Using Operations Research to Understand an Energy Efficiency Paradox
Apologies for the diminished rate of postings recently. I am trying to get the word out about the smaller updates and items via twitter instead of through the blog, so be sure to check out my or4green feed over there. I still intend to keep putting longer pieces on this blog. Here is a new one:
Back in April, I attended a Green Campus Conference in Connecticut. It was an interesting meeting. A frequently repeated statement was that Connecticut is aiming to become the most energy efficient state in the Union. There are rankings and the last time I checked it was third (see this post). So, many of the talks at the conference were geared towards increasing energy efficiency on college campuses.
The knee-jerk question that comes to mind for me when I hear about energy efficiency is: “What about Jevons’ paradox?” That is, as energy efficiency increases, whatever the energy was being used to obtain (cooling, heating, production, etc.) can now be now be obtained using less energy. So from a sustainability perspective, that’s good – less energy means fewer emissions, and lower general life cycle impact (e.g. production and transportation of fuel, equipment, etc.). But looked at in reverse, you could decide to keep your energy use constant, and utilize the efficiency gains to bring greater cooling, heating, production, etc. Therein lies Jevons’ paradox – energy efficiency in the latter case has not led to reduced energy usage. The environmental impact is neutral – no more energy is being consumed – but the opportunity for energy efficiency to reduce environmental impact was not utilized. Note we have assumed the unit cost of energy remains the same.
As I listened to the speakers talk about various ways to implement and finance energy efficiency on college campuses, I began to think Jevons’ paradox would not really be an issue in most of their cases. To see why, let’s look it at from an operations research perspective. We’ll use a very simple example to get the point across in which we only consider energy used to cool a building (home, campus building). The formulation follows:
x = energy used for cooling, in units of kilowatt-hours (kWh)
p = measure of energy efficiency in units of cooling level per kWh
L = p * x = cooling level
c = per unit energy cost (in $ per kWh)
C = c* x = total energy cost (in $)
B = budget (in $), total energy cost will be constrained to be less than B
m = per unit level of emissions (in lbs CO2 per kWh)
E = m * x = total emissions (in lbs CO2)
c x <= B — budget constraint
1 <= p x <= 10 — assume the comfort level must be between 1 and 10 (think levels on the AC unit)
Min C = c x — Cost
Max L = p x — Comfort
Min E = m x — Emissions
Now we will look at a few different scenarios. The guiding question is: as p (energy efficiency) increases, does energy consumption decrease? If so, does it decrease as much as it possibly could? Then note that environmental impact directly tracks energy consumption in this model.
Scenario 1 – Homeowner – would like to maximize comfort, but is constrained by a budget
Max p x
st c x <= B
1 <= p x <= 10
The scenario description implies that the budget is the binding constraint, which means the optimal solution is x = B/c.
If p increases, x stays fixed (assuming cost remains the binding constraint as p increases, which it will as long as p < 10 (c/B)). So energy consumption does not decrease as may have been intended – Jevons’ paradox is in effect. The efficiency gains have been used to increase comfort.
Scenario 2 – College campus – also constrained by a budget, wants comfort set at some reasonable level (we’ll say 5), but not necessarily to maximize it. So since comfort need not be maximized, it seems plausible that campus officials would want to try to minimize cost (or emissions or a combination of the two but let’s start with cost)
Min c x
st c x <= B
p x = 5
By inspection we see that the comfort constraint forces the solution x = 5 / p (assuming this is within budget, which we will assume).
So if p increases, clearly x decreases. In other words, in this scenario, increased energy efficiency leads to reduced energy consumption, and it is because the amount of what is being obtained with the energy is held constant.
Other scenarios – Given the three objectives, another natural way to look at this situation is as a multiple-objective linear program (MOLP). So far, we saw that when max cooling was the objective, Jevons’ paradox was in full effect. When cooling was constrained and min cost was the objective, Jevons’ paradox was not in effect at all. In the trade-off space explored by MOLP, you can probably guess what happens. The energy efficiency gains are split – less energy is consumed, though not as little as would be possible, because some of those gains are used to increase cooling without increasing energy use. Another variation would be to convert the cooling hard constraint in the campus scenario to a goal and solve the goal program. And of course, the setting can be changed to another type of energy use, such as heating, electricity, transportation, etc.
Coming back to my original question at the conference, I think the speakers were primarily in the college campus scenario. Most likely, they are already keeping their buildings at a comfortable level of heating or cooling, so any energy efficiency gains allow them to consume less energy, spend less money, and pollute less. In the transition from trying to maximize whatever quantity the energy is being used for (e.g. comfort), to not needing or not wanting to do so, Jevons’ paradox switches off. Campus electricity could be a little more complicated. Most likely individual buildings, labs, or dorm rooms for that matter do not have their own electricity bills. But if they did, I can imagine the paradox coming into effect. For example, a high energy lab that might have had to limit its operations due to high electricity costs, might consume more as energy efficiency increases.
In all, it was an interesting meeting. Thanks to Bill Leahy, Director of the Institute for Sustainable Energy at Eastern Connecticut State University for organizing it. For more information about the conference, go to the conference web site.