Mats Bohman and Roland Andersson
5.3. Optimal price structure
5.3.2. Pricing with binding constraints
Let us now turn to the analysis of optimal prices during periods (k) of full capacity utilization, i.e. where X* > 0, indicating a higher short-run marginal cost than ce. In such cases the quantity of electricity produced will be equal to hQ.
Different cases may occur depending on where this capacity limit hQ happens to fall in relation to the quantity of electricity demanded. With reference to expression (5.17), we can establish the optimal price for all such cases.
When interpreting condition (5.17), it should be kept in mind that the scarcity value Xk refers to the joint product which, in optimum, will be distributed between the two goods. The share assigned to electricity Q&) depends on how hQ is related to the ranges in the benefit function for electricity, i.e., whether the public utility is a buyer or a seller or no trade occurs with the grid. The share that is assigned to district heating is denoted X£.
The utility is a seller to the grid.
We start with the regime where the public utility is a seller to the grid, i.e., hQ>q%, which is quite common in peak periods; cf. Camm (1981).
As is evident from (5.17), the optimal local price for electricity is then />£. Further, ifpf > ce, it is immediately clear that the share of X* which
will be assigned to electricity ispf - ce = Xl > 0.
This is actually an equilibrium condition for the electricity market. As the value X* is determined by (5.17), it is also evident that the share of the scarcity value which is assigned to district heating (X£) is determined residually as X* - X| = X£ (see Figure 5.1). It is also quite possible that X* = XI and thus that XJ? = 0. This implies that production of heat is so high that marginal kWh of heat will not be of any value to district-heat-ing consumers. Then the cogeneration plant is on the margin used solely as a thermal-generation plant. Thus there may be peak periods in which it is optimal to have a zero energy price for heat.
On the other hand, if pk = ce, then the scarcity value X* will be assigned entirely to district heating so that X* = X*. Finally, if pf < ce, then we have X| < 0. As a result, XJ? = X* + X|, which means that the scarcity rent which should be referred to district heating X£ will be greater than the scarcity value X* which refers to the cogeneration plant. This outcome simply means that district-heating consumers will have to pay part of the marginal operating cost ce.
In order to establish the optimal price structure, it is important to find out whether or not the capacity constraints are binding in consecutive periods with lower demand. Indicating the peak period by index k and the consecutive period by k+1, we always have thatpf >pf+i. If the public utility sells to the grid in both periods, it is clear that the optimal electricity price will decrease. The effect on optimal energy prices for heat depends on whether or not the capacity is binding in period k+1. If it is binding, and since ce is constant, it is obvious that X& > \%+\. Of course, X* > X*+i as k is the peak period. But since X| - X|+i can be greater than X* - X*+i, it is possible to get the result Xj? < X*+i. It is therefore quite possible that the price structure for heat will retain the reverse price pattern as compared to the price structure for electricity, ex en when the capacity constraints are binding in some periods.
However, there are i-wo other possibilities, i.e., the case where i) th;*>
capacity is not binding in k+l, and ii) XJ? > X&4. In both cases, the optimal price for heat will be higher in period k than in period k+l, thus modifying the previous result. But the price structure over the off-peak periods will still vary in a reverse pattern from the electricity prices during these periods.
The utility is a buyer from the grid
Another regime of practical interest is when the public utility buys from the grid during the relevant periods, i.e., where hQ < q%. From (5.17) we can conclude thatpjf is the relevant peak price for electricity. Aspf and ce are constants, Xj? = X* - X& will still hold. Thus, in this case, the optimal energy price for heat is also determined residually. All of the price structures already dealt with for hQ > q% may also occur in this case.
Prices, marginal costs A
P hQ Ik
n
hQ Quantity of electricity MB = marginal benefit function
ce = constant marginal operating cost per kWh of electricity pen = price or value of heat in electricity terms
= price obtained by the utility when selling electricity to the grid - binding electricity capacity constraint
= scarcity value for the joint product
= share of X* assigned to electricity
= share of X* assigned to district heating
Figure 5.1: Optimal structure when capacity constraints are binding
No trade with the grid
A third possibility is that q% < hQ < q%, indicating that no trade will occur with the grid in peak periods. The optimal prices for electricity and heat are still obtained from (5.17). But in this case the price for heat cannot be residually determined.
The optimal price pattern still depends on the price structure on the grid and on whether the public utility is a seller or a buyer or does not trade with the grid, even in off-peak periods. If it becomes a seller, then the pattern will be similar to the one dealt with under the heading "The utility switches from buying to selling" presented below. On the other hand, if the utility becomes a buyer, then the analysis below concerning a switch from seller to buyer is relevant. If the public utility does not trade even in off-peak periods, then the prices for either heat or electricity may decrease, increase or remain constant over consecutive periods with decreasing load.
The utility switches from selling to buying.
Regimes where the utility switches from selling in the peak period to buying from the grid in off-peak periods may also occur. It is not necessary to discuss determination of the scarcity value and its distribu-tion, as this procedure is the same as in the cases already treated. The new aspect concerns the level pt as compared to
ps+i-In principle p£ can be greater than, less than or equal to pif+i. If pk+\ Sp£, then the optimal price structure for heat will include a higher price in peak than off-peak periods; i.e., a normal peak-load pattern. The factors which determine such an outcome are the size of the interval
R,nS Pm Pm
the size of PrriPmB. K
, and the size of the difference pB -pB+\. The greater and the smaller the difference pB -pB+\, the more likely the outcome specified above will occur. However, when this out-come does occur, the optimal electricity price pattern will not show the usual peak-load structure.
B **
If, on the other hand,p£+i < pY., that is if the price structure for electricity exhibits the usual pattern, then the optimal price structure for heat may exhibit any one of the patterns observed when the utility sells to the grid.
The utility switches from buying to selling.
However, it may also be optimal for the utility to buy electricity in the peak period and then switch to becoming a seller in subsequent periods.
Under such circumstances, it is absolutely certain thatpf > p*+i, regard-less of whether or not the capacity constraint is binding in period k+1.
The reverse price structure for heat is therefore ceteris paribus more likely to occur in this case.
5.4. Conclusions
By use of the conditions for optimal pricing and standard peak-load pricing analysis, we have derived the price structures under different regimes. The results obtained can be summarized as follows:
i) Capacity constraints are not binding:
• whenever the utility is either selling or buying on the grid during two consecutive periods, the optimal local electricity prices are given by the prices on the grid. This means that these prices will show the normal peak/off-peak pattern. The corresponding optimal price structure for heat shows a reverse pattern.
• whenever the utility does not trade with the grid in two consecutive periods, local electricity prices do not necessarily show the normal peak/off-peak pattern, as they are no longer determined by the prices on the grid. Whatever pattern the electricity prices show, the optimal heat price structure will be a mirror-image of it.
ii) Capacity constraints are binding:
• whenever the utility is either selling or buying on the grid during two consecutive periods, the scarcity rents of the joint product and the local electricity prices will vary in the normal peak/off-peak pattern. But the price for heat may under these circumstances be higher in peak than in some off-peak periods as the demand for heat has to be kept compatible with the capacity. However, during off-peak periods, the prices for heat will show a reverse pattern to that of electricity prices.
• whenever the utility does not trade with the grid or switches from one rogime to another between two consecutive periods, different price structures can be found for each specific configuration. They depend on the structures of prices for buying and selling on the national grid and on the distribution of the scarcity value between the two goods over the different periods.
Let us finally interpret the main result of oar analysis, i.e., that the optimal price structure for heat can be the reverse of the price structure for electricity both on the grid and on the local market. The price on the grid is high in peak periods indicating that electricity during such periods has a high value. In the simple case when cogeneration capacity isnotbindingin peak periods, then thepriceforheatislow.Theeconomic
significance of this is that the price sensitivity of heat demand is
"exploited" in order to increase the heat production and so the production of electricity during peak periods when an increase in electricity produc-tion has a high value. During periods with a lower price for electricity, the value of "exploiting" the heat demand is less. It is therefore worthwhile to decrease the heat load during such periods. However, in cases where (»generation capacity is binding in peak periods, the price for heat may yet have to be higher than during some off-peak periods in order to keep district-heating demand compatible with the capacity.