Risks for energy shortage

The Swedish power industry has assumed that both of the random variables, supply of electrical energy and demand for electrical energy, are normally distributed (CDL, 1978, section 5). This assumption simplifies calculations, and no reason has been found to assume that the distributions are of another form. We will use the same assumption in our calculations. However, we should discuss the suitability of doing so.

As for demand, the assumption of a normal distribution should be a reasonable approximation because of the central limit theorem of statis-tical theory. This should also be the case for the influx of water into the reservoirs that provide hydroelectric energy. For this assumption to be reasonable for thermal power, there must be a large number of relatively small units. However, several large nuclear power plants have recently been put into use in Sweden. This casts some doubt on the suitability of assuming that the supply of electrical energy is normally distributed. It also means that there may be important discontinuities in the distribu-tion, even in the extremities. However, this factor should be of less importance when trying to determine the optimal size of the energy reserve margin, as we are doing here, than when trying to determine the optimal reserve margin for generating capacity.

Demand and supply are shown in Figure 8.1A in the form of normal distributions. The expected value of supply is E(s) and of demand E(d)..

To be able to show the risk for shortfall as an area in the diagram, we combine supply and demand to form excess supply (supply less demand).

The distribution of excess supply is shown in Figure 8. IB. The standard

Probability

probability distribution for,

demand iqd) probability

distribution for supply (q⁸)

TiWlf

E(d) E(s)

Figure 8.1 A The probability distributions for demand and supply of electrical energy.

deviation for s-d is equal to the square root of the sum of the squares of the standard deviations for s and d, and E(s-d) is equal to E(s) - E(d).

This makes the distributionof s-d flatter than the distributions of s and d. The distribution of s is shown, displaced, in Figure 8. IB for referenc The risk for shortfall is the area under the distribution curve to the I .

i-Probability

probability , distribution

* for excess supply (qs-qd) risk for shortfall

E(s-d)

Figure 8. IB. The probability distribution for the excess supply (supply demand) of electrical energy.

Probability for shortage of at least this size

1

--0 . . . .

5% size of shortage Figure 8.2. The risk of shortages of different sizes when the overall risk

of shortage is 3 percent.

of the origin, where supply is just equal to demand, shown as the shaded area in the figure.

The Swedish power industry now uses the rule of thumb that the risk of a shortage of energy should be at most 3 per cent - that is, that all the energy demanded be delivered in at least 32 years out of 33. This rule applies without regard to the size of the shortfall. Obviously, if the risk of any shortfall is 3 percent, then the risk that there will be a shortage of, say 5 per cent is considerably less, namely 0.6 per cent (if s-d is normally distributed and its standard deviation is 8.4 percent). The relationship between the size of the shortage and the risk that it will occur for this case is shown in Figure 8.2. Different degrees of shortage are shown on the horizontal axis. Assuming that the overall risk of a shortfall is 3 per cent, the risk of shortfalls of any particular magnitude can be read off the vertical axis.

The value of having an additional kWh in the energy reserve depends on the willingness to pay for an additional kWh when there is a shortfall, as well as the probability there will be a shortage of energy. The willingness to pay in turn depends on which user of electricity is hit by the shortage, and thus on the supplier's options for directing the avail-able energy to different users. To illustrate this, we start with a simplified picture of the situation in which the marginal willingness to

~i

P0 .

"fifij 1*15 TWh Figure 8.3. The value of a reserve margin for various degrees of

dis-crimination.

pay for energy is constant at SEK 1.00. This represent the average marginal willingness to pay, as shown by the straight-line demand curve in Figure 8.3. Assuming that the price of electricity is SEK 0.14, the normal damand for electricity will be 100 tWh per annum. If the price should increase to SEK 1.86 or above, then no electricity would be demanded, according to this demand curve. Let us also assume the expected reserve margin is 15 per cent. This means the risk of a shortage of energy is 4 per cent. We assume that it is affected by the shortage, so that the user willing to pay, for example, SEK 1.85 is as likely to be hit as one willing to pay only SEK 0.14. This means that the expected willingness to pay for an extra kWh of electricity will be SEK 1.00, the average willingness to pay. In this case, the willingness to pay for keeping one kWh of energy in reserve will be SEK 1.00 weighted by the risk of shortfall, 0.04, or SEK 0.04.

However, it is unlikely that the suppliers are completely unable to direct the available energy to users with higher willingness to pay. Let us assume that the power company can always see to it that the shortage affects only those whose willingness to pay is in the lower half of the scale - those willing to pay less than SEK 1.00 for the last kWh of energy they wish to use rather than do without it. The average willingness to pay of this category of customer is (SEK 1.00 + 0.14)/2 = SEK 0.57. The average willingness to pay of the remaining users (those whose willing-ness to pay is greater than SEK 1.00) will be (SEK 1.86 + 1.0CO/2 = SEK 1.43. The risk that the shortfall will be greater than 50 terawatt hours

Willingness to pay (SEK/kWh)

0.04-, 0.03

0 _{1 1 1 1 1}

Figure 8.4. Willingness to pay as a function of the degree of discrimination

is practically zero in this case. The willingness to pay to maintain an additional hour in reserve will thus be:

SEK 1.43 x 0 + SEK 0.57 x 0.04 = SEK 0.023

Even this limited degree of control reduces the value of the reserve margin considerably. We can continue by dividing the users into four categories - assuming we can direct the shortage to the quarter with the least willingness to pay to begin with, and then to the second quarter, and so on. The average willingness to pay of the four categories of users will the be:

1. (1.86 +1.43)/2= 1.645 2. (1.43+1.00)/2= 1.215 3. (1.00 +0.57)/2 = 0.785 4. (0.57+ 0.14)/= 0.355

The willingness to pay for an additional kWh of energy in reserve will then be:

1.645 x 0 + 1.215 x 0 + 0.785 x 0 + 0.355 x 0.04 = SEK 0.014

since the risk that a shortage will exceed 25 terawatt hours is practically zero. If the power company can distinguish eight categories of customers according to willingness to pay, the resulting willingness to pay will be

SEK 0.010. As the number of categories is increased, the willingness to pay to maintain one kWh of energy in reserve approaches the limit of SEK 0.008. Figure 8.4 illustrates the relationship between the number of categories and the marginal value of reserve energy.

If the demand curve represents the demand of energy and the power company cannot distinguish among the various categories of customers, then the value of an additional kWh on reserve in the given situation would be about SEK 0.04. However, if the power company can reasonably discriminate according to willingness to pay, the value of an additional kilowatt hour of energy on reserve would be only about SEK 0.01.

In the simplified case above, we assumed that the demand curve was linear. Results from certain econometric studies indicate that demand curves become steeper as the size of the shortage increases. This may be an indication that an assumption of a constant-elasticity demand would provide a better approximation, at least in a limited interval. However, such an assumption leads to unreasonably great willingness to pay for an extra kilowatt hour if the energy shortage i s great. A reasonable upper limit for the willingness to pay might be the costs for users to maintain their own energy reserve. Therefore, we use such an upper limit in our calculations below.

An inverse constant-elasticity demand curve can be formulated as fol-lows:

f

(e

where p(qd) is the marginal willingness to pay at the quantity qd; T\ is a constant indicating the elasticity of demand (which is -i\); and z is a normally distributed error term with zero mean.

To calculate the willingness to pay to increase the reserve margin by one kWh, we divide the users of electricity into n categories as above. We then calculate the average willingness to pay for an extra kWh (given that there is a shortage) for each category and then multiply it by the risk that the category of user will be hit by a shortage. The sum of the corresponding values for all the categories will be the ex ante willingness to pay for the maintenance of an additional kWh of energy in reserve.

This can be expressed mathematically

| p(q)dq • \f{q)dq

average willingness to pay of user the risk that user category category i for an additional kWh if there i will be hit by a shortage

is a shortage that hits that category

where/"is the normal distribution function:

D is the expected demand for electrical energy, and K is the expected total capacity (= D + reserve margin); and a is the standard deviation of the excess supply.