FOURTH LECTURE LOAD FACTOR AND COST OF POWER The cost of the power supplied at the customer's meter consists of three parts. A. A fixed cost, that is, cost which is independent of the amount of power used, or the same whether the system is fully loaded or carries practically no load. Of this character, for instance, is the interest on the investment in the plant, the salaries of its officers, etc. B. A cost which is proportional to the amount of power used. Such a proportional cost, for instance, is that of fuel in a steam plant. C. A cost depending on the reliability of service required, as the cost of keeping a steam reserve in a water power trans- mission, or a storage battery reserve in a direct current dis- tribution. Since of the three parts of the cost, only one, B, is propor- tional to the power used, hence constant per kilowatt output, — the other two parts being independent of the output, — hence the higher per kilowatt, the smaller a part of the capacity of the plant the output is ; it follows that the cost of power delivered is a function of the ratio of the actual output of the plant, to the available capacity. Interest on the investment of developing the water power or building the steam plant, the transmission lines, cables and distribution circuits, and depreciation are items of the character A, or fixed cost, since they are practically independent of the power which is produced and utilized. Fuel in a steam plant, oil, etc., are proportional costs, that is, essentially depending on the amount of power produced. 52 GENERAL LECTURES Salaries are fixed cost, A ; labor, attendance and inspection are partly fixed cost A, partly proportional cost B, — economy of operation requires therefore a shifting of as large a part thereof over into class B, by shutting down smaller substations during periods of light load, etc. Incandescent lamp renewals, arc lamp trimming, etc., are essentially proportional costs, B. The reserve capacity of a plant, the steam reserve main- tained at the receiving end of a transmission line, the difference in cost between a duplicate pole line and a single pole line with two circuits, the storage battery reserve of the distribution system, the tie feeders between stations, etc., are items of the character C ; that is, part of the cost insuring the reliability and continuity of power supply. The greater the fixed cost A is, compared with the propor- tional cost B, the more rapidly the cost of power per kilowatt output increases with decreasing load. In steam plants very frequently A is larger than B, that is, fuel, etc. not being the largest items of cost; in water power plants A practically al- ways is far larger than B. As result thereof, while water power may appear very cheap when considering only the proportional cost B — which is very low in most water powers — the fixed cost A usually is very high, due to the hydraulic development required. The difference in the cost of water power from that of steam power therefore is far less than appears at first. As water power is usually transmitted over a long distance line, while steam power is generated near the place of consumption, water power usually is far less reliable than steam power. To insure equal reliability, a water power plant brings the item C, the reliability cost, very high in comparison with the reliability cost of a steam power plant, since the possibility of a break- down of a transmission line requires a steam reserve, and LOAD FACTOR AND COST OF POWER 53 where absolute continuity of service is required, it requires also a storage battery, etc. : so that on the basis of equal reliability of service, sometimes very little difference in cost exists between steam power and water power, unless the hydraulic develop- ment of the latter was very simple. The cost of electric power of different systems therefore is not directly comparable without taking into consideration the reliability of service and the character of the load. As a very large, and frequently even the largest part of the cost of power, is independent of the power utilized, and therefore rapidly increases with decreasing load on the system, the ratio of average power output to the available power capac- ity of the plant is of fundamental importance in the cost of power per kilowatt delivered. This ratio, of the average power consumption to the available power, or station capacity, has occasionally been called "load factor." This definition of the term "load factor" is, however, undesirable, since it does not take into consideration the surplus capacity of the station, which may have been provided for future extension; the reserve for insuring reliability C, etc. ; and other such features which have no direct relation whatever to the character of the load. Therefore as load factor is understood, in accordance with the definition in the Standardization Rules of the A. I. E. E., the ratio of the average load to the maximum load; any excess of the station capacity beyond the maximum load is power which has not yet been sold, but which is still available for the market, or which is held in reserve for emergencies, is not charged against the load factor. The cost of electric power essentially depends on the load factor. The higher the load factor, the less is the cost of the power, and a low load factor means an abnormally high cost 5+ GENERAL LECTURES per kilowatt. This is the case in steam power, and to a still greater extent in water power. For the economical operation of a system, it therefore is of greatest importance to secure as high a load factor as possible, and consequently, the cost — and depending thereon the price — of electric power for different uses must be different if the load factors are different, and the higher the cost, the lower the load factor. Electrochemical work gives the highest load factor, frequently some 90%, while a lighting system shows the poorest load factor — in an alternating current system without motor load occasionally it is as low as 10 to 20%. Defining the load factor as the ratio of the average to the maximum load, it is necessary to state over how long a time the average is extended ; that is, whether daily, monthly or yearly load factor. "" F f9 h ^■ , f(, n M ~(f / r* n^ z^ ^ — — — :p — _ — / 2 4 >™ ~i t" ~ > ~] r "7 5" / 2 — ; r 4 t ( 9 ^ r / 0 / a — _L _ _ 1 1 il Fig. 14. Summer Lighting Load Curve. For instance, Fig. 14 shows an approximate load curve of a lighting circuit during a summer day : practically no load LOAD FACTOR AND COST OF POWER 55 except for a short time during the evening, where a high peak is reached. The ratio of the average load to the maximum load during this day, or the daily load factor, is 22.8%. Fig. 15 shows an approximate lighting load curve for a winter day : a small maximum in the morning, and a very high evening maximum, of far greater width than the summer day curve, giving a daily load factor of 34.5%. ■■" ■^ "" "" w^m { \ ■" ""' " "" \ \ 1 1 \ ' \ ~^ 1 i- i - — \ rft, ^ / /f V7 P ?i IG m 'II G \ z 0/ D Ci 'R\ /s \ 1 1 s. \ / y \ 1 \ / I i V 1 1 1 \ .^ 1 \ V \ / V V / V, •■» / \ \ / \ 1 1 / v\ n 1 1 / \ I 1 1! / > \ 1 \\ / ^ \ i VJ 1 // r V 1 / \ |V \ / ^ / ^ v^. /> / \ ^c "' ,• / / h~" ^ — *^ / 2 A f X ' i • / 0 / ? ^' i > c \ / 0 / L 1 1 Fife. 15. Winter Ufehtlnfe Load Curve. During the year, the daily load curve varies between the extremes represented by Figs. 14 and 15, and the average annual load is therefore about midway between the average load of a summer day and that of a winter day. The maximum yearly load, however, is the maximum load during the winter S6 GENERAL LECTURES day; and the ratio of average yearly load to maximum yearly load, or the yearly load factor of the lighting system, therefore is far lower than the daily load factor : if we consider the aver- age yearly load as the average between 14 and 15, the yearly load factor is only 23.6%. One of the greatest disadvantages of lighting distri- bution therefore is the low yearly load factor, resulting from the summer load being so very far below the winter load ; econ- omy of operation therefore makes an increase of the summer lighting load very desirable. This has lead to the development of spectacular lighting during the summer months, as repre- sented by the various Luna Parks, Dreamlands, etc. 1 1 I f\ ?f /6 Fa :r< Oft r^ ^M^/T/? I 0/ \d Lfi ?n - -X ^ ,^ "~ -\ -7^ \ < t \ \ } \ ' 1 k J \ J f s ~" ■^ /^ 1 ^ : 6 ^ / 0 k e i t 6 s / f ^ 1 1 1 JL 1 1 Fl^. 16. Factory Power Load Curve. The load curve of a factory motor load is about the shape shown in Fig. 16: fairly constant from the opening of the factories in ithe morning to their closing in the evening, with perhaps a drop of short duration during the noon hour, and a low extension in the evening, representing overtime work. It gives a daily load factor of 49.5%. LOAD FACTOR AND COST OF POWER 57 This load curve, superimposed upon the summer lighting curves, does not appreciably increase the maximum, but very greatly increases the average load, as shown by the dotted curve in Fig. 14; and so improves the load factor, to 65.4% — thereby greatly reducing the cost of the power to the station, in this way showing the great importance of securing a large motor load. During the winter months, however, the motor load overlaps the lighting maximum, as shown by the dotted curve in Fig. 15. This increases the maximum, and thereby increases the load factor less, only to 41.7%. This is not so serious in the direct current system with storage battery reserve, as the overlap extends only for a short time, the overload being taken care of by storage batteries or by the overload capacity of generators and steam boilers; but where it is feasible, it is a great advantage if the users of motors can be induced to shut them down in winter with beginning darkness. It follows herefrom, that additional load on the station during the peak of the load curve is very expensive, since it increases the fixed cost A and C, while additional load during the periods of light station load, only increases the proportional cost B; it therefore is desirable to discriminate against peak loads in favor of day loads and night loads. For this purpose, two-rate meters have been developed, that is, meters which charge a higher price for power consumed dur- ing the peak of the load curve, than for power consumed dur- ing the light station loads. To even out load curves, and cut down the peak load, maximum demand meters have been developed, that is, meters which charge for power somewhat in proportion to the load factor of the circuit controlled by the meter. Where the circuit is a lighting circuit, and the maximum demand therefore coincides with the station peak. 58 GENERAL LECTURES this is effective, but on other classes of load the maximum de- mand meters may discriminate against the station. For in- stance, a motor load giving a high maximum during some part of the day, and no load during the station peak, would be pref- erable to the station to a uniform load throughout the day, including the station peak, while the maximum demand meter would discriminate against the former. By a careful development of summer lighting loads and motor day loads, the load factors of direct current distribution systems have been raised to very high values, 50 to 60% ; but in the average alternating current system, the failure of developing a motor load frequently results in very unsatisfac- tory yearly load factors. " " 0 (\ V J 3 t *f r 1 J 1 1 f \ 1 \ . '^i J / \j \ f r f 5 i s / S V / \, / ^ /^ i ^ c d i ? fO i 2 ^ J^ {. f a /f /ie 1 / hi/^ 1 ^ IL^OAD \LoAjd Cuff\/ H 1 1 Fl§. 17. Railroad Load Curve. The load curve of a railway circuit is about the shape of that shown in Fig. 17 : a fairly steady load during the day, with a morning peak and an evening peak, occasionally a smaller noon peak and a small second peak later in the evening, then tapering down to a low value during the night. The average LOAD FACTOR AND COST OF POWER 59 load factor usually is far higher than in a lighting circuit, in Fig. 17: 54.3%. In defining the load factor, it is necessary to state not only the time over which the load is to be averaged, as a day, or a year, but also the length of time which the maximum load must last, must be counted. For instance, a short circuit of a large motor during peak load, which is opened by the blowing of the fuses, may momentarily carry the load far beyond the station peak without being objectional. The minimum dura- tion of maximum load, which is chosen in determining the load factor, is that which is permissible without being ob- jectionable for the purpose for which the power is distributed. Thus in a lighting system, where voltage regulation is of fore- most importance, minutes may be chosen, and maximum load may be defined as the average load during that minute during which the load is a maximum; while in a railway system, a half-hour may be used as a duration of maximum load, as a railway system is not so much aflFected by a drop of voltage due to overload, and an overload of less than half an hour may be carried by the overload capacity of the generators and the heat storage of the steam boilers ; so that a peak load requires seri- ous consideration only when it exceeds half an hour.