems, with each subproblem corresponding to a realization of

the uncertain parameters (Li et al., 2007a). According to Huang and Loucks (2000), this

TCFP model can be converted into the following model by letting the random parameter

r in constraint r take discrete values rh with the probability level ph (the probability of

occurrence for scenario h, where ph is positive and the sum of ph for all scenarios is equal

to 1):

Max 1 1

1

2 21

v

h hh

v

h hh

C X p D Yf

C X p D Y

(4.5a)

63

subject to:

Pr{ ( )} 1 , 1,2,...,s s h s sA X A Y b q s S (4.5b)

, 1, 2,..., ; 1, 2,...r r h rhA X A Y r m h v (4.5c)

, 0, 1, 2,...hX Y h v (4.5d)

where h = index for the probability levels (h = 1, 2, ..., v), where v is the number of possible

realizations for random parameters r (usually being 3, 5, or 7 in most cases).

According to Huang (1998), if coefficients in model (4.5) are uncertain in both left-

and right- hand sides, constraints (4.5b) is generally nonlinear, and the set of feasible

constraints may become very complicated. When the left-hand coefficients [elements of

As(τ) and sA ] are deterministic and the right-hand coefficients [bs(τ)] are random (for

all qs values), constraints (4.5b) become linear and the set of feasible constraints is convex

(Huang, 1998; Zare and Daneshmand, 1995):

( ) , 1,2,...sqs s h sA X A Y b s S (4.5e)

where 1( ) sqs s sb F q , given the cumulative distribution function of bs, i.e. Fs(bs), and

the probability of violating constraint s (i.e. qs). Therefore, the TCFP model (4.5) could be

transformed into a linear fractional programming (LFP) model through converting

constraints (4.5b) into a deterministic version, i.e. constraints (4.5e).

Moreover, according to the LFP method presented in Chadha and Chadha (2007), if

64

(i) 2 21

0v

h hh

C X p D Y

for all feasible X and Y, (ii) the feasible region is non-empty and

bounded, and (iii) the objective function is continuously differentiable, then the TCFP

model (4.5) could be solved through a linear programming approach.

The detailed solution process for the TCFP model can be summarized as follows:

Step 1: Formulate the original TCFP model, i.e. model (4.4).

Step 2: Convert model (4.4) into model (4.5) through letting the random parameter

r take discrete values rh with the probability level ph.

Step 3: Given a significance level (qs) for each constraint s, convert stochastic

constraints (4.5b) into deterministic constraints (4.5e).

Step 4: Solve the transformed model through the LFP method.

Step 5: Repeat steps 3 to 4 under different qs levels.

4.3. Case study

4.3.1. Overview of the study system

The developed TCFP method is applied to a stream water quality management system

with representative data within a Chinese context (Li et al., 2014; Li and Huang, 2009; Li

et al., 2013b; Qin and Huang, 2009). A planning horizon of 15 years, which is divided into

three 5-year periods, is considered. The local authority is willing to make a water quality

management scheme over the planning horizon. A schematic diagram of the study system

65

Industry

23 4

1

5

Municipality

Wastewater treatment plant

Y1 Y2L0

L2L3

L4 L5L6

Y3 Y4Y5

Y6

Paper millTannery Tobacco

Recreation

L1

Figure 4.1 Schematic diagram of the study system

66

is shown in Fig. 4.1. Specifically, there are five wastewater dischargers along the stream,

which belong to industrial, municipal, and recreational sectors. The BOD concentration at

the head of reach 1 (L0) is 2 mg/L; L1 to L6 are BOD concentrations at the ends of reaches

1 to 6 (mg/L); the lengths of reach 1 to 6 (Yj, j = 1, 2, ..., 6) are respectively 4, 3.5, 2, 2, 4.5

and 3 km; the first-order reaeration and deoxygenation rates (ka and kd) are respectively

0.63 and 0.50 day-1 when stream temperature is about 20 ; the stream flow (Qr) is

325,000 m3/day; the average flow velocity (u) is 8.0 km/day; the initial DO deficit of the

stream (D0) is near zero. With the purpose of meeting the environmental requirements,

specific facilities will be applied to treat the raw wastewater from the industrial and

municipal sectors before discharge. The facility operating costs are related to the

wastewater inflows and their treatment levels. Moreover, the discharging pollutants from

these sources would affect stream water quality. To achieve sustainable water quality

management, the system managers desire suitable plans of production and wastewater

discharge.

There are significant variations in water utilization and discharge conditions. Table

4.1 provides the water consumption and wastewater discharge rates with the associated

probabilities. In order to guarantee the stream water quality, wastewater treatments are

needed for the industrial and municipal sectors. Table 4.2 shows the treatment efficiencies

as well as the raw BOD concentrations at different discharge sources. Table 4.3 presents

the allowable BOD loads for dischargers as regulated by the local authority. Generally, the

BOD concentration and DO deficit in the stream should be lower than 6 and 3 mg/L,

respectively (Haith, 1982). Table 4.4 provides the related economic data and the pre-

regulated targets, as well as minimum and maximum product demands for each economic

67

Table 4.1 Water consumption and wastewater discharge rates with the associated

probabilities

Probabilities Water consumption rates Wastewater discharge rates

k = 1 k = 2 k = 3 k = 1 k = 2 k = 3

Wastewater treatment plant (m3/m3 produce)

h = 1 (low) 0.2 0.65 0.65 0.65 0.63 0.63 0.63

h = 2 (medium)

0.6 0.69 0.69 0.69 0.67 0.67 0.67

h = 3 (high) 0.2 0.74 0.74 0.74 0.72 0.72 0.72

Paper mill (m3/tonne)

h = 1 (low) 0.2 291.5 269.9 248.3 276.9 256.4 235.9

h = 2 (medium)

0.6 306.0 283.4 260.7 290.7 269.2 247.7

h = 3 (high) 0.2 322.9 298.9 275.1 306.8 284.0 261.3

Tannery plant (m3/tonne)

h = 1 (low) 0.2 116.3 107.6 99.0 111.6 103.3 95

h = 2 (medium)

0.6 122.1 113.5 104.0 117.2 109.0 99.8

h = 3 (high) 0.2 128.1 125.2 109.2 123 120.2 104.8

Tobacco factory (m3/tonne)

h = 1 (low) 0.2 202.4 202.4 202.4 190.3 190.3 190.3

h = 2 (medium)

0.6 212.6 212.6 212.6 199.8 199.8 199.8

h = 3 (high) 0.2 223.1 223.1 223.1 209.7 209.7 209.7

68


Probabilities Water consumption rates Wastewater discharge rates

k = 1 k = 2 k = 3 k = 1 k = 2 k = 3

Recreation sector (m3/ha/yr)

h = 1 (low) 0.2 2836.5 2578.7 2344.2 2808.1 2552.9 2320.8

h = 2 (medium)

0.6 3120.1 2836.5 2578.7 3088.9 2808.1 2552.9

h = 3 (high) 0.2 3432.1 3120.1 2836.5 3397.8 3088.9 2808.1

69

Table 4.2 BOD concentrations of wastewater discharged and treatment efficiencies


Paper

mill

Tannery plant

Tobacco factory

Recreational sector

Efficiency (%),ηi 89 84 81 92 -

BOD concentration, Cik (kg/m3)

0.21 0.33 1.2 2.2 0.06

70

Table 4.3 Allowable BOD loading for each source

Allowable BOD loading, BAik(τ) (kg/day)

Economic activity Period qs = 0.01 qs = 0.05 qs = 0.10 qs = 0.25

Wastewater treatment plant 1 798.47 812.10 819.37 831.51

2 878.47 892.10 899.37 911.51

3 953.47 967.10 974.37 986.51

Paper mill 1 321.74 328.55 332.18 338.26

2 316.74 323.55 327.18 333.26

3 306.74 313.55 317.18 323.26

Tannery plant 1 301.74 308.55 312.18 318.26

2 316.74 323.55 327.18 333.26

3 306.74 313.55 317.18 323.26

Tobacco factory 1 103.37 106.78 108.59 111.63

2 88.37 91.78 93.59 96.63

3 78.37 81.78 83.59 86.63

Recreational sector 1 331.74 338.55 342.18 348.26

2 331.74 338.55 342.18 348.26

3 331.74 338.55 342.18 348.26

71

Table 4.4 Pre-regulated targets, product demands, benefit, and costs analysis for the

sectors

Time period

k = 1 k = 2 k = 3

Pre-regulated target:

Water production (m3/day) 42500 50000 55000

Paper (tonne/day) 23 25 27

Leather (tonne/day) 13 14 15

Tobacco (tonne/day) 3.0 3.0 3.0

Recreational activity (ha/yr)

730 839.5 912.5

Minimum product demand:



Leather (tonne/day) 9 9 10



534 620 693

Maximum product demand:



Leather (tonne/day) 14.5 16.0 16.0



800 912 985

72


Time period

k = 1 k = 2 k = 3

Net benefits from differentproducts, NBik

Water production ($/m3) 4.7 5.1 5.7

Paper ($/tonne) 403.0 443.3 487.6

Leather ($/tonne) 1320 1386 1413.8

Tobacco ($/tonne) 12500 12000 11500

Agricultural ($/ha) 145.1 152.3 155.4

Cost for wastewater treatment,Ei (103$/yr)

Wastewater treatment plant 33.07 35.05 37.16

Paper mill 34.54 36.61 38.65

Tannery plant 36.00 38.16 39.64

Tobacco plant 31.89 33.80 35.83

73

activity.

4.3.2. Water quality simulation model

As effective tools for stream water quality management, water quality models have

been extensively developed (Zhu and Huang, 2013). Several water quality models were

proposed in the past decades, such as the Streeter–Phelps, O’Conner, Dobbins, and

Thomas models (Rauch et al., 1998). In this study, the Streeter–Phelps model is used for

quantifying water quality constraints related to BOD and DO discharges as well as

reflecting deoxygenation and reaeration dynamics within the stream (Li and Huang, 2009).

Therefore, the BOD load and DO deficit related to the wastewater discharge sources

could be predicted as follows (Li and Huang, 2009; Thomann and Mueller, 1987):

0 1 11 2

1 1

(1 )BOD

+ (1 )BOD (1 )BOD

d j d j

d n

n nk t k t

nj j

k tm m m m

L e L e

e

(4.6a)

1 1 11 1

d n a n a nk t k t k tdn n n

a d

kD L e e D e

k k

(4.6b)

where:

BODi = the total amount of BOD to be disposed of at source i (kg/day);

Dn = the oxygen deficits at the beginnings of reaches n;

ka = the first-order reaeration rate constant (day-1);

kd = the first-order deoxygenation rate constant (day-1);

L0 = the initial BOD in the stream immediately after discharge (mg/L);

74

Ln = the respective BOD loads in the river at the beginnings of reaches n (mg/L);

tj = the length of reach j expressed in time units;

ηi = the wastewater treatment efficiency at discharge source i (%).

4.3.3. TCFP model for water quality management

The problem under consideration is how to effectively plan the production levels for

multiple water users, where the water quality managers place great emphasis on the

environmental resources preservation. The optimization objective is to maximize the

expected net benefits per unit of water consumption subject to the environmental

requirements under uncertainty over the planning horizon. Generally, the complexities of

the study problem include: (a) how to maximize the expected net benefits with possible

low water consumptions and environmental impacts; (b) how to analyze the tradeoff

between the system efficiency and constraint-violation risk; (c) how to reflect features of

many uncertain parameters available as probability distributions; (d) how to reflect

complex features of uncertain economic implications (i.e. penalties) under the violation of

environmental requirements; and (e) how to analyze dynamic interactions between the

pollutant loading and water quality. Therefore, the proposed TCFP method is considered

suitable for tackling such a problem. According to model (3) and its transformed form, i.e.

model (4.5), we have:

75

4 3 3 4 3

5 51 1 1 1 1 1

3 4 3

5 5 5 51 1 1 1 1

4 3

1 1 1

Net benefitsMax

1825 5 1825

5 5

1

Water consumption

825

v

ik ik k k ih ik ikh iki k k i k h

v v

h k kh k ih ik ikh ikhk h i k h

v

ih ikh ikhi k h

f

NB T NB T p NB X T

p NB X T p E W X

p M X

3

1 1

5v

ih ikh ikhk h

p M X

(4.7a)

subject to:

(1) BOD discharge constraint:

Pr 1 ( ) 1 , , ,ik ikh ikh ik ikX W C BA q i k h (4.7b)

(2) Maximum allowable BOD discharge constraints:

1 1 1 1 1 1.558 1 / , ,k kh kh k r k BODX W C Q R k h (4.7c)

1 1 1 1

2 2 2 2 2

1.252 0.803 1 /

1 / , ,

k kh kh k r

k kh kh k r k BOD

X W C Q

X W C Q R k h

(4.7d)

1 1 1 1

2 2 2 2

3 3 3 3 3

1.105 0.709 1 /

0.883 1 /

1 / , ,

k kh kh k r

k kh kh k r

k kh kh k r k BOD

X W C Q

X W C Q

X W C Q R k h

(4.7e)

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4 4

0.975 0.626 1 /

0.779 1 /

0.883 1 /

1 / , ,

k kh kh k r

k kh kh k r

k kh kh k r

k kh kh k r k BOD

X W C Q

X W C Q

X W C Q

X W C Q R k h

(4.7f)

76

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 5

0.736 0.472 1 /

0.588 1 /

0.666 1 /

0.755 1 /

/ , ,

k kh kh k r

k kh kh k r

k kh kh k r

k kh kh k r

kh kh k r k BOD

X W C Q

X W C Q

X W C Q

X W C Q

X W C Q R k h

(4.7g)

(3) Maximum allowable DO deficit constraints:

1 1 1 1 2 0.552 0.171 1 / , ,k kh kh k r k DOX W C Q R k h (4.7h)

1 1 1 1

2 2 2 2 3

0.607 0.233 1 /

0.108 1 / , ,

k kh kh k r

k kh kh k r k DO

X W C Q

X W C Q R k h

(4.7i)

1 1 1 1

2 2 2 2

3 3 3 3 4

0.638 0.276 1 /

0.188 1 /

0.108 1 / , ,

k kh kh k r

k kh kh k r

k kh kh k r k DO

X W C Q

X W C Q

X W C Q R k h

(4.7j)

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4 5

0.647 0.322 1 /

0.291 1 /

0.256 1 /

0.205 1 / , ,

k kh kh k r

k kh kh k r

k kh kh k r

k kh kh k r k DO

X W C Q

X W C Q

X W C Q

X W C Q R k h

(4.7k)

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

5 5 5 6

0.622 0.326 1 /

0.319 1 /

0.303 1 /

0.276 1 /

0.152 / , ,

k kh kh k r

k kh kh k r

k kh kh k r

k kh kh k r

kh kh k r k DO

X W C Q

X W C Q

X W C Q

X W C Q

X W C Q R k h

(4.7l)

(4) Product demand constraints:

min max , ,ik ik ikT T T i k (4.7m)

(5) Non-negative constraints:

77

0, , ,ikhX i k h (4.7n)

where the coefficients in constraints (4.7c) to (4.7l) are calculated from equations (4.6a)

and (4.6b) with the system inputs; Tik and Xikh are decision variables; i = 1 for the municipal

wastewater treatment plant, i = 2, 3, 4 for the industrial sectors (a paper mill, a tannery

plant, and a tobacco facility), and i = 5 for the recreational sector; j = 1 for the upstream

end, and j = 6 for the downstream end; wastewater from these sources would enter the

stream at the beginnings of reaches 2 to 6; Eik = the treatment cost coefficients for source

i during period k (Eik > 0); Mikh = coefficients for water consumption at source i in period

k with level h; pih = the probability of random parameters at source i with level h (%); Qr

= the stream flow (103 m3/day), with 1

,I

ik riQ Q

where Qik is the amount of discharged

wastewater from source i during period k (103 m3/day); Wikh = the wastewater discharge

rate at source i in period k with level h (m3/unit product); Xikh = the production level of

source i during period k with level h, i.e. the second-stage decision variable (unit/day or

ha/yr).

The TCFP model for water quality management (4.7) can be solved through the

solution algorithm as detailed in the section of Solution Methods. The model decision

variables Xikh are the production levels of different sources during the planning periods. In

practical implementation, EXCEL and LINGO were used to process data and solve model

(4.7). The optimal solutions corresponding to various constraint-violation levels can be

obtained through taking different qs levels. The computational runtime would be several

seconds.

78

Table 4.5 Solutions obtained from the TCFP model

Economic activity

k

Wastewater-discharge

rate

Ph (%) Target Tik

Planned production level xikh

0.01 0.05 0.10 0.25


1 Low 20 42500 30000 30000 30000 30000

Medium 60 42500 30000 30000 30000 30000

High 20 42500 30000 30000 30000 30000

2 Low 20 50000 52000 52000 52000 52000

Medium 60 50000 35000 35000 35000 35000

High 20 50000 35000 35000 35000 35000

3 Low 20 55000 55000 55000 55000 55000

Medium 60 55000 55000 55000 55000 55000

High 20 55000 55000 55000 55000 55000

Paper mill 1 Low 20 23 19 19 19 19

Medium 60 23 19 19 19 19

High 20 23 19 19 19 19

2 Low 20 27 21 21 21 21

Medium 60 27 21 21 21 21

High 20 27 21 21 21 21

3 Low 20 25 22 22 22 22

Medium 60 25 22 22 22 22

High 20 25 22 22 22 22

79


Economic activity

k


rate

ph (%) Target Tik


0.01 0.05 0.10 0.25

Tannery

1 Low 20 13 11.9 12.1 12.3 12.5

Medium 60 13 11.3 11.5 11.7 11.9

High 20 13 10.8 11.0 11.1 11.3

2 Low 20 14 13.4 13.7 13.9 14.1

Medium 60 14 12.7 13.0 13.2 13.4

High 20 14 11.6 11.8 11.9 12.2

3 Low 20 15 14.2 14.5 14.6 14.9

Medium 60 15 13.5 13.8 13.9 14.2

High 20 15 12.8 13.1 13.3 13.5

Tobacco 1 Low 20 3.0 3.1 3.2 3.2 3.3

Medium 60 3.0 2.9 3.0 3.1 3.2

High 20 3.0 2.8 2.9 2.9 3.0

2 Low 20 3.0 2.6 2.7 2.8 2.9

Medium 60 3.0 2.5 2.6 2.7 2.7

High 20 3.0 2.4 2.5 2.5 2.6

3 Low 20 3.0 2.3 2.4 2.5 2.6

Medium 60 3.0 2.2 2.3 2.4 2.5

High 20 3.0 2.1 2.2 2.3 2.3

80


Economic activity

k


rate

ph (%) Target Tik


0.01 0.05 0.10 0.25

Recreation

1 Low 20 730 803 803 803 803

Medium 60 730 803 803 803 803

High 20 730 803 803 803 803

2 Low 20 839.5 912.5 912.5 912.5 912.5

Medium 60 839.5 912.5 912.5 912.5 912.5

High 20 839.5 912.5 912.5 912.5 912.5

3 Low 20 912.5 985.5 985.5 985.5 985.5

Medium 60 912.5 985.5 985.5 985.5 985.5

High 20 912.5 985.5 985.5 985.5 985.5

81

4.3.4. Results and discussion

Table 4.5 shows the solutions of optimal production levels (Xikh) obtained through the

TCFP model under different significance levels (qs). For example, when qs = 0.01 and

wastewater discharge rate is low, the production level during period 1 for the municipal

wastewater treatment plant, paper mill, tannery plant, tobacco facility, and recreation

would respectively be 42,500 m3/day, 23, 14, 3.0 tonne/day, and 803 ha/yr. Similarly, the

production plans for the three periods under different qs levels can be interpreted.

Moreover, the TCFP results in Table 4.5 can provide the desired wastewater discharge

(Wikh • Xikh) and water consumption (Mikh • Xikh) patterns for different water users. For

example, when qs = 0.01 and wastewater discharge rate is low, the discharged wastewater

from the municipal wastewater treatment plant, paper mill, tannery, tobacco, and

recreation in period 1 would respectively be 18900, 5261.1, 1353.30, 606.68, and 6177.82

m3/day; the relevant water consumptions would respectively be 19500, 5538.5, 1410.29,

645.26, and 6240.3 m3/day.

The results in Table 4.5 also indicate that probabilistic deficits would occur if the

planning level does not meet the pre-regulated target. Correspondingly, the probabilistic

deficits would lead to excess water consumption, wastewater discharge, and economic

penalty due to the violation of environmental requirements. For example, the optimized

production target for the municipal wastewater treatment plant would be 42,500 m3/day in

period 1. However, the planned production levels under low, medium, and high wastewater

discharge rates would be 30,000 m3/day. Correspondingly, there would be 12,500 m3/day

of production deficits (subject to penalties) under these three discharge rates. The penalties

82

would present in terms of raised treatment costs and/or the punishments due to excess

wastewater discharges. Those for periods 2 and 3 can be similarly interpreted based on the

results in Table 4.5. The optimal production patterns under different water availabilities

and target levels are presented in Figs. 4.2 to 4.6. Generally, a higher target level would

lead to a higher benefit, but at the meantime, a higher risk of production shortage (and thus

a higher penalty cost) would occur when the wastewater discharge rate is low. In contrast,

a lower target level would lead to a lower benefit, a lower risk of violating the previously

regulated targets, and thus a lower penalty.

In addition, the TCFP results in Table 4.5 also indicate that a higher qs level would

correspond to a higher ratio objective. For example, when qs is raised from 0.01 to 0.25,

the ratio objective would be increased from 13.65 to 13.79 $/m3. The ratio objective

denotes the efficiency of water utilization, and the qs level denotes the probability at which

the constraints can be violated. Thus, the relationship between the ratio objective and

uncertain conditions demonstrates a tradeoff among efficiency, constraint violation, and

policy scenarios. An increased qs level means an increased admissible risk, which leads to

a decreased strictness for the constraints thus an expanded decision space. As a result, a

higher risk of production shortage causes a higher penalty. Under a higher qs level, an

alternative corresponding to a higher efficiency of water utilization (i.e. less use of water

and higher net benefits) will be obtained. However, the reliability of meeting water

availability constraints and environmental requirements would decrease at the meantime.

On the contrary, planning under a lower qs level would result in an alternative

corresponding to an increased reliability but a lower benefit.

83

Figure 4.2 Target and planning production level for the wastewater treatment plant

30

35

40

45

50

55

60

1 2 3 1 2 3 1 2 3

Wat

er s

uppl

y le

vel 1

03

m3 /

day

Probability Level

Planning level Pre-regulated target


84

Figure 4.3 Target and planning production level for the paper plant

15.0

20.0

25.0

30.0

1 2 3 1 2 3 1 2 3

Pap

er (

t/day

)

Probability Level



85

Figure 4.4 Target and planning production level for the leather plant

10.0

11.0

12.0

13.0

14.0

15.0

16.0

1 2 3 1 2 3 1 2 3

Lea

ther

(t/

day)

Probability Level



86

Figure 4.5 Target and planning production level for the tobacco plant

2.0

3.0

4.0

1 2 3 1 2 3 1 2 3

Tob

acco

(t/

day)

Probability Level



87

Figure 4.6 Target and planning area for the recreational sector

600

700

800

900

1000

1100

1 2 3 1 2 3 1 2 3

Are

a (

ha)

Probability Level



88

When the water quality managers place more emphasis on the economic aspect and

aim towards a maximized system benefit, the optimal ratio problem shown in model (4.7)

can be changed into a maximum benefit problem by replacing (4.7a) with the following

objective:

4 3 3 4 3

5 51 1 1 1 1 1

3 4 3

5 5 5 51 1 1 1 1

Max Net benefits

1825 5 1825

5 5

s

nk nk k k nh nk nkh nkn k k n k h

s s

h k kh k nh nk nkh nkhk h n k h

f

NB T NB T p NB X T

p NB X T p c w X

(4.7o)

The generated model is a two-stage chance-constrained linear programming (TCLP)

problem subject to the constraints (4.7b) to (4.7n). Therefore, results under different qs

levels can be obtained from the TCLP model by using the chance-constrained

programming method with the same parameter settings of stochastic uncertainty. The

TCLP solutions under different qs levels are provided in Table 4.6. The production plans

obtained from the TCLP and TCFP models are generally different. It is indicated that, due

to the simplification of the objective, the TCLP model cannot optimize the system

efficiency of water utilization. Consequently, the system net benefits under all qs levels

from the TCLP solutions are higher than those of the TCFP solutions because of utilizing

more water. Figs. 4.7 to 4.9 compare the results obtained from both the TCLP and TCFP

models. As shown in Fig. 4.7, the solutions of net benefits from TCLP are $1778.09×106

when qs = 0.01, $1789.00×106 when qs = 0.05, $1794.82×106 when qs = 0.10, and

$1804.54×106 when qs = 0.25. Obviously, the TCLP model leads to higher system benefits

than the optimal ratio model under a range of qs levels. However, in Fig. 4.9, the net

benefits per unit of water consumption obtained from TCLP is around 13.37 $/m3 under qs

89

Table 4.6 Solutions obtained from the TCLP model

Economic activity

k

Wastewater discharge

rate

ph (%) Target Tik


0.01 0.05 0.10 0.25


1 Low 20 42500 50000 50000 50000 50000

Medium 60 42500 50000 50000 50000 50000

High 20 42500 48008 48828 49265 49995

2 Low 20 50000 52000 52000 52000 52000

Medium 60 50000 52000 52000 52000 52000

High 20 50000 52000 52000 52000 52000

3 Low 20 55000 55000 55000 55000 55000

Medium 60 55000 55000 55000 55000 55000

High 20 55000 55000 55000 55000 55000

Paper mill 1 Low 20 23 22.0 22.5 22.7 23.1

Medium 60 23 21.0 21.4 21.6 22.0

High 20 23 19.9 20.3 20.5 20.9

2 Low 20 27 23.4 23.9 24.2 24.6

Medium 60 27 22.3 22.8 23.0 23.4

High 20 27 21.1 21.6 21.8 22.2

3 Low 20 13 24.6 25.2 25.5 26.0

Medium 60 13 23.5 24.0 24.3 24.7

High 20 13 22.2 22.7 23.0 23.4

90


Economic activity

k


rate

ph (%) Target Tik


0.01 0.05 0.10 0.25

Tannery

1 Low 20 14 11.9 12.1 12.3 12.5

Medium 60 14 11.3 11.5 11.7 11.9

High 20 14 10.8 11.0 11.1 11.3

2 Low 20 15 13.4 13.7 13.9 14.1

Medium 60 15 12.7 13.0 13.2 13.4

High 20 15 11.6 11.8 11.9 12.2

3 Low 20 3.0 14.2 14.5 14.6 14.9

Medium 60 3.0 13.5 13.8 13.9 14.2

High 20 3.0 12.8 13.1 13.3 13.5

Tobacco 1 Low 20 3.0 3.1 3.2 3.2 3.3

Medium 60 3.0 2.9 3.0 3.1 3.2

High 20 3.0 2.8 2.9 2.9 3.0

2 Low 20 3.0 2.6 2.7 2.8 2.9

Medium 60 3.0 2.5 2.6 2.7 2.7

High 20 3.0 2.4 2.5 2.5 2.6

3 Low 20 3.0 2.3 2.4 2.5 2.6

Medium 60 3.0 2.2 2.3 2.4 2.5

High 20 3.0 2.1 2.2 2.3 2.3

91


Economic activity

k


rate

ph (%) Target Tik


0.01 0.05 0.10 0.25

Recreation

1 Low 20 730 803 803 803 803

Medium 60 730 803 803 803 803

High 20 730 803 803 803 803

2 Low 20 839.5 912.5 912.5 912.5 912.5

Medium 60 839.5 912.5 912.5 912.5 912.5

High 20 839.5 912.5 912.5 912.5 912.5

3 Low 20 912.5 985.5 985.5 985.5 985.5

Medium 60 912.5 985.5 985.5 985.5 985.5

High 20 912.5 985.5 985.5 985.5 985.5

92

Figure 4.7 The comparison of net benefits between optimal-ratio and TCLP models

1778.09

1789.00 1794.82

1804.54

1483.65

1492.07

1496.56

1504.06

1480.00

1485.00

1490.00

1495.00

1500.00

1505.00

1510.00

1700

1720

1740

1760

1780

1800

0.01 0.05 0.1 0.25

Net

ben

efits

for

TC

LP (

$106

)

qs level

TCLP Optimal ratio

Ne

t be

nefit

sfo

r o

ptim

al-r

atio

($1

06 )

93

Figure 4.8 The comparison of water consumption between optimal-ratio and TCLP

models

133.12

133.80 134.17

134.77

108.70

108.86

108.94

109.08

108.30

108.50

108.70

108.90

109.10

109.30

109.50

130

131

132

133

134

135

136

0.01 0.05 0.1 0.25

TC

LP w

ate

r co

nsu

mpt

ion

(106

m3)

qs level

TCLP Optimal ratio

Opt

ima

-ra

tio w

ater

co

nsum

ptio

n (1

06m

3 )

94

Figure 4.9 The comparison of system efficiency between optimal-ratio and TCLP models

13.36 13.37 13.38 13.39

13.65

13.71 13.74

13.79

13.10

13.20

13.30

13.40

13.50

13.60

13.70

13.80

13.90

0.01 0.05 0.1 0.25

Net

ben

efits

/Wat

er s

uppl

y ($

/m3 )

qs level

TCLP Optimal ratio

95

= 0.01 to 0.25, which is significantly lower than that from the optimal ratio model.

Moreover, when qs is raised from 0.01 to 0.25, the efficiency of water utilization

obtained from TCLP would be increased from 13.36 to 13.39 $/m3, while that from the

optimal ratio model would be increased from 13.65 to 13.79 $/m3. Furthermore, Fig. 4.8

indicates that the total water consumption obtained from TCLP is from 133.12 to 134.77

million m3 under qs = 0.01 to 0.25, which is significantly higher than that from the TCFP

model. When the qs level is raised, the net benefits would be increased with a sacrifice of

increased constraint-violation risk; however, the water consumption for five water users

would also be increased. In contrast, the optimal ratio model results in a lower level of

water consumption and a higher system efficiency of water utilization. Compared with

TCLP, the optimal ratio model could more effectively address the sustainable water quality

management problem and provide more information regarding tradeoffs and

interrelationships among multiple system factors.

Generally, with the comprehensive consideration of water availability, stochastic

demands, and multiple policy scenarios, the developed TCFP approach has the following

advantages over the other optimization methods. Firstly, it can balance conflicting

objectives without modifying their original magnitudes; secondly, it can provide an

effective connection between environmental regulations and economic implications

presented as penalties due to improper policies; thirdly, it can account for randomness in

both the objective and constraints; fourthly, it can support an in-depth analysis of the

interrelationships among efficiency, policies and constraint-violation risk. Therefore, the

proposed TCFP method can also be applied to other resources and environmental

management problems, such as energy systems planning, waste management, and air

96

quality management.

4.4. Summary

A two-stage chance-constrained fractional programming (TCFP) approach has been

developed for supporting water quality management systems under uncertainty. This

method can handle ratio optimization problems associated with policy analysis and

uncertainties expressed as probability distributions, where two-stage stochastic

programming (TSP) is integrated into a stochastic linear fractional programming (SLFP)

framework. An effective solution method is proposed to tackle this integrated model. The

TCFP method has advantages in: (1) balancing the conflict of two objectives; (2) reflecting

different policies; (3) tackling uncertainty available as probability distributions; and (4)

presenting optimal solutions under different constraint-violation conditions.

Through a case study of a water quality management system, the applicability of the

proposed method has been demonstrated. The solutions obtained from the TCFP model

are effective for identifying sustainable water quality management schemes with

maximized system efficiency under various constraint-violation risks and different policy

scenarios. The results also indicate that reasonable solutions can incorporate valuable

uncertain information into the decision making process and generate flexible water quality

management schemes under policy scenarios and different levels of constraint violation.

Moreover, it can provide detailed analysis of the interrelationships among efficiency,

different policies, and constraint-violation risk. In practice, through employing advanced

water quality simulation models for dealing with multiple pollutants, the TCFP model can

be extended to tackle more complicated water quality management systems.

97

This study attempts to provide a TCFP modeling framework for solving ratio

optimization problems involving analysis of policies and random inputs. Thus economic

penalties were taken into consideration as corrective measures against any arising

infeasibility caused by a particular realization of uncertainty, such that a linkage to

previously regulated policy targets was established. Although the proposed method is

applied to water quality management for the first time, the results suggest that it is also

applicable to other environmental and resources management problems. The TCFP could

be further intensified through incorporating methods of fuzzy set, interval analysis, and

game theory into its framework.

98

CHAPTER 5

DYNAMIC CHANCE-CONSTRAINED TWO-STAGE FRACTIONAL

PROGRAMMING FOR PLANNING REGIONAL ENERGY SYSTEMS IN THE

PROVINCE OF BRITISH COLUMBIA, CANADA

5.1. Background

Management of energy resources is essential to regional economic development and

environmental protection throughout the world (Ma and Nakamori, 2009; Mavrotas et al.,

2008). However, there are many challenges in effective resource management due to

issues of energy demand, supply, and allocation among various users (Lin and Huang,

2011). In addition, a variety of uncertainties are associated with these issues and the related

parameters such as future electricity demands, resource availabilities, energy allocation

targets, facility capacity-expansion options, and economic costs, as well as their

interrelationships (Cai et al., 2009). This leads to many complexities in decision-making

processes (Luhandjula, 2006). Moreover, these complexities will be further intensified by

multiobjective features that involve balancing a trade-off between environmental

protection and economic development (Zhou et al., 2014). Therefore, efficient system

analysis techniques, which can systematically consider environmental, energy, and

economic issues, are desired for multiobjective planning regional energy systems under

complexities.

Over the past decades, a large number of energy system analysis methods were used

for planning and management of energy systems (Cormio et al., 2003; Dicorato et al., 2008;

99

Endo and Ichinohe, 2006; Iniyan and Sumathy, 2000; Jebaraj and Iniyan, 2006; Khella,

1997; Lee and Chang, 2007; Pękala et al., 2010; Pohekar and Ramachandran, 2004;

Ramachandra, 2009; Ramanathan, 2005; Remmers et al., 1990; Srivastava and Misra,

2007; Wene and Ryden, 1988). For example, the Market Allocation model (MARKAL)

was proposed as a large-scale energy activity analysis model and was widely applied to a

number of regions for planning of energy management systems (Fishbone and Abilock,

1981; Henning et al., 2006; Howells et al., 2005; Unger and Ekvall, 2003). The Energy

Flow Optimization model (EFOM) was established and broadly applied in European

countries for regional energy systems planning (Cormio et al., 2003; Grohnheit and Gram

Mortensen, 2003; Howells et al., 2005). Among them, multiobjective optimization

methods were widely used to provide desired management schemes under various system

conditions (Ahmadi et al., 2012; Fadaee and Radzi, 2012; Koroneos et al., 2004; Liu et al.,

2010; Ren et al., 2010). For example, Ren et al. (2010) proposed a multiobjective goal

programming approach to analyze the optimal operating strategy of a distributed energy

resource system which minimizes both energy costs and environmental impacts that is

assessed in terms of CO2 emissions. Zhang et al. (2012) developed a short-term

multiobjective economic environmental hydrothermal scheduling model, where the

objective was to simultaneously minimize energy costs as well as the effects from pollutant

emissions. However, multiobjective optimization methods could not effectively tackle

practical energy management problems due to its need to transform multiobjectives into a

single measure based on unrealistic or subjective assumptions. Linear fractional

programming (LFP), which can compare objectives of different aspects directly through

their original magnitudes, was widely employed in various management problems for

100

dealing with the above concern (Charnes et al., 1978; Mehra et al., 2007; Stancu-Minasian,

1997a, 1999). Recently, Zhu and Huang (2013) developed a dynamic stochastic fractional

programming (DSFP) approach for capacity-expansion planning of electric power systems

under uncertainty.

However, most of the previous studies were unable to reflect linkages existing among

energy activities, emission mitigation, and economic developments, particularly in the

province of British Columbia in Canada. In addition, the previous studies could not

effectively provide desired planning schemes in practical multiobjective energy

management problems that involve inputs of random information, complexities of multi-

stage decisions, and dynamic variations of system behaviors. One potential approach for

better addressing these issues is to integrate chance-constrained programming (CCP), two-

stage stochastic programming (TSP), and mixed-integer linear programming (MILP) into

a fractional programming (FP) framework for supporting multiobjective energy systems

planning and air pollution alleviation. Thus, the corresponding solutions could be used for

generating decision alternatives and helping decision makers gain insight into interactions

among multiobjectives, system violations, multi-stage decisions, and dynamic variations.

Therefore, in this study, a dynamic chance-constrained two-stage fractional regional

energy model (DCTFP-REM) will be developed to support regional energy systems

planning and environmental management under uncertainty through integrating CCP, TSP,

and MILP techniques within a FP optimization framework. The development of DCTFP-

REM entails the following elements:

101

(i) integration of CCP, TSP, and MILP techniques to formulate a dynamic chance-

constrained two-stage fractional programming (DCTFP) method for dealing with issues of

multiobjective tradeoffs and dynamic variations, as well as uncertainties presented as

random variables in the objectives and constraints;

(ii) development of a dynamic chance-constrained two-stage fractional regional

energy model (DCTFP-REM) based on the proposed DCTFP method; and

(iii) application of DCTFP-REM to the province of British Columbia to demonstrate

its applicability in supporting energy system planning and environmental management.

Desired regional energy system management schemes under different constraint-

violation levels will be obtained. They are helpful for (a) facilitating the dynamic analysis

of capacity-expansion decisions; (b) identifying energy allocation patterns of generating

and heating technologies; (c) examining a linkage between predefined policies and the

implied economic penalties; (d) analyzing complex interrelationships among renewable

energy utilization efficiency and different subsystems (i.e., energy resources supply and

demand) under different system violation levels; and (e) addressing conflicts between

environmental and economic objectives in regional energy system planning.

5.2. Overview of the British Columbia energy system

5.2.1. The province of British Columbia

British Columbia is the westernmost province in Canada, which has a total area of

944,735 km2 (Statistics Canada, 2005). It is bordered by the Province of Alberta to the

102

east, the Yukon and the Northwest Territories to the north, the US state of Alaska to the

northwest, the Pacific coast to the west, and the US states of Washington, Idaho, and

Montana to the south (British columbia's destination site, 2014). The current population is

4.631 million, approximately 13% of Canada’s population and is responsible for roughly

13% of the national gross domestic product (Statistics Canada, 2014). The annual

population growth rate was 1.51% during the period of 2005-2010, which increased from

4.197 million to 4.524 million (National Energy Board, 2013). Table 5.1 presents the

population, labour force, employment, and households in the province of British Columbia

from 2005 to 2035 (National Energy Board, 2013). Compared to the previous year, the

GDP of the province of British Columbia increased 2.7%, reaching to $159,330 million in

2014 (National Energy Board, 2013). Table 5.2 lists the goods GDP, and services GDP for

the province of British Columbia from 2005 to 2035 (National Energy Board, 2013).

The province of British Columbia has an abundance of hydropower capability.

Electricity in the province is mainly generated from BC Hydro, which is one of the largest

electric utilities in Canada and serves 95% of the province’s population (BC Hydro, 2013).

A total of 31 hydroelectric facilities and three thermal generating plants are operated by

BC Hydro, accounting for 12,000 MW of installed generating capacity (BC Hydro, 2013).

Also, about 95% of the electricity converted by BC Hydro is produced from hydroelectric

facilities, which are situated throughout the Peace, Columbia and Coastal regions in the

province of British Columbia (BC Hydro, 2013). BC Hydro is capable of generating over

43,000 gigawatt hours of electricity annually to meet the demand of more than 1.9 million

residential, commercial, and industrial customers. BC Hydro also continues to explore

alternative energy sources, such as wind and wave power (BC Hydro, 2013).

103

Table 5.1 Population, labour force, employment, and households in the province of

British Columbia

year Population (thousand)

Labour Force (thousand)

Employment (thousand)

Households (thousand)

2005 4197 2264 2130 1724

2010 4524 2502 2324 1863

2015 4806 2707 2506 2018

2020 5064 2857 2670 2157

2025 5308 2986 2806 2276

2030 5529 3099 2930 2381

2035 5724 3213 3052 2480

Source: Statistics Canada, NEB

104

Table 5.2 GDP, goods GDP, and services GDP in the province of British Columbia

Year

Total GDP

(million $1997) Goods GDP

Services GDP

2005 131993 33428 98565

2010 144052 30876 113176

2015 163810 36707 127103

2020 188720 40464 148255

2025 213122 43904 169217

2030 237118 48233 188885

2035 260961 52801 208160

Source: Statistics Canada, NEB

105

5.2.2. British Columbia energy system

British Columbia’s energy system would designed to cover the entire province. The

study time is from 2010-2040, which is further divided into six planning periods to reflect

the dynamics of British Columbia’s energy system. The major sources of electricity and

heat generation in British Columbia include fossil fuels such as natural gas, coal, diesel,

and fuel oil, as well as renewable energies such as biomass, hydro, wind, solar, geothermal,

wave/tide. Specifically, these energies including electricity and heat are consumed by the

residential, commercial, industrial, and transportation sectors. Population growth and

economic development in the province over the last few decades are contributing to

increments of energy demands. According to National Energy Board (2013), the total

amount of energy consumption by end-users in British Columbia increased from 1,221.1

PJ in 2010 to 1274.7 PJ in 2010. The residential, commercial, industrial, and transportation

sectors consumed approximately 12.7%, 11.5%, 46.44%, and 29.36% of total energy

during the period from 2001 to 2010, respectively (National Energy Board, 2013). In 2010,

the residential sector accounted for almost 31.44% and 20.49% of all electricity and natural

gas used, respectively; and the industrial sector consumed 45.46% and 62.34% of total

electricity and natural gas consumption, respectively (National Energy Board, 2013).

Renewable or green energy resources are encouraged under resources conservation

and environmental protection. Currently, the total capacity of electricity generation in

British Columbia is 16554 MW, including natural gas-fired, coal-fired, diesel-fired, fuel

oil-fired, hydropower, wave power, tide power, geothermal energy, biomass energy, solar

energy, and wind power. Hydropower remains the dominant source of renewable energy

in British Columbia. In 2010, the ratio of renewable utilization to local consumed energy

106

was 93.7% (National Energy Board, 2013) . The average ratio of renewable energy

utilization was 93.83% during the 2005 to 2010 period.

The primary sources of pollutant emissions in the province are the consumption of

fossil fuels. For example, the amount of carbon dioxide (CO2) emission by energy sector

reached 54, 500 kt, which accounts for 82.45% of the total emission of CO2 in British

Columbia (Environment Canada, 2004). It is expected that the ratio of renewable energy

utilization to total energy consumed will increase due to the adoption of renewable energy

resources and improvement of conversion technologies. Electricity generated by

renewable energy resources such as hydro and wind power is a low emitted pollutant

particularly when compared with non-renewable energies. Enhancement of renewable

energy utilization will result in lower energy-related CO2 emissions in British Columbia.

5.2.3. Statement of problems

According to the above information and discussion, the energy management system

in the province of British Columbia is complicated. Decision makers should systematically

consider a number of complex processes such as energy activities, emission mitigation,

and economic development. In addition, such complexities would be further intensified

due to the uncertainties associated with parameters in the objective function and

constraints, as well as capacity expansions for energy conversion facilities to meet the

continuing increments of demand. In British Columbia’s energy system, various

uncertainties may exist in numerous system parameters (such as energy demand, allocation

target, technological efficiency, emission policy, and utilization factors) as well as their

interrelationships. For example, the random characteristic of resource availability, energy

107

production efficiency, energy demand, and allocation target can be possible sources of

uncertainties. Furthermore, decision makers in British Columbia will face challenges to

balance the conflicting objectives of maintaining rapid economic growth and reducing

environmental pollution. Thus, an effective long-term planning of energy systems is highly

desired with comprehensive consideration given to these complexities and uncertainties.

The problem under consideration is how to effectively identify energy allocation

plans and capacity expansion schemes. The complexities of the study problem include:

(a) how to effectively allocate energy demands to end-users and supplies to

production facilities;

(b) how to deal with the uncertainties existing in both the objective and constraints;

(c) how to identify reasonable capacity expansion schemes for facilities under

uncertainty;

(d) how to maximize renewable energy resources utilization with potential low

system costs and environmental impacts;

(e) how to reflect latter economic penalties of corrective measures due to the violation

of previously regulated environmental policies; and

(f) how to capture the tradeoff between system efficiency and reliability.

108

5.3. Development of DCTFP-REM model

5.3.1. Dynamic chance-constrained two-stage fractional programming (DCTFP)

method

A dynamic chance-constrained two-stage fractional programming (DCTFP) method

is proposed in this study. The related modeling components will be depicted in the

following.

(1) Linear fractional programming

Linear fractional programming (LFP) involves the optimization of two conflicting

objective functions subject to a decision space delimited by a set of constraints. A general

LFP problem can be defined as follows (Zhu and Huang, 2011):

Max CX

f xDX

(5.1a)

subject to:

AX B (5.1b)

0X (5.1c)

where X and B are column vectors with n and m components respectively; A is a real m ×

n matrix; C and D are row vectors with n components; α and β are constants. According to

Charnes and Cooper (1962), if (i) the objective function is continuously differentiable, (ii)

the feasible region is non-empty and bounded, and (iii) D X + β > 0 for all feasible X, the

109

LFP model can be transformed to the following linear programming problem under

transformation X r X :

Max ,g X r CX r (5.2a)

subject to:

AX r B (5.2b)

1DX r (5.2c)

0X (5.2d)

0r (5.2e)

(2) Two-stage stochastic programming

Two-stage stochastic programming (TSP) is an effective method for dealing with

optimization problems where an analysis of multi-stage decisions is required while the

relevant data are mostly uncertain. In the TSFP model, two subsets of decision variables

are included: initial variables that must be determined before the random short-term events

are resolved, and recourse variables that will be determined when the events are later

disclosed. (Li et al., 2008c). A general TSP model can be formulated as follows (Birge and

Louveaux, 1988):

Min [ ( , )]TQf C X E Q X (5.3a)

subject to:

x X (5.3b)

110

with

( , ) min ( )TQ X f y (5.3c)

subject to:

( ) ( ) ( )D y h T x (5.3d)

y Y (5.3e)

where 1 1, ,n nX R C R and 2 ,nY R is a random variable from space ( , , F P ) with

kR , 2 2 2 2: , : , : ,n m m nf R h R D R and 2 1: .m nT R By letting

random variables (i.e. ) take discrete values h with probability levels hp

( 1,2,...,h v and 1hp ), the above TSP can be equivalently formulated as a linear

programming model as follows (Ahmed et al., 2004; Li et al., 2007a):

Min1 2

1

v

T h Th

f C X p D Y

(5.4a)

subject to:

1, 1, 2,...,r rA X B r m (5.4b)

2, 1, 2,..., ; 1, 2,...t t thA X A Y t m h v (5.4c)

10, , 1, 2,...,j jx x X j n (5.4d)

20, , 1, 2,..., ; 1, 2,...,jh jhy y Y j n h v (5.4e)

111

where 1TC are coefficients of first-stage variables (X) in the objective function;

2TD are

coefficients of recourse variables (Y) in the objective function; rA and tA are coefficients

of X in constraints r and t; tA are coefficients of Y in constraints t; th is random variables

of constraints t, which is associate with probability level ph.

Obviously, model (5.4) can tackle uncertainties in right-hand sides presented as

probability distributions when coefficients in the left-hand sides and in the objective

function are deterministic (Li et al., 2008c).

(3) Chance-constrained programming

In real-world management problems, the uncertainty associated with various right-

hand-side parameters also needs to be reflected. Chance-constrained programming (CCP)

method can be employed to effectively deal with optimization problems where some right-

hand-side parameters are of stochastic features and can be represented as probability

distributions (Zhu and Huang, 2011). In the CCP model, it is required that the constraints

be satisfied under a given probability level. A typical CCP model can be formulated as

follows (Huang, 1998):

Min ( )f X (5.5a)

subject to:

Pr[ ( ) ( )] 1 , 1, 2,...,i i iA t X b t p i m (5.5b)

0X (5.5c)

112

where ( ) ( ), ( ) ( ), ;i iA t A t b t B t t T A(t) and B(t) are sets with random elements defined

on a probability space T; ( [0,1])i ip p is a given level of probability for constraint i (i.e.

significance level, which represents the admissible risk of constraint violation); m is the

number of constraint.

If coefficients in model (5.5) are uncertain in both left- and right- hand sides,

constraints (5.5b) is generally nonlinear, and the set of feasible constraints may become

very complicated (Huang, 1998; Zare and Daneshmand, 1995). When the left-hand-side

coefficients [elements of A(t) ] are deterministic and the right-hand-side coefficients [bi(t)]

are random (for all pi values), constraints (5.5b) become linear and the set of feasible

constraints is convex:

( )( ) ( ) , 1, 2,...ipi iA t X b t i m (5.5d)

where ( ) 1( ) ( ),ipi i ib t F p given the cumulative distribution function of bi [i.e. Fi(bi)] and

the probability of violating constraint i (pi).

(4) Development of the DCTFP method

Therefore, one potential approach to improve the existing method is to integrate two-

stage programming, chance-constrained programming, and mixed-integer linear

programming techniques into the LFP framework. This leads to a dynamic chance-

constrained two-stage fractional programming method as follows:

113

Max 1 1

1

2 21

v

h hhv

h hh

C X p D Yf

C X p D Y

(5.6a)

subject to:

Pr 1 , 1,2,...,s s h sA X A Y b q s S (5.6b)

, 1, 2,..., ; 1, 2,...i i h ihA X AY i m h v (5.6c)

10, , 1, 2,...j jx x X j k (5.6d)

20, , 1, 2,... ; 1, 2,...jh jh hy y Y h v j k (5.6e)

1 10, , and integer variables, 1,...,j j jx x X x j k n (5.6f)

2 20, , and integer variables, 1, 2,... ; 1,...,jh jh h jhy y Y y h v j k n (5.6g)

where ( ) ( ), sA A ( )sA A , ( ) ( ),sb B and ; As(τ), and ( )sA are

random coefficients in the constraint s; A(τ), ( )A , and B(τ) are sets with random

elements defined on a probability space Г; iA and iA are coefficients in the constraints.

If the denominator in model (5.5) is strictly positive on the feasible region, The TSFP

model can be equivalently reformulated as the following linear programming problems:

Max * *1 1

1

v

h hh

g C X p D Y r

(5.7a)

subject to:

* * ( ) sqs s hA X A Y r b (5.7b)

114

* * , 1, 2,..., ; 1,2,...i i h ihA X AY r i m h v (5.7c)

* *2 2

1

1v

h hh

C X p D Y r

(5.7d)

* *, , 1, 2,...h hX r X Y r Y h v (5.7e)

* * *10, , 1,2,...j jx x X j k (5.7f)

* * *20, , 1,2,... ; 1,2,...jh jh hy y Y h v j k (5.7g)

1 10, , and integer variables, 1,...,j j jx x X x j k n (5.7h)

2 20, , and integer variables, 1, 2,... ; 1,...,jh jh h jhy y Y y h v j k n (5.7i)

0r (5.7j)

Model (5.7) can be solved according to the algorithm of branch and bound. The

optimization solutions of xj (j = 1, 2, …, k1) and yjh (j = 1, 2, …, k2) can be obtained through

the transformations of *j jx x r (j = 1, 2, …, k1) and * /jh jhy y r (j = 1, 2, …, k2, and h

= 1, 2, …, v), while the solutions for integer variables of xj (j = k1 + 1, k1 + 2, …, n1) and

yjh (j = k2 + 1, k2 + 2, …, n2, and h = 1, 2, …, v) can be obtained directly.

The developed dynamic chance-constrained two-stage fractional programming

method can thus deal with multiobjective and capacity-expansion issues, as well as

uncertainties described as probability distributions in the objectives and constraints.

5.3.2. Development of the DCTFP-REM model

Based on the developed DCTFP method, a dynamic chance-constrained two-stage

fractional regional energy model (DCTFP-REM) is developed in this study for supporting

115

energy management in the province of British Columbia. The objective is to maximize

total renewable energy utilization per unit of system cost, while a set of constraints define

the interrelationships between the system factors/conditions and decision variables. In

detail, the system cost of the DCTFP-REM model is formulated as a sum of the following

elements:

(1) Costs for primary energy supply

5 6 2 6 2 6 6 6

11 1 1 1 1 1 1 1

5 6 3 2 6 3

1 1 1 1 1 1

2 6 3

1 1 1

rjt jt rkt rkt ct ct rnt rntj t k t c t n t

rjt rjt h rjth rkt rkt h rkthj t h k t h

rct rct h rcthc t h

f PE DE PH DH PC DC PD DD

PE PPE p ZE PH PPH p ZH

PC PPC p ZC

6 6 3

1 1 1rnt rnt h rnth

n t h

PD PPD p ZD

(5.8)

(2) Costs for power generation

10 6 10 6 3

21 1 1 1 1

jt jt h jt jt jthj t j t h

f CP TP p CP PCP XP

(5.9)

(3) Costs for capacity expansions of power generation

10 3 6

31 1

jmt jmt jtj m t

f EP YP COP

(5.10)

(4) Costs for heating

3 3 3 3 3

41 1 1 1 1

kt kt h kt kt kthk t k t h

f CH TH p CH PCH XH

(5.11)

(5) Costs for capacity expansions of heat generation

116

3 3 3

51 1

kmt kmt ktk m t

f EH YH COH

(5.12)

(6) Fixed and variable costs for cogeneration

2 3 2 3 3

61 1 1 1 1

2 3 3

1 1

ct ct h ct ct cthc t c t h

cmt cmt ctc m t

f CC TC p CC PCC XC

EC YC COC

(5.13)

(7) Costs for controlling contamination

5 3 3 2 3 3

71 1

2 3 3 5 3 3 3

1 1 1

2 3 3 3

1 1

jgt jgt jt jt kgt kgt kt ktj g t k g t

cgt cgt ct ct h jgt jgt jth jtc g t j g t h

h kgt kgt kth kt h cgt cgtk g t h

f GP SP TP NP GH SH TH NH

GC SC TC NC p GP SP XP NP

p GH SH XH NH p GC SC

2 3 3 3

1 1cth ct

c g t h

XC NC

(5.14)

Thus, the ratio objective of the DCTFP-REM model can be formulated as follows:

10 3

5 3

1 2 3 4 5 6 7

renewable power generation renewable heat generationMax

system cost

jt ktj k

f

XP XH

f f f f f f f

(5.15)

The constraints of DCTFP-REM are defined as follows:

(1) Mass balance constraints of energy sources

117

, , ,jt jt jth rjt rjthNP TP XP DE ZE r j t h (5.16a)

, , ,kt kt kth rkt rkthNH TH XH TH ZH r k t h (5.16b)

, , ,ct ct cth rct rcthNC TC XC TC ZC r c t h (5.16c)

4

,Pr 1 , , ,rnt rnth ndth nth DMd

TD ZD DM q r n t h

(5.16d)

(2) Electricity demand constraints

10 2 4

,1 1 1

Pr 1 ,jt jth ct cth dth dth DMEj c d

TP XP TC XC DME q t h

(5.17)

(3) Heat demand constraints

3 2 4

1 1 1

,

Pr

1 ,

kt kth ct ct cth dthk c d

dth DMH

TH XH HP TC XC DMH

q t h

(5.18)

(4) Capacity constraints

3

1

,

Pr

1 , ,

t

jt jth j jmt jmt jthm t

jth UPcap

TP XP RP EP YP UPcap

q j t h

(5.19a)

3

1

,

Pr

1 , ,

t

kt kth k kmt kmt kthm t

kth UHcap

TH XH RH EH YH UHcap

q k t h

(5.19b)

3

1

,

Pr

1 , ,

t

ct cth c cmt cmt cthm t

cth UCcap

TC XC RC EC YC UCcap

q c t h

(5.19c)

118

3

1

,t

j jmt jmt jm t

RP EP YP VP j t

(5.19d)

3

1

,t

k kmt kmt km t

RH EH YH VH k t

(5.19e)

3

1

,t

c cmt cmt cm t

RC EC YC VC c t

(5.19f)

(5) Technique constraints

3

1

,

Pr

1 , ,

t

jt jth j j jmt jmt jthm t

jth UPcap

TP XP RAP RP EP YP UPcap

q j t h

(5.20a)

3

1

,

Pr

1 , ,

t

kt kth k k kmt kmt kthm t

kth UHcap

TH XH RAH RH EH YH UHcap

q k t h

(5.20b)

3

1

,

Pr

1 , ,

t

ct cth c c cmt cmt cthm t

cth UCcap

TC XC RAC RC EC YC UCcap

q c t h

(5.20c)

(6) Energy resources constraints

6

,1

Pr 1 , ,rjt rjth rjh rjh UPEt

DE ZE UPE q r j h

(5.21a)

6

,1

Pr 1 , ,rkt rkth rkh rkh UPHt

DH ZH UPH q r k h

(5.21c)

6

,1

Pr 1 , ,rct rcth rch rch UPCt

DC ZC UPC q r c h

(5.21e)

6

,1

Pr 1 , ,rnt rnth rnh rnh UPDt

TD ZD UPD q r n h

(5.21g)

119

(7) Environmental constraints

5 3 3 3 3 3

1 1

2 3 3 5 3 3

1 1

3 3 3

1

Pr 1 1

1 1

1

1

jgt jgt jt jt kgt kgt kt ktj g t k g t

cgt cgt ct ct jgt jgt jth jtc g t j g t

kgt kgt kth ktk g t

cgt cgt ct

SP AP TP FP SH AH TH FH

SC AC TC FC SP AP XP FP

SH AH XH FH

SC AC XC

2 3 3

1

,1 , , ,

h ct gthc g t

gth SE

FC SE

q g t h

(5.22)

(8) Expansion option constraints

3

1

1 ,jmtm

YP j t

(5.23a)

0 or 1 , , tjmtYP j m (5.23b)

3

1

1 ,kmtm

YH k t

(5.23c)

0 or 1 , , tkmtYH k m (5.23d)

3

1

1 ,cmtm

YC c t

(5.23e)

0 or 1 , , tcmtYC c m (5.23f)

(9) Non-negativity constraints

, , , 0 , , , , , ,rjth rkth rcth rnthZE ZH ZC ZD r j k c n t h (5.24a)

, , 0 , , , ,jth kth cthXP XH XC j k c t h (5.24e)

120

where:

c = the type of cogeneration, c = 1, 2 (where c = 1 for natural gas-fired, 2 for coal-fired);

d = demand user, d = 1, 2, 3, 4 (where d = 1 for residential user, d = 2 for commercial user,

d = 3 for industrial user, d = 4 for transportation user);

g = the type of pollutant, g = 1, 2, 3, 4 (where l = 1 for CO2, 2 for SO2, 3 for NOx, 4 for

PM);

h = energy resource demand level, h = 1, 2, 3;

j = the type of electricity generation, j = 1, 2, ..., 10 (where j = 1 for natural gas-fired, 2 for

coal-fired, 3 for diesel-fired, 4 for fuel oil-fired, 5 for biomass-fired, 6 for hydro

power, 7 for wind power, 8 for solar energy, 9 for wave/tide power, 10 for geothermal

energy);

k = the type of heat generation, k = 1, 2, 3 (where k = 1 for natural gas-fired, 2 for coal-

fired, 3 for geothermal energy);

m = the capacity expansion option, m = 1, 2, 3, every technology of power generation,

heating, and cogeneration are provided with three expansion options;

n = primary resource demand by end-users, n = 1, 2, ..., 6 (where n=1for natural gas, n =

2 for diesel, n = 3 for fuel oil, n = 4 for LPG, n = 6 for biomass);

r = energy production places, r = 1, 2 (where r = 1 for local energy supply, 2 for imported

energy supply);

t = time period, t = 1, 2, 3, 4, 5, 6 (where t = 1 for years 2010 – 2015 t = 2 for years 2016–

2020 t = 3 for years 2021 – 2025, t = 4 for years 2026 – 2030, t = 5 for years 2031 –

2035, t = 6 for years 2036 – 2040);

121

DCrct = target supply of primary energy resource from production place r for cogeneration

technology c in period t (PJ);

DDrnt = target supply of primary energy resource n from production place r for end-users

in period t (PJ);

DErjt = target supply of primary energy resource from production place r for conversion

technology j in period t (PJ);

DHrkt = target supply of primary energy resource from production place r for heating

technology k in period t (PJ);

TCct = target activity of cogeneration technology c in period t (PJ);

THkt = target heat generated by heating technology k in period t (PJ);

TPjt = target electricity generated by conversion technology j in period t (PJ);

XCcth = excess activity of cogeneration technology c in period t under demand level h (PJ);

XPjth = excess electricity generated by conversion technology j in period t under demand

level h (PJ);

XHkth = excess heat generated by heating technology k in period t under demand level h

(PJ);

YCcmt = binary variable, identifying whether or not capacity expansion option m for

cogeneration technology c needs to be undertaken in period t;

YHkmt = binary variable, identifying whether or not capacity expansion option m for heating

technology k needs to be undertaken in period t;

YPjmt = binary variable, identifying whether or not capacity expansion option m for

conversion technology j needs to be undertaken in period t;

122

ZCcth = excess supply of primary energy resource for cogeneration technology c in period

t under demand level h (PJ);

ZDnth = excess supply of primary energy resource n from production place r for end-users

in period t under demand level h (PJ);

ZEjth = excess supply of primary energy resource from production place r for conversion

technology j in period t under demand level h (PJ);

ZHkth = excess supply of primary energy resource from production place r for heating;

ph = probability levels (i.e. 20%, 60% and 20% corresponding to low, medium and high

levels of energy demand, respectively);

,dth DMHq , ,dth DMEq , ,nth DMq = constraint-violation probability for heat demand constraints,

electricity demand constraints, mass balance constraints;

,jth UPcapq , ,jth UPcapq , ,kth UHcapq , and ,cth UCcapq = constraint-violation probability for capacity

and technique constraints;

,gth SEq = constraint-violation probability for environmental constraints;

,rjh UPEq , ,rkh UPHq , ,rch UPCq , and ,rnh UPDq = constraint-violation probability for energy

resources constraints;

ACcgt = the average emission abatement efficiency of pollutant g for cogeneration

technology c in period t;

AHkgt = the average emission abatement efficiency of pollutant g for heating technology k

in period t;

APjgt = the average emission abatement efficiency of pollutant g for conversion technology

j in period t;

CCct = fix cost for cogeneration technology c in period t ($106/PJ);

123

CHkt = fix cost for heating technology k in period t ($106/PJ);

CPjt = fix cost for power generation technology j in period t ($106/PJ);

DMndth =primary resource n demand by end-user d in period t under demand level h (PJ);

ECcmt = capacity expansion option m for cogeneration technology c in period t (GW);

EHkmt = capacity expansion option m for heating technology k in period t (GW);

EPjmt = capacity expansion option m for conversion technology j in period t (GW);

GCcgt = elimination cost of pollutant g for cogeneration technology c in period t ($106/kt);

GHkgt = elimination cost of pollutant g for heating technology k in period t ($106/kt);

GPjgt = elimination cost of pollutant g for power generation technology j in period t

($106/kt);

HPct = the thermoelectric ratio;

NCct = cogeneration efficiency for technology c in period t;

NHkt = heating efficiency for technology k in period t;

NPjt = generation efficiency for technology j in period t;

PCrct = cost for energy supply from production place r for cogeneration technology c in

period t ($106/PJ);

PDrnt = cost for energy supply n from production place r for end-users in period t ($106/PJ);

PErjt = cost for energy supply from production place r for conversion technology j in period

t ($106/PJ);

PHrkt = cost for energy supply from production place r for heating technology k in period

t ($106/PJ);

RCj = residual capacity for cogeneration technology c (GW);

RHj = residual capacity for heat generation technology k (GW);

124

RPj = residual capacity for power generation technology j (GW);

SCcgt = the average emission rate of pollutant c for cogeneration technology c in period t

(kt/PJ);

SHkgt = the average emission rate of pollutant k for heating technology j in period t (kt/PJ);

SPjgt = the average emission rate of pollutant g for conversion technology j in period t

(kt/PJ);

UCcapcth = conversion coefficient from capacity to energy for cogeneration technology c

in period t under demand level h (PJ/GW);

UHcapkth = conversion coefficient from capacity to energy for heat generation technology

k in period t under demand level h (PJ/GW);

UPcapjth = conversion coefficient from capacity to energy for power generation

technology j in period t under demand level h (PJ/GW);

VCc = allowable capacity for cogeneration facility c (GW);

VHk = allowable capacity for heat-generation facility k (GW);

VPj = allowable capacity for power-generation facility j (GW);

COCct = variable cost for cogeneration technology c in period t ($106/GW);

COHkt = variable cost for heating technology k in period t ($106/GW);

COPjt = variable cost for conversion technology j in period t ($106/GW);

DMEdth = electricity demand by end-user d in period t under demand level h (PJ);

DMHdth = heating demand by end-user d in period t under demand level h (PJ);

PCCct = penalty costs for the excess activity of cogeneration technology c in period t

($106/PJ);

PCHkt = penalty cost for excess heat generated by technology k in period t ($106/PJ);

125

PCPjt = penalty costs for the excess electricity generated by conversion technology j in

period t ($106/PJ);

PPCrct = penalty costs for the excess energy supply from production place r for

cogeneration technology c in period t ($106/PJ);

PPDrnt = penalty costs for the excess energy supply n from production place r for end-

users in period t ($106/PJ);

PPErjt = penalty costs for the excess energy supply from production place r for conversion

technology j in period t ($106/PJ);

PPHrkt = penalty costs for the excess energy supply from production place r for heating

technology k in period t ($106/PJ);

RACc = capacity utilization rate for cogeneration technology c in period t;

RAHk = capacity utilization rate for heating technology k in period t;

RAPj = capacity utilization rate for power generation technology j in period t;

UPCrch = available imported resource for cogeneration facility c under level h (PJ);

UPDrnh = available imported resource n for end-users (PJ);

UPErjh = available imported resource for power generation facility j (PJ);

UPHrkh = available imported resource for heat generation facility k (PJ).

5.4. Result analysis

The solutions of primary energy suppliers obtained from the DCTFP-REM model

under different qs = 0.01 in the province of British Columbia are provided in Tables 5.3-

126

Table 5.3 Solutions of primary energy suppliers for power generation under qs = 0.01

Local supply (PJ) Import supply (PJ)

Primary energy supply

Period Low Medium High Low Medium High

Natural gas (j = 1)

t = 1 7.90 7.90 13.66 6.21 6.96 1.95

t = 2 8.40 8.46 15.09 6.78 7.51 1.68

t = 3 13.67 14.68 15.50 1.96 1.77 1.77

t = 4 15.10 15.99 16.89 1.85 1.85 1.85

t = 5 14.94 15.82 16.71 1.93 1.93 1.93

t = 6 15.86 16.80 17.74 2.00 2.00 2.00

Coal (j = 2) t = 1 0.00 0.00 0.00 0.00 0.00 0.00

t = 2 0.00 0.00 0.00 0.00 0.00 0.00

t = 3 0.00 0.00 0.00 0.00 0.00 0.00

t = 4 0.00 0.00 0.00 0.00 0.00 0.00

t = 5 0.00 0.00 0.00 0.00 0.00 0.00

t = 6 0.00 0.00 0.00 0.00 0.00 0.00

Diesel (j = 3) t = 1 0.08 0.08 0.71 0.85 0.85 0.22

t = 2 1.00 1.00 1.00 0.00 0.00 0.00

t = 3 0.09 0.33 0.69 0.93 0.69 0.32

t = 4 1.03 1.03 1.03 0.00 0.00 0.00

t = 5 1.21 1.21 1.21 0.00 0.00 0.00

t = 6 1.33 1.33 1.33 0.00 0.00 0.00

127



Primary energy supply Period Low Medium High Low Medium High

Fuel oil (j = 4) t = 1 0.16 0.16 4.60 4.78 4.78 0.34

t = 2 2.78 4.11 5.25 2.47 1.14 0.00

t = 3 5.53 5.53 5.53 0.00 0.00 0.00

t = 4 5.80 5.80 5.80 0.00 0.00 0.00

t = 5 6.04 6.04 6.04 0.00 0.00 0.00

t = 6 6.25 6.25 6.25 0.00 0.00 0.00

Biomass (j = 5) t = 1 0.79 0.79 0.79 4.64 4.92 5.21

t = 2 0.84 0.84 0.84 4.92 5.23 5.53

t = 3 0.88 0.88 0.88 5.19 5.51 10.62

t = 4 2.15 2.44 11.20 4.22 4.26 2.54

t = 5 6.63 6.98 11.25 0.00 0.00 0.00

t = 6 17.15 17.93 10.88 0.00 0.00 0.00

128

Table 5.4 Solutions of primary energy suppliers for heat generation under qs = 0.01



Natural gas (k = 1)

t = 1 7.90 7.90 7.90 213.47 225.12 236.77

t = 2 8.40 8.40 8.40 226.78 239.16 251.53

t = 3 102.02 109.07 20.03 145.78 151.77 253.85

t = 4 247.70 268.77 282.44 4.64 4.64 4.64

t = 5 265.72 279.96 294.20 4.83 4.83 4.83

t = 6 79.49 72.69 283.17 5.00 5.00 5.00

Coal (k = 2) t = 1 0.60 2.85 3.00 2.12 0.00 0.00

t = 2 2.88 3.03 3.18 0.00 0.00 0.00

t = 3 3.03 3.19 3.35 0.00 0.00 0.00

t = 4 3.18 3.35 3.52 0.00 0.00 0.00

t = 5 1.10 1.19 3.66 0.00 0.00 0.00

t = 6 3.43 0.22 0.38 0.00 0.00 0.00

129

Table 5.5 Solutions of primary energy suppliers for cogeneration under qs = 0.01



Natural gas (c = 1)

t = 1 99.06 107.22 169.14 3.16 3.16 3.16

t = 2 69.38 94.85 156.17 3.36 3.36 3.36

t = 3 40.69 83.53 144.29 3.54 3.54 3.54

t = 4 17.57 72.83 133.07 3.71 3.71 3.71

t = 5 36.94 64.49 122.90 3.86 3.86 3.86

t = 6 166.18 202.56 130.33 4.00 4.00 4.00

Coal (c = 2) t = 1 0.00 0.00 0.00 0.00 0.00 0.00

t = 2 0.00 0.00 0.00 0.00 0.00 0.00

t = 3 0.00 0.00 0.00 0.00 0.00 0.00

t = 4 0.00 0.00 0.00 0.00 0.00 0.00

t = 5 0.00 0.00 0.00 0.00 0.00 0.00

t = 6 0.00 0.00 0.00 0.00 0.00 0.00

130

Table 5.6 Solutions of primary energy suppliers for end-users under qs = 0.01



Natural gas (n = 1)

t = 1 1386.70 1386.70 1386.70 203.38 302.43 698.63

t = 2 1606.50 1606.50 1606.50 167.13 396.63 855.63

t = 3 1802.85 1802.85 1802.85 94.45 480.78 995.88

t = 4 1986.60 1986.60 2710.64 0.00 559.53 403.09

t = 5 2118.55 2304.84 3339.93 283.16 429.78 0.00

t = 6 2478.44 2861.13 3491.73 98.92 0.00 0.00

Diesel (n = 2) t = 1 387.80 387.80 387.80 99.72 99.72 195.38

t = 2 423.61 518.88 638.98 35.22 0.00 0.00

t = 3 457.80 457.80 457.80 88.46 114.58 245.38

t = 4 491.75 491.75 698.93 0.00 129.13 62.44

t = 5 525.70 525.70 525.70 61.07 143.68 293.88

t = 6 558.25 605.23 875.38 85.85 110.65 0.00

Fuel oil (n = 3) t = 1 312.20 312.20 389.21 80.28 80.28 80.28

t = 2 331.45 331.45 417.26 85.23 85.23 85.23

t = 3 339.50 339.50 428.99 87.30 87.30 87.30

t = 4 340.20 340.20 430.01 87.48 87.48 87.48

t = 5 348.60 392.94 492.54 39.35 39.35 39.35

t = 6 396.11 441.79 543.29 0.00 0.00 0.00

131




Gasoline (n = 4) t = 1 625.80 625.80 780.16 160.92 160.92 160.92

t = 2 618.10 618.10 768.94 158.94 158.94 158.94

t = 3 607.25 607.25 753.13 156.15 156.15 156.15

t = 4 625.10 625.10 779.14 160.74 160.74 160.74

t = 5 644.35 644.35 807.19 165.69 165.69 165.69

t = 6 660.45 700.45 889.15 111.33 111.33 111.33

LPG (n = 5) t = 1 40.53 43.06 53.16 0.00 0.00 0.00

t = 2 43.41 49.06 60.36 0.00 0.00 0.00

t = 3 43.98 53.06 65.16 0.00 0.00 0.00

t = 4 44.45 56.06 68.76 0.00 0.00 0.00

t = 5 51.74 59.06 72.36 0.00 0.00 0.00

t = 6 55.80 62.06 75.96 0.00 0.00 0.00

Biomass (n = 6) t = 1 714.77 759.30 937.40 0.00 0.00 0.00

t = 2 702.20 794.80 980.00 0.00 0.00 0.00

t = 3 636.35 771.80 952.40 0.00 0.00 0.00

t = 4 628.25 766.30 945.80 0.00 0.00 0.00

t = 5 662.23 760.30 938.60 0.00 0.00 0.00

t = 6 666.41 745.30 920.60 0.00 0.00 0.00

132

5.6. In detail, Tables 5.3-5.5 present the solutions of energy suppliers for electricity

generation, heat generation, and cogeneration. Energy demands by sectors are presented

in Table 5.6. For example, when qs = 0.01 and the demand level is low, the primary energy

supplies of natural gas, coal, diesel, fuel oil, biomass for electricity generation would

respectively be 7.9, 0, 0.08, 0.16, and 0.79 PJ, and imported of those would be 6.21, 0.

0.85, 4.78, and 4.64 PJ; the primary energy supplies of natural gas, coal, imported natural

gas, and imported coal would be 7.9, 2.85, 213.47, and 2.12 PJ, respectively; the utilization

of natural gas, diesel, fuel oil, gasoline, liquefied petroleum gas (LPG), and biomass for

demand sectors would respectively be 1386.7, 387.8, 312.2, 625.8, 40.53, and 714.77 PJ.

Similarly, the energy supply schemes for all of the technologies under different qs levels

over the planning horizon can be obtained and interpreted.

Moreover, the results in these tables indicate that the local energy supplies for power

generation would grow steadily over the 30-year planning horizon due to rapid population

growth and economic development. In contrast, the utilization of imported energy supplies

for power generation would decrease. For example, when qs = 0.01, and the demand level

is medium, the natural gas supply for electricity generation would increase from 7.9 PJ in

period 1 to 16.8 PJ in period 6. In comparison, the imported natural gas supply would

decline from 6.96 PJ in period 1 to 2.0 PJ in period 6.The primary reason for this is that

the allowance of local supply is higher and the price of imported supply is more expensive.

Coal supply from local and import sources for energy facilities under different qs and

demand levels over the planning periods would remain at low amounts due to British

Columbia’s coal-fired plant capacity and emission control policy. Figures 5.1 to 5.4

illustrate diverse energy resource supply schemes for power generation, heat generation,

133

Figure 5.1 Primary energy suppliers for power generation technologies under qs = 0.01

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

20.00t=

1t=

2t=

3t=

4t=

5t=

6t=

1t=

2t=

3t=

4t=

5t=

6t=

1t=

2t=

3t=

4t=

5t=

6t=

1t=

2t=

3t=

4t=

5t=

6t=

1t=

2t=

3t=

4t=

5t=

6

Ene

rgy

allo

catio

n (P

J)

Period

Low Medium HighLow (import) Medium (import) High (import)

Biomass NG Coal Fuel oil Diesel

134

Figure 5.2 Primary energy suppliers for heat generation technologies under qs = 0.01

0

50

100

150

200

250

300

t=1 t=2 t=3 t=4 t=5 t=6 t=1 t=2 t=3 t=4 t=5 t=6

Ene

rgy

allo

catio

n (P

J)

Period

Low Medium HighLow (import) Medium (import) High (import)

Nature Gas Coal

135

Figure 5.3 Primary energy suppliers for cogeneration technologies under qs = 0.01

0

50

100

150

200

250

t=1 t=2 t=3 t=4 t=5 t=6 t=1 t=2 t=3 t=4 t=5 t=6

Ene

rgy

allo

catio

n (P

J)

Period

Low Medium High

Low (import) Medium (import) High (import)

Nature Gas Coal

136

Figure 5.4 Primary energy suppliers for end-users under qs = 0.01

0

500

1000

1500

2000

2500

3000

3500

4000

t=1

t=2

t=3

t=4

t=5

t=6

t=1

t=2

t=3

t=4

t=5

t=6

t=1

t=2

t=3

t=4

t=5

t=6

t=1

t=2

t=3

t=4

t=5

t=6

t=1

t=2

t=3

t=4

t=5

t=6

t=1

t=2

t=3

t=4

t=5

t=6

Ene

rgy

allo

catio

n (

PJ)

Period

Low Medium High

Low (import) Medium (import) High (import)

LPG NG Diesel Gasoline Fuel oil Biomass

137

and cogeneration under qs = 0.01 over the planning horizon.

Figures 5.5 and 5.6 provide the results of electricity productions from both non-

renewable and renewable power generation technologies under qs = 0.01 over the entire

planning horizon. The results indicate that electricity generated through different power

generation technologies would increase steadily in order to satisfy the future electricity

demand. Over the planning period, electricity generated through hydropower is the

primary electricity produced in the energy system due to its high availability and large

capacity in the province of British Columbia. For instance, under qs = 0.01, the electricity

generated through hydropower at the low demand level would be 931.09, 989.13, 1042.23,

1092.45, 1137.93, and 1178.06 PJ, respectively. Moreover, owing to its location on the

west coast of Canada, British Columbia has an abundance of wave/tide power.

Correspondingly, electricity produced from the wave/tide power could play an important

role in supplying electricity to British Columbia’s energy system. For example, when the

demand rate is low and qs is 0.01, the wave/tide power facility at different periods would

generate 263.82, 280.26, 295.31, 309.54, 322.43, and 333.80 PJ, respectively. In addition,

electricity production of natural gas-fired plants, which is important to the non-renewable

electricity supply, would be 12.28, 13.05, 13.75, 14.41, 15.01, and 15.54 PJ respectively

over the planning period.

Figure 5.7 presents generation schemes for the heating technologies including natural

gas-fired, coal-fired, and geothermal power under qs = 0.01. Figure 5.8 shows the

electricity generated from cogeneration technologies including natural gas-fired and coal-

fired thermal plants under qs = 0.01. Due to a small capacity and environmental protection

138

Figure 5.5 Electricity productions from different non-renewable power generation

technologies under qs = 0.01

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

20.00

t=1 t=2 t=3 t=4 t=5 t=6 t=1 t=2 t=3 t=4 t=5 t=6 t=1 t=2 t=3 t=4 t=5 t=6

Pow

er

gene

ratio

n (P

J)

Period

Low Medium High

NG Fuel oil Diesel

139

Figure 5.6 Electricity productions from different renewable power generation


0

200

400

600

800

1000

1200

1400

t=1

t=2

t=3

t=4

t=5

t=6

t=1

t=2

t=3

t=4

t=5

t=6

t=1

t=2

t=3

t=4

t=5

t=6

t=1

t=2

t=3

t=4

t=5

t=6

t=1

t=2

t=3

t=4

t=5

t=6

t=1

t=2

t=3

t=4

t=5

t=6

Pow

er g

ener

atio

n (

PJ)

Period

Low Medium High

Ocean Biomass Hydro Solar Wind Geothermal

140

Figure 5.7 Heat generation from different generation technologies under qs = 0.01

0

50

100

150

200

250

300

t=1 t=2 t=3 t=4 t=5 t=6 t=1 t=2 t=3 t=4 t=5 t=6 t=1 t=2 t=3 t=4 t=5 t=6

Hea

t Gen

erat

ion

Period

Low Medium High

NG Geothermal Coal

141

Figure 5.8 Electricity generation from different cogeneration technologies

under qs = 0.01

0

20

40

60

80

100

120

140

160

180

200

t=1 t=2 t=3 t=4 t=5 t=6 t=1 t=2 t=3 t=4 t=5 t=6

Ele

ctric

ity g

nera

tion

(PJ)

Period

Low Medium High

Nature Gas Coal

142

factors, coal-fired cogeneration technology would not produce power and heat.

Furthermore, under different demand rates and various constraint-violation levels,

electricity and heat generation patterns obtained from the DCTFP-REM model can be

illustrated similarly.

Tables 5.7-5.9 present the capacity-expansion schemes for power generation, heat

generation, and cogeneration during the planning period under qs = 0.01. For example, for

non-renewable power generation, the natural gas-fired facility would be expanded with

the first option (a capacity of 0.05 GW) in period 6. The results also indicate that capacity

expansions for power generation would mostly be undertaken with the largest capacity

option in the first period in order to increase renewable electricity production. For instance,

a capacity of 3.5 GW would be added to the hydropower facility at eh beginning of period

1 when qs = 0.01. However, capacity expansions for heat generation facilities would be

taken with the third option at the beginning of period 1due to small residual capacities and

more demanding conditions. In comparison, natural gas-fired cogeneration technology

would be expanded with a capacity of 0.1 GW at the beginning of period 6, while coal-

fired cogeneration technology would be expanded with a capacity of 0.005 GW at the

beginning of period 2. The capacity-expansion schemes for non-renewable and renewable

power generation, heat generation, and cogeneration facilities under qs = 0.01 are provided

in Figures 5.9-5.12. Capacity expansion solutions can be interpreted under various

constraint-violation levels in the same way.

The DCTFP-REM results also indicate that a higher qs level would correspond to a

greater ratio objective value and a higher system cost. For example, when the constraint-

143

Table 5.7 Binary solutions for capacity expansions of power generation under qs = 0.01

Power-generation facility

Capacity expansion option t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

Natural gas-fired (j = 1) m = 1 0 0 0 0 0 1

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Coal-fired (j = 2) m = 1 0 0 0 1 0 0

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Diesel-fired (j = 3) m = 1 0 0 0 0 1 0

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Fuel oil-fired (j = 4) m = 1 0 0 0 0 1 0

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Biomass-fired (j = 5) m = 1 0 0 0 0 0 1

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Hydropower (j = 6) m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

Wind power (j = 7) m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

144




Solar power (j = 8) m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

Wave/tide power (j = 9)

m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

Geothemal power (j = 10)

m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

145

Table 5.8 Binary solutions for capacity expansions of heat generation under qs = 0.01

Heat-generation facility


Natural gas-fired (j = 1)

m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

Coal-fired (j = 2)

m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

Geothemal (j = 3)

m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

146

Table 5.9 Binary solutions for capacity expansions of cogeneration under qs = 0.01

Cogeneration facility



m = 1 0 0 0 0 0 1

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Coal-fired (j = 2)

m = 1 0 1 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

147

Figure 5.9 Capacity expansion schemes for different non-renewable power generation


0

0.01

0.02

0.03

0.04

0.05

0.06

1 2 3 4 5 6

Cap

acity

exp

ans

ion

(GW

)

Period

Natural gas-firedCoal-fired

Diesel-fired Fuel oil-fired

148

Figure 5.10 Capacity expansion schemes for different renewable power generation


0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

1 2 3 4 5 6

Ca

paci

ty e

xpan

sion

(G

W)

Period

Biomass-firedHydropower

Wind powerSolar energy Geothermal energy

Wave/tide

149

Figure 5.11 Capacity expansion schemes for heat generation facilities under qs = 0.01

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

1 2 3 4 5 6

Cap

acity

exp

ansi

on (

GW

)

Period

Natural gas-fired Coal-fired Fuel oil-fired

150

Figure 5.12 Capacity expansion schemes for cogeneration facilities under qs = 0.01

0

0.02

0.04

0.06

0.08

0.1

0.12

1 2 3 4 5 6

Cap

acity

exp

ansi

on (

GW

)

Period


151

violation level is raised from 0.01 to 0.25, the ratio objective value would be increased

from 11.45 to 12.14 PJ per 109 $, while the system cost would be increased from $887.79

× 109 to $951.73 × 109. The qs level indicates the probabilities at which constraints can be

violated, and the ratio objective value represents the renewable energy utilization per unit

of system cost. Therefore, the interactions among system reliability and efficiency can be

demonstrated through the relationship between the constraint-violation level and ratio

objective. An increased qs level, which represents an increased system uncertainty, leads

to a decreased strictness for the constraints and thus expands the decision space. As a result,

decisions under a higher qs level would result in an alternate of higher system efficiency

but a decreased system reliability of meeting end-user demand, environmental protection,

and resources availability constraints.

The above scenario is considered maximizing the renewable energy utilization per

unit of system cost. Another scenario is to put more concerns on the economic aspect to

merely minimize the system cost. A conventional two-stage chance-constrained mixed-

integer linear programming (TCMIP) model is analyzed to further demonstrate the

advantages of the developed DCTFP-REM model. The optimal-ratio objective (5.15) can

be changed into a least-cost problem through replacing the following objective:

1 2 3 4 5 6 7Min system costf f f f f f f f (5.25)

Thus, the obtained model under least-cost scenario can be solved within a TSP

framework through introducing the CCP and MILP techniques. Under various qs levels,

the production schemes of facilities that rely on renewable energy resources are

significantly different between two scenarios. Tables 5.10-5.13 present the solutions of

152

Table 5.10 Solutions of primary energy suppliers for power generation

from TCMIP under qs = 0.01


Primary energy supply

Period Low Medium High Low Medium High

Natural gas (j = 1)

t = 1 7.90 7.90 13.66 7.90 7.90 13.66

t = 2 8.40 8.46 15.09 8.40 8.46 15.09

t = 3 13.67 14.68 15.50 13.67 14.68 15.50

t = 4 15.10 15.99 16.89 15.10 15.99 16.89

t = 5 14.94 15.82 16.71 14.94 15.82 16.71

t = 6 15.86 16.80 17.74 15.86 16.80 17.74

Coal (j = 2) t = 1 0.00 0.00 0.00 0.00 0.00 0.00

t = 2 0.00 0.00 0.00 0.00 0.00 0.00

t = 3 0.00 0.00 0.00 0.00 0.00 0.00

t = 4 0.00 0.00 0.00 0.00 0.00 0.00

t = 5 0.00 0.00 0.00 0.00 0.00 0.00

t = 6 0.00 0.00 0.00 0.00 0.00 0.00

Diesel (j = 3) t = 1 0.08 0.08 0.71 0.08 0.08 0.39

t = 2 1.00 1.00 1.00 1.00 1.00 1.00

t = 3 0.09 0.33 0.69 0.09 0.33 1.02

t = 4 1.03 1.03 1.03 1.03 1.03 1.03

t = 5 1.21 1.21 1.21 1.21 1.21 1.21

t = 6 1.33 1.33 1.33 1.33 1.33 1.33

153




Fuel oil (j = 4) t = 1 0.16 0.16 4.60 0.16 0.16 4.60

t = 2 2.78 4.11 5.25 2.78 4.11 5.25

t = 3 5.53 5.53 5.53 5.53 5.53 5.53

t = 4 5.80 5.80 5.80 5.80 5.80 5.80

t = 5 6.04 6.04 6.04 6.04 6.04 6.04

t = 6 6.25 6.25 6.25 6.25 6.25 6.25

Biomass (j = 5) t = 1 0.79 0.79 0.79 0.79 0.79 0.81

t = 2 0.84 0.84 0.84 1.72 1.78 6.37

t = 3 0.88 0.88 0.88 6.07 6.39 6.71

t = 4 2.15 2.44 11.20 6.37 6.70 7.04

t = 5 6.63 6.98 11.25 6.63 6.98 7.33

t = 6 17.15 17.93 10.88 6.86 7.23 7.59

154

Table 5.11 Solutions of primary energy suppliers for heat generation




Natural gas (k = 1)

t = 1 7.90 7.90 7.90 7.90 7.90 7.90

t = 2 8.40 8.40 8.40 8.40 8.40 8.40

t = 3 102.02 109.07 20.03 102.02 109.07 8.85

t = 4 247.70 268.77 282.44 248.69 268.77 272.22

t = 5 265.72 279.96 294.20 265.72 279.96 294.20

t = 6 79.49 72.69 283.17 78.50 72.69 304.58

Coal (k = 2) t = 1 0.60 2.85 3.00 2.71 2.85 3.00

t = 2 2.88 3.03 3.18 2.88 3.03 3.18

t = 3 3.03 3.19 3.35 2.79 3.11 3.35

t = 4 3.18 3.35 3.52 2.05 2.18 2.71

t = 5 1.10 1.19 3.66 1.10 1.19 1.53

t = 6 3.43 0.22 0.38 0.19 0.22 0.37

155

Table 5.12 Solutions of primary energy suppliers for cogeneration




Natural gas (c = 1)

t = 1 99.06 107.22 169.14 99.06 107.22 169.14

t = 2 69.38 94.85 156.17 69.38 94.85 170.56

t = 3 40.69 83.53 144.29 40.84 83.58 144.29

t = 4 17.57 72.83 133.07 17.57 73.52 133.54

t = 5 36.94 64.49 122.90 36.94 64.49 124.15

t = 6 166.18 202.56 130.33 168.75 202.56 115.93

Coal (c = 2) t = 1 0.00 0.00 0.00 0.00 0.00 0.00

t = 2 0.00 0.00 0.00 0.00 0.00 0.00

t = 3 0.00 0.00 0.00 0.00 0.00 0.00

t = 4 0.00 0.00 0.00 0.00 0.00 0.00

t = 5 0.00 0.00 0.00 0.00 0.00 0.00

t = 6 0.00 0.00 0.00 0.00 0.00 0.00

156

Table 5.13 Solutions of primary energy suppliers for end-users




Natural gas (n = 1)

t = 1 1386.70 1386.70 1386.70 1386.70 1386.70 1386.70

t = 2 1606.50 1606.50 1606.50 1606.50 1606.50 1606.50

t = 3 1802.85 1802.85 1802.85 1802.85 1802.85 1802.85

t = 4 1986.60 1986.60 2710.64 1986.60 1986.60 2710.64

t = 5 2118.55 2304.84 3339.93 2118.55 2304.84 3339.93

t = 6 2478.44 2861.13 3491.73 2478.44 2861.13 3491.73

Diesel (n = 2) t = 1 387.80 387.80 387.80 387.80 387.80 387.80

t = 2 423.61 518.88 638.98 423.61 518.88 638.98

t = 3 457.80 457.80 457.80 457.80 457.80 457.80

t = 4 491.75 491.75 698.93 491.75 491.75 698.93

t = 5 525.70 525.70 525.70 525.70 525.70 525.70

t = 6 558.25 605.23 875.38 558.25 605.23 875.38

Fuel oil (n = 3) t = 1 312.20 312.20 389.21 312.20 312.20 389.21

t = 2 331.45 331.45 417.26 331.45 331.45 417.26

t = 3 339.50 339.50 428.99 339.50 339.50 428.99

t = 4 340.20 340.20 430.01 340.20 340.20 430.01

t = 5 348.60 392.94 492.54 348.60 392.94 492.54

t = 6 396.11 441.79 543.29 396.11 441.79 543.29

157




Gasoline (n = 4) t = 1 625.80 625.80 780.16 625.80 625.80 780.16

t = 2 618.10 618.10 768.94 618.10 618.10 768.94

t = 3 607.25 607.25 753.13 607.25 607.25 753.13

t = 4 625.10 625.10 779.14 625.10 625.10 779.14

t = 5 644.35 644.35 807.19 644.35 644.35 807.19

t = 6 660.45 700.45 889.15 660.45 700.45 889.15

LPG (n = 5) t = 1 40.53 43.06 53.16 40.53 43.06 53.16

t = 2 43.41 49.06 60.36 43.41 49.06 60.36

t = 3 43.98 53.06 65.16 43.98 53.06 65.16

t = 4 44.45 56.06 68.76 44.45 56.06 68.76

t = 5 51.74 59.06 72.36 51.74 59.06 72.36

t = 6 55.80 62.06 75.96 55.80 62.06 75.96

Biomass (n = 6) t = 1 714.77 759.30 937.40 714.77 759.30 937.40

t = 2 702.20 794.80 980.00 702.20 794.80 980.00

t = 3 636.35 771.80 952.40 636.35 771.80 952.40

t = 4 628.25 766.30 945.80 628.25 766.30 945.80

t = 5 662.23 760.30 938.60 662.23 760.30 938.60

t = 6 666.41 745.30 920.60 666.41 745.30 920.60

158

Figure 5.13 Electricity productions from hydropower under qs = 0.01

900

950

1000

1050

1100

1150

1200

1250

1300

1350

t=1 t=2 t=3 t=4 t=5 t=6

Pow

er g

ener

atio

n (P

J)

Period

Low (DCTFP-REM) Medium (DCTFP-REM) High (DCTFP-REM)

Low (TCMIP) Medium (TCMIP) High (TCMIP)

159

Figure 5.14 Electricity productions from wave/tide power under qs = 0.01

0

50

100

150

200

250

300

350

400

t=1 t=2 t=3 t=4 t=5 t=6

Po

wer

ge

nera

tion

(PJ)

Period

Low (DCTFP-REM) Medium (DCTFP-REM) High (DCTFP-REM)Low (TCMIP) Medium (TCMIP) High (TCMIP)

160

energy suppliers for electricity generation, heat generation, and cogeneration from the

TCMIP model. Specifically, Figures 5.13 and 5.14 compare the electricity produced from

two primary renewable energy resources in the province of British Columbia (i.e.

hydropower, wave/tide power). Tables 5.14-5.16 show the binary solutions obtained from

the TCMIP model under qs = 0.01. As revealed in Table 5.14, the geothermal and the

wave/tide facilities would be expanded with a lower capacity under the least-cost scenario.

For example, when qs = 0.1, the geothermal energy facility would be expanded with the

second option (a capacity of 0.15 GW) in period 6 under the least-cost scenario. In

comparison, it would have the third capacity expansion (i.e. a capacity of 0.25 GW) at the

beginning of period 1. The capacity-expansion schemes of non-renewable and renewable

power generation, heat generation, and cogeneration facilities from the TCMILP model

under qs = 0.01 are provided in Figures 5.15-5.18.

Moreover, Figure 5.19 compares the system cost corresponding to DCTFP-REM and

least-cost scenarios under various constraint- violation levels. As indicated in Figure 5.19,

the system cost solutions from the least-cost model are $846.64 × 109 when qs = 0.01,

$856.88 × 109 when qs = 0.05, $860.95 × 109 when qs = 0.1, and $885.78 × 109 when qs =

0.25. As the results shown, the system costs corresponding to the least-cost scenario are

slightly lower than the DCTFP-REM model under a range of qs levels. However, as shown

in Figure 5.20, the renewable energy utilization per unit of cost obtained from the DCTFP-

REM model is around 11.86 PJ per 109 $, which is significant higher than 9.3 PJ per 109

$ under the least-cost scenario.

The solutions obtained from the above two scenarios could provide useful decision

161

Table 5.14 Binary solutions from TCMIP for capacity expansions of

power generation under qs = 0.01



Natural gas-fired (j = 1) m = 1 0 0 0 0 0 1

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Coal-fired (j = 2) m = 1 0 0 0 0 0 1

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Diesel-fired (j = 3) m = 1 0 0 0 0 0 1

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Fuel oil-fired (j = 4) m = 1 1 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Biomass-fired (j = 5) m = 1 0 0 0 0 0 1

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Hydropower (j = 6) m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

162



capacity expansion option t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

wind power (j = 7) m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

solar power (j = 8) m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

wave/tide power (j = 9)

m = 1 0 0 0 0 0 1

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

geothemal power (j = 10)

m = 1 0 0 0 0 0 0

m = 2 1 0 0 0 0 0

m = 3 0 0 0 0 0 0

163


heat generation under qs = 0.01

Heat-generation facility



m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

Coal-fired (j = 2)

m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

Geothemal (j = 3)

m = 1 0 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 1 0 0 0 0 0

164


cogeneration under qs = 0.01

Cogeneration facility



m = 1 1 0 0 0 0 0

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

Coal-fired (j = 2)

m = 1 0 0 0 0 0 1

m = 2 0 0 0 0 0 0

m = 3 0 0 0 0 0 0

165

Figure 5.15 Capacity expansion schemes of non-renewable power generation

technologies from TCMIP under qs = 0.01

0.00

0.01

0.02

0.03

0.04

0.05

0.06

1 2 3 4 5 6

Cap

acity

exp

ansi

on (

GW

)

Period

Natural gas-fired Coal-firedDiesel-fired Fuel oil-fired

166

Figure 5.16 Capacity expansion schemes of renewable power generation technologies


0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

1 2 3 4 5 6

Cap

acity

exp

ansi

on (

GW

)

Period

Biomass-firedHydropower

Wind powerSolar energy Geothermal energy

Wave/tide

167

Figure 5.17 Capacity expansion schemes of heat generation facilities from TCMIP

under qs = 0.01

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

1 2 3 4 5 6

Cap

acity

exp

ansi

on

(GW

)

Period


168

Figure 5.18 Capacity expansion schemes of cogeneration facilities from TCMIP under qs

= 0.01

0.00

0.02

0.04

0.06

0.08

0.10

0.12

1 2 3 4 5 6

Cap

acity

exp

ansi

on

(GW

)

Period

Natural gas-fired Coal-fired

169

Figure 5.19 The comparison of system costs between DCTFP-REM and TCMIP models

887.79

907.20

919.54

951.73

846.64 856.88860.95

885.78

840

860

880

900

920

940

960

0.01 0.05 0.1 0.25qs level

DCTFP-REM TCMIPS

yste

m C

ost

($10

9 )

170

Figure 5.20 The comparison of system efficiencies between DCTFP-REM and TCMIP

models

11.45

11.8412.01 12.14

9.35 9.33 9.359.19

9

10

10

11

11

12

12

13

0.01 0.05 0.1 0.25qs level

DCTFP-REM TCMIPR

enew

able

ene

rgy

utili

zatio

n/co

st (

PJ/

$109 )

171

alternatives under different pre-regulated policies and various energy availability.

Compared with the least-cost model, the DCTFP-REM model could be an effective tool

for providing environmental management schemes under various system conditions.

Generally, the developed DCTFP-REM model has the advantages in (a) balancing

conflicting objectives, (b) reflecting multi-stage decisions, (c) providing desired capacity

expansion schemes, (d) accounting for randomness in both the objective and constraints,

and (e) analyzing interrelationships among efficiency, policy scenarios, economic cost,

and system reliability. Moreover, techniques of the DCTFP-REM model could also be

applied to other practical problems such as waste management, water quality management,

and air quality management.

5.5. Summary

In this study, a dynamic chance-constrained two-stage fractional regional energy

model (DCTFP-REM) was developed for supporting the planning of the energy

management system under uncertainty through the integration of two-stage programming

(TSP), chance-constrained programming (CCP), and mixed-integer linear programming

(MILP) techniques into a fractional programming framework. The DCTFP-REM model

could effectively solve multiobjective problems involving issues of multi-stage decision,

capacity expansion, and random information. The advantages of the developed DCTFP-

REM model include (a) balancing conflicting objectives, (b) addressing uncertainties in

both the objective and constraints, (c) identifying reasonable capacity-expansion strategies,

(d) reflecting multi-stage decisions, and (e) providing desired management schemes under

different constant-violation conditions.

172

Through a real-world case study of the British Columbia’s energy management

system, the applicability of the developed DCTFP-REM model has been demonstrated.

The proposed model, which maximizes system efficiency under various constraint-

violation conditions, could successfully identify energy-resource allocation and capacity-

expansion schemes over a long-term planning period. Results of the case study reveal that

the production and capacity-expansion schemes for the facilities relying on renewable

energy resources are sensitive in the DCTFP-REM model. The results also suggest that

both hydropower and wave/tide power are notable renewable energy resources for the

electricity supply.

Moreover, conflicts between environmental protection that maximizes the renewable

energy resource utilization and economic development that minimizes the system cost can

be effectively addressed through the DCTFP-REM model without setting a factor for each

objective. Such a capability will help facilitate effective exploration and reflection of

trade-offs between conflicting objectives, which implies a significant improvement in

terms of multiobjective environmental systems planning. The results also indicate that the

DCTFP-REM model can facilitate dynamic analysis of the interactions among efficiency,

policy scenarios, economic cost, and system reliability.

This study attempts to provide a two-stage regional British Columbia energy model

for tackling practical mutilobjective optimization problems involving policy scenario

analyses, constraint violation conditions, and capacity expansion issues. Although the

DCTFP-REM model was applied to British Columbia’s energy management system for

the first time ever, it can also be an effective tool for supporting other practical

environmental management problems. However, owing to data availability and system

173

complexity, there are still numerous factors that need to be systematically considered in

the future study, such as uncertainties expressed as intervals. Extensions of the DCTFP-

REM method through integrating other methods of fuzzy set and interval analysis within

its framework would be an interesting topic that also deserves future research efforts.

174

CHAPTER 6

CONCLUSIONS

6.1. Summary

In this dissertation research, a set of two-stage fractional programming methods were

developed and applied to hypothetical and real-world cases of multiobjective

environmental management under uncertainty. The elements of the methods include: (a) a

two-stage fractional programming method for managing multiobjective waste

management systems; (b) two-stage chance-constrained fractional programming for

sustainable water quality management under uncertainty; and (c) a dynamic chance-

constrained two-stage fractional programming method for planning regional energy

systems in the province of British Columbia, Canada. The developed methods could help

provide decision alternatives for supporting various multiobjective environmental

management under uncertainty. A brief summary of this dissertation research is provided

as follows.

In chapter 3, a two-stage fractional programming (TSFP) method was developed and

applied to solid waste management. The TSFP method is based on an integration of the

existing two-stage programming and mixed-integer linear programming techniques within

a fractional programming framework. It could not only address the conflicts between two

objectives (e.g. economic and environmental effects) without the demand of subjectively

setting a weight for each objective, but could also provide a linkage between pre-regulated

policies and economic implications expressed as penalties. Moreover, TSFP could account

175

for the dynamic variations of system capacity due to the expansions of waste-management

facilities and support an in-depth analysis of the interactions between system efficiency

and economic cost.

In chapter 4, a two-stage chance-constrained fractional programming (TCFP)

approach was developed for supporting water quality management systems under

uncertainty. This method can handle ratio optimization problems associated with policy

analysis and uncertainties expressed as probability distributions, where two-stage

stochastic programming (TSP) is integrated into a stochastic linear fractional

programming (SLFP) framework. In addition, an effective solution method is proposed to

tackle this integrated model. The TCFP method has advantages in: (1) addressing the

conflict of two objectives; (2) reflecting different policies; (3) tackling uncertainty

available as probability distributions; and (4) presenting optimal solutions under different

constraint-violation conditions. The obtained solutions effectively identified reasonable

water quality management schemes with maximized system efficiency under various

constraint-violation risks and different policy scenarios.

In chapter 5, a dynamic chance-constrained two-stage fractional (DCTFP) method

was developed. Techniques of two-stage programming (TSP), chance-constrained

programming (CCP), and mixed-integer linear programming (MILP) were integrated into

a linear fractional programming (LFP) framework. It could effectively solve

multiobjective problems under different policy scenarios and various levels of constraint

violation. Moreover, it could facilitate dynamic analysis for decisions of system-capacity

expansions over a long-term planning period. Based on the proposed DCTFP method, a

dynamic chance-constrained two-stage fractional regional energy model (DCTFP-REM)

176

was developed for planning of regional energy systems in the province of British Columbia,

Canada. Results of the case study for the province of British Columbia provided desired

decision alternatives for managing the province’s energy system within a long-term

context; they also reflected the interactions among efficiency, policy scenarios, economic

cost, and system reliability.

6.2. Research achievements

The main contribution of this dissertation research is the development of a set of

innovative methods for supporting multiobjective environmental management under

uncertainty. The developed methods were applied to multiobjective environmental

problems including solid waste management, water quality management, and energy

system management. The developed methods addressed conflicts between two objectives

(e.g. economic and environmental effects) within an environmental management system,

without the demand of subjectively setting a weight for each objective. Such a capability

facilitated effective exploration and reflection of trade-offs between conflicting objectives,

which implied a significant improvement in terms of multiobjective environmental

systems planning. Moreover, economic penalties were taken into consideration as

corrective measures against any arising infeasibility caused by a particular realization of

uncertainty, such that a linkage to pre-regulated policy targets was established.

Furthermore, the methods facilitated an in-depth analysis of the interactions between

system efficiency and economic cost.

Based on the developed methods, a dynamic chance-constrained two-stage fractional

regional energy model (DCTFP-REM) was developed for planning regional energy

177

systems in the province of British Columbia, Canada. Results of a real-world case study

could help energy managers and decision makers analyze complex energy-related factors

and issues within a long-term planning period, which were useful for supporting the

planning of regional energy system management in the province.

6.3. Recommendations for future research

(1) In this dissertation research, a set of multiobjective optimization methods were

developed. However, many practical environmental decision-making problems may not

simply involve two conflicting objectives. Therefore, extensions of the proposed methods

through considering three or more objective problems would be an interesting topic that

deserves future research efforts.

(2) Due to data availability and system complexity in environmental management

problems, there are still numerous factors that need to be systematically considered in

future study, such as uncertainties expressed as intervals. Therefore, the proposed

fractional optimization methods could be further enhanced through incorporating methods

of interval analysis, fuzzy set, and game theory into its framework.

(3) Although the DCTFP-REM model was successfully developed, further enhancing

the quality of input data will help improve the reliability of the regional energy system

planning.

(4) The developed innovative mathematical programming methods can be potentially

extended to other multiobjective management problems, such as water resource

management and air pollution control planning.

178

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