Bioethanol supply chain system planning under supply and demand uncertainties

Transportation Research Part E 48 (2012) 150–164

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Bioethanol supply chain system planning under supply and demand uncertainties Chien-Wei Chen a, Yueyue Fan b,⇑ a b

Institute of Transportation Studies, University of California, Davis, CA 95616, United States Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, United States

a r t i c l e

i n f o

Article history: Received 23 February 2011 Received in revised form 3 June 2011 Accepted 24 July 2011

Keywords: Energy supply chain planning Stochastic programming Decomposition Cellulosic biofuel Biowastes

a b s t r a c t A mixed integer stochastic programming model is established to support strategic planning of bioenergy supply chain systems and optimal feedstock resource allocation in an uncertain decision environment. The two-stage stochastic programming model, together with a Lagrange relaxation based decomposition solution algorithm, was implemented in a realworld case study in California to explore the potential of waste-based bioethanol production. The model results show that biowaste-based ethanol can be a viable part of sustainable energy solution for the future. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction A sustainable energy future calls for a more diversified energy portfolio that could alleviate the pressing issues of oil dependence and greenhouse gas emission. Bioenergy has been strongly promoted by US federal policy as part of the solution (US Congress, 2007). However, the challenge of realizing cost-effective energy solutions with minimal impact on food and other natural resource supplies has not been thoroughly investigated (International Energy Agency, 2006; United Nation, 2007). In this study, we emphasize on lignocellulosic biomass as an ideal feedstock source compared to corn grain for its following advantages (Farrell et al., 2006; Hill et al., 2006; Jenkins et al., 2007): better efficiency in terms of life-cycle environmental performance, higher per-acre ethanol yields, lower impact on land use and agriculture, and the variety of resources. A biofuel pathway concerns all the facilities and operations involved in the supply chain, including feedstock resources, production and delivery infrastructures, and the end users. The true potential of bioenergy at a sustainable level needs to be sought through rigorous system analyses for the entire energy supply system. Such a system approach requires an integrated knowledge in alternative energy technologies, spatial economics, and operations research. Some existing studies attempt to separately analyze individual process of a bioenergy pathway, such as cost estimation for feedstock processing and transportation (Atchison and Hettenhaus, 2004; Graham et al., 2000; Hamelinck et al., 2005; Kumar et al., 2005; Mahmudi and Flynn, 2006) and economic feasibility analysis of the conversion technologies (Kaylen et al., 2000; Kumar et al., 2003; Petrolia, 2008; Wallace et al., 2005; Zhan et al., 2005). However, it has become evident that the cost-effectiveness and life-cycle-impact of biofuel production depends on the design of the entire biofuel supply chain (Farrell et al., 2006; Hill et al., 2006). The efficiency of the entire supply system depends on the geography of the feedstock

⇑ Corresponding author. Tel.: +1 530 754 6408; fax: +1 530 752 7872. E-mail addresses: [email protected] (C.-W. Chen), [email protected] (Y. Fan). 1366-5545/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.tre.2011.08.004

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resources, the layout and operation of the biorefineries, and the cost of accessing the energy market. These factors are not independent of each other. For example, a larger-size biorefinery may provide better energy conversion efficiency due to economy of scale, but may impose higher transportation cost due to the need for accessing a more dispersed biomass resource supply. A few studies emphasized on optimizing biofuel supply problems from a supply chain perspective considering both strategic- and tactical-level decisions (Eksßiog˘lu et al., 2009, 2010; Gunnarsson et al., 2004; Sokhansanj et al., 2006; Tembo et al., 2003; Zhu et al., 2011). Most of these studies focused only on the upstream of the supply chain from biomass feedstock to refineries. The models developed in Eksßiog˘lu et al. (2009, 2010) included the supply chain from biomass resource all the way to biofuel terminals, which are probably the most comprehensive (in terms of supply chain echelon) studies available in the literature so far. Besides establishing system-oriented decision tools for biomass and biofuel logistics, these studies also contributed to the literature by establishing multi-period models to incorporate seasonal variation of biomass supplies. In addition to the system dependencies, uncertainty is another major challenge in long-term strategic planning of biofuel supply systems. Cellulosic biofuels, compared with conventional fuels, face more uncertainties in future feedstock supply and biofuel demand, due to unpredictable weather conditions (Persson et al., 2009) and changing regulations and policies. For example, Fig. 1a shows how some of the biomass yields in California fluctuate over 1999–2008 (normalized by the 10year average). Fig. 1b shows different demand projections under different environmental policy scenarios (Yeh et al., 2008). Despite of the importance of addressing uncertainties in biofuel supply system planning as identified in Eksioglu et al. (2009) and IEA (2006), there is only one stochastic model in biofuel supply chain literature (Cundiff et al., 1997), which focused only on storage facilities for herbaceous biomass. The goal of this study is to establish a stochastic model that can be used to provide reliable solutions for the design of the entire biofuel supply chain under potential future supply and demand uncertainties. To handle uncertainties, a commonly used engineering approach is to examine each scenario separately. This is also called wait-and-see approach (Birge and Louveaux, 1997), as if one could wait and see the actual realization of random events and then make decisions accordingly. Another simple approach is to aggregate all scenarios to a single scenario (such as using expected value) and then solve the corresponding deterministic problem. Solution produced by this approach is called expected-value solution. These deterministic approaches are conceptually and computationally simple, but may generate unreliable solutions. For example, a wait-and-see solution may perform well in one scenario, but may cause extremely bad consequence (very costly or even infeasible) in other possible scenarios. In this study, we emphasize on developing a stochastic approach that hedges well against a wide range of future possibilities. A mixed integer stochastic programming model is developed to achieve the least expected system cost. Optimal strategies on bioethanol production, feedstock procurement, and fuel delivery are solved simultaneously within the integrated system. The stochastic mathematical model is used to evaluate the economic feasibility and system robustness in a case study of California. Specific questions to be answered via the model include:  Can ethanol converted from wastes be part of a sustainable energy solution that is economically viable and environmentally acceptable?  What are the infrastructure requirements to support the production and delivery of such a bioethanol system?

Fig. 1a. 1999–2008 California biomass yields (normalized by the 10-year average).

Fig. 1b. Transportation energy demand projection under different environmental policy scenarios.

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 What is the potential risk caused by imperfect information of the future supply and demand, and how might we reduce such risk? Stochastic programming was first introduced by Dantzig (1955) to handle uncertainty with mathematical programming, and was further developed by other researchers with contributions in both theory and computation (Van Slyke and Wets, 1969; Wets, 1966, 1974). As pointed out in a recent comprehensive review of strategic supply chain management (Melo et al., 2009), realistic models that address uncertainties and dynamics are few. This study provides a real-world case study addressing uncertainties involved in strategic planning of renewable energy supply systems. The rest of the paper is organized as follows. Model structure and formulation are presented in Sections 2. Section 3 focuses on the implementation of a decomposition-based solution method. The background of California case study, results and sensitivity analyses are provided in Section 4. Conclusions and future research will be discussed in Section 5. 2. Mathematical model In the biofuel supply system planning problem, the planning decisions, such as refinery and terminal locations and sizes, need to be made before the uncertainty is revealed. These planning decisions are usually capital intensive and cannot be easily adjusted once implemented. The operational decisions, such as feedstock procurement, ethanol production and transportation, can be adjusted (with a recourse cost) depending on the actual realization of uncertain parameters. To distinguish the different natures of planning and operational decisions and to allow recourse for operational decisions, we adopt a stochastic programming modeling (Birge and Louveaux, 1997). Under the standard two-stage stochastic programming paradigm, the first-stage (planning) decision has to be made before the actual realization of system uncertainties. Therefore, a reasonable solution to the planning decision should not be scenario dependent since the future scenario is not known at the time when the planning decision is made. This concept is denoted as ‘‘non-anticipativity’’ by Rockafellar and Wets (1991). The second-stage (operational) decisions are allowed to have recourse after a random event occurs and affects the outcome of the first-stage decision. A recourse decision made in the second-stage is typically interpreted as corrective measures against any infeasibility or inconvenience caused by a particular uncertainty realization. Since the recourse decision is scenario-dependent, the second-stage cost is also a random variable. The objective of a typical two-stage programming model is to make the first-stage decisions in a way that minimizes the sum of the first-stage costs and the expected value of the random second-stage costs. A discrete set of possible scenarios and their associated probabilities are used to describe the random event. In this study, we separately consider two major sources of uncertainties, feedstock supply and fuel demand. Let f represent the random parameters, H be the set of possible scenarios for f, and h (h 2 H) denote an individual scenario. The following decisions are sought through the proposed two-stage stochastic programming model:    

refinery and terminal locations and sizes, feedstock resource allocation strategy, ethanol production plan; and, feedstock and fuel transportation plan.

Note that the demand and technology are assumed static. The objective is to optimize the bioenergy infrastructure system layout in a planning context by focusing on long-term steady-state conditions. Hence, the problem studied here is a steady-state planning problem, where system transitional issues are not considered. For convenience, all notations used in the model formulation are summarized in Table 1. The complete model formulation (using demand uncertainty as an example) is presented in (1), (1.a), (1.b), (2)–(18). Minimize

X j2J

RC fj zj þ 0

þ Eh @

X

RC vj Sizej þ

j2J

X

X

TC k yk

k2K

RP  xj ðhÞ þ

j2J

XXXX f 2F if 2If

j2J

p2P

PRpf wpif j ðhÞ þ FTCðhÞ þ ETCðhÞ þ q

X

1 shortl ðhÞA

ð1Þ

l2L

where

FTCðhÞ ¼

XXX f 2F if 2If

ETCðhÞ ¼

XX j2J

k2K

j2J

  TDBS þ TTBS  dist if j SPBS CapBS

! þ LUBS 

P

p p2P wif j ðhÞ

ð1  MC f Þ

  0 1 0 1  dist jk  dist kl TDLQ þ TTLQ TDLQ þ TTLQ XX SPLQ SPLQ @ A @ þ LULQ  sjk ðhÞ þ þ LULQ A  tkl ðhÞ CapLQ CapLQ k2K l2L

ð1:aÞ

ð1:bÞ

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Table 1 Formulation notation table. Parameters F: index f, set of feedstock types If: index if, set of feedstock fields for feedstock type f J: index j, set of candidate refinery locations K: index k, set of candidate terminal locations L: index l, set of demand cities P: index p, set of price level H: index h, set of uncertain scenarios RC fj : annualized fixed refinery capital cost at location j RC vj : variable refinery capital cost at location j RP: ethanol production cost ($/gallon) CapRj: maximum refinery capacity at location j (gallon) TCk: terminal capital cost at location k CapTk: the maximum terminal service capacity at location k (gallon) PRpf : procurement cost of feedstock type f at price level p ($/dry ton) distij: distance between node i and j TDBS: distance dependent transportation cost ($/mile/truckload) of bulk solids (including fuel, insurance, maintenance, and permitting cost) TTBS: travel time dependent transportation cost ($/h/truckload) of bulk solids (including labor and capital cost) LUBS: truck loading and unloading cost of bulk solids ($/wet ton) CapBS: truck capacity of bulk solids (wet ton) SPBS: average truck travel speed for transporting bulk solids (mile/h) TDLQ: distance dependent transportation cost ($/mile/truckload) of liquids TTLQ: travel time dependent transportation cost ($/h/truckload) of liquids LULQ: truck loading and unloading cost of liquids ($/gallon) CapLQ: truck capacity of liquids (gallon) SPLQ: average truck travel speed for transporting liquids (mile/h) MCf: moisture content of feedstock type f, used to convert the feedstock dry ton to wet ton Cf: conversion rate of feedstock type f (gallon/dry ton) q: penalty cost for demand shortage ($/gallon) p Supplyif : the annual yields of biomass feedstock f in its field if at price level p Demandl: fuel demand at city l Decision variables zj: 1 if a refinery is located at j; 0 otherwise yk: 1 if a terminal is located at k; 0 otherwise Sizej: The design size of refinery j (gallon) xj(h): The total amount of ethanol produced at refinery j (gallon) under scenario h wpif j ðhÞ: The total amount of feedstock with type f transported from field if to refinery j at price level p (dry ton) under scenario h sjk(h): The amount of ethanol transported from refinery j to terminal k (gallon) under scenario h tkl(h): The amount of ethanol transported from terminal k to city l under scenario h shortl(h): The amount of demand shortage under scenario h

The objective function (1) minimizes the total expected system cost, which consists of the first-stage and expected second-stage costs. The operator Eh fg denotes the mathematical expectation with respect to the random parameter h. The first-stage cost includes the refinery fixed and variable capital cost and the terminal capital cost. These cost terms are invariant with the realizations of random parameters. In contrast, the second-stage costs are scenario dependent, including costs of ethanol production, feedstock procurement, feedstock transportation (FTC), ethanol transportation (ETC), and possible penalty on fuel shortage. Note that the model does not force all demand to be satisfied by in-state production. In case of fuel shortage, a penalty cost will be imposed, which can be interpreted as the expenses of importing ethanol from outside of the state. The fuel shortage penalty function is assumed to be piecewise linear in terms of fuel shortage quantity levels. The feedstock procurement cost function is assumed to be piecewise linear in terms of procurement quantity levels. The specific functional forms of these piecewise functions will be presented in the case study in Section 4. The feedstock and fuel transportation costs are defined by (1.a) and (1.b), respectively. Both transportation costs consist of loading/ unloading cost, time dependent travel cost, and distance dependent travel cost.The model is subject to the following constraints. Constraints on feedstock sites:

X

p

wpif j ðhÞ 6 Supplyif

8if 2 If ; p 2 P; h 2 H

ð2Þ

j2J

Constraint (2) states the total procured feedstock of type f cannot exceed its availability of that type, at any given price p level p. Note that when feedstock supply uncertainty is considered, Supplyif ðhÞ will be used to incorporate possible scenarios of feedstock supply type f at each location i at any given price level p. Constraints on ethanol refineries:

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XXX f 2F if 2If p2P

X

wpif j ðhÞ  C f ¼ xj ðhÞ 8j 2 J; h 2 H

sjk ðhÞ ¼ xj ðhÞ 8j 2 J; h 2 H

ð3Þ ð4Þ

k2K

X

sjk ðhÞ 6 CapRj  zj

8j 2 J; h 2 H

ð5Þ

k2K

8j 2 J; h 2 H

ð6Þ

Sizej P xj ðhÞ 8j 2 J; h 2 H

ð7Þ

Sizej 6 CapRj

Eq. (3) calculates the total ethanol amount converted from feedstock at refinery j under scenario h. Eq. (4) is a flow-conservation constraint, which states that total ethanol produced at refinery j equals the amount delivered to terminals under any scenario h. Constraint (5) is a logic constraint stating that no ethanol can be produced unless there is a refinery operating at this location. Constraint (6) states that the design size of a refinery should not exceed the maximum allowable capacity at that location. Constraint (7) defines that the design size of refinery j is the maximum production over all scenarios. Constraints on terminals:

X

sjk ðhÞ 

j2J

X

X

t kl ðhÞ ¼ 0 8k 2 K; h 2 H

ð8Þ

l2L

tkl ðhÞ 6 CapT k  yk

8k 2 K; h 2 H

ð9Þ

l2L

Eq. (8) sets a flow-conservation constraint on terminals. Constraint (9) is a logic constraint for terminals, which can be similarly explained as for constraint (5). Constraints on demand centers:

X

t kl ðhÞ þ shortl ðhÞ ¼ Demandl ðhÞ 8l 2 L; h 2 H

ð10Þ

k2K

Eq. (10) quantifies the shortage at each city in each scenario as the amount of demand that is not satisfied by the total fuel supplied to the city. Integer and non-negativity constraints:

zj ¼ f0; 1g 8j 2 J

ð11Þ

yk ¼ f0; 1g 8k 2 K xj ðhÞ P 0 8j 2 J; h 2 H

ð12Þ ð13Þ

Sizej P 0 8j 2 J

ð14Þ

wpif j ðhÞ

ð15Þ

P 0 8if 2 If ; 8j 2 J; p 2 P; h 2 H

sjk ðhÞ P 0 8j 2 J; 8k 2 K; h 2 H

ð16Þ

tkl ðhÞ P 0 8k 2 K; 8l 2 L; h 2 H

ð17Þ

shortl ðhÞ P 0 8l 2 L; h 2 H

ð18Þ

Constraints (11) and (12) set binary integer restriction to the location decision variables z and y, and the rest are non-negativity restrictions. 3. Decomposition method Although stochastic modeling approaches provide more reliable results, they often come with heavier computational burden for problems of non-trivial size. In some numerical experiments, we were not able to solve the stochastic model directly using commercial solvers. Therefore, decomposition methods were exploited to overcome the numerical difficulties. 3.1. Progressive hedging (PH) algorithm There are a handful of decomposition methods in handling large scale stochastic programming problems (Ruszczynski, 1997). Some well documented and widely implemented methods include L-shaped method (also called vertical decomposition) (Van Slyke and Wets, 1969) and progressive hedging (PH) method (also called horizontal decomposition) (Rockafellar and Wets, 1991). Based on our previous research experience and numerical experiments, we found that the PH method is well suitable for solving this problem. PH method decomposes a stochastic problem across scenarios by partitioning the original problem into manageable scenario sub-problems. Recent successful applications of PH in solving stochastic mix-integer problems can be found in Fan and Liu (2010) and Watson and Woodruff (2010). For the convenience of the readers, the basic scheme of PH method is briefly described below by using a generic mathematical model described in (19) and (20):

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Minimize ðc  xÞ þ

X

Ps ðt s  ys Þ

ð19Þ

8s 2 S

ð20Þ

s2S

Subject to :

ðx; ys Þ 2 F s

where S is the set of possible scenarios for random event, sðs 2 SÞ denotes an individual scenario, x denotes the first-stage decisions with a cost coefficient vectors c, and ys represents the second-stage decisions with associated cost coefficient vectors ts. For each scenario s 2 S, we denote the probability of the occurrence as Ps. The objective is to minimize the total cost of the first- and second-stages as described in (19). The decisions are subject to the constraints defined by the feasibility set Fs for each scenario s as described in Constraint (20). The model defined by Eqs. (19) and (20) can be simply separated into many scenario sub-problems. Solving the scenario sub-problems defined in all sðs 2 SÞ will give us different s-dependent first-stage solutions, denoted as xs for each s 2 S. However, these solutions cannot be directly implemented, because at the time when the location decision solutions are implemented, one does not know yet which scenario is going to happen. In order to consolidate the s-dependent solutions to an implementable solution, we must impose the following condition:

xs ¼ x0s

8s 2 S; 8s0 2 S; s0 –s

ð21Þ

or equivalently

xs  z ¼ 0 8 s 2 S

ð22Þ

where z is a vector of free variables. This condition is called a non-anticipativity constraint defined by Rockafellar and Wets (1991), which states that a reasonable policy should not require different actions relative to different scenarios if the scenarios are not distinguishable at the time when the actions are taken. Therefore, the overall stochastic program can be formulated as:

Minimize

X

Ps Q s ðxs ; ys Þ

ð23Þ

s2S

Subject to :

ðxs ; ys Þ 2 F s xs  z ¼ 0

8s 2 S 8s 2 S

ð24Þ

Function Qs(xs, ys) is the total first- and second-stage cost in a given scenario s, which depends on the decisions xs and ys. The PH method decomposes a stochastic problem across scenarios and partitions the problem into manageable sub-problems. Define

Lr ðX; Y; z; WÞ ¼

X s2S

1 P s Q s ðxs ; ys Þ þ ðws Þ0  ðxs  zÞ þ ckxs  zk2 2

ð25Þ

as the augmented Lagrangian, where W is the vector of dual variables for the constraints in (22), and c > 0 is a penalty parameter associated with violation of the non-anticipativity constraints. Therefore, the augmented Lagrangian integrates the nonanticipativity constraints with the original objective function. The stochastic problem becomes

Minimize Lr ðX; Y; z; WÞ over all ðxs ; ys Þ 2 F s :

ð26Þ

2

1 2

Due to the nonseparable penalty term ckxs  zk in Expression (26), the problem cannot be decomposed directly. The PH method achieves decomposition by alternately fixing the scenario solutions (x, y) and the implementable solution z in (26). The detailed procedures are described below. PH algorithm procedure Step 1 Set the iteration index k = 0.

  ð0Þ ð0Þ Solve for each scenario sub-problem and then obtain xs ; ys ; 8s 2 S. P ð0Þ ð0Þ ð0Þ Initialize zð0Þ :¼ s2S Ps xs and ws :¼ cðxs  zð0Þ Þ ð0Þ

If xs

¼ zð0Þ ; 8s 2 S then the optimal solution is found; otherwise continue with step 2.

Step 2 k=k+1 Solve for each scenario 8s 2 S xs þ 2c kxs  zk1 k2 Þ : ðxs ; ys Þ 2 F s xs :¼ arg minx ðQ ðxs ; ys Þ þ wk1 s P ðkÞ ðkÞ ðk1Þ ðkÞ ðkÞ Update z :¼ s2S Ps xs and ws :¼ ws þ cðxs  zðkÞ Þ; 8s 2 S ðkÞ

Step 3 Check whether the termination criterion

h

e ¼ kzðkÞ  zðk1Þ k2 þ

P

s2S P s kx

ðkÞ

 zðkÞ k2

optimal solution is found, otherwise, go to step 2 and continue the iterations.

i1=2

 0 is reached; if yes, an

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3.2. Implementation of PH algorithm Rigorous convergence proof of the PH algorithm in convex and continuous problems is given by Rockafellar and Wets (1991). These convergence theorems are valid unconditional to any particular choice of the penalty parameter c. However, it was also pointed out in Rockafellar and Wets (1991) that parameter c plays an important role in convergence in practice. In addition, several studies (Løkketangen and Woodruff, 1996; Mulvey and Vladimirou, 1991, 1992; Watson and Woodruff, 2010) reported some important factors that may influence the setting of penalty parameter c, which provided valuable numerical results for our research. For example, it was suggested that the choice of c should depend on the cost coefficient of the decision variable that an effective c value should be close in magnitude to the coefficient of decision variable (Watson and Woodruff, 2010). Numerical experiments related to parameter tuning will be reported in the case study in Section 4.

4. Case study The model is applied to California as a case study. California has taken aggressive actions to reduce GHG emissions in the transportation sector and there exist many types of lignocellusloic biomass, thus providing an ideal case study region. In California, the Low Carbon Fuel Standard (LCFS) requires nearly 3.7 billion gallons per year (BGY) of ethanol fuels from low carbon sources (e.g., bio-wastes including agricultural and forest residues and municipal, sugarcane, or cellulosic energy crops) by 2020 (Yeh et al., 2009). The Bioenergy Action Plan (California Bioenergy Interagency Working Group, 2006) has targeted specific goals of increasing the share of in-state biofuel production from 20% in 2010 to 40% by 2020 and to 75% by 2050. It requires about 1.48 BGY in-state biofuel production in 2020.

4.1. Model inputs Model input data are described below. Technology assessment: The LignoCellulosics Ethanol (LCE) via hydrolysis and fermentation conversion technology with specific Dilute Acid pretreatment process was considered in this study. This biofuel conversion technology is expected to be ready for commercialization between 2015 and 2025 (Parker et al., 2007). It is assumed that multiple types of feedstock could be converted to ethanol with single conversion technology through advanced uniform-feedstock preprocessing prior to the conversion process at refineries, based on the report published by Idaho National Laboratory (Bioenergy Program, 2008). Feedstock resources: Eight major types of bio-wastes are considered in the study, including corn stover, rice straw, wheat straw, forest residues, MSW (Municipal Solid Waste) wood, MSW paper, MSW yard, and cotton residual. The existing feedstock annual yields and location data are adopted from the Western Government Association (WGA) report (Parker et al., 2007) and have been aggregated at county or city centroids in GIS. These feedstocks are widely dispersed across the state, and different feedstocks tend to cluster in different regions. In general, the agricultural residues are concentrated in central valley area, MSW is available in metropolitan areas (i.e., San Francisco and Los Angeles areas), and forest residues are mainly located in Northern California. Associated parameters including total yields, moisture content and conversion rate are summarized in Table 2. The conversion rates are measured by the gallons of ethanol converted from one dry ton of the feedstock. The feedstock procurement cost is the expense of acquiring and transporting feedstocks from fields to the roadside in a transportable form (Parker et al., 2007). The procurement cost function is piecewise linear to the procured quantity, as shown in Fig. 2 (Parker et al., 2007). Use forest as an example, about 12.5 million dry tons of forest feedstock is available at a price of $10/dry ton, and an additional 17.5 million tons can be available with a price increment of $5/dry ton. This figure also clearly denotes the maximum availability of each feedstock type. For instance, MSW-yard can maximally provide about 2.1 million dry tons.

Table 2 Feedstock parameters. Feedstock types

Total yield (dry ton)

# of nodes

LEC conversion rate (gallons fuel/dry ton biomass)

Moisture content (% weight)

Geographic resolution

Cornstover Wheat straw Forest Rice straw Cotton residual MSW-paper MSW-wood MSW-yard

562,667 368,059 4268,238 866,605 439,088 1743,082 898,776 2124,381

27 32 47 14 10 57 57 57

80.6 76.8 90.2 76.8 71.4 86.0 78.9 70.0

15 15 50 15 50 10 50 50

County County County County County Municipality Municipality Municipality

Resource: Parker et al. (2007).

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Fig. 2. Feedstock supply by procurement cost.

Refineries, terminals, and demand clusters: In this study, there are 28 potential sites for refineries, 29 potential sites for terminals, and 143 demand clusters (cities). Their geographical representations are given in Fig. 3. Detailed location criteria and data processing are given in Parker et al. (2007). For refineries, the cost includes fixed capital cost (facility setup cost) and variable capital cost (facility size-dependent cost). The fixed capital cost was annualized assuming a real discount rate of 10% and a lifetime of 20 years. Based on the mid-term technology performance (Parker et al., 2007), the annualized fixed capital cost of refineries is $6.2 million (M) for all potential locations with maximum capacity at 100 million gallons per year (MGY), and the variable capital cost and ethanol production cost are set to be $0.314 and $0.26 per gallon respectively. The capital cost associated with ethanol storage and blending at terminals is estimated to be $1.57 M for a daily handling capacity up to 100 thousand barrels (equivalent to an annual capacity of 1533 million gallons) (Reynolds, 2000). Operating cost at terminals is neglected. Cities with a population of 50,000 are considered as demand sites, which mainly cluster in metropolitan area. The ethanol demand for each city is estimated by distributing the total ethanol demand of the state proportionally to the city populations. Transportation costs: California road network was considered for transporting feedstock and fuels, which contains local, rural, and urban roads as well as major highways. Transportation costs are separately measured for bulk solid and liquid by three major cost components – loading/unloading costs, time- and distance-dependent costs, as shown in Table 3. Uncertain future demand: In California, ethanol has been widely used in blending with gasoline at a rate of 5.7% (i.e., E5.7), which requires nearly 900 million gallons of ethanol per year (State Board of Equalization, 2010). However, the existing instate production capacity is only 81 MGY (Jenkins et al., 2007), which is less than 10%. For future ethanol demand estimation, the California Energy Commission has projected the in-state ethanol demand using future fuel demand forecast and different bland rate cases (Jenkins et al., 2007). The in-state ethanol production for E5.7 blend rate would need about 300 MGY in 2015 and 400 MGY in 2020. For E10 blend rate, the in-state ethanol production is expected be around 500 MGY and 700 MGY for 2015 and 2020 respectively. In this study, we consider a set of discrete demand scenarios of 300, 400, 500, 600, and 700 MGY with even probabilities.

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Fig. 3. Potential sites for refineries and terminals, and demand clusters.

Table 3 Trucking cost.

Loading/unloading Time dependent Distance dependent Truck capacity

Liquids

Bulk solids

$0.02/gallon $32/h/truckload $1.30/mile/truckload 8000 gallons

$5/wet ton $29/h/truckload $1.20/mile/truckload 25 wet tons

Resource: Parker et al. (2007).

Uncertain feedstock supply: Ten supply uncertain scenarios are generated based on 1999–2008 historical data aggregated at county level. Agricultural residue data are based on the USDA National Agricultural Statistics Service (NASS). We assumed that the agricultural residue yields are proportional to the production of crops. Municipal solid wastes are based on California Waste Stream County Profiles (CalRecycle). Forestry fires are considered to be the main cause of forestry yields fluctuation. The annual wildfire information between year 1999 and 2008 is available from the California Department of Forestry and Fire Protection’s Fire and Resource Assessment Program (FRAP). We assumed that loss of forestry productivity is proportional to the size of fires. Penalty cost: The stochastic model does not force all demand to be satisfied by in-state production. In case of shortage, a penalty cost will be imposed, which can be interpreted as the expenses of importing ethanol from outside of the state, such as corn ethanol from mid-west or sugarcane ethanol from Brazil. The total shortage penalty cost is set to be a piecewise linear function. Fig. 4 plots the unit penalty cost at different shortage levels. The cost of $2.25 per gallon represents the actual current market price for ethanol. The gradually increasing penalty cost (from $2.25 to $2.75) reflects extra cost associated with acquiring fuels from further distance or non-contracted venders. The extremely high penalty cost of $5 per gallon of ethanol beyond 20% shortage is imposed to encourage in-state production and to control the maximum amount of imported ethanol for energy security purpose. 4.2. Numerical results 4.2.1. Case study I: demand uncertainty In the baseline scenario, we consider four discrete demand scenarios of 300, 400, 600, and 700 MGY with equal probabilities. Optimal solution: The main feedstock resources recommended by the optimal solution are forest residues (contributing to about 50% of produced fuel), MSW-wood (about 20–36% depending on different demand scenarios), and MSW-paper (about 14–22%). In high demand scenario (700 MGY), a small portion of cotton residue and MSW-yard is used. The minimum total system cost is found at $603.3 M, of which about 44% is attributed to refinery capital cost, 22% to fuel production cost, 15% to biomass procurement cost, 14% to biomass transportation cost, 5% to fuel transportation, and less

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Fig. 4. Unit penalty cost at different shortage levels.

than 1% to terminal cost. The transportation cost (for biomass and fuel) is about 19% of the total system cost, indicating the importance of logistics planning for biofuel supply systems. Table 4 summarizes the refinery and terminal location solutions obtained from different (stochastic and deterministic) models. The solution to the stochastic program model is labeled as the ‘‘SP solution.’’ The solution to the deterministic model considering only the average demand (500 MGY) is labeled as the ‘‘expected-value solution.’’ The solution to each scenariodependent problem is called ‘‘wait-and-see solution’’ as if one could wait until the uncertain information is revealed to make a decision. The geographic layout of the SP solution and the expected-value solution is plotted in Fig. 5. Apparently, the system layouts of the two solutions differ significantly. Performance evaluation of stochastic and deterministic solutions: In Table 5, the performance (in terms of the total system cost) of the SP, wait-and-see, and expected value solutions (column 2–7) are evaluated in the four possible demand scenarios (row 2–5). Note that wait-and-see solution to a particular scenario necessarily performs the best among all solutions in that particular scenario, as italicised along the diagonal of the table. This is because a wait-and-see solution is generated with perfect information of the scenario. However, a wait-and-see solution may perform poorly in other scenarios as the assumed value for the random parameter is no longer valid. For example, the wait-and-see solution from 400 MGY scenario achieves the minimum cost of 418.21 $M in demand scenario of 400 MGY, but a very high cost in demand scenarios of 600 and 700 MGY. The last three rows of Table 5 present the expected total system cost, the expected value of perfect information (EVPI), and the amount of unsatisfied demand following a specific solution. As expected, the SP solution returns a low expected total system cost comparing to deterministic solutions. It can also be observed that the SP solution results in a small range of the total cost across all possible demand scenarios, indicating a better reliability. In addition to the expected total system cost, the value of stochastic programming solution (VSS) and the EVPI are also commonly used performance measures for evaluating the quality of a stochastic model solution. EVPI is defined as the maximum value of having perfect information about the future (Birge and Louveaux, 1997). EVPI is quantified as the average difference between the objective values from the SP and the wait-and-see solutions. A lower EVPI value implies that the model is less sensitive to imperfect information of model parameters, or that the potential regret of not having perfect information of future is small. In this case study, the EVPI of the stochastic programming model is 69.89 $M, much less than the EVPI of the deterministic models. On one hand, the stochastic modeling approach is favorable for decision making under uncertainties because it relies less on the knowledge of the future decision environment. On the other hand, the non-trivial value of EVPI suggests that effort in improving estimates of uncertain parameters is worthwhile, even though stochastic programming model may be less sensitive to imperfect information than its deterministic counterparts.

Table 4 Summary of stochastic and deterministic solutions. Strategies

Refinery location

# of refinery

Terminal location

Stochastic programming solution

14, 16, 17, 20, 21, 24, 28

7

4, 27, 28

Deterministic models Wait-and-see solution 300 MGY 400 MGY 600 MGY 700 MGY

14, 20, 21 5, 14, 20, 21 5, 14, 17, 20, 21, 24, 28 14, 16, 17, 19, 20, 21, 24, 28

3 4 7 8

4, 5, 4, 4,

Expected-value solution (500 MGY)

5, 14, 20, 21, 24, 28

6

4, 27, 28

Location ID number are from GIS map.

5 11, 21 27, 28 6, 27, 29

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(a) Stochastic Model Solution

(b) Expected Value Solution

Fig. 5. The optimal system configurations of SP and expected value solutions.

Table 5 Performance evaluation of stochastic and deterministic solutions. Scenarios evaluated on

Strategies Stochastic programming ($M)

Wait-and-see 300 MGY ($M)

Wait-and-see 400 MGY ($M)

Wait-and-see 600 MGY ($M)

Wait-and-see 700 MGY ($M)

Expected-value (500 MGY) ($M)

Scenario 1: 300 MGY Scenario 2: 400 MGY Scenario 3: 600 MGY Scenario 4: 700 MGY

451.44

310.82

344.12

421.66

458.27

384.77

522.27

564.23

418.21

491.63

528.05

455.30

675.51

1140.96

945.12

643.52

677.73

764.69

764.01

1429.55

1233.01

873.08

761.12

1046.37

Expected total cost

603.31

861.39

735.11

607.47

606.29

662.78

EVPIa Total unsatisfied demand (MGY)

69.89 0

327.97 800

201.70 500

74.05 100

72.87 0

129.37 300

a EVPI is computed as, taking stochastic model as an example: 69.89 = 0.25 * (451.44  310.82) + 0.25 * (522.27  418.21) + 0.25 * (675.51  643.52) + 0.25 * (764.01  761.12).

Stochastic programming models usually impose more data and computing challenges. The concept of Value of Stochastic programming Solution (VSS) (Birge and Louveaux, 1997) can be used to justify whether extra modeling and computing effort involved in stochastic programming is worthwhile. The VSS compares the objective value obtained from the expected-value solution with the stochastic programming solution, defined by VSS = EEV  SP. In general, a bigger VSS indicates higher benefit from using stochastic programming approach. In this case study, the VSS is 59.47 million dollars (662.78  603.31 = 59.47), which justifies use of more sophisticated modeling techniques and the extra computational efforts.

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Designed Size of Refinery (MGY) # of plant

SP solution

expected-value solution

14

100.00

100.00

16

72.45

72.45

17

75.72

78.65

19

89.93

88.12

20

100.00

100.00

21

100.00

100.00

24

100.00

100.00

28

61.90

60.78

Fig. 6. The optimal system layout considering feedstock supply uncertainty.

4.2.2. Case study II: feedstock supply uncertainty In this case study, we consider biomass feedstock yields as the main source of uncertainty. Ten scenarios (from year 1999 to 2008) of feedstock supply are considered with equal probability. The future demand is set as a constant value of 700 MGY. The optimal refinery and terminal locations obtained from the SP model are plotted in Fig. 6. It is observed that the planning decisions of the SP solution and the expected-value solution (which considers only the average biomass yields of the 10 scenarios) are similar. The results indicate that, in a steady-state yearly-based model, the optimal system layout (planning decision) of the bioethanol supply chain is not sensitive to the uncertainty caused by feedstock supply fluctuation. Through adjusting the procurement quantities of different biomass types and redistributing feedstock flows from biomass fields to refineries, the system can achieve most efficient operation without having to changing its physical layout. We notice that the diversity of feedstock types considered in this study made such easy adjustment possible. 4.3. Computational experiments 4.3.1. Effects of penalty parameter c on convergence Previous research has demonstrated in convex cases that the setting of c is strongly influenced by the sensitivity of the objective function with regard to changes in the first-stage decision variables. The coefficients in the objective function vary significantly with different categories of decision variables. For example, the coefficient associated with the refinery location variable is the fixed refinery capital cost ($6.2 million), while the coefficient associated with the refinery size variable is the variable capital cost ($0.314 per gallon). Therefore, different c values should be set for different first-stage decision variables: c_p for refinery location variable, c_t for terminal location variable, and c_c for refinery size variable. The sequences of convergence resulted from different sets of c values are plotted in Fig. 7. The x axis denotes the number h i1=2 P . In Fig. 7a, given of iterations, and the y axis denotes the value of the error term e ¼ kzðkÞ  zðk1Þ k2 þ s2S P s kxðkÞ  zðkÞ k2 the c_c and c_t equal to 0.1 and 1 respectively, the performance of the algorithm varies significantly with c_p values, in terms of number of iterations and acceptable errors. For example, with an extremely high value of c_p (=1000), the sequence converges fastest but with large error. In contrast, a small value of c_p (=1) results in noticeable oscillation and slow convergence. Additional numerical experiments on c_c have been conducted by decreasing c_c further down to 0.01, as shown in Fig. 7b. These numerical experiments revealed that the selection of c is crucial, which will affect the solution efficiency and the quality of the final results. 4.3.2. Effects of number of scenarios In this experiment, we examine how the performance of PH algorithm might be affected by the number of scenarios in the model. Fig. 8 shows the execution time (CPU second) of running the stochastic model with different number of scenarios. All experiments were carried out on a Dell Precision Model 650 Workstation with 4 GB RAM and Dual-Xeon 3.06 GHz processor under Windows XP environment. The ‘‘non-decomposed’’ curve represents the execution time from directly solving the

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Fig. 7. Sequences of convergence results from different values of the penalty parameter c.

Fig. 8. Comparison of computing time.

Table 6 Comparison of solution accuracy. # of scenario

Non-decomposed (optimal value) ($M)

PH algorithm ($M)

Gap (%)

4 5 6 7 8

603.31 601.99 677.98 749.85 709.40

604.06 602.15 679.05 751.17 710.32

0.124 0.027 0.158 0.176 0.131

stochastic models using AMPL-CPLEX. As shown in the figure, the advantage of PH algorithm does not stand out until the number of scenarios becomes non-trivial. In Table 6, the objective value by the PH algorithm is compared to the optimal value obtained from directly solving the problem using AMPL-CPLEX. The difference between the two values in percentage is computed as the ‘‘gap’’ of numerical accuracy. The gaps are negligible, indicating the potential of PH algorithm in improving computational efficiency without scarifying solution quality.

5. Conclusions and discussions The key methodological contribution of this study to the literature is on the development of an integrated modeling framework that can be used to support future biofuel system planning under uncertainties. As discussed in Section 1,

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stochastic models considering the entire biofuel supply chain systems are lacking in the literature. Through integrating modeling and algorithm design efforts, we are able to implement the proposed stochastic model in problems of real-world sizes. Based on a case study of converting bio-wastes to ethanol in California, it was found that ethanol can be produced at an average delivered price of $1.20 per gallon through optimally planning the entire bioethanol supply chain, which demonstrates the feasibility of waste-based cellulosic ethanol as part of sustainable future energy solution. In bioenergy research community, feedstock fluctuation has been a major concern in bioenergy supply system planning. We found that when the entire supply chain is considered and when the system accommodates diversified feedstock types, annual feedstock supply fluctuation has little effect on the physical layout design of a bioethanol supply chain. Of course here we only considered recurrent risk caused by supply fluctuation. The observations reported here may not be applicable if nonrecurrent risks (such as catastrophic events) are considered. We should also point out that a yearly-aggregated model is only suitable for long-term planning where a steady-state condition is assumed. In studies where biomass seasonality and storage operation are the main concerns, this kind of yearly-aggregated model would not be appropriate. Rather, a multi-period model that distinguishes monthly (or even finer time resolution) inventories and operations should be adopted. In this study, we focused on finding the optimal bioenergy supply infrastructure system layout under steady-state condition. An equally important question in long-term planning that has not been addressed is how to transit to a future desired system state from current state. To address dynamic transient issues under uncertainties, one would need to construct a multi-stage stochastic model, which imposes even more computational challenges. One of our ongoing efforts is to develop effective modeling and computational methods to incorporate dynamics caused by evolving technology growth in bioenergy supply system planning under uncertainty. Renewable energy system design is a relatively new field. However, relevant topics such as supply chain management and infrastructure system planning have been extensively studies and shall shed light on studies in renewable energy planning and operations. Advanced system-oriented approaches may lead to more efficient energy resource allocation and system design strategies. Meanwhile, renewable energy research needs may present new modeling and computational challenges. We hope this work would inspire/encourage more effort in bridging the two communities in operations research and renewable energy. 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