### Introduction

A great deal of literature has been devoted to calculating the EROI of various energy sources. However, little explanation has been offered for why we should care, or what effect a lower EROI would have on civilization. Why even bother tracking EROI?

Some authors have speculated that declining EROI would imply less net energy for other purposes in the economy. For example, a decline in EROI from 100 down to 10 might imply a 90% loss of net energy for civilization. However, that conclusion was incorrect. EROI does not determine the amount of net energy obtained by civilization as a whole. This is because our global industrial civilization has been increasing the amount of energy (and therefore the amount of energy for investment) exponentially over time, and this effect has greatly outpaced any decline in EROI. For example, increasing the amount of energy (and energy investment) by a factor of 1.33 would compensate for a decline in EROI from 1 billion down to 4, with no loss of net energy. (An increase in the amount of total energy by 1.33x would allow 1/4th (or 0.33 / 1.33) of total energy to be devoted to obtaining the energy, yielding a net energy factor of 1.0 (or 1.33-0.33), or no change). Since the amount of energy for investment can increase exponentially, increasing it by a factor of 1.33 is obviously within the capability of industrial civilization, thereby more than compensating for any plausible decline in EROI. As an example, it has been pointed out repeatedly that the EROI of oil has declined from 100 in 1930 to less than 10 now, but there hasn’t been a decline in vehicle traffic by 90% worldwide since 1930. Quite the opposite, vehicle travel has increased tremendously since that time, because the amount of oil has increased by a such a large factor that net energy from oil increased, despite massive declines in EROI.

Other possible effects of lower EROI would include greater land usage for solar farms, or higher costs of energy. However, those effects would be minor, as a matter of arithmetic, as long as EROI remains above some very low threshold. For example, a low EROI of 10 would imply that a solar farm would need to have ~11% greater land area, compared to a solar farm with an EROI of 1000, in order to obtain the same amount of net energy ( (1-1/1000) / (1-1/10) ~= 1.11) . Prices for solar panels would also be higher by a similar amount (11%), all other things being equal. However, the price of solar panels has dropped rapidly in recent years and is far below the price of electricity from coal-fired plants. The lower EROI of solar panels would have little effect on price compared to recent declines in price for other reasons.

One effect of lower EROI, which has not been investigated, is the effect on growth. A lower EROI implies a larger upfront investment of energy. A larger upfront investment of energy requires a greater sacrifice of net energy now in order to obtain a desired level of exponential growth. It will take time before exponential growth compensates for the original sacrifice of net energy to initiate that growth. As a result, a lower EROI implies a temporary sacrifice when initiating or accelerating growth.

A simple mathematical example may demonstrate this point. Assume a civilization which gets all its energy from solar panels with an EROI of 10 and a lifetime of 10 years. Also assume that generations of solar panels do not overlap; new panels are kept in a dark warehouse until old ones expire, at which point, all the old panels are replaced at once with new ones. In which case, our hypothetical civilization must invest 10% of its gross energy (or 1/eroi) on building new panels just to replace those that expire, leaving 90% of all gross energy for all other purposes. If that civilization decided to embark upon exponential growth, and double the amount of energy it obtains in each generation of solar panels, then it must double the investment energy from 10% to 20%, leading to a reduction of energy for all other purposes from 90% to 80%. This reduction would persist for one generation of panels (10 years) until exponential growth overcame it. Thus, our hypothetical civilization must undergo a temporary reduction of net energy by ~10% for 10 years, in order to accelerate exponential growth from 0%/year to ~7%/year. This reduction by 10% of net energy is determined by EROI. If the civilization had solar panels with an EROI of 100, then only a 1% reduction in net energy would be required to accelerate growth by the same amount.

Of course, reality is more complicated, for several reasons. First, generations of solar panels do actually overlap -- new panels are installed before all the old ones have expired. As a result, we cannot represent the compound growth with a simple exponential formula. Second, any real civilization obviously would not accelerate its growth from 0%/year to 7%/year overnight, because the sudden loss of net energy would be disruptive. Instead, any actual civilization would obviously ramp up growth more slowly, over a period of a few years, at least.

In this paper, a simple computer model is used to calculate the effect of EROI on growth of energy obtained. A simple python program is presented which repeatedly calculates re-investment of energy over time, and which ramps up growth slowly enough that net energy never falls below some threshold level.

The result is that even low EROI values (such as the EROI of 10 for solar panels) have only a modest effect on growth, as will be shown below.

### The Model

The model is implemented as a simple python program which calculates energy re-investment over time. It executes in a single loop which calculates energy obtained and energy re-investment, over and over again, for each year. It takes input parameters such as EROI, target growth rate, and a minimum threshold for net energy. The minimum threshold parameter determines how low net energy can go (relative to the original value) before growth must be curtailed, in order to ramp up growth more slowly. When growth is curtailed (during the ramp-up period), all excess energy beyond the minimum threshold is re-invested for exponential growth.

When this program is run with an EROI of 10.0, a target growth rate of 4%/year, and a minimum threshold of 97%, we obtain the following results. The civilization requires six years of ramping up growth in order to reach the desired growth rate of 4%/year. During the ramp-up period, 3% of net energy is sacrificed in order to accelerate growth. During the ramp up period, all exponential growth is re-invested, according to the assumptions above. The output of the program is as follows:

Initial net:0.900000 minimumNetAmount:0.873000

year | total | retired | toBeAdded | net |

0 | 1.000 | 0.040 | 0.0508 | 0.873 |

1 | 1.011 | 0.040 | 0.0551 | 0.873 |

2 | 1.026 | 0.040 | 0.0612 | 0.873 |

3 | 1.047 | 0.040 | 0.0696 | 0.873 |

4 | 1.077 | 0.040 | 0.0815 | 0.873 |

5 | 1.118 | 0.040 | 0.0981 | 0.873 |

STOP RAMPING UP | ||||

6 | 1.176 | 0.040 | 0.0871 | 0.959 |

7 | 1.223 | 0.040 | 0.0889 | 1.000 |

8 | 1.272 | 0.040 | 0.0909 | 1.045 |

As we can see, net energy must decline from 0.9 to 0.873, for six years, in order to initiate growth. After which, the civilization has enough surplus gross energy to grow at 4%/year with no further sacrifice.

The source code for the model is as follows:

# This is free and unencumbered software released into the public domain. # # THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, # EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF # MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. # IN NO EVENT SHALL THE AUTHORS BE LIABLE FOR ANY CLAIM, DAMAGES OR # OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, # ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR # OTHER DEALINGS IN THE SOFTWARE. eroi = 10.0 lifetime = 25 targetGrowth = 1.04 minimumNetFactor = 0.97 total = 1.0 yearsRun = 30 #panelsYearly keeps track of the panels installed each year; one element # per year. Each element refers to the amount of energy those panels # will return EACH YEAR. panelsYearly = [] oldSumPanels = 0.0 rampUpPeriod = True # Populate pre-existing panels for year in range(0,lifetime): panelsYearly.append(total/lifetime) # Run simulation originalNet = total - total/eroi minimumNetAmount = sum(panelsYearly) * minimumNetFactor * originalNet print("Initial net:%f minimumNetAmount:%f" % (originalNet, minimumNetAmount)) for year in range(0, yearsRun): sumPanels = sum(panelsYearly) retiredYearly = panelsYearly.pop(0) if (rampUpPeriod == True and oldSumPanels > 0 and sumPanels > oldSumPanels * targetGrowth): rampUpPeriod = False print("Stop ramping up") if (rampUpPeriod): investment = sumPanels - minimumNetAmount yearlyToBeAdded = (investment * eroi) / lifetime else: yearlyToBeAdded = sumPanels * targetGrowth - sumPanels + retiredYearly investment = (yearlyToBeAdded * lifetime) / eroi panelsYearly.append(yearlyToBeAdded) print ("year:%i total:%f retired:%f toBeAdded:%f net:%f" % (year, sumPanels, retiredYearly, yearlyToBeAdded, sumPanels-investment )) oldSumPanels = sumPanels

### Conclusion

A decline in EROI worldwide down to 10 would impose a delay of 6 years when accelerating growth from 0% to 4% for the world economy as a whole.

Of course, the model above is simplified in some ways. The algorithm used to determine how to ramp up growth would almost certainly be more sophisticated in reality. In which case, the above model is

**sub-optimal**. Results in reality would be slightly better.
Repeatedly throughout this paper, an EROI of 10 was assumed because that is the EROI of crystalline silicon photovoltaics, which has the lowest EROI of any common source of generating electricity. Still, such a low EROI imposed only a modest delay when accelerating growth. As a result, any plausible decline in EROI in the future would have only modest effects on growth.

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