The model

5.2.1 Statement of the problem.

The problem is to find the prices for both heat and electricity locally consumed that will maximize the net social benefits of the use of a cogeneration plant with given capacity. This may be expressed as:

maxW=TB-TC (5.1) where TB is total willingness-to-pay for the cogeneration output and TC is the total cost function.

The maximization of (5.1) is subject to a capacity constraint. This problem has to be solved by considering two given parameters:

• buying and selling prices for electricity on the national grid, and

* the distribution capacity for district heating.

Budget constraints are not considered. Thus the problem dealt with may be referred to as solely an efficiency problem.

The joint feature of the supply can be specified by a simple propor-tionality factor as:

^ i=l,...,n (5.2) where qi indicates the hourly quantity of electricity, measured in kWh, during a given period i and Qi denotes the hourly amount of steam for district heating measured in kWh during the same period. The coeffi-cient a is a constant independent of the volume of electricity produced and of the period³. This makes it possible to use electricity output as the variable of optimization.

The value oi the coefficient a may differ from one production unit to another and usually falls somewhere in the range 0.25 £ a £ 0.6. Small production units (in MW) often have a lower a-value than larger units. In reality a cogeneration \ lant may consist of several production units of different sizes with different a-values.

Such complications are disregarded in this analysis, i.e., it is assumed that the a-: actor is fixed.

5.2.2. The total benefit function.

The total benefits of a cogeneration plant to a local community will depend on the quantities produced. Formally TB in (5.1) can be ex-pressed as

i=l,...,n, ( 5 3 a )

or by the use of (5.2) as

TBi = tfffi). * = I.-.», ( 5 3 b )

where TBi in (5.3a) or (5.3b) represents the hourly gross benefit in each period i. [If substitution for each of the goods over periods were taken into account, the arguments in (5.3a) could be expressed as output vectors qi = (q\,...,qn) and qf = (q^,...,qn). Furthermore, substitution between the two goods in each period could be included by letting

= q?(Qi), i = 1,-> n. Thus (3a) could be written as and (3b) as

TB = cjXtf).]

In order to keep the number of indices at a manageable level, we assume that the benefit schedules will be constant in each period i, but may vary between periods.

If there were no possibilities of trading with the grid, the marginal benefit function for electricity would be the inverse demand function attributable to local consumers. But, as it is possible to sell any amount of electricity the cogeneration plant can produce in a given period i to the grid at the energy price for sales in the high-voltage tariff (p?), demand becomes perfectly elastic at this price. It is also possible to buy any amount of electricity from the grid in a given period i at a fixed price ipf). Thus for each period, the marginal benefit of electricity must lie in a closed interval bounded from above bypf and from below hyp?.

We write the gross hourly benefits of the electricity supplied in each period as

The Bi function in (5.4) is one part of (5.3a) or (5.3b). The second stems from the production of heat. Provided that the utility behaves so that excess demand never occurs in the district-heating market, the total willingness to pay for various quantities of heat produced will provide a measure of the second part. In symbols this value is given by

TV? = TV?(q?) i = I,.»,» ^{( 5}'^{5 )} For simplicity, the periods used for decomposition of the benefits of heat production are identical to the periods given by the high-voltage tariff.

Using (5.2), (5.5) can be rewritten as

TVPiq?) = T V f W = TVffof) i = l,...,n (5.6) The total hourly benefit is thus

i = l,...,n (5.7) The marginal hourly benefit, MBi, of a variation in output level is

which amounts to a vertical summation of the parts.⁴ The joint aspect of supply thus gives a public-good flavor to determination of the marginal value of the cogeneration output.

In order to obtain the total annual benefits, we have to sum over all periods:

n n

TB = ^UTBiiqf) = J[jli[TVf(qf}+Bf(qt)] ^.9)

»=l i = i

where U represents the exact duration of the subperiods as specified in the high-voltage tariff.

5.2.3. Cost conditions and capacity constraints.

The technologies for both cogeneration production and local electricity and heat distribution are assumed to give rise to constant unit cost-rigid capacity cost conditions.⁵ As the problem addressed here is to determine prices at the cogeneration plant, the operating costs for local electricity F.nd heat distribution can be se it zero in the subsequent analysis without loss of generality.

Thus, the production costs of cogenerated output in terms of electricity can generally be expressed as:

4 If substitution between heat and electricity were taken into account, the demand functions would not be additively separable. But it would, of course, still be possible to obtain a marginal benefit function directly from (5.3b).

5 Such a cost assumption is frequently employed in this kind of context; see e.g. Crew

& Kleindorfer (1979) and Rees (1984).

.r^en1 (5-10)

where P is the marginal capacity cost per year, Q is the cogeneration capacity, and c^e is a constant operating cost per kWh.

The cogeneration capacity can be limited by several facers such as the steam-production capacity of the boiler, electrical condensation capacity, the district-heating network capacity and design, etc. In order to keep the exposition simple, we let the condensation production capacity in MW electricity represent the capacity constraint Q. The capacity con-straint can thus be formulated as:

hQ>qi i=l,...,n (5.11) where h indicates one hour.

5.2.4. Optimum conditions.

The Lagrangian for the problem stated in (5.1) can, using (5.9), (5.10) and (5.11), be expressed as:

max L = X UiTVfiqi) + Bf(gf) - c ql + k(hQ-qi)} - PQ (5.12) qf,Q, U ''=1

The following conditions hold at an optimum:

dqf dqi dqf

and i=l,...,n (5.13)

dTVf dB! \

= 0

and (5.14) Q

n ^

i=\

= 0

and i = l,...,n (5.15) c^e in (5.13) is t^Tie constant marginal operating cost. A* is the dual variable to the capacity constraint. This term expresses the scarcity value at-tributable tv, the capacity limit in each specific period. The function dTVf/dqf in condition (5.13) indicates a district-heating consumer's marginal willingness-to-pay for a marginal kWh of heat in terms of the corresponding quantity of electricity. According to (5.8), this term can be expressed aspiiqi). BBi /dqi is the derivative of the gross benefit function for electricity sold to local or national consumers. This derivative is given by:

for qi<qi

fi(qi) for qi < qi < qi i = \,...,n (5 ^{1 6}) pf forqi<qi

depending on where optimal production qi happens to fall. The limits qi and qi show how much electricity is demanded locally at an energy price equal to the price given in the high-voltage tariff for power bought from <pf) and sold to (pf) the grid during the given period. fi(qi) shows the local consumers' marginal willingness-to-pay for optimal electricity production.

Given this specification of the terms in (5.13) and provided that qi > 0, we can rewrite (5.13) as^#

= l n (5.17) dq!

Expression (5.17) defines not only the optimal electricity production (qi) in every period, but also the optimal prices for heat and locally distributed electricity in each period.

According to (5.15), the scarcity rents X* > 0 in all periods in which the capacity constraint is binding. Jn all periods./ where the capacity con-straint is not binding, expression (5.13) simplifies x>:

Jc:i

Condition (5.18) tells us that the optimal quantity will be given at the production level where the sum of district-heating and electricity consumers' marginal willingness to pay equals the marginal operating cost.

Condition (5.14) defines optimal capacity, provided that there are no indivisibilities in capacity expansion or construction. Summing condi-tion (5.13) over all n periods weighing each term with the duracondi-tion li and substituting this sum in (5.14), the following optimal capacity condition is obtained:

p = h £ l[pt((??) + ^ - c

] = hj^Uki (5.19)

«=i °Q* ;=i

This condition simply means that at optimum, the weighted sum of capital rentals across pricing periods equals the marginal cost of capacity.

However, even if capacity is perfectly divisible in the long-run perspec-tive, it is usually not the case in the short run due to the lead times of capacity formation. In the following, we confine the analysis to the short-run problem of using existing capecity, regardless of whether it is at the long-run optimum level or not.