### Can solar roads be cost effective?

6/22/2014 03:51:00 PM
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Here's a slightly mathematical doodle on the feasibility of solar roads.

The cost-benefit comparison most commentators are implicitly doing here is asking whethter the value of the electricity generated by the solar road exceeds the extra cost of building the solar road versus a standard asphalt road. This depends on a lot of variables: how long the roads last, the difference in cost between solar roads and asphalt roads, how much electricity the solar panels produce, how much electricity the solar panels consume (you can't paint on glass, so the lines have to be displayed by electric-powered LEDs; you can't plow glass, so the ice has to be melted off by electric-powered heaters), and how much energy is lost during transmission (most of our interstate surface area isn't near cities!), what times people consume the most electricity (we can't store unused electricity, nor can we control when solar panels generate the most electricity!), and related to the last point, what the price of electricity is.

But, this is not the right calculation to do. This kind of math is what led people to erroneously think that proton-beam therapies were a cost effective cancer treatment in adults. To explain my analogy, proton-beam therapies are enormously expensive, but do save lives by killing cancer, meaning that the total benefits they provide would be worth the high cost. BUT, it turns out that we already have an alternative cancer treatments--traditional radiation therapies--that provide the same benefits at a fraction of the cost. The lesson here is that you can't compare total benefits to total cost, you have to compare added benefit to added cost compared to the best alternative.

Which brings me back to solar roads. Solar roads are a difficult and expensive option in a world that has tons and tons of cheap, more effective alternatives. For example, hardly any buildings in the US have solar panels on their roofs. Putting solar panels on roofs would eliminate all of the technological limitations of the solar road: they wouldn't need to have tempered-glass surfaces capable of holding semi-trucks; they wouldn't need LED displays; they wouldn't need to be heated to clear the ice; they wouldn't need to be textured to provide traction for cars; they could be angled to maximize the amount of direct sunlight; they'd be located closer to where the electicity is needed, so less transmission losses. All of these are things that would not only allow the panels to generate more electricity, but also reduce the cost of making the solar panels. And this retains the main advantage of solar roads: we wouldn't need to spoil unused land to install these panels.

So let's do a little math. Let $C_{sr}$ represent the cost of, say, a square meter of solar road, while $C_{r}$ is the cost of a square meter of asphalt, non-solar road, and $C_{s}$ is the cost of a square meter of solar panel installed on a roof (for our purposes, it will suffice to assume that these quantities are the summed discounted costs over all future periods--including both upfront building and future maintenence). Furthermore, let $W_{sr}$ be the energy produced by a sqare meter of solar road, compared to $W_s$ units of energy produced by a square meter of solar panels installed on a roof (for our purposes it will suffice to say that energy is the usuable power output over the course of, say, a representative year). There are a couple of postulates that I think should be uncontroversial:
1. $W_{sr} \lt W_s$
2. $C_{sr} \geq C_r+C_s$
The first postulate is the direct statment of all the things summarized in the previous paragraph--for lots of reasons, solar roads generate less electricity per unit of area than a solar panel on a roof. The second postulate is a bit less obvious but an approximate proof would be as follows:
From the first postulate we know that $\frac{W_{sr}}{W_s} \lt 1$ which implies by way of the second postulate that $$C_{sr} \gt \frac{W_{sr}}{W_s}\left(C_r+C_s\right)$$ which implies that $$\frac{C_{sr}}{W_{sr}} \gt \frac{C_r+C_s}{W_s}.$$ Note that the left hand side is the cost per unit of energy of the solar road, and the right hand side is the cost per unit of energy under the best alternative. Solar roads are cost inefficient whenever my postulates are true. Case closed.