Expenses and income are both functions of time. Costs start accruing when a power plant is proposed and initial studies are made, for instance, on environmental impact. Land is purchased, the plant is designed, equipment is ordered, and con­struction begins. This takes many years. The plant is finished and begins producing power. Profits begin to be made, year by year. At the same time, there are expenses for operating the plant, and for repairing and replacing equipment. To get a reasonable number for the COE, one has to adjust all the expenses and income forward or backward to the same date. Time is money. This is called discounting. It is done with another formula:

I, (C + OM + F + R + D) t /(1 + r) ‘
I,,Et /(1 + r)’

This is a formula unfamiliar to physicists but may be more familiar to readers involved with business or finance. Here C is the capital cost, OM is operation and maintenance, F is the fuel cost, R is the cost of replacements, D is the cost of decommissioning at the end of life, and r is the discount rate. In the denominator, E is for earnings. The sum is over time t. To derive a value at time zero for an expense or income occurring at another time, a discount has to be applied. The discount rate is like an interest rate but includes also expectations of what the market will be like, how much inflation there will be, and factors like those. Financiers normally assign a discount rate between 5 and 10%.

Suppose we want to calculate the COE as of the start of planning. We set that as t=0. For simplicity, let us do the accounting annually, not monthly or daily. Suppose it takes five years to get ready, five years to build the plant, and it has been operating for another five years. For years t = 1-5, we have the money Cl — C5 spent in those years, which is only interest on money borrowed, salaries, and rental for office space. For years t=6-10, C will be much larger, as the plant is built. For years 11-15, we have C+OM+F for those years, and also E earned in those years. Each year’s amounts are divided by (1 + r) raised to the power t in order to get the value as of t=0. Both the numerator and the denominator are summed over the years, and the ratio is the COE. In later years, there will also be values for R and D.

To get a better idea of what discounting means, let us consider a simple example. Suppose you borrow $1M to build a machine, taking five years to do so. At the end of the five years you sell the machine for $1M. However, you could not have sold that machine for $1M at Year 0, since that machine did not exist yet and you could not make any money with it. It has a smaller discounted value at Year 0, given by C/(1+r)5, according to the formula above. If C=$1M and the discount rate is r=5%, we have a value at t=0 of C/(1.05)5, which works out to be only $0.784M. The reason is that you had to pay compound interest during the five years. One million dollars compounded annually at 5% is $1M times (1.05)5, which is $1.276M. You had to pay $0.276M in interest, so you made only $0.724M, and that is closer to the value of the machine at t=0. Actually, you did not have to borrow all the money at once, so the discounted, or levelized, value is $0.784M, which is exactly the reciprocal of $1.276M.

This exercise points out that a large part of the cost of any power plant, regardless of its power source, is interest during construction. If the discount rate is 7.5% (halfway between 5 and 10%), and the plant takes five years to construct, summing over the discounted value of one-fifth of the capital cost for each of five years shows that 20% of the cost is interest and other financial factors. The levelized COEs of all different kinds of power plants (except fusion) in many different countries have been analyzed in exhausting detail by the International Energy Agency.6