00589791 Dynamic Economic Dispatch Sced

IEEE Transactions on Power Systems, Vol. 12, No. 2, May 1997

961

Dynamic Economic Dispatch Using The Extended Security Constrained Economic Dispatch Algorithm Wayne R. Barcelo*, Member, IEEE

Parviz Rastgoufard, Member, IEEE

Department Of Electrical Engineering Tulane University New Orleans, LA 701 18 USA

Department Of Electrical Engineering Tulane University New Orleans, LA 70 1 18 USA

Abst~ct-An algorithm for solving the multi-stage dynamic economic dispatch (MDED) problem in real-time is presented. The MDED problem is formulated by formally adding ramp rate constraints to the extended security constrained economic dispatch (ESCED) problem for all stages beyond the first stage. The MDED problem is then so,ved using the ESCED a,gorithm Ill] with ramp rate constraint sensitivity coefficients. A new two-component method of observing regulating margin constraints is also introduced and test results are presented comparing the one and two-stage dynamic EDC results.

ed in [ I I]. Thus all the advantages of the ESCED algorithm as noted in [ 1 11 are preserved in the solution of the multi-stage dynamic economic dispatch problem. The speed and convergence reliability of the ESCED algorithm with its ability to coordinate regulating margin and constraints in conjunction with network constraints makes it ideally suited for real-time economic dispatch operation over multiple stages. The capabilities of the ESCED algorithm in multi-stage operation were demonstrated and tested using the Bulk Power Management System (BPMS) and power system data of a major power company. A sample of the test results obtained are presented in this paper.

I. INTRODUCTION Dynamic Economic Dispatch may be considered as the latest development in economic dispatch. In its purest form, Dynamic Economic Dispatch combines the areas of static economic dispatch, including optimal power flow, Load Frequency Control (LFC) and Automatic Generation Control (AGC) into ajoint design process considering the dynamics of the problem. This is the most accurate formulation of the economic dispatch problem but also the most difficult to solve because of its large dimensionality. The first paper in this area appeared in 1972 [ l ] by Bechert and Kwatny. Additional papers in this area did not appear until the 1980's [2-91 covering various approaches to solving the dynamic prcblem. Papers by Innorta [9] in 1988 and Somuah and Khunaizi [IO] in 1990 include nehvork security constraints in the problem formulation. The paper by Innorta also introduced a discretization approach for solving the problem. The research results presented in this paper use the discretization approach of Innorta and add regulating margin constraints to the network security and generating unit ramp rate constraints. The algorithm used is the Extended Security Constrained Economic Dispatch (ESCED) algorithm present96 SM 580-1 PWRS A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the 1996 IEEEIPES Summer Meeting, July 28 - August 1, 1996, in Denver, Colorado. Manuscript submitted July 28, 1995; made available for printing June 25, 1996.

*Dr. Wayne R. Barcelo i s presently an Assistant Prokssor of Electrical En. . ginecrtng at the Universie ofNew Orleans. New 0rle:lns. L A 70145 USA.

11. THE DYNAMIC ECONOMIC DISPATCH PROBLEM WITH NETWORK SECURITY, REGULATING MARGIN AND RAMP RATE CONSTRAINTS In [ 111 the Extended Security Constrained Economic Dispatch (ESCED) problem was defined as including the security constrained problem plus the addition of system regulating margin constraints up and down and the addition of generating unit ramp rate constraints up and down. It was further defined to be the problem of determining the optimum trajectories required to follow "Base Generation Requirement" where Base Generation Requirement is the low frequency component of Generation Requirement which equals Control Area Generation minus ACE where ACE < 0 implies a generation deficit. The addition of ramp rate constraints required estimating the change in base generation requirement at a future time t + Te where t is the time of execution of the last EDC and Te is the period of the EDC execution frequency. Neglecting wear and tear costs due to ramping a unit up and down, the objective function to be minimized was the integral of the production cost function from t to t - Te. However, taking the unit trajectories to be linear bemeen t and t + Te , since the cost function of a unit is an increasing function, minimization of the integral is equivalent to minimization of the production cost at time t Te. Thus the ESCED problem is essentially a one-stage dynamic economic dispatch problem. By considering additional stages, that is the production cost at times t + 2Te , t + 3Te ... t + kTe and imposing ramp rate constraints between the stages we obtain the multi-stage dynamic economic dispatch problem. The major difference between the one-stage and the multi-stage

0885-8950/97/$10.00 0 1996 IEEE

962

problems is that in the multi-stage problem the ramp rate constraints beyond the first stage must be added formally since the generation level for the second and succeeding stages is not known a priori. ?&tation Convention: To simplify the formulation of the multi-stage problem we indicate the stage dependence of a variable by adding a capital I to all variable names where I varies from 1 to K, K equaling the number of stages. For example, the vector P becomes PI and for the first, second and third stages, PI is represented by P1, P2 and P3, respectively. The jth component of PI is represented by PIj where I = 1,2, 3, ..., K for a K-stage problem. Using this convention, we can formulate the multi-stage problem as follows:

Minimize 1=1 jeEI

I=l

Subject to the following constraints for all stages 1=1 to I=K: 1. Power Flow :

(sI)GPI= 6GI,

(2)

2. Unit Dispatch Limits: P I - I PI IPI+

3. Ramp Rate Constraints (1st stage Only): (Pld') 5 I(Pld')

(4)

4. Ramp Rate Constraints (after 1st stage):

Fir- IPJ - PI 5

FI,?

, J= I+1

(5)

5 . Network Security and Regulating Margin Constraints:

Where j'

PIj PI

= Generating Unit Cost Curve Of The j* Unit =

Generating Unit Power Output Of The jth Unit I* Stage

= n-Dimensional Column Vector Of Grnerating Unit Power For

6PI

The I* Stage = PI, = Set Of Control Area Units Participating In E X At Stage I = n-Dimensional Row Vector Of Inverse Penalty Factors, Stage 1 = PI - P1,

6GI,

=

PI-

=

PI+

=

Pld-

=

(Including The Effect Of Variations In sf From SI) Generating Unit Low Dispatch Limit Column Vector. Stage I Generating Unit High Dispatch Limit Column Vector, Stage I Generating Unit Low Dynamic Dispatch Limit Column Vector,

PI,+

=

First Stage Generating Unit High Dynamic Dispatch Limit Column Vector,

fl0

=

First Stage m-Dimensional Network Security And Regulating htargin

11I

=

Constraint Initial Value Column Vector, Stage I (see Appendix) The i n s 11 Constraint Sensitivity Coefficient hlatris, Stage I

PI0 E1 SI

= Initial Value Of PI

FI+

= The Network Security And Regulating Margin Constraint

Fir+

=

FI;

= -TeP-

TeP+

Upper Limit Column Vector, Stage I = The Generating Unit Rate Limit Up, Stage I =

The Generating Unit Rate Limit Down, Stage I

In the last two definitions given above, P' and P - are column vectors of the upper and lower generating unit ramp rate limits, respectively. Network security constraints represented by (6) are Line Mva, Line Mw, Line Group Mw and Area Mw Import constraints [ 1 11. Sensitivity coefficients @Ii) for these constraints are easily calculated from a real-time power flow Jacobian matrix which is available on a modem BPMS. Generating unit ram^ rate constraints given by PI; I P I j IPI; ,where P I j is the ramp rate of the jrhunit (Irh stage) and P I ; , PI; are the lower and raise ramp rate limits respectively, are most efficiently implemented for the first stage as dynamic high/low dispatch limits (Pld+/Pld-) as in (4) to avoid adding a large number of additional functional constraints. Using Te as previously defined, the dynamic dispatch limits are calculated as in (7)

For stages beyond the first, ramp rate constraints are implemented as in ( 5 ) . If we let fIJ, where J = 1-1, be the generating unit ramp rate vector between stages I and I+l, then flJ = PJ - PI = SPJ - &PI. The ramp rate constraints can then be written in terms of the ramp rate constraint sensitivity mamces pIJ and pJI between stages I and J as flJ = (p1J)GPI f (uJI)6PJ where pIJ and pJI are the sensitivities with respect to PI and PJ, respectively. Note that pIJ consists only of-1's and 0's and pJI consists only of +l's and 0's. See the definition of p in (10) for further clarification. Regulating margin constraints are expressed in the standard form of (6) as explained in [ l l ] with sensitivity coefficients of either 0, -1 or +1 depending on the following conditions: dMIuj/dPIj= 0 ,

PIj I BIujf

(sa)

dMIuj/dPIj = -1 ,

PIj

(8b)

BIuj"

Estimated Change In Generation Requirement At Stage I

where MIuj is the regulating margin up for the jth unit, I* stage and BIuj'= PI.'1 - Tm-pl; is a new limit called the regulating margin up sensitivity coefficient breakpoint for the jth unit as explained in [ 1 I]. T, is the time interval over which the regulating margin is calculated, usually I O minutes. Similar results will be obtained for the regulating margin down sensitivity coefficients except for a change of s i n . When the number of stages is small. the power flow constraint sensitivity vector s and the nenvork constraint

963

sensitivities may be assumed to remain constant with little

pl

0

..'

0

FI'

0

p2

"'

0

F2'

..

..

error. Similarly, 6G, the change in generation requirement may be estimated by a simple regression analysis using previous generation requirement data. When a large number of stages is considered, resulting in a significant look-ahead period, a short term load forecast program may be required to estimate SG for stages significantly in the future. Also, the s and network constraint sensitivities for future stages may either have to be calculated by a study mode power flow at the start of the day and updated periodically or saved from the previous day's calculations. Either approach is considered satisfactory considering the uncertainty in all data when the look ahead period is large. 111. SOLVING THE MULT-STAGE DYNAMIC EDC PROBLEM USING THE ESCED ALGORITHM

0

.

..

0 ... pK ... ........................

FK

+

... . .

....

.l12

p21 .'.

0

F12'

0

p 2 3 ...

0

F23'

0

.

. ..

..

0

... pKJ

FJK'

A . Derivation Of The Multi-Stage Hessian Matrix

An algorithm for solving the Extended Security Constrained Economic Dispatch (ESCED) problem was presented in [ I 11. The only difference between the ESCED problem as formulated in [ I I ] and the multi-stage dynamic EDC problem is the formal introduction of ramp rate constraints between adjacent stages. To use the ESCED we need to formulate the multi-stage problem as an equivalent 1stage problem. This equivalence can be obtained fairly easily by writing the Lagrangian for the multi-stage problem in terms of vectors and matrices including the terms for constraint relaxation [ 1 I]. The resulting equivalent one-stage Lagrangian is

L = C( P ) + X{paP + f,- F'

} + 3Ltmax (pR6P+ fOH

-F

i}

(9) where we define the following vector and matrix quantities:

?

-

P1

,

P2

P=

,

- PK -

h=

fo =

PIR

...

0

F',

0

...

0

F2

.

.

.

,

.

FK;

0

.

.

.

.

0

...

pKJ,

More details of this development are presented in the .lppendix. Substituting the equivalent one-stage sensitivity coefficient matrix p as defined in ( I O ) and C" = dC/SP'. a diagonal matrix, into (12) produces the multi-stage Hessian matrix. The sensitivity coefficient matrix p is arranged so h a t each PI includes the power flow constraint sensitivity vecror -SI as its first row and each column vector hI includes systzm lambda (Asl) as its first element.

B. Derivation Of The Pre-Eliminated Sollition Eqiration From (9) we find the gradient of L to be

964

constraints are added to the ramp rate constraints using a onestage dynamic EDC. In the one-stage formulation, production cost is minimized over one interval at a time. Table 3 defines the meanings of the symbols used in Table 1 to facilitate interpretation of the data. The area Mw import constraint was set at 1800 Mw with units 5 , 7, 9 and 20 inside the closed area constraint boundary. This constraint is mapped into the original constraint set as a network security constraint via (6). where zo = (Po, h,) and all other vectors and matrices are as The regulating margin (RM) up and down constraints defined in (10). By Newton's method we wish to solve HAz = were both established using a two component method. In this -g or using (1 1) and (12) we get method the base regulating margin requirements are set equal to the sum of the expected base generation requirement change over the next 10 minutes (for the ramp load profile this would be 400 Mw up) and an operator entered regulating margin reserve requirement which was set at 200 Mw for this test. So during the load ramp the regulating margin up (down) requirement was 400+200 = 600 (-400+200 = -200) Mw. where Cot= X / a P evaluated at Po. Multiplying the first block In Table 1, the first three time intervals t = 0, t = 0-1, in (13) by (C'4)-1 , eliminating p and making the other and t = 0-2 are used to establish steady state conditions before substitutions given in [ 1 11 leads to the Hessian pre-elimination starting the load ramp. Time t = 0 corresponds to the unconequation strained case, that is, it is the dispatch that results with the area and regulating margin constraints removed. Time t = 0- 1 is the dispatch that results when the regulating margin and area constraint are first applied using the dispatch at t = 0 as the starting point. Note that U3 is rate limited. To remove this rate limit condition a second dispatch was n e c e s s q corresponding to the t = 0-2 case which resulted in the desired steady state where subscript v emphasizes we are writing L with respect to conditions. Note the shift in generation behveen U3 and U4 free units only [ l l ] . For a problem with a few stages (14) can and the relaxation of the Regulating Margin Down constraint be solved using standard Guassian elimination techniques. If at times t = 0- 1 and t = 0-7. Notice also in Table 1 that various units are rate limited the problem to be solved contains many stages. py(Cyo")-lpyt will be sparse and so sparse matrix techniques should be used at several times during the load ramp, that the area constraint is binding from t=5 through t=25 minutes, and that the reguto solve (1 4) quickly. lating margin up constraint is binding at t=25 and 30 minutes and then relaxed for the remainder of the load ramp. IV. SIMULATION RESULTS Table 2 shows the results obtained for a ?-stage d>namic Test results for a large scale problem were compiled on EDC using the identical constraints and test configuration. In the software development machine of a major power compa- the 2-stage formulation, production cost is minimzed over 2 ny's BPMS. The Test System was a model of the company's intervals but only the dispatch results for the first sta,oe are actual power system which is on the order of 1200 buses, used and presented in Table 7 . To highlight the differences. 1700 lines and 85 generating units. As noted in [ 111, timing only variations from the results shown in Table 1 are tests indicated that the algorithm took about 6 msec wall clock presented and rows without any differences are omitted. time per iteration on a 5 to 6 MIPS machine. Comparison of Tables 1 and 2 shows that for this test case To test the ramp rate constraints a 40 Mw/Min load there is very little difference between the 2-stage and rhe 1profile was developed starting at 1900 Mw and ramping up to stage results. the major difference being that the 2-stage EDC 3900 Mw in 50 minutes. After selecting eight units to starts ramping U20 a little sooner. These results primarily participate in the test, starting at 1900 Mw, the ESCED was demonstrate that the proposed method works. executed every 5 minutes corresponding to load increments of In general one expects the addition of extra stages to the 700 Mw until 3900 Mw was reached and the unit trajectories EDC to enhance performance since the further ahead the EDC stabilized. This test also simulates realistic conditions when can see, the more opportunity it has to make adjustnimts in one considers the load profile as occurring on top of a the present to avoid problems in the future. In addition. there sufficiently large base load supplied by fixed base generation. are some constraints. fuel constraints for instance, thst can Table 1 shows a sample of the resulting baseline only be observed b\. considering multiple stages. trajectories when network security and regulating margin and the Hessian matrix H to be

965 TABLE I OPTIMUM TRAJECTORIES BY I-STAGE DYNAMIC EDC WITH SECURITY. REGULATING MARGIN AND RAMP RATE CONSTRAINTS ~

~~~~

~

Note: In the above table, a blank entry means the last entry in the row is repeated.

TABLE 2 OPTIMUM TRAJECTORIES BY 2-STAGE DYNAMIC EDC WITH SECURITY. U M P RATE & REGULATING MARGIN COSSTRAIIU’TS

4. CONCLUSIONS

I

I

TABLE 3 TABLE 1 SYMBOL CONVENTION DEFINITIONS Convention I Definition Superscript + (-) I High (Low) Dispatch Limit Superscript +u (+d) I Regulating Margin U p (Dn) Breakpoint Superscript +r (-r) Ramp Rate Up (Dn) Limit R Constraint Relayed Superscript +R Ramp Rate Up Limit Relaxed LA ($ / Mwh) Area Constraint Lambda Regulating Margin Up Constraint Lambda k f in ($ / Mwh) kni ($ / Mwh) Regulating Margin Dn Constraint Lambda A( I ) ($ / Mwh) System Lambda including the penalty cost I o f r e l m i n r a constraint

1

I

I

A method of formulating the multi-stage dynamic EDC problem with network security, regulating margin and ramp rate constraints as a single stage problem so that it could be solved using the ESCED algorithm [ l I ] was presented. The ESCED algorithm was used to solve a 2-stage dynamic EDC problem and simulation results were compared to the I-stage solution results. For the load profile used, the ?-stage results showed a small improvement over the I-stage results. -4twocomponent method of implementing regulating margin constraints was introduced and appears to be the ideal way to implement regulatins margin constraints in real-time. The ESCED algorithm appears to be a viable method for perform-

966

ing multi-stage dynamic EDC in real-time. Exploiting the symmetry and sparsity in the Hessian Pre-Elimination solution approach was not performed in this research but should be investigated to optimize solution time for dynamic EDC problems with a large number of stages.

T. E. Bechert and H. G. Kwatny, "On the Optimal Dynamic dispatch of

VII. APPENDIX

The Lagrangian for the multi-stage problem is:

L = C(P1) + C(P2) + ... + C(PK)

+ h2t (ao+ p26P2-

F2+) + hztmax

Real Power", IEEE Trans PAS, May/June 1972, p 889.

[ a ,+~p2R36132 - F ~ R + ]

D.W. Ross and S. Kim, "Dynamic Economic Dispatch of Generation", IEEE Trans PAS, Nov/Dec 1980, p 2060. W. G. Wood, "Spinning Reserve Constrained Static and Dynamic Economic Dispatch", IEEE Trans PAS, Feb 1982, p 381.

H. Mukai, "A Reevaluation of the Normal Operating State Control of the Power System Using Computer Control and System Theory Part 111. Dispatch Targeting", IEEE Trans PAS, Jan 1981, p 309. J. Zaborszky, et al, "Stabilizing Control in Emergencies Part 1. Equilibrium Point and State Determination", IEEE Trans PAS, May 1981, p 2374.

P. Kambale, et al, "A Reevaluation of the Normal Operating State Control (AGC) of the Power System Using Computer Control and System Theory Part 111. Tracking the Dispatch Targets with Unit Control", /E€€ Trans P.-lS, June 1983. p 1903.

P. P. J . Van Den Bosch. "Optimal Dynamic Dispatch Owing to Spinning-Reserve and Power-Rate Limits"./EEE Trans PAS. Dec

+ hKt ( fKo + pK6PK - FK+ )

+ hKtmax [ f K o +~ ~

K R 1~- FK,+ P

]

+ h12t [ (p12)6P1 + (p21)6P2 - F12+]

+ h12',,,

[ (p12)R6P1

(P21)R6P2 - F 1 2 ~ + ]

+ h 2 j t [ (p23)SP2 + (p32)6P3 - F23+ ] + h23tmax [ (p23)R6P2 + (p32)R6P3 - F23Rf ]

1985, p 3395.

.

A. Kuppurajulu and P. Osso&ki, "An Integrated Real-Time ClosedLoop Controller for Normal and Emergency Operation of Power Systems". /EE€ Trans PlVRS, Feb 1986, p 242.

+ hJKt [ (pJK)GPJ

M. Innorta, et al. "Security Constrained Dynamic Dispatch of Real Power for Thermal Groups".IEEE Trans PIVRS. May 1988. p 774.

+hJKfm,

C. B. Somuah and N. Uunaizi, "Application of Linear Programming Redispatch Technique to Dynamic Generation Allocation", /€€E Tram P WRS. Feb 1990, p 20.

where hIJ is the Lagrange multiplier vector for the ramp rate constraints between stages I and J; and FIJ+ = FI,' or -FI; is the rate limit vector . When FIJ' = -FI/. the siens of uIJ and pJI are reversed. Combining P1, P2: h'l. h2: et;. ( A l j can be writ- ten in the form of(AJ) which is reducible to (9). A constraint fl at stage I is obtained as follows:

W. R. Barcelo and

P.Rastgoufard. "Control Area Performance

Improvement By Extended Security Constrained Economic Dispatch ", Paper No. 96 W M 1971-PWRS, Presented at the 1996 IEEE Winter

- (pKJ)GPK - FJK+ ]

[ (~JK)RGPJ+ (~KJ)RGPK- FJKR']

(A1)

Power Pvleeting Januar). 24, 1996.

fl

=

fl,+

pl(Pl-Pl,)

+ p2(P2-P1)

VI. BIOGRAPHY + . . . + pJ(PJ-PJ-1) + pI(PI-PJ) Wayne R. Barcelo (BSEE. hlSEE. MBA, MS-Math. Ph.D.) is currently an Assistant Professor of Electrical Engineering at The University Of New Orleans. H e has over 75 years experience in electric power systems control, engineering, planning, operations and research with a major power company.

where J = 1-1. Assuming only that adjacent pM's are equal [pM+l = pM], (A?) is a telescoping sum reducing to:

fl Parviz Rastgoufard received his BSEE from SUNY/Buffalo in 1976. He received his MS and Ph.D from Michigan State University in 1978 and 1983 respectively. From I983 - 19S7 he was a faculty member ofthe EE Dept. at North Carolina State University. In 1957 he joined the EE Dept. faculty of Tulane Universtiy where he presently serves as the Director of the Center for Electrodynamics Systcms Research. His research interests are in the area of I : q c scale systcnis decision =id control in general and p o w r systcms

stcurit!' nnd rcliabilit! i n particular.

(A')

=

f l o - ( p I ) ( P l o ) + (pI)(PI)

=

flo+(pl)(SPI)

(A3)

where GPI=PI-PI, and flo = f l o + ( P I - p1)PIo which reduces to flo = f l o if it is assumed that all the pM's are equal. For the power flow constraint a similar result is obtained with (sI)GPI= 6GIr, - (sl-sl)Pl, = &GIr as in (2) where 6GIra is the actual estimated change in generation requirement.

L = C ( P I ) + C ( P 2 ) + ... + C ( P K ) f

' 0

f20

fK0

+[11' 1 2 ' ... hK' i h12'

123' ... UK']

"'

0

FI'

p2

'.'

0

F2'

.

...

..

6P 1

0

... pK

6P2

p1

0

0

0

FK' ........

......

0 0

0

...

0

p23

...

0

.

..

.

112 p21 0

0

F12'

5PK

F23'

FJK'

0

floK

f20 R

K O ,

F1;

'

0

0

F';

J.llR

.

.

0

...

6P 1 pKR

6P2

FK ;

........

.........

0

F12;

6PK

0

0

0

0

...

pKJ,

F2Z;

FJK;