Table 1 compares the gravimetric energy density (amount of energy per unit of weight) of pre-industrial biomass fuels with the fossil resources of the industrial era and the nuclear fuel resource of the atomic era. Air-dry wood has energy density around 16 MJ/kg and most other biomass fuels have energy densities below 20 MJ/kg, while good quality bituminous coal is over 24 MJ/kg. The energy density of crude oil is just below 42 MJ/kg and that of refined oil products is 43-46 MJ/kg. Moreover, the energy density of uranium is over 3·10⁶ MJ/kg. Solid and liquid fuels have an even greater advantage in terms of volumetric density (amount of energy per unit of volume) in comparison to biomass and gaseous fluids: natural gas rates around 35 MJ/m³ while crude oil has approximately 35 GJ/m³, i.e. its volumetric density is one thousand times higher [17-19].

The historic transitions from biomass to coal and then from coal to petroleum entailed a movement towards more concentrated sources of energy. Higher energy density carriers present significant advantages in terms of their extraction, portability, shipping and storage costs, and conversion options [17]. The greater the energy density (gravimetric and volumetric) the more energy transported or stored for the same amount of weight or volume. The changeover to a high energy density supply infrastructure took place not only at the resource harvest links in the supply chain but also at the conversion nodes and delivery links.

Table 2 compares the power densities (energy flux per unit of horizontal surface) of alternative electricity supply infrastructures. All renewable generation technologies have power densities that are substantially lower (2-3 orders of magnitude) than the fossil-fuelled modes. The modest energy density of renewable sources and the very low power density of renewable energy extraction imply that these new technologies will require much larger infrastructures, spread over significantly greater areas, relative to today's infrastructure of fossil fuel extraction, combustion and electricity generation, to produce an equivalent quantity of energy [17,20]. Renewable technologies will generally require larger energy expenditures for the initial emplacement of their facilities–i.e. they will entail higher emplacement energy costs per unit of nameplate capacity.

Advances in most human endeavors—transportation, agriculture, commerce, science and technology, health care, household life—were driven directly or indirectly by the changes in society's underlying energy systems and the availability of surplus energy. Indeed, the extraordinary expansion of the human population, economic growth and rising standards of living were powered by high-EROI, high energy surplus fossil fuels [2,20-21].

In the pre-industrial age, the attainable energy density at the output of a converter node was constrained by practical capacities of the harvest and shipping channels which supplied it. This in turn constrained the population density that could be supported. Food collection and delivery constrained the concentration of population to what its hinterland could support–primarily small villages embedded in the hinterland itself, to reduce delivery distances to a day's travel or so. Waterwheels driving grain mills and sawmills of the 1800's delivered very modest amounts of power. As a result, the cities of the preindustrial age were relatively small and societies were predominantly rural and agricultural [22].

Throughout the pre-industrial era, not only population densities, but overall populations were constrained by the infrastructure for food (energy) delivery. Malthus stated the constraint on a nation's overall population as a function of arable land availability (i.e., food/energy supply). Sustainability was maintained for many centuries preceding the late 1700's as a quasi-steady state balance of energy supply and population – but it was maintained at a small world population and at a medieval lifestyle of stagnant (and small) GDP per capita [23].

The transition to high energy density carriers and converters, where it has taken place in the Western industrialized nations, has dramatically changed the character of society. Population is now concentrated in cities, many of which are huge compared to the pre-industrial era. Population migrated to the factory towns of England and America during the 1800's to exploit the concentrated energy density from coal-fueled steam power. Factory production rapidly replaced the earlier cottage industry regime of societal organization.

There are several underlying causes for these historical changes in society that accompanied the evolution in energy supply infrastructure. They happened in part because the new energy supply infrastructure delivered an increased net surplus energy relative to that required to maintain the earlier medieval steady state. Also, institutional arrangements were made which facilitated corresponding concentration of capital and labor to match the concentration of energy.

Several figures of merit are in use to characterize the net (excess) energy output to be derived from emplacing or enlarging an energy supply infrastructure under an energy plowback constraint. One example is the energy return on (energy) investment (EROI). It is defined as [2, 26-28]: EROI = gross quantity of energy delivered over the infrastructure lifetime quantity of energy expended to emplace and operate the infrastructure overs its lifetime

The numerator is given by: ( Numerator ) = P n p ⋅ ψ ⋅ Twhere

P_np is the nameplate power capacity of the infrastructural facility;

The denominator is equal to: the sum of the energy, E, expended to initially emplace (and ultimately decommission) the infrastructural facility, plus the energy expended for the operation and maintenance (O&M) of the facility over its lifetime: ( Denominator ) = E + P n p ψ h Twhere

E is the energy expended to emplace and ultimately decommission the facility;

Thus, (1) EROI = P n p ψ h T E + P n p ψ h T .

Note that in this definition:

the energy content of the fuel resource harvested and processed is not included in the denominator – i.e., only the energy needed for the infrastructure per se is accounted for;

Inspection of the formula shows that EROI will be increased by:

reducing E P n p, the emplacement energy/nameplate capacity;

When determining the numerical value of EROI for an energy supply infrastructure, it is necessary to consider the entire supply chain from harvesting the resource to delivering energy services to end users – summing up all the energy expended to emplace and operate that supply chain. This involves a life cycle analysis (LCA) to evaluate energy consumption elements in every stage of the supply chain—the resource collection, energy conversion, and the energy distribution and delivery links of the infrastructure.

The extent of upstream and downstream energy expenditures to emplace new infrastructure is not standardized, and as a result, ranges of values for EROI are found in the literature. LCAs are extensive undertakings when performed individually for a specific infrastructure, and are even more complex when done to enable a comparison of alterative infrastructures under consistent assumptions. Still, there are several comparative LCAs in the literature that have taken care to report values for EROI under consistent assumptions for alternative candidate infrastructures.

Given an EROI value for a candidate infrastructure, the excess energy available for societal use is [3,29]

As Hall [30] points out, while much of human progress can be attributed to technology, much of that technology has been a means for using more energy for human ends. Surplus energy is what facilitates economic growth, technological progress, and most human endeavors. If energy supply infrastructures with high EROI are deployed then only a small portion of society's energy budget would be required by the energy sector itself. The rest could be utilized to support all economic, commercial, and social activities that are so critically dependent upon energy [20].

The EROI as a figure of merit pertains to “efficiency” of an infrastructure–its ability to deliver excess energy to society–integrated over its lifetime (given an assumption about the availability of a resource input). However, it does not address:

scalability (because it is a ratio);

An informative way to think about EROI is in terms of the fraction of lifetime gross energy output that is expended for initial emplacement and that which is needed for operation and maintenance: (2) EROI = P n p ψ T E + P n p ψ h T = 1 E P n p ψ T + hi.e., EROI = 1 ( fraction of gross production expended for initial emplacement ) + ( fraction of gross production expended for Q & M )

The first term represents an initial “capitalization” expenditure of energy. And the second term represents an ongoing “variable” expenditure. However, unless the details of the LCA are available, this breakdown is not evident because when EROI values are reported, the two components of the denominator become subsumed and indistinguishable within a single number.

A second figure of merit, the “energy payback period”, τ₁, is used to characterize the “capital” component. The energy payback period is a measure of the time interval required for the infrastructure – once it is installed – to deliver net energy sufficient to cover the initial energy investment [31,32] (3) τ 1 P n p ψ ( 1 − h ) = Eor (4) τ 1 = E P n p ψ ( 1 − h )

Writing

it can be seen that the ratio of energy payback period to infrastructure lifetime, τ₁/T, will be much less than 1.0 (which is obviously desirable) if:

E P n p the energy expended for emplacing a unit of nameplate capacity is small;

If the detailed reporting of a LCA includes values for both EROI and τ₁, then it is possible to”back calculate” the two subcomponents of EROI.

The doubling time metric, τ₂, is a measure of the time interval required to accumulate enough excess energy to deploy new infrastructure sufficient to double power output. It is necessary to develop equations for the infrastructure's dynamic response in order to find a mathematical expression for the doubling time figure of merit [33].

Consider the growth of an energy park where at time t:

The annual-average power available for societal use is related to installed nameplate power as: (6) P ( t ) = P n p ( t ) ψ ( 1 − h ) ( 1 − β )where β is the fraction of produced power that is plowed back into the construction of new plants and their associated resource supply and delivery infrastructures.

Based on the availability of energy diverted from societal use to build new infrastructure, the incremental capacity dC(t) under construction during time period dt is: (7) d C ( t ) = C ( t + d t ) − C ( t ) = d t [ ( 1 − h ) ψ β P n p ( t ) q − C ( t ) λ ]where:

C(t) = nameplate plant capacity under construction at time t;

The net capacity of nameplate power coming on line during time interval dt is the net of new build and decommissioning: (8) d P n p ( t ) = P n p ( t + d t ) − P n p ( t ) = d t [ − 1 T P n p ( t ) + C ( t ) λ ]

Collecting the above results, the system equations that define the dynamics of growth for the energy supply under an energy plowback constraint are: (9) d d t P n p ( t ) = − 1 T P n p ( t ) + C ( t ) λ d d t C ( t ) = ( 1 − h ) ψ β P n p ( t ) q − C ( t ) λor in matrix notation (10) d d t { P n p ( t ) C ( t ) } = A { P n p ( t ) C ( t ) }where (11) A = { − 1 T 1 λ ( 1 − h ) ψ β q − 1 λ }

The initial boundary conditions (assuming steady state) are (12) P n p ( t 0 ) = P 0 C ( t 0 ) = λ ( 1 − h ) ψ β q P 0 .

The most positive eigenvalue of the state transition matrix, A, sets the upper bound on the rate of energy supply growth attainable for a given reinvestment factor, β. The eigenvalues, α, of A are solutions of the quadratic equation: (13) det | A − α I | = 0 det | − 1 T − α 1 λ ( 1 − h ) ψ β q − 1 λ − α | = 0or (14) α 2 + α ( 1 λ + 1 T ) + 1 λ T − 1 λ ( 1 − h ) ψ β q = 0

This equation can be solved for the eigenvalues, α, using the quadratic formula: (15) α = − ( 1 λ + 1 T ) ± ( 1 λ + 1 T ) 2 + 4 λ T [ ( 1 − h ) ψ T β q − 1 ] 2

There are two eigenvalues. By inspection, at least one is positive if ( 1 − h ) ψ T β q > 1. Calling the most positive eigenvalue α*, the persisting solution of the state equations is simple exponential growth at an annual rate of α*, starting from the steady state initial condition: (16) P n p ( t ) = P 0 e α ∗ t C ( t ) = λ ( 1 − h ) ψ β q P 0 e α ∗ t

The doubling time figure of merit τ₂ – applicable to growing infrastructures under an energy plowback constrained is defined as (17) τ 2 = ln 2 α ∗and is the time interval it takes to accumulate the energy needed to double the emplaced infrastructure–given a specified energy reinvestment fraction, β.

The upper bound on achievable growth rates, α*, constrained by energy plowback, is determined by the following infrastructural characteristics:

λ : licensing and construction time period;

The sources for numerical values for these parameters are as follows: T, ψ, h,and q are usually documented in the life cycle analyses that produce values for EROI and/or τ₁; licensing and construction time period, λ, is known from actual plant construction practice; and the energy plowback fraction, β, is a parameter to be assumed in parametric scoping studies.

To meet rapid growth in energy demand a high value of α* (a short doubling time) is desirable. Examination of (14) indicates that the infrastructure's indigenous growth α* will be larger when (18) ( 1 − h ) ψ T β q > 1 and λ Τ < 1

With the exception of the energy plowback fraction, β, all the other parameters determining the above inequalities reflect the infrastructure's underlying physical and technological characteristics. Substantial differences between fossil fuel-based and renewable infrastructures in terms of these underlying characteristics have very significant implications for their differential ability to sustain high rates of indigenous growth.

One of the most fundamental attributes of renewable technologies is intermittency, which refers to the fraction of time that a given energy source/facility is available to society [20]. An important consequence of the intermittency of these technologies (i.e., the fact that wind does not blow all the time and the sun does not shine all the time) is their low capacity or load factor–i.e., low ψ values. By contrast, because of the continuous nature of fossil fuel extraction, most conventional (fossil-fueled and nuclear) generating technologies have very high load factors (high ψ values) and are “dispatchable.”

Fossil-fueled and renewable technologies also have substantially different energy and power densities. The lower energy density of renewable sources as compared to fossil fuels implies that the former require significantly larger infrastructures–labor, capital, materials and energy–to produce an equivalent amount of energy [20]. Similarly, the low power density of renewable energy extraction implies that for renewable infrastructures large quantities of energy must be expended to emplace a unit of nameplate power capacity–i.e., for renewable conversion nodes, q is large.

The fact that renewable technologies have low ψ and high q values while fossil-fueled generating modes have high ψ and low q values (and hence the ratio ψ q has much larger values for fossil-fueled as compared to renewable technologies), has important consequences for their respective abilities to achieve high rates of indigenous growth. What matters to doubling time is the time phasing of the initial capital vs the ongoing O&M components of EROI. Consider the case of a renewable technology whose EROI is similar to that of a fossil-fueled generation mode. Given a value for EROI, it is easy to see from (3) that (19) ( 1 − h ) ψ T β q = { ψ T q [ 1 − 1 EROI ] + 1 } βEven with the same value of EROI, the renewable technology could still have a much smaller value ψ q (relative to the fossil fueled technology) because of its intermittency and low power density. Assuming that the two technologies have similar T values (and the energy plowback fraction β is the same), then (18) implies that the value of ( 1 − h ) ψ T β q for the renewable technology will be much smaller relative to that of the fossil fueled technology. This in turn implies, according to (14), that the renewable technology will have a much lower achievable growth rate, α* (and correspondingly longer doubling time τ₂) relative to the fossil fueled generating mode.

IEA [34] performs a life cycle analysis (LCA) of different electricity generation sources (coal, oil, LNG, nuclear, wind, PV, solar thermal, hydro, and geothermal) in Japan. And it applies a consistent set of net energy formulas across the different generation options. The study's analysis is based on power outputs and annual capacity factors for the most typical generation plants in Japan–for fossil fuels and nuclear the reported values for annual capacity take into account periodic inspections while for renewables they are the maximum obtained under normal operating conditions in Japan. The estimates of the net supplied energy by each power generation system are based on a standardized power plant with nameplate capacity of 1,000 MW and an assumed life expectancy of 30 years for each plant. From the net supplied energy data, the energy payback period of each generation option is being estimated.

The IEA study was published in 2002. Thus, the study's reported estimates of LCA parameters are considerably outdated-especially for wind which has been experiencing very rapid technical change. Moreover, capacity factors and consequently net energy returns for renewables are highly site-dependent. Clearly, the wind and solar resources of Japan are not necessarily comparable to those found in the best sites around the world. However, the objective of our illustrative analysis is not to obtain the most accurate point estimates or representative values of net energy parameters. Instead, whet we seek to show is that the single numerical values of EROI are not by themselves sufficient to evaluate the potential of alternative energy supply infrastructures for indigenous growth.

The study provides estimates of EROI, ψ, τ₁ and assumes that T = 30. For both coal-fired generation and wind power, EROI = 6. Coal has a much larger capacity factor (ψ = .75) relative to wind (ψ = .20). Moreover, coal has a much shorter estimated energy payback period (τ₁ = 0.15 years) in comparison to wind (τ₁ = 3.39 years). From these values we can back-calculate E P n p ψ T, q, and h. These estimates are presented in Table 3. For coal, q = .094 and thus ψ q = 7.98. For wind, q = .637 and ψ q = 0.31. With a 20% plowback (i.e. β = .2), coal-fired plants can attain 73% annual expansion growth rate while for wind power the computed annual growth rate is only 2%.

Thus, coal-fired generation shows potential to support rapid indigenous growth. Wind, on the other hand, seems quite constrained. This at first might appear to be surprising in light of wind's EROI being as large as coal's and its O&M plowback fraction h being smaller than that of coal. To understand this outcome it is necessary to recognize the time phasing of the initial capital energy input vs the ongoing O&M energy inputs making up EROI in (3). The initial capital component for wind, representing the fraction of gross production expended for initial emplacement, is over 25 times larger than that of coal's-or equivalently for coal the ratio ψ q is over 25 times larger than that of wind's. According to (18), this implies that the ratio ( 1 − h ) ψ T β q has a much larger value of 40.08 for coal relative to wind for which the corresponding value is just 1.75. The doubling time τ₂, again with a 20% plowback, is 1.3 years for coal and 28.5 years for wind. Thus, the ability of wind to rapidly scale up its production by bootstrapping its own energy appears to be limited relative to coal.