An interval full-infinite mixed-integer programming method for planning municipal energy systems – A case study of Beijing

Applied Energy 88 (2011) 2846–2862

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An interval full-infinite mixed-integer programming method for planning municipal energy systems – A case study of Beijing Y. Zhu a,1, G.H. Huang a,2, Y.P. Li a,⇑, L. He a,3, X.X. Zhang b,4 a MOE Key Laboratory of Regional Energy Systems Optimization, S-C Energy and Environmental Research Academy, North China Electric Power University, Beijing 102206, China b State Grid Energy Research Institute, XiCheng District, Beijing 100052, China

a r t i c l e

i n f o

Article history: Received 23 October 2010 Received in revised form 3 January 2011 Accepted 25 January 2011

Keywords: Decision making Energy systems Functional interval Planning Mixed-integer Uncertainty

a b s t r a c t In this study, an interval full-infinite mixed-integer municipal-scale energy model (IFMI-MEM) is developed for planning energy systems of Beijing. IFMI-MEM is based on an integration of existing intervalparameter programming (IPP), mixed-integer linear programming (MILP) and full-infinite programming (FIP) techniques. IFMI-MEM allows uncertainties expressed as determinates, crisp interval values and functional intervals to be incorporated within a general optimization framework. It can also facilitate capacity-expansion planning for energy-production facilities within a multi-period and multi-option context. Then, IFMI-MEM is applied to a real case study of energy systems planning in Beijing. The results indicate that reasonable solutions have been generated. They are helpful for supporting (a) adjustment of the existing demand and supply patterns of energy resources, (b) facilitation of dynamic analysis for decisions of capacity expansion and/or development planning, and (c) coordination of the conflict interactions among economic cost, system efficiency, pollutant mitigation and energy-supply security. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Undoubtedly, energy plays an increasingly significant role in economic development and human activities for achieving sustainable development of energy and environmental systems. The growth of global energy demands, the scarcity of resources, and the limits to local and global environmental damage are questioning whether energy supplies can meet the increasing electric demands. Development of optimization models for energy systems planning has attracted considerable interest over past decades. Decision makers are in a quandary on how to balance increasing energy demands with population growth and economic development, mandated requirements [1–4]. However, in energy management systems, uncertainties exist in a number of system components as well as their interrelationships [5,6]. The uncertainties are often associated with various complexities in terms of information quality. These uncertainties among various uncertain parameters may lead to a variety of adverse impacts on energy ⇑ Corresponding author. Tel.: +86 10 6177 2977; fax: +86 10 5197 1255. E-mail addresses: [email protected] (Y. Zhu), [email protected] (G.H. Huang), [email protected] (Y.P. Li), [email protected] (L. He), zhangxiaoxuan @sgeri.sgcc.com.cn (X.X. Zhang). 1 Tel.: +86 10 5197 1215; fax: +86 10 5197 1255. 2 Tel.: +86 10 6177 2980; fax: +86 10 5197 1255. 3 Tel.: +86 10 6177 2978; fax: +86 10 5197 1255. 4 Tel./fax: +86 10 6341 1664. 0306-2619/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2011.01.058

systems management. Energy systems planning models are desired to address such uncertainties and complexities, which is crucial to deal with these challenges to secure technically safe, economically efficient and environmentally friendly at multiple scales [7–10]. Previously, a large number of research works were undertaken to reflect these uncertainties and complexities in planning of energy systems [11–20]. For example, Nowak and Römisch [21] developed an optimization model using multistage stochastic programming for the weekly cost-optimal generation of electric power in a hydro-thermal generation system under uncertain demands. The model involves a large number of mixed-integer (stochastic) decision variables and constraints linking time periods and operating power units. Liu et al. [22] proposed a dynamic optimization approach for managing nonrenewable energy resources under uncertainty, where approaches of interval-parameter programming (IPP) and chance-constrained programming were incorporated into an integer programming framework to deal with uncertainties ex} rnberg and Römisch pressed as intervals and probabilities. Nu [23] developed a two-stage stochastic programming model for short- or mid-term cost-optimal electric power production planning, considering power generation in a hydro-thermal generation system under uncertainty in demand and fuel prices. Mavrotas et al. [24] proposed a fuzzy linear programming model for facilitating investment decisions to meet energy demand efficiently, through handling uncertainties expressed as continuous, integer variables and fuzzy sets. Sadeghi and Hosseini [25] used

Y. Zhu et al. / Applied Energy 88 (2011) 2846–2862

fuzzy linear programming approach for planning energy systems in Iran, where uncertainties of investment costs in objective function coefficients were considered; their study indicated that uncertainties could significantly affect the results of energy model through comparing crisp and fuzzy models. Lin and Huang [26] proposed an energy systems planning model using IPP for planning energy allocation and capacity expansion within a regional jurisdiction, where interval solutions allowed for detailed interpretation of the trade-offs between environmental pollution risks and economic objectives. More recently, Li et al. [27] developed an integrated fuzzy-stochastic optimization model for planning energy systems in association with GHG mitigation; in this study, multiple uncertainties presented as probability distributions, fuzzyintervals and their combinations were allowed to be incorporated within an optimization framework. Among optimization methods that can handle uncertainties, interval-parameter programming (IPP) approach is effective for handling uncertainties expressed as interval numbers without known probability distributions and membership functions, which can exist in model’s objective function and constraints. IPP allows uncertainties to be directly communicated into an optimization process and resulting solutions; it also does not lead to more complicated intermediate models, and thus has a relatively low computational requirement [28–31]. For example, Huang et al. [28] developed a grey linear programming approach for municipal solid waste management planning under uncertainty. Fang et al. [31] proposed a linear programming with fuzzy coefficients in constraints, where relations between optimal solutions and extreme points of the linear semi-infinite program were established. However, conventional IPP can only solve the problems containing crisp interval coefficients [a, b], whose lower- and upper-bounds (i.e. a and b) are both deterministic and definitely known. This is based on the assumption that these interval coefficients are unchanged even if they could be affected by associated impact factors. In actual systems, this definition is not suitable for all cases where the two bounds may be associated with the external impact factors [32]. For example, in energy systems planning problems, if the energy purchased cost (CN) is affected by the interest rate, the lower- and upper-bounds can vary since any variation in interest rate will lead to a corresponding change in CN. Therefore, the concept of crisp interval may not be suitable for describing such an uncertainty. An effective way of describing this uncertainty is functional intervals; this can be defined as a lower- and an upper-bound, which are both functions of its associated impact factor. For example, if CN is expressed as a functional interval of [152.60(1 + a)t, 157.20 (1+a)t]  106 RMB/PJ; the a and t denote the interest rate and interval in each period time, CN is a function of interest rate, ranging between 152.60(1 + a)t  106 RMB¥/PJ and 157.20 (1 + a)t  106 RMB¥/PJ. An attractive technique that could tackle functional intervals was full-infinite programming (FIP), which was proposed to deal with the uncertainties expressed as crisp intervals and functional intervals. The FIP can deal with infinite objectives and constraints under uncertainty, which improves upon the conventional semiinfinite programming (SIP) that can only reflect problems with merely infinite constraints in a deterministic environment [33,34]. Previously, there were plenty of studies on SIP methods and its application [35–46]. For example, Weber [35] formulated a semi-infinite programming approach for vector optimization problems which extends multiple objective linear programming problems. Wang and Kuo [36] developed a perturbation method for solving linear semi-infinite programming problems. León and Vercher [39] proposed semi-infinite programming techniques for solving a class of fuzzy linear programs. The coefficients in constraints are modeled as LR-fuzzy numbers with different

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shapes, and illustrate it by means of several examples. Wang et al. [41] formulated a list of functions to generalize semiinfinite min–max programming. First order optimality conditions were established in this study for unconstrained generalized semi-infinite programming problems. Guo et al. [44] developed an inexact chance-constrained semi-infinite programming approach for energy systems planning under uncertainty. In this study, uncertainties expressed as interval parameters, distribution information and functional intervals were taken into account. He et al. [46] proposed an interval-parameter mixed-integer linear semi-infinite programming model for municipal solid waste management under uncertainty. In this study, uncertainties including crisp intervals and functional intervals in constraints. Nevertheless, there are few studies related to FIP with infinite objectives and constraints under uncertainty. He et al. [47] proposed an interval full-infinite programming method to support solid waste system management which could effectively deal with uncertainties expressed as interval values and functional intervals within infinite objectives and constraints. However, the developed method was incapable of reflecting dynamic complexities, such as the timing, sizing and siting in planning capacity-expansion schemes. From a planning point of view, municipal energy demands may keep increasing due to population increase and economic development. Therefore, the available capacities of energy-generation facilities may also vary among different periods. This tendency could often result in insufficient capacities of facility to meet the overall energy demands. Moreover, dynamics of system-capacity expansions under uncertainty were complicated in municipal energy management systems. Such dynamic issues were not effectively addressed in the previous studies. Consequently, a related optimization analysis will require the use of integer variables to indicate whether a particular facility development or expansion option needs to be undertaken. Mixed-integer programming (MILP) is a useful tool for this purpose. Therefore, the objective of this study is to advance an interval full-infinite mixed-integer municipal-scale energy model (IFMIMEM) to support energy systems planning and environmental management under uncertainty. The development of IFMI-MEM necessitates sub-tasks include: (1) incorporation of FIP, MILP and IPP techniques to formulate an interval full-infinite mixedinteger programming (IFMIP) method for dealing with dynamics of capacity-expansion issues and uncertainties presented as crisp intervals and functional intervals; (2) development of an interval full-infinite mixed-integer municipal-scale energy model (IFMIMEM) based on the proposed IFMIP method, and (3) application of IFMI-MEM to supporting energy systems planning and environmental management in the city of Beijing, China. The solutions obtained are helpful for supporting (a) adjustment of the existing demand and supply patterns of energy resources, (b) facilitation of dynamic analysis for decisions of capacity expansion and/or development planning, and (c) coordination of the conflict interactions among economic cost, system efficiency, pollutant mitigation and energy-supply security.

2. Study system The City of Beijing (39°560 N, 116°200 E) is located in the northern part of the North China Plain, with Tianjin City on its southeastern border and Hebei province surrounding the city on its other three sides. Beijing occupies an administrative area of around 16,808 km2. The city makes up of 11 zones and 7 counties, as shown in Fig. 1. Beijing is surrounded by mountains on the west, the north, and the northeast, with the southeastern plain sloping gradually downward towards the Bohai Sea. At the end of 2008,

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Y. Zhu et al. / Applied Energy 88 (2011) 2846–2862

Fig. 1. Geographical position and districts of Beijing.

the permanent population of Beijing was16.95 million, and the GDP of Beijing reached to 1186.59 billion RMB in 2009 [48].

5 billion tonnes of coal equivalent in 2020.This will lead to serious issues about energy security and environment pollution [49]. 2.2. Energy flow

2.1. Energy demand and supply With the increasing development of society and economy, the total amount of energy demand has a rapid growth in the last few decades. Population growth suggested significant increments of energy demands in Beijing over the past years. Also, variations of energy activities due to population growth are further intensified by transportational mode shift, metropolitan re-urbanization, and economic development in Beijing. According to Statistics Bureau of Beijing, total energy consumption was 42.29 million tonnes of coal equivalent in 2001, and increased to 63.44 million tonnes of coal equivalent in 2008. In detail, electivity, coal, oil and natural gas account for 35.40%, 16.30%, 20.00% and 7.40%. If energy supply is lower than the energy demand, then either the supply must be expanded or additional energy must be imported. It is estimated that the total primary energy consumption of China will reach

Energy supply of the municipality relies on coal, which can be from import and mining sources. Energy import from other regions usually has higher costs; this recourse is always limited by the availability of electricity from an adjacent power grid and the cost-barrier of transmission infrastructure. The majority of energy supplies of Beijing depend on imports, such as importing coal, crude oil, and natural gas from other sources. Energy import from other regions usually has higher costs. Therefore, the import energy always was limited by the availability of electricity from adjacent power grids and the cost-barrier of transmission infrastructure. Moreover, the export energy is refined petroleum products, including gasoline, diesel, fuel oil and other refined petroleum products for benefit. Due to the resource limitations of Beijing, Beijing’ energy supply partly depends on energy inputs from the other regions such as Shanxi, Inner-Mongolia and Hebei provinces.

Y. Zhu et al. / Applied Energy 88 (2011) 2846–2862

Coal accounts for a significant part in Beijing’s energy supply, although it is relatively ‘‘dirty’’. The amount of coal purchased from the other regions accounts for 94% of the total amount of coal consumption in Beijing. Most of the coal production is used for electricity generation, while a few part of the production is exported. The consumption of coal reached 27.42 million tonnes, and the total amount of energy consumption reached 6285.04 million tonnes of standard coal in 2008 [50,51]. All crude oil would import from other provinces. Oil products are mainly used for transportation activities, while part of them is used for power generation. The refinery capacity amounts to 8.50 million tonnes by refining crude oil per year. In detail, diesel and gasoline are mainly used for transportation activities, while part of them is used for power generation. There are no natural-gas resources in Beijing. An implemented project named West-to-East Gas Transport (WEGT) helps deliver natural gas to Beijing. Consumption of natural gas grew dramatically in the 1990s with some continued growth into the early 2000s, with the amount of 4.66 billion m3 in 2007. The city’s majority of natural gas is used for municipal electricity generation and industrial heat production, while the minority is exported for gaining economic profit. Moreover, the domestic electricity production is far from meeting the increasing demands of Beijing. The amount of electricity input from North China Power Grid accounts for 67% of the total amount of electricity consumption of Beijing. Many generating facilities and technologies are available to meet the city’s electricity demand. These facilities and technologies work on the basis of different energy input requirements, capital, operating and maintenance costs, and emission coefficients. By 2008, the total electricity consumption was 67,510 GW h, and the electric consumption from thermal power plant amounted to 22,440 GW h. At present, total electricity capacities are only 4.60 GW, including 2.90 GW of coal-fired generation, 0.80 GW of Hydro-pumped-storage generation, 0.18 GW of hydropower, and 0.66 GW of oil-fired generation, as shown in Fig. 2. A total of 4.59 GW of clean-coal-fired, naturalgas-fired, landfill-gas-fired and wind-power generation facilities will be installed in the next decade. Electricity from these new facilities will mitigate the contradiction between energy supply and demand, and increase the higher robustness of the municipal power grid. Renewable energy resources have been considered as new electricity sources to address the crisis of energy shortage, air pollution, and climate change. Renewable energy resources are mainly employed for power generation of Beijing. Beijing has a good condition to develop and make use of renewable energy. For example, average annual sunshine reaches to 2763 h per year. However, there is still a great gap among China and some developed countries for the exploitation and utilization of renewable energy resources. For limited availabilities, low level of technology and weak capacity of equipment manufacturing, a small amount of

Fig. 2. Distribution of electricity generation capacity in 2008 (GW).

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renewable energy resources was commissioned in Beijing compared to conventional fossil fuel source. These disadvantages limit the development of the extensively use of renewable energy (e.g. solar, waste and wind power), which perform obviously for wind and biomass power generations. 2.3. Air pollution control In the recent years, Beijing’s pollution emissions are generated primarily from energy production and consumption, agricultural activities, waste disposal and industrial processes. For example, power plants are the main source of SO2 (i.e. sulfur dioxide) emission of Beijing. The amount of SO2 emission reached 123.21  103 tonnes in 2008, which account for 48% of the total emission of SO2 in Beijing. The pollution sources of PM10 (particulate matter with particle size below 10 lm) mainly contain industry, construction and traffic. The amount of PM10 emission reached 48.33  103 tonnes in 2008. The pollution sources of NOx (i.e. nitrogen oxides) mostly include artificial sources, such as the burning of coal and fuel oil. To improve the environmental quality, Beijing has made major significant achievement by developing many energy policies. Furthermore, Beijing has determined to cut 10% of the total emission amounts of major pollutants (the 2005 level) by year 2010. By 2015, the utilization of new energy and renewable energy will reduce more than 30 million tonnes of CO2 (i.e. carbon dioxide) and 200 tonnes of SO2. Actually, there exist complexities and uncertainties in the pollutant emission process. Previously, few studies examined uncertain and complex responses of energy management systems to various economic-development strategies and emission-mitigation measures. 2.4. Energy participants Energy systems of Beijing have complex interaction with a number of subsystems. Fig. 3 shows interactive relationships between different system sectors, including energy production, processing, transmission and utilization, and relevant activities/ services. Energy conversion industries of Beijing consist of power generation, heating generation, coking, and oil refining. The local production of secondary energy types consist of the products of coal, coke, gasoline, diesel, kerosene, fuel oil, liquefied petroleum gas and other oil products. The city’s coke production industries contain Beijing coking chemical plant and Shougang Group. In order to reduce pollutant emission, Beijing coking chemical plant has been shut down in 2006. Shougang Group also plans to stop production in 2010. So coke production would be reduced in the next decade. The main oil production industry in Beijing is Sinopec Beijing Yanshan Company. Sinopec Beijing Yanshan Company plans to expand its capacity, and the petroleum products will increase accordingly. In the energy and environmental systems of Beijing, multiple conventional and energy resources/technologies need to be allocated to multiple end-users, including primary industry, secondary industry and tertiary industry. The primary industry includes agriculture, forestry, animal husbandry and fishery, which products directly from nature resources. The secondary industry consists of industry and construction business, which is a processing sector of primary products. The tertiary industry is the service industry. In 2008, the energy consumption of primary industry, secondary industry, tertiary industry and household consumption accounted for 1.6%, 48.9%, 34.8% and 14.7% of the whole energy consumptions, respectively. Generally, energy activities/services are responsible for relevant infrastructural investments and pollutant/GHG emissions, and have impacts on local ecosystems. However, institutional measures and socio-economic activities would also have effects on the energy systems through various policies, actions and strategies, which would then have indirect impacts

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Energy Exports

Energy Inputs

Processing Technology

Energy Supply

Electric Power Plant

Domestic Production

Thermal Power Plant

Heating Plant

City Power Grid System

Residents

Commerce

Industry

Transportation

Atmospheric Pollutants

Air Pollution Control Technology

SO2

NOx

PM

Environment Fig. 3. Diagram of energy management system in Beijing.

on the other components within the community. Therefore, energy systems planning will help to design a variety of activities under these limited ‘‘allowances’’ for energy allocation and resource consumption, in order to realize desired socio-economic and environmental objectives. 2.5. Statement of problems According to above discussions, there are many complex processes in the energy systems of Beijing that should be considered by decision makers, such as energy production, conversion, transmission and utilization as well as the resulting greenhouse gas (GHG) and pollutant emissions. Moreover, many system parameters (such as resource availability, facility capacity, production efficiency and allocation target, as well as their interrelationships) may appear uncertain. Such uncertainties might be further complicated by not only the natural variations of energy resources of Beijing, but also the associated economic and environmental implications for their utilization. These complexities and uncertainties could affect the related optimization processes and the generated decision schemes. In energy systems, uncertainty may express as crisp intervals and functional interval is presented. These parameters are described with their lower- and upperbounds being both functions, which are based on the relationships among various energy systems components. To address such complexities and uncertainties, systematical energy systems planning are desired under comprehensive study and application. In the study system, multiple energy resources/technologies need to be

allocated with limited availabilities for meeting the energy demands. This leads to an application of an interval full-infinite mixed-integer programming (IFMIP) to municipal energy systems planning. If energy supply cannot sufficiently meet the end-users’ demands, decision makers will face a dilemma of either investing more funds in capacity expansions of the existing facilities or turning to other production options, or putting extra funds into energy imports at raised prices. No matter which way would be adopted in response to the deficiencies of energy productions, bad influence would be incurred. The problem under consideration is how to effectively allocate energy and decide how to expansion to minimize the net system costs. Decision makers need to identify desired energy-flow allocation, facility-expansion and air pollutants mitigation schemes with a minimized system cost. Therefore, the problems existing in energy systems of Beijing should be considered: (a) how to effectively allocate the energy demands and supply to production departments, (b) how to deal with the uncertainties expressed as crisp intervals and functional intervals in the objective function and constraints, and (c) how to generate capacity expansion schemes with sufficient considerations as to whether or not increase increments of facility capacities. 3. Development of IFMI-MEM 3.1. IFMIP method According to Li et al. [27], an IPP model can be expressed as follows:

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Y. Zhu et al. / Applied Energy 88 (2011) 2846–2862 k X

Minimize f  ¼

cj xj þ

j¼1

n X

cj xj

ð1aÞ

j¼kþ1

Subject to: n X

k X

n h i h i X  jaij ðsi Þjþ Sign aij ðsi Þ xj þ jaij ðsi ÞjSign aij ðsi Þ xþj 6 bi ðsi Þ

j¼1 

aij xj 6 bi

j¼kþ1

ð4bÞ

ð1bÞ

j¼1

xj P 0

ð1cÞ 

 mn  1n  n1 where a ; bi 2 fR gm1 ; c , and x ; ij 2 fR g j 2 fR g j 2 fR g R± denote a set of interval numbers; x± are decision variables that can be sorted into two categories: continuous and binary.  c j ðj ¼ 1; 2; . . . ; kÞ are positive coefficients; c j ðj ¼ k þ 1; k þ 2; . . . ; nÞ are negative coefficients. An interval can be defined as a number with known lower- and upper-bounds. Based on an interactive algorithm [30], model (1) can be transformed into two deterministic submodels that correspond to the lower- and upper-bounds for the desired objective function value. Despite the effectiveness of the above methods in solving models containing interval-parameters, they are still limited because functional interval may exist in practice can hardly be handled. Functional interval is defined as an interval with its lower and upper bounds being functions of an independent variable. It is indicated that the lower- and upper-bounds of the crisp interval can vary with its independent variable. When functional interval parameters need to be addressed, an interval full-infinite mixedinteger programming (IFMIP) method can be proposed to solve these problems. Therefore, functional intervals are incorporated within the IPP framework, an IFMIP method can be formulated as follows:

k X

Minimize f  ¼

cj ðsi Þxj þ

j¼1

n X

cj ðsi Þxj ;

for all

si 2 ½sl; su

j¼kþ1

ð2aÞ

xj 6 xþjopt ;

j ¼ 1; 2; . . . ; k

ð4cÞ

xþj P xjopt ;

j ¼ k þ 1; k þ 2; . . . ; n

ð4dÞ

xþj P 0; xj P 0;

8j



aij ðsi Þxj 6 bi ðsi Þ

ð2bÞ

j¼1

xj P 0

ð2cÞ 

 where a ij ðsi Þ; bi ðsi Þ, and c j ðsi Þ are functional interval parameters. þ c ð s Þ and c ð s Þ (j = 1, 2, . . . , k) are positive for all sivalues; c i i j j ðsi Þ j and cþ j ðsi Þ (j = k + 1, k + 2, . . . , n) are negative functions for all si values. Model (2) can be converted into two submodels: Submodel 3:

þ

Minimize f ¼

k X

cþj ð i Þxþj

s

þ

j¼1

n X

cþj ð i Þxj ;

s

for all

si 2 ½sl; su

j¼kþ1

ð3aÞ

 where xþ jopt and xjopt are solutions of submodels (3) and (4), respec-

tively; Sign() is defined as:

8   <1 h i > aij ðsi Þ P 0 for all si 2 ½sl; su   Sign aij ðsi Þ ¼ > : 1 aij ðsi Þ < 0 for all si 2 ½sl; su

Based on the IFMIP method, an interval full-infinite mixedinteger municipal-scale energy model (IFMI-MEM) will be formulated for planning energy systems of Beijing. The objective of IFMI-MEM is to minimize the system costs over a long-term planning horizon. The constraints are related to availability of energy consumption allocated amount for each user and the capacity of converting technology and environmental regulations for SO2, NOx, and PM10 emissions. Thus, we have: Objective function:

Minimize f  ¼

13 X 3 X

6 X 3 X     CNik ak  XN ik  COjk ak  XOjk

i¼1 k¼1

þ

2 X 3 X

j¼1

PRmk  PC mk ðgmk Þ þ

s

þ

n X

ð3cÞ

þ

"

PDRnk  PDC nk ðbnk Þ #

c

12 X 3 X



Minimize f ¼

k X j¼1

cj ð i Þx

s

þ

n X

þ cj ð i Þxþ ;

s

for all

PRRnk  PRC nk ðdnk Þ þ HRRnk  HRC nk ðenk Þ

þ ð4aÞ

#

YHqnk  PRAqnk  PRInk ðfnk Þ

L

q¼1

si 2 ½sl; su

j¼kþ1

3 X

L

"

n¼10 k¼1

Submodel 4:

YM qmk  PAqmk

q¼1

YPqnk  PDAqnk  PDInk ð nk Þ

q¼1

8j

9 X 3 X

n X

n¼1 k¼1

ð3bÞ

xþj P 0; xj P 0;

#

PImk ðhmk Þ  L þ

j¼kþ1 þ bi ð i Þ

j¼1 k¼1

"

m¼1 k¼1

n h i h i X jaij ðsi Þj Sign aij ðsi Þ xþj þ jaij ðsi Þjþ Sign aij ðsi Þ xj

6

ð5Þ

When some of the decision variables in model (2) are integers, the IFMIP can help tackle the facility expansion issue in energy systems. The upper and lower bounds of the optimal objective and decision variables can be obtained through solving the IFMIP model. When the upper and lower bounds of a functional interval maintain constants, the parameters of model can be converted to interval forms, making the model solvable. This shows the functional interval is a more general definition for interval uncertainty than the interval number. The significance of this definition is its capability in reflecting modeling uncertainties with more complexities and describing the real-world conditions with more effectiveness. Consequently, IFMIP method inherits the advantages of FIP and IPP, and allows uncertainties expressed as determinates, crisp interval values and functional intervals to be incorporated within a general optimization framework.

Subject to: k X

ð4eÞ

3.2. IFMI-MEM modeling formulation

Subject to: n X

Subject to:

18 X 3 X  n¼13 k¼1

 PGRnk  PGC nk ðlnk Þ þ HGRnk  HGC nk ðpnk Þ

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Y. Zhu et al. / Applied Energy 88 (2011) 2846–2862

L þ

9 X 3 X

PDRnk þ

n¼1 k¼1

3 X

!

q¼1 12 X 3 X

ð1  SLk Þ  Cuk  L þ

12 X 3 X

YP qnk  PDAqnk  hk " PRRnk þ

3 X

n¼10 k¼1

PRRnk

þ

n¼10 k¼1

3 X

! YHqnk

12 X 3 X

PRAqnk  ð1  PRBk Þ  hk  ð1  SLk Þ  Cuk  L þ

½PGRnk  ð1  SLk Þ  Cuk

" HRRnk

þ

n¼10 k¼1

3 X

þ

3 X

PDRnk þ

n¼1 k¼1

3 X

! YHqnk

YP qnk  PDAqnk  hk

PDRnk

n¼10 k¼1

þ

#

 ð1  SF pk Þ  L þ

! YP qnk

3 X

 PDAnk  hk

 AMRnpk

!

YHqnk  PRAqnk  hk

18 X 3 X 3 X 

 AMHnpk

ð6aÞ

(1) Energy availability

 HGRnk þ PGRnk  AMGnpk

n¼13 p¼1 k¼1

 ð1  SF pk Þ  L 6 ESpk

n¼13 k¼1

8p; k

ð6kÞ

(5) Expansion cost constraints 2 X 3 X 3 X

YM qmk  PAqmk  PImk ðgmk Þ  L

m¼1 q¼1 k¼1

XNik 6 ZOik

8k

ð6bÞ

i¼1 k¼1

3 X

  8n; k 6 MNmk ak

9 X 3 X 3 X

(2) Mass balance constraints

n¼10 k¼1

ð6jÞ

q¼1

Subject to:

HRRnk þ

8n; k

12 X 3 X 3 X

 PRRnk þ HRRnk þ

18 X 3   X   PGRnk þ HGRnk  CPpk þ CEpk =hk  SU k  L þ

18 X 3 X

 hk

n¼10 p¼1 k¼1

q¼1

13 X 3 X

#

q¼1

 ð1  SF pk Þ  L þ

YHqnk  PRAqnk  ð1  PRBk Þ  hk

   CPpk þ CEpk =hk  SU k  L

3 X

n¼1 p¼1 k¼1

q¼1

þHRRnk þ



PRAqnk SRnk

q¼1

9 X 3 X 3 X

!

" 12 X 3   X  CPpk þ CEpk =hk  SU k  L þ PRRnk 3 X

ð6iÞ

(4) Environmental constraints

 L

n¼13 k¼1 9 X

 hk

8n; k

 ð1  SHk Þ  L P SHRnk 18 X 3 X

 PRAqnk 

#

SRnk

q¼1

 ð1  SLk Þ  L P SRRnk

YHqnk

q¼1

#

"

! YHqnk  PRAqnk  PRBk  hk þ HGRnk

ð6lÞ

YP qnk  PDAqnk  PDInk ðcnk Þ  L

n¼1 q¼1 k¼1

  8n; k 6 MP nk ak

ð6mÞ

q¼1

 L P ð1 þ SHk Þ  DHk

8k

ð6cÞ

12 X 3 X 3 X

YHqnk  PRAqnk  PRInk ðgnk Þ  L

n¼10 q¼1 k¼1 13 X 18 X 3 X 

PDRnk

þ

PRRnk

þ

HRRnk

þ

HGRnk





XNik



 XOik

i¼1 n¼1 k¼1

P 0 8k

ð6dÞ "

13 X 18 X

sum3k¼1

PDRnk

þ

i¼1 n¼1

þ

3 X

YPqnk

 PDAnk  hk þ

q¼1

3 X

YHqnk



¼ 1; if capacity expansion is undertaken ¼ 0;

if otherwise

8n; k

;

#

 PRAqnk  ð1  PRBk Þ  hk þ

L P ð1 þ SLk Þ 

ð6nÞ

(6) Constraints for capacity expansion of electricity-generation facilities:

YPqnk

PRRnk

ð6oÞ

PGRnk

q¼1

DEk

  8n; k 6 MRnk ak

8k

ð6eÞ

YM qnk



¼ 1; if capacity expansion is undertaken ¼ 0; if otherwise

;

ð6pÞ

(3) Demand constraints 13 X 18 X 3 X 

XNik



XOik



P

DIk

8k

ð6fÞ

YHqnk

i¼1 n¼1 k¼1 2 X 3 X

PRLmk þ

m¼1 k¼1

P

YM qmk  PAqmk  SPmk

PDRnk

þ

n¼1 k¼1

ð6gÞ 3 X

! YPqnk

 PDAqnk 

SDnk

#  hk

q¼1

 ð1  SLk Þ  L P SPDnk

8n; k

¼ 1; if capacity expansion is undertaken ¼ 0; if otherwise

;

8n; k

(7) Nonnegative constraints

 hk  L

8m; k

"



ð6qÞ

!

q¼1

SNmk

9 X 3 X

3 X

8n; k

ð6hÞ

XN ik ;

XOik ;

PRik ;

PDRik ;

PRRik ;

HRRik P 0

ð6rÞ

Multiple energy resources, multiple conversion technologies and multiple end-uses are involved in the study system. Many kinds of energy sources (coal, crude oil, gasoline, diesel, kerosene, fuel oil, LPG, natural gas, imported electricity, and imported heat) are considered. The types of energy consumptions contain electricity and heat. The detailed nomenclatures for the variables and

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Y. Zhu et al. / Applied Energy 88 (2011) 2846–2862 Table 1 Energy, electric and heat demand.

4. Results analysis

Time period k=1 Energy demand (PJ) Local coal [200.80, 210.80] Raw coal [2606.40, 2706.40] Washed coal [200.00, 210.00] Coke [5.00, 6.00] Crude oil [2628.00, 2748.00] Gasoline [402.80, 417.80] Diesel [2.00, 3.00] Kerosene [664.80, 680.80] Fuel oil [4.00, 7.00] LPG [15.00, 25.00] Natural gas [1780.00, 1800.00] Electric [1840.00, power 1860.00] Heat power [396.20, 516.20]

k=2

k=3

[200.80, 210.80] [3088.40, 3188.40] [160.00, 170.00] [4.90, 5.90] [2880.00, 3000.00] [469.80, 484.80] [5.00, 6.00] [709.00, 725.00] [6.00, 9.00] [19.20, 29.20] [2136.00, 2156.00] [1880.00, 1900.00] [441.80, 561.80]

[218.00, 228.00] [3005.00, 3105.00] [200.00, 210.00] [17.00, 18.00] [3280.00, 3400.00] [588.40, 603.40] [10.00, 11.00] [705.00, 721.00] [6.00, 9.00] [22.00, 32.00] [2492.00, 2512.00] [1920.00, 1940.00] [508.60, 628.60]

parameters are provided in Appendix A. Table 1 shows the energy demands in different period times. With the population growth and economic development in Beijing, energy demands increase gradually in planning periods. However, the consumptions of coal decrease, such a drop is due to the practicing of energy policies made by the local government of Beijing. Tables 2–4 provide the economic and technological data of each power conversion technology. In energy systems, energy prices are closely related to the volatility of interest rates, just as the expansion costs and operating costs are closely related to the values of depreciation rate. The depreciation rate is the ratio of the fixed assets depreciation and the original value of fixed assets, which reflects the costsharing degree of fixed assets, such as power generating equipment and facility. The functional intervals can be identified through interest rate and depreciation rate series using historical data. Moreover, there will be a set of functional intervals corresponding to interest rate and depreciation rate value within its range to generate variations in objective functional and varied constraints.

Solutions obtained through IFMI-MEM are presented in Figs. 4–9. Most of the results for the objective function value and most of the non-zero decision variables are interval numbers. Generally, solutions presented as intervals demonstrate that the related decisions should be sensitive to the uncertain modeling inputs. 4.1. Energy allocation Fig. 4 conations the optimized results of energy allocation. Specifically, coal would always be the largest source among all supplies in all time periods. The supply of coal in periods 1–2 would increase significantly to meet the increasing demands for electricity. Nevertheless, its domestic production would drop from [534.91, 563.41] PJ in period 2 to [524.18, 539.35] PJ in period 3. Such a drop is due to the closure of small-scale mining facilities from period 1. The gap would be filled by imports from other energy systems, mostly from the Shanxi and Inner-Mongolia provincial energy systems. Beijing would gradually reduce the use of coal, so the change of import coal supply demonstrated that significant decrease would be emerged. Crude oil supplies would basically be determined by refinery activities in the municipality of Beijing. The amount would increase from [455.21, 473.71] PJ in period 1 to [570.01, 588.51] PJ in period 3 partly due to the increase of private passenger vehicles. Unfortunately, there is no direct resources supply to meet the domestic demands in Beijing. Instead, they are provided by refined oil. The total supply of gasoline would increase from [84.34, 105.00] PJ in period 1 to [121.41, 122.41] PJ in periods 3, and kerosene supply would increase from [128.01, 129.17] PJ in period1 to [140.85, 148.00] PJ in period 3. As the dramatic increase of transport tool, gasoline would increase 23.29% in period 3. Gasoline is consumed mostly by passenger vehicles and kerosene is consumed by chemical fuel. Generally, gasoline is dominant fuels for running vehicles, although a number of innovative technologies have been developed to use alternative fuel, such as hybrid fuel, hydrogen, and solar power. The traffic department of Beijing includes public transportation, transportation rental, road transportation, and civilian transportation. In fact, the traditional energy (gasoline and diesel oil) is the dominant energy consumption of traffic department,

Table 2 Energy purchased cost and selling cost. Time period k=2

k=3

Energy purchased cost (106 RMB¥/PJ) Local coal [14.40(1 + a)t, 16.80(1 + a)t] Raw coal [22.90(1 + a)t, 25.80(1 + a)t] Washed coal [29.80(1 + a)t, 33.30(1 + a)t] Coke [36.80(1 + a)t, 47.40(1 + a)t] Crude oil [123.40(1 + a)t, 128.20(1 + a)t] Gasoline [152.60(1 + a)t, 157.20(1 + a)t] Diesel [143.30(1 + a)t, 148.00(1 + a)t] Kerosene [148.00(1 + a)t, 151.90(1 + a)t] Fuel oil [131.10(1 + a)t, 133.50(1 + a)t] LPG [116.60(1 + a)t, 118.70(1 + a)t] Natural gas [46.40(1 + a)t, 50.00(1 + a)t] Electric power [79.20(1 + a)t, 87.50(1 + a)t] Heat power [30.00(1 + a)t, 32.00(1 + a)t]

k=1

[18.10(1 + a)t, 20.00(1 + a)t] [25.80(1 + a)t, 28.60(1 + a)t] [29.80(1 + a)t, 33.30(1 + a)t] [47.40(1 + a)t, 58.00(1 + a)t] [131.40(1 + a)t, 135.00(1 + a)t] [158.80(1 + a)t, 164.00(1 + a)t] [153.00(1 + a)t, 154.30(1 + a)t] [153.00(1 + a)t, 157.50(1 + a)t] [136.20(1 + a)t, 140.70(1 + a)t] [118(1 + a)t, 122.60(1 + a)t] [60.41(1 + a)t, 68.00(1 + a)t] [84.00(1 + a)t, 97.20(1 + a)t] [30.00(1 + a)t, 32.00(1 + a)t]

[20.10(1 + a)t, 23.00(1 + a)t] [28.90(1 + a)t, 33.50(1 + a)t] [29.80(1 + a)t, 33.30(1 + a)t] [56.10(1 + a)t, 68.50(1 + a)t] [135.00(1 + a)t, 138.50(1 + a)t] [165.80(1 + a)t, 172.00(1 + a)t] [156.20(1 + a)t, 163.20(1 + a)t] [161.00(1 + a)t, 169.10(1 + a)t] [142.10(1 + a)t, 145.70(1 + a)t] [119.80(1 + a)t, 125.70(1 + a)t] [74.40(1 + a)t, 84.00(1 + a)t] [101.40(1 + a)t, 107.20(1 + a)t] [30.00(1 + a)t, 32.00(1 + a)t]

Energy selling cost (106 RMB¥/PJ) Local coal Gasoline Diesel Kerosene Fuel oil LPG

[20.10(1 + a)t, 22.50(1 + a)t] [161.00(1 + a)t, 165.80(1 + a)t] [151.00(1 + a)t, 154.60(1 + a)t] [155.00(1 + a)t, 161.70(1 + a)t] [119.00(1 + a)t, 124.40(1 + a)t] [137.30(1 + a)t, 141.00(1 + a)t]

[22.40(1 + a)t, 24.80(1 + a)t] [170.50(1 + a)t, 175.00(1 + a)t] [158.00(1 + a)t, 167.40(1 + a)t] [167.50(1 + a)t, 172.00(1 + a)t] [124.00(1 + a)t, 130.30(1 + a)t] [143.00(1 + a)t, 151.70(1 + a)t]

[18.10(1 + a)t, 20.10(1 + a)t] [154.00(1 + a)t, 157.60(1 + a)t] [146.40(1 + a)t, 151.00(1 + a)t] [150.00(1 + a)t, 152.60(1 + a)t] [117.00(1 + a)t, 118.60(1 + a)t] [130.00(1 + a)t, 133.70(1 + a)t]

Note: The symbols a and t denote the interest rate and interval in each period time, t = 5 years.

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Y. Zhu et al. / Applied Energy 88 (2011) 2846–2862

Table 3 Cost of operating and expansion in electric power plant. Time period k=2

k=3

Operating cost of electric power plant (106 RMB¥/PJ) Hydropower [68.11(1 + b), 68.70(1 + b)] Pumped storage [1.88(1 + b), 66.90(1 + b)] wind power [18.30(1 + b), 155.40(1 + b)] Photovoltaic power [62.50(1 + b), 618.00(1 + b)] Waste generation [25.00(1 + b), 133.30(1 + b)] Biomass power 50.00(1 + b) Coal-fired power [4.70(1 + b), 15.00(1 + b)] Oil-fired power [1.00(1 + b), 22.00(1 + b)] Natural gas power [4.10(1 + b), 14.80(1 + b)]

k=1

[67.30(1 + b), 81.20(1 + b)] [1.88(1 + b), 66.90(1 + b)] [12.70(1 + b), 123.30(1 + b)] [23.50(1 + b), 478.80(1 + b)] [11.10(1 + b), 125.00(1 + b)] 38.90(1 + b) [1.80(1 + b), 12.00(1 + b)] [1.00(1 + b), 2.00(1 + b)] [4.10(1 + b), 14.80(1 + b)]

[64.50(1 + b), 75.70(1 + b)] [1.88(1 + b), 66.90(1 + b)] [11.30(1 + b), 125.00(1 + b)] [77.90(1 + b), 333.20(1 + b)] [08.40(1 + b), 116.70(1 + b)] [25.00(1 + b), 133.40(1 + b)] [0.85(1 + b), 11.10(1 + b)] [1.00(1 + b), 2.00(1 + b)] [4.10(1 + b), 14.80(1 + b)]

Expansion cost of electric power plant (106 RMB¥/PJ) Hydropower [1.00(1 + c), 1.05(1 + c)] Pumped storage [4.80(1 + c), 5.00(1 + c)] wind power [8.00(1 + c), 10.00(1 + c)] Photovoltaic power [43.00(1 + c), 45.00(1 + c)] Waste generation [12.00(1 + c), 13.60(1 + c)] Biomass power [4.10(1 + c), 4.50(1 + c)] Coal-fired power [4.95(1 + c), 5.00(1 + c)] Oil-fired power [5.00(1 + c), 5.30(1 + c)] Natural gas power [4.50(1 + c), 4.50(1 + c)]

[0.95(1 + c), 1.05(1 + c)] [4.57(1 + c), 5.00(1 + c)] [7.20(1 + c), 8.00(1 + c)] [40.00(1 + c), 43.00(1 + c)] [10.00(1 + c), 12.00(1 + c)] [4.00(1 + c), 4.20(1 + c)] [4.95(1 + c), 5.00(1 + c)] [5.00(1 + c), 5.30(1 + c)] [4.00(1 + c), 4.50(1 + c)]

[0.90(1 + c), 1.00(1 + c)] [4.35(1 + c), 5.00(1 + c)] [6.80(1 + c), 7.50(1 + c)] [38.50(1 + c), 40.00(1 + c)] [9.75(1 + c), 11.25(1 + c)] [3.50(1 + c), 3.95(1 + c)] [4.95(1 + c), 5.00(1 + c)] [5.00(1 + c), 5.30(1 + c)] [4.00(1 + c), 4.50(1 + c)]

Note: The symbols b and c denote the old and new equipment depreciation rate of energy conversion technology type.

Table 4 Operating cost and expansion cost of thermal power plan. Time period k=2

k=3

Electric operating cost of thermal power plant (106 RMB¥/PJ) Coal-fired cogeneration [14.20(1 + d), 16.45(1 + d)] Oil-fired cogeneration [17.80(1 + d), 21.88(1 + d)] Gas-fired cogeneration [12.95(1 + d), 29.70(1 + d)]

k=1

[11.30(1 + d), 13.40(1 + d)] [17.50(1 + d), 22.58(1 + d)] [12.95(1 + d), 29.70(1 + d)]

[10.43(1 + d), 12.55(1 + d)] [17.20(1 + d), 23.28(1 + d)] [12.95(1 + d), 29.70(1 + d)]

Heat operating cost of thermal power plant (106 RMB¥/PJ) Coal-fired cogeneration [2.14(1 + e), 2.94(1 + e)] Oil-fired cogeneration [3.12(1 + e), 3.92(1 + e)] Gas-fired cogeneration [1.15(1 + e), 1.95(1 + e)]

[2.14(1 + e), 2.94(1 + e)] [3.12(1 + e), 3.92(1 + e)] [1.15(1 + e), 1.95(1 + e)]

[2.14(1 + e), 2.94(1 + e)] [3.12(1 + e), 3.92(1 + e)] [1.15(1 + e), 1.95(1 + e)]

Expansion cost of thermal power plant (106 RMB ¥/PJ) Coal-fired cogeneration [5.00(1 + f), 5.20(1 + f)] Oil-fired cogeneration [5.22(1 + f), 5.30(1 + f)] Gas-fired cogeneration [4.00(1 + f), 4.50(1 + f)]

[5.00(1 + f), 5.30(1 + f)] [5.22(1 + f), 5.40(1 + f)] [4.00(1 + f), 4.60(1 + f)]

[5.00(1 + f), 5.50(1 + f)] [5.22(1 + f), 5.50(1 + f)] [4.00(1 + f), 4.90(1 + f)]

Note: The symbols d, e, and f denote the depreciation rate of old electric equipment, old heat equipment and new equipment.

lower bound

upper bound

increased percentage 0.3

600

0.25 500

Energy supply (PJ)

400 0.15

300

0.1

0.05 200 0 100 -0.05

0

-0.1 i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 i=9 i=10 i=11 i=12 i=13 i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 i=9 i=10 i=11 i=12 i=13 i=1 i=2 i=3 i=4 i=5 i=6 i=7 i=8 i=9 i=10 i=11 i=12 i=13

period 1

period 2 Fig. 4. Solutions of energy supply.

period 3

Increased percentage

0.2

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Y. Zhu et al. / Applied Energy 88 (2011) 2846–2862 upper bound

increased percentage

0.4

250

0.2

200

0

150

-0.2

100

-0.4

50

-0.6

0

m=1

m=2

m=1

period 1

m=2

m=1

period 2

Increased percentage

Eenergy processing (PJ)

lower bound

300

-0.8

m=2

period 3

Fig. 5. Solutions of energy processing.

Fig. 6. Total energy allocation amount.

lower bound

upper bound

increased percentage

40

1.2

35

1 0.8 0.6

25 0.4 20 0.2 15 0

Increased percentage

Electric genetration (PJ)

30

10 -0.2 5

-0.4

0

-0.6 n=1

n=2

n=3

n=4

n=5

n=6

period 1

n=7

n=8

n=9

n=1

n=2

n=3

n=4

n=5

n=6

n=7

n=8

n=9

n=1

n=2

n=3

n=4

period 2

n=5

n=6

n=7

n=8

n=9

period 3

Fig. 7. Electric generation of conversion technology type in electric power plant.

supplemented by natural gas and electricity. In 2007, the amount of gasoline and diesel oil from the traffic consumption reached up to 7.94 million tonnes. Natural gas would be untraceable in terms of production. Its supply would rely on imports from western China.

After completion of a project named ‘‘Shan-Jin Second Pipeline’’, the total natural gas supply in periods 1–3 would increase significantly to meet the demands for the increasing end-user demands for spacing heating and other uses. The imports grew quickly,

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Y. Zhu et al. / Applied Energy 88 (2011) 2846–2862 lower bound

upper bound

increased percentage

4

60 3 50 2

40 30

1

20 0 10 0

Increased percentage (%)

Electric generation (PJ)

70

-1 n = 10

n = 11

n = 12

n = 10

period 1

n = 11

n = 12

n = 10

period 2

n = 11

n = 12

period 3

Fig. 8. Electricity generation of conversion technology type in thermal power plant.

lower bound

upper bound

increased percentage

30

0.6

Heat generation (PJ)

0.4

20 0.2 15

0 10 -0.2

5

Increased percentage (%)

25

-0.4

0 n = 10

n = 11

period 1

n = 12

n = 10

n = 11

n = 12

period 2

n = 10

n = 11

n = 12

period 3

Fig. 9. Heat generation of conversion technology type in thermal power plant.

increasing from [313.87, 315.71] PJ in period1 to [438.68, 440.51] PJ in period 2. The significant raise would be primarily attributed to the expanded natural-gas-fired electricity generation capacities and the increasing residential and commercial demands. Natural gas would mostly contribute to power generation, industrial raw materials, fuel of urban life, and transportation in Beijing. Otherwise, power purchased supply hover between [323.87, 379.90] PJ in period 1and [343.35, 339.90] PJ in period 2. It is demonstrated that power purchased would be the primary type in meeting the demands for lighting, residential appliances, commercial equipment and motors, as well as space cooling. The LGP as one of the clean energy supply would have a significant increase, and the amount would rise from [19.71, 19.87] PJ in period 1 to [27.01, 27.18] PJ in period 3. In addition, as an important fuel in Beijing, washed coal supply would decline from an interval of [50.54, 50.71] PJ in period 1 to [46.84, 47.01] PJ in period 2. Such variations would be caused by the moving of some large energy consumption factories in Beijing. Washed coal would decrease 7.31% in period 2 and 2.0% in period 3. Fig. 5 contains the solutions for energy processing generation by various technologies in three periods, which is produced by the existing processing generation facilities. Primly, there is a stationary phase in periods 1 and 2, the coking generation would account for [16.67, 20.00] PJ in period 1 and [16.49, 19.82] PJ in period 2, respectively. In period 3, there would be a significant drop in coking production, so the coking generation would reach [6.60, 9.93] PJ. The coking generation would decrease 1.08% in period 2 and 59.97% in period 3. Such drops would due to the steel works-Shougang Group moving to Tangshan city, so the coking production capacity would decline sharply. Besides, the amount

of oil refining would have a significant growth, the amount would reach [154.5, 184.59] PJ in period 1, [183.33, 216.67] PJ in period 2, and [216.67, 250.00] PJ in period 3, respectively. Oil refining would increase 18.66% in period 2 and 18.18% in period 3. In Beijing, crude oil is all need to be input to meet the increasing traffic, residential and commercial requirements. In fact, fossil fuels would be the major resources for electricity generation; this is attributable to their relatively low operating and expansion costs. The total production of energy consumption would increase steadily over the 15-year horizon. In the planning horizon, the cumulative purchase volume of crude oil would contribute the largest part, which would be [1525.40, 2020.90] PJ, or 29.53% of total production. Crude oil products are mainly used for the large traffic demand in Beijing, while part of them is used for power generation. In detail, coal purchased consumption by end-users would reach [1510.70, 1596.20] PJ, which would account for 29.25%; natural gas would reach [1128.73, 1134.23] PJ; and electricity power would reach [1000.73, 1169.70] PJ, as shown in Fig. 6. Therefore, the reduction of using fossil fuel in Beijing cannot only encourage the use of renewable energy, but also improve the urban atmospheric environment. Development and utilization of renewable energy is a major way to protect the environment and address climate change. Renewable energy can ensure long-term stable energy supply, as well. To ease the contradiction between energy supply and demand, renewable energy is actively developing in Beijing. At the same time, the model could reflect the dynamics of decisions for energy allocation, electricity generation and facility expansion under various periods. Therefore, according to above analysis, import coal, crude oil, gasoline, kerosene, natural

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Y. Zhu et al. / Applied Energy 88 (2011) 2846–2862

gas, and power purchased would be the dominant energy in Beijing. There is a steadily increasing tendency of energy consumption due to the growing demands for electricity and heat generation and the increasing end-user demands. The dispatches of coal, oil product and natural gas supplies would be attributed to their economic and technical advantages in meeting electricity and heat requirements.

social and economic development, energy with lower pollution emission (natural gas) can conforms to the social and economic needs compares to coal. As the technology of cogeneration can highly increase the energy efficiency, it obtained great support from government.

4.2. Electricity and heat generation

When the existing capacity cannot meet the energy and electric demands, facility expansion would be obtained through the IFMIMEM to guarantee the municipal requirement use. The capacity expansion and the operation of each electricity-generation facility are planned for the three periods. The patterns of energy supply would vary greatly in the three periods because of capacity expansion in period 1. Capacities of relevant facilities (e.g. coking, coal, and natural gas) would be expanded. In period 2, the existing capacities might be insufficient to meet end-user demands, invoking capacity expansion of relevant facilities (i.e. coking, photovoltaic power and oil-fired power facilities). Coking capacity would be expanded to 76 PJ since the related capital cost is reduced; an additional capacity of 30 MW would be required by new photovoltaic power facilities in electric power plant; and an additional capacity of 300 MW from oil-fired power facilities would be installed in thermal power plant. In period 3, energy supply cannot meet the traffic demand of Beijing, and the renewable energy would be desired to support the sustainable development of society and economy. Correspondingly, the oil refining would be increased to 817 PJ and an additional capacity of 50 MW from hydropower facilities would be installed. In fact, the price of coal would still maintain a low level. Moreover, as the advance of technology, pollutant emission would be decrease in electric power plant in period 3. As a result, an additional capacity of 300 MW would be required by new coal-fired cogeneration facilities in electric power plant. Such increase is significantly high. The productions of based on other technologies would not be changed. Consequently, such patterns of facility expansion could be mainly dependent on: (a) availability limits of renewable energy, (b) price competitiveness of related facilities after capacities after capacity expansion, and (c) price is comparatively high for electricity import. After expansions in the three periods, the total capacity of the facilities would be sufficient to dispose the increased energy and electric demands. 4.4. Air pollution emission control Pollutant emission associated with energy-related activities, and the coal-fired capacities would still contribute the most of emissions. In this study, the emission amounts of SO2, NOx and PM10 would be significantly increased along with the ever increasing electricity demand-levels. Fig. 10 shows the detail result of lower bound

upper bound

6000

Emission amount (103 tonne/year)

Fig. 7 represents the solutions of electricity generation of different conversion technologies in electric power plant. The total electric generation of electric power plant would be [137.65, 170.38] PJ. As coal is considered a ‘‘dirty fuel’’, the consumption of coal would be reduced in electric power plant. The coal-fired power would decrease from [15.05, 17.72] PJ in period 1 to [6.69, 6.97] PJ in period 3. Due to the advantage of environmentally clean, economical and efficient, and has an enviable safety record, natural-gas power would account for the largest contribution among all energy resources. According to the figure, electricity would be generated primarily by natural gas-fired facilities. Natural gas electricity generation would soar from [21.69, 27.12] PJ in period 1 to [28.71, 36.14] PJ in period 3 onward. Such an abrupt increase can be attributed to the utilization of newly expanded generation capacities, when large amount of natural gas is available from western China. Meanwhile, oil would be another important resource that can provide large amounts of electricity. The oil-generated electricity would rise from [4.01, 4.31] PJ in period 1 to [4.67, 7.33] PJ in period 3 onwards. Besides, the renewable energy such as consumption of hydropower, wind power, photovoltaic power and biomass power would remain at a low level, and the existing consumption of renewable energy has not been fully utilized. Fig. 8 demonstrates the electric generation of conversion technology type in thermal power plant. Oil-fired power cogeneration would be [0.42, 5.65] PJ in period 1, [1.81, 10.84] PJ in period 2, and [2.50, 11.60] PJ in period 3, respectively, which is partially due to high transportation demands. The coal-fired cogeneration has a stable tendency in the planning periods, the amount would be [23.56, 38.71] PJ in period 1, [29.60, 50.97] PJ in period 2, and [28.37, 57.14] PJ in period 3, and coal-fired cogeneration would increase 25.63% in period 2 and decrease 4.16% in period 3. There would be limited to coal using in thermal power plant. Gas-fired combined-cycle power generation has a stable tendency as well, it would take [28.97, 49.48] PJ in period 1, [28.82, 63.71] PJ in period 2 and [28.67, 48.89] PJ in period 3, and Gas-fired combined-cycle power generation would decrease 0.53% in period 2 and 0.53% in period 3. Oil-fired power cogeneration would be obviously lower than coal-fired cogeneration and gas-fired combined-cycle power generation in thermal power plant. It is indicated that gas-fired power cogeneration would be the largest electric generation in thermal power plant, corresponding to socio-economic and energy policy of Beijing. Coal and oil would be inferior to electricity and natural gas in terms of fuel cost and technology performance, this would result in stable or declining trends of their consumption in periods 1–3. Fig. 9 shows the heat generation in thermal power plant. There would have slight difference in the heat generation number between the coal-fired cogeneration and the gas-fired combinedcycle power in the three periods. Heat generation of coal-fired cogeneration would amount to [15.30, 16.59] PJ in period 1, [21.49, 21.84] PJ in period 2, and [20.41, 24.49] PJ in period 3, respectively. Heat generation from gas-fired power cogeneration would be [20.86, 21.20] PJ in period 1, [26.97, 27.31] PJ in period 2 and [20.61, 20.95] PJ in period 3, respectively. Heat generation from oil-fired power cogeneration would decline in the planning periods as well. It is indicated that, with the increasing growth of

4.3. Capacity expansion scheme

5000 4000 3000 2000 1000 0 SO2

NOx

period 1

PM10

SO2

NOx

period 2

PM10

SO2

NOx

period 3

Fig. 10. Solutions of pollution emission.

PM10

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pollutant emission obtained through the IFMI-MEM. It is indicated that the power sectors would be one of the largest pollutant emission sources. With an environmental protection goal, electricity generated from coal-fired and gas-fired power conversion technologies would be reduced accordingly. In addition, with closing low efficiency of energy consumption and high pollutant emission power plant in the run-up to the pollutant emission standards, the pollution emissions were significantly reduced accordingly. Therefore, with the implementation of energy saving policy and the development of energy technology, the pollutant emission has a steady downward tendency. For example, the emission amount of SO2 would decrease from [5061.76, 5347.39]  103 tonne/year in period 1 to [1807.26, 2424.69]  103 tonne/year in period 3. The emission amount of NOx would decline from [4034.12, 5449.08]  103 tonne/year in period 1 to [2006.33, 2657.57]  103 tonne/year in period 3. This drop mainly because the policy of transportation reduction in Beijing. And the emission amount of PM10 would be [324.05, 625.31]  103 tonne/year in period 1, [540.19, 807.02]  103 tonne/year in period 2, and [400.66, 627.77]  103 tonne/year in period 3. Thus, the results would provide useful bases for generating decision alternatives with a desired technology combination that would lead to a satisfied environmental quality as well as a minimized abatement cost. In future, more and more environment-friendly electricityconversion technologies would be chosen for energy generation to satisfy the ever-increasing demand. 4.5. System cost The objective is to minimize the system cost according to the optimized demands in this study. Solution of the objective function    fopt ¼ RMBU½709:14; 3208:25  1012 provides two extremes of net system cost over the planning horizon. As the actual value of each continuous variable varies within its lower- and upperbounds, the net system cost would change correspondingly  þ between fopt and fopt . In this model, most uncertain parameters are expressed as intervals with known lower- and upper-bounds.  Besides, the lower bounds of cost coefficients correspond to fopt ; these imply that the manager has an optimistic attitude for estimating the system cost. Conversely, a plan with a higher system cost would better resist energy and electricity shortage. Besides, it corresponds to a conservative estimation towards the economic effects (i.e. upper-bound cost coefficients). Furthermore, according to above analysis, environmental factors will bring some impacts for the total system cost. Considering the environmental factors, the expected system costs would decrease obviously. For instance, the system cost would be RMB¥[105.37, 316.10]  1012 in period 1, RMB¥[136.28, 1160.60]  1012 in period 2, and RMB¥[46.74, 1731.55]  1012 in period 3. If environmental factors are not taken into account, the resulting plans of expected cost would vary within its relevant solution interval: the system cost would be RMB¥[101.25, 310.50]  1012 in period 1, RMB¥[127.03, 1121.33]  1012 in period 2 and RMB¥[460.09, 1722.35]  1012 in period 3. It is demonstrated that, with the consideration of pollutant emission, the behavior of environment management would bring increased system costs. Currently, coal-dominated energy structure in Beijing has led to serious air pollution. Therefore, the government should adopt more suitable measures to promote the renewable energy using and reduce the negative efforts of environmental pollution. 4.6. Discussion In this study, the developed IFMI-MEM integrates intervalparameter programming (IPP), mixed-integer linear programming (MILP) and full-infinite programming (FIP). The IFMI-MEM can

effectively reflect the uncertainties presented as crisp interval values and functional intervals related economic factors to be incorporated within a general optimization framework. Due to instantaneous effects from fluctuations of the interest rate and depreciation rate, the lower- and upper-bounds of several parameters vary correspondingly. Different from conventional crisp intervals, a functional interval reflects both internal (by interval parameters themselves) and external (by functional relations to external factors) uncertainties of the system. If the functional intervals are substituted by determine values and crisp intervals, the study problem can also be solved through an interval mixedinteger linear programming (IMILP) method. When the independent variable is fixed, the IFMI-MEM is converted into an interval mixed-integer programming municipal-scale energy (IMILPMEM) problem. Objective function solution of IMILP-MEM (RMB¥[2.16, 2.72]  109) provides two extremes of net system cost over the planning horizon as well. Compared with IFMIMEM, the system cost relatively low in the IMILP-MEM solution. Since the fluctuating market factors are not addressed in the IMILP-MEM model, the relevant constraints are relaxed. Thus the IMILP-MEM solution do not really optimal since it implies a reduced system reliability level. To ensure the constraints to be satisfied under all interest rate and depreciation rate levels, stringent constraints are used in IFMI-MEM, although this might raise the system cost. The conventional IMILP-MEM model can only tackle problems with coefficients of the objective function and constraints being crisp intervals; this is based on an assumption that the coefficients are not affected by any external factors. Thus IMILPMEM is only applicable to problems where effects of the external factors are too minor to be considered. If such effects are significant, the solutions from the IMILP-MEM would become unreliable with a raised risk of constraint violation. In comparison, this IFMI-MEM can reflect such effects. The conventional IMILPMEM method can only deal with problems with fixed lowerand upper-bounds. In the IFMI-MEM, the constraints can be adjusted in response to the fluctuations of external factors. For example, with the raised interest rate and depreciation rate, the energy cost and facility expansion costs are increased correspondingly. More expenses are needed for energy cost and facility expansion. Although the numbers of objectives and constraints are infinite in the IFMI-MEM, the solution algorithm will not significantly increase computation time. The infinite objectives are converted into a single one and the infinite constraints are also converted into finite ones by combining constraints. Thus the solution algorithm is applicable to real-world energy systems. The IMILP-MEM can only deal with single-objective programming problems as it merely allows the number of constraints to be finite. As an extension of the IMILP, the IFMIP contains functional intervals in its infinite objective function. It can deal with inexact programming problems with both infinite objectives and infinite constraints, and generate a link of facility expansion schemes. In its solution process, the IFMI-MEM is firstly converted into a single-objective programming problem. An IFMI-MEM applies to Beijing. Compared with the previous study of municipal energy systems, the applied value of IFMIMEM includes: (a) IFMI-MEM can better reflect the uncertainties and tackling the trade-offs among system reliability and objectivity expressed as crisp intervals and functional intervals. The continuous variable solutions are related to decisions of energy supply and allocation, the interval solutions can help decision makers obtain multiple decision alternatives, and the binary-variable results represent the decisions of facility expansion.

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(b) IFMI-MEM effectively reflects interrelationships between energy availability and its economic implications (interest rate and depreciation rate) through the adoption of dependencies between energy availability due to energy deficiency within energy systems. The model also incorporates existing energy policies directly into its optimization process. It improves upon the existing approaches for municipal energy systems planning, such that robustness of the optimization process can be enhanced. (c) Through planning of municipal energy systems, cost-effective options are obtained based on a least-cost strategy with a minimized economic cost and a maximized energy security. Then, the obtained solutions of IFMI-MEM provide more practical decision bases for formulating energy allocation pattern, expansion schemes, and emission reduction policies. So, decision makers can adjust the existing demand and supply patterns of energy resources, generate facilitate dynamic analysis for capacity expansion, and coordinate the conflict interactions among economic cost, system efficiency, pollutant mitigation and energy-supply security. Meanwhile, several assumptions are made for the IFM-MEM model in Beijing, which may bring some limitations of energy systems of Beijing. For instance, only typical energy types are considered; energy consumption just involves ‘‘electricity and heat supply’’; and the capacity expansion plans are formulated based on hypothesis, which may not be true for practical problems. Furthermore, the peak power generation and demands are not reflected in this study. Therefore, it is necessary to advance more realistic model to tackle above problems and enhance the practical applicability. In addition, owing to the complex situation of the energy system in Beijing, the required data for the planning study are extensive. Although most of the acquired data are relatively accurate (deterministic numbers, or interval numbers with relatively narrow ranges), others are less so (highly uncertain). Improving the quality of the input data through further investigation and verification would help enhance the reliability of the generated solutions. Finally, with the advanced air pollution mitigation techniques to be adopted, extensions of administrative approaches for controlling pollution emission are desired. As an advisable alternative, emissions trading may be an interesting topic by providing economic incentives for achieving reductions, which deserves future research efforts. In addition, the study is a new attempt to develop an IFMIMEM and apply it to energy systems of Beijing. However, there is still much space for improvement of the proposed method. First, the interval full-infinite mixed-integer programming (IFMIP) method may require future improvements in considering more complexities expressed as fuzzy set, possibilities, and stochastic to support system for real-world applications. Thus, it can be used as an efficient tool for analyzing and visualizing impacts of energy and environmental policies. Second, a major limitation of IFMIP is that it can merely deal with functional intervals linearly related to its factors. These relationships could be nonlinear in many practical environmental management problems, so it is necessary to advance more sophisticated methods to tackle these uncertainties. In addition, the parameters may be associated with multiple factors in energy systems. In this study, the lower- and upper-bounds of the functional intervals are assumed to be associated with a single independent variable in this study. However, functional intervals may be affected simultaneously by many variables in real word. How to deal with the IFIP problems containing multiple independent variables will be our future studies.

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5. Conclusions Functional intervals represent more complex uncertainties in real-world systems. Their lower- and upper-bounds are functions of some independent external factors, with characteristics of both intervals and functions. By introducing functional intervals into the objective and constraints of interval mixed-integer programming municipal-scale energy (IMILP-MEM) problem, interval full-infinite mixed-integer municipal-scale energy (IFMIMEM) model is formulated. The IFMI-MEM can tackle the inexact programming problems that contain both infinite objectives and constraints due to the effects from some external factors. With the proposed concept of functional intervals, more complicated inexact programming problems can be tackled. The developed method is applied to energy systems of Beijing under a variety of uncertainties. Reasonable solutions have been generated as both binary and continuous variables. The generated solutions can provide desired energy resource allocation, electricity generation, heat generation, expansion schemes and emission reduced policy with a minimized system cost and a maximized energy security. Tradeoffs between system costs and reliability can be tackled. Higher costs will increase system stability. The model can be used for examining the relations between system cost and resources availability. Moreover, the obtained interval solutions of IFMI-MEM can also be used for generating decision alternatives and thus help decision makers identify desired policies under various economic and system constraints. Although this study is the first attempt for energy systems planning through the developed IFMI-MEM, the model results suggest that this method is applicable for other problems that involve uncertainties expressed as crisp intervals, functional intervals, as well as dynamic complexities (e.g. capacity-expansion planning).

Acknowledgments This research was supported by the Major State program of Water Pollution Control (2009ZX07104-004). The authors are grateful to the editors and the anonymous reviewers for their insightful comments and suggestions.

Appendix A. List of symbols

Subscripts i = energy purchased type i = 1 local coal i = 2 coal purchased i = 3 washed coal i = 4 coke i = 5 crude oil i = 6 gasoline i = 7 diesel i = 8 kerosene i = 9 fuel oil i = 10 liquefied petroleum gas i = 11 natural gas i = 12 electricity power i = 13 heat power j = energy selling cost type j = 1 local coal j = 2 gasoline j = 3 diesel (continued on next page)

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j = 4 kerosene j = 5 fuel oil j = 6 liquefied petroleum gas k = period time, k = 1, 2, 3 m = processing technology type m = 1 coking m = 2 oil refining n = conversion technology type n = 1 hydropower n = 2 pumped storage n = 3 wind power n = 4 photovoltaic power generation n = 5 waste generation (waste incineration, waste gas system) n = 6 biomass power generation n = 7 coal-fired power generation n = 8 oil-generated electricity n = 9 natural gas power n = 10 coal-fired cogeneration n = 11 oil-fired power cogeneration n = 12 gas-fired combined-cycle power n = 13 coal-fired heating n = 14 oil-fired heating n = 15 gas-fired heating n = 16 electric heat storage n = 17 heat pump technology n = 18 geothermal heating For electric power plant, n = 1, 2, . . . , 9 For thermal power plant, n = 10, 11, 12 For heating plant, n = 13, 14, . . . , 18 p = pollution type, p = 1 (SO2), p = 2 (NOx),p = 3 (PM10) q = expansion program Decision variables XN ik = supply amount of local energy type i in period k XO jk = supply amount of transferred local energy type i in period k PR mk = generation amount of energy processing technology m in period k PDR nk = electric generation amount of conversion technology type nin electric power plant in period k PRR nk = electric generation amount of conversion technology type n in thermal power plant in period k HRR nk = heat generation amount of conversion technology type nin thermal power plant in period k YP  qnk = capacity-expansion option of conversion technology type nunder different expansion programq in electric power plant in period k YM qmk = capacity-expansion option of energy processing technology m under different expansion program q in period k YH qnk = capacity-expansion option of conversion technology type nunder different expansion program q in thermal power plant in period k Parameters CN ik = purchased cost of energy type iin period k ak = interest rate in period k CO jk = selling cost of transferred local energy type i in period k PC  mk = operating cost of energy processing technology m in period k gnk = old equipment depreciation rate of energy processing technology type m in period k

PAqmk = expansion capacity of energy processing technology m under different expansion program q in period k PI mk = investment cost of energy processing technology m in period k hnk = new equipment depreciation rate of energy conversion technology type n electric power plant in period k PDC  nk = operating cost of conversion technology type n in electric power plant in period k bnk = old equipment depreciation rate of energy conversion technology type n electric power plant in period k PDAqnk = expansion capacity of conversion technology type n under different expansion program q in electric power plant in period k PDI nk = investment cost of conversion technology type n in electric power plant in period k cnk = new equipment depreciation rate of energy conversion technology type n electric power plant in period k PRC  nk = electric operating cost of conversion technology type n in thermal power plant in period k dnk = old electric equipment depreciation rate of energy conversion technology type n in thermal power plant in period k HRC  nk = heat generation operating cost of conversion technology type n in thermal power plant in period k k nk = old heat equipment depreciation rate of energy conversion technology type n in thermal power plant in period k PRAqnk = expansion capacity of conversion technology type n under different expansion program q in thermal power plant in period k PRI nk = investment cost of conversion technology type n in thermal power plant in period k fnk = new equipment depreciation rate of energy conversion technology type n in thermal power plant in period k PGR nk = electric generation amount of conversion technology type n in heating plant in period k

PGC  nk = electric operating cost of conversion technology type n in heating plant in period k HGR nk = heat generation amount of conversion technology type nin heating plant in period k HGC  nk = heat operating cost of energy conversion technology type n in heating plant in period k L = 5 years lnk = old electric equipment depreciation rate of energy conversion technology type n in heating plant in period k pnk = old heat equipment depreciation rate of energy conversion technology type n in heating plant in period k hk = electric generation hours in period k SLk = transmission loss in period k Cuk = transportation cost of electricity in period k PRB k = heat proportion in thermal power plant in period k CPpk = environmental facilities cost for pollution type p in period k CE pk = cost of emission of pollution type pin period k SUk = financial subsidy in period k ZO ik = energy availability of energy type i in period k DE k = total electric demand in period k DH k = total heat demand in period k DI k = total energy demand in period k SP  mk = the loss of processing technology type m in period k SN mk = energy processing demand of processing technology type m in period k

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SD nk = electric loss of energy conversion technology type n in electric power plant in period k SPD ik = electric demand for energy conversion technology type n in electric power plant in period k SR nk = electric loss of energy conversion technology type n in thermal power plant in period k SRR ik = electric demand for energy conversion technology type n in thermal power plant in period k SHR ik = heat demand for energy conversion technology type n in heating plant in period k AMR npk = emission amount of pollution type p in electric power plant in period k AMH npk = emission amount of pollution type p in thermal power plant in period k AMG npk = emission amount of pollution type p in heating plant in period k SFpk = emission efficiency of pollution type pin period k ES pk = allowed amount of pollution type pin period k MN  mk = funding of electric power plant expansion for processing technology type min period k MP nk = funding of electric power plant expansion for energy conversion technology type n in period k MR nk = funding of thermal power plant expansion for energy conversion technology type n in period k

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