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Proceedings of the 34thConference on Decision & ControlNew Orleans, LA - December 1995M07 1:30 ADVANCED GENERATION CONTROLWITH ECONOMIC DISPATCH

D. Brian Eidson Marija D. IliCDept. of Electrical Engineering and Computer Science

Massachusetts Institute of TechnologyCambridge, M A 02139

AbstractA new approach for the Automatic Generation Control (AGC)of electric power networks is introduced in this paper. Thi sapproach also supports an Economic Dispatch feature whichoperates at a slower time scale and interacts with the AGCfrequency stabilization function.

AGC is separated into two subtasks: frequency regulationand tie-line flow control, and a hierarchy of controls are formu-lated t o realize these two functions. Th e frequency regulationlevel, called secondary control, is decentralized, and optimizeswith respect to both frequency offset and change in generatorgovernor controls. This innovation allows an administrativearea to readily choose the cost-performance mix it desires. Thetie-line flow control level, called tertiary control, is centralized,and works on a slower time scale than secondary control. Ter-tiary control compensates for inadvertent flows between areasby momentarily offsetting generator frequencies. These fre-quency offsets shift th e phase angles found at tie-line terminals,and eventually realize the desired line flows.

The role of Economic Dispatch (E D) is to reschedule the en-tire system to minimize overall generation cost. The proposedversion of ED is centralized, and invoked at an even slowerrate than tertiary tie-line regulation. When applied, EconomicDispatch sends signals directly to secondary controllers, andestablishes new interface flow settings for tertiary level ti eli necontrol.

1 IntroductionIn this paper, we introduce models and control tech-niques for Automatic Generation Control (AGC) andEconomic Dispatch on electric power systems.

In what follows, we develop a discrete-time modelwhich tracks the succession of frequency steady-statesevolving from periodic governor set-point changes,slow load disturbances, and slow movements in tie-line flows. With this model, a decentralized frequencycontrol is proposed which allows a regional systemoperator to choose a cost-performance mix most suit-able to his needs. This frequency control function iscalled secondary level control.

Next, to address the issue of inadvertent tie-lineflows between areas, an interconnected system modelrelating tie-line flows to frequency set points is de-

veloped. Then, a slower-acting, centralized controlscheme is proposed to effect this tie-line regulation.This control, called tertiary control, acts by sendingfrequency (set-point) goals to every secondary con-troller in the interconnected system.

In developing the control models, a recursion forthe evolution of generator power is also developed.h e 1 cost curves are then associated with generatorpower outputs, the result being an equation govern-ing the evolution of AGC cost. Given this new costmodel, controls can be applied, and an EconomicDispatch-type function realized.

This version of Economic Dispatch attempts tominimize overall system fuel cost in meeting load de-mand. Like the tie-line regulator of AGC, EconomicDispatch is realized as a tertiary level control, whichsends governor frequency set points to all secondarylevel machines participating in frequency regulation.One difference, however, is that Economic Dispatchis applied at a much slower rate than tie-line control,so when invoked, it also sends a signal to the ter-tiary level (tie-line) controller. This control updatescontrol flow set points used by the tie-line control op-eration.

2 Model DevelopmentWe begin by developing a model relevant for sec-ondary level frequency control. The model isconstructed by linking machine droop characteris-tics (section 2.1)-which describe the steady-statefrequency-versus-power behavior of generators-withnetwork power flow constraints (section 2.2).2.1 Droop CharacteristicAssume real/reactive power (frequency/voltage) de-coupling. For Automatic Generation Control (AGC)applications, this implies that governor actions onlyinduce changes in frequencylreal power; no changes

0-7803-2685-7/95$4.000 995 IEEE 3450

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in terminal voltage occur. This decoupling assump-tion is commonly made in AGC studies 14,71.Consider a single machine consisting of a governor,

turbine and generator. Under the decoupling assump-tion, a simple state-space model for the linearized'dynamics of this machine is

7A,,,."

1+ [ - PG+ [ i ] a c ] .(1)

M , and D , are, respectively, the moment of inertiaof the generator and its damping coefficient. T, andTg are the time constants of the turbine and gover-nor. The state variables are the generator frequency,WG; the turbine mechanical power, Pt , ; and the gov-ernor valve opening, a, which serves to regulate tur-bine power. w z f [ k ] , = 0,1, . . . s the (frequency) setpoint value for the governor-a set point which canbe updated every kT, econds. PG, he real (electri-cal) power output of by the generator, appears hereto be an exogenous variable; however, it is is not anindependent input since it depends on constraints im-posed by interconnections with other generators.

Assume from here forward that the machine dy-namics of (1) are stable for the anticipated range ofPG. This is normally the case if the governor (primarycontrol) is designed properly, such that the systemmatrix designated A P T ; as eigenvalues with negativereal parts.Consider now the steady s tate to which the systemsettles between updates for the governor set point.Setting the derivatives in (1) to zero and solving forWG yields

W G [ ~ ] (1- ~ D ) w ; ~ [ k ]T p ~ [ k ] ( 2 )where (T is the (steady-state) droop constant, which,for the model in (1) is

To emphasize that the WG here is the steady stateresponse at t = kT,, note that WG in (2) is expressedas w ~ [ k ] .

An equivalent definition for thedroop characteristic-a definition which allows facile

lNote that the linearization assumption is not restrictive.Since the models developed within are causal, the ou r tech-niques can be extended to to stepwise linearization-for thecase of 1-step finite-horizon control.

incorporation of a governor-turbine-generator modeldifferent from the one chosen in (1)-is

(4)For further use, stack the droop relations fora num-

ber of machines to form a vector of generator frequen-cies:* [ k ] = ( I - c D ) g ; f [ k ]- E&[k] , ( 5 )

where * [ I C ] , g z f [ k ] , and &[k] are vectors de-scribing the frequencies, frequency set points, andelectrical power outputs of all the generators in asubsystem; b = diag[ D' D 2 D" 1 , C =diag[ o1 u2 . . . ( T ~, and I , an identity matrix ofdimension m.

These droop relations are coupled through &,which in turn is governed by network interconnectionconstraints.2.2 Network ConstraintsConsider now the connections between machineswithin an area, or sub-network, of a large intercon-nected system. Network constraints are typicallyexpressed using nodal equations which require thecomplex-valued power into the sub-network, S , tobe equal to the complex-valued power 3 = + Qinjected into each node. In other words,

- N

- Nwhere 5 = -N + jQ N is the vector of netcomplex power injections into all nodes, and ?iusis the admittance matrix of the sub-network.- = [ Vl ejs l , V2ej62,. . . is the vector of all nodalvoltage phasors, each with magnitude V , and phase6; (collectively, with magnitude

The real part of ( 6 ) can be further partitioned intoreal power network constraints at the generator andload terminals

and phase 4).

(7)where& epresents injections into the network fromgenerator buses, E L represents power removals fromload buses, and .FG nd E L represent tie-line flows ofpower from adjacent systems into the network at thegenerator and load buses, respectively.

Linearizing (7) under the decoupling assumption(af"/ay= 0), one obtainsP G + & JGG JGL

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where J; j = alr fahj , i, = G, . Observe that,upon linearization, the variables appearing in (8) areall incremental quantities: their values indicate off-sets from the nominal value at which the linearizationoccurred.J L L should be invertible under normal operatingconditions. Make the definitions

(9)(10)(11)

l & = K p & + - F + D p P L . (12)

AAA

Dp = -JGLJ;iKP = J G G + D P J L G- = &+Dp& .

Solving the partitioned expression (8) for& yields

2.3 Frequency ModelTo derive a state-space description for frequency dy-namics, the network constraints on& n (8) are firstsubstituted into the multi-machine droop character-istic (5):

@G[k] = (1 C D ) G f [ k ]( K P b [ k l- @I + DpP,[kl) 413)

Then (13) is evaluated at two successive AGC sam-pling instances, kT s and (k+ l)Ts, hese results sub-tracted, and the approxiniation

WG[k + 1]Ts = &[ k + 13 -&[k] (14)incorporated. Note the backward Euler approxima-tion in (14), which is important because this leads toa (causal) model different from that found in [3].

The subsystem frequency model which evolves fromthese operations isWG[k+ 11 = (1 CKPTS)-~%[k]+(I- ED) u[k]+ (f[k] Dpdlk])} . (15)-[k] & f [ k + 13 -&f[k] (16)-[k] = E [k+ 11 - F[kld[kI = P L [ k + 11 - P L [ k l .

where

(17)(18)

AA

2.4 Generator Power ModelsA model which follows the evolution of generator out-put power, P G , in response to governor controls) canbe derived using a similar approach. The resultingmodel is-,[k + 11 = ( I+KpCT,)-l{ P&]+

KP (1 ED) ~ s L G f [ k 1 (@ I - D,d[kI)}(lS)

Another useful generator power model is one thatrelates generator output power to generator fre-quency. This model, derived by evaluating (12) attwo sample points and making the backward Eulerapproximation of (14), is

& [k + 13 = &[k] +KpT&G[k + 13-f[kI + DPdPl . (20)This model will be manipulated to develop tertiarylevel control schemes for both tie-line flow manage-ment (section 4.1) and , when combined with the gen-eration costing methods of section 2.5, Economic Dis-patch (section 6 . 3 ) .2.5 Generation (Fuel) CostAGC cost can be formulated using (19) and marginalfuel cost. Typically, curves for the fuel cost of indi-vidual generators are available. These curves are, ingeneral, nonlinear, and due to the variability in theprice of &el, time-varying:

I? = h(PG,t) (21)where PG is a large signal generator power output-rather than the incremental quantity PGwith whichwe have dealt previously-h(.) is a nonlinear functionwhich maps generation into a cost, and t is time.

In this paper we choose to model h(.) as time in-variant, although our methods can easily be extendedto account for slow temporal variability (e.g. season-ality) in the cost curves.

Typically, generator fuel cost curves are modeledas quadratic functions of P G : for a single generationunit, th is implies

(22)= a0 + alPG + a z P , ,where the a0 parameter accounts for the nonzero costof maintaining spinning reserve.

Since the frequency control model is linearized, welinearize this cost curve about an operating point Pz.The incremental model for the offset in generator costfrom this operating point is

C=mcPG , (23)where c is the incremental cost, and

is the marginal fuel cost.The incremental cost for all of the generators on the

system is found by stacking the equations governingcost into vectors, c. The resulting expression is

c = M c & , (25)where M , is a diagonal matrix of marginal fuel costs.

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2.6 AGC non-participationNot all generators in an area need participate in fre-quency regulation. Define a participation matrix, Cas a matrix with dimension numpartgen numgen

the objective of the finite horizon control is to mini-mize

J [ k ]= E { g T [ ki-]Qg[k + 11+uT[k]Ru[k]}27)composed of Ones and zeros such that only partici- with respect to 14, subject to the model constraintpating generators are selected. For example, the par-ticipation matrix for a three-generator system wherethe-second generator doesn't participate would be the2 x 3 matrix and

E {.} in (27) represents the expected value operator,

Thus, the participating controls for such a model be-come g p = CZL; n the model of (15), 21 is replacedwith CTgP.

Note that the frequency at non-participating gen-erator buses will still be directly regulated given thisadjustment. If no direct stabilization is desired atnon-participating buses, the controlled states of (15)are reduced via appropriate multiplication by C andCT .

3 Secondary Level Control:

A f ( I+CKpTs ) - l (29)B e A ( 1 - C b ) (30)

d,[k] 2 AX (f[4D,d[k]) (31)The weighting matrices Q and R in (27) are diago-nal, with nonnegative entries. Their relative valuestrade off-for each participating generator-the im-portance of frequency regulation versus the coupledfuel cost and undesirability of changing a governor setpoint.

The optimal g [ k ] s of the form- [ k ] -K , (A%[k] - gg t [ K ] ) , (32)

Regulating Area Frequency where K, = ( R +B ~ Q B ) - ' B T Q . (33)The goal of secondary level control is to stabilize gen-erator frequencies, %[ k + 11, within a region so that 3.2 Infinite Horizonthey reach desired set values, gg t [ K ] ,espite the in-troduction of a slow load disturbance, &c], which isassumed to have zero mean. For this application, we

For the infinite horizon case, 24k] is chosen so thatthe performance criterion

examine two types of control: finite (1-step) and infi-nite horizon control. Finite horizon control facilitatesfrequent parameter updating or relinearization, whileinfinite horizon control should offer better asymptoticperformance if the model parameters are quite accu-rate. The effects of plant uncertainty are not studiedin this paper.

In both approaches, the expected increment in tieline flows, f[k],s considered small, and is neglectedin the computation of the controls. A decentralizedcontrol which incorporates some flow dynamics is cur-rently being developed, but will not appear here.Non-interactive flow compensation like that foundin [3] cannot performed here since measurements forE[k + 11= f [ k ]+E[k] are not available at the timethe control computed.

J , = J[kI (34)k=Ois minimized. The control which minimizes J , is~ [ k ] -Ks gG[k]- &3'[K])+B-' ( I - A ) )@'[IT] ,

(35)(36)

where

and S is the solution to the familiar (see [5, 61) dis-crete algebraic Riccati equation

K, = ( R+BTSB)-lBTSA,o = A ~ S As +Q - A ~ S B ( RB T S B ) - ' B ~ S A .

(37)

4 Tertiary Control: Regulat-3.1 Finite Horizon ing Tie Line FlowsGiven the assignment In the secondary level frequency control application,arealsubsystem control is decentralized. As a con-sequence, the dynamics of tie-line flows between the[ k+ 11= W G [ k+ 11- ,=t[K] (26)

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areas are not modeled. Each area uses its own sec-ondary control to regulate frequency, and this actionchanges the phase angles at its end of tiel ines. Sincethe party at the other end of a tie-line is performing asimilar, uncoordinated action, tie-line flows can driftfrom their scheduled values.

One responsibility of tertiary level control shouldbe to occasionally update the governor set pointsgrt[K]n each otherwise autonomous area such thatprescribed tie-line agreements are maintained. An-other goal of tertiary control is to reschedule tie-line flows in response to the expected load variations,or new contracts between areas. At present, thisrescheduling task is not done in a systematic way; itis, instead, carried out by agreement between severalareas at a time when needed.

Since tertiary level models are by necessity central-ized, and thus quite large in scale, the time intervals,Tt,between updates should be much longer than theTs or secondary control. This presents no problemhowever, because in practice, tie-line rescheduling isdone on a fairly slow time scale.

4.1 Centralized Model for TertiaryControlWe now derive a model relevant for tie-line flow con-trol. The goal is to find a relationship between gen-erator frequency and tie-line flows for the entire in-terconnected system.

Assume that the interconnected system consists ofR areas, enumerated 1, 2, . .. R. Assemble a vec-tor of the frequencies of all participating generators,from every area, within the interconnected system;call this vector fig'. Denote the associated vector ofgenerator power outputs PG.Take the new samplingrate to be Tt, and index these longer time incrementswith [K],ather than [IC].

Now treat this new global, interconnected systemas a very large subsystem, and apply (20)-whichdescribes the power dynamics of a subsystem. Thisyields

Note that although the form of (38) may resemble(20), some important differences do exist. Since thisis the entire interconnected system,1. no external tie-line flows exist, so f[K]2. d[K] s a vector describing the increment in load

0 ;

demand throughout the interconnected system;

3 . K p and Dp are interconnected system equiva-lents of the secondary level Kp, and Dp,, =1, 2, .. , R (but, due to area interconnections,neither can be be easily assembled from the Kp,or DR).

An equivalent expression for interconnected systemdynamics can be obtained by stacking the state-spaceequations for each area's secondary dynamics (15)oneatop the other:PG[K + 11 = PG[K]+K g k T t Q z t [ [ K+ 11

-f[K] +Dzkd[K] (39)For this case, f[K] F[K 11 - F[K]s a vectorconcatenation of the increments in flow measurementsfrom all areas, and DFkand Kgk are block diagonalmatrices formed from theDb nd Kj, associated witheach area.Equating (38) and (39), and solving forF[K 11 =f[K]+F[K]ields the tie-line flow model for whichtertiary level controls will be designed:F[K 11 = F[K] (Kgk- Kp ) TtC?zt[K+ 11+ (D!k -Dp) d[K] . (40)

4.2 Tertiary Level Control DesignThe goal of tertiary control is to derive a frequencyset point vector OFt[K]-which is later broken upand sent to the secondary level controllers in eacharea-such that a desired (mapped)2 flow schedule,p e t is eventually realized, perhaps over several Ttsampling periods. Fsettself can be occasionally resetto reschedule interface flow agreements between theadministrative areas, perhaps made by means of arecent Economic Dispatch calculation.

The finite and infinite horizon proportional controlsolutions for the feedback gain matrix Kt are struc-turally identical to those found at the secondary level:the only difference is that, for this control problem,A ---+and B + Kjtfk- Kp ) T t . Relative valuesof the matrix weights, R and Q, determine the trade-off of frequency deviation versus the rate at whichtie-line flows settle to their scheduled values.The structure for the infinite horizon controller is

Ogt[K+ 11 = - K t (F,[K] - F s e t )-where E { d [ K ] }s the load deviation expected overthe tertiary sampling period. Notice the addition ofan noninteracting (anticipative) control term.

'We denote a c t u a l flows as FG and F L ; m a p p e d flows are afunction of the actual flows: F = FG + DrkFh .

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Another subtlety associated with tertiary control isthat F = FG+DF'FL maps the actual ti e line flowsFG and FL back to the generators. This means thatone cannot use our models to directly control FGandFL.However, these flows can be indirectly controlled(but not uniquely specified) via F. Most important,though, is that the number of mapped flows to becontrolled should never exceed the total number oftie lines. The control problem is overconstrained oth-erwise. Therefore, multiplication by a participationmatrix C of ones and zeros must be used to reducethe number of flow states to be controlled to the num-ber of tie lines. The number of controls, though neednot be reduced.

5 Economic Dispatch: Mini-mizing Cost

Denote another form of tertiary control, which againsends frequency set point signals C4gt,down to sec-ondary controllers, Economic Dispatch. EconomicDispatch reschedules the generation within the en-tire system to minimize total cost, while at the sametime att empting to minimize the frequency deviationincurred on the system to realize this function. Con-ventional (stat ic) Economic Dispatch is performed ata much slower rat e than AGC, typically every 15 min-utes in the United States.

5.1 Economic Dispatch ModelThe cost model used for Economic Dispatch is simplyderived by evaluating (38) at a sampling rate of T e d ,and multiplying both sides by the diagonal matrixof marginal fuel costs, M,, of the generators in theinterconnected system. The derivation of M, is foundin section 2.5. The resulting model for c , the cost ofgeneration throughout the system, is

c [ ~11 = C [ N ] +M , X K ~ T , ~ C ~ ~ ~ [ N11+M,Dpd[N] . (42)

Note that [NI is chosen as a time index. This distin-guishes the slower rate at which Economic Dispatchis invoked from the (relatively) faster rate associatedwith the tie-line regulation function of AGC.

5.2 ED Control DesignControls are similar to tie-line flow design, except thecost set point is c f l o o r .This set point determines thefloor for the deviation in cost (from the linearization

point); in essence it keeps the generation of any par-ticipating unit from falling beyond a preset limit. Forthe infinite horizon controller,

(43)The weighting matrices Qe d and Redused in calculat-ing Ked allow a system operator to pick the speed ofcost reduction to the deviation in frequency necessaryto achieve that amount of reduction.

Since Economic Dispatch perturbs area interfaceflows to minimize cost, a new tie-line set point, FE;,must computed and sent t o tert iary flow control, sinceit is invoked much more often. The set point is com-puted using (40), where one solves for FZL = F[K+11 with F[K] := F;;: and C4gt[K+ 11 := C4Et[N+ 11.

Unlike the tertiary level tie-line control, cost statescan be minimized throughout the system; none needb e ~ m i t t e d . ~his is a desirable characteristic, be-cause the power industry tends to dispatch all unitswhen at tempt ing to minimize costs. As far as theauthors know, this formulation is the first which hastreated Economic Dispatch as a non-static optimiza-tion process, and the first that has tied EconomicDispatch to the frequency control function found inpower systems. Moreover, through participation ma-trices, not all generators need part icipate in Economicdispatch. This is an important feature, since not allgenerator owners in a deregulated environment-forexample, Independent Power Producers-may wantto participate in a coordinated cost reduction scheme.If the number of states to be stabilized is reducedto less than the total number of generators, for exam-ple, by using a st ate selection matrix, C, of dimensionnumpart tatesx numge,, then a noninteracting antic-ipative control term may be added4 The controlledstate then becomes cp= C c p ,and the controls are

QE'[[K + 11 = -Ked ( c p [ K ] cPfIoor)-(CMcx:KpTed)-lMcDpf { d [ ~ ] (44)

6 Simulation StudyThe 3-generator, 5-bus system depicted in Figure 1was used in our simulations. In thi s system, thereare two ti e lines, one between generators #2 and #3,and another between loads #4 and #5.

The generator parameters and initial operatingpoints for this system are found in Table 1. All intra-area transmission lines had impedance 0.01 +jO.l pu;

3Performance tends to improve, however, if one of the di-4the interconnected system version of Kp has a rank-onerectly regulated states is omitted.

deficiency because phase angles must have a reference.

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all tie lines were weaker, with impedance 0.1 + j lpu. Secondary controls sampling time, T,, was 2seconds; tertiary control, when invoked, acts every20 ~econds.~

N.-3

/._.....-..._.......

4 / *

Figure 1: An interconnected 5-bus power system.

-31-,k

I Load Flow Data 1

- dash..senl,sol&gen2,dots=gen3

bus 1 #1 I #2 I #3 1 #4 I #5V I 1 I 1 1 1 I .9909 1 .9820I I I I IP I .424 I .9995 I .9992 I -1.200 I -1.199

~I Generator Parameters I#1 5%#2 5%HzH3 5%

Table 1: Per-unit data for the 5-bus example.

6.1 Secondary (Frequency) ControlIn Figure 2, we demonstrate the results of applyingsecondary control (only) to a situation where a stepload increase of 0.2 pu occurs at load bus #4 (Area I)30 seconds into the simulation. For this simulation,Q = R = I , where I is the identity matrix. Plots offrequency offset, generator power outputs, and devi-ation in actual (not mapped) tie line flows into AreaI are found in this figure.Note that at the outset of the load disturbance inArea I, the frequency offset decreases in both areas,but less in Area I1 due to high impedance of the tielines between the areas. Note tha t there are net flowsinto Area 11,however.

51n actual practice, tertiary control would invoked on thetime scale of minutes; however, for illustrative purposes here,we invoke it much faster.

0 10 20 30 40 50 60Tie-line flows into Area I:solid=gen lie,dots=load ie

S O MP-

Figure 2: Bus #4 0.2 pu step disturbance t = 30 sec:Secondary (infinite horizon) control, Q = R = I .

Figure 3 (top) compares the frequency regulationperformance at generator #2 (which is representa-tive) for finite and infinite horizon secondary control.Note that infinite horizon approach offers a slightlyquicker stabilization response. Figure 3 (bottom) de-picts the case where no secondary control (only pri-mary/mechanized control) is applied. Notice tha ta longer time scale is used to demonstrate that allgenerator frequencies approach the same equilibriumpoint, but this equilibrium is nonzero.6.2 Tertiary (Tie-Line) ControlWe next demonstrate that tertiary control can beused to drive the sum of inadvertent flows to zero.6Figure 4 displays the result of invoking tertiary con-trol on top of secondary control to regulate themapped flows for generators #2 and #3. As the top(frequency) plot reveals, this is achieved by increas-ing the frequencies (phase angles) in Area I, whiledecreasing the frequencies (phase angles) in Area 11.This intentional frequency offsetting is particularlyapparent in the 40-60 second interval.6.3 Economic DispatchWe will assume quadratic generator fuel cost curvesc = .5+ PG+ P& for generators #1 and #2, andc = 1+ 5pG + 2Pz for generator #3. These curves

6Note: for these simulations, we choose the rightmost termof (35) o be zero although in this simulation, the set point fedsecondary control is non-zero. Including tha t term also drivesthe sum of inadvertent flows to zero.

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x lo4 Frequencyoffset 0 gen U2 . nf and 1-step horizon

-e ocB-826 28 30 32 34 38 38Time (sec)

Frequency offset (all m s ) or no secondary controlY jn-'- d a h . ; ( a f l l 3 d i U = g ~ 2 W p n ~--

-1 c \ 4i - 2 1- 3 \ ---4close to same)

35 30 35 40 45 50 55

Figure 3: Top: Frequency offset at gen #2 for infiniteand 1-step horizon secondary control. Bottom: Fre-quency offsets at all generators, no secondary controlapplied.

are linearized about the operating point for the sys-tem, yielding marginal costs of 3.0, 1.8, and 4.5 $/pupower.

We choose to directly minimize only two of the coststates (associated with generators #2 and #3)-soour results can be contrasted with those for tie-linecontrol. The cost deviation floor for all generators isset at 0. A step disturbance at t = 30 sec is initiatedas w as done in the previous sections, and like tie-line control, Economic Dispatch is invoked every 20seconds.

Figure 5 displays the frequency offsets, power out-puts, and tie-line flow response resulting from an Eco-nomic dispatch simulation. Observe tha t since gen-erator #3 has the highest marginal cost, its poweroutput is eventually driven to its (linearization) floorvalue. Also, as one would expect the net tie-lineflows into Area I do not sum to zero: since poweris (marginally) cheaper in Area I, power flows intothat area to compensate for losses.

Figure 6 compares the generation costs of usingvarious control strategies to respond to a load dis-turbance. Remark th at not only does use of no sec-ondary control result in poor frequency regulation, italso is more expensive. As one can see, combiningEconomic dispatch with secondary control offers thelowest system cost. Tertiary-level tie-line regulationis cheaper than just secondary control only becauseit minimizes tie line flows of power from the moreexpensive generator in Area 11.

60

5 x 10.' Frequency deviation from 60Hz- 1 -

_J-10; i o 20 j, 40 i o 60 ; eo 90 to o

ii -0.01 L lo i o 20 30 40 50 BO 70 80 m looTime (sec)Figure 4: Same as Figure 1, with tertiary (Q = R =I) and secondary Q = R = I . Both controls infinitehorizon.

7 ConclusionsWe have introduced models and optimal control tech-niques for the tasks of Automatic Generation Control(AGC) and Economic Dispatch on electric power net-works. Unlike current implementations of AGC [ l , 1,the our AGC design w as composed of a hierarchyof controls, which we called secondary and tertiarycontrol. Secondary control, which was decentralized,demonstrated the capability to adjust the power gen-erated within an administrative area so as to drivefrequency offsets to zero. Tertiary control, which wascentralized, returned the sum of interface power flowswith neighboring systems to their contract values.

Another, even slower-acting tertiary controlparadigm, called Economic Dispatch, was introducedto reschedule generators so as to minimize overall sys-tem fuel cost. The efficacy of thi s technique, alongwith tha t of the AGC techniques, w a s demonstratedvia simulation.

Future work includes demonstrating the perfor-mance advantages of this advanced generation con-trol over conventional AGC, comparing this new, dy-namic version of Economic Dispatch with static Eco-nomic Dispatch approaches, and incorporating gen-eration constraints into 1-step controls for the var i-ous algorithms. Also, some efforts will be undertakento design controls more robust to errors in the plant(system) models.

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Frequency dewaton from M) z

DU -5p!U

20 40 60 80 100 120Generator real Dower outDuts ldevlatm from lmearmtronf

Tie-line flows into Area I: s o l i ie, &&=load tie

.....................

20 40 60 80 100 120 o . lO 20 40 60 80 100

3 0.02n%B 0 -nBE -0.02

Y

0

. .0.2, I

120

Sum f generationcostsusmg dlfteremcontrols

sec only_ _ _ _ _ _ _ _ - - -- - - - _ _ _5 dot dash.Eco Disp

.-..a-g 0-a genr;&jfi&ipa;Was=sen5... . . . . . . . . . . . . . . . . . . . . . . . -

Figure 5: Economic Dispatch with (Q = R = I) andsecondary Q = R = I . Both controls infinite horizon. Figure 6: Total (systemwide) incremental cost asso-ciated with different controls.8 AcknowledgementsThe authors gratefully acknowledge support for thisresearch from EPRI/PECO collaborative projectRP3954-1 and DOE project grant DEF641-92ER-110447. Assef Zobian of MIT also deserves specialthanks for supplying a load flow program and assist-ing in its integration into our simulation software.

References[l] J. Carpentier, To be or not t o be modern that

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