Pricing in an interconnected system

3.4.1 Production - transmission

The space dimension of the pricing problem has not yet been explicitly taken into account in this chapter. The question of how to determine optimal prices in different nodes on a transmission line has been explicit-ly addressed by Boiteux & Stasi (1952). Turvey (1968a) has treated a related problem, viz. the problem of minimizing the costs of generation and transmission along a line. Bohn, Caramanis and Schweppe (1984) have treated the problem of establishing optimal prices in electrical networks over space and time. These contributions are not reviewed here. Instead we will in a simple analytical framework show how marginal supply costs vary at different locations and therefore how varying prices over space can be determined. Our analysis was inspired by the Weber-Alonso location model (1928 and 1964).

Assume the existence of thermal power plants at one node called T and hydro power plants at another node called H. These two nodes are connected by a high-voltage transmission line. This line contains a number \M) of other nodes of given location, where power can be withdrawn from the transmission line and delivered to energy con-sumers in the neighborhood. As there are concon-sumers also localized at the production nodes T and H, we have M + 2 nodes where energy is demanded but only 2 nodes where energy can be produced. We assume that the installed generation capacity in both thermal and hydro plants is large enough to serve the demand from the entire line provided there is sufficient fuel or water available.

Given the localization of production plants and consumers, the problem is to determine how much energy should be produced in the different plants and how energy should be shipped on the transmission line. More specifically, the problem is to minimize the marginal cost of supply at each node on the line The marginal cost of supply is simply the sum of two components: marginal operating costs in the plant delivering the

energy and marginal costs of transmitting the energy from the produc-tion node to a consumpproduc-tion node.

The marginal operating costs in hydro production are in a sense indeter-minate. During periods with abundant water inflow when the production potential of the plant is in excess of demand, the marginal operating costs are virtually zero. At all other times when precipitation is less, the marginal operating costs are equal to the scarcity rent that creates equilibrium between demand and available supply. In thermal produc-tion, however, the marginal operating costs are well defined, as has been lucidly expressed by Boiteux (1957, p 137):

"The cost of the additional kilowatt-hour is therefore equal... to the cost of the coal required for the production of this kilowatt-hour by the marginal station".

The second cost item is transmission costs. These costs are due to the fact that some energy is lost during transmission. The amount lost when transmitting a marginal kilowatt-hour a unit of distance on a given line depends on the voltage and the area of the transmission cable. As long as the transmission line is not overloaded, the marginal transmission costs are given solely by these losses. But when the capacity of the transmission line is approached, declines in power tension, known as

"voltage drops", can occur. These voltage drops represents deterioration in the quality of service which, during overload periods, represents an additional cost item as part of marginal transmission costs (quite similar to the marginal congestion costs in urban traffic).

The two production nodes T and H and distance between them are indicated in Figure 3.2. The marginal operating costs for thermal produc-tion are b per kilowatt-hour. The marginal transmission loss per unit of distance is x which is a technically given constant. The marginal cost of energy supply stemming from T at any node up to H will thus be given by the function 6(1 + xdi), i - T,..., H, where dh indicates the distance between node i and T. The slope of this marginal cost function is constant and given by bx.

In the same way, Figure 3.2 illustrates different marginal cost functions Pi, Pi and Pz of energy supply stemming from node H. Each one of these will show the costs at each node j depending on the distance dk from node H, on precipitation R and thus on the scarcity rent for hydro

Relaxing the assumption of sufficient thermal generation capacity would lead to the possibility of a scarcity value of thermal capacity X^H>0. The marginal coat of supply stemming from T would then be (Ä+Ä.wXl-MtfV) * = T...H.

v=T,...fl

N2 N3 H Distance Figure 3.2. Optimal price structure along a transmission line

production. R^w indicates a very wet year, R^ a year with intermediate precipitation and Rr a dry year. It is worth noting that not only the intercept, but also the slope, will be determined by this scarcity rent.

Now, given any specific configuration of marginal supply cost functions for thermal and hydro it is easy to determine how to use the different operating plants and how to ship energy on the transmission line to serve the demand from the different nodes on the line in an efficient way, For instance, in a very wet year when condition R prevails, the scarcity rent for hydro production will be Pi. Under these circumstances the marginal cost of hydro supply at each node j is given by the function PY(1 +%dii)J = H,...,T. As the marginal cost of hydro supply at node T will be Pi < b, no thermal production is warranted. This will also be true for years with less precipitation (Ä^) than Ä as long as (1 +xdii) P^\R )<b. In the limiting case where equality holds, some thermal production may be needed in order to secure equilibrium at node T at an energy price equal to b.

When condition R^ prevails, the scarcity rent for hydroproduction is Pf. Although Pz < b, both the hydro plant and the thermal plant are

required. This is because hydro capacity Q^fir^J will be totally ex-hausted by the demand arising at nodes between N2 and H at the scarcity rent F$. If more demand is directed towards H, the scarcity rent would increase, indicating a higher marginal cost of supply at every node. Thus the remaining JV2 - T nodes must be supplied from T. At the equilibrium nodeN2 we have that 6(1 + xS²) = p¥{l +

In a dry year, indicated by Br hydro capacity will be reduced to (^]R I As a consequence the equilibrium node will lie closer toH than under condition / r \ In Figure 3.2. this node is indicated by N3 and the scarcity rent at node H by P3.

The cost functions presented in Figure 3.2 do not only determine how to operate the system and the localization of an equilibrium node A/*

somewhere along the line between T and H. They also show the optimal price structure over the line as a function of the parameters R, b, z and the demand functions at each node. Under periods when both thermal and hydro production are warranted the optimal energy price at node T will be 6. If the equilibrium node iV* is situated between T and H the optimal prices will increase from b to b(l + xSr), i = T,...^*, as we move from one node to another between T and iV*. From iV* the optimal energy price at each node will be declining as we proceed towards node H. Thus, there is a strong case for differentiating electricity prices over space and between years with different precipitation, even if peak-load problems on the demand side do not arise.

3.4.2 Distribution of medium and low voltage electricity

Unlike the situation in the production and the transmission part of the supply system there are certain fixed costs in the local distribution system that can be allocated in an objective way among the different consumers. As Boiteux-Stati, (B-S), have pointed out "the capacity to be provided for the connection of each client C. is directly determined by the maximum demand of this client" (1952, p 111). The customer will

"rent" this connection exclusively for himself, over its economic lifetime.

Thus, as B-S suggest, the individual customer can be made responsible for the total costs of such an investment (e.g. by a fixed connection charge corresponding to an annuity of cost). If this fixed connection charge is higher than this, the customer has an incentive to underdimension the connecting cable (and conversely).

However, the importance of such costs has decreased in densely popu-lated areas as the length of those individual connection cables has been

reduced. Nowadays, a "circle cable" joining several consumers is quite often used. In such cases it is not possible to define an individual cost for this cable. The situation is then like any collective network, since the load depends on the number of customers and their different consump-tion patterns. Thus, the investment costs for such a segment cannot be distributed among customers except in an arbitrary way. (Of course, investment costs have to be financed somehow).

Customers usually have different consumption patterns; they demand different amounts of energy in different periods. If peak consumption differs systematically and can be staggered to some extent, the maxi-mum demand of all customers will not occur at the same time. Thus, by using this kind of semi-individual (or semi-collective) network, its capacity can be dimensioned to be less than the consumers' aggregated maximum demand. How much less depends on the customers' demand diversity, i.e., the irregularity of their demand in relation to a collective peak. Contrary to B-S, we believe that any attempts to determine customers' "cost responsibility" for such semi-collective installations in an overall tariff will always be arbitrary.

The costs for an extra kilowatt-hour distributed on such a medium or low voltage network are equal to the costs of energy losses in transfor-mation and distribution during off-peak periods and the scarcity rents either in the transformation stations or in the distribution network during peak hours. In addition, extra demand will cause voltage drops.

At the margin, extra demand would then lead to deterioration in the service supplied to all customers. Thus, the demand for an extra kilowatt-hour at peak hours will have a negative external effect. Of course, this negative external effect represents a cost which has to be considered along with other real costs.