IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No. 7 Julv 1981

ECONOMIC FUEL DISPATCH

Floyd J. Trefny, Member, IEEEHouston Lighting & Power Company

Houston, Texas

Abstract - The economical use of all types of fuelsavailable for the generation of power has become a majorconcern of electric utilities. Present methods do notfully take advantage of standard economic load dispatchminimization techniques because the fuel constraints oneach type of fuel are not considered -in the classicalcalculation. Presented herein is a minimization tech-nique that includes the standard load constraints aswell as the applicable fuel constraints. The techniqueis as it would be applied to real-time control of a pow-er system participating in short-term fuel management byautomatic means.

The minimization technique was programmed for useon a Xerox Sigma 5 computer and applied to twenty-fourhours of actual power system data. The generation sched-ule is compared to that which would result if fuel con-straints were not considered. The comparison shows thatthe fuel consumed from each supplier can be adequatelycontrolled by adjusting the power output of various gen-erating units to the point of always operating the powersystem within its fuel limitations and within contrac-tual constraints. By applying this method, small addi-tional amounts of fuel will be required to serve thesame power system energy demand but the cost of thisfuel is much less than the penalty imposed by not main-taining each fuel contract, thus greatly minimizing thetotal cost of power production.

INTRODUCTION

Since 1974, some power utilities have encountered anew dispatch problem more significant than automaticgeneration control and economic dispatch calculations.Because of the sudden concern over fuel shortages, manyfuel suppliers increased the constraints of their fuelsupply contracts to the point that many utilities wereforced to curtail some units' power production and limitcontinuous operation under economic dispatch criteria.This came about because certain fuels were no longeravailable, were available only in a limited supply orwere cut off from certain power plants. Thus, stricteconomic loading of generation became impossible. Therewere no automatic ties between unit fuel availabilityand desired power production for that unit.

The new problem was given the name "fuel dispatch-ing", as it concerns scheduling of boiler and nuclearreactor fuel requirements in accordance with fuel pro--curement contracts, all of which are overlayed witheconomic and stability considerations. Fuel suppliesvary in quantity and type, forcing utilities to burn acomplete spectrum of nuclear, gas, oil and coal fuels.Dispatching these fuel supplies is a function of fueltype constraints, boiler fuel input configuration, thecost of each type, the system load and generation sched-ule; and therefore fits into a class of dispatchingnormally done in the utilities' energy control center.

Paper A80 033-1, recommended and approved by the Power SystemsEngineering Committee of the IEEE Power Engineering Society for presenta-tion at the IEEE/PES 1980 Winter Meeting, February 3-8, New York City.This paper was recommended for TRANSACTIONS status 81WM 102-3, andpresentation by title for written discussion at the IEEE/PES 1981 WinterMeeting, February 1-6, Atlanta, GA. Manuscript submitted September 4,1979; made available for preprinting November 14, 1980. This paper has beenpublished in 1980 Winter Meeting Text of Abstracted Papers, copyright 1980by the IEEE. Made available for publication in TRANSACTIONS November14, 1980.

Kwang Y. Lee, Member, IEEEUniversity of Houston

Houston, Texas

The present practice of some utilities is to gross-ly schedule blocks of fuel for various units (usually bytype of fuel) and base load units to consume that fuel.Only the largest blocks of fuel are used by units parti-cipating in economic dispatch. Thus, units that are baseloaded cannot participate in reducing the total fuelconsumption but can very effectively accommodate thecurrent fuel schedule. This is not a fuel economicsolution, but is usually the most monetarily economicsolution, given the available method.

A form of fuel dispatching is done to analyze thelong term effect of fuel procurement contracts. In thistype of study, averages and assumptions are used whichlimits its application when applied to the immediateneeds of an operating power system. Concerns of animmediate nature require a mathematical formulation con-sisting of minimization functions that can be solved ina real-time process controller similar to those alreadyin use by power utilities. This can be accomplished byadapting the classical economic dispatch calculation toconsider the numerous fuel constraints for each supplierand each generating unit. The solution to an economicfuel dispatch calculation should be similar to the clas-sical calculation in order for an automatic generationcontrol program to use the results for load frequencycontrol purposes.

POWER SYSTEM OPERATING REQUIREMENTS AND DATA

The objective of economic fuel dispatching is toconsider fuel constraints in addition to the total powerrequirement and unit power generation limits in thedetermination of unit set points. These constraints aredefined by the unit fuel consumption limitations and thedesired system fuel consumption schedule.

Unit Fuel Constraints

At Houston Lighting & Power Company, most oil andgas burning units have the capability to burn fuel frommore than one supplier. A typical unit may have gaslines to two or three suppliers and the ability to burnfuel oil. Not all suppliers will necessarily be in ser-vice at a given time. However, at least two are usuallyoperational to maintain a more reliable fuel supply.

When more than one supplier is serving a boiler,the amount of fuel flow from each supplier is controlledby the fuel regulator alignment. Figure 1 shows a singleboiler being supplied from two gas companies. The amountof fuel flowing to the boiler from Supplier B is at afixed rate determined by the manually adjusted set pointon the regulator. The fuel from Supplier B is said tobe on flow control. The balance of the fuel required bythe boiler comes from Supplier A, which is said to be onpressure control. Any change in boiler fuel requirementswill be totally reflected in gas Supplier A. Units aredesigned such that any supplier may be on pressure orflow control. However, only one supplier is allowed tobe on pressure control at any one time.

Some generating units are configured so that theyshare a fuel regulation system. Thus, the change inrequirements of two or more boilers is reflected in thesingle supplier on pressure control for that set ofunits.

A multiple supplier fuel input configuration re-quires four limits for each fuel supplier. A high sus-tained limit (HSL) is the maximum flow rate allowable bythe custody metering system. A low sustained limit (LSL)is defined similarly. An additional low limit above the

© 1981 IEEE

3468

AUONTRLCCONTOLS

GASSUPPLY A

GASSUPPLY

I

MANUALFLOW CONTROL

Fig. 1. Typical boiler fuel input configuration

3469from many different sources. These fuels vary in type,but all can be equivalently specified in Btu's. Themethod by which fuel is put into the system forms thebasis for fuel dispatching. Obviously, generating unitsnot equipped to burn certain fuels must be consideredwhen determining fuel schedules. It was found thatnormally, a typical generating unit can only burn fuelfrom three suppliers, although as many as ten fuels areused in a given power utility. Thus, various combina-tions of units burning overlapping fuel types make up atypical power utility's fuel input configuration.

The amount of fuel on flow control for a given uniti and fuel supplier j will be later referred to as fifThe amount on pressure control is fi . Note that if fi3is greater than zero for a given unit i and fuel suppli-er j, then f.f must be zero for the same i and j. Con-versely, if fif is greater than zero for a given i and jthen fi3 is deiined to be zero for the same i and j.

DETERMINATION OF PRESSURE/FLOW CONFIGURATION

LSL, is applicable only when the supplier is on pressurecontrol. This is a low pressure limit (LPL), set toprovide adequate flow through the pressure regulator toinsure its proper operation. A step limit (SPL) foreach fuel supplier is required to describe the minimummanual change on the flow controlled fuel. For computa-tional convenience, these limits have units of MKB/Day( 109 Btu/Day).

System Fuel Constraints

System fuel constraints are dictated by the physi-cal facilities used to deliver the fuel and by fuel sup-ply contracts. Fuel supply contracts are characterizedby one or more of the following:

A. Maximum allowable volume per period of time, rang-ing from three minutes, one hour, one day, a month,a year or up to several years.

B. Minimum volume per period of time similar to theabove.

C. Absolute volume per hour, per day, per month or peryear.

D. Maximum allowable change in rate of take.

For short-term fuel management, a simple limit gen-erator algorithm can be used to define the high and lowlimits for each applicable fuel contract for the periodof the control cycle. The routine selects the fuelsupplies that should be exhausted for the control cycleand the supplies that should be used only as necessary.

For later purposes, the names given to these limitsare LFCL and HFCL, for low and high contract limits res-pectively, and have units in MKB/Day. Note that forabsolute rate limits, LFCL = HFCL. Table I shows a typ-ical set of fuel contract limits.

Table I. Typical Fuel Contract Limits

ORDER OFASCENDING COST

GAS1GAS2NUCOILlOIL2COAL

LFCL*

400.0800.0

0.00.00.0

80.0

HFCL*

600.01300.0400.0500.0200.080.0

*MKB/Day (10 Btu/Day)

System Fuel Input Configuration

The fuel used in a power system typically comes

A determination of which fuel supplies for eachunit should be on flow control must be made in order tominimize the effort of the economic fuel dispatch calcu-lation. This is best accomplished by using an off-linemethod by which system fuel requirements are correlatedwith usual unit fuel requirements. Although most genera-ting units have the capability to change any fuel supplyto pressure control, arbitrary changing should be avoid-ed because of the effort required and the possibility ofupsetting the unit while the switch is being made.

Problem Definition

The problem of determining the pressure/flow stateof each fuel supply for a generating unit is one of ana-lyzing the usual loading characteristics of each unit asit responds to system load changes. If a particularfuel supply needs to change at a particular time of dayand a unit following economic dispatch criteria tends tochange in the same direction, one could justify thisunit on pressure control for that supply. It is obviousthat no unit will economically respond to load changesexactly as a system fuel supply optimally would change.However, patterns of unit load changes are somewhatconsistent with desired fuel supply changes. Since allunits may have no more than one fuel supply pn pressure,it then becomes reasonable to set that supply on pres-sure control which best follows the optimum fuel sched-ule.

Technique and Approach

To analytically determine the pressure/flow stateof the fuel supply for each unit, a mathematical corre-lation between a unit's output changes and a fuel sup-ply's desired schedule changes is required. A samplecorrelation defined as follows best serves this purpose[7]:

N

d Li=lIx*i.yj - yl

I (x -Px) I (yi - -y)]

[ IX1) i= ( p)

(1)

where Xi and Yi are elements of their respective sets Xand Y containing N elements. The values of p1x and p yare the sample means of each respective set and thevalue r is the correlation coefficient.

It should be realized that for sets X and Y whereXi or Yi are constant for all i, the value of r is unde-fined. Computationally for this case, r is set to zero.

For the problem of determining pressure/flow sta-tus, the set X contains the fuel requirements by hourfor a generating unit and the set Y is used to containthe value of the hourly desired fuel supplier to be

3470

correlated. To determine values for X and Y, an economicgeneration dispatch calculation is required to determinethe total amount of fuel required to operate the powersystem. This is obtained by using a forecasted hourlyload demand curve for a power system as input, andsolving the economic dispatch problem. The total systemfuel demand is calculated hourly, then put into an opti-mum fuel schedule which determines the amount of fuel tobe consumed from each supplier. The determination ofthe desired fuel schedule for each supplier is made byassigning amounts of fuel according to cost, constrainedby high and low limits on each supplier.

Numerical Results

The total amount of fuel required for each hour ofa typical daily load curve was computed by the economicdispatch program. Using the typical fuel contract limitsgiven in Table I, the desired fuel for each supplier wasdetermined.

The economic dispatch calculation also determinesthe desired average output of each unit each hour. Foreach unit, these values are correlated with the fuelsuppliers' desired fuel volume according to (1).

Figure 2 shows the desired fuel curve for GASI andGAS2 and that which is desired from unit CU-3. Note thesimilarities of the GAS2 curve and CU-3 curve. The cor-relation coefficient for CU-3 and GAS2 is 0.9570, whilethe coefficient for CU-3 and GAS1 is 0.8355. Thus, CU-3should be on pressure control on GAS2.

MKB/DAY

1100.CORRELATION COEFFICIENT, 0.9570

1000

900-800

4 1GAS DESIRED800'./

700-

GAS DESIRED ,,CORRELATION COEFFICIENT: 0.8335600 / /

500 X

400-

200- CEDAR BAYOU 3 FUEL REQUIREMENT

1002 4 6 8 0 1'2 14 16' 18 i20224

HOUR

Fig. 2. Cedar Bayou 3 performance curve

The pressure/flow status for the other units fol-lows similarly. There are exceptions to always using thebest correlation for determining fuel control status;such as when two units have a common fuel input regula-tor. Here, the selection of pressure/flow status has tobe made by correlating the sum of the requirements ofall units served by a fuel control point.

The status of each unit's pressure/flow configura-tion must be reconsidered when significant changes inthe operating power system occur. Seasonal variations inthe load curve, changes in generation unit availability,changes in transmission capability and changes in thefuel contract limits can greatly influence the correla-

tion analysis. However, for normal power system opera-tion, only monthly switches of pressure/flow status onsome units are required.

Fuel Control Matrix Representation

To simplify the mathematics, it is convenient todefine a matrix representing the particular status ofeach generating unit's fuel supply configuration. Thematrix C is defined with its elements as follows:

C1C. tO

if f.P' > 0ijif f.P = 0

13

Thus, by definition of f-Y, the element Cij corres-ponding to a unit i and'its luel supply j is unity ifand only if a change in fiv with respect to Xi, the pow-er output, reflects the exact same total change in thatfuel supply. It should be noted that only one elementof any row of the C matrix can be unity while all othersmust be zero. Thus, unity entries denote which unitsare on pressure control for that supplier's column.

ECONOMIC FUEL DISPATCH

The economic generation dispatch calculation isformulated in the standard non-linear programming (NLP)problem format as follows [8]:

Find a vector x* to:

Minimize:

N N 3e(x) = I Ei = E (Ai + BiXi + DiX)

i=1 i=l(2)

with respect to x,

subject to:N

p(x) = LOAD - I x. = 0,i=l

(3)

where x is the vector of real variables X1, X2 ... XN,and feasibility range:

LGL. < X. < HGL..1- i- 1

(4)

The objective function e(x) is shown as the sum ofall unit's fuel inputs, each evaluated at the unit'soutput of Xi megawatts. The objective then is to mini-mize the total fuel consumption for a given power systemgeneration configuration. Note that the input-outputequations that describe each unit's fuel consumptionwere found such that the third term of the general poly-nominal form is forced to zero. This greatly minimizesthe computational time, while maintaining sufficientaccuracy. The constraint on the objective function p(x)requires that the total power production of the systemis equivalent to the demand (LOAD). The feasibilityrange given in (4) is a set of special'constraints thatimposes an upper and lower bound on each variable in x

such that the maximum and minimum power productionlimits of_each unit are not violated. The set of allpossible x satisfying the constraints given in (3) and(4) determines_ the feasible solutions from _which asingle vector x* will be found such that e(x) willachieve its minimum value.

The special constraints of (4) can be easily satis-fied by limiting the selection of x such that (4) isalways satisfied. Computationally, this becomes simplya limit check on the values of variables X1, X2 ... XNbefore they are considered to be feasible candidates ofx*. Therefore, only the general constraint p(x) will beconsidered in the following development.

Let

L(x,X) = e(x) + -p(x).

L(x,X) is a Lagrangian function with a multiplier Xused by a multitude of authors who further developed thebasic multiplier methods [1, 2, 3, 6 and 8].

Let x* be a regular point of the constraint of (3).Then, a necessary condition that e(x*) be a constrainedminimum is [6]:

VL(x*,X*) = 0 (6)

Expanding (5) gives:

N NL(x,X) I (Ai + MX +D)X? ) + X(LOAD - I Xi). (7)

i=1 ~~~~~~~i=1

Invoking the necessary condition (6) insures thefollowing solution to be at least a local minimum pro-vided that is chosen such that (3) is satisfied. Thus,

DE. 2

Bi + 3D.X. = X for-all i. (8)

The solution in (8) implies that all active gener-ating units should be operating at the same incrementalproduction cost, i.e., each unit's input-output curveshall have equal derivatives at xi*.

Economic Fuel Dispatch Calculation

Thelated ineconomic

economic fuel dispatch calculation is formu-the standard NLP problem format similar to thegeneration dispatch calculation as follows:

Find a vector x* to:

Minimize:

N Nf.(x)= Fi = (A. + BiXi + DiXi )

i=l i=l

with respect to x, where

(9)

3471

The special constraints of (13), (14a) and (14b)form the feasible areas of search allowed in selection

(5) of candidates for x*. In order to minimize search time,(13) and (14) can be reduced to an equation of the form:

LDL. < X. < HDL.,1- 1- 1

(15)

where LDLi and HDLi are the dispatch low and high limitsobtained by analyzing the special constraints to deter-mine which are active and which are inactive for a givenfuel configuration.

The HDLi is found by first solving the followingfor Xi:

M 3I HSL., = A. + B.X. + D,X

j=l 13 1 1 1 1 1

and then checking to determine if the solution Xi isbound by (13). If so, HDLi = HGLi. Otherwise, HDLi = Xi.Thus, the generation production is upper bound toinclude the given unit fuel constraints.

The LDLi is found similarly by solving the follow-ing for Xi:

M 3X LSL.. + PL. = A. + B.X. + D X,

j=l 13 1 1 1 1 11

where PLi = LPLij - LSLij for that j where fil' > 0.13 ~~~~~3In this form, the special constraints of (13) and

(14) can be handled by limiting the selection of xaccording to (15). This form of constraint is again asimple limit check on values of variables X1, X2 ... XNbefore they are considered to be feasible candidates ofx*.

Using the objective function (9) and the con-straints (11) and (12), define the Lagrangian functionas follows:

ML(x, ,) = f(x) + Ap(x) + i [jg. (x)I +

M (16)

X= 0jMj+M(X]

M M4

ij=

ij j=1 i3 i1

subject to:

Np(x) = LOAD - Xi= 0

i=lN

gi(x)= I f.. - HFCL. < 0i=l 13 3 -

N

gj+M (x) = LFCL.- I f..<i0j+M ~3 =13

i = 1,2,3.. .N,

j = 1,2,3...M, (

j = 1,2,3...M1(

and feasibility range:

LGL. < X. < HGL.1 - 1 - 1

LSL.. < f.. < HSL..13 - 13 - 13

LPL.. < f.P < HSL.,13 - 13 - 13

for all i,

where A, a. and aj+M are Lagrangian type multipliers.(10) For (a) to be minimized, the Lagrangian function

(16) must be expanded such that the necessary conditionsfor a minimum at the solution point x* are realized.For constrained minimization involving inequality con-straints, Kuhn and Tucker developed such a set of condi-tions [6, pp. 233].

(11) The following development invokes these necessaryconditions to insure the minimization of f(x*).

Expanding (16) gives:'12a)

M 3 +N

C12b) L = (A. + BX. + DiX.3) + A[LOAD i X) +

M N

I- i (I- fij HFCL) +

(13)

for all i and j, (14a)

for all i and j. (14b)

The objective function f(x) and constraint p(x) arethe same as those of the economic dispatch calculation.However, additional constraint equations have been addedas shown in (12a) and (12b) to force consideration of Msystem fuel contract limitations.

M

. aj+Mj=1

N(LCL i=l ij).

Taking the partial derivative with respect to Xitand noting that

3f. af:P f.f13

=13 + 13

ax. ax. DX.1 1 1

af .p13

DX.1

3472

since by definit$on

af..ax.-

gives

aL 2 M a N

-B, + 3D.X.- A+ I K- (a, fax. J. 3x~fj). J i

for j such that gj+M(x*) is active and for i such thatf.P > 0 or13

aF.

aX. A iji(20c)

for j where both i(x*) and gj+M(x*) are inactive andfor i such that fi > °.

Thus, (2Q) can be written:

(17) dF. df.PIX =X13=C j$ for all i and j.dX. dX. lij j1 1

(21)

But since for j such that fiP > o

N2i (i i- i - Cij a (B, + 3DX3

where C- is the element of the control matrix previous-ly defined. Then (17) becomes:

B., + 3D,X. A + I C., (B. + 3D X )1 -

1x jJi 1

(18)

Cij aj+M (Bi + 3DXi2).

Setting (18) equal to zero gives:

a=F 2

a~x. i +Dii=1 M M

1 + cijojC - I Cij j+M

In summary, the result in (21) implies that alloperating units on pressure control for a given suppliershould operate at the same incremental production cost,provided that the cost is found subject to the givensupplier's constraints.

The optimal solution to the economic fuel dispatchprobl(pi is (21), given the constraints and the values ofall fig. A* was discussed previously, the fuel supplyfor a given unit which has more than one supplier inservice is defined as the sum of fil and fit for all j3J 1)given the unit i. The solution given in (21) is a func-tion of only fi-, since this particular supply iscontrollable by changing the output of the unit Xi. Thefuel represented by fi is controllable but only by man-ual means, which usually requires sending an operator tothe fleld control house to adjust the fuel regulator.

It is therefore proposed to operate using the fueldispatch calculation until it becomes economical to makea manual change in a given unit fif for a particularsupplier, This can be represented in the followingformat:

Find a set z* to;

Minimize:

But for any given unit

M

X C..c. = C. .c.,j=l 13 3 13i

since for any row of Cij, only one non-zero entity isallowed. Rewriting (19) gives:

aF A

ax. l+C. ., C1 1)+ ijaj+M

for all i and j. (19a)

N NP - I F.- E E.

i=1 i1=(22)

with respect to fij and z is the set of all feasiblesubsets of the real variables _('flf, f15, fA3, f21f25f f25,..*fif) subject to f(x*) being the constrainedminimization as previously presented with the additionalconstraint

f.f > SPL...1)- is

(23)

On close examination of the constraints given in(12), one can see that for any constraint in g (xY at x*which is active, the corresponding constraintgj+'(xY rtx* is precluded from being active. Conversely, anyactive constraint in gj+M_x) at_x* precludes the corres-ponding constraint in gj(x) at x*' from being active. Inaddition, one set of constraints gj(x and the corres',ponding gj+M(x) must be inactive to allow for theconstraint of (11).

The Lagrangian multiplier for inactive constraintsmust be identically zerQ to satisfy the necessaryminimization conditions.

Thus, (19a) can be written:

aF. A d

ax. 1 + C.ijA ij.1 13)3

(20a)

for j such that gj(x*) is active and for i such that f13> 0 or

=__ 1_ = i (20b)

i ij,j+M i

Therefore, the feasible change in z is limited to astep of SPLij for any fi -

In determining a suitable z to be considered for acandidate of z*, one needs to examine the nature of thepenalty P. The value of P must always be greater thanor equal to zero since all $j are not equal to A, atwhich point the absolute minimum fuel consumptionoccurs. It should be expected then, that as «j PA, Pwould be reduced. The value of is controlled by theconstraints (12) which are functions of f.z which isassumed to be given when determining J

The t6tal amount of fuel controiled by the con-straints can be calculated by

N NFUELC. = I (f + fi]) - i (e + fi,)ri-l . i=l

(24)

where eiv is similar to f.P except that it was found atiJ iJ,the economic generation dispatch solution.

It should be observed that those fuel supplies withthe largest FUELCj are causing the differences in Sj andA.

Since

3473

M M fe. e.-=Ei .1 fi

fo j-l i i j-l Lj

for j on pressure control on unit i, (24) becomes:

NFUELC. = IfPi i=l 1j

N

I (f;i=l

N N- XEi +

i=l i=l

M

j=l

D-E + f)

j i

for j on pressure control on unit i._Therefore, a new candidate for z* can be found if a

new f>~can be calculated to insure that for all i

M M fI E. + I fi. < SPL.., (26)J ij

I

j-.l13 j

and that FUELCj in (25) is reduced. If no_other candi-dates for z* can be found, then the given z must be z*.If a new z is calculated, the solutions given in (8) and(21) must be recalculated until such time as no new

candidates for z* can be found.

SIMULATION

In order to promote a complete test of the tech-nique presented, it was felt that a software modelshould be constructed which would emulate the actual

operation of a power system participating in automaticcontrol of its generation and fuel. Since it is feltthat application of the presented methods of economicfuel dispatching apply only to the immediate control ofa power system, routines were developed to generate li-mits and control problems found in a real-time operatingsystem. The power system was simulated for a singletwenty-four hour period using the actual hourly demand,generation schedule and fuel consumption schedule thatoccurred on January 15, 1979. It was found that thisparticular day had characteristics typical of mostwinter load conditions.

Model Structure

The power system model is a simple routine to inputthe generation, unit status, the system gross load, thefuel contract requirements and the load output of anyunits that, because of size or mechanical problems,could not participate in remote automatic control. Themodel was constructed to handle up to forty generatingunits consuming as many as six types of fuel, through up

to thirty fuel control points. It is felt that theselimits can easily be changed to suit any power system,although computation times may increase.

The basic approach of the model is to first calcu-late a classical economic generation dispatch using as

input all variables at an initilized state, from which a

value for the desired output of all units is known. Fromthese values, the total amount of fuel consumed by eachunit is calculated and distributed according to thepressure/flow status to give the amounts of fuel by sup-

plier at each controlled fuel input. As part of the in-

Table II. Economic Generation Dispatch Solution - Hour 6

Lambda = 4.8032 Load = 6310 MWNAME STATUS HGL LGL DES TBTU GAS1 GAS2 NUC OIL1 OIL2 COAL

BN 1 OFF 183.00 60.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0BN 2 BLK 183.00 60.00 88.00 21.76 5.3 F 16.5 P 0.0 0 0.0 0 0.0 0 0.0 0BN 3 REG 245.00 90.00 207.53 47.54 50.8 P 48.6 F O.O 0 0.0 0 0.0 0 0.0 0BN 4 REG 245.00 90.00 226.95 51.79 50.8 P 48.6 F 0.0 0 0.0 0 0.0 0 0.0 0CU 1 REG 770.00 400.00 506.22 116.99 41.8 F 75.2 P 0.0 0 0.0 0 0.0 0 0.0 0CU 2 BLK 770.00 400.00 400.00 92.85 44.0 P 48.9 F 0.0 0 0.0 0 0.0 0 0.0 0CU 3 REG 770,00 400.00 648.51 144.44 31.2 F 113.3 P 0.0 0 0.0 0 0.0 0 0.0 0DW 7 REG 183.00 60.00 60.00 15.51 0.0 0 15.5 P 0.0 0 0.0 0 0.0 0 0.0 0GB 1 BLK 75.00 40.00 31.00 10.06 0.0 0 26.9 P 0.0 0 0.0 0 0.0 0 0.0 0GB 2 OFF 75.00 40.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0GB 3 BLK 118.00 40.00 65.00 16.85 0.0 0 26.9 P 0.0 0 0.0 0 0.0 0 0.0 0GB 4 OFF 118.00 40.00 0.00 0.00 0.0 0 0.0 0 -0.0 0 0.0 0 0.0 0 0.0 0GB 5 REG 400.00 125.00 293.98 66.41 18.3 F 48.1 P 0.0 0 0.0 0 0.0 0 0.0 0PH 1 REG 183.00 60.00 138.42 31.46 16.1 F 46.1 P 0.0 0 0.0 0 0.0 0 0.0 0PH 2 REG 183.00 60.00 134.40 30.74 16.1 F 46.1 P 0.0 0 0.0 0 0.0 0 0.0 0PH 3 REG 300.00 125.00 131.18 30.70 0.0 0 30.7 P 0.0 0 0.0 0 0.0 0 0.0 0PH 4 OFF 585.00 300.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0PH 5 REG 585.00 300.00 488.88 108.94 20.3 F 8.6 P 0.00 0.00 0.0 0 80.0 FPH 6 REG 585.00 300.00 488.88 108.94 45.9 F 63.0 P 0.0 0 0.0 0 0.0 0 0.0 0RN 1 REG 450.00 250.00 358.26 79.60 46.7 F 32.9 P 0.0 0 0.0 0 0.0 0 0.0 0RN 2 REG 450.00 250.00 425.55 95.90 32.2 P 63.7 F 0.00 0.00 0.00 0.00RN 3 REG 585.00 290.00 517.95 117.61 52.0 P 65.6 F 0.00 0.00 0.00 0.00RN 4 OFF 770.00 400.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0WH 1 BLK 75.00 30.00 39.00 10.13 8.3 0 42.4 P 0.0 0 0.0 0 0.0 0 0.0 0WH 2 REG 245.00 85.00 174.41 40.54 8.3 F 42.4 P 0.00 0.00 0.00 0.00WH 3 BLK 300.00 50.00 232.00 52.26 74.3 P 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0WH 4 BLK 300.00 50.00 77.00 22.06 74.3 P 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0Wi 1 BLK 40.00 20.00 20.00 11.11 0.0 0 52.5 P 0.0 0 0.0 0 0.0 0 0.0 0WJ 2 BLK 40.00 20.00 20.00 11.11 0.0 0 52.5 P 0.0 0 0.0 0 0.0 0 0.0 0WJ 3 BLK 80.00 40.00 44.00 15.14 0.0 0 52.5 P 0.0 0 0.0 0 0.0 0 0.0 0WJ 4 BLK 80.00 40.00 44.00 15.14 0.0 0 52.5 P 0.0 00 0.0 0.00 0.0 0WS 1 OFF 118.00 45.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.00 0.00WS 2 BLK 118.00 45.00 65.00 16.41 0.0 0 16.4 P 0.0 0 0.0 0 0.0 0 0.0 0WS 3 REG 390.00 180.00 384.00 87.81 64.6 P 23.2 F 0.0 0 0.0 0 0.0 0 0.0 0EDC TOTALS 6310.12 1469.8 551.8 838.0 0.0. 0.0 0.0 80.0DESIRED FUEL BY SUPPLIER 589.8 800.0 0.0 0.0 0.0 80.0PENALTY FUEL: GAS1

3474

put of pressure/flow status, amounts of fuel on flow are

entered as starting points for further calculations. Theindividual fuel control values are summed to give thetotal system fuel and its distribution to each fuelsupplier. Subtracting these values from those calculatedon the generation dispatch gives the amounts of fuel tobe controlled by the economic fuel dispatch. Althoughthe amount to be controlled on each supplier is notrequired for calculation purposes, it is produced forinformational reasons.

The total amount of fuel required as calculated bythe economic generation dispatch represents the absoluteminimum of fuel usage which can be expected. Any pertur-bation-of the unit loads from this point increases thetotal fuel requirement. Therefore, any result of theeconomic fuel dispatch different from the generationdispatch calculation will require more fuel. A fuelsupplier must be selected to provide this extra fuel.For simplicity, this small amount of additional fuel isgiven to a supplier whose constraints are inactive.

The calculation of the economic fuel dispatch ismade according to the methods described previously. Thisgives a resulting power output for each unit which thencan be used to calculate the unit's total fuel require-ment and hence, each fuel control point's consumptionlevel. It should be noted that the power output of eachunit is constrained by the limits on the allowable fuelflows from each supplier at each fuel control point.However, it was determined that the total HSLi. for eachunit and each supply was not attainable since designcriteria in the power system called for total fuelcapability to be provided by a single supplier.

The tolerance allowed in the iteration technique is+ 0.5 MKB/Day for fuel constraints and + 1 MW for loadconstraints. In order to compare the result of theeconomic generation dispatch and the economic fuel dis-patch, the mismatch in the load constraints must beevaluated. Since the incremental production cost at thesolution point is known, the mismatch can be readilyconverted to fuel and added to the appropriate fuel sup-plier. Hence, a common basis for comparison is providedand the penalty for operating under economic fueldispatch criteria can be obtained.

The next step is to verify that no further minimi-zation of the penalty can be obtained by changing theamounts on flow control at each fuel input point. Thefeasible changes in fif are evaluated and then checkedto determine if a fuel how amount change is desirable.The amount of change made equals the value which wouldreturn the unit to operation at the power output calcu-lated by the economic generation dispatch.

If a change in fuel status is made, the economicgeneration dispatch and fuel dispatch routines are rerun

for the given power system load. Reruns are made untilno feasible change in fif is found.

The model was programmed for use on a Xerox Sigma 5computer system using 31.5 KBytes of core storage. The.execution time, without printing hourly results, aver-

aged 24 seconds per load point. With all print options,average execution time was 43 seconds per load point.

Results

The above model was used to determine the ability

Table III. Economic Fuel Dispatch Solution Without Penalty Minimization - Hour 6

Lambda = 4.8032 Load = 6310 MW Beta_= 10.00 4.69 0.00 0.00 0.00 5.00

NAME STATUS HDL LDL FDES TBTU GAS1 GAS2 NUC OILl OIL2 COALBN 1 OFF 183.00 60.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0BN 2 BLK 183.00 84.98 88.00 21.76 5.3 F 16.5 P 0.0 0 0.0 0 0.0 0 0.0 0BN 3 REG 245.00 196.56 245.00 56.25 63.6 P 48.6 F 0.00 0.0 0 0.0 0 0.0 0BN 4 REG 245.00 211.09 245.00 55.90 63.6 P 48.6 F 0.00 0.0 0 0.0 0 0.0 0CU 1 REG 770.00 400.00 442.96 102.69 41.8 F 60.9 P 0.0 0 0.0 0 0.0 0 0.0 0CU 2 BLK 770.00 426.02 400.00 92.85 44.0 P 48.9 F 0.0 0 0.0 0 0.0 0 0.0 0CU 3 REG 770.00 400.00 586.99 130.64 31.2 F 99.5 P 0.0 0 0.0 0 0.0 0 0.0 0DW 7 REG 183.00 60.00 60.00 15.51 0.00 15.5 P 0.00 0.0 0 0.0 0 0.0 0GB 1 BLK 75.00 40.00 31.00 10.06 0.0 0 26.9 P 0.0 0 0.00 0.0 0 0.0 0GB 2 OFF 75.00 40.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0GB 3 BLK 118.00 40.00 65.00 16.85 0.0 0 26.9 P 0.0 0 0.0 0 0.0 0 0.0 0GB 4 OFF 118.00 40.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.00 0.00GB 5 REG 400.00 159.38 272.00 61.44 18.3 F 43.1 P 0.0 0 0.0 0 0.0 0 0.0 0PH 1 REG 173.00 115.73 119.64 27.29 16.1 F 44.5 P 0.0 0 0.0 0 0.0 0 0.0 0PH 2 REG 183.00 145.52 145.52 33.28 16.1 F 44.5 P 0.0 0 0.0 0 0.0 0 0.0 0PH 3 REG 300.00 125.00 125.00 29.34 0.0 0 29.3 P O.o 0 0o. 0 0.0 Q 0.00PH 4 OFF 585.00 300.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0PH 5 REG 585.00 530.45 530.45 118.65 20.3 F 18.4 P 0.0 0 0.0 0 0.0 0 80.0 FPH 6 REG 585.00 300.00 466.32 103.85 45.9 F 57.9 P 0.0 0 0.0 0 0.0 0 0.0 0RN 1 REG 450.00 315.63 344.95 76.59 46.7 F 29.9 P 0.0 0 0.0 0 0.0 0 0.0 0RN 2 REG 450.00 406.25 450.00 101.55 37.9 P 63.7 F 0.0 0 0.0 0 0.0 0 0.0 0RN 3 REG 585.00 412.15 585.00 133.05 67.5 P 65.6 F 0.0 0 0.0 0 0.0 0 0.0 0RN 4 OFF 770.00 400.00 0.00 0.00 0.0 0 0.00 0.0 0 0.0 0 0.0 0 0.0 0WH 1 BLK 75.00 41.25 39.00 10.13 8.3 0 39.0 P 0.0 0 0.0 0 0.0 0 0.0 0WH 2 REG 245.00 85.00 159.09 37.14 8.3 F 39.0 P 0.0 0 0.0 0 0.0 0 0.0 0WH 3 BLK 300.00 89.06 232.00 52.26 74.3 P 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0WH 4 BLK 300.00 89.06 77.00 22.06 74.3 P 0.0 0 0.00Q 0.0 0 0.0 0 0.0 0WJ 1 BLK 40.00 20.00 20.00 11.11 0,0 0 52.5 P o.O 0 0.0 0 0.0 0 0.0 0WJ 2 BLK 40.00 20.00 20,00 11.11 0.00 52.5 P 0.0 0 0.0 0 0.0 0 0.0 0WJ 3 BLK 80,00 40.00Q 44.00 15.14 .O 00 52.5 P 0.00 0.0 0 0.0 0 0.0 0WJ 4 BLK 80.00 40.00 44.00 15.14 0.0 0 52.5 P 0.0 0 0.0 0 0.0 0 0.0 0WS11 OFF 118.00 45.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0WS 2 BLK 118.00 45.00 65.00 16.41 0.0 0 16.4 P 0.0 0 0.0 0 0.0 0 0.0 0WS 3 REG 390.00 180.00 390.00 89.16 66.0 P 23.2 F 0.0 0 0.0 0 0.0 0 0.0 0EFD TOTALS 6292.92 1471.1 591.0 800.1 0.0 0.0 0.0 80.0HOUR 6 CHANGE FLOW ON BN 3 SUPPLIER GAS2 FROM 22.67 TO 13.97HOUR 6 CHANGE FLOW ON RN 3 SUPPLIER GAS2 FROM 65.58 TO 50.14PENALTY = 1.37 MKBtu/DAY

3475

Table IV. Economic Fuel Dispatch With Penalty Minimization - Hour 6

Lambda = 4.8032 Load = 6310 MW Beta = 4.85 4.74 0.00 0.00 0.00 5.00-

NAME STATUS HDL LDL FDES TBTU GAi GAS2 NUC OILi 01L2 COAL

BN 1 0FF 183.00 60.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0BN 2 BLK 183.00 84.98 88.00 21.76 5.3 F 16.5 P 0.0 0 0.0 0 0.0 0 0.0 0BN 3 REG 245.00 156.60 213.94 49.00 63.1 P 39.9 F 0.0 0 0.0 0 0.0 0 0.0 0BN 4 REG 245.00 211.09 236.69 54.00 63.1 P 39.9 F 0.0 0 0.0 0 0.0 0 0.0 0CU 1 REG 770.00 400.00 474.39 109.75 41.8 F 68.0 P 0.0 0 0.0 0 0.0 0 0.0 0CU 2 BLK 770.00 426.02 400.00 92.85 44.0 P 48.9 F 0.0 0 0.0 0 0.0 0 0.0 0CU 3 REG 770.00 400.00 617.27 137.39 31.2 F 106.2 P 0.0 0 0.0 0 0.0 0 0.0 0DW 7 REG 183.00 60.00 60.00 15.51 0.0 0 15.5 P 0.0 0 0.0 0 0.0 0 0.0 0GB 1 BLK 75.00 40.00 31.00 10.06 0.0.0 26.9 P 0.0 0 0.0 0 0.0 0 0.0 0GB 2 0FF 75.00 40.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0GB 3 BLK 118.00 40.00 65.00 16.85 0.0 0 26.9,P 0.0 0 0.0 0 0.0 0 0.0 0GB 4 OFF 118.00 40.00 0.00 0.00 0.,0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0GB 5 REG 400.00 159.38 282.76 63.86 18.3 F 45.6 P 0.0 0 0.0 0 0.0 0 0.0 0PH 1 REG 183.00 115.73 128.99 29.35 16.1 F 46.5 P 0.0 0 0.0 0 0.0 0 0.0 0PH 2 REG 183.0O 145.52 145.52 33.28 16.1 F 46.5 P 0.0 0 0.0 0 0.0 0 0.0 0PH 3 REG 300.00 125.00 125.00 29.34 0.0 0 29.3 P 0,o 0 0.0 0 0.0 0 0.0 0PH 4 0FF 585.00 300.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0PH 5 REG 585.00 530.45 530.45 118.65 20.3 F 18.4 P 0.0 0 0.0 0 0.0 0 80.0 FPH 6 REG 585.00 300.00 477.27 106.30 45.9 F 60.4 P 0.0 0 0.0 0 0.0 0 0.0 0RN 1 REG 450.00 315.63 351.40 78.04 46.7 F 31.3 P 0.0 0 0.0 0 0.0 0 0.0 0RN 2. REG 450.00 406.25 438.39 98.86 35.2 P 63.7 F 0.0 0 0.0 0 0.0 0 0.0 0RN 3 REG 585.00 343.01 546.50 124.14 74.0 P 50.1 F 0.0 0 0.0 0 0.0 0 0.0 0RN 4 0FF 770.00 400.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0WH 1 BLK 75.00 41.25 39.00 10.13 8.3 0 40.6 P 0.0 0 0.0 0 0.0 0 0.0 0WH 2 REG 245.00 85.00 166.62 38.80 8.3 F 40.6 P '*O. 0 0.0 0 0.0 0 0.0 0WH 3 BLK 300.00 89.06 232.00 52.26 74.3 P 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0WH 4 BLK 300.00 89.06 77.00 22.06 74.3 P 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0143 1 BLK 40.00 20.00 20.00 11.11 0.0 0 52.5 P 0.0 0 0.0 0 0.0 0 0.0 0.WJ 2 BLK 40.00 20.00 20.00 11.11 0.0 0 52.5 P o.o 0 0.0 0 0.0 0 0.0 0143 3 BLK 80.00 40.00 44.00 15.14 0.0 0 52.5 P 0.0 0' 0.0 0 0.0 0 0.01434 BLK 80.00 40.00 44.00 15.14 0.0 0 52.5 P 0.0 0 0.0 0 0.0 0 0.0 0Ws 1 OFF 118.00 45.00 0.00 0.00 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0 0.0 0WS 2 BLK 118.00 45.00 65.00 16.41 0.0 0 16.4 P 0.0 0 0.0 0 0.0 0 0.0 0W4S3 REG 390.00 180.00 390.00 89.15 66.0 P 23.2 F 0.0 0 0.0 0 0.0 0 0.0 0EFD TOTALS 6310.20 1470.3 590.4 799.9 0.0 0.0 0.0 80.0PENALTY =0.50 MKBtu/DAY

Table V. Economic Fuel Dispatch With Penalty Minimization -Twenty-four Hour Summary*MBtu (1o6 Btu/Day) TOTAL DOLLAR COST OF PENALTY: $602.52 @ $1.75 PER MBtu

JAN 15, 1979 BTHOUR LOAD FUEL GASi GAS2 NUC OILi 01L2 COAL PENALTY* GASi GAS2 NUC 0IL1 01L2 COAL

1 6040 1409.4 529.5 799.9 0.0 0.0 0.0 80.0 537.8 4.7125 4.7107 0.0000 0.0000 0.0000 5.00002 6075 1417.2 537.3 799.9 0.0 0.0 0.0 80.0 492.7 4.7406 4.7107 0.0000 0.0000 0.0000 5.00003 5990 1398.3 518.2 800.1 9.0 0.0 0.0 80.0 588.6 4.7034 4.6912 0.0000 0.0000 0.0000 5.00004 5960 1391.7 511.5 800.1 0.0 0.0 0.0 80.0 648.4 4.6765 4.6912 0.0000 0.0000 0.0000 5.00005 6075 1417.3 537.2 800.1 0.0 0.0 0.0 80.0 556.5 4.7638 4.6912 0.0000 0.0000 0.0000 5.00006 6310 1470.3 590.4 799.9; 0.0 0.0 0.0 80.0 496.8 4.8492 4.7449 0.0000 0.0000 0.0000 5.00007 6750 1570.5 599.8 890.8 0.0 0.0 0.0 80.0 92.3 4.8828 4.9121 0.0000 0.0000 0.0000 5.00008 7140 1662.2 599.8 982.5 0.0 0.0 0.0 80.0 271.5 4.8828 5.0848 0.0000 0.0000 0.0000 5.00009 7190 1674.2 599.8 994.5 0.0 0.0 0.0 80.0 337.6 4.8828 5.1120 0.0000 0.0000 0.0000 5.0000

10 7280 1696.0 599.8 1016.3 0.0 0.0 0.0 80.0 466.1 4.~8828 5.1642 0.0000 0.0000 0.0000 5.000011 7150 1664.6 599.8 984.9 0.0 0.0 0.0 80.0 285.6 4.8828 5.0903 0.0000 0.0000 0.0000 5.000012 6900 1605.4 599.8 925.7 0.0 0.0 0.0 80.0 109.4 4.8828 4.9756 0.0000 0.0000 0.0000 5.000013 6750 ~1570.5 599.8 890.8 0.0 0.0 0.0 80.0 92.3 4.8828 4.9121 0.0000 0.0000 0.0000 5.000014 6650 1547.5 599.8 867.8 0.0 0.0 0.0 80.0 167.7 4.8828 4.8712 0.0000 0.0000 0.0000 5.000015 6660 1549.8 599.8 870.0 0.0 0.0 0.0 80.0 152.1 4.8828 4.8755 0.0000 0.0000 0.0000 5.000016 6600 1536.1 599.8 856.3 0.0 0.0 0.0 80.0 230.0 4.8828 4.8520 0.0000 0.0000 0.0000 5.000017 6620 '1540.6 599.8 860.9 0.0 0.0 0.0 80.0 205.1 4.8828 4.8596 0.0000 0.0000 0.0000 5.000018 6830 1589.1 599.8 909.3 0.0 0.0 0.0 80.0 93.3 4.8828 4.9457 0.0000 0.0000 0.0000 5.000019 7145 1663.4 599.8 983.7 0.0 0.0 0.0 80.0 282.2 4.8828 5.0873 0.0000 0.0000 0.0000 5.000020 6970 1621.9 599.8 942.1 0.0 0.0 0.0 80.0 140.9 4.8828 5.0061 0.0000 0.0000 0.0000 5.000021, 6650 1547.5 599.8 867.8 0.0 0.0 0.0 80.0 167.7 4.8828 4.8712 0.0000 0.0000 0.0000 5.000022 6300 1468.0 588.1 799.9 0.0 0.0 0.0 80.0 463.6 4.8413 4.7449 0.0000 0.0000 0.0000 5.000023 5950 1389.5 509.7 799.8 0.0 0.0 0.0 80.0 668.0 4.6765 4.6899 0.0000 0.0000 0.0000 5.000024 5650 1323.2 443.1 800.0 0.0 0.0 0.0 80.0 718.0 4.6010 4.6350 0.0000 0.0000 0.0000 5.0000

TOTAL 6568 1530.2 573.4 876.8 0.0 0.0 0.0 80.0 344.3/AVERAGE SYSTEM GROSS HEAT R/AlL 9.707 MBtu/kWh

3476

of the economic fuel dispatch calculation to controlfuel by suppliers as it is consumed in a power system.The model used the actual hourly average power produc-tion for a period of twenty-four hours. Of particularinterest is the sixth hour in the study which was thefirst hour of the day where a significant load changeoccurred. Table II shows the result of the firsteconomic generation dispatch calculation.

The result of the calculation shows the status ofall units where REG is the unit participating in dis-patch calculations, BLK is that unit's output fixed atthe current load level and OFF is the unit's off line.The fuel control status is designated with "P" for pres-sure control, an "F" for flow control and "O" for theoff state.

The target load of 6310 MW was realized with 0.12MW mismatch. The total amount of fuel required was1469.8 MKB/Day (109 Btu/Day). For this hour, the gener-ation dispatch allocated 551.8 MKB/Day to GAS1, 838.0MKB/Day to GAS2 and 80.0 MKB/Day to COAL. The desiredfuel schedule would have GAS1 at 589.8, GAS2 at its min-imum of 800.0 and COAL at 80.0. Thus, it is desirableto move 38.0 MKB/Day from GAS2 to GAS1. Note that ifGAS2 was intrastate gas and GAS1 was interstate, thiswould amount to a savings of $57,000 per day if thedeviation persisted throughout the day.

Note that for units that share a common fuel con-trol point, the unit fuel allocation is shown as the sumof all units using that control point.

Since the GAS1 constraint is not active, it hasbeen flagged to receive any penalty fuel.

The results of the first calculation of the econom-ic fuel dispatch are shown in Table III. Solving theactive fuel supply constraint first reduced the outputsof all units on GAS2 pressure control such that thedesired fuel of 800 MKB/Day was consumed. In calculationof the fuel for GAS1, which has inactive constraints,the model extended all GAS1 pressure controlled units totheir upper limit, at which point only 6292.22 MW of thedesired 6310 MW load was scheduled. This mismatch isintolerable and therefore requires further iterations.

On analysis of the values of flow control, it wasfound that unit BN-3 could reduce its setpoint on GAS2from 22.67 to 13.97 MKB/Day without violating any fuellimits and, similarly, RN-3 could reduce GAS2 flow by15.44 MKB/Day. On the change of fuel limits, note thatthe low dispatch limit on BN-3 changed from 196 MW to156 MW and RN-3 changed from 412 to 396 MW. Reducingthe total amount on flow control from GAS2 by 24.14 MKB/Day increases GAS1 by a like amount. The results ofthis second calculation are summarized in Table IV.

In this solution, the load mismatch obtained wasonly 0.2 MW, while each fuel was within tolerance of itsdesired setpoint. Note that the original lambda calcu-lated for the economic generation dispatch was 4.8032.(Note: Lambda and beta are scaled between 0.0 and 10.0.)The beta constants calculated where 4.85 and 4.74 forGASI and GAS2 respectively. This means those units withGAS1 on pressure control are operating at an equalincremental production cost different from the originallambda and thus, are not operating fuel efficiently.Units on GAS2 pressure control are operating similarlybut at an equal incremental cost also different fromlambda. Because of the mode of operation, 0.5 MKB/Dayof additional fuel will be burned, but the desired fuelschedule has been realized.

Table V is a twenty-four hour summary of the opera-tion of the model including penalty minimization. Thepenalty minimization technique has reduced the cost ofoperating under economic fuel dispatch criteria whilemaintaining the desired fuel schedule for each supplier.Note, however, that the total system gross heat rate wasincreased to 9.707 MBtu/kWh, in comparison with the eco-nomic generation dispatch heat rate of 9.705 MBtu/kWh.

In an effort to verify the conclusions on thedetermination of the pressure/flow status, an identicalday was run except that the unit CU-3 was switched from

GAS2 on pressure coptrol to GAS1 on pressure control.This economic fuel dispatch solution yielded an increasein the total day's penalty of $388.30 at $1.75 per MBtu.However, the number of fuel flow changes was reducedsince more fuel control is obtained on GAS1.

IMPLEMENTATION AND CONCLUSION

The economic fuel dispatch calculation was devel-oped specifically for real-time control of power systemsfuel consistent with the standard methods now in use.

The basis of the method produces the desired output or

setpoints of all generating units, which would be passedto an automatic generation control routine to use in itscontrol of power system frequency and power interchange.The automatic generation control routine would, whensystem conditions require, move all units' powerproduction to the economic fuel dispatch desired output,thus controlling fuel as well as other power systemparameters. It is felt that the standard transmissionline loss minimization and sulphur oxide emissioncontrol techniques can be incorporated into the economicfuel dispatch calculation, although this may signifi-cantly increase the computation time [5].

The changes in fuel flow amounts required in orderto minimize the fuel dispatch penalty should be reportedto the system operator by outputting a message on CRTdevices and/or logging them on operations typers. Thesystem operator would then telephone the appropriatepower plant operator and request him to manually makethe change. After the system operator receives verifi-cation that the flow has been changed, he would enterthis information into the economic fuel dispatch database.

Another aspect of the fuel dispatch technique isthat it will tend to tune itself as it continuously pro-cesses data from day to day. Once the fuels on flow are

set, only a minimal number of change requests will bemade.

The result of using the simulation model, which isthe application of the methods described, shows conclus-ively that fuel by type or supplier can be adequatelycontrolled to a desired schedule by using the economicfuel dispatch calculation. The ability to control fuelto within + 0.5 MKB/Day of each contract provides thepower utility a practical and economic method of

insuring full consumption of the least expensive fuels.The actual cost savings to a power utility will varydepending on the cost differences in its fuel contracts.

Considering that a two percent error is likely whendispatching fuel by manual methods, the economic fueldispatch will reduce this error to less than 0.25percent. Using a nominal amount for a fuel contract andinterstate and intrastate gas prices, approximately$480,000 a month could be saved by automatic economicfuel dispatch, although $18,000 will be spent in penal-ties incurred due to inefficient scheduling of power.

REFERENCES

[1] R. E. Collins, Mathematical Methods for Physicistsand Engineers, Reinhold, 1968.

[2] R. Courant, Differential and Integral Calculus,Volume II, Wiley-Interscience, 1936.

[3] H. H. Happ, "Optimal Power Dispatch - A Comprehen-sive Survey", IEEE Transactions on Power Apparatusand Systems, May/June, 1977, pp. 841-854.

[4J L. K. Kirchmayer, Economic Operation of Power Sys-tems, John Wiley and Sons, Inc., 1958.

[5] L. K. Kirchmayer, Economic Control of Interconnect-ed Systems, John Wiley and Sons, Inc., 1959.

[6] D. G. Luenberger, Introduction to Linear and Non-Linear Programming, Addison-Wesley, 1973.

[7] P. L. Meyer, Introductory Probability and Statisti-cal Applications, Addison-Wesley, 1965.

[8] D. A. Pierre, M. J. Lowe, Mathematical ProgrammingVia Augmented Lagrangians, Addison-Wesley, 1975.

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DiscussionM. E. El-Hawary (Faculty of Engineering and Applied Science,Memorial University of Newfoundland, St. John's, Newfoundland,Canada): This is a worth-while contribution to the area of optimaleconomic operation of electric power systems. The authors have cor-rectly focused their attention on a problem of current significance con-cerning fuel management by incorporating its aspects in the economicdispatch procedure. The authors should be commended for their work.From the authors' statement concerning the nature of their objective

function (2) it appears that the choice of a cubic cost expression withzero second order term is based on simplifying the computational com-plexity in comparison with a "full" cubic cost. It does not appear,however, that the authors' have considered the more popular [9]quadratic cost as a possible alternative. The quadratic cost expressionhas many clear advantages. Would the authors clarify this point.The formulation given assumes a fixed number of suppliers M for all

units i = 1, -, N. Presumably this is not a restriction and the authors'procedure can handle the situation where the number of suppliers isunit-dependent, i.e. M is replaced by Mi. Is it legitimate to make thisstatement?

Floyd J. Trefny and Kwang Y. Lee: The authors appreciate the interestshown and wish to thank the discussor for his comments and questions.A "full" cubic cost curve to represent the input-output relationship

for each generator becomes complex to numerically solve because thefirst derivative is a full quadratic equation. Solving this equation thenrequires a subiteration loop in the overall solution method. This addi-tional computation can be negated by using either a quadratic equationfor an input-output curve or cubic equation with a zero second orderterm. At Houston Lighting & Power Company, a better correlationwith field data is obtained by using the cubic equation with a zero se-cond order term.The method presented can handle the case where N units is equal to

the number of fuel supplies M. However, when this is the case, themethod presented would reduce to a minimization method with fuelequality constraints on all but one unit. For most utilities, this wouldprove impractical, that is to have only one unit being economicallydispatched as system load changes. The authors believe that mostutilities' fuel supplies are much less in number than their number ofunits. IfM is greater than N, this method will not converge because thematrix C cannot be defined properly.

Manuscript received April 3, 1981.REFERENCE

[9] M. E. El-Hawary, and G. S. Christensen, "Optimal EconomicOperation of Electric Power Systems", Academic Press, NewYork, 1979.

Manuscript received February 23, 1981.