3 4 Po The rates are charged for monthly use of power. Thus the maximum demand is . read monthly and the same is done with reference to the consumption of energy for each month. Thus, the first Wright system is applied with a rate, P, for an 83 hours' use per month, then a second rate, P2, up to 125 hours' use; from 126 to 166 hours' use a third rate, Pz, a fourth rate, PA, up to 208; a fifth rate, P5, up to 250, and a sixth rate, above 250 hours' use. A customer has consumed in a month 3000 kilowatt-hours withi a maximum demand of 10 kilowatts. The actual schedule of the Edison Company will show the following bill: 3000 = 300 hours IO $35.25 14.10 II.30 420 kilowatt-hours at 2.16 cents. 6.56 2.50 h = 9.08 Total $78.79 = 2.62 cents average rate per kilowatt-hour. Although the clerical work, in the actual application of such a schedule, is not very great, as the clerks make use of tables and curves, the schedule appears to be somewhat complicated and clumsy. Nevertheless, there is advantage in it, and this is the reason why I thought to put it before you, as I believe that the principles on which it is based are rational and sound. But the form given to it is not the best and can be better, as I shall endeavor to explain. Before doing so I should say that a similar method of charging was adopted by several other companies in Italy. A few adopted a plain meter system and others a Hopkinson's or Manchester system, that is, a fixed charge per unit of maximum demand and a rate per kilowatt-hour of consumption.* I have said that the principles underlying this method of charging are rational and sound. In fact, the schedule considers two ways of variation of the rate—the maximum demand of power made by the consumer and the hours' use of such demand. In general, the kilowatt-hour will cost more to the producer if sold to a consumer who takes one kilowatt than to a consumer who takes 10 kilowatts. Moreover, the first consumer can pay a higher rate and still have an advantage on other systems of obtaining power, and a wise seller of electrical energy has to take advantage of this. Hence a sliding scale of rates is justified. With reference to taking into account the hours' use of the demand, I must say that the application of the maximum-demand system is better warranted in the case of power than in regard to lighting. It is easier for a consumer to equalize the rate at which lie takes the energy for motors than to do so with lainps. You can not light your lamps in the daytime and put them out at night, but you can run your machinery in many different ways and choose the one which gives you a lower maximum demand. From this point of view, the maximum-demand system can be inade useful to the consumer as well as to the producer or seller. Again, with respect to motor service, one of the main drawbacks of the maximum-demand system is that the cost of the kilowatt-hour can not be ascertained until the year's or half year's end. With reference to power, the results of each month are more or less the same; there are no dark or light months to bother about; you can apply to each quarter or to each month its own maximum demand. You arrive then at the same goal, that is to say you can cut down the peak and at the same time you can at the end of each month send in the exact bill to the customer. I will now endeavor to show how easily a schedule of rates can be worked out on the same principles, not as a new thing, but only as a rational application of known principles. Let us draw out a curve to represent the variations of the * The Manchester-Hopkinson system was applied for the first time by the Milan Edison Co. in 1884 with a rate of 30 lire ($6) per lamp installed and .025 lira (0.5 cent) per lamp-hour of 16 candle-power. rate p of the kilowatt-hour in accordance with the hours' use h of the Wright system. The formula giving the relation between p and h is : PiH + (n - H) P, h or + B H (P - P2) (1) h Where P, and P, are the first and second rates and H the hours of use of the first rate, Equation (1) can be briefly expressed by the equation A y = (2) which, when properly plotted, is a branch of an equilateral hyperbola referred to an asymptote (the vertical) and to a line parallel to the other asymptote. This curve is drawn in full lines in Fig. 1. In a schedule of rates based on the principles enounced, in order to apply a sliding scale for the different values of the maximum demand, we ought to have for each of them a special curve corresponding to the equation H(Pi -- P.) + P2 h where H, P1, P, together or two or one of these values, should be special to each value of the maximum-demand M; which means that the schedule of rates ought to be formed by a group of curves having for analytical expression the equation (2). Now, the simplest way to arrive at the group of curves is to start from one of the curves—say the one relating to the maximum demand of one kilowatt (M = 1), which we will call the fundamental curve—and obtain the others by multiplying A and B by a certain coefficient of reduction C, this coefficient being a certain function of M, say C=f (M) thus for M=M C=C, A + C, B or yı = G P= GA = G( X from which we draw Yı = C y This equation allows us to greatly simplify the schedule. In fact, let us on one side fix the Wright system for a maximụm-demand Ni = 1 and prepare on the other side a curve of the coefficient C, namely, a curve containing for each value of Ma value of C; then suppose a customer's maximum demand to have been II IO. 2 We will work out his bill in accordance with the fundamental Wright system M= I and simply multiply the result by the value of C corresponding to M = 10. = P=2 cents H83 hours and that the customer has taken 3000 kilowatt-hours in a month with M IO. We have 83 (5 — 2) + 2 = 2.83 300 C == 0.57 hence P=2.83 x 0.57 = 1.61 price per kilowatt-hour, or in other , words, the bill should be worked out as follows: 3000 = 300 hours IO 83 h X 10 kilowatts x 5=$41.50 217 h X 10 kilowatts X 2= 43-40 IO C=0.57 $84.90 M $84.90 X 0.57=$48.39 and so it is for any value of M. The function C=f(M) may be any one, although it would be preferable if it embodied a simple law thus allowing us to obtain easily the values of C for fifths or tenths of units of M. I shall return to this point farther on, as I want now to bring in some additional simplifications of the proposed schedule. We have seen that the expression of a Wright system is: p = H (P - P) + P (1) On the other hand, let us consider a rate-system of the type called the Manchester-Hopkinson, in which a fixed charge A has to be paid per kilowatt of maximum-demand, and a rate B for each kilowatt-hour consumed. We want to draw out a curve according to this system on the same basis as for the Wright system. We see that if W is the |