in

Reginc Pow

Keywords:

tionr sy

focused on providing a sufcient energy supply at minimum cost. Different from using least-cost models,

issues of capacity expansion and random information. It has advantages in balancing conicting objec-

gy syst

methods have been explored, it is still considered difcult to iden-

with uncertainties in the input information, such as future electric-ity demands and resource availabilities. The third is the reectionof dynamic characteristics of facility capacity issues. Therefore,efcient mathematical programming techniques for planning theelectric power systems under these complexities are desired.

were directly quantied by economic indicators and encom-e function. Foras (GHG)-odel, whe

trading of GHG-emission credit was represented throueconomic measure. However, equating the conventionalobjective least-cost optimization framework to real-worldlems involving social and environmental considerations mightlead to difculties in obtaining solutions from a sustainabilityperspective.

For better reecting the multi-dimensionality of the sustain-ability goal, it was increasingly popular to represent the energymanagement problems within a Multiple Objective Programming(MOP) framework [30,31,18,45,1,33]. Thus environmental impactswere also elected as explicit objective functions in the models

Corresponding author. Address: S-C Institute for Energy, Environment andSustainability Research, Resources and Environmental Research Academy, NorthChina Electric Power University, Beijing 102206, China. Tel.: +1 391 146 8225;fax: +1 306 585 4855.

Electrical Power and Energy Systems 53 (2013) 553563

Contents lists available at

Electrical Power an

.e lE-mail address: huang@iseis.org (G.H. Huang).tify sustainable management plans for hybrid electrical systems.The rst challenge is to identify a trade-off between conictingeconomic and environmental concerns. The second is associated

passed in the aggregated least-cost objectivinstance, Li et al. [22] proposed a greenhouse gtion oriented energy system management m0142-0615/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijepes.2013.05.022mitiga-re thegh ansingle-prob-role in the social and economic development of urban communi-ties. At the present, the major energy sources for electricity gener-ation are non-renewable fossil fuels, which could have seriousconsequences for the local and global environment [27]. With thegrowing health and environmental awareness of the people, devel-oping renewable energy sources has gained much attentionthroughout the world [11,44]. Although numerous optimization

lated as single-objective linear programming (LP) problems aim-ing at the minimization of total cost under specic levels ofenvironmental requirements [23]. For example, Zhu et al. [51]developed a municipal-scale energy model for the City of Beijing,where the objective was to minimize the system cost over theplanning horizon, and the constraints included the restrictionsfor air pollutant emissions. Occasionally environmental concernsSustainabilityGeneration expansion planningFractional programmingChance-constrained programmingRenewable energyStochastic uncertainty

1. Introduction

Sustainable management of enertives, handling uncertainty expressed as probability distributions, and generating exible capacity-expansion strategies under different risk levels. The method is applied to an expansion case study ofmunicipal electric power generation system. The obtained solutions are useful in generating sustainablepower generation schemes and capacity-expansion plans. The results indicate that DSFP can supportin-depth analysis of the interactions among system efciency, economic cost and constraint-violationrisk.

2013 Elsevier Ltd. All rights reserved.

ems plays a signicant

Previously, many integrated models were proposed in design-ing environmentally responsible energy management systems[43,28,25,34,38,39]. Classically quite a few models were formu-Accepted 10 May 2013 a more sustainable management approach is to maximize the ratio between renewable energy genera-tion and system cost. The proposed DSFP method can solve such ratio optimization problems involvingDynamic stochastic fractional programmmanagement of electric power systems

H. Zhu a, G.H. Huang b,c,a Institute for Energy, Environment and Sustainable Communities, University of Regina,b S-C Institute for Energy, Environment and Sustainability Research, North China ElectricUniversity of Regina, Regina, Canada S4S 0A2

a r t i c l e i n f o

Article history:Received 9 January 2013Received in revised form 29 April 2013

a b s t r a c t

A dynamic stochastic fracplanning of electric powe

journal homepage: wwwg for sustainable

a, Saskatchewan, Canada S4S 0A2er University, Beijing 102206, China

al programming (DSFP) approach is developed for capacity-expansionstems under uncertainty. The traditional generation expansion planning

SciVerse ScienceDirect

d Energy Systems

sevier .com/locate / i jepes

andbesides the least-cost desire [20]. Correspondingly, multi-criteriadecision making (MCDM) approaches, such as goal programming[36], weighted sum [30,19] and fuzzy multi-objective techniques[33], were widely employed to determine satisfactory compromisesolutions. In general, the approaches to deal with multi-objectiveproblems could be classied into two categories: compromise-pro-gramming and aspiration-analysis approaches. In compromisingprogramming, the importance of each objective was delineatedby weighting factors, and the feasible solution with the shortestoverall distance to the ideal values of objectives is considered mostdesirable [33]. Thus, the solutions determined by compromisingprogramming were highly dependent on the preferences of deci-sion-makers. In contrast, the substance of aspiration approacheswas to develop a single objective programming model throughoptimizing one objective and converting the others into con-straints under certain aspiration levels. This manipulation wasespecially practical for large-scale problems; nevertheless, thetrade-offs of multiple objectives were neglected and the systemcomplexities could not be adequately reected. In general, theseMCDM methods had two major limitations. First, they usuallycombined objectives of multiple aspects into a single measure onthe basis of subjective assumptions. The work of setting weightingfactors or economic indicators inevitably entailed additional dif-culties. Second, they merely focused on system inputs and outputs,and none of them could facilitate analysis of system efcienciesrepresented as output/input ratios.

For the optimization of system efciency, Fractional Program-ming (FP) was used in many management problems[10,41,42,15], in which the optimization of ratio between twoquantities (e.g. output/input, cost/time, or cost/volume) was de-sired. The use of ratio objectives in FP problems assures that onlythe solutions with better achievements per unit of inputs (e.g. cost,resource, time) would be selected. In addition, the scenarios whereFP techniques can be applied are the same as those for LP andMCDM techniques [37,21]. Thus FP can be a natural and powerfultool for studying and analyzing the issues related to sustainabilityin resources management problems. Although applications of FPwere reported in various areas ranging from engineering to eco-nomics [32], the method was seldom applied to energy systemsplanning. Moreover, few studies of FP under uncertainty were re-ported [7,17].

In real-world problems of energy systems management, renew-able energy resources are normally subject to spatial and/or tem-poral uctuations; electricity demands are merely impreciselyestimated [29,2]. The type of uncertainty that attracts major atten-tion is randomness existing in right-hand-side parameters[14,24,50,49]. For example, the availability of solar and wind ener-gies are uncertain and can be expressed as probability distributions[4]. The chance-constrained programming (CCP) is an attractivetool to deal with problems associated with such uncertainties.Zhu and Huang [48] developed a stochastic linear fractional pro-gramming (SLFP) method for supporting sustainable waste man-agement, which incorporated the CCP technique within a FPframework, and could solve ratio optimization problems associ-ated with random information. However, SLFP was not able to dealwith the capacity expansion issues in electric power systems. Onepotential approach to improve SLFP is to integrate the mixed inte-ger linear programming (MILP) technique within its framework.

Therefore, the objective of this study is to develop a dynamicstochastic fractional programming (DSFP) approach for sustainablemanagement of electric power systems. The CCP and MILP tech-niques will be incorporated into a linear fractional programming(LFP) framework. The integrated DSFP approach can not only deal

554 H. Zhu, G.H. Huang / Electrical Powerwith ratio objective, but also reect dynamics of facility expansionunder stochastic uncertainties. The developed method will then beapplied to a case study to demonstrate its advantages. Desiredmunicipal power system management schemes under differentconstraint-violation levels will be obtained, which will help deci-sion makers analyze the interrelationships among renewablepower generation efciency, system risk and many related factors.

2. Methodology

2.1. Mixed integer linear fractional programming

A general linear fractional programming (LFP) problem can beformulated as follows:

Max f x1; x2; . . . ; xn Pn

j1cjxj aPnj1djxj b

1a

s:t:Xn

j1aijxj 6 bi; i 1;2; . . . ;m 1b

xj P 0; j 1;2; . . . ;n 1cwhere aij, bi, cj, dj 2 R; a and b are scalar constants. Assume that thesolution set of model (1) is nonempty and bounded, and the objec-tive function is continuously differentiable.

Charnes and Cooper [8] showed that if the denominator is con-stant in sign (assuming that

Pnj1djxj b > 0) for all X = (x1,x2,

. . . ,xn) on the feasible region, the LFP model can be transformedto the following linear programming problems under transforma-tion xj r xj ("j):

Max gx1; x2; . . . ; xn; r Xn

j1cjxj a r 2a

s:t:Xn

j1aijxj 6 bi r; i 1;2; . . . ;m 2b

Xn

j1djxj b r 1 2c

xj P 0; j 1;2; . . . ;n 2d

r P 0 2eModel (2) can be solved through the usual simplex algorithm.

Thus the optimal solution of Model (1) can be obtained throughtransformation xj xj =r ("j).

In a mixed integer linear fractional programming (MILFP) prob-lem, some decision variables are dened as integers. Thus theproblem can be formulated as:

Max f Pp

j1cjxj Pn

jp1cjyj aPpj1djxj

Pnjp1djyj b

3a

s:t:Xp

j1aijxj

Xn

jp1aijyj 6 bi; i 1;2; . . . ;m 3b

xj P 0; j 1;2; . . . ;pp < n 3c

yj P 0 and yj integer variable; j p 1; . . . ;n 3dIf the denominator is constant in sign (assuming thatPp

j1djxj Pn

jp1djyj b > 0) on the feasible region, the MILFPmodel can be transformed to:

Energy Systems 53 (2013) 553563Max g Xp

j1cjxj

Xn

jp1cjyj a r 4a

ands:t:Xp

j1aijxj

Xn

jp1aijyj 6 bi r; i 1;2; . . . ;m 4b

Xp

j1djxj

Xn

jp1djyj b r 1 4c

yj r yj; j p 1; . . . ;n 4d

xj P 0; j 1;2; . . . ;pp < n 4e

yj P 0 and yj integer variable; j p 1; . . . ;n 4f

r P 0 4gModel (4) can be solved through the branch-and-bound algo-

rithm. The solution of integer variables yj (j = p + 1, . . . ,n) can be ob-tained directly, and the optimal solution of xj (j = 1,2, . . . ,p) can stillbe obtained through transformation xj xj =r (j = 1,2, . . . ,p).

The MILFP model can tackle ratio optimization problems andcapacity-expansion issues; however, it may not be able to dealwith uncertain parameters.

2.2. Dynamic stochastic fractional programming

When some of the right-hand-side parameters in model (3) areof stochastic features and can be represented as probability distri-butions, a dynamic stochastic fractional programming (DSFP) mod-el can be formulated as follows:

Max f Pp

j1cjxj Pn

jp1cjyj aPpj1djxj

Pnjp1djyj b

5a

s:t:PrXp

j1aijxj

Xn

jp1aijyj 6 bitP 1 pi; i 1;2; . . . ;m 5b

xj P 0; j 1;2; . . . ;pp < n 5c

yj P 0 and yj integer variable; j p 1; . . . ;n 5dwhere t 2 T; bi(t) is a random right-hand-side parameter (in con-straint i) dened on a probability space T; pi (pi 2 [0,1]) is a givenlevel of probability for constraint i (i.e. signicance level, which rep-resents the admissible risk of constraint violating).

According to the CCP methods [9], when the left-hand-sidecoefcients [Ai] are deterministic and the right-hand-side coef-cients [bi(t)] are random (for all pi values), the constraintPrAiX 6 bitP 1 pi can be converted as:AiX 6 bitpi 6where bitpi F1i pi, given the cumulative distribution functionof bi [i.e. Fi(bi)] and the probability of violating constraint i (pi).

Constraint (6) is linear and the set of feasible constraints is con-vex. Thus, the CCP method can be used to solve the DSFP model (5)by converting them into deterministic versions through: (i) xing acertain level of probability pi (pi 2 [0,1]) for uncertain constraint i,and (ii) imposing that constraint should be satised with at least aprobability level of 1 pi.

The detailed solution process for DSFP can be summarized asfollows:

Step 1: Formulate the original DSFP model [i.e. model (5)].Step 2: Given a signicance level (pi) for each constraint i, for-

H. Zhu, G.H. Huang / Electrical Powermulate the deterministic MILFP model [i.e. model (3)] by con-verting stochastic constraints [i.e. constraints (5b)] intodeterministic ones:Xp

j1aijxj

Xn

jp1aijyj 6 bitpi; i 1;2; . . . ;m 5b0

where bitpi represents corresponding values given the cumula-tive distribution function of right-hand-side parameter bi and theprobability of violating constraint i (pi), bitpi F1i pi .

Step 3: Formulate and solve the transformation model [i.e.model (4)], and obtain solutions for xj (j = 1, 2, . . ., p), yj(j = p + 1, . . ., n) and r.Step 4: Calculate the solutions for xj: xj xj =r (j = 1,2, . . . ,p).Step 5: Repeat steps 24 under different pi levels.

3. Case study

3.1. Overview of study system

The developed DSFP method is applied to a case study of long-term planning of a regional electric power system with representa-tive data within a Chinese context [4,5,26,46]. In the study system,there is an independent municipal electricity grid, where the elec-tricity can be converted from both fossil fuels (coal and natural gas)and renewable energies (hydro, solar and wind). With these energycarriers, ve types of electricity-generation facilities (coal-red,natural gas-red, hydropower, solar energy and wind power) areemployed by a wholly state-owned electric power company inthe regional power supply system. With the economic develop-ment, the total electricity demands of residential, commercialand industrial end-users would rise accordingly. When the existingpower-generation capacities are insufcient to meet the increasingenergy demands, capacity expansion will become necessary. Thedecision makers are responsible for planning the capacity expan-sions for these facilities and allocating energy resources/servicesfor electricity supplies through multiple technologies. To reectthe dynamic features of the study system, three time periods (5-year for each period) are considered in a 15-year planning horizon.

Currently, the electricity generation in the study system pri-marily relies on non-renewable fossil fuels, which are only avail-able by extraction or renery with limited supplies. When thedomestic supplies cannot satisfy energy demands, additionalamount of coal and natural gas may be imported with higherprices. Moreover, carbon dioxide (CO2) from fossil fuels red powerplants is a signicant source of greenhouse gas (GHG) emissions,which may contribute to climate change and global warming[47]. Burning fossil fuels also produces other air pollutants, suchas sulfur dioxide (SO2), nitrogen oxides (NOx), volatile organic com-pounds (VOCs) and heavy metals. In contrast, most of renewableenergy sources are environmentally friendly, and allow the regionto increase its independency on the primary energy supplies. Withthe increasing concerns of environmental protection and resourcesconservation, the power generation technologies based on renew-able energy resources are encouraged. Therefore, the planners de-sire optimal capacity expansion plans and sustainable resourcesallocations, which can achieve maximized renewable power gener-ation with minimized system cost.

Table 1 presents the average market prices for domestic or im-ported fossil fuels and operation costs for electricity generation inthe three periods. The capital investment costs and capacity expan-sion options for power-generation facilities are listed in Table 2.Every facility could expand its capacity once per period, wherethree capacity expansion options are provided for each facility.Once capacity expansions of the facilities are decided for a period,

Energy Systems 53 (2013) 553563 555they would be implemented at the beginning of the period. Thus,the increased facility capacities could be employed for the currentand the following periods. Table 3 provides electricity demands

i: Primary energy resources (i = 1,2,3,4);j: Power-generation facilities (j = 1,2, . . . ,5);k: Time periods (k = 1,2,3);m: Capacity-expansion options (m = 1,2,3).

XjkYjmk

Table 1Costs for energy supply and electricity generation.

Time period

k = 1 k = 2 k = 3

6

Hydropower (j = 3) 0.485 0.485 0.485Solar (j = 4) 0.00975 0.00975 0.00975

556 H. Zhu, G.H. Huang / Electrical Power and Energy Systems 53 (2013) 553563Wind (j = 5) 1.3745 1.3745 1.3745

Table 2Capacity expansion options and costs for power-generation facilities.

Time period

k = 1 k = 2 k = 3

Capacity expansion options, Vjmk (GW)Energy supply cost, Pik ($10 /PJ)Coal (i = 1) 2.56 2.76 2.96Imported coal (i = 2) 3.20 3.50 3.80Natural gas (i = 3) 4.89 5.09 5.29Imported natural gas (i = 4) 6.29 6.59 6.89

Electricity generation cost, Cjk ($106/PJ)

Coal (j = 1) 0.1465 0.1465 0.1465Natural gas (j = 2) 0.5755 0.5755 0.5755over the planning horizon, which can be presented as probabilitydistributions under different pi levels. Table 4 presents the existingand allowable capacities of power-generation facilities and theavailabilities of energy resources. Due to the natural variations ofrenewable energy resources, the distribution information of allow-able capacities for solar energy and wind power facilities under dif-ferent probability levels of constraint violation (pi) is given inTable 5.

3.2. The DSFP model for regional power systems planning

In the study region, multiple primary energy sources (domesticand imported production), multiple technologies (conventionaland renewable electricity generation), and multiple facilities exist.Thus the following indices will be used:

Theunit olationships among the decision variables and system conditions/

follow

(a)

Coal (j = 1) m = 1 0.04 0.04 0.04m = 2 0.06 0.06 0.06m = 3 0.08 0.08 0.08

Natural gas (j = 2) m = 1 0.012 0.012 0.012m = 2 0.026 0.026 0.026m = 3 0.036 0.036 0.036

Hydropower (j = 3) m = 1 0.006 0.006 0.006m = 2 0.008 0.008 0.008m = 3 0.01 0.01 0.01

Solar (j = 4) m = 1 0.004 0.004 0.004m = 2 0.008 0.008 0.008m = 3 0.012 0.012 0.012

Wind (j = 5) m = 1 0.006 0.006 0.006m = 2 0.01 0.01 0.01m = 3 0.016 0.016 0.016

Capital cost, Ejmk ($106/GW)

Coal (j = 1) 771 721 671Natural gas (j = 2) 826 776 726Hydropower (j = 3) 3240 3040 2840Solar (j = 4) 4500 4300 4100Wind (j = 5) 1464 1364 1264

Table 3Electricity demands [Dk(t), PJ] in period k under different pi levels.

pi level 0.01 0.05 0.1 0.25

D1(t) 9.450 12.171 13.624 16.052D2(t) 17.715 20.441 21.894 24.322D3(t) 25.890 28.611 30.064 32.492X4

i1

X3

k1Pik Zik

(b) xed and variable operating costs for power generation:

X5

j1

X3

k1Cjk Xjk

(c) capital (investment) costs for facility expansions:

X5

j1

X3

m1

X3

k1Ejk Vjmk Yjmk

where

Pik = average cost for primary energy supply i in period k ($106/PJ);Cjk = average variable and maintenance cost for generating elec-tricity from facility j in period k ($106/PJ);Ejk = capital cost for capacity expansion of facility j within per-iod k ($106/GW);Vjmk = capacity expansion for facility j with option m in period k(GW).Thus, the ratio objective of DSFP model can be formulated as

follows:

0.5 0.75 0.9 0.95 0.99

18.750 21.448 23.876 25.329 28.050

27.0235.19ing:

costs for primary energy supply:factors. In detail, the system cost is formulated as a sum of theobjective is to maximize renewable power generation perf system cost, while a series of constraints dene the interre-for facility j with option m in period k will be installed or not;Zik: Supply of primary energy resource i in period k (PJ).: Electricity generation at facility j in period k (PJ);: Integer variable (=1 or 0) representing capacity expansionFor effective management of various energy resources and facil-ities, these system components and their interactions need to beconsidered in an integrated modeling framework. The difcultiesof planning this regional power system include: (a) how to inden-tify optimal capacity-expansion schemes for power-generationfacilities; (b) how to maximize renewable power generation witha possible low system cost; (c) how to reect the stochastic fea-tures of uncertain parameters (e.g. electricity demands, allowablecapacities for solar-energy and wind-power facilities); and (d)how to analyze the tradeoff between the system efciency andthe risk of system reliability. Thus, the developed DSFP method willbe applied to the planning of such a system. The decision variablesinclude:0 29.718 32.146 33.599 36.3250 37.888 40.316 41.769 44.490

Table 4Capacities of power-generation facilities and availabilities of energy resources.

Power-generation facilities Existing capacity (GW) Allowable capacity (GW) Energy resources Availability (PJ)

Coal (j = 1) 0.06 0.18 Coal (i = 1) 100Natural gas (j = 2) 0.04 0.092 Imported coal (i = 2) 60Hydropower (j = 3) 0.02 0.036 Natural gas (i = 3) 40Solar (j = 4) 0.01 VC4(t) Imported natural gas (i = 4) 30Wind (j = 5) 0.01 VC5(t)

Table 5Allowable capacities (GW) for solar energy and wind power facilities under different pi levels.

pi level 0.01 0.05 0.1 0.25 0.5 0.75 0.9 0.95 0.99

Solar energy, VC4(t) 0.0322 0.0334 0.0340 0.0350 0.0362 0.0374 0.0384 0.0390 0.0402Wind power, VC5(t) 0.0301 0.0333 0.0350 0.0379 0.0411 0.0443 0.0472 0.0489 0.0521

Table 6Results of the DFSP model.

Period pi = 0.01 pi = 0.05 pi = 0.10 pi = 0.25

Power generation, Xjk (PJ)Coal-red (j = 1) k = 1 13.543 9.461 8.424 5.995

k = 2 13.642 14.993 12.909 9.461k = 3 21.153 22.075 18.922 15.768

Natural gas-red (j = 2) k = 1 3.154 3.570 3.154 3.154k = 2 7.546 3.154 3.154 3.543k = 3 8.199 4.242 5.311 5.406

Hydropower (j = 3) k = 1 4.730 4.730 4.730 4.730k = 2 5.676 5.676 5.676 5.676k = 3 5.676 5.676 5.676 5.676

Solar energy (j = 4) k = 1 3.469 3.469 3.469 3.469k = 2 4.730 4.730 5.361 5.361k = 3 4.730 4.730 5.361 5.361

Wind power (j = 5) k = 1 3.154 4.100 4.100 4.100k = 2 4.730 5.046 5.046 5.676k = 3 4.730 5.046 5.046 5.676

Primary energy supply, Zik (PJ)Coal (i = 1) k = 1 0 0 0 16.493

k = 2 29.982 26.931 37.369 31.315k = 3 70.018 73.069 62.631 52.192

Imported coal (i = 2) k = 1 44.829 31.315 27.882 3.352k = 2 15.171 22.696 5.360 0k = 3 0 0 0 0

Natural gas (i = 3) k = 1 0.321 8.995 7.947 7.947k = 2 19.016 7.947 7.947 8.929k = 3 20.662 10.689 13.384 13.623

Imported natural gas (i = 4) k = 1 7.626 0 0 0k = 2 0 0 0 0k = 3 0 0 0 0

Capacity expansion,P3

m1Vjmk Yjmk (GW)Coal-red (j = 1) k = 1 0.04 0 0 0

k = 2 0 0.04 0.06 0k = 3 0.04 0.04 0 0.04

Natural gas-red (j = 2) k = 1 0 0 0 0k = 2 0.012 0 0 0k = 3 0 0 0 0

Hydropower (j = 3) k = 1 0.01 0.01 0.01 0.01k = 2 0.006 0.006 0.006 0.006k = 3 0 0 0 0

Solar energy (j = 4) k = 1 0.012 0.012 0.012 0.012k = 2 0.008 0.008 0.012 0.012k = 3 0 0 0 0

Wind power (j = 5) k = 1 0.01 0.016 0.016 0.016k = 2 0.01 0.006 0.006 0.01k = 3 0 0 0 0

Renewable power generation (PJ) 41.628 43.204 44.466 45.727Non-renewable power generation (PJ) 67.237 57.495 51.874 43.327System cost ($106) 1019.724 878.128 817.760 711.100Renewable power generation / system cost (PJ per $109) 40.822 49.200 54.375 64.305

H. Zhu, G.H. Huang / Electrical Power and Energy Systems 53 (2013) 553563 557

pi = 0.01

PoweCoal

21.153Natu 3.154

7.5468.1994.730 4.730 4.730 4.730

3.4694.7304.7303.154

558 and Energyk = 3Hydropower (j = 3) k = 1

k = 2k = 3

Solar energy (j = 4) k = 1k = 2k = 3

Wind power (j = 5) k = 1k = 2k = 3Max f

The co(1)

(2)

PrimCoal

Impo

Natu

Impo

CapaCoal

Natu

Hydr

Solar

Wind

ReneNon-SysteRenePeriod

r generation, Xjk (PJ)-red (j = 1) k = 1

k = 2k = 3

ral gas-red (j = 2) k = 1k = 2Table 7Results of the least-cost model.H. Zhu, G.H. Huang / Electrical Power renewable power generationsystem cost

P5

j3P3

k1XjkP4i1P3

k1Pik ZikP5

j1P3

k1Cjk XjkP5

j1P3

m1P3

k1Ejk Vjmk Yjmk7a

nstraints of DSFP are dened as follows:Mass balance constraints of fossil fuels:

X1k CR1 6X2

i1Zik; 8k 7b

X2k CR2 6X4

i3Zik; 8k 7c

Capacity constraints for electricity generation:

ECj X3

m1

Xk

p1Vjmp Yjmp UcapP Xjk; 8j; k 7d

ary energy supply, Zik (PJ)(i = 1) k = 1

k = 2k = 3

rted coal (i = 2) k = 1k = 2k = 3

ral gas (i = 3) k = 1k = 2k = 3

rted natural gas (i = 4) k = 1k = 2k = 3

city expansion,P3

m1Vjmk Yjmk (GW)-red (j = 1) k = 1

k = 2k = 3

ral gas-red (j = 2) k = 1k = 2k = 3

opower (j = 3) k = 1k = 2k = 3

energy (j = 4) k = 1k = 2k = 3

power (j = 5) k = 1k = 2k = 3

wable power generation (PJ)renewable power generation (PJ)m cost ($106)wable power generation / system cost (PJ per $109)4.730 5.046 5.046 5.6764.730 5.046 5.046 5.676(3)

(4)

(5)

(6)

029.98270.01844.82915.17100.32119.01620.6627.62600

0.0400.0400.01200.010.00600.0120.00800.010.010

41.62867.2371019.7240.8223.469 3.469 3.4693.469 4.730 4.7303.469 4.730 4.7304.100 4.100 4.1005.676 5.676 5.676 5.6765.676 5.676 5.676 5.67622.075 18.922 15.7683.570 3.154 3.1543.640 3.154 3.1545.503 5.942 6.03713.543 9.461 8.424 5.99513.642 15.768 13.540 10.481pi = 0.05 pi = 0.10 pi = 0.25Systems 53 (2013) 553563Electricity demand constraints:

PrX5

j1Xjk P DktP 1 pk;D; 8k 7e

Energy resources constraints:

X3

k1Zik 6 UPi; 8i 7f

Technical constraints of electricity generation:

Xjk P dj ECj Ucap; 8j; k 7gCapacity expansion constraints:

ECj X3

k1

X3

m1Vjmk Yjmk 6 VCj; j 1;2;3 7h

0 0 13.11626.931 37.370 34.69273.069 62.630 52.19231.315 27.882 6.72925.261 7.448 00 0 08.995 7.947 7.9479.173 7.947 7.94713.868 14.974 15.2120 0 00 0 00 0 0

0 0 00.04 0.006 0.040.04 0 00 0 00 0 00 0 00.01 0.01 0.010.006 0.006 0.0060 0 00.012 0.012 0.0120 0.008 0.0080 0 00.016 0.016 0.0160.006 0.006 0.010 0 0

40.681 43.204 44.46660.017 53.136 44.589

4 876.857 816.718 711.06946.395 52.900 62.534

9506 )

andOptimal-ratio Least-cost(7)

(8)

(9)

whereCR1CR2

Fig. 1.models700p = 0.01 p = 0.05 p = 0.1 p = 0.25 750800

850

900

Syst

em c

ost (

$1010001050

H. Zhu, G.H. Huang / Electrical PowerAvailability constraints of renewable energy resources:

PrECjX3

k1

X3

m1Vjmk Yjmk6VCjtP1pj;VC ; j4;5 7i

Expansion option constraints:

X3

m1Yjmk 6 1; 8j; k 7j

Yjmk 0 or 1; 8j;m; k 7kNon-negativity constraints:

Zik P 0; 8i; k 7l

Xjk P 0; 8j; k 7m

= converting efciency from coal to electricity;= converting efciency from natural gas to electricity;

(A) System cost

(B) Renewable power generation per unit of cost

p = 0.01 p = 0.05 p = 0.1 p = 0.25

Optimal-ratio Least-cost

35.0

45.0

55.0

65.0

Ren

ewab

le p

ower

gen

erat

ion

/ cos

t (PJ

/ $1

09)

Comparison of system performances from optimal-ratio and least-cost.ECj = existing capacity of power-generation facility j (GW);Ucap = conversion coefcient from power-generation capacityto energy (GW to PJ);Dk(t)= total electricity demand in period k (PJ);UPi = available resource for primary energy i (PJ);dj = facility utilization rate (%);VCj = allowable capacity for power-generation facility j (GW);VCj(t) = allowable power-generation capacity for solar energy(j = 4) and wind power (j = 5) facilities (GW);pk,D = constraint-violation probability for electricity demandconstraints;pj,VC = constraint-violation probability for availability con-straints of solar energy (j = 4) and wind power (j = 5).

4. Results and discussion

Table 6 shows the results of the DSFP model under different pilevels, which include the power generation patterns, primary en-ergy supply strategies and capacity expansion plans. For example,when pi = 0.01, the electricity generation through coal-red, natu-ral gas-red, hydropower, solar energy and wind power technol-ogies in period 1 would respectively be 13.543, 3.154, 4.730,3.469 and 3.154 PJ; the primary energy supplies of coal, importedcoal, natural gas and imported natural gas in period 1 wouldrespectively be 0, 44.829, 0.321 and 7.626 PJ; the capacity-expan-sion levels for coal-red, natural gas-red, hydropower, solar en-ergy and wind power facilities at the beginning of period 1 wouldrespectively be 0.04, 0, 0.01, 0.012 and 0.01 GW. Similarly, theplans for the three periods under different pi levels can beinterpreted.

The results of power generation indicate that renewable elec-tricity supply would be insufcient for the future energy demands.Over the planning horizon, coal-red electricity would increasesteadily and still play an important role in the power system dueto its high availability and competitive price. In comparison, gener-ating electricity from natural gas is more expensive; thus the nat-ural gas-red facility would only be expanded under somedemanding conditions (i.e., when the electricity demands cannotbe guaranteed by expansions of the other energy generation facil-ities). For instance, a capacity of 0.012 GW would be added to thenatural gas-red facility at the beginning of period 2 whenpi = 0.01, while no capacity expansion would be taken for this facil-ity under pi = 0.05, 0.1 and 0.25. For the facilities relying on renew-able energy resources, capacity expansions would be conducted inthe rst two periods for efcient increase in sustainable electricitygeneration. Under various constraint-violation risk levels, thehydropower facility would not only have similar capacity expan-sion plans (i.e. expanded by 0.01 GW at the beginning of period1, and 0.006 GW at the beginning of period 2), but also have similarelectricity generation patterns (i.e. 4.730, 5.676 and 5.676 PJ inperiods 1, 2 and 3). These points of similarity among the energymanagement schemes corresponding to different pi levels indicatethat the hydropower-related results are not sensitive to the uncer-tain system inputs. In contrast, the results related to solar energyand wind power are very sensitive due to natural variations inthe availabilities of these energy resources.

The DSFP results in Table 6 also reect that a higher ratioobjective and a lower system cost would be obtained under ahigher pi level. For example, when pi is raised from 0.01 to0.05, the ratio objective would be increased from 40.822 to49.200 PJ per $109, and the system cost would be decreasedfrom $1019.724 106 to $878.128 106. The pi level representsthe probability of constraint violation, and the ratio objective

Energy Systems 53 (2013) 553563 559indicates the system efciency (i.e. renewable power generationper unit of system cost). Thus, the relationship between theratio objective and pi can reect the interrelationships among

l

l

al-f

andOptimal-ratio mode

ansi

on (G

W)

Cap

acity

exp

ansi

on (G

W)

k = 1 k = 2 k = 3 k = 1

Optimal-ratio mode

Co

p = 0.05

p = 0.05

560 H. Zhu, G.H. Huang / Electrical Powersystem efciency, economic cost, and system reliability (i.e.energy-supply security and constraint-violation risk). A lowerpi level means a lower admissible risk of violating the con-straints, which leads to an increased strictness for the con-straints and thus a shrunk decision space. When theavailabilities of renewable energy resources are restricted undera lower pi level, more electricity would be generated from non-renewable resources to meet the increased electricity demands,leading to larger capacity expansions for conventional facilities.Thus an alternative of lower system efciency (i.e. higher sys-tem costs and lower renewable power generation) would be ob-tained, but the reliability of meeting demanding conditionswould also increase. On the contrary, decisions under a higherpi level would loosen the electricity demands and allow morepower generation from renewable energy resources, leading toan alternative of decreased reliability but higher system ef-ciency. Therefore, the DSFP approach can provide useful solu-tions associated with different risk levels, which will helpdecision makers identify desired electricity production andcapacity-expansion plans under various system conditions. Gen-erally, planning under a lower pi level would be suitable fordemanding conditions, whereas planning under a higher pi canbe used under advantageous conditions.

Besides the scenario of maximizing the system efciency, an-other scenario of minimizing the system cost is also analyzed toevaluate the effects of different energy supply polices. The opti-mal-ratio problem presented in Model (7) can be converted intoa least-cost problem with the following objective:

Solar e

Cap

acity

exp

k = 1 k = 2 k = 3 k = 1

Fig. 2. Comparison of capacity expansion schemeLeast-cost model

k = 2 k = 3 k = 1 k = 2 k = 3

Least-cost model

ired facility

p = 0.1 p = 0.25

p = 0.1 p = 0.25

Energy Systems 53 (2013) 553563Min f system cost X4

i1

X3

k1Pik Zik

X5

j1

X3

k1Cjk Xjk

X5

j1

X3

m1

X3

k1Ejk Vjmk Yjmk 7a0

Thus, the obtained least-cost model is a mixed-integer linearprogramming (MILP) problem containing stochastic right-handparameters, which can still be solved by CCP method. Table 7 pre-sents results of the least-cost model under different pi levels, whichcan be compared with those of the optimal-ratio model as shownin Table 6. When pi = 0.01, the obtained energy managementschemes under these two scenarios are totally identical; however,when pi takes other values (i.e. pi = 0.05, 0.10 or 0.25), differencescan be found between the results of two scenarios. Obviously,the optimal solutions corresponding to two different objectives[i.e. (7a) and (7a0)] are generally different except when the feasiblesolutions dened by the constraints [i.e. (7b)(7m)] are controlledunder an extremely low constraint-violation risk (i.e. the solutionspace are severely restricted).

Fig. 1 compares the system performances corresponding to theoptimal-ratio and least-cost scenarios under various pi levels. Asindicated in Fig. 1A, the system costs obtained from these twomodels would both decrease with a raised pi level. Although theleast-cost model achieves slightly lower system costs whenpi = 0.05, 0.10 and 0.25, the differences in system cost betweentwo scenarios are not as obvious as those in system efciency. Asindicated in Fig. 1B, the optimal-ratio model can lead to signicant

nergy facilityk = 2 k = 3 k = 1 k = 2 k = 3

s from optimal-ratio and least-cost models.

andH. Zhu, G.H. Huang / Electrical Powerhigher system efciencies under these pi levels. For example, whenpi = 0.05, the least-cost model leads to a system cost being 0.145%lower than the optimal-ratio model; while the optimal-ratio modelresults in a system efciency being 6.046% higher than the least-cost model.

Moreover, when pi = 0.05, 0.10 and 0.25, the differences be-tween the obtained results under two energy supply scenariosare demonstrated in Figs. 2 and 3. Fig. 2 indicates that, even withthe same conditions, a higher capacity of the solar energy facilitywould be achieved in periods 2 and 3 under the optimal-ratio sce-nario. For example, when pi = 0.05, the solar energy facility wouldhave no capacity expansion in periods 2 and 3 under the least-cost

Fig. 3. Comparison of power-generation patterns froEnergy Systems 53 (2013) 553563 561scenario; whereas it would have the second capacity expansion inperiod 2 under the optimal-ratio scenario. In comparison, the coal-red facility would have similar capacity expansion schemes underthe two scenarios, except the case when pi = 0.25 (a higher capacitywould be necessary for period 2 under the least-cost scenario).Accordingly, as shown in Fig. 3, with the same pi level, the opti-mal-ratio model leads to a relatively higher percentage of solar en-ergy supply; in comparison, the least-cost model leads to relativelyhigher percentages of coal-red and natural gas-red electricitysupplies. In addition, these two models lead to same energy supplyschemes for hydropower and wind power facilities. Thus the ratiosof renewable electricity to the total supplied electricity under the

m (A) optimal-ratio and (B) least-cost models.

solved via a linear fractional goal programming model. Forest Ecol Manage

use planning in a tourism district of India. J Environ Inf 2011;17(1):1524.[20] Karaki SH, Chaaban FB, Al-Nakhl N, Tarhini KA. Power generation expansion

andoptimal-ratio scenario (i.e. 42.90%, 46.16% and 51.35% whenpi = 0.05, 0.10 and 0.25) would be signicantly higher than thoseunder the least-cost scenario (i.e. 40.40%, 44.85% and 49.93% whenpi = 0.05, 0.10 and 0.25).

The above scenario analysis could help the energy system man-agers to identify desired electricity supply options under variouselectricity supply policies. When the manager aims to increaserenewable power generation while keeping a relatively low systemcost, the DSFP model can be a particularly effective tool for identi-fying schemes with optimal system efciency under various condi-tions. The DSFP results can effectively support energy systemsmanagement in (a) balancing conicting objectives, (b) providingdesired capacity-expansion plans, and (c) analyzing the interac-tions among system efciency, economic cost and energy-supplyreliability. To plan a practical power system expansion, detailedanalytical studies are generally necessary to simulate the opera-tion of the projected system under the future conditions [3]; multi-criteria decision analysis (MCDA) techniques [12] would also beuseful for comparing the attributes (e.g. reliability levels, price ofelectricity) obtained/estimated from various plans and thus select-ing the robust plan.

As the rst attempt to develop a new fractional programmingapproach for long-term capacity-expansion planning of electricpower systems under uncertainty, DSFP still has some limitations.Because additional uncertainties in the objective parameters (e.g.fuel prices, capital costs) would make the system planning consid-erably more complex, DSFP only considers the random right-hand-side parameters in constraints (e.g. electricity demands, renewableenergy resource availability). In the future study, DSFP can be fur-ther improved by integrating other uncertain optimization meth-ods within its framework, such as interval programming [16,13]and fuzzy mathematical programming [40]. Furthermore, whenDSFP is applied to more complicated cases in a competitive elec-tricity market environment, the electricity prices should be inputalong the planning horizon, and advanced decision-making tech-niques such as game theory [6] and genetic algorithms [35] mayneed to be incorporated into the modeling frame to deal with theinteraction between the decisions of the different generationagents.

5. Conclusions

A dynamic stochastic fractional programming (DSFP) approachhas been developed for supporting sustainable electric power sys-tems planning under uncertainty. The DSFP method can solve ratiooptimization problems involving issues of capacity expansion andrandom information, where techniques of chance-constrained pro-gramming (CCP) and mixed integer linear programming (MILP) areintegrated into a linear fractional programming (LFP) framework.An effective solution method has also been explored to tackle thisintegrated model. The proposed DSFP approach has advantages in:(a) balancing conicting objectives, (b) optimizing system ef-ciency, (c) handling stochastic information expressed as probabil-ity distributions, and (d) generating exible capacity-expansionstrategies under different risk levels.

Applicability of the DSFP method has been demonstratedthrough an expansion case study of municipal electric power gen-eration system. The obtained solutions are effective in generatingsustainable power generation schemes and capacity expansionplans with maximized system efciency under various con-straint-violation risks. The results indicate that DSFP can incorpo-rate valuable stochastic information into the process of decisionmaking, and generate dynamic management schemes with differ-

562 H. Zhu, G.H. Huang / Electrical Powerent levels of system reliability. Furthermore, it can supportin-depth analysis of the interactions among system efciency, eco-nomic cost and constraint-violation risk.planning with environmental consideration for Lebanon. Int J Electr PowerEnergy Syst 2002;24:6119.

[21] Lara P, Stancu-Minasian IM. Fractional programming: a tool for the assessmentof sustainability. Agric Syst 1999;62:13141.

[22] Li GC, Huang GH, Lin QG, Zhang XD, Tan Q, Chen YM. Development of a GHG-mitigation oriented inexact dynamic model for regional energy systemmanagement. Energy 2011;36:338898.

[23] Li YP, Huang GH, Chen X. Planning regional energy system in association withgreenhouse gas mitigation under uncertainty. Appl Energy2011;88(3):599611.2006;227:7988.[16] Huang GH, Niu YT, Lin QG, Zhang XX, Yang YP. An interval-parameter chance-

constraint mixed-integer programming for energy systems planning underuncertainty. Energy Sources Part B: Econ Plann Policy 2011;6(2):192205.

[17] Hladik M. Generalized linear fractional programming under intervaluncertainty. Eur J Oper Res 2010;205:426.

[18] Hsu GJY, Chou FY. Integrated planning for mitigating CO2 emissions in Taiwan:a multi-objective programming approach. Energy Policy 2000;28:51923.

[19] Jeganathan C, Roy PS, Jha MN. Multi-objective spatial decision model for landThis study attempts to provide a modeling framework for solv-ing ratio optimization problems involving issues of capacity expan-sions and random inputs. Although the proposed method is for therst time applied to the energy management led, it can be a pow-erful decision tool for other resources management problems. TheDSFP could be further enhanced through incorporating intervalanalysis methods and fuzzy set theory into its framework.

Acknowledgements

This research was supported by the Program for Innovative Re-search Team (IRT1127), the MOE Key Project Program (311013),and the Natural Science and Engineering Research Council ofCanada.

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H. Zhu, G.H. Huang / Electrical Power and Energy Systems 53 (2013) 553563 563

Dynamic stochastic fractional programming for sustainable management of electric power systems1 Introduction2 Methodology2.1 Mixed integer linear fractional programming2.2 Dynamic stochastic fractional programming

3 Case study3.1 Overview of study system3.2 The DSFP model for regional power systems planning

4 Results and discussion5 ConclusionsAcknowledgementsReferences