Sugerencias

Solar panels generate revenue through sales of electricity, and electricity is typically sold to electrical utilities and transmitted to end users. Electricity prices fluctuate stochastically in time. The manner in which they fluctuate can be modeled quite differently depending on the choices of time frames. For example, Knittel and Roberts ( 2005) provides an empirical investigation into the hourly electricity spot prices. Hourly prices are characterized by heavy seasonality as well as occasional spikes during times of high electricity demand. Hourly prices would be very useful for a power plant which has variable costs and which have the capability to ramp up production quickly.

However, solar panels provide electricity at no marginal cost, and it is therefore beneficial to run them at full capacity at all times. Therefore, we can opt to model wholesale prices of electricity over longer periods of time, which smoothes over short term fluctuations and produces average electricity prices that fluctuate less dramatically.

Modeling monthly electricity prices is relatively straightforward. Figure 4-1 shows the Q-Q plot of the natural log of monthly wholesale electricity prices, obtained from the Independent

Electricity System Operator IESO, a regulatory institution in Ontario. The monthly wholesale electricity prices fit the lognormal distribution quite well. We provide the model for the price of annual electricity prices using the Geometric Brownian Motion (GBM) as described in equation

(4-1).

. (4-1)

The in equation (4-1) stands for the wholesale electricity price, and and are its annual rate of appreciation and standard deviation respectively. is a standard Wiener process. The revenue generated from solar panels is also a function of the amount of sunlight ( ) panels receive, as well as the rate of decline ( ) in the efficiency of the panels. We assume that this rate of decline in efficiency is exponential, and we assume that and are independently

distributed. The expected discounted revenue over the lifetime of the panels is expressed mathematically as follows.

.

Figure 4-1: Q-Q plot of the natural log of monthly wholesale electricity prices as reported by the Independent Electricity System Operator (IESO).

is the discount rate required by the owner, and is introduced to simplify

notation. Note that since we only care about the expectation of , we can remain agnostic about the distribution of . Since

is constant, follows the same GBM process that

does.

Installation costs of solar panels have also fluctuated over the years. Increasing economies of scale, learning curves and technological progress are expected to contribute towards lower installation costs in the long run. Figure 4-2 shows the Q-Q plot of the natural log of monthly

global solar module prices, which we use as a proxy for overall solar installation costs. The data was provided by Solarbuzz, an international solar energy research and consulting company.

Figure 4-2: Q-Q plot of monthly global solar module prices, as provided by Solarbuzz.

While installation costs don’t fit the lognormal distribution as closely as electricity prices do, we believe that the data fits closely enough for us to justify the usage of the GBM as an initial model, leaving the usage of alternate distributions to possible future work. Equation (4-2)

describes the movement of installation costs.

. (4-2)

stands for installation costs, whereas and stand for annual rate of appreciation and standard deviation, respectively. is a standard Wiener process.

Once panels are installed, they must be maintained with an associated cost. We model these costs to rise with inflation in the long run, but not to fluctuate stochastically. The resulting

maintenance cost model for time is given by , where is the rate of appreciation. The maintenance cost over the lifetime of the panels is expressed as follows.

.

Here, is introduced to simplify notation.

Governments may provide subsidies to encourage solar panel installations. These subsidies may also vary over time. In this chapter, we consider subsidies which help lower the cost of

installations, and denote them by . We assume is deterministic.

When the owner of a resource decides to install the solar plant, the owner pays an up-front installation cost, and receives revenue from selling electricity plus any subsidies, and pays maintenance fees. The value of a solar installation is expressed in equation (4-3).

– . (4-3) The Bellman equations for the option to build solar plants are shown in equations (4-4). The top equation describes the evolution of the value of the option when the option to build the plant remains unexercised. The bottom equation describes the value of the option when the option is exercised. Equation (4-5) describes the final condition. The equations are similar to that presented for a spread option, except our model must incorporate an increasing strike price.

(4-4) . (4-5)

signifies the exercise boundary, in which for , it is optimal to exercise the option – i.e. install the solar panels. If is below the boundary, the owner is better off waiting.

Note that there can only exist one value for each pair. The continuation value of (top equation of (4-4)) with respect to is a convex function, which slope varies from greater than 0 to less than 1, and is always positive in value. The payoff (bottom equation of (4-4)) is either 0 or have a slope of 1 with respect to . Such functions can only intersect once, and that value is .