A mathematical analysis of full fuel cycle energy use

e Net energy analysis nvi gy , be l fu s s uel the defi mat to provide precise and intuitively reasonable definitions of other energy production metrics such as EROI nment elopin use fu t path es cha costs, n app alysis, of energy consumes energy in different forms, so a full description of the fuel cycle requires an accounting system that tracks the direct use of energy and material losses in each step of the production chain, and a systematic method for converting different forms of energy to comparable units. This type of approach has been widely used in transportation energy analysis, where it is often referred to as ”well-to-wheels” produced widely varying estimates of the net energy for the same process [11,23], largely for two reasons: (1) some approaches only count fossil-based fuels as energy, and ignore energy inputs from renewables or biomass residues, and (2) some approaches allocate a fraction of the process energy use to by-products, which reduces the apparent energy intensity of production of the fuel itself. This is a case where a modeling concept which is valid in econom- icsdusing the value of a by-product to subsidize the cost of productiondis incorrectly applied to physical data. Another Contents lists available at er els Energy 37 (2012) 698e708 E-mail address: [email protected]. production chain from extraction to delivery, and estimates the total energy required to deliver one unit of energy to the point of use. FFC analysis can be used to compare the resource productivity of different fuel sources, such as conventional vs. unconventional gas production, and to compare the efficiency of devices that use different fuels, such as gasoline vs. electric vehicles. It also provides a more accurate estimate of the full impacts of energy conservation and efficiency policies [6]. The extraction, processing and delivery [1,18]. There have been numerous attempts to apply LCA to biofuel production pathways [20] but in this case the results have been more problematic [11,4,15]. In these applications the goal is to determine the “net energy” output from a biomass-based produc- tion chain. Loosely, the net energy is equal to the energy content of one unit of output fuel, minus the energy consumed by all the processes steps required to produce it. Different authors have 1. Introduction Given concerns about the enviro use, there is a keen interest in dev sources and new technologies that broad range of energy developmen with more choices, but also creat metrics that reliably quantify the trade-offs of different alternatives. A popularity is full fuel cycle (FFC) an 0360-5442/$ e see front matter � 2011 Elsevier Ltd. doi:10.1016/j.energy.2011.10.021 al impacts of fossil fuel g both alternative fuel els more efficiently. A ways provides society llenges in developing benefits, and potential roach that is gaining in which looks at the full analysis [10,27,29,30]. Many of these applications use the GREET model developed at Argonne National Laboratories [28,2]. A conceptually similar, but more general, approach is LCA (life-cycle- analysis). LCA can be applied to either manufactured products or commodities, and attempts to account more broadly for the use of energy, materials andwater in the production, distribution, lifetime use and disposal of the product. LCA has been used to compare the lifetime energy use and emissions of electricity generation using different fuels and technologies [17,21,25,26], and to estimate the emissions impacts from non-conventional fossil fuel production parameters that depend only on directly observable physical data. � 2011 Elsevier Ltd. All rights reserved. (energy return on energy invested). The multiplier is a non-linear function of a set of energyeintensity A mathematical analysis of full fuel cycl Katie Coughlin Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA a r t i c l e i n f o Article history: Received 22 February 2011 Received in revised form 4 September 2011 Accepted 14 October 2011 Available online 15 December 2011 Keywords: Full fuel cycle EROI a b s t r a c t Given concerns about the e a broad range of new ener reliably quantify the costs present a definition of a ful production chains, and ha them. The term FFC (full f processing, conveyance to applications, the metric is consumption, gives an esti En journal homepage: www. All rights reserved. energy use ronmental impacts of fossil fuel use, there is a keen interest in developing sources and technologies. This in turn creates a need for metrics that can nefits, and potential trade-offs of different alternatives. In this paper, we el-cycle metric that is flexibile enough to describe a wide variety of energy ufficient mathematical rigor to allow meaningful comparisons between cycle) refers to the complete fuel production chain including extraction, retail distribution center and delivery to final consumers. For ease of use in ned as an FFC multiplier which, when applied to the point-of-use energy e of the FFC energy use. We also show that the FFC multiplier can be used SciVerse ScienceDirect gy evier .com/locate/energy concept, closely related to net energy, is the energy return on energy invested or EROI [22]. EROI has been primarily used to estimate changes in the productivity of oil and gas fields [3,13,14]. The definition of EROI for a broader range of fuel sources and produc- tion processes is hampered by the same lack of methodological consistency as net energy calculations [22], which makes it very difficult to use these metrics in practice. The issue here is not so much the calculation of a true value, but rather the need for a methodology with the flexibility to describe a wide variety of production chains, and the mathematical coherence that allows meaningful comparisons to be made. In this paper, we present a definition of a full fuel-cycle metric that addresses this need for rigor and consistency. For ease of use in applications, fuel-cycle impacts are expressed through an FFC multiplier which, when applied to the point-of-use energy consumption, gives an estimate of the FFC energy use. The multiplier is a function of a set of energyeintensity parameters that depend only on directly observable physical data. While the data requirements and calculations needed to define the param- some very simple examples, which are used to illustrate the underlying logic, then present the fuel cycle equation for the general case. Additional mathematical detail is provided in the appendices. Section 4 provides the formula for the fuel cycle multiplier, and several other metrics including EROI. In Section 5 we present a simple application of the multiplier formula to model the impact of changes in the energy intensity of fuel production on energy prices. The final section presents some conclusions and discussion of future work. 2. Approach As illustrated in Fig. 1, the productive economy can be divided into two sections, the energy production sector and the ROE (rest of the economy). Here we refer to energy use by the rest of the economy as final consumption; it is exclusively an energy sink. The energy production sector also consumes energy, so it is both a source and a sink. This means that, given a decrease in demand for energy in the ROE, the energy needed to produce that energy is also K. Coughlin / Energy 37 (2012) 698e708 699 eters can be extensive, the parameter definitions are very simple in physical terms. The methodology distinguishes between primary fuels (such as coal or natural gas) and energy carriers (such as electricity or steam), with both included in the FFC multiplier. The approach can account for both geographic variability and time dependence in the parameters, but for simplicity these aspects of the problem will not be discussed here. We will also show that the FFC multiplier can be used to provide precise and intuitively reasonable definitions of EROI and other resource efficiency metrics. One of the benefits of this approach is that it provides a clear definition of what needs to be calculated, in physical terms, without getting lost in the details of how to do the calculation. We believe this is a necessary step in establishing a consistent framework for the analysis of resource efficiency in energy production. Another advantage is that the effect of physical constraints on production can be examined independently of economic phenomena. This is potentially very useful, as the demand-driven models currently used to represent the productive economy do not function well under realistic limits on resource availability [19]. The paper focuses primarily on establishing the mathematical framework, with an example numerical calculation given in the Appendix. The rest of the paper is organized as follows: In Section 2 we discuss the approach in more detail and define our terminology. Section 3 contains the mathematical derivations. We begin with Fig. 1. The economy is divided into the energy production sector and all other activity. Th produced. decreased. But this induced decrease in energy sector demand induces a further decrease in energy sector demand, leading to the problem of “infinite regress” [22]. This recursive relationship is illustrated by the circular arrow in Fig. 1. Mathematically, it implies that the amount of energy required for energy production must be a non-linear function of the parameters describing the energy intensity of energy production. We will show in the next section that this formula can be derived by summing up the sequence of small changes in energy sector demand that are induced by a change in final consumption. The energy production chain consists of extraction, processing, transportation to distribution centers, further processing as needed, and distribution to final consumers, as illustrated in Fig. 2. The existence of multiple steps in the production process is important to the numerical calculation of the energy intensity parameters, but does not affect the way these parameters appear in the multiplier formula. Hence, in deriving the multiplier formula, we can ignore the details of the production process. We provide a mathematical treatment of energy use and material losses for each of the process steps in Appendix 2. Similarly, our approach makes no a priori assumptions about system boundaries; these have no impact on the definition of the formula for the multiplier. In practical calculations, the definition of system boundaries depends on the degree of precision desired in the final answer. Given a desired numerical precision, or a quantitative estimate of e amount of energy needed for energy production depends on the amount of energy defined so that any aspect of the process that affects the answer at scrib y 37 this degree of precision is included. The energy used to build the infrastructure used in energy production is not included in the calculation, for two reasons. First, LCA studies consistently show that this contribution is small when amortized over the lifetime of the infrastructure [21,25,26], so the additional modeling effort doesn’t greatly improve the precision of the calculation. Second, and more fundamentally, changes to the level of fuel production occur over very different time scales than material changes to infrastructure. Infrastructure lifetime is typi- cally several decades, and it may take up to a decade or longer to plan, approve and implement infrastructure projects. While production levels should scale with infrastructure capacity over the long term, they also vary significantly over the short term. The fuel cycle multiplier defined here is intended to capture the impact of changes to final consumption in the rest of the economy that may occur over time scales of one to a few years. It should therefore be based on the component of energy use by the energy production sector that scales directly with fuel output. The multiplier itself is a dimensionless number that will be different for different types of energy. We use the term energy to refer both to primary fuels and to energy carriers such as electricity. Given the importance of electricity in the economy, we provide a detailed analysis of the derivation of a multiplier that can be applied to grid electricity. Our definition of the multiplier is based on energy content, i.e. the ratio of the energy content of all fuels used in the FFC to the energy content of a unit of fuel at the point of use. To convert different fuels to equivalent energy units we use heat content (MJ per physical unit). 3. Mathematical derivations the uncertainty in the multiplier, system boundaries should be Fig. 2. Categories used to de K. Coughlin / Energ700 The demand for energy in the rest of the economy is what drives activity in the energy production sector, so logically the latter should be a well-defined function of the former. In this section we derive mathematical formulae that express this rela- tionship, for a set of increasingly complicated examples. To begin with, the conceptual framework is outlined using a simple example and no algebra. We then repeat this example and intro- duce the iterative approach used to solve the general case. For concreteness, the calculation is structured as follows: we assume a base case which defines the total energy production for the whole economy, and for which the required parameters can be defined. We then analyze a scenario in which final consumption is altered by some amount (which could be either positive or negative, and is not necessarily small). The goal of the calculation is to define the total annual energy production under the new scenario. 3.1. Conceptual framework Consider the simplest possible situation, in which we have only one fuel and this fuel supplies all the energy used in the economy. The baseline variables are � F is the total energy use by the whole economy � G is final consumption, i.e. the amount used by the rest of the economy � c is the energy intensity of fuel production, defined as the quantity of fuel used per unit of fuel output; by definition 0 < c < 1 It follows from these definitions that the energy use by the fuel production sector is cF, and therefore F ¼ cF þ G or equivalently F ¼ G=ð1� cÞ. If c is a constant, then for this simple case we can define the multiplier as m ¼ F G ¼ 1 1� c; (1) m relates total production to total final consumption. More gener- ally c may be a function of total output; if this is the case, we can still use equation (1) to define the change in the full fuel cycle energy induced by a small change in G. The formula for m is a non-linear function of the parameter c. In a linear approach, given the final consumption G, the energy required to produce it is estimated as cG, which leads to a total energy production of Flin ¼ Gð1þ cÞ, and thus mlin ¼ 1þ c. Values of m and the linear approximation mlin are plotted in Fig. 3. The two are very close for small values of c, but diverge rapidly when c becomes larger. This indicates that the quantitative difference e the fuel production chain. (2012) 698e708 between using a linear vs. non-linear approach to calculating full fuel cycle effects is not significant if energy use by the energy sector is small, but could be very important when it increases above 20% or so. Fig. 3 also shows that the multiplier diverges as c approaches one. This is a logical physical limit; if cwere larger than one the fuel production process would be a net consumer of fuel, i.e. would produce no output at all. 3.2. Example calculations Before continuing with the examples, we present a brief outline of the mathematical approach. The notation used is summarized below; as the examples get more complicated this notation will be modified as needed. � The base case total energy use for the whole economy is F (physical units) imat gy 37 � The base case final consumption, outside the energy sector, is G (physical units) � The increment to final consumption is g (physical units) � The increment to total energy use is f (physical units) � the total energy use in the modified case is F 0 ¼ F þ f (physical units) The formula we want to calculate is the relationship between f and g. By definition f =g > 1, i.e. f has the same sign as g and is larger in magnitude. To determine the solution, we consider a sequence of approxi- mations, F0, F1,. Fnwith Fn converging to F 0 as n/N. Each term in the sequence is calculated by adding the appropriate linear incre- ments. The corresponding sequence of approximations to f are denoted fn, with fn/f as n/N. The calculation is initialized by setting F0 ¼ F and f0 ¼ f . At the first iteration we set F1 ¼ F0 þ f0; (2) and in general, for the nth increment Fnþ1 ¼ Fn þ fn: (3) The expression for Fnþ1 can be rewritten as Fnþ1 ¼ F0 þ ðf0 þ f1 þ.þ fnÞ; (4) which will be used below to derive the closed-form expressions Fig. 3. Multiplier m and the linear approx K. Coughlin / Ener in the limit n/N. For each example, the calculation proceeds in three basic steps: initialization of F0 and f0, derivation of the equation for the increment fn, and solution for F 0 based on the sum in equation (4). 3.2.1. Example 1 We return here to the example of a single fuel, with c equal to the fuel intensity of fuel production. The starting increment f0 ¼ g, and the starting approximation to the solution is F0 ¼ F. The next term f1 is defined by adding the fuel needed to produce f0: f1 ¼ cf0; (5) which leads to F1 ¼ F0 þ f0 þ f1 ¼ F þ f þ cf : (6) Similarly, in the nth termwe add the fuel used in the production of fn�1, so fn ¼ cfn�1: (7) Based on equation (7) the general term can be written (MWh) Fnþ1 ¼ F0 þ f0 � 1þ cþ c2 þ.cn � : (8) The sum in brackets is the Taylor series expansion of the func- tion 1=ð1� cÞ, which leads to F 0 ¼ F þ g 1� c (9) or equivalently f ¼ g 1� c (10) The sum in brackets converges because 0 < c < 1, as noted above. 3.2.2. Example 2 In the second example we look at the case of an energy carrier rather than a primary fuel. The canonical example is electricity, but this approach can also be used for energy carriers such as steam or heat. As in example 1, we assume for simplicity that only one primary fuel is used; we also assume that only electricity is used to extract and process this fuel. This calculation requires several new variables: � The base case total electricity use for the whole economy is E � The base case final electricity consumption, outside the energy ion mlin as a function of the parameter c. (2012) 698e708 701 sector, is R (MWh) � The increment to final consumption of electricity is r (MWh) � The increment to total consumption of electricity is e (MWh) � The total electricity use by the whole economy in the modified case is E0 ¼ E þ e (MWh) � The quantity of fuel required to produce one MWh of grid electricity is a (physical units per MWh) � The electricity intensity of fuel production is b (MWh per physical unit) Here a refers to the fuel required to deliver a unit of electricity to the consumption site, which we call the burn rate. Because b describes an electricity end-use, the parameters a and b need to be calculated using consistent definitions of the consumption site. The increment r leads to changes in both electricity and primary fuel use, so the iterative approach now requires two sequences, E0, E1,. En with En converging to E0 as n/N, and the Fn defined as in Example 1. The calculation is initialized by setting E0 ¼ E, e0 ¼ r and F0 ¼ F. An increment to electricity production implies an y 37 increment in the amount of fuel used, so the initial value of f is equal to f0 ¼ ae0: (11) At the next iteration, the change f0 to fuel production in turn implies a change in the electricity use, so e1 ¼ bf0 ¼ aben�1: (12) The general sequence of increments is en ¼ bfn�1 ¼ aben�1; (13) for electricity and fn ¼ aen ¼ abfn�1 (14) for the primary fuel. As in example 1, as long as ab < 1, the sum over n is equal to the function 1=ð1� abÞ. The general solution is e ¼ r 1� ab (15) and f ¼ ar 1� ab: (16) Rearranging gives the relationship e ¼ r þ bf : (17) When the full fuel cycle is taken into account, the additional electricity use is just the amount required to produce the fuel increment f. The product ab is a dimensionless number repre- senting the quantity of fuel used to produce a unit of fuel output, through conversion to electricity. This reflects the fact that ulti- mately only primary fuels are used or conserved; energy carriers act as conversion technologies only. The efficiency of conversion is physically constrained to be less than one, so ab < 1 as required for convergence. The definition of a multiplier for this case, and for the remaining examples, is a little more subtle and is left to Section 3. 3.2.3. Example 3 In this example we illustrate how to combine multiple energy sources. We again assume only one primary fuel in the economy, and allow both this fuel and electricity to be used in fuel produc- tion.We calculate the total change in electricity and fuel production for the whole economy that result from a change r in final consumption of electricity. The notation is as defined in Examples 1 and 2. In this case the initial values are e0 ¼ r and f0 ¼ ar. As above, at each successive iteration, we add the increment to fuel/electricity required to produce the previous increment. For electricity, each successive increment is induced by the increment to fuel produc- tion through the equation enþ1 ¼ bfn: (18) For the primary fuel, the nþ 1st increment is the fuel needed to supply the nth increment to both electricity and fuel production: fnþ1 ¼ aenþ1 þ cfn ¼ ðabþ cÞfn: (19) Following the steps outlined in the previous examples leads to the solution ar K. Coughlin / Energ702 f ¼ 1� ðabþ cÞ: (20) Solving the equations for E0 leads once again to equation (17). These simple examples have been used to demonstrate the method and clarify the logic of the approach. In the next sectionwe present the equations for the general case. 3.3. The general case In the most general case multiple fuels are used in the energy sector, both to generate electricity and directly in the fuel produc- tion chain. The problem is conceptually no different from the simple examples presented above, but the accounting requires a slightly more elaborate notation. The definitions of the variables R, r, E0 and e are as in Examples 2 and 3 above. The rest of the notation is modified as follows: � x and y are used to label different fossil fuels (coal, natural gas, petroleum-based fuels, etc.) � Fx is the total annual production of fuel x in the base case � gx is an increment to final consumption of fuel x � F 0xis the total annual production of fuel x in the modified scenario � fx is the total (FFC) increment in the modified scenario, with F 0x ¼ Fx þ fx � ax is the average burn rate of fuel x in electricity production; if fuel x is not used for electricity then ax ¼ 0 � bx is the electricity intensity of production for fuel x � cxy is the quantity of fuel y used directly in production of fuel x (both measured in physical units) In the calculations below we introduce matrix and vector notation for these variables; a is the vector with components ax etc., and C is the matrix with components cxy. The symbol I is used for the unit matrix. It is important to define ax in a way that is consistent with how the electric grid operates. Electricity is withdrawn from a grid within which a large number of power plants using different fuels and generation technologies are interconnected over a specific region. Hence, consumption of grid electricity can’t be attributed to a particular type of fuel or plant. This is handled here by defining ax as an average for the region; ax is the total quantity of fuel x consumed per year divided by the total MWh output per year. This introduces a geographic dependence of the parameters, which needs to be considered in practical calculations, but which can be ignored in this methodology discussion. In practice the value of ax depends on three factors which may vary independently: the fraction of electricity in the region that is produced by fuel x, the power plant heat rates (MJ/MWh), and the fuel heat content qx. Of these three, the most important source of variability is the fraction of generation allocated to each fuel. This step of the calculation accounts for the use of renewables to generate electricity. Tech- nologies based on renewable energy fluxes such as wind or solar do not use fuel per se, so the presence of renewables will lower the fraction of electricity generated from any primary fuel, and so lower the value of ax. Note we do not include energy from biomass here as a renewable, because biomass feedstock is a material quantity with its own production chain, not a naturally occurring energy flux. The approach defined here can be applied to biomass energy, but needs to be generalized to account for the use of other resources such as land and water [4]. To clarify the notation, in the equations below the iteration index n is moved to a superscript. To distinguish between the iteration index and exponents, we use parentheses on the iteration index. In the most general case, we assume an arbitrary increment to both electricity use and final consumption of fuel x; the calcu- (2012) 698e708 lation is initialized using the starting increments gx and r: x gy 37 The total increment to the production of fuel x depends on the increment to electricity use, and the change in demand for any other fuel which uses fuel x in its production chain, i.e. if cxy is not zero, then a change in the production level of fuel y implies a change in fuel x. Thus, the fuel increments must satisfy the matrix equation f ðnþ1Þx ¼ X y � axby þ cxy � f ðnÞy : (23) Here the product axby represents the indirect use, via elec- tricity generation, of fuel x in the production of fuel y, and cxy represents the direct use of fuel x to produce fuel y. Defining the new matrix vxy ¼ axby þ cxy, and using equation (4), the equation for F is Fðnþ1Þ ¼ Fð0Þ þ � Iþ V1 þ.þ Vn � ,fð0Þ: (24) As in the simpler examples, we get the analytic solution by taking the limit as n/N of the sum in brackets. The mathematical details are discussed in Appendix A; here we will just assume that the limit exists. We define the matrix of coefficients mxy as MhIþ XN n¼1 Vn: (25) Structurally, the important point is that M is a non-linear function of the physical parameters a, b and C. Substituting equa- tion (25) into equation (24) gives the solution f ¼ M,fð0Þ ¼ M,ðar þ gÞ (26) The sum of g and ar represents the total increment to final consumption with electricity use expressed in fuel terms. The equation for the total increment to electricity production is Eðnþ1Þ ¼ Eð0Þ þ eð0Þ þ b, � fð0Þ þ fð1Þ þ.þ fðn�1Þ � ; (27) which leads to the generalization of equation (17), e ¼ r þ b,f: (28) The parameters bx and cxy are defined for the entire production chain, which in practice will consist of a number of successive steps. Appendix B provides a brief overview of how calculations for a series of process steps should be combined into a single param- eter value. Detailed calculations using this approach for electricity and natural gas are presented in [5]. 4. Definition of the FFC multiplier and other metrics The equations presented above define the relationship between full fuel cycle energy use and final consumption. In the simplest case of a single fuel, the FFCmultiplier m is equal to the ratio f =g, but for more complicated cases we need to add another step, where eð0Þ ¼ r; f ð0Þx ¼ gx þ axr: (21) At each iteration, electricity use is modified by the amounts used in production of various fuels: eðnþ1Þ ¼ X bxf ðnÞ x for n � 0: (22) K. Coughlin / Ener primary fuels are converted to equivalent energy units and sum- med. For this step we need the following variables: � qx is the heat content of fuel x (MJ per physical unit) � ε (epsilon) is the energy content of the set of final consumption increments gx plus r (MJ) � ε0 is the energy content of the set of total fuel increments fx (MJ) The multiplier m is defined as the ratio m ¼ ε0=ε: (29) Note that, in calculating ε0, only the primary fuels fx should be used, as the change in electricity use is already included through the ar term in equation (26) 4.1. FFC multiplier The energy content of the fuels fx and gx is given by ε ¼ X x qxðgx þ axrÞ ¼ q,ðgþ arÞ: (30) and ε 0 ¼ X x qxfx ¼ q,f ¼ q,M,ðgþ arÞ: (31) Hence, the multiplier is m ¼ ε 0 ε ¼ q,M,ðgþ arÞ q,ðgþ arÞ : (32) Its important to be careful when using these equations to analyze energy carriers such as electricity. What matters for the FFC accounting is the fuel consumed in electricity production, and the associated nonlinearities in the fuel cycle. It is not correct to use the ratio e=r as the FFC multiplier for electricity. In a scenario in which only the final consumption of electricity is altered, the multiplier is defined by setting f ¼ 0 in equation (32), which gives melectricity ¼ q,M,a q,a : (33) If we’re interested in a case where only the final consumption of fuel x is modified, we can define a multiplier for this fuel as mx ¼ P yqymyx qx : (34) These special cases apply, for example, to the full fuel cycle savings associated with electricity or natural gas conservation in buildings [6]. 4.2. Other metrics Depending on the context, it may be useful to convert the multiplier m to some other metric. In this section we define several variants of the FFC multiplier, and provide explicit functions for the case described in Example 1. � FFC adder a The “fuel cycle adder” a (alpha) represents the quantity of energy used in fuel production per unit of fuel made available to the rest of the economy. The amount of energy used in fuel production is ε0 � ε, and the amount made available for final consumption is ε, so ε 0 � ε (2012) 698e708 703 a ¼ ε ¼ m� 1: (35) Plin ¼ Sc: (41) The factor 1=ð1� cÞ2 in equation (40) represents an amplifica- tion of the impact on price of changes in the energy intensity parameter c, which occurs because of the nonlinearity of the fuel cycle. While the linear approximation is blind to the existence of a physical limit at c ¼ 1, the non-linear formula ensures that this singularity appears in the economics. Even for moderate values of c the difference between the non-linear and linear expressions can be significant. To the extent that fuels can be substituted for one another in applications, an increase in the energy intensity of production for one fuel can lead to price increases in other fuels. To make a similar type of argument in themore general case defined by equation (32), one would need to look at the behavior of the eigenvalues of the y 37 � Conversion efficiency p The notion of efficiency can have a variety of specific metrics associated with it, but the general idea is to define some kind of output-to-input ratio for a process. For example, the thermal effi- ciency of a heat engine is the percentage of input heat energy that is output in the form of work. Mathematically, an efficiency is required to be a dimensionless number that takes on values between zero and one. Here, we define a conversion efficiency p (pi) for the fuel cycle as the ratio of “energy out/energy in”, where energy out is the energy provided to the rest of the economy, and energy in is the energy in the total fuel extracted by the energy production sector. With this definition, p ¼ 1 m ¼ ε ε 0: (36) Given that m � 1, it follows that p˛½0;1� as required. � Energy profit g The term “energy profit” g (gamma) is used here to describe our variant of a net energy metric. It is equal to the ratio of the energy that ismade available to the rest of the economy to the energy that is used to produce it. This is just the inverse of the fuel cycle adder, so g ¼ 1 a ¼ ε ε 0 � ε: (37) � Energy return on energy invested (EROI) Loosely defined EROI is the ratio of the energy “delivered to society” [3,22] by a process to the total energy used in that process. Here we interpret this textual definition as follows: A property of EROI that is commonly cited is that, when EROI ¼ 1 there is no net gain of energy from the production process. We can define a metric that has this property if we interpret “delivered to society” as total energy production (final consumption plus what is used in the energy sector), and assume that the energy “used in the process” is the energy used by the energy production sector. With these definitions the ratio of the two is m=a : EROI ¼ ε 0 ε 0 � ε ¼ m a ¼ m m� 1: (38) Since m � 1, mathematically EROI � 1, and the value of EROI will approach one as m becomes large. The reason for this is clear if we return to Example 1 - in this case the ratio m=a reduces to EROI ¼ 1=c, so an EROI of one implies that c ¼ 1. This is the physical limit of zero production; in this limit both m and a become infinite. Equations for each of the above metrics are summarized in Table 1, along with the appropriate formulae for the case of a single fuel described in Example 1. Definitions for the more general case described by equation (32) are given in Appendix A. Three of the metrics are also plotted as a function of the param- eter c in Fig. 4 (the curve for energy profit g looks similar to EROI, and the curve for the fuel cycle adder a is similar to that for m). Although all the metrics contain the same information, the figure shows that they display very different levels of sensitivity to variation in c. The efficiency metric p is a linear function of c, and is well-behaved over the entire interval c˛½0;1�. The others are all non-linear functions, which diverge either as c/1 or as c/0. This should be borne in mind when using these metrics in K. Coughlin / Energ704 applications. 5. Nonlinearity and dynamics In this section we construct a very simple model of the relationship between the FFC multiplier and fuel prices, and use it to illustrate the importance of non-linear dependence of m on the energy intensity parameters. It may be intuitively obvious that, as the energy intensity of primary fuel production increases, the price of fuel must also increase. The key point is to realize that production costs scale with the amount of raw material handled, while revenues scale with the amount of marketable output, i.e. the amount available for final consump- tion. The relationship between these two is governed by the FFC multiplier. To explore these ideas in more detail, we work through the Example 1 case of a single fuel. Let the cost of production per unit of raw material be S, and the market price per unit sold be P. For simplicity, we assume S contains the expected profit and ignore fixed costs. For each unit of raw material that enters the fuel production chain, the amount sold is 1=m, and so the revenues obtained are P=m. On average, for the industry to be viable, costs must equal revenues which implies that P ¼ mS. It follows that P depends on the energy intensity parameter c through the equation P ¼ S 1� c: (39) Now consider a situation in which the value of c changes over time and induces a corresponding change in price. Using the notation dc=dt ¼ _c etc., we have _P ¼ S _c ð1� cÞ2 : (40) The same calculation using the linear approximation to the fuel cycle multiplier (mlin ¼ 1þ c) leads to _ _ Table 1 Summary of FFC metrics. Name Symbol Energy equation Example 1 formula Energy available for final consumption ε Full fuel cycle energy ε0 Fuel cycle multiplier m ε0=ε 1=ð1� cÞ Fuel cycle adder a ðε0 � εÞ=ε c=ð1� cÞ Energy profit g ε=ðε0 � εÞ ð1� cÞ=c Conversion efficiency p ε=ε0 1� c Energy return on energy invested ε 0=ðε0 � εÞ 1=c (2012) 698e708 matrixM as a function of the different physical parameters (see also ROI gy 37 Fig. 4. Multiplier m, conversion efficiency p, and E K. Coughlin / Ener Appendix A). The more general case includes the possibility of producing one form of fuel using other fuels as input and taking a net loss in energy terms; equation (40) clarifies that this cannot be done profitably without subsidies or other special pricing arrangements. In the real world, many factors influence the total expendi- tures on energy by final consumers. However, market indicators may be strongly influenced by the production costs associated with marginal suppliers, i.e. those suppliers whose output fluc- tuates in response to market demand. These are also likely to be suppliers with the highest costs and highest fuel cycle energy use; this is the case for example for oil production from tar sands, deep water drilling or oil shale. Although the argument presented above is extremely simplified, equation (40) points the way towards a mathematically clear link between high volatility of energy prices and the energy intensity of production for marginal sources. 6. Discussion In this paper, we have derived a mathematically precise formula that captures the full fuel-cycle impacts of changes to consumption of energy at the point of use. These impacts are expressed through the full fuel-cycle multiplier and related metrics, which are easy to use in applications. The FFC energy provides a more accurate measure of the full impacts of energy policies and resource devel- opment paths that affect the demand for a particular form of energy. The approach focuses on the relationships between physi- cally meaningful variables, and applies irrespective of the details of particular production processes. It can be generalized to include spatial variation in the parameters, and applied in a temporally dynamic framework, as needed. as a function of the energy intensity parameter c. (2012) 698e708 705 The analysis of physical relationships is not easy, but they are much more heavily constrained than are economic relationships, and so may provide a more solid basis for extrapolating the outcomes of uncertain energy development pathways. A focus on the physical picture also ensures that the underlying processes are represented in a way that is complete and consistent. In contrast, for cost-benefit analysis, any component of the system that doesn’t have a price essentially drops out of the picture. While economists have made progress in expanding the concept of price to cover a wider range of resources and services [12], this approach can be quite laborious compared to the relatively straightforwardmethods that use physical variables. We believe the approach developed here can help to improve the rigor of energy policy analysis while keeping the modeling assumptions relatively clear. This accounting framework supple- ments, rather than displaces, more general approaches such as LCA. The physical parameters defined here can be calculated using a variety of engineering or economic simulation tools, and so provide a means of comparing the output of these more complex models. Ultimately, if the energy requirements of a very broad range of energy production systems can be described within a single mathematical formalism, this will ensure that the numbers used to compare different options are truly meaningful. Acknowledgments This work was supported by the Assistant Secretary for Energy Efficiency and Renewable Energy, Building Technologies, U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Wewould like to thank Gabrielle Wong-Parodi, David Fridley, Andy Sturges and Lisa Thompson for early reviews and comments on this paper. generalized to any number of steps, and a similar approach used for y 37 Appendix A. Linear algebra In this appendix we provide some mathematical support for the assumption in equation (25) that the sum definingM exists. Strictly speaking, thematrix algebra should use dimensionless numbers. As defined in the text, the elements of V have physical units attached to them; they are non-dimensionalized here using the heat content factors qx. We define � ~f x ¼ fxqx � ~ax ¼ axqx � ~by ¼ by=qy � ~cxy ¼ cxyqx=qx � ~vxy ¼ vxyqx=qy � ~mxy ¼ mxyqx=qy The matrix elements ~cxy and the products ~ax~by are now dimensionless, while ~f x has the dimensions of energy (MJ). Returning to equation (23), it becomes ~f ðnþ1Þ x ¼ X y � ~ax~by þ ~cxy � ~f ðnÞ y : (42) which leads to ~MhIþ XN n¼1 ~V n : (43) If the square matrix ~V is diagonalizable, then the sum exists and the relationship can be rewritten formally as ~M ¼ � I� ~V ��1 (44) An m by m square matrix is diagonalizable if it has m distinct eigenvalues. All the entries of the matrix ~V are positive or zero, and because all of the entries are independent, it has full rank. Thus, in general we expect the matrix to have m distinct eigenvalues [24]. Assuming the m eigenvalues of ~V, defined as lk, k ¼ 1;.;m are distinct, it can be shown that ~V can be represented in the form ~V ¼ Z�1LZ (45) where Z is an invertible matrix, Z�1 is the inverse of Z, and L is a diagonal matrix whose diagonal entries are the eigenvalues of ~V. This relationship allows us to easily evaluate ~V n : ~V n ¼ � Z�1LZ �� Z�1LZ � . � Z�1LZ � ¼ Z�1ðLÞnZ (46) The elements of ðLÞn are just ðlkÞn ; the usual algebraic steps allow us to write ~M ¼ Z�1DZ (47) where D is a diagonal matrix with diagonal entries equal to dk ¼ 1 1� lk : (48) Equation (48) shows that the fuel cycle multiplier will diverge if any of the eigenvalues approaches one; this is the generalization of the limit c/1 in Example 1. Note that the values dk are still defined for lk > 1, so formally the multiplier can describe a production process that is actually an energy sink. Such a process could exist in a sub-sector of the energy production chain. Returning to the original matrices V and M, we can derive an K. Coughlin / Energ706 expression for a as follows: M satisfies M� I ¼ XN n¼1 Vn ¼ V Iþ XN n¼1 Vn ! (49) so M ¼ Iþ VM (50) Substituting into equation (32) leads to m� 1 ¼ a ¼ q,VM,ðgþ arÞ q,ðgþ arÞ : (51) The other metrics discussed in Section 4 can be defined in a similar fashion. Appendix B. Multi-step processes and loss rates The coefficients bx and cxy represent the total electricity and fuel use in the production process, per unit of marketable fuel output. The production chain generally consists of a series of steps, each of which has its own energy requirements and loss rates. In this appendix, we provide a brief summary of how the formalism can be extended to represent a series of process steps. To indicate successive steps in a process, we use the superscript index ½k�. In general, each step can have its own values for the energy intensity coefficients, which we denote b½k�x and c ½k� xy . As some process steps may also involve losses, to aggregate all the steps into a single coefficient, we need to keep track of the fraction of material that passes from step k to step kþ 1. We define this fraction as g½k� � 1 ; the percent loss at step k is therefore equal to 1� g½k�. The energy use coefficients are always defined as fuel or electricity use per physical unit of material input to that process step. If necessary, the ratios g[k] can be used to convert these coefficients to energy intensity per unit of material output. As an example, consider the electricity use for a 3-step process consisting of extraction, processing and distribution to the final consumers (in the equations below, to lighten the notationwe drop the subscript x). We define b½0�, b½1� and b½2� as the electricity use per unit of material handled in each step, g½1� as the fraction of the material extracted that can be processed (step 0 to step 1), and g½2� as the fraction of the material processed that can be distributed (step 1 to step 2). The fuel that is distributed is the marketable product. It follows that � the electricity use in step 0 is b½0� per unit of material extracted � the electricity use in step 1 is b½1� per unit of material processed, or b½1�g½1� per unit of material extracted � the electricity use in step 2 is b½2� per unit of material distrib- uted, or b½2�g½2�g½1� per unit of material extracted. The total electricity use per unit of material extracted is the sum b½0� þ b½1�g½1� þ b½2�g½2�g½1�. The coefficient b, which is defined as the electricity use per unit of marketable material, is equal to this sum divided by g½2�g½1� : b ¼ b ½0� þ b½1�g½1� þ b½2�g½2�g½1� g½2�g½1� : (52) The denominator is equal to the amount of marketable material obtained per unit of material extracted. This equation can be (2012) 698e708 the coefficients cxy. Appendix C. Multiplier for natural gas In this appendix we provide a detailed example calculation of the fuel cycle multplier for natural gas. This calculation is approx- imate and meant to be illustrative only. For simplicity, we assume that the only fuel used in natural gas production is natural gas. This is approximately true for conventional gas production [16] and is consistent with the accounting methods used for the natural gas industry by the EIA (Energy Information Agency) [9,8]. The major steps in the production chain are extraction (k ¼ 0), separation ðk ¼ 1Þ, transmission through the interstate pipeline system (k ¼ 2Þ and distribution from the city gate to final consumers could be significant relative to demand growth, which is also on the order of 1e2% a year. ction. Net pipeline imports Lease and plant Adjusted lease and plant Pipe- line use Final consumption Multiplier E F G ¼ F(1 þ E/D) H I ¼ D þ EeGeH (G þ H þ I)/I 2.81 0.73 0.84 0.57 20.0 1.070 3.05 0.76 0.88 0.58 19.6 1.075 2.94 0.78 0.91 0.58 20.0 1.075 3.06 0.86 1.00 0.62 20.7 1.078 2.71 0.86 0.98 0.65 21.2 1.077 Table 3 Forecast of the natural gas multiplier and EROI for different values of the shale gas energy intensity parameter s. The fraction of supply coming from shale, as forecast by the EIA, is z. Year z Multiplier EROI s ¼ 1 s ¼ 1:5 s ¼ 2 s ¼ 1 s ¼ 1:5 s ¼ 2 2010 0.226 1.130 1.139 1.147 8.70 8.22 7.79 2015 0.321 1.118 1.129 1.139 9.48 8.78 8.17 2020 0.350 1.115 1.126 1.137 9.72 8.94 8.28 2025 0.404 1.111 1.123 1.136 10.0 9.12 8.36 2030 0.436 1.108 1.121 1.134 10.3 9.28 8.46 2035 0.465 1.107 1.121 1.135 10.3 9.27 8.41 K. Coughlin / Energy 37 (2012) 698e708 707 (k ¼ 3Þ. The EIA historical data series include time series for production from different sources distinguished as off-shore or on-shore, associated (with oil production) or non-associated, and uncon- ventional (shale gas and coal bed methane). The data is not suffi- ciently detailed to determine whether the energy intensity of production varies significantly with the source type, so the only distinction we retain in this analysis is shale gas vs. all other. The natural gas used in natural gas production is tabulated as “Lease and Plant” and “Pipeline and Distribution” use. Lease and plant use includes both extraction and separation steps. Pipeline use includes transmission, but it is unclear whether it also includes local distribution to the final consumer. Material loss rates are also tabulated by EIA. Losses at the extraction stage include venting and flaring, gases used for repressurizaton, and non-hydrocarbon gases removed. At the separation stage, the fraction removed as natural gas liquids is tabulated as extraction loss. The natural gas delivered to consumers includes net pipeline imports from Canada and Mexico, and net imports of liquified natural gas (LNG). The production chain for LNG includes additional steps that consume significant energy, but these steps are not modeled here. LNG currently provides less than 2% of US consumption, and EIA forecasts that this proportion will remain small through 2035 [8], so neglecting these details will not have a large impact on the numerical estimate of themultiplier. The production of shale gas also requires additional energy consuming steps, for example the transport and treatment of produced wastewater. In this case EIA forecasts rapid growth, with shale gas providing over 45% of US supply in 2035 [8], so the additional energy use is likely to become numerically important. Given the lack of data on the details of shale gas production, we model the additional energy use as if it were an increment to lease and plant consumption, with the size of the increment treated as a variable. The input data and calculations are summarized in Table 2, which shows data for 2004e2009. All volumes are in trillion cubic feet (Tcf). In calculating the multiplier, net pipeline imports are included as part of US supply, and it is assumed that the energy use in extraction and processing is the same as for domestic production. Hence, lease and plant consumption is scaled up proportionally, as Table 2 Summary of natural gas data and calculation of the multiplier for natural gas produ Year Extraction Losses in extraction (k ¼ 0Þ g0 Losses in separation (k ¼ 1Þ g1 US dry gas production A B 1-B/A C 1-C/(A-B) D ¼ AeBeC 2004 23.97 4.45 0.81 0.93 0.95 18.59 2005 23.46 4.53 0.81 0.88 0.95 18.05 2006 23.54 4.13 0.82 0.91 0.95 18.50 2007 24.66 4.47 0.82 0.93 0.95 19.27 2008 25.64 4.52 0.82 0.95 0.95 20.16 2009 26.01 4.41 0.83 1.02 0.95 20.58 2.26 shown in the Adjusted Lease and Plant column. The EIA data for pipeline use presumably already include imports, so these numbers are not adjusted. The loss factors defined in Appendix B are calculated for the extraction and processing steps; we assume there are no losses in the transmission and distribution steps. Because the EIA data do not map directly onto our definition of production steps, it isn’t possible to compute the parameters for each step k, but it is straightforward to calculate the multiplier. The required algebraic steps are noted in the second row of Table 2. The calcu- lation gives a multiplier of about 1.075 with little variation over time. For comparison, the multiplier output by the GREET model in their study for DOE is 1.063 [7]. The same calculations can be repeated using forecasts from the Annual Energy Outlook [8]. For these forecasts, we model the possible impact of increased energy use for shale production as follows: Let z be the fraction of supply that comes from shale gas, and s a parameter defining the energy intensity of shale gas production relative to conventional gas; s ¼ 1:5 indicates that shale gas is 50% more energy intensive. The additional energy requirement is modeled by multiplying the lease and plant consumption by the factor 1� zþ sz: (53) The output of these calculations for several values of s are given in Table 3. The table also includes the EROI defined in equation (38). The calculation based on the EIA forecast assuming no difference in energy intensity for shale gas is given in the s ¼ 1 column. The EIA data effectively forecast a declining multiplier, as total dry gas production increases over time while the lease and plant use actually decreases slightly [8,5]. In 2035, if shale gas is 50% more energy intensive to extract, the multiplier increases from 1.107 to 1.121. If shale gas production is 100% more energy intensive, the multiplier is 1.135. These are not particularly large changes, but 0.91 1.01 0.60 21.2 1.076 References [1] Brandt R. Converting oil shale to liquid fuels with the alberta taciuk processor: energy inputs and greenhouse gas emissions. 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A full fuel-cycle analysis of energy and emissions impacts of transportation fuels produced from natural gas. Technical Report ANL/ESD-40. Argonne National Laboratory; January 2000. [30] Xie X, Wang M, Han J. Assessment of Fuel-Cycle energy use and greenhouse gas emissions for Fischer-Tropsch diesel from coal and cellulosic biomass. K. Coughlin / Energy 37 (2012) 698e708708 Report 43, James A. Houston, TX: Baker III Institute for Public Policy, Rice University; January 2010. Environmental Science & Technology April 2011;45(7):3047e53. A mathematical analysis of full fuel cycle energy use 1. Introduction 2. Approach 3. Mathematical derivations 3.1. Conceptual framework 3.2. Example calculations 3.2.1. Example 1 3.2.2. Example 2 3.2.3. Example 3 3.3. The general case 4. Definition of the FFC multiplier and other metrics 4.1. FFC multiplier 4.2. Other metrics 5. Nonlinearity and dynamics 6. Discussion Acknowledgments Appendix A. Linear algebra Appendix B. Multi-step processes and loss rates Appendix C. Multiplier for natural gas References