An interval full-infinite programming approach for energy systems planning under multiple uncertainties

Electrical Power and Energy Systems 43 (2012) 375–383

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

An interval full-infinite programming approach for energy systems planningunder multiple uncertainties

Y. Zhu a,1, G.H. Huang a,⇑, L. He a,b, L.Z. Zhang c,2

a MOE Key Laboratory of Regional Energy Systems Optimization, S-C Resources and Environmental Research Academy, North China Electric Power University, Beijing 102206, Chinab Department Civil Engineering, Faculty of Engineering, Architecture and Science, Ryerson University, 350 Victoria Street, Toronto, Ontario, Canada M5B 2K3c Chifeng Municipal Research Institute of Environment, Linhuang Street, Songshan District, Chifeng City, Inner Mongolia 024000, China

a r t i c l e i n f o

Article history:Received 7 February 2012Received in revised form 28 May 2012Accepted 29 May 2012Available online 4 July 2012

Keywords:UncertaintyFull-infiniteFunctional intervalEnergy systemsDecision making

0142-0615/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijepes.2012.05.066

⇑ Corresponding author. Tel.: +86 13911468225; faE-mail addresses: [email protected] (Y

(G.H. Huang), [email protected] (L. He),(L.Z. Zhang).

1 Tel.: +86 10 5197 1255; fax: +86 10 5197 1306.2 Tel./fax: +86 476 8283 303.

a b s t r a c t

An interval full-infinite programming regional energy model (IFIP-REM) is developed in this study forsupporting energy systems management under uncertainty. IFIP-REM integrates full-infinite program-ming (FIP) into an interval linear programming (ILP) framework. The IFIP is capable of addressing multi-ple uncertainties existing in related costs, impact factors and system objectives (expressed asdeterminates, crisp interval values and functional intervals). The modeling approach inherits the advan-tages of ILP and FIP, and allows uncertainties and decision-makers’ aspirations to be directly communi-cated into the optimization process and resulting solutions. The developed method is applied to anenergy planning system, where pollutant emissions are desired to be controlled. The results indicate thatreasonable solutions can be generated and used to support the obtained interval solutions of IFIP-REMmodel, and the solutions can be used for generating decision alternatives and thus help decision makersidentify desired policies under various economic and system constraints to coordinate the conflict inter-actions among economic cost, system efficiency, pollutant mitigation and energy-supply security.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Energy planning is crucial for securing technology safe, eco-nomically efficient and environmentally friendly energy manage-ment systems [32,8,31,27,34,18]. Undoubtedly, with economicdevelopment, population growth and urban expansion, energyplays an increasingly significant role in modern economic develop-ment and human activities for achieving sustainable developmentof society and economy. It is well known that energy consumptioncan result in air pollutants, which are extremely harmful to bothpublic health and environment. Besides, climate change is consid-ered to be one of the most challenging problems in the 21st cen-tury [38]. Electricity generation sectors are considered the majorcontributor of greenhouse gases (GHG) due to the consumptionof fossil fuel [36]. Decision makers are thus in a quandary onhow to balance increasing energy demands with the populationgrowth and the economic development, mandated requirementsfor GHG emissions reduction, and less fossil fuel consumption. Inthe past decades, there have been a number of studies on energy

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systems management and planning to reach the goal of sustainableeconomic development and effective environmental management[41,12,43,21,33,19,26,1,5,2,6,13,22,28].

However, energy system is subject to a variety of social, eco-nomic, political, environmental and technical factors, which arehighly uncertain and interrelated. Therefore, various studies havebeen undertaken to address parameter uncertainties presented asvarious forms. This would be useful in analyzing system reliability,assessing associated risks and supporting the formulation of soundmanagement policies [10,23,35]. Therefore, energy systems plan-ning models are desired to secure technically safe, economicallyefficient and environmentally friendly at multiple scales [40].However, it is still difficult to address the complex linkages that ex-ist among different energy activities and their socio-economic andenvironmental implications in a multi-sector, multi-period, andmulti-objective context; and handling uncertainties associatedwith dynamic changes of system conditions [16,29,13,7,9].

To address such uncertainties, a number of programming meth-ods were developed in energy planning problems, which were gen-erally based on interval linear programming (ILP) [5]. ILP approachis effective for handling uncertainties expressed as interval num-bers without known probability distributions and membershipfunctions, which can exist in the model’s objective function andconstraints. ILP improved upon the conventional approachesthrough direct integration of uncertain information into the mod-eling formulations [17]. For example, Kanudia and Loulou [20]

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376 Y. Zhu et al. / Electrical Power and Energy Systems 43 (2012) 375–383

introduced stochastic programming methods for supporting globalenergy systems planning under uncertainty. In this study, stochas-tic programming method was introduced into an ILP framework.Liu et al. [30] proposed a fuzzy-stochastic robust programmingmodel for regional air quality management under uncertainty,where approaches of ILP and stochastic programming methodswere incorporated into an ILP framework to deal with uncertain-ties expressed as intervals and probabilities. Sadeghi and Hosseini[39] applied a fuzzy linear programming model for optimizing sup-ply energy systems in Iran, where uncertainties of investment costsin the objective functional coefficients were considered. In thisstudy, ILP and fuzzy linear programming model were introducedinto a general optimization framework. More recently, Mavrotaset al. [32] developed an ILP model comprised of fuzzy parametersto handle uncertainties of energy costs. Lin et al. [25] proposedan interval-fuzzy two-stage stochastic optimization model forregional energy systems planning under uncertainty, where uncer-tainties expressed as fuzzy sets, stochastic numbers and intervals.

Li et al. [24] planning regional-scale electric power system un-der uncertainty by proposing multistage interval-stochastic inte-ger linear programming approach. Lin and Huang [28] developedan interval-fuzzy stochastic optimization for regional energy sys-tems planning and greenhouse-gas emission management underuncertainty. The proposed method then applied to a case studyfor the Province of Ontario, Canada. Zhu et al. [45] formulated amunicipal-scale energy systems model under functional intervaluncertainties.

However, ILP can only deal with inputs expressed as crisp inter-val [a, b], the lower and upper bounds (a and b) are definitelyknown. To illustrate, electricity price (Pijk) can be estimated to bea crisp interval of [947, 1047] $/kW h. This means that the real Pijk

value is imprecise. However, we know that it falls between 947 $/kW h and 1047 $/kW h. This expression reflects the inherent uncer-tainty affected by the difficulty in characterizing the system fea-tures [17,27]. In actual systems, this definition is not suitable forall cases where the two bounds may be associated with the externalimpact factors. For instance, if the Pijk value is affected by the energyprice, the lower and upper bounds will not be unchanged since anyvariation in energy price will lead to a corresponding change in Pijk.As a consequence, the concept of the crisp interval may not be suit-able for describing such an uncertainty. An effective way of describ-ing this uncertainty would be the functional intervals; this can bedefined as a lower and an upper bound, which are both functionsof its associated impact factor. When the coefficients of the objec-tive functions and constraints are both allowed to be functionalintervals and/or intervals, the ordinary linear programming prob-lem becomes a more complicated problem [11,42,14,15,44,45].Thus, interval full-infinite programming (IFIP) can be generatedby introduced full-infinite programming (FIP) into an interval linearprogramming (ILP) framework. The IFIP is capable of addressingmultiple uncertainties existing in related costs, impact factors andsystem objectives (expressed as determinates, crisp interval valuesand functional intervals). The modeling approach inherits theadvantages of ILP and FIP, and allows uncertainties and decision-makers’ aspirations to be directly communicated into the optimiza-tion process and resulting solutions.

Therefore, to address the above mentioned issues, developmentof suitable systems analysis approaches to integrate a variety ofcomponents (objectives, constraints and activities) into a generalmodeling framework would be necessary. The objective of thisstudy is to advance an interval full-infinite programming regionalenergy model (IFIP-REM) to support the energy systems planningand environmental management under uncertainty. To illustratethe applicability of the energy model, IFIP-REM is applied to a hypo-thetical case study for helping relevant managers identify the de-sired energy policies. In detail, the IFIP-REM model necessitates

sub-tasks include: (a) development of a municipal energy systemsmodel to formulate the processing of energy generation, conversionand consumption; (b) application of the municipal-scale energymodel to a hypothetical case study for tackling the competitiveand uncertainties among various energy sectors and processes;and (c) generation of decision alternatives through coordinatingthe conflict interactions among economic cost, system efficiencyand energy-supply security. Its outputs would be interpreted togenerate desired planning alternatives for a number of humanactivities, as well as the relevant policies and strategies.

2. Methodology

When the objective and constraints of an ILP problem containcrisp intervals and functional intervals, an IFLP problem can beformulated. To begin, consider an ILP problem as follows:

Max f� ¼Xk

j¼1

c�j x�j þXn

j¼kþ1

c�j x�j ð1aÞ

subject to:

Xn

j¼1

a�ij x�j 6 b�i ð1bÞ

x�j P 0 ð1cÞ

where a�ij ,b�i are interval coefficients; c�j (j = 1,2, . . . ,k) are positivecoefficients; c�j (j = k + 1, k + 2, . . . ,n) are negative coefficients. Mode(1) can be divided into the following two submodels, which ensuresthat stable and continuous solution intervals can be formed [17]:

submodel 2:

Max fþ ¼Xk

j¼1

cþj xþj þXn

j¼kþ1

cþj x�j ð2aÞ

subject to:

Xk

j¼1

ja�ij j�Signða�ij Þxþj þ

Xn

j¼kþ1

ja�ij jþSignða�ij Þx�j 6 bþi ð2bÞ

xþj P 0; x�j P 0 8j ð2cÞ

submodel 3:

Max f� ¼Xk

j¼1

c�j x� þXn

j¼kþ1

c�j xþ ð3aÞ

subject to:

Xk

j¼1

ja�ij jþSignða�ij Þx�j þ

Xn

j¼kþ1

ja�ij j�Signða�ij Þxþj 6 b�i ð3bÞ

x�j 6 xþjopt; j ¼ 1;2; . . . ; k ð3cÞ

xþj P x�jopt; j ¼ kþ 1; kþ 2; . . . ;n ð3dÞ

xþj P 0; x�j P 0; 8j ð3eÞ

where a�ij and aþij are lower and upper bounds of a�ij ; b�i and bþi arelower and upper bounds of b�i ; c�j and cþj are lower and upperbounds of c�j ; x�jopt

and xþjoptare lower and upper bounds of x�j ; Sign(�)

is defined as:

Signðx�Þ ¼1 ðx� P 0Þ�1 ðx� < 0Þ

(ð4Þ

Y. Zhu et al. / Electrical Power and Energy Systems 43 (2012) 375–383 377

When the coefficients in the constraints and objectives are func-tional intervals (instead of crisp intervals), model (1) can be formu-lated as the following IFIP problem:

Max f� ¼Xk

j¼1

c�j ðh0Þx�j þXn

j¼kþ1

c�j ðh0Þx�j ; for all h0 2 ½hl0;hu0� ð5aÞ

subject to:

Xn

j¼1

a�ij ðhiÞx�j 6 b�i ðhiÞ for all hi 2 ½hl; hu� ð5bÞ

x�j P 0 ð5cÞ

where a�ij ðhiÞ, b�i ðhiÞ, and c�j ðhiÞ are function interval parameters.The IFIP model can be converted into two FIP submodels:

submodel 6:

Maxfþ ¼Xk

j¼1

cþjhl0 þ hu0

2

� �xþj þ

Xn

j¼kþ1

cþjhl0 þ hu0

2

� �x�j ð6aÞ

subject to:

Xk

j¼1

ja�ij ðhiÞj�Sign½a�ij ðhiÞ�xþj þXn

j¼kþ1

ja�ij ðhiÞjþSign½a�ij ðhiÞ�x�j 6 bþi ðhiÞ ð6bÞ

xþj P 0; x�j P 0; 8j ð6bÞ

submodel 7:

Max f� ¼Xk

j¼1

c�jhl0 þ hu0

2

� �x� þ

Xn

j¼kþ1

c�jhl0 þ hu0

2

� �xþ ð7aÞ

subject to:

Xk

j¼1

ja�ij ðhiÞjþSign½aþij ðhiÞ�x�j þXn

j¼kþ1

ja�ij ðhiÞj�Sign½a�ij ðhiÞ�xþj 6 b�i ðhiÞ ð7bÞ

x�j 6 xþjopt; j ¼ 1;2; . . . ; k ð7cÞ

xþj P x�jopt; j ¼ kþ 1; kþ 2; . . . ;n ð7dÞ

xþj P 0; x�j P 0; 8j ð7eÞ

where x�joptand xþjopt

are solutions of submodels (6) and (7), respec-tively; c�j ð

hl0þhu02 Þ and cþj ð

hl0þhu02 Þ (j = 1,2, . . . ,k) are positive for all h

Fig. 1. Framework of a repres

values; c�j ðhl0þhu0

2 Þ and cþj ðhl0þhu0

2 Þ (j = k + 1, k + 2, . . . ,n) are negativefunctions for all h values; Sign(�) is defined as:

Signðx�Þ ¼1 ðx� P 0Þ�1 ðx� < 0Þ

�ð8Þ

Then, the functional intervals of the model can be converted tointerval forms, which can be solved through conventional solutionmethods.

3. Case study

3.1. Overview of the study system

A hypothetical case is used to illustrate the applicability of theIFIP approach, based on the data from the literatures published[4]; Beijing [3,37]. There are four types of electricity power plants(i.e. coal-fired, fuel power, gas power and nuclear power), fourtypes of energy allocations (i.e. coal, gasoline, gas, nuclear), andthree electric users (i.e. industry, municipality, agriculture). Theelectricity users are the largest contributors to the regional econ-omy. These electric power plants consume fuels for process electricand heating generation. Fig. 1 gives the overview of an energymanagement system. The planning horizon is assumed to 30 years,three time periods are divided over the planning horizon, witheach period having a time interval of 10 years. Period 1 is from1st to 10th years, period 2 is from 11th year to 20th years, Period3 is from 21st to 30th years. At the beginning, each user has anexisting system associated with air pollution control facilities.Due to the limited availability, electricity demand, its fluctuatingprice (Table 1) and the environmental impacts, the users wouldhave to consider clean and efficient fuel even fuel substitutionsin the near future to improve their economic and environmentalefficiencies. The problem under consideration is how to effectivelyplan the energy supply and consumption patterns in the region un-der a number of environmental, economic, and availability con-straints in order to maximize the overall system benefit. Table 2presents conversion costs of different periods over the planninghorizon. They are expressed as present value dollars, and the costescalated to reflect prospective conditions and then discountedto generate present value cost in the objective function.

3.2. Modeling formulation

An IFIP-REM model can be formulated to solve the resourcesallocation problem, four kinds of energy sources i (coal, gasoline,

entative energy system.

Table 1Unit energy price in planning period.

Lower bound Upper bounder Mean value

Coal ($/103 kg) 44.00 65.00 54.50Oil ($/m3) 471.00 629.00 550.00Natural gas ($/103 m3) 500.00 650.00 575.00Nuclear ($/kg) 5123.00 7035.00 6079.00

Table 2Energy conversion cost in planning horizon.

Time period

k = 1 k = 2 k = 3

The conversion cost (104 $/kW h)Coal [26.56, 27.93] [26.32, 27.90] [27.66, 28.08]Oil [23.16, 24.02] [22.56, 23.70] [22.37, 23.85]Natural gas [27.37, 23.14] [22.75, 23.25] [20.16, 21.96]Nuclear [27.37, 30.21] [29.15, 31.50] [29.35, 31.25]


gas, nuclear) are allocated to energy user j (industry, municipality,agriculture) over a planning horizon, with the objective of optimiz-ing the system performance. In the system, the continuous decisionvariables are related to energy resources allocation. The objective isto maximize overall benefit in the region over a planning horizon,which is divided into three periods. Otherwise, several aspects suchas energy supply, energy availability, and environmental conditionsfor SO2, CO2, NOx (nitric oxides) and TSP (total suspended particu-late) emissions are considered in the constraints.

In this study, a new type of uncertainty expressed as functionalinterval. The traditional interval can be defined as a lower and anupper bound, which are both functions of its associated impact fac-tor. For example, if Pijk is expressed as a functional interval of[947 + 106a, 1047 + 107a]. Where a denotes energy price, and thenwe know that Pijk is a function of the energy price, ranging between947 + 106a and 1047 + 107a. When the coefficients of the objectivefunctions and constraints are both allowed to be functional inter-vals and/or intervals, the ordinary linear programming problem be-comes a more complicated interval full-infinite programming (IFIP)problem. Subject to the variation of energy price (Table 1), electricprice and power purchased are assumed to be functional intervalsinstead of crisp intervals (Tables 3 and 4). Energy price is consideredto be the most important factor in energy planning system since italways fluctuates with the variation of the electric price. Thisfluctuation of the energy price will correspondingly determine thefinal system costs. Since many parameters are uncertain, the IFIP

Table 3Average electricity price during planning period.

Time period

k = 1 k =

Average market prices of electricity to industry (106 $/kW h)Coal [94.70 + 10.60a, 104.7 + 10.70a] [21Oil [41.70 + 1.30a, 124.70 + 1.40a] [4.3Natural gas [19.40 + 1.10a, 109.5 + 1.2a] [82Nuclear [496.00 + 0.10a, 514.00 + 0.10a] [27

Average market prices of electricity to municipality ($/kW h)Coal [94.00 + 10.30a, 103.00 + 10.60a] [19Oil [41.00 + 1.30a, 100.00 + 1.30a] [4.2Natural gas [19.00 + 1.10a, 109.40 + 1.10a] [82Nuclear [495.80a + 0.10a, 516.40 + 0.10a] [27

Average market prices of electricity to commerce ($/kW h)Coal [94.80 + 10.70a, 104.70 + 10.70a] [21Oil [42.00 + 1.30a, 124.70 + 1.40a] [4.4Natural gas [19.50 + 1.10a, 109.50 + 1.20a] [83Nuclear [496.10 + 0.10a, 516.50 + 0.10a] [27

Note: The symbols a denotes the price of unit energy resource.

approach is considered suitable for the study problem. Accordingly,the problem under this consideration can be formulated into anIFIP-REM as follows:

Objective function:

Max f� ¼X4

i¼1

X3

j¼1

X3

k¼1

X4

n¼1

zi � P�ijkðaiÞ � x�ijk � Lk

þX4

i¼1

X3

j¼1

X3

k¼1

X4

n¼1

bi � gi � HE�ijk � x�ijk � Lk

�X4

i¼1

X3

j¼1

X3

k¼1

X4

n¼1

dc � Coik � CoCK � x�ijk � Lk

�X4

i¼1

X3

j¼1

X3

k¼1

X4

n¼1

ds � Sik � SCK � x�ijk � Lk

�X4

i¼1

X3

j¼1

X3

k¼1

X4

n¼1

dn � Noik � NoCK � x�ijk � Lk

�X4

i¼1

X3

j¼1

X3

k¼1

X4

n¼1

dt � TSik � TCK � x�ijk � Lk

�X4

i¼1

X3

j¼1

X3

k¼1

X4

n¼1

ðzi � Gu�ijk � zi � T�ijk � O�ijkÞ � x�ijk � Lk

For all : a�i 6 ai 6 aþijk ð9bÞ

Subject to:(1) Supply constraints:X4

i¼1

X3

j¼1

X3

k¼1

zi � x�ijk P D�k ð9cÞ

X4

i¼1

X3

j¼1

X3

k¼1

zi � P�ijkðaiÞ � x�ijk � Lk 6 Q�ikðaiÞ ð9dÞ

X4

i¼1

X3

j¼1

X3

k¼1

Gu�ijk � zi � x�ijk � Lk 6 N�k ð9eÞ

X3

i¼1

X3

j¼1

X3

k¼1

x�ijk 6 R�ik ð9fÞ

2 k = 3

.70 + 10.00a, 23.70 + 10.20a] [50.00 + 9.00a, 60.00 + 9.00a]0 + 1.20a, 29.40 + 1.20a] [356.50 + 0.70a, 361.90 + 0.80a].80 + 1.00a, 88.00 + 1.00a] [672.00 + 0.10a, 770.00 + 0.20a]6.00 + 0.100a, 301.70 + 0.10a] [174.30 + 0.10a, 183.00 + 0.10a]



Table 4Power purchased in planning horizon.

Time period

k = 1 k = 2 k = 3

Power purchased to industry (106 $)Coal [390.20 + 0.05a, 390.2 + 0.58a] 0.04 + 0.74a [441.60 + 0.97a, 441.70 + 0.97a]Oil [396.30 + 0.17a, 396.30 + 0.17a] [334.90 + 0.03a, 334.90 + 2.46a] [390.30 + 0.20a, 390.3 + 0.20a]Natural gas 402.20 + 0.05a [463.70 + 0.45a, 463.80 + 0.05a] 585.90Nuclear [437.40 + 0.099a, 437.4 + 0.10a] [345.00 + 0.03a, 345.20 + 0.03a] [277.50 + 0.04a, 277.9 + 0.041a]

Power purchased to municipality (106 $)Coal [390.20 + 0.05a, 390.30 + 0.58a] 415.10 + 0.74a [441.60 + 0.97a, 441.70 + 0.97a]Oil 396.40 + 0.17a [334.90 + 0.24a, 334.90 + 0.25a] 390.30 + 0.20aNatural gas [402.70 + 0.49a, 4023 + 0.49a] [463.70 + 0.05a, 463.80 + 0.05a] [585.90, 585.90 + 0.001a]Nuclear [437.40 + 0.001a, 437.40 + 0.10a] [345.00 + 0.03a, 345.20 + 0.03a] 277.50 + 0.04a

Power purchased to commerce (106 $)Coal [390.20 + 0.05a, 390.20 + 0.59a] [415.10 + 0.74a, 415.10 + 0.74a] [441.60 + 0.97a, 441.70 + 0.97a]Oil [396.40 + 0.17a, 396.40 + 0.18a] [334.90 + 0.25a, 334.90 + 0.25a] [390.30 + 0.20a, 390.30 + 0.20a]Natural gas [402.30 + 0.05a, 402.30 + 0.05a] [463.70 + 0.05a, 463.80 + 0.05a] [585.90,585.90 + 0.01a]Nuclear [437.40 + 0.10a, 437.40 + 0.01a] [345.00 + 0.03a, 345.20 + 0.03a] [277.50 + 0.04a, 277.50 + 0.04a]

Note: The symbols a denotes the price of unit energy resource.


(2) Environmental constraints:X4

i¼1

X3

j¼1

X3

k¼1

Coik � ð1� CoukÞ � x�ijk � Lk

6 DH �Wk � h2k � CoE�k ð9gÞ

X4

i¼1

X3

j¼1

X3

k¼1

Sik � ð1� SxkÞ � x�ijk � Lk 6 DH �Wk � h2k � SE�k ð9hÞ

X4

i¼1

X3

j¼1

X3

k¼1

Noik � ð1� NoukÞ � x�ijk � Lk 6 DH �W�kh2k � NoE�k ð9iÞ

X4

i¼1

X3

j¼1

X3

k¼1

TSik � ð1� TxkÞ � x�ijk � Lk 6 DH �Wk � h2k � TE�k ð9jÞ

(3) Nonnegative constraints:x�ijk P 0 ð9kÞ

wheref± = maximum profit of planning scope;i = type of energy resources, i = 1, 2, 3, 4, where i = 1 for the coal,2 for fuel oil, 3 for natural gas, 4 for nuclear;j = type of users, j = 1, 2, 3, where j = 1 for municipality, 2 forindustry, 3 for agriculture;k = time period, k = 1, 2, 3;Lk = length of period k (10 year);DH = length of period time (1 year)x�ijk = allocated amount of energy resource i to user j during per-

iod k (decision variables);zi = electric generation of per unit energy resource i;P�ijkðaiÞ = average market prices of electricity for energy

resources i to user j during period k, which are functions of ai;ai = price of unit energy resource i;bi = proportion of energy resources i used for heating;gi = amount of heat generated by per unit energy resources i;HE�ijk = market price of heat of energy resource i to user j during

period k, which are functions of Vijk;dc = amount of CO2 generated by each 1 kW h electricity;ds = amount of SO2 generated by each 1 kW h electricity;dn = amount of NOx generated by each 1 kW h electricity;dt = amount of TSP generated by each 1 kW h electricity;

n = type of pollutants, n = 1, 2, 3, 4; where n = 1for CO2, n = 2 forSO2, n = 3 for NOx, n = 4 for TSP;Gu�ijk = average capital cost of boiler system during period k;T�ijk = cost for converting of energy resource i to user j during

period k;O�ijk = operating cost of energy resources i to user j during

period k;Dk = total electricity demand during period k;Q�ikðaiÞ = Power purchased of electric quantity during period k ,

which are functions of ai;N�k = maximum allowable budget for boiler system substitution

during period k;R�ijk = resource availability from resource i to user j during period

k;Coik = average emission rate of CO2 from resources i during per-iod k;CoCk = cost for CO2 emission abatement during period k;Couk = average efficiency for CO2 emission abatement duringperiod k;CoE�k = maximum allowable emission concentration level for

CO2 during period k;Sik = average emission rate of SO2 from resources i during periodk;SCk = cost for SO2 emission abatement during period k;Suk = average efficiency for SO2 emission abatement duringperiod k;SE�k = maximum allowable emission concentration level for SO2

during period k;Noik = average emission rate of NOx from resources i during per-iod k;NoCk = cost for NOx emission abatement during period k;Nouk = average efficiency for NOx emission abatement duringperiod k;NoE�k = maximum allowable emission concentration level for

NOx during period k;TSik = average emission rate of TSP from resources i to user jduring period k;TCk = cost for TSP emission abatement during period k;Tuk = average efficiency for TSP emission abatement during per-iod k;TE�k = maximum allowable emission concentration level for TSP

during period k;Wk = average pollution diffusion distance;hk = effective average height of the pollution diffusion.


4. Result analysis

The problem is solved through the IFIP-REM. Fig. 1 presents theoptimized energy allocation process for energy resources i to user jduring period k. In IFIP-REM, the optimization solutions for objec-tive and decision variables are still intervals, though the systemparameters and the solutions are expressed as functional intervals.The results indicate that different electricity demands and energyavailability would yield different energy converting technologyand different operating costs. This is useful for decision makers ingaining insight into a tradeoff among different environmental, eco-nomic and system criteria. It is also observed that the electricityconsumption, energy resource allocation patterns would vary dueto the temporal and spatial variations of energy systems conditions.

4.1. Energy allocation

In the entire planning periods, the energy allocation has arelatively balanced distribution for municipality, industry and agri-culture. Fig. 2 conations the results of energy allocation of munici-pality. Specifically, coal would be the dominate source among allsupplies in all time periods. The supply of coal in periods 1–3 wouldincrease significantly to meet the increasing demands for end-users.The amount of coal would increase from [571.54, 627.61] �106 kW h in period 1 to [898.53, 915.28] � 106 kW h in period 3.With the increase of private passenger vehicles, fuel oil is dominantfuels for running vehicles, although a number of innovative technol-ogies have been developed to use alternative fuel. In this study, fueloil would be untraceable in terms of production. The amount of fueloil would be [546.64, 650.12] � 106 kW h in period 1, [676.85,733.42] � 106 kW h in period 2, and [621.08, 671.07] � 106 kW hin period 3. Nevertheless, the drop of its production in period 3 isdue to the large pollutant emission. Natural gas would have a signif-icant increase, and the amount would be [475.91, 660.42] �106 kW h in period 1, [671.46, 771.88] � 106 kW h in period 2 and[601.75, 804.11] � 106 kW h in period 3. Natural gas would mostlycontribute to power generation, industrial raw materials, fuel of ur-ban life, and transportation. The amount of nuclear would be[443.70, 579.13] � 106 kW h in period 1, [569.17, 588.13] �106 kW h in period 2 and [667.11, 676.22] � 106 kW h in period 3.Currently, nuclear is widely used for generating electric power inmunicipal energy systems. Nuclear power would be another majoroption to be selected. Nuclear plant will not only meet the increasingelectricity demands but also compete with coal-fired plants, result-ing in a significant reduction of pollutant emissions. Similarly, Fig. 3presents the energy allocation of industry and Fig. 4 shows theenergy allocation of agriculture.

Fig. 2. Energy allocatio

4.2. Electricity allocations

The electricity would assign to municipality, industry and agri-culture respectively. Fig. 5 shows the results of electricity alloca-tion through IFIP-REM model.

In period 1, the distribution of various energy allocations hasuniformity. The electricity generated from coal would be [1.72,1.89] � 109 kW h, and allocate to municipality, industry and agri-culture respectively. The electricity generated from fuel oil wouldbe [1.70, 1.95] � 109 kW h. The amount of electricity generatedfrom natural gas would be [1.72, 1.98] � 109 kW h. The electricitygenerated from nuclear would be [1.69, 1.73] � 109 kW h. Accord-ing to the solutions, the electricity used to industry would be [2.20,2.53] � 109 kW h, which account for 34.97% and 34.47% of the totalelectric allocation in this period. Therefore, industry is the highestelectricity user, which should be allocated at maximums value. It isindicated that industry is the priority energy consumer for energyallocation. The electricity generated from municipality would be[2.04, 2.52] � 109 kW h, and the electricity generated from agricul-ture would be [2.05, 2.51] � 109 kW h.

In period 2, the electricity generated from coal would raise to[2.38, 2.41] � 109 kW, it also has a high growth rate. The electricitygenerated from coal assigned to industry at the maximums, withthe amount of [99.10, 99.30] � 106 kg. The electricity generatedfrom fuel oil would be [2.10, 2.21] � 109 kW h, and the amountof electricity generated from natural gas would be [2.08, 2.33] �109 kW h. Although, the nuclear power generation in this periodis still the lowest with the total amount of [1.70, 1.77] �109 kW h, it also has a rapid growth over the previous period forits low pollutant emission. However, industry is still the maximumelectricity consumer compared to period 1. Electricity allocated toindustry would be [2.85, 2.98] � 109 kW h. This indicates thatindustry would be the priority energy consumer for energyallocation.

In period 3, the amount of electricity generated from coal wouldbe [2.54, 2.60] � 109 kW, compare with previous two periods. Theelectricity generated from coal assigned to municipality at themaximums, with the amount of [110.50, 112.60] � 106 kg. Theelectricity generated from fuel oil would be [1.87, 2.83] �109 kW h, and the amount of electricity generated from naturalgas would be [1.84, 2.15] � 109 kW h. However, due to the results,electricity allocated to industry would be [2.76, 3.07] � 109 kW h.Otherwise, due to the solutions, the electricity generated from coalassigned to industry would be [2.79, 3.67] � 109 kW h. The elec-tricity allocation of municipality and agriculture would be [2.79,3.07] � 109 kW h and [2.72, 2.99] � 109 kW h, respectively. Theabove variations accord with the fact that coal has the highest

n to municipality.

Fig. 3. Energy allocation to industry.

Fig. 4. Energy allocation to agriculture.


availability and the most competitive price. As the oil and gas re-sources keep dropping down in period 3, the coal’s role will pro-gressively get resumed.

4.3. Air pollution control

Pollutant emissions associate with energy-related activities,and the coal-fired capacities would still contribute the most of

Fig. 5. Electricity allocation

emissions. In this study, the emission amounts of CO2, SO2, NOx

and TSP would have a significantly increase with the increasingelectricity demand-levels. Fig. 6 presents the pollution emissionsobtained through IFIP-REM model. The amount of CO2 would in-crease from [9.24, 188.09] � 103 kg/year in period 1 to [15.03,193.51] � 103 kg/year in period 3. The amount of SO2 would be[7.81, 455.89] � 103 kg/year in period 1, [12.57, 557.29] � 103 kg/year in period 2, and [14.79, 602.73] � 103 kg/year in period 3,

s in different periods.

Fig. 6. Optimized emissions of pollutants and system benefit.


respectively. The amount of NOx would increase from [9.09,129.38] � 103 kg/year in period 1 to [15.26, 100.23] � 103 kg/yearin period 3. The amount of TSP would be [0, 66.09] � 103 kg/yearin period 1, [0, 71.19] � 103 kg/year in period 2 and [0, 77.73] �103 kg/year in period 3, respectively. This indicates that the pollu-tant emission has a steady upward tendency. Furthermore, Fig. 6shows the system cost in different periods. With the considerationof pollutant emission, the behavior of environmental managementwould decrease system benefits. Considering the pollutant emis-sion, the system benefit would be $[17.28, 22.18] � 109 in period1, $[21.70, 23.67] � 109 in period 2, and $[22.58, 25.24] � 109 inperiod 3.

4.4. System benefit

The optimized objective function value (f�opt = $[61.56,71.08] � 109) provides two extremes of net system benefits overthe planning horizon. As the actual value of each of the decisionvariables varies within its lower and upper bounds, the net systembenefit correspondingly changes between f�opt and fþopt . In this case,three IFIP-REM problems are formulated. In this study, a- repre-sents the most optimistic perception, in which the highest pricesof energy resources are estimated. a+ is the pessimistic situation,in which the lowest prices of energy resources are estimated. apresents the medium perception, in which the averaged prices ofthe P levels are provided. Fig. 7 represents optimized system ben-efit under different energy prices. It indicates that a lower energyprice corresponds to a higher system benefit. Although the systemparameters are functional intervals, the results of IFIP-REM are stillconstants crisp intervals.

Fig. 7. System benefit under different energy price.

IFIP inherits the conventional ILP’s advantages in providing flex-ible schemes to decision makers, but not raising the complexity ofthe schemes. Particularly, lower decision variables values withintheir solution intervals should be used under optimistic systemconditions, and higher decision variable values within their solu-tion intervals should be used under pessimistic system conditions.Thus, flexible decision alternatives can be generated from the IFIPsolutions according to projected planning situations.

5. Conclusions

In this study, interval full-infinite programming energy model(IFIP-REM) method has been developed and applied to a hypothet-ical case study of energy management in a regional energy system.IFIP-REM integrates full-infinite programming (FIP) into an intervallinear programming (ILP) framework. The approach can effectivelyreflect uncertainties expressed as crisp intervals and functionalintervals of energy systems. As an extension of the interval num-bers, functional intervals are proposed and incorporated into theILP framework in this study. This becomes a new type of intervalfull-infinite programming problem since both of the objectivesand constraints are infinite. The model is suitable for dealing withenergy systems programming problems. Moreover, the obtainedinterval solutions are useful for generating decision alternatives,which represent various options for environmental-economictradeoffs. IFIP-REM is also useful for supporting decision analysisof energy-related emission control initiatives.

The study is the first attempt to apply an IFIP method to a hypo-thetical energy systems management. It may require futureimprovements in considering more complexities. A major limita-tion of the model is that it can merely deal with functional inter-vals linearly related to its factors. These relationships could benonlinear in many practical environmental management problems,so it is necessary to advance more sophisticated methods to tacklethese uncertainties. In addition, the parameters may be associatedwith multiple factors in energy systems. However, the lower andupper bounds of the functional intervals are assumed to be associ-ated with a single independent variable in this study. In practicalsystems, they may be simultaneously affected by many variables.How to deal with the IFIP problems containing multiple indepen-dent variables will be our future studies.

Acknowledgments

This research was supported by the Major Science and Technol-ogy Program for Water Pollution Control and Treatment(2009ZX07104-004). The authors are grateful to the editors andanonymous reviews for their insightful comments and suggestions.


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An interval full-infinite programming approach for energy systems planning under multiple uncertainties