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
recycling facilities ............................................................................................................ 29
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
recycling facilities ............................................................................................................ 42
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
technologies under qs = 0.01 .......................................................................................... 139
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
technologies under qs = 0.01 .......................................................................................... 147
Figure 5.10 Capacity expansion schemes for different renewable power generation
technologies under qs = 0.01 .......................................................................................... 148
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
WG2kh (District 2) (tonne/day)
h = 1 (low) 0.2 135 215 295
h = 2 (medium) 0.6 170 230 320
h = 3 (high) 0.2 205 245 355
WG3kh (District 3) (tonne/day)
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
recycling facilities
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
Optimized 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
Table 3.7 Continued.
Waste flow (tonne/
day)
i j k
Level of
waste
generation
Probability (%)
Target waste flow
Excess waste flow
Optimized 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
recycling facilities
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
Optimized 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
Table 3.8 Continued.
Waste flow (tonne/
day)
i j k
Level of
waste
generation
Probability (%)
Target waste flow
Excess waste flow
Optimized 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
Table 3.8 Continued.
Waste flow (tonne/
day)
i j k
Level of
waste
generation
Probability (%)
Target waste flow
Excess waste flow
Optimized 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
Table 4.1 Continued.
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
Wastewater treatment plant
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:
Water production (m3/day) 30000 35000 40000
Paper (tonne/day) 19 21 22
Leather (tonne/day) 9 9 10
Tobacco (tonne/day) 2.8 2.3 2.1
Recreational activity (ha/yr)
534 620 693
Maximum product demand:
Water production (m3/day) 50000 52000 55000
Paper (tonne/day) 24 27 29
Leather (tonne/day) 14.5 16.0 16.0
Tobacco (tonne/day) 3.5 3.0 2.8
Recreational activity (ha/yr)
800 912 985
72
Table 4.4 Continued.
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
Wastewater treatment plant
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
Table 4.5 Continued.
Economic activity
k
Wastewater-discharge
rate
ph (%) Target Tik
Planned production level xikh
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
Table 4.5 Continued.
Economic activity
k
Wastewater-discharge
rate
ph (%) Target Tik
Planned production level xikh
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
Period 1 Period 3 Period 2
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
Planning level Pre-regulated target
Period 1 Period 3 Period 2
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
Planning level Pre-regulated target
Period 1 Period 3 Period 2
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
Planning level Pre-regulated target
Period 1 Period 3 Period 2
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
Planning level Pre-regulated target
Period 1 Period 3 Period 2
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
Planned production level xikh
0.01 0.05 0.10 0.25
Wastewater treatment plant
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
Table 4.6 Continued.
Economic activity
k
Wastewater discharge
rate
ph (%) Target Tik
Planned production level xikh
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
Table 4.6 Continued.
Economic activity
k
Wastewater discharge
rate
ph (%) Target Tik
Planned production level xikh
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
Table 5.3 Continued.
Local supply (PJ) Import supply (PJ)
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
Local supply (PJ) Import supply (PJ)
Primary energy supply Period Low Medium High Low Medium High
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
Local supply (PJ) Import supply (PJ)
Primary energy supply Period Low Medium High Low Medium High
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
Local supply (PJ) Import supply (PJ)
Primary energy supply Period Low Medium High Low Medium High
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
Table 5.6 Continued.
Local supply (PJ) Import supply (PJ)
Primary energy supply Period Low Medium High Low Medium High
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
technologies under qs = 0.01
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
Table 5.7 Continued.
Power-generation facility
Capacity expansion option t = 1 t = 2 t = 3 t = 4 t = 5 t = 6
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
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 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
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 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
technologies under qs = 0.01
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
technologies under qs = 0.01
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
Natural gas-fired Coal-fired Fuel oil-fired
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
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 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
Table 5.10 Continued.
Local supply (PJ) Import supply (PJ)
Primary energy supply Period Low Medium High Low Medium High
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
from TCMIP under qs = 0.01
Local supply (PJ) Import supply (PJ)
Primary energy supply Period Low Medium High Low Medium High
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
from TCMIP under qs = 0.01
Local supply (PJ) Import supply (PJ)
Primary energy supply Period Low Medium High Low Medium High
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
from TCMIP under qs = 0.01
Local supply (PJ) Import supply (PJ)
Primary energy supply Period Low Medium High Low Medium High
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
Table 5.13 Continued.
Local supply (PJ) Import supply (PJ)
Primary energy supply Period Low Medium High Low Medium High
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
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 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
Table 5.14 Continued.
Power-generation facility
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
Table 5.15 Binary solutions from TCMIP for capacity expansions of
heat generation under qs = 0.01
Heat-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 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
Table 5.16 Binary solutions from TCMIP for capacity expansions of
cogeneration under qs = 0.01
Cogeneration facility
Capacity expansion option t = 1 t = 2 t = 3 t = 4 t = 5 t = 6
Natural gas-fired (j = 1)
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
from TCMIP under qs = 0.01
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
Natural gas-fired Coal-fired Fuel oil-fired
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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