Anyone who believes that exponential growth can go on forever in a finite world is either a madman or an economist.
Kenneth E. Boulding
Is Wind Energy an Effective Substitute for Coal and Natural Gas?
Wind energy is the fastest growing renewable energy source, and its cost are surprisingly low if one ignores the issue of integration into the grid. The latest version of NREL’s Windpowering America report claims that wind electricty costs are currently $0.03/kWh for high wind locations and $0.04/kWh for low wind locations, and by 2010 costs are project to go down to $0.02/kWh and $0.03/kWh. On the surface of things these estimates appear to be very ecouraging; Wind generated electricty is already cheaper than fossil fuel generated electricty and costs are continuing to drop. Lester Brown of the Earth Policy Institute has jumped on the wind band wagon and believes believes that this energy source can effectively solve the potential energy crisis arising from the depletion of fossil fuels.
The problem with this rosy picture of wind energy’s potential is that the quoted costs are for electric energy coming out of the generators on the wind tower. When one considers the cost of integrating the highly variable ouput of those generators into a 24/7 grid whose voltage must maintained within very tight limits, then the cost picture becomes quite different. Wind power is highly variable on both short and long time scales. In order to maintain the grid voltage within specified limits fossil fuel power plants must be turned up and down to compensate for these variations. Promoters of wind power point out (correctly I believe) that these variations in wind power will become smaller when widely dispersed widely dispersed wind farms are integrated into a power generation system, since low power at some locations will be compensated by higher power at other locations. It is widely quoted on the internet that for up to 20% penetration of wind into the generation mix integration costs are reasonable, although generally this number is not attributed to a specific source.
The United Wind Utility Group (UWIG) has published some simulation studies of the costs of wind integration on their website. They do indeed claim that for up to 20% wind penetration into the generation mix, excess operating cost are moderate (less than $0.01/kWh), but probably acceptable. Two important points should be noted about this claim, however. The first is that 20% means 20% of the rated power capacity of the system. However, the capacity factors of windfarms are typically less than half that of conventional power plants so that wind penetration in terms of energy delivery is less than 10%. The second and more important point is that the UWIG study only addresses the excess operating costs associated with turning fossil fuel plants up and down to compensate for variable wind power and says nothing about excess capital costs. What do I mean by excess capital costs, you ask? I just said that wind energy costs were less than $0.04/kWh, which number naturally includes capital costs. So how can there be a capital cost issue?
Electric generating systems are sized to meet peak load requirements (This frequently occurs in the Summer when air conditioning use is at a maximum.). The questions is, can wind capacity be counted on for meeting peak load requirements? A number of people have argued that, because it is possible for low wind conditions to exist over a wide geographical area, the answer to this question is no. In point of fact in many locations winds are much lower in the Summer, just at the time when the largest loads are needed. If this argument is valid then adding wind generation to a system always increases capital costs above a system which uses conventional generation only.
Ten percent wind energy penetration into the grid would be nice, but it is far from a solution to the problem of CO2 emissions, and the problems of excess operating and capital costs will get worse as the percent penetration rises. Furthermore, in the long run natural gas and coal are going to run out, and what will we use for power balancing then? We really need to start taking a long view of things rather than spending all of our time and energy trying to find ways to keep the ‘growth and greed economy’ going for a few more decades.
One can attempt to solve the problems of wind power’s intermittancy by adding energy storage to the system, but energy storage has proved to a be thorny technical problem, and proposed storage systems will add far more cost to the generated electricty than the turbines themselves. This fact does not necessarily mean that such systems are useless, but the idea that wind power is ready to go substitute for fossil fuels which can keep the stock market rising for decades to come is not correct. I will have more to say about energy storage options in furture posts.
 See for example: Outgrowing the Earth, W.W. Norton & Company, 2004 or Plan B 2.0, W.W. Norton & Company 2006.
 This is Ted Trainer’s terminology. See: The Conserver Society, Zed Books, 1995.
Feb. 15, 2007
EROI Is a Poor Measurement of the Economic Quality of Energy Production
Fossil fuels are said to have higher economic quality as energy sources than renewable sources such as wind turbines and solar photo-voltaic panels. A parameter frequently used to characterize the economic quality of energy production is EROI (Energy Return On Investment) which is defined as the ratio the energy output of a given energy production process to the energy consumed during that process. Thus if the gross energy output of the process is EG, and the energy consumed during the process is f×EG then EROI is given by:
EROI = EG/(f×EG) = 1/f
where f is the fraction of the gross energy ouput consumed during the production process.
For example conventional oil from very large reservoirs is said to have EROI in the range of 30 to 100, while corn ethanol is said to have EROI in the range of .7 to 1.7. When EROI<1 (or f>1) more energy is consumed than is produced. However, in spite of its widespread adoption, EROI is not a good measurement of the economic quality of energy production as a few simple examples will show.
Suppose that we have a choice between building a hydroelectric project in two different locations. At both locations the expected gross energy output over the lifetime of the installation is an identical number EG. At one location the energy consumed during the construction of the project is 0.01×EG (f=0.01), while at the second location the energy consumed during construction is 0.10×EG (f=0.10). Thus the EROI values for the two installations are 100 and 10 respectively. Should we assume that the first location is economically ten times better than the second? The answer is clearly no. The net energy output (EN) is given by:
EN = EG-(f×EG) = (1-f)×EG
For the first location EN=0.99×EG while for the second location EN=0.90×EG. The difference in net energy output is only 10%, not a factor of 10. The factor 1-f=EN/EG which appears in the above equation, I call the energy efficiency. It is the fraction of the gross energy output EG which is available for other economic purposes other than producing energy. The energy efficiency is a better (though by no means sufficient) parameter for characterizing the economic quality of energy production than is EROI.
One might think that the above example, which involves a choice between discrete energy producing projects, is different than a case where an indefinitely large amount of energy can be produced depending on the amount of resources dedicated to such production. For instance consider the case of the production of solar photovoltaic panels. Again we consider two different production processes which produce PV panels with identical functionality. That is a solar panel of a given area is expected to produce the same gross output of energy EG over its life time regardless of which production process was used. Again one of the two processes consumes an amount of energy 0.01×EG (f=0.01), and the other process consumes an amount energy 0.10×EG (f=0.10). Thus the first type of PV panel has an energy efficiency of 0.99 (EROI=100) and the second type of PV panel has an energy efficiency of 0.90 (EROI=10).
Suppose one has enough initial energy to build ten PV panels of the first type. The same amount of energy invested in building solar panels of the second type would produce only one panel. Doesn’t this imply that the panel with EROI=100 is ten times better than the panel with EROI=10? Again the answer is no. If an energy production process has a positive energy balance (more energy is produced than consumed) then thinking about this process in terms of spending a fixed energy budget does not make sense. If all forms of energy are economically equivalent (and it is only when this assumption is made that simple minded calculations of EROI make sense) then we should be able to run our factory from the output of our own solar panels. If the PV panels have an EROI of 100 then for each 100 panels built we must divert one to the roof of our factory, so that we are only able to ship 99 panels. If the PV panels have an EROI of 10 then for each 100 panels built we must divert 10 panels to the roof of our factory, so that we are able to ship only 90 panels. Again the net energy output of the two process is proportional to energy efficiency and not to EROI.
Although energy efficiency is a better parameter than EROI for characterizing the economics of the energy production process, it is by no means a sufficient one. Energy efficiency allows one to account for the energy costs of energy production, but other costs are important as well. Two further example will make this clear.
Let us consider the production of two different types solar panels both of which have an energy efficiency of 0.90 (EROI=10). For either type of panel we must divert one of every 10 panels manufactured to the roof of our factory. However, in this case one of the two types of panels produces energy EG over its expected life time, while the other type of panel produces energy 2×EG over its life time. It is clear that if all other manufacturing costs are equal, then the second type of panel has twice the economic quality of the first. This difference in quality cannot be described by the energy efficiency or by EROI. The phrase ‘if all other manufacturing costs are equal ‘ is important. The energy cost of producing energy is described by the energy efficiency, but the cost in terms of other production resources such as labor, land, fresh water, etc are important (capital equipment can be though of as embedded energy, labor etc rather than as a separate category).
The importance of non-energy related factors of production in determining the economic quality of a given energy source can be seen most clearly in the case of bio-fuels, where the amount of land required to produce a given output of energy is a key economic parameter independent of the energy efficiency. Let us suppose that the gross energy output per hectare (2.47 acres) of land for a given bio-fuel is EGH. Then the net energy output per hectare is given by:
ENH = (1-f)×EGH
where f is the fraction of the energy EGH which is consumed during the production process. It is clear that even in the limit that f=0 (or energy efficiency=1), EGH is still an important economic parameter. As a fanciful example suppose we found an energy crop that grew and harvested itself without human intervention. Imagine some variety of tree which produces an energy containing sap which spontaneously flows out of the tree in the springtime and percolates its way through the ground to a subterranean chamber. Every Spring the chamber fills up without any human effort or intervention. If the energy output of this magical crop was EGH = 1 liter of gasoline equivalent per hectare then it would be economically useless in spite of its perfect energy efficiency (or infinite EROI). Pure energy theories of economic value are not tenable.
Feb. 14, 2007
The cost of renewable electricity generation is frequently cited in cost per unit of generating capacity. For example wind generation is frequently cited as costing U.S. $1000/kW. Crystalline silicon PV installations are quoted as costing U.S. $ 7000/kW. People working on tracking solar concentrator PV systems using high efficiency, triple junction, thin film PV cells cite target costs of U.S. $3000/kW. However, the consumer of grid generated electricity does not care about the capital cost of gas turbines or coal fired steam power plants. All she cares about is the price per kWh. How can these cost for installed generating capacity be converted to costs per unit of energy delivered?
Consider a particular source of renewable electricity generation R. We define the following parameters:
CR = Capital cost per kW of generating capacity
MR = Capital cost per kW multiplication factor (>= 1, depends on loan length and interest rate)
F = Capacity factor (Effective fraction of generating capacity used averaged over 24hr/day 365days/year)
N = Generator lifetime in years
OMR = Operation and and maintenance cost per year per kW
The electricity generation cost per kWh is then given by:
(1) GR = [(MR×CR) / (N×F×365×24)] + OMR
Consider the example of wind generation where the capital cost is CR = U.S. $1000/kW, the capacity factor is F=0.3 and the turbine lifetime is N=20 years. As a simple case let us assume that operating and maintenance costs are negligible (OMR=0), and let us ignore the increase in capital costs due to interest payments (i.e. assume MR=1). Then the cost of electricity/kWh is given by:
GR = $1000/(20×0.3×365×24) = $0.019/kWh
This cost is cheaper than the $0.03 to $0.04 quoted by the NREL in the most recent version of their Wind Powering America report. To get an idea of the effect of interest payments on the cost consider the case where the capital is borrowed at 8% interest for a period of 15 years. In this case the capital multiplication factor is MR=1.75 and the total cost per kWh is $0.034 which is close to NREL’s estimate of wind electricity costs. The assumption that wind OM costs are negligible relative to capital costs is also frequently made.
I am not going to derive the formula for calculating the capital multiplication factor M, but I will give it here for reference. I define the following parameters:
r = Annual interest rate expressed as a fraction (e.g. for 8% interest r=0.08)
n = Length of loan in years
Then the capital multiplication factor is given by:
(2) M = (n×r)/[1-(1+r)-n]
If the payments are monthly rather than yearly n is the number of monthly payments and r is the monthly interest rate. However, M does not change by much using monthly payments compared to yearly payments (1.72 compared to 1.75 for the example given above), so for simplicity I just use the assumption of annual payments.
The so called “hydrogen economy” in which intermittent renewable generation sources are converted into portable, storable chemical potential energy in the form of hydrogen has been much discussed and written about. The most common method discussed for producing hydrogen from renewable generation is the electrolytic decomposition of water. Since electrolyzer capital costs are quoted per kW of input power we can use a method similar to that described above to calculate hydrogen cost per unit of chemical potential energy. I am going to give the hydrogen energy content in kWh. This is not a normal energy unit for describing the energy content of chemical fuels. For one thing the number of kWh of hydrogen energy content is not the same thing as the amount of electrical energy you would get if you used the hydrogen for producing electricity, since the conversion process is not 100% efficient. Nevertheless, for convenience I use kWh as hydrogen chemical potential energy units. Electrolysis is not 100% efficient. That is only a fraction of the electrical energy consumed by the electrolyzer ends up in the form of chemical potential energy stored as hydrogen. This fraction is defined as the energy efficiency of electrolysis.
In order give a formula for hydrogen production costs I define the following parameters:
ßE = Electrolysis Efficiency (~0.7)
CE = Electrolysis Capital Cost/kW over life time of the renewable generation source providing power
ME = Electrolysis Capital cost multiplication factor (>= 1, depends on loan length and interest rate)
OME = Electrolysis Operating and Maintenance Cost per kW per year
If the electrolyzer lifetime is different than that of the renewable generator the then calculating CE and ME can be more complicated than calculating CR and MR for the renewable generator itself, but nevertheless one can in principle separate the cost into a direct capital cost and an interest multiplication factor.
With these definition the cost of hydrogen production per kWh is then given by:
(3) HY = (GR/ßE) + [(ME×CE)/(F×N×365×24×ßE)] + (OME/ßE)
The first term in this equation is just the electricity generation cost given in formula (1) divided by the electrolysis efficiency. The second two terms give the electrolysis costs of hydrogen production. They are of the same form as the terms of the expression for electricity cost except that they are divided by the electrolysis efficiency. Notice that I am using the same lifetime N for the electrolyzer as for the renewable generator. If this assumption is not true, we account for it by adjusting CE and ME. Since the electrolyzer is operating off power provided by the renewable generator it necessarily has the same capacity factor F. Equation (3) can be rewritten as:
(4) HY = (GR/ßE) + (EL/ßE)
where EL=(ME×CE)/(F×N×365×24) is the electrolysis related cost of turning 1kWh of electricity into hydrogen. Without the factor of 1/ßE equations (2) and (3) are the total cost of generating 1kWh of electricity and and converting it into hydrogen at efficiency ßE. The factor of 1/ßE reflects the fact that more than 1kWh of electricity must be generated in order to produce 1kWh worth of chemical potential energy in the form of hydrogen.
Again let us take wind generation as an example and make the simplifying assumption that operation an maintenance costs are negligible (OMR=0 and OME =0) and that interest costs are zero (MR=1 and ME=1). We assume CR = U.S. $1000, N=20 years, F=0.3, and ßE=0.7. What is the electrolyzer capital cost per kW CE? Unfortunately electrolyzers are expensive pieces of equipment. I will assume CE = U.S. $ 4000/kW over a 20 year time period. In this post I am primarily trying to discuss methodology of energy cost calculations, and so will not spend time trying to justify this cost assumption (I suspect that it is actually low.). Under these assumptions the cost of hydrogen production per kWh is given by:
HY = (0.019/0.7) +[4000/(0.3×20×365×24×0.7)] = 0.027 + 0.109 = $0.136/kWh
If we assume an interest multiplication factor of 1.75 for both the wind turbines and the electrolyzers then the cost becomes:
HY = 1.75×0.136 = $0.236/kWh
A gallon of gasoline contains 36.7kWh so this price is equivalent to $8.66/ gallon.
This kind of analysis can be extended to cover a variety of cases. I will give two more examples. Suppose you believe (as I do) that hydrogen is a poor choice as an energy carrier. Then one could chemically react the hydrogen with some other substance to produce an energy carrier with more favorable properties. One possibility of such an energy carrier is ammonia (NH3). The cost of capital equipment for producing ammonia can not be given in units of kW so so that the production cost will have to be estimated by some other methodology than that describe above. However, ammonia production from hydrogen will have an energy efficiency associated with it. That is only a fraction of the chemical potential energy contained in the hydrogen will end up contained in the chemical potential energy of the resultant ammonia. This fraction is the energy efficiency of ammonia synthesis. This efficiency (which is less than 1) must divide the cost of hydrogen production since we must synthesize more than 1kWh of hydrogen in order end up with 1kWh of ammonia. Thus if the efficiency of ammonia synthesis is ßA and the production cost/kWh (again these are not normal units for computing ammonia production costs) is CA (this number is the total production cost, not capital cost) then the cost of ammonia syntheses via the renewable generation/electrolysis/chemical synthesis route is give by:
(5) AM = (HY/ßA) + CA = (GR/ßE×ßA) + (EL/ßE×ßA) + CA
If we take the previous wind/hydrogen example in which hydrogen costs were $0.236/kWh and assume that that the ammonia synthesis efficiency is ßA=0.9 and that ammonia synthesis costs are $0.005/kWh then we find:
AM = (0.236/.9) + .005 = $0.257/kWh
which is equal to $9.44/gallon of gasoline energy equivalent. Again since this post is about methodology I will not discuss the basis of the assumptions for ammonia production costs, but if you are interested stay tuned for a later post.
I do not know if anyone will ever find this post and if they do findit whether they will stay awake long enough to reach this point. However, I am nearing the end. I want to give one more example of an energy cost calculation. Let us consider using electrolytic hydrogen as a means of storing electricity. In this case we have to have a means of reconverting the hydrogen into electricity. The most often discussed means of doing this re-conversion is fuel cells, but gas turbines or reciprocating internal combustion engines could be used a well. Whatever technology is used will have a cost/kW and conversion efficiency less than 1 (Does this story sound familiar?). Let us suppose that the cost/kW of our hydrogen conversion technology is CH, the capital multiplication factor is MH, the operation and maintenance costs are OMH, the conversion efficiency is ßH, the capacity factor is FH, and the lifetime is NH. In addition we may have to consider hydrogen storage costs/kWh = SH. Then the cost of electricity going through the chain of renewable generation, electrolysis, and re-conversion to electricity is given by:
(6) GH = (HY/ßH) + [(MH×CH)/(FH×NH×365×24)] + OMH + SH
Where HY is the hydrogen cost which has to be divided by ßH since we have to produce 1/ßH kWh of hydrogen in order to end up with 1kWh of electricity. HY in formula (6) no longer has the same value that it did in formula (1). In the previous examples we were considering the process of converting electricity into chemical fuel and calculating the cost of the fuel. Once we start using the chemical fuel as a temporary storage for electricity the economic calculations get more complicated. If electricity from the renewable generator is predictably available at a time when electricity demand is high (e.g. weather forecasters are predicting steady, strong afternoon winds for the next three days) it would be foolish to send that electricity through an electrolyzer and then back through a re-conversion device losing more than 50% of the energy in the process. Therefore some fraction of the time the renewable generator would supply power directly to the grid. This fact implies that the effective capacity factor of the electrolyzer would drop and its contribution to hydrogen costs would rise. Therefore HY in formula (5) is higher than its value in formula (2).
As a specific example let us suppose our re-conversion technology is a combined cycle gas turbine with capital cost CH=$550/kW and conversion efficiency ßH=0.7. Let us further assume that OMH=0 ,SH=0, and N=20 years. The capacity factor of the re-conversion device no longer has to match that of the renewable generator since we can use the stored hydrogen at whatever rate we desire. I will assume, without any real justification, that the capacity factor FH=0.5. I take the hydrogen cost from the earlier wind/electrolysis example as HY=$0.236/kWh and ignore the increased cost due to the lower capacity factor for the electrolyzer. Finally I take the capital multiplication factor to be MH=1.75. Then the electricity generation cost going through the complete cycle of hydrogen storage and re-conversion is given by:
GH = ($0.236/0.7) + [(1.75×$550/(0.5×20×365×24)] = $0.337 + $0.011 = $0.348/kWh
Note that under this particular set of assumptions the cost of electricity production is dominated by the cost of hydrogen production. This domination occurs because large scale combined cycle gas turbines have very low capital costs compared to the wind turbine/electrolyzer combination. Again I am not claiming that this set of assumptions would apply to real world electricity production. I am just trying to describe the methodology for doing these kind of calculations.
This cost of course only applies to that fraction of the electricity which is stored in the form of hydrogen. The total cost of electricity from this system would be the weighted average of the electricity which goes directly into the grid from the renewable generator and the fraction which went through hydrogen storage. If the fraction of electric energy which comes directly out of the renewable generator is f and the cost per kWh GR then the total cost of electricity from this system is given by:
(7) GT = (f×GR) + [(1-f)×GH]
May 3, 2007
A Super Grid for Smoothing Out the Power Fluctuations of Intermittent Renewable Energy Sources
The Irish wind power company, Airtricity has recently announced a proposal for building a 10GW array of wind farms over a wide area of the North Sea which will be connected to a high voltage direct current (HVDC) undersea ‘super grid’. HVDC transmission of electricity is a relatively recent innovation enabled by high power semiconductor based transformers which allow high voltage DC current to be converted to AC and fed into traditional AC power grids. HVDC transmission lines have much lower losses (three time less) than AC lines and, unlike AC lines, can be operated bi-directionally (That is current can be transmitted in either direction along the line.). Other advantage of HVDC transmission include the ability to easily connect grids operating at different frequencies and the ability to have very long underwater cables without the need to actively compensate for signal phase shift due to cable capacitance.
If the variable nature of wind power is ignored then wind generated electricity is relatively cheap. But the fact that wind may not be blowing when you need it, or the fact it may be blowing hard when you do not need the electricity limits its usefulness. Wind farms dispersed over a wide geographical area collectively have less energy output variability than any single wind farm or than a group of wind farms located relatively close together. In order to take advantage of the smaller variation output you have to be able to transmit the power from wherever it is available to wherever it is needed. Thus if your wind farms are spread out over hundreds or even thousands of kilometers you need a high capacity long distance network of power transmission cables. This network is going to be expensive. Even HVDC cables are going to be expensive, but they will be significantly cheaper than an equivalent network of AC cables, plus they will have lower transmission losses.
Airtricity envisions this wind/super grid system eventually being extended to include wind farms in the Atlantic Ocean, the Mediterranean Sea, North Africa, and central Asia. Potentially solar energy farms, wave farms, tidal farms etc. could eventually be integrated into the same super grid. Of course smoothing out the fluctuations of wind energy still does not turn it into a dispatchable energy source. If you have lots of wind at night time when demand is low, a problem still exists in utilizing this energy. Pumped hydro storage is one suggestion for dealing with this problem. The existence of a super grid would give more flexibility in using such a storage method since the storage reservoirs could be a long distance from the wind farms they were serving. Norway’s large hydropower system has been suggested a potential storage location for such an integrated European power system.
The super grid concept for integrating intermittent renewable energy sources is hardly new. Buckminster Fuller proposed this idea on a global scale in his book Critical Path published in 1981. The recent emergence of HVDC transmission technology substantially improves the economics of such a project. Nevertheless formidable barriers exist to the construction of such a system. HVDC or not the construction of thousands of kilometers of DC transmission lines is going to be expensive. Furthermore, such a project requires coordination of electric power distribution on scale much larger than anything previously attempted. In order for the full economic benefit of smoothing out the fluctuations of renewable energy power sources to be realized the whole system has to be completed. This situation is different than that of other large scale public projects of past such as the interstate highway system in the U.S. which was built over a period of 35 years at cost of 425 billion dollars (inflation adjusted to 2006 dollars). For the interstate highway system each stretch of road became fully useful as it was completed.
Furthermore, whatever the estimated costs of building such a system may be today, increasing oil prices are likely to inflate those costs as various aspects of mining, construction, transportation etc. which are dependent on oil become more costly. I am personally skeptical that the super grid concept is a path to maintaining business as usual functioning of private finance capitalism (i.e. maintaining exponential growth of the stock market) for decades to come.
Some people maintain that if we do not build these kind of large scale infrastructure project while fossil fuels are still relatively cheap then they will never get built at all. Personally I doubt that any infrastructure that we can build only while fossil fuels are plentiful will be sustainable in a post fossil fuel future. I have no doubt that if we could intelligently anticipate the consequences of fossil fuel depletion and plan and act with long term foresight, the transition to a new economic order would be far easier than otherwise. But I think that the evidence is clear that such foresight does not exist and the real adaptation to the consequences of resource depletion will only occur after it becomes manifest that our economic system is badly failing. It may still be possible to bootstrap our way up to such a large scale super grid if we develop the ability to apply production resources intelligently to truly useful economic activity rather than leaving manufacturing infrastructure investment in the hands of private finnaciers pursuing indiscriminate short term growth.
Jan 17, 2008
Energy Payback Time and Energy Production Boot Strapping
Energy has to be expended to build and install renewable energy generators such as PV panels, solar concentrator systems, wind turbines etc. The length of time required after installation for all of the energy consumed during the manufacturing and installation to be returned from the output of a given renewable energy source is called the energy payback time. If energy paybacks times are long then initial outlays of energy required to install renewable generation can have negative economic consequences.
Let's attempt to quantify these consequences. I define the following variables:
C0 : Initial yearly net energy supplied from finite energy sources which are being depleted.
r : Yearly depletion rate as a fraction of C0. That is in year number two the available energy is (1-r)×C0.
f : The fraction of the net energy C0 that is used to install renewable energy capacity
P : Energy payback time in years of the newly installed renewable energy generators
For mathematical simplicity I will assume that all of the renewable energy generators are turned on at the beginning of the year. So during the first year we spend energy f×C0 building and installing renewable energy generators which we turn on at the beginning of the second year.
During the first year the energy available for producing goods and services other than energy (I call this 'useful' energy) is given by:
U1 = (1-f)×C0
During the second year the useful energy is given by:
U2 = (1-f)×[(1-r)×C0+(f/P)×C0]
The first term inside the square brackets is the energy output from the depleting energy sources and the second term is the energy output from the newly installed renewable generators. We have to multiply by a factor of 1-f since we need to go on installing new generation to make up for the continuously depleting conventional energy sources. If we wish to avoid energy descent then we require that U1=U2. Inspection of the equations above reveal that value of f which makes U1=U2 is given by:
To be concrete suppose that the energy lost from conventional sources is 0.05×C0, and the energy payback time of the renewable generators is P=5 years. Then f=0.05×5=0.25. That is we would have to sacrifice 25% of our net energy output to install new renewable generation. If we install the same amount of renewable generation every year and if the energy descent of the conventional sources was strictly linear so that the losses remain 0.05×C0 per year, then after 20 year the conventional sources would be exhausted and they would have been replaced by an equivalent capacity of renewable generators. The price for avoiding energy descent is a 25% decrease in useful energy over a period of 20 years.
If the energy payback time is greater than or less than 5 years then the energy price of avoiding energy descent goes up or down. For example if the energy payback time were 1 year then we would only have to sacrifice 5% of our useful energy to avoid energy descent. If energy payback time were 20 years then it would physically impossible to avoid energy descent since 100% of our energy would have to be dedicated to installing new capacity.
Similarly if the rate of enery depletion were slower or faster the energy price of avoiding energy descent would be smaller or larger.
June 4, 2009
The real science of political economy, which has yet to be distinguished from the bastard science, as medicine from witchcraft, and astronomy from astrology, is that which teaches nations to desire and labor for the things that lead to life; and which teaches them to scorn and destroy the things which lead to destruction.