US2836358A - Calculating tables and the like - Google Patents

Description

May 27, 1958 J. H. STARR Erm. 2,836,358

CALCULATING TABLES, AND THE Lm:

May 27, 1958 .1. H. STARR ETAL CALCULATING TABLES, AND THE LIKE 4 Sheets-Sheet 2 Filed April 6. 1953 May 27, 1958 J. H. STARR Erm. 2,836,358

' cALcULATING TABLES, AND THE LIKE Filed April 6,1953 4 Sheets-Sheet 3 Invervl'or: James H. 5km* 8c May 27, 1958 J. H. STARR ETAL 2,836,358

CALCULATING TABLES, AND THE LIKE Filed April 6, 1953 4 Sheets-Sheet 4 Am fier xlilo. l.

lll l InVehTorS: Zomes Horr 3v Rober? ACiQrkQr,

by AH-y United States Patent O 2,836,358 CALCULATING TABLES AND THE LIKE .lames H. Starr, La Grange, and Robert A. Clark, Jr., River Forest, Ill.

This invention relates to improvements in calculating tables, and the like. Generally, the invention has to do with calculating tables which are specially adapted to the solution of problems relating to power generating and distributing systems, including transmission line problems and other problems of like nature. It will appear, however, that various of the features herein disclosed may also be used in calculating tables for other kinds of determinations than those suggested above.

The invention may also be said to relate to improvements in the art of power generation and distribution since the disclosures to be hereinafter set forth relate generally to an improved method of analyzing and determining the conditions of operation of power generating units and the tie lines connecting said units, with the objective of producing and delivering the delivered energy at a minimum unit cost; and the invention further relates to the means to practice said method. More specifically, the invention relates to improvements in such method and in the means to practice the same, whereby the effects of losses in the tie lines connecting the generating units and the loads, and losses in the transmission lines, will be taken into account in making the determinations of opertaion for production and delivery of the energy at a minimum cost.

At this point we call attention to the fact that the present disclosures may and do apply to use in making determinations in the case of multi-unit power generating and delivering systems, whether the several units of such multi-unit system be housed under a single roof or comprise a single group, and wherein the units are connected by tie lines such as bus bars or other ties, such as transformers and connecting lines, and `constitute a single power station, or whether the several units be distantly separated from each other lbut connected by transmission or other tie lines. The features of the present invention are usable to effect the desired determinations, or to facilitate the making of such determinations, with accuracy and expedition in either of the general classes of problems outlined above. In all cases, however, proper account is or may be taken of the losses in the tie lines, transmission lines, and other network elements interposed between the generating units and the points at which energy is delivered, and interposed between various generating or supplying units. These losses will be much smaller in the case of the tie lines or elements connecting the generating units of a single power station than will be the losses in tie lines and network lines and elements, especially when the several generating stations or supplying units are located distant from each other, but in principle the eiiects of the tie line and otherlosses are the same in both categories of cases. The differences are primarily of degree, not kind.

It is well known that when a demand for delivery of a stated amount of power at a given point is to be satistied by supply from two or more generating or supplying units, the determination of the most economical division of the demanded power as between said units must take into account the forms and the values of the input-output 'i ice curves of the several generating or supplying units in question, and must take into account the forms and the values of the incremental rate curves based on said inputoutput curves for said units. The input-output curve for each unit shows the variation of input (whether in fuel or stated in dollars per unit power or energy) needed to satisfy varying output demands over the power range covered Iby the curve. The incremental rate curve shows the rate of variation of the input-output curve as the output is varied. Thus, the incremental rate curve shows the amount by which the increments of input must change as the output is successively changed by equal increments of output.

It is well known that when two or more generating or supplying units are acting to satisfy the demand, the most economical division of the total demand between said units will be that division in which all of the supplying units are operated at loads such that all of the units are operated at equal incremental rates. That is, for the most economical division of the total demand, all of the supplying units should be operated at the same incremental rate conditions (disregarding for the time being losses inV the lines connecting said supplying units). When the total demand for power changes the new total power value should be re-divided between all of the supplying units in such a way that all of said units are again (or still) operating at equal incremental rate conditions and loadings. These new incremental rates may or may not be the same as those existing prior to the change of total power demand. This is true, disregarding for the time being the losses before referred to.

The practical eect of the above statement is that the most economical total power output is obtained when each of the power contributing units is operating at such a load that throughout a minute change of its proportionate contributed power the cost rate of its contributed power will be the same as the cost rate of each of the other contributing units when their respective contributed loads are also changed by proportionate minute amounts.

The foregoing statement disregards the effects of losses, as already explained.

The truth of the foregoing statements is evidenced by proper mathematical analysis and is well known, and we shall hereinafter make reference to published works showing the mathematical derivation of the above conclusions. In applying the above stated principles to any specific set of units, proper account must be taken of any conditions of discontinuity in the curves of incremental rates plotted against outputs, for the several contributing units under consideration. Also, the curves of incremental rates `for the several contributing units are conveniently plotted on a cost or dollars and cents basis so that the final results of the analysis shall reveal the operating conditions which will enable delivery of energy at the delivery point or points at a minimum total or overall cost in dollars. Furthermore, the incremental rate curves for the several units must in any case be based on certain underlying assumptions, and they mayk show Idelivered cost incrementsv at the bus bars or the generator terminals or elsewhere. In any case such curves should represent determinations of incremental costs including all proper elements of such costs from the coal (or other fuel supply) to the assumed point of energy delivery for the respective generating or supplying units. v When more than one supplying unit feeds power into the system, said units are tied together within theV generating station by bus bars, switches, transformers, and

other elements. Energy losses will occur in these tie' elements and these losses will necessarily aiect the cost of energy delivered to the system being fed. Frequentlyv i contributing units affect the total cost of the useful energy delivered bythe Y station due KYto'division Vof vthe powerV between the several in a -ma-nner which -is not A'the most economical.vr l .Y Y 'Y We shall presently consider the effects of the 'tie and othergne'twork orfline orv like losses lon the solutionjof lthe factors determiningsuch distribution. vAccordingly'the*-Y principle already referred-to of equality of incremental rates ofthe several power contributing units in order to ensure most economical: delivery of the useful'power Y must be modied in proper manner in order to takecare Y of the 'effects of the losses occurring in the connecting lines and other connecting-elements. We shall presently show how such modifications are to bemade, and shall disclose the'means whichwe have disclosed herein to facilitate` the determination of the exact" division `of power between the contributing -units in order to ensure Y the most economical o r lowest costrdeliveryof thezuseful problem of effecting Ythe ymost .economical division of theV Y demanded power among the contributing units', -and shall disclose a method and the means which'we have devised for Aincluding saidrlo'sses in the determinations of division Vof power between the contributing unitsiso* as to ensure delivery of the useful energy tothe delivery points'at a 'minimum total ycost perr'unit energy. Y`Said method and said means may be:usefully'applied whether the contributinglunits. be located Yclose together, as' under a common roof, comprising a single generating station with itsY units feeding into'common buszbars, in which case the effects of the losses occurring in said bus bars and in any y connecting transformers will be properly included in the solution of 4the problem; `or `whether said contributing units comprise separate steam driven Ygenerating units withlseparate l.transformers individualV to said generating units with Vties connecting .the `outgoing lines; orwhether the contributing units be located distant from'each other. In 'cases where all Vof the contributing units are Vlocated close together .and comprise in eiect ai single power station, the .tielosses occurring within such station may be includedin making the determinationsof best divisions ,of load Yas between the units' of such .completestatiom andincremental .costcurvesforsuch entire station Ymay then bemade based .on such best conditions/of operationY of the units .of such station as lanentity. vThese incrementalV cost curves may then be used Vin connection with further'determnations of division of load between svi Paper 52-l12g JanuaryV V1952, pages,

eral stations feeding intoV thecomplete. network in .order v such elements or'by Yeinpircal.tests.

to determine the conditions of operation which'will en? sure most economical delivery vofY the total .load Vtothe points of use. The losses occnrringijwithinthe station itself above, referredto may he; conveniently termedlnternal "lossesand Vthe losses occurring outside `of the contributing stations' :may be Vconveniently termed external .or network .or `.line losses. i 1

Y We have already 'shown thatgwhen .the losses are disregarded the several unitsV contributingto `thetotal ldemanded power should be so operatedthat their incremental ratesVV are'fthe isameyino'rderV to ensure delivery -of the total energy most economically. This is also true Y in the case of contributing stations distant from each other, when th'e `losses Aare disregarded. However, the networkand line lossesmay andjgenerally will bejcon siderableand constitute a Considerable portion of the energy'delivered by the contributing stations or units. These losses constitute'an additional Vload which .must be carried VVby thejcolntributing stations .Acc ordingly,VVV this loss load will require that Vall of the contributing sta-VV tions together be loaded by an amount equal to the de livered load plus this loss load in order to dcliverthe specified useful or delivered load. However, lit'does not necessarily followy that Vthisadditional loss load Ywillflcne carried by the several contributingV units proportionately to the most economical loadsA which they shouldcarry. This istrue sinceV the manner inrwhich lthis loss load-will Y bedistrihuted over rthe network Will ldepend on Vmanyenergy to the delivery points. Y The above relationships have been known and analyzed mathematically in published explorations including the following: y

The theory of incremental rates andtheir-'practical application to load division, vbyl M. J. Steinberg v'and Theodore H. Smith, part L fElectrical Engineering? for March 1934, pages 432-445. Same,` part 'IL' Electrical l-EngineeringV for April 1934, pages 571-584.V Y l Y' Intrasy'stem transmission losses, by `E. E. George., Electrical Engineering Transactions for March 1943,

volume 62, pages 153-158. f

CoA-ordination of fuel costV and ,transmission lossby Y use of the network analyzer `to determine plantloading schedules, by george, H.VW. Page, and J. B. Ward, A .l Proceedings, .Section T .9242, 1949, volume 68, pages 1512. VEvaluation of methods' of co-ordinating Yincremental fuelcostsY and incremental transmission losses by L.. VK. Kirchmaye'r; and GL W. Stagg, A. I. E. E. Technical- 1.10 and accom? panying tables and figures; x Y 'The mathematicaladeterminations `and solutions .of optimum' loading v'conditions for the elementsrof any given generating plant' gare. necessarily based on someV assump'V tions and,rwhere necessary, on suitable test perfomance data forjthe varionselements and :units of such plant, assumed fuel costs, etc.. Primarily such assumptions .arielV test `performance data .are'necessary to determinegand Y plot input-output curves and inc rernental4 ratecu es for i the variousY units ofeachcenerating statioaortotherunpyinaunit. Performance .data .of various elements, such as transformers, etc... affecting incremental transmission Y losses may besecured @titiller from the manufacturers 0f With the Irieressary .test .and other-perfomance .data at .hand thef'mathematical :solution 'of the .theoretically correct division of the total network load between hef buting units .feeding kthe'network,fortiene Leal overall conditions 0f operatnfis very involved and laborious :and timewnsumins. `Even so, the solutions thus A,evolvedare usually based .on certain several contri Y simplifying assumptions, .and corresponding errors may be ,expected to creep intofrthe mathematical solutions.V

Such schemes have also included various cut and -tryop- Y comprising a special type o f'network analyieror c :g/ilc-ulrat-v. Y

erations, and insome ,cases network analyzers Lhave been used to simplifythe operations. All of theseipreviously, known means ofsolution yof theextremely involved prob-.Y lems presented bythe-network and -linejlosses have been, so complex, `or vhave .involvedsuch timeY consuming op-V erations, and 'possibilities `of error, `that theyY have found but limited application ,tothe real. prolbemsv PGSfnted f in the operationslofipowergeneration.and distribution systems.- 'Y 7' f f Specially, .onefeature of our invention includes means ing board by which `powerlosses ocungjinjthe. varionsnetwork elements whichrare, simulated by Qlcalcnlatins board may heldirectly Arealcolla'suitable.instrument 0r instruments such. as a roltmeter or voltmeters- 1.1111@ arrangement is such thatsaid Yinstrmuirlents maybe caused to read directly in .powerloss `stroll as 'megan/atte; not;

'assass withstanding that said losses are for the most part a squared function of the line current values (being principally resistance losses).

The loss in each line element of the network may be individually determined by the disclosures above referred to. A further feature of the invention relates to the provision of means whereby the losses occuring in any selected group (or all) of the line elements may be shown by a single reading of the same instrument as is used for showing the individual losses of the various line elements or network elements; or another and special sunimarizinfJ instrument may be used for showing the desired summation. lrovision is made in the calculating board for quickly connecting the power loss indicating instrument or instruments to those selected line elements or groups of line elements concerning which it is desired to know the power losses under various loading and load division conditions. Thus it is possible to make very quick determinations of said power losses by direct readings of the instrument or instruments, and to make such determinations for any given set of stable conditions of network operation, or to determine and show how the power losses vary under regularly varied condi tions of loading, etc., at different network locations, or for varying selected conditions of division of total load delivered by the contributing supplying units.

The calculating board is provided with suitable line and other element simulating elements or sections. Each of these preferably and conveniently includes means to simulate both the resistance and inductance (or capacitance) values of the real network element being simulated. Both the resistance simulating element and the inductive (or capacitive) simulating element for each real network simulating section may be adjustable through the necessary range of values to simulate the real values of the corresponding real network elements.

The loss in each real network element is primarily a function of the current flowing in said element and is a resistance loss. As such, the resistance loss is a squared function of the current value. In the case of such an element as a transformer included in a network section to be simulated or as an element thereof itself, the resistance loss occurring in such transformer will also vary as a squared function of the current; the iron losses may be disregarded as they are substantially constant and may be ignored in solving for the most economical division of load between the stations.

In order to determine the loss in the real network element it is therefore necessary to provide means in the simulating network element which corresponds to said real network element by which the loss occurring in the real network element as a squared function of the current, may be simulated. We have also provided each simulating network section with the means to simulate the resistance and the reactance already referred to; it being understood that thereby provision has been made in the simulating network section to simulate the resistance drop occuring in the real network, the reactance effects, and the loss, this latter function being indicated or shown specially. This loss indicating element of the simulating network includes a special resistance element, presently to be referred to, and which special resistance element is supplemental to the resistance element previously referred to, which simulates the resistance of the real network element.

Although the resistance loss is a squared function of the current flow in the network element in question, the means for indicating this loss is such, and is so arranged, that'the indicated loss shall be according to a linear function of the actual loss. It will be indicated as a D. C. voltage appearing across a pair of terminals. The conversion from the condition of a loss whose Value is a squared function of the current, to an indication in the form of a D. C. voltage across a pair of terminals, is very important to the performance of the objectives and the functions to be herein disclosed. For example, by the use of such linearity of indication, and especially when the indications are given as voltages appearing at the terminals which correspond to the simulated real network elements, it is possible to immediately and at a single operation show the additive losses of a number of selected network elements, or all of the network elements of an entire system. This result can be and is obtained in our present disclosures by providing the means to connect into a series circuit all of the voltage indicating terminals of the simulating network elements of which the summed loss is to be ascertained. Then the total voltage from such series circuit will be an indication of the total loss occuring in all of said simulated network elements under the conditions of current flow then existing. The change of total loss in the group of network elements (or in all of the network elements) whose voltage indicating terminals are at the time connected into such series, which change of total loss accompanies a re-division of the supply of power from the several contributin g power units, may be at once ascertained from the new reading of such loss as compared to the loss indicated by the previous reading for the previous conditions of division of power supply.

The ability to thus bring all of the network element loss indicating voltages of a group of the simulating elements into a single additive reading or indication is also dependent on the electrical separation of the circuits which give the loss indications, from the simulating network elements themselves. We have provided means and arrangements whereby this result is obtained. The need and importance of this characteristic will be seen when it is remembered that the additive A. C. voltages appearing across a number or group of simulating network elements, which voltages simulate the impedance drops across said elements, do not necessarily, and in fact almost invariably do not, bear any direct relation to the additive D. C. voltages indicating losses occuring in the network elements referred to.

The means whereby the loss in each network element is shown as a D. C. voltage reading varying linearly with loss at a pair of terminals includes a special non-inductive resistance element included in the simulating section of the miniature network, and through which special resistance element substantially all of the current of the miniature network section flows. The A. C. drop across this special resistance is then used to function a local circuit of such nature and characteristics that the desired linear power-loss-measuring voltage indication will appear at the terminals of such local circuit. Since this special resistance acts as a portion of the total resistance of the simulating network section, the other resistance element of such section must be adjusted to a value such that the sum of the resistances of such special resistance element and such other resistance element will correspond to and simulate the actual resistance of the real network element which such section simulates. We have made provision for eecting such correction in setting up the simulating elements of the miniature network section.

We have above referred to the special resistance element which comprises a portion of the means to determine the loss in the network section. The drop across this special resistance element will be proportional to the current which it carries. Therefore said drop would be considerably larger when carrying a large current (such as when said section simulated a heavily loaded real network element) than when carrying a small current (as when said section simulated a lightly loaded real network element). Now the local circuit by which the current owing in such special resistance element is caused to deliver a D. C. voltage across a pair of terminals, which D. C. voltage is linearly related to the squared functional loss occuring in the network section, Vincludes special devices which have a peculiar current vs. voltage D. @voltage output from such a device varies asa nonlinear functionV of the'A. C. current owing to such device. But Ythis relationship isY only true over a comparatively'limited range of current values (for a given such device). Among such Vdevices having this desired characteristic is a class of devices known in the electrical arts as varisto'rsf yAmong these are some which include the elemental material oertrianiurn, and also some crystal i preaches linearity.VV We have providedmeans whereby wel are able to ensure operation according to the squared Y ylaw function'over a range of D. C. voltage outputs sufri- `cient to meet all requirements of the device as used in our present disclosures. Y Y

f The above explanation concerns the means which we have provided to indicate and measure the losses whichV occur in the individual network elements, or in selected Y groups or all of said elements. These losses must be characteristic or relation;V This characteristicA isthat the Y supplied byV the severalpower generating or supplying Y ,sources feeding into the network. It now remains to show the means which -wehave provided to enable us tol determine .the divisionof the total power fed into the network between the several contributing units in such m'anner'asV to ensure themost economical supplyV ofrsaid total power from such contributing units. At this pointA it must be remembered thatthe total supplied power includes both the power delivered to the consuming units `connected to the network (the useful delivered power), and Valso the losses occurring Vthroughout the network it- I self. It must also be noted thatY said total losses are distributed over the network sections or units according to .the resistances or impedances Vof said Yelements andthe floads which they severally carry whenV the various de- A' Vlivered loads, are drawing power Vfrom Vtheir respective points of connection into theV network. VIt is alsoand at once apparentV that theY distribution of the lcurrents Vthrough the network will depend upon the manner in which the total load-(including the'losses) is divided among the several contributing units themselves, and that any shift or readjustment, of the total Ypower between the Vvarious network elements. Thisin'itself will cause many 'changes to occur in the values of the losses occuring in the network elements so that a redistribution and re- Vevaluation of said losses overl the'entire network'will occur. These changes may of themselves, and generally will, result in either an increase or a decrease of the total losses, with accompanying change in the total power which must be fed into the network to supply both the 4useful load and the losses.V This Achange in total required power supply or input will of itself call for aV redistribution or kre-.division of the total power between' the contributing units. Thug-each change of one kind re- 'quires a correctional change to be made at various points of Vthe system, in order to finally arrive at a balance of conditions which shall represent the desired solution'of the problem. In the case of a complex network, having many loadsV receiving power, and ofvarious load values, and having many network elements, and a number of a great Vmany tests and readjustments would almostwin- Y variably .be requiredrbeforereven a close approximation to the desired conditions could be attained. Of course that approximation, as exact as can possibly be obtained, isjthe, determination of the manner in which the several contributing power supply units .shall divide the VVtotal several contributing units is immediately accompanied Y Vby a redistribution of the current flows through the power between themfor e Vsurring maximum economy of y overall operation. Y

However, We shall presently show tributors may be quicklyjand accurately determined by use of the facilities which'we have previously outlined herein.Y Y Y Here it may be noted'that all o f the facilities and the methods of solution of the problem of determining 'the Ymost economical division of load between a plurality of power supplying units when theeffects of the losses are Y `to be included inthe determination, are applicable to theV case of a multi-unit generating station in whichv tie lines, tie transformers, bus bars, and other power loss creating elements areY present. 'f vWe haveamade referencegto this fact hereinbefore. Accordingly, 'when desired or justified, the methods and `facilitieshereinbefore outlined may beV applied Vto thesolutions of problems involving determina-Y tion of incremental rates (of an entire powerY station) Vto Y plot the curve of incremental costs or rates of delivered energy for an entire supplying station; In such 'case the useful load orY loads would be the points of deliver; of the energy to the transmission line onlines, or the point of connection into the network being supplied byV such station. Y

from each contributing station into the network, so that therseveral powers contributed by the stationsV maybe measured. Such means takes the form Vof suitablekwattmetersincluded in the connections ofthe stations tothe network, ora wattmcter whichV maybe `successively switched into the connection of'eachrcontributing station to the network. The'determination V of the most economical division of load between the severalcontributing vstations, with proper account .taken of Ytheeiects Voffthe line losses, requiresra determination offtne etects pro duced'by small or incrementalchanges inthe powerssupplied Vby the several stations. These incremental changes in supplied power are` of small values compared to the actual power inputs themselves, of 4the order of 5% of the respective powerinputs from the several Vstations.. The accuracy ofa determination of theprnost economical division vof power between the contributors requires that these power Vincrements be themselvesraccnrate within Vsmall percentages ofsuc'h incremental values.V Thus, if

the incremental change of power of any Astation should be known withinran accuracy of 5% of Such incremental change, the Yactual power change to be thus accurately Vdetermined would be of the order of 0.725% of the total` supplied power, a very'small change of total power determination. Nevertheless it is important .that these incremental power changesV be accurately known in order to Veffect accurate determinations of the final solution `ot the problem of most economicalY division of power between the contributing stations. f v

. An importantrfeature of our present invention relates totheprovision of an incremental power or wattmeter which Vshall be capable of indicating the increments of power change with a high degree of accuracy. Such an instrument is hereinafter disclosed and illustrated herein.

When it is desiredto determine the values of incremental losses occuring in the network Velements or certain ,generatintJ or supplying units contributing to the network,

of said elements, the` ditlerences in voltage readings of the loss indicating voltmeter may be used for this purpose; but since these incremental changes in value of the losses will be very'small as'compared to the totai losses indicated, it is seen that accuracyy of determination of the incremental changes mayrequire special provision to enable such incremental change values to'be made.

Accordingly we have made provision for determiningrthe Y. Y

incremental loss changes which are to be determined, within a small percentage'of error or" suchincremental values.; v

how the necessary Y e determination ofthe division of totallpowery (including the losses) Vbetweenxthe several contributing power Vc'on- We have provided means Yto measure the power input in the drawings:

Figure l shows typically the relation between input and output for a typical generating station or plant (plant a) and also shows the corresponding incremental rate curve for said plant, both of these curves being determined according to well known and understood principles and tests;

Figure 2 shows typically the three incremental rate curves for three plants of assumed characteristics (including the incremental rate curve for plant a, which curve is taken from Figure l), the curves of Figure 2 being all drawn on the same scale for ready comparison of the incremental rates under various test conditions;

Figure 3 shows schematically a simple calculating board and its sections, under the conditions of a set-up for power input by `three contributors, and ll points of delivery of load from said network; and this figure also shows schematically the series circuit for providing a summation of the individual losses of the several network sections extending substantially parallel to the various network sections for convenience of schematic showinv;

Figure 4 shows schematically more in detail a typical simulating network section, including the variable main resistance, the variable reactance, the variable special resistance used for actuating the local loss determining circuit, and said local loss determining circuit, and one form of local loss determining circuit shown schematically according to s'unple form, and including varistors as rectiers and in order to secure the desired conversion from the squared relation to the linear condition of loss indications;

Figure 5 shows typically the characteristic curve of relation between current and voltage for a varistor included in the rectifying portion of the circuit shown in Figure 4; and the curve of Figure 5 shows how said relation between current value and voltage drop across the varistor is according to a non-linear relationship over a limited range of current (and voltage) values, but approaches linearity for higher values of said functions; and this curve also shows that it is desirable to retain the conversion operations or effects of the local circuit within the non-linear relation portion of this curve;

Figure 6 shows schematically how the local loss determining circuits of a number of simulating network sections may be connected into a summarizing loss determining circuit for showing simultaneously the combined losses of all or selected simulating network sections, by addition of the several loss indicating voltages of the sections; and this gure also shows means, local to each of the local circuits for adjusting the per unit value of the loss indications indicated by the voltages appearing at the terminals of such local circuit; and this igure also shows additional means to enable the use of a multira'nge voltmeter in connection with the series loss indicating circuit to enable summation of the losses of many selected simulating network sections from which the additive value of the loss indicating voltages would be so large as to make it desirable to use a higher range voltmeter than for readings of voltage indicating losses under other test conditions, the arrangement shown in Figure 6 including a simple changeover circuit and switch to enable use of the voltmeter of either of its ranges of readings;

Figure 7 shows schematically a modified form of one of the rectier sections of the unit 70 shown in Figure 4, by the use of which modified form of rectifier section it is possible to secure the delivered D. C. potential according to the squared law over an increased range of applied A. C. potentials applied to the points 68 and 69 of said Figure 4, it being understood that the section shown in Figure 7 is one of the four sections shown in Figure 4, and that in making the substitution of the form of Figure 7 into the showing of Figure 4 each of the sections or rectiers shown at 80, S1, 82 and 83 shall be of the form shown in Figure 7; and

Figure 8 shows schematically a form of incremental CFI l0 load wattmeter suitable for use in measuring the power supplied to each of the points of power supply into the network, it being understood that each of the wattmeters shown at 102, 103 and 104 in Figure 3 may be of the type shown in Figure 8. Y

The features of our present invention are applicable to various forms of calculating tables and like devices, and the present improvements are especially intended for use in connection with alternating current calculating tables. Various forms of such calculating tables are well known in this art, including, by way of example, those shown in Patents Nos. 2,523,453, issued to I ames H. Starr September 26, 1950, and 2,613,237, issued to lames H. Starr October 7, 1952. It is unnecessary to illustrate and escribe herein the full details of a calculating board incorporating the improvements of our present invention, and accordingly we have herein shown said improvements more or less schematically as applied to a simple form of calculating board. However, we have also shown in more detail the essential elements of one simulating network section incorporating our present improvements for full understanding of said improvements. Before referring to said schematic calculating board showings, however, we shall rst refer to Figures 1 and 2 showing the general forms of the input-output and incremental rate curves previously referred to herein, for better understanding of our improvements.

ln Figure l the output and input values are shown by the horizontal and vertical readings, respectively; and the incremental rate values are shown by vertical values. The curve 20 shows the relation of input to output over the range of loadings covered by the chart. The ratio of increment of input for each increment of output is the incremental rate for the output in question and may be expressed as al1/d0. This ratio also is a measure of the slant of the curve 20 at any selected point. The curve 2l, on the foregoing basis, is the incremental rate curve for the unit of which Figure l shows the characteristics. We are here especially concerned with the values and operational characteristics supplied by the incremental rates, such as shown by curve 21. The values shown by such incremental rate curves may be given as incremental costs of delivered energy or power for assumed fuel costs, and assumed operating costs of the unit or plant for which a given incremental rate curve is representative. This is a convenient manner of developing such incremental rate curves, as thereby the determinations may be made both on a comparative basis and also on an absolute cost basis.

lt will also be evident that when a given generating station or contributor contains a number of power supplying units there may be developed a combined curve of incremental rates, representing the overall per unit cost of energy delivered by such plant to the transmission lines, taking account of all of the cost factors involved in the production of the energy and properly to be considered in developing such plant curve. ln developing such a plant incremental rate curve use may be made of the improvements herein disclosed in order to take account of the eects of losses when such effects are appreciable, and when a high degree of accuracy and reliability of the final results is desired. We have previously referred to this fact.

ln Figure 2, presently to be considered more in detail, we have brought the incremental rate curves for three typical contributing plants together on a single chart, and on the same scale of values, for purposes of convenience in reference of the incremental rates of the several plants to each other and to the loads carried by the several plants. These are the curves 22, Z3 and 24 for the plants a, b and c, respectively. These incremental rate curves are purposely shown as non-parallel, and of widely dilerent rate values for stated loads.

Now, when the losses are disregarded the best and Yent problem.

most economical division of the total load between thev plants a, b, and c would bethat in which theincr'emental rates for all three cfg-these plants, Vas shown *by curves 122, 23 land 24S-.were equal. For example, Yifthe total Vload to be divided'were substantially 193.5 megawatts,

down to the position where the new total vvalue would be produced bythe summation ofthe three points of kintersection of such lineZS at'its'new Ycurves 22, 23 and 24. y Y

We have already,` shown that the inclusion of the effects of the losses in determining the devision of the loads may be very important to actual operation under the most economical operating conditions. We have also shown that Yitcan be mathematically demonstrated that the total network losses-have a definite mathematical effect on the solution of the problem of most economical division when the eiects of losses-are toV be included in the determinations.

position, `with the three in Figure 3 we have shown, schematically a Ysimple net-V Vwork supplied by the three. stations a, b, andcgnurnbered as v26, 27and'23, respectively in said figure. The' network ilustrated includesrthe network sections 29 to 45,inclusive, feeding the loads 46 to 56, respectively. The general arrangements of simulating networks ofgminiature size are well known,V and thereforeiweY do not herein showin detail the various sockets and plugs, and other detailed devices byrwrhich the ynetwork maybe set up to simulate, in its connection, the connectionsV of the real network. lt will also'be understood that suitable am- Y meters, voltmeters, and other elements or units are provided for measuring theY desired current flowsgfvoltages, etc., at various points of the set-up of the miniature simulating network, and that. the necessary ground and return connections are provided in the .calculating table to completethe circuits. Y Y j Y Furthermore, each of the network sections 29 to 45, inclusive may be of convenient form and arrangement, provided it complies with the requirements hereinbefore stated, and incorporates the needed provisions for' determining and indicating'the loss occuring in such section.

In Figure 4 we have shown schematically one Vsimulating network section meeting the requirements ofthe presln this ease We have shown -the main resistance velement 57 which is adjustable" through V'the required range of values kvariabile and adjustable inductance (or capacitance) 59, and the special resistance element (non-conductive), 6i)

connected in series relationship. This special resistance element@ is also adjustable as'to value as yshown by the arrow. The terminals of the Vsectionare shown at 61 and 62, and the Y currentV flowing through this section simulates the current flowingthrough'the corresponding section of the real network being simulated.

The section also 'includes a small transformer whose primar je# is connected to the terminals of the special resistance 6l? so that the potential imposed on this primary isequal to .the drop'occurring over the special resistance 66 at the Vadjusted value of said resistance. This transformer is or" such small size yand design that its primary Vcurrent (substantially only magnetizing current) is so small in comparison to the current ow vthrough the resistanceV 6G' that, for` all practical purposesthe current through said `special resistance is the. sameas that owing through the section Yof the simulating network. Thus, the drop limposed on the transformer primary is substantially proportionalY to the `current through the network lsection. -Accordingly the deliveredpotential of the transformer secondary 65 is also substantially Vproportional to the current tlowing through the simulating network section.` This is, of course an alternating potential.

The terminals' and 67 of the transformer secondaryV 65 connect to the two diagonally 'opposite points 68 andV 69 of the rectifier unit 70, andthe intermediate points of this rectifier unit, 71 and 72 deliver rectified current to the lines 73 and 74. The potential of rthis rcctiied current will pulsate,rsince it corresponds to the rectified half-waves of the potential delivered over the lines`66 and 67 from the transformer secondary. YIny order to lmgely eliminate these pulsations, and to deliver a relatively smooth delivered potential, we have Ishown, the capacitor 75 bridged across the lines 73 and 74.VV With this arrangement a substantially smooth potential and f current will be Vdelivered beyond the capacitor. 1 The lines '73 and 74 connect to the terminals of a potentiometer 76 which must absorb the entire potential coming over said lines 73 andr74. We have provided the adjustable contact 77 on this potentiometer so that the delivered voltage at-the terminals 78 and 79 will bel proportionate to the voltage delivered over the lines 73 and 74,:as modifiedV by the potentiometer setting.

The recti'er unit '7,0 includes the rectifying elements 80, 81,82 and 83 which are one-way current ow elements. These elements are such as to possess the peculiar characteristic to which we have previously referred herein. namely, that the permissible current flow through them is not a linear function of the potential toV which they are Y subjected, over the range of current values needed for theV present purposes. We will presently disclose herein a modied arrangement ofelements whereby the range of current values within which the squared function will obtain may be substantially increased to meet whatever range of said values may be needed to meet the requirements of the present combinations of elements'and uses thereof.

In Figure'S we have shown a typical characteristic ofv one such device as those shown at S0, 81', 82 and 83, by the curve `84-showing variation of'current ow permitted through the device kfor varying potentials imposed across its terminals. It will beseen that over aY portion of the range of current variation the ycurve 84 is Nnon-linear,

and that over a small portion of this curve between Ythev potential, values of l and r4V (on the arbitrary scale of;

Figure 5), the current varies substantially according t0 the relation Vthat'the current value is proportional to theV squared value of the potential imposed across the .element.k For potentials lessv than l (on said arbitrary scale), therelationship above referred to is of au order higher than the squared relation. For potential values higher .than substantially 4 (on said arbitrary scale) the by the movable contact-58; the Y curve approaches linearity, as shown by the dashed portion 84a of said curve. vSuch devices Yare included inthe f Vterm ,varistorsf and we have previously mentioned certain of them which arereadily obtainable. Others are also useful for the present purposes. The essential characteristic of such devices as used in the rectifying circuit is that the voltage delivered at Vthe terminals 78, 79 shall vary according to the square law relation, as already explained.

With such a rectifying device included in the circuit arrangementsV the following operations will obtain:

The drop across the special resistance 60 is substantially proportional to the work section. Thel potential developed across the secondary g terminals of the-transformer is therefore also proportionalY tothe value of the current now through the network sect-um.V But the loss occurring in the network section is prpertionalxtolthe Y current Alowing through? thatrsection. Accordingly Ait is needful that-the linearly proportional A. C. notential acrossfthe transformer 'sccondaryelines and .67be converted :into a D. C. voltage across the terminals78 and 79 'which Voltage Yshall be'lproportionalto the square of the, value of the current HOW thlQugh Vthe resistance .60,

squared function of ,the

. 13 and therefore proportional to the square of the value of the A. C. potential delivered between th lines 66 and 67. Although the potential delivered at the terminals 71 and 72 of the rectifying unit 70 is a rectified potential, the permissible value of current iiow permitted by the rectifying unit through such unit mustbe according to the characteristics of the local circuit of Figure 4, including the characteristics of the rectifying elements 80, 81, 82 and 83. This current flows through the resistance of the potentiometer 76, so the D. C. voltage which will be developed across that potentiometer must be proportional to the above mentioned current value. Therefore as the D. C. potential across the terminals 71 and '72 of the unit 70 varies, the D. C. current flowing through the resistance 76 will also vary, but according to the requirements that the relation dened by the square law previously referred to shall obtain.

in Figure 6 we have shown four of the Voltage and loss indicating elements such as just described, but brought into a series circuit arrangement to integrate the several loss measuring indications for the four network sections. These sections are shown at 85a, 85h, 85c and 85d, respectively. The terminals for the loss indicating voltages of these four elements are shown at 78a and 79a, 78b and 79h, 7&9c and 79", and 7813l and 79d, respectively. These several terminals are suitably connected into the series circuit shown. The loss indicating voltmeter 89 is of suitable design and is provided with suitable scales, to read losses in values convenient for the conduct of the intended operations. Conveniently this voltmeter is a multiscale, vacuum tube instrument, having two or three scales in comparison to which the pointer may be read, and suitable internal switching arrangements, so that the instrument may be used for several different voltage ranges, to enable use of the same instrument over a wide range of losses. Conveniently also, additional range enabling facilities may be provided as shown in Figure 6. in this case we have shown the two throw, two pole switch 90, with the two pairs of stationary contacts 91 and 92, and 93 and 94, the switch blades being designated 9S and 96. The contacts 91 and 92 are connected by the jumper 97. A potentiometer element 9S may he tied across a suitable D. C. supply, such as the battery 99 by closing the switch lili). The movable contact 101 enables securing any desired value of bucking potential between the stationary terminals 93 and 94, the connections being properly made so that when the switch 95 is thrown to engagesaid terminals 93 and 94 the voltage as adjusted by the setting of the contact 101 will buck the accumulated voltage delivered by all of the loss indicating elements of the network then connected into the poss testing local circuit. By this means a single voltmeter may be used to measure small incremental voltages, i. e., suchas a few percent of the total.

It is also noted that the foregoing or equivalent arrangement makes it possible to bring the amount of the bucking voltage to a value such that the totalized sum f all of the loss voltages in the series circuit is neutralized. When this is done the net voltage shown by the instrument Si? will be zero, the instrument having been brought to a suppressed zero position. Having done this, any change in the network system which produces a change in the total amount of the losses, can be read as a direct reading of the instrument 89, and without need of subtracting one reading from another. Thus the instrument under such suppressed zero setting becomes a direct reading incremental loss meter. Without further change of the instrument the incremental losses would then be read on the same scale of said instrument as would be used for reading the total losses. Such a scale, covering a considerable range of voltages as needed for indicating the total loss, would be dicult to read accurately for the small incremental loss voltage indications. Furthermore, such small incremental losses could not be accurately indicated on the instrument without change of instrument characteristics. We have made provision for change iof the instrument characteristics as follows:

The instrument 89 may normally be used for the total loss voltages, which are shown by reading the position of the pointer 89e on the high range scale S9". By depressing the button 892 the characteristics of the instrument are changed so that it becomes a low range instru-v ment and continues as such low range instrument as long as the button 89 is depressed. During such low range condition the voltage of small amount will be accurately shown by the position of the pointer S9EL on the scale 89d, the full range of which scale corresponds to a small voltage as compared to the full range of the scale 89". When the button is released the instrument characteristics are .again restored to the high voltage range characteristics.

Using the foregoing instrument $9, or equivalent device, when the bucking voltage has been adjusted to set the instrument reading back to zero (by full bucking) so as to suppress the zero reading, the instrument button 59 may be depressed so as to bring into use the low voltage elements of the instrument. While said button is depressed the incremental change in power supplied to the network by one of the contributors 26, 27 or 28, as shown by the corresponding instrument iti, 103 or 104, may be determined, and at the same time the incremental loss corresponding to such incremental power supply may be noted on the instrument S9. Such incremental loss may thus be very accurately determined, corresponding to the incremental power indication of the instrument 102, w3 or 1%4 (or a modified form of incremental wattmeter presently to be disclosed herein, or equivalent device). Both incremental power and incremental loss may thus be very accurately determined.

The accurate incremental data, both of power supply and of losses, may be used for various determinations, some of which have been previously referred to herein.

The indicating wattmeters 102, 1-93 and 104 shown in Figure 3 are included in the supply leads connecting the contributing stations'25, 27 and 28 to the network. Some of the uses and functions of these wattmeters have already been disclosed herein. We will presently disclose one special form of such wattmeters, which we term incremental wattmeters, and which are so arra-nged and designed that very small incremental power measurements may be made by them with a high degree of accuracy. We have already referred to the desirability of provision of suchraccurate wattmeters in connection with the present equipment.

1r'eferring to Figure 7 we have therein shown schematically a modified form of rectier element which may be substituted for each of the elements Si?, Si, S2 and 83 of the unit 76 of Figure 4. By way of simplicity of description we shall refer to this element of Figure 7 as being in substitution for the element Sil of Figure 4, and shall describe its characteristics and connections on that assumption; but in so doing it will be understood that other elements similar to that of Figure 7 will likewise be substituted for the elements S1, S2 and S3, and with their terminals properly connected into the network 70 to ensure properly related rectifying effects for the currents delivered to the terminals 71 and 72 of said unit 70.

The unit shown in Figure 7 and theredesignated in its entirety by the numeral includes one or more varistors 1536 connected in series across the elemental terminals iii? and 16S. `When the element 1% is substituted for the element 8@ of Figure 4 these terminals 15W and 198 connect to the points '71and 68 of the unit 70 of Figure 4; other elements like that of Figure 7 would also be connected with their terminals 167 and 1% connected respectively as follows; element 1l5 substituted for element 81 would have its terminals 1M and 1:38 connected to the terminals 71 and 69, respectively; element 165 substituted for element S2 would have its i terminals 107 andV 10S connected to the terminals i v 6s, and 72, respectively; and element 105 substituted for the element S3 Vwould yhave'it's terminals 107 and 108 connected to the terminal- s 69 and 72,.re spectively.

In Figure 7 we have shown' three of the varistors 106 connected in series and designated 106e, 106band 17069,` respectively. Y A greater or less number of these varistors will be' used as requiredrby the particular installation, and depending on the characteristics of the varistors themselves. When the varistors 105 are all Vof like characteristics to a close tolerance it will be evident that the Ytotal potential applied between the terminals'107 and 108 will bervequally divided between all said varistors j(in Vthe showing Vof Figure 7 each varistor being subjected toY one-third such potential). Accordingly, the voltage range or variation which must be taken care ofV by each such varistor will be only one-third of the total applied potential. i Thus, too, each such varistor will operate onV a much more limited portion of the curve 84 of Figure 5 than would otherwise be the case. By this means it" is possible to bring the varistor operations into a portion of asados@ Y -16 fue required errer iu' the measurement ef Y is Syfofv such increment, thepermissible error sl l*only 0.25% of the ieedl eefuelly beine .Supplied by, the Station. whose load' is being measured. 'ny suitable' wattmeter er otherV pou/er measuring iiletrufueut 0r. sievieevwhieh.

Y ieieepabi'e of: measuriuswifhiu the required` eeeufeey v instruments are available olf-'aj designjandjcapacity whichV the curve S4 which will be of form substantially meeting i the requirements of thersquare law.V Y e Y However, we have made'still Yfurthereprovision for bringing the operations Vinto the square law relation, as follows: Y Y

Another varistor (orV series ofY varistors) 109 is also connected between the terminals 107 and 108 by the con-V nectionsV 1107and 111,su`eh varistor 109'beingproperly connecte-dwith respect to the vkvaristors 106 sorthat it permits current flow between'the terminals 107 and 108 inthe same direction as the .faristors v106. Such fact is of the several varistors Thus we have' provided Ytwo paralleipaths for the current ow between the terminals 10'?- and 103, one path including a plurality of the varistors* in series, the other path including one or more Vevidenced by the Vpointing of the arrows at the locations et theload being earried muy be used for the Purposes of the present iuyeution; buf-,Siiree it is V'dcmbtfulthat sueh wouldimeetthe present requirements We new diseloee an instrument which,V will fully meer Sueh requirementsY as are imposed by the present equipmeut` For thisV purent purposes, and vthe following description thereof is nowV given: .v l l t, v l l The wattmeter'shown inl-ligure 8 is what'lnay be designated asa suppressed zero wattmeter.' It is fso arranged that normally it indicatesY the Vactual powerbeing supplied over' .the Ilines to. which it is connected, suchAA power being read as the' indication Qi a suitable pointer e or needle working over a scale of suitable divisions and power indications. hou/evene small' increment of power is to .be read bythe instrument, the circuits may beltemporarilyrnodiued by e suitablefbutton or" switch in such armer that the amount oi the iueremeutal Y f value may be .t in eoruparisou to o suitable seele, oonuenieutly another seele, than. that normally used, since tije incremental values Aare' small, ,but are nevertheless spread over sueh scale of Size vappr.feriiueuus'the Size ofthe normally: used scale. e Upon depressionof such Vbutton or actuation of such switch the circuit modifications may be adjustedso as to bring the pointer reading of the instrument to zero position (priorY to the incremental change), thus suppressing the instruments zero reading;

and then, while the circuits'are 'still changed, and vupon making the incremental Vchange of poweiggthe' Ypointer otherr'varistors, also in serieswith each other-V but in parallel with the i'irst group.4 We have also provided the resistor 12m series'with the varstor (or Yvaristors) 109,

' so that the current dow ythrough the yleft-hand branch of the circuit is affected by the presence of this resistor 112. lfrdesired this resistor 112 may be adjustableV as shown by the arrow 113 of Figure 7. 'Y

By'the use of the combination Yshown in Figure 7, or such variations thereof as may be found'desirable, as to number of varistors, etc., Vit is found possible to assure operation according to the requirementsl of the square lawV over a wide range of applied voltages, and completely sucient to meet all requirements of the operations herein contemplated, ,and With an accuracy or" a few percent error throughthe required range of opera.-

tions.

lt is also to be noted that although the unit shown in Figure 4 by the numeral l70, and also such unit as modified by the disclosures of Figure 7 is a full wave rectifying unit, still it is" completely possible to use half-wave rectifying units for the purposes of our present disclosures, provided that the square law to which we have referred, is applicable and operates at all `times during the .rectifyins operations, so that the desired and intended results are obtained.

We have previously made reference to the Yneed of providing for accurate measurements of slight changes in load supplied to the networli by the contributing staments should be made in load division between said contions, during the process of determining what adjusttributing stations in order tol ensure most economical overall system operation. We have also pointed out that close accuracy in Vdetermination of such ASmall or incremental load changes (within a small-percent of error in the measurement ofv the increment itselflnecessarily corresponds to extremely smallV error ofrmeasurernent when compared to the actual value'of the load itself. Thus, when the incremental changeY amounts to, say, 5 yand of the instrument will show (the value ofsuch increment by pointer swing over a range of the scale far largerthan would have been the pointer movement, had therinstrument circuits not been modiied; .I n'otherf Wordsfthe circuit changes include provision for multiplyingY the pointer swing .by a ypre-determined multiplying'factor.V After the incremental value has been thus` accurately determined, the circuits of the instrument will be restored to their normal condition byreleasing such ,button or restoringrsuch `switch to `its normal position, the zero suppression being kalso discontinued. Specifically, the instrument shown in Figure 8 is as follows:

To measure powerl owingin the circuit Vau, `the current l and the voltage `E maybe Vamplified Aby the ameY pliiiers 114 and 11S,` respectively, theampliier outputs supplying the current and voltage coils, respectively, of

a rwattmeter 116. lf the amplifier gains are vcalibrated i and the amplifiers have no'phase shift error, Ythe vwattmeter indicationV will Ybe where K is aproportionality constant, and 0 Vis the phase angle'between'E and If Y The above described structure and method are `well known Vand are frequently employed in network calculating `tables and in other applications.

VIt is desired to measure a small increasel in the power carried in the circuit azz.. Such'an increase may be smallv compared to the power Vcarried'in the `circult prior to thev increase, audit-he increase inthe indication of lthe Ywattmeter may not `be Yaccuratelyreadable. To provide an accurate reading-of a small increase in `power the .structure here Vdescribed has beenV developed. Iitconsists of the increment Normally the output voltage of the transformer 120, which is proportional to the current I and in phase therewith, is impressed between the grid of tube 117 and ground; while the grid of the tube 118 is maintained effective at ground potential, the contacts of the push button 121 being open. Tube 117 acts as a conventional resistance coupled amplifier, resistor 122 being the load resistor, although of low gain value due to the parallel plate resistance of tube 118. The amplitied output of tube 117, if the circuit is properly designed with established principles of resistance coupled amplifier design well known in the art followed, is a voltage which is substantially exactly proportional in magnitude to the magnitude of current l, and which is substantially exactly in phase with current l except for a possible deliberate reversal of phase in the polarity of transformer 12u, as will be mentioned hereinafter. This voltage is impressed on the grid of amplifier 114, and, with proper adjustment of proportionality factors, results in an indication by the wattmeter of P=KEI cos 0, as previously described.

The voltage E of circuit aa is impressed on the voltage divider 123, and a predetermined fraction of E is impressed on the input grid of ampliiier 115 and also on the grid of tube 119. Tube 119 is connected as a cathode follower, a form of amplier having a gain of less than unity which is well known in the art. As is well known, the input impedance of the cathode follower grid is very high, approaching infinity at relatively low frequencies as may be employed in a network calculator. Hence, the connection of the grid of tube 119 does not measurably alter the voltage impressed on the input grid of'ampliiier 115. Tube 119, acting as a cathode follower, developes a voltage across its cathode resistor 124, which is substantially exactly proportional in magnitude to the magnitude of the voltage E, and substantially exactly in phase therewith. The cathode resistor 124 is shown as a potentiometer having a sliding contact connected, through the push button 121 and the condenser 125 for the purpose of blocking the direct current cathode voltage, to the grid of tube 118. By closing the contacts of the push button, and adjusting the position of the sliding contact of the potentiometer 124, a voltage in phase with and any desired fraction (between zero and an upper limit controlled by the circuit constants) of the magnitude of E, may be impressed on the grid of tube 113.

lf the current I is zero, and the push button contacts are closed, tube 11S acts as a conventional resistance coupled ampliiier, resistor 122 being the load resistor, and its output voltage is impressed on the input grid of ampliiier 114. If polarity of the output of amplifier 114 is reversed with respect to its input, while that of ampliiier 115 is not so reversed, the output of amplier 114 is a voltage in phase opposition to E and adjustable in magnitude through movement of the sliding contact ofthe potentiometer 124. Under this condition, the wattmeter will indicate -where K' is a constant depending upon the position of the sliding contact, (l) is the cosine of angle O degrees, being the angle between E and the input to amplifier 114, and where the minus sign indicates the reversal of polarity through amplifier 114.

fiers, but as amplifiers having a common load resistor, 122. Neglecting for the moment certain secondary eriects, the voltage developed across. 122 will be the sum of the instantaneous voltages due to the plate currents of tubes 117 and 11S. Keeping in mind the polarity reversal of transformer 120, the voltage developed across 122 by plate current of tube 117 will be Recalling the phase reversal provided in amplifier 114, and with suitable correction for the gain of that amplier, the output of amplifier 114 will be e'=KI/g-KE and the wattmeter will indicate P=E(KI/-K'E) :KIE cos f--K'E' Since K is adjustable by the sliding potentiometer contact, the wattmeter indication may be adjusted to P=0 when the push button is closed; and is P=KEI cos 0, the true power, when the push button is open. If the gain of amplifier 114i is now changed from value G to value G', the output of amplifier 114 changes in the same ratio to become e'=G'/G(K1/g-K'E) andthe wattrneter will indicate P=G"/G(K1E cos 9-KE2) =G/(0) =0 if K has been adjusted to produce a zero power reading before the amplifier gain was increased.

If, now, the current in circuit aa is caused to increase by a small increment (Il/ where 0 may have any Value not necessarily equal to Q, the voltage applied to the input grid of ampliiier 114 becomes G'/G(Kl/g+KdI/ q-KE) and the wattmeter indicates P=G/ G (KEI cos -I-KEdl cos 6'KE2) This equals P=G/G(KEdI cos 6') This is true since KEI cos 0 equals K'E2 by adjustment.

Thus the wattmeter indicates the increment in power in circuit aa and at gain ratio of G/G compared to the indication of the larger power ow in the same circuit before increment.

The use of a plate load resistor in any resistance coupled amplifier produces a change in voltage applied at the plate as the grid voltage changes. The use or" a common load resistor impresses on the plate of each of tubes 117 and 118 a change in plate voltage due to the signal on the grid of the other. Using triodes and circuit constants normal for high gain, this circumstance would produce harmonic voltages in the combined output which under the circumstances could produce errors in the watt- 1 the contributing stations.

. the estimated meter indication. One method of `'rieiducingi'this secondary effect is to employ a relatively low value ofY load resistor, which is equivalent to a low gain. ,Since the Output of the tubes H7 and 118 is amplified in ampliiier 114, a very low gain at this point is not a disadvantage. Another and possibly more effective remedial measure is to employ pentode tubes as tubes 1117 and 118 as the performance of pentodes is practically unalfected, over a reasonable range, by variations in plate voltage.V rriodes are shown in Figure 8a3 being simplerV to describe and to explain, but it should be understood that pentodes will normally result in improved performance. j

Therequipment hereinbefore disclosed may be used to supply information concerning effects of change of distribution or division of total-load .between the several contributing stations, including the effects of such changes 'on losses in the various network elements, and including effects of incremental changes or loading supplied by each contributing station.y It will be evident that said equipment may be used in various ways'to secure desired information on the basis of which to effect solutionrof various problems related to load division and load distribution over the network. We have already pointed outV the factY that changes in division Vof load between contributing stations, while maintaining the delivered loads at correct values of load, voltage, and power factor, eiect changes in network losses,which in turn affect the total power required to be input by the contributing. stations. The equipment herein disclosed may be used Vfor study of Vall of theseV relationships'and eects, and for facilitating solution of the basic problems involved in determining the most economical division of total power between We do not limit ourselves to the use of such equipment according to any specied or particular procedure, nor to any speciiic method of solution ofthe basic problem of load division. Generally the informationY and data which may be secured by use of this equipment will be used for facilitating mathematical solution of variousV related to such division of power. statement in mind we shall state hereunder one simple procedureV which may be followed for determining a close solution of the problem of most economical load division, under network conditions frequently encountered.

j A first approximation of the load divisionV is made, based on the judgment and experience of the engineers, and using, if desired, lsuch incremental rate curves of the'various contributingv stations as are generally shown in Figure 2. By this means a rather close approximation to the correct division may be made, taking into account losses, which losses are added to the useful delivered load on the network; The operator adjusts the load at each load center to correspond to the known conditions, andhe divides both the real and the reactive loads among the contributing stations as his judgment indicates they should be dividedfor best overall .performance. He checks the voltages'at the load' centers and makes such voltage adjustments at Vthe contributing sta Ytions as may be necessary to bring all load center voltages within acceptable limits.

The operator then increases the load on one station Vby a small or incremental amount, by adjustment of the phase position of that stations Voutput voltage, and reads fromk the wattmeters provided for'the contributing stations, not only the incremental increase of the load thus increased on the one station, Vbut also the decreases in output of the other contributing stations. ln some cases, depending on thek form of the network and the loads connected thereto, itV may be known beforehand that increase at one station will not affect one or more of the loads on other stations, the effects being limited to one or two, only, of the other, stations. Similar procedure is followed for each contributing station, the original conditions being restored at all of the stations, before proceedving to each succeeding station of the group` problems directly or indirectly` With the foregoingV Y iesbis *i The change in output Y tests may be expressed v ,(I),Y AA, AP, m1 .Y It can be shown by rigorous derivation that, withthe .riet delivered :loads remainingV constant .is slightly greater than zerol in the positive direction at 'optimum loading Vof station nL (it lbeing noted that dFl, dF2

cost-s at the station buses). Ygreater than zero, the station loading is greater than optimum; and if less than zero, the loading is less than i optimum. `The operation isrepeated for allv Ystations contributing to the network, and theloading of all of the stations is compared. AnY indication isA thereby obtained of the necessary' readjustnent in each station, both as to direction' and amount of suchy readjustment. A number of tests may be conducted Ybased on the above general procedure, until finally an acceptably close approximation to the exact division desired for most economical load Ydivision is obtained Vto meet4 the requirements.V j

It is not always possible to attain the theoretically most economical load division. it may be necessary to employ some portion of the capacity of some khighly economical station to supply reactive in order to maintain voltage at some load center within acceptable limits, or any one of several other practical considerations may restrict the system to something less than the most economical division. The system operator, guided vby the values determined according to the foregoing procedure, readjusts the division of both real and reactive power among the several stations, and by using the facilities hereinbefore disclosed, is able at once to determine whether or not the revised loading results in any voltage condition or other Vsituation Vwhich would not be acceptable'operatingV practice.y if no such condition is indicated by the presentlydisclosed facilities, the operator makes a second series of readings and obtains a second series or" values. These new values will Vshow the extent of the improvement in hourly fuel cost which the re-division of loads has accomplished, and will indicate the amount and direction of any further re-division which may be desirable. The refinement to which these corrections will be Vcarried is subject to the judgment of the operator, but he is always able to quickly check that judgment by reference to the present facilities. Furthermore, the system operator is able to completely Yduplicate the system performance under lthe conditions which he proposes to achieve y.before Yordering any re-adjustment in either load'or busbar voltageV at any contributing station. v

Another and very important yuse ofthe presentV facilities is in determination of Vpower costs for. billing purposes at interchange points. where :power is.Y exchanged betweenseparate systems. To this end it'is possibleto compute from the station operating costs and the line loss data, the cost of'delivered power. Thus, the proposed delivered load, or the load as actually delivered, may be placed onV the network at the proper point, and then a new solution for most economical generation `of the total power may be made,V Then a computation may be made'from the Vgenerating station power cost andthe line loss, of the cost of delivered power. The difference will obviously be the cost of deliveringtherrproposed exchange power. Y Y.

Another use 4of the present" facilities is in computing the cost of power trunked'for an Vinterconnecting company from one point toauother point on the system. -T he network may first be set up for the loads carried by the network, and the most economical generating schedule for the contributing stations may be determined. The system loss is also noted and the cost of delivered power is computed. The trunked poweris then added tothe network at one point and 'taken from the network for each of thefabove seriesof Fn represent the incremental `generating If the quantity is appreciably .at the point 'of delivery. Then the most economical generation schedule for the contributing stations is determined and the system losges again noted. The total cost of power to the network owning system is then computed, and the difference between this amount and the cost when not trunking will be the true cost of trunking the power through the network.

The foregoing illustrations will briefly indicate some of the beneficial uses of the facilities hereinbefore disclosed. Many other useful applications of said facilities will be evident to users of such equipment.

We claim:

l. A calculating table including in combination a plurality of sections to simulate real network sections, and means to connect said simulating sections together to simulate the real network, together with means to determine and give a direct indication of power loss occurring in at least one network section due to current flow through said network section, at least said network section including a resistance element, simulating network connections to said resistance element, current supply connections to said network for flow of current through the network sections including said section which includes a resistance element, said simulating network connections being constituted for ow of substantially the full current of such section through said resistance element to cause potential drop across said resistance element substantially proportionate to the full current ow through such network section, together with means to translate the drop of potential across said resistance element into a dierence of D. C. potential which is proportional to the squared value of said drop of potential across said resistance element, and means to give an indication of power loss proportional to said diierence of D. C. potential.

2. Means as specied in claim l, wherein said means to translate the drop of potential across the resistance element into a diierence of D. C. potential which is proportional to the squared value of the drop of potential across the resistance element includes a transformer, together with connections between the reistance element and the primary of such transformer, the potential across the primary of said transformer being substantially proportional to the current owing through the network section to which said transformer corresponds, and rectifying means between the power loss indication giving `means and the secondary of the transformer.

3. Means as specified in claim 2, wherein the primary and secondary of said transformer are insulated from each other.

4. Means as specied in claim 3, wherein each of a plurality of the simulating network sections is provided with a resistance element, and a transformer having a primary, and connections between each resistance element of a section and the primary of the transformer corresponding thereto, and wherein each of said plurality of the simulating network sections is provided with means to translate the drop of potential across its resistance element into a difference of D. C. potential which is proportional to the squared value of the drop of potential across its resistance element, together with connections to summarize said differences of D. C. potential for said plurality of sections, and means to give lan indication of the summation of said differences of D. C. potential for said plurality of sections.

5. Means as specified in claim 4, wherein said plurality of simulating network sections includes all of the simulating network sections of the calculating table.

6. Means as specified in claim l, wherein the means to translate the drop of potential across the resistance element into a difference of D. C. potential which is proportional to the squared value of said drop of potential across said resistance element, includes varistor current rectifying means.

7. A calculating table including in -combinatioda plurality of sections to simulate real network sections, and means to connect said simulating sections together to simulate the real network, and current supply connections for flow of current through the network sections of the calculating table, each section including impedance simulating means to simulate the impedance of the corresponding real network section, and also including means responsive to the current ow through such section to produce a difference of D. C. potential which is proportional to the power loss occurring in such section due to current ow through such section.

8. Means as specified in claim 7, wherein said means which is responsive to the current flow through each simulating network section to give an indication proportionate to the power loss due to the current flowing through such section includes a power loss determination circuit element electrically independent of the simulating network sections, together with terminals for said power loss determination circuit element, and wherein the diiference of D. C. potential between the terminals of each power loss determination circuit element is linearly proportionate to the power loss due to the current flow through such section of the simulating network.

9. Means as specified in claim S, together with an indicating device to indicate power loss, and connections between the terminals of the power loss determination circuit elements of selected simulating network sections and said indicating device, including means to cause the indicating device to give its indication proportionally to the summation of the D. C. potentials across the terminals ot the power loss determination circuit elements of said selected simulating network sections.

lo. Means as specified in claim 9 wherein said indicating device comprises an instrument of the multi-range D. C. voltmeter type and includes means to select the range on which said instrument gives its indications, together with means to introduce into said series circuit an opposing D. C. potential of selected amount.

ll. Means as specified in claim l0, wherein the said opposing D. C. potential is adjustable to an amount to balance the series additive circuit potential corresponding to a selected loading of the simulating network.

l2. A calculating table including in combination a plurality of sections to simulate real network sections, and means to connect said simulating sections together to simulate the real network, together with means to determine and give a direct indication of power loss occurring in at least one network section due to current flow through said network section, at least said network section including a resistance element, simulating network connections to said resistance element, A. C. current supply connections to said network for flow of current through the network sections including said section which includes a resistance element, said simulating network connections being constituted for flow of substantially the full current of such section through said resistance element to cause potential drop across said resistance element substantially proportionate to the full current flow through such network section, together with means to translate the drop of potential across said resistance element into a difference of D. C. potential which is proportional to the squared value of said drop of potential across said resistance element, and means to give an indication of power loss proportional to said difference of D. C. potential, wherein the simulating network current flowing in said section is an alternating current supplied between said A. C. current supply conr ctions and the drop across said resistance element is an A. C. potential difference, and wherein the means to translate said drop of potential across said resistance element into a difference of D. C. potential which is proportional to the squared value of said drop of A. C. potential across said resistance element includes a rectifye 23 ing element of the-varistor type, a pair of D.' terminals, means and connections between the resistance element and said D. C. terminals to produce a diierence @saaie of D.V C. potential between said terminals, said means including at least one such varistor element having the characteristic that thevalue of the current flow through said varistor element is substantially proportional to the square Vof the value ofthe A. C. potential imposed onV said varistor element through a limited range ofsaid A. C. potential values, and a loading resistor element in connection with said terminals to absorb current ow delivered through said varistorrelement, the D. C. potential between said terminals being proportional to the current ow through said loading resistance element aspermitted by said varistor element.

13. Means as specified in claim 12, wherein the means to translate the drop of A. C. potential across said resistance element into a difference of D. C. potential which is proportional to the squared value of said A. C.

potential impressed across said resistance element intors, wherein each branch of said full wave rectifyingrY unit includes at least one of said varistors. V

14. Means as specilied in claim 13, wherein Yeach branch of said full wave rectifying unitincludes a plurality of said varistors together with connections between said varistors, said connections being constituted to establish a series circuit in each branch of said rec tifying unit, the series circuit in each branch including the varistors of said branch and being constituted to permit current ilow in a given direction, only, in said branch.

` 15. Means as specified in claim 14, together with an other varistorv and a resistor for each branch of the full wave rectifying unit, and connections for said other varistorV and resistor of such branch, said connections being constituted to establish'a series circuit including the varistor and the resistor of such branch, and to connect such series circuit inparallel connectionV with the series circuit of said branch whichseries circuit includes the pluralityrof varistors, one of said circuits including a plurality of varistors in series with each other andthe other of said circuits including a varistor and a resistor in series with each other,'and both of said circuits which include the varistor means permitting current iiow through said branch in the same direction, only.

` 16. Means as specified in claim 12, wherein the means to translate said drop of potential across said resistance Velement .into a difference of D. C. potential which is proportional to the squared value of said A. C. potential which is impressed Vacross said resistance element n- Fvalue of current ow through the corresponding current cludes a rectifying unit including a plurality of rectifying elements of the .varistor type, and connections between` said varistor type rectifying` elements and the points of A. C. potential difference, said connections being constituted to establish a series circuit including said plurality lof varistor elements permitting current iiow in afgiven direction, only, between said points of A. C. potential difference.

17.Y Means as specified in clairn 16, togetherwith another rectifying element of the varistor type and a resistor, and connections to placeV said varistor and Ysaid resistor in series circuit with each other, and connections between the terminals of said series circuit and the terminals of the series lcircuit which includes the plurality of varistor` elements, to thereby establish parallel circuits between the points of A. Cspotential difference,

one of -said circuits including the plurality of varistors inseries with leach other and the other of said circuits including the varistor and the resistor in series with each other, and both Vof saidcircuits including `varistor means permittingY current tlow through said .branch in the same direction, only.

18. A calculating table for use in the analysis oftheV elects produced on total Vcost of supply of power to load demands on a network producedY by change of'distribu- -tion of the total current supply to said network among a plurality of contributors of power to suchY network, said`calculatingv table including Vin combinationA a plurality of sections to simulate real network sections,- and means to connect said simulating sections'rtogether to simulate the real network, each section including impedance simulating means to simulate theimpedance of the corresponding real networksection and including Vmeans to adjust the impedance of such section, a plurality of connections for input of current to said network at selected points of said network, means to adjust the value ofV current input by each of said connections, and power metering means in conjunction with each current inputv connection aforesaid, Veach of said' power metering means vincluding a movable power indicating element and also including means under control of the operator to change -the, amount of movement of the movable power indicating element per unit change of power in the current in- Y y put connection to which said power metering means corresponds. Y

19. Means as specified in claim'lS, wherein there is means under control of the operator to cause the indicating element of the power metering' means Vto read at theV zero power Vindicating position for ari` established input connection to the network. Y

20. A calculating table for4 use inthe analysisV of the effects produced on total cost 'of supply of power to ,load demands on a networkproduced by change of distribution of the current Ysupply to said network among a plurality of contributors of power to such network, said calculating table includingin combination a plurality of sections to simulate real network sections, and means to connect said simulating sectionsY together to simulate the real network, each section including impedance simf ulating means to simulate the impedance of the corre-V sponding real network section and including meansto Vadjust the impedance of such section, a plurality of connections for input of current to said network at selected points of said network, means to adjust the Valueiof current inputrbyreach ofH said connections, and power 4metering means in conjunction with each current input connection aforesaid, each'of said power metering means including a .movable power indicating element and also including means under. control of the operator `to cause the indicating element Aof therpower metering `means y,to read at the zero power indicating position for an established value of current ow through the corresponding current input connection to the network.

References Cited in the iile of this patent Y UNITED STATES PATENTS 2,244,369

NetworkY Calculator (Haupt),The Review of Scientific lnstruments,vol. 2l, No. 8, August 1950, pages 683-686. A AGeneral-Purpose Electronic Wattmeter (Garrett), Proceedings of the I. VR. E., vol. 40, February, 1952, pages -171. Y

Mean Square Vacuum-Tube Voltmeter (Rosenthal and Badoyannis), Electronics, vol.`25, pagesV 12S-131.

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