Economic Load Dispatch

Optimum scheduling problem in power system generally can be partitioned into two sub problems namely optimum allocation of units (generator) called unit commitment (U.C) and optimum allocation of generation to each unit called economic load dispatch (E.L.D) or economic load scheduling (E.L.S).They both combined together called optimal unit commitment.Unit commitment should be solved first before go for economic load scheduling problem.

Optimal Unit Commitment

U.C E.L.D O.U.C

(OUT)

UNITS

(IN)

METHODS OF GEN SCHEDULING

Base loading to capacity: -Loaded to capacity according to their Efficiencies

Base loading to most efficient load:-Loaded to their most efficient load according to their heat rates

Loading proportional to capacity:-Loaded

proportional to their capacities Loading proportional to most efficient load: -

Loaded to their most efficient load.If loaded all units, additional load is distributed proportional to the difference between the rated capacity and the most efficient load

Incremental loading: -The load is divided to keep the units operating at equal incremental cost.

Thermal plant modeling:

Heat rate curve:-Heat Energy needed to generate one unit

of Electrical Energy. Input / output curve:-Input energy rate (Mcal/h) or cost

of fuel used per hour (Rs/h) as a function of generator output. It is a concave curve.

Incremental cost:-It is the ratio of small change in input to small change in output. Mathematically it is defined as below &Expressed as (Rs/Kwh)

Incremental cost= Δ input / Δ output i.e.dCi / dPGi

Heat rate curve:- Heat rate curveHi(PGi) which is

the heat energy (Mkcal) needed to generate one unit of electrical energy (MWh)

Input / output curve:-Input energy rate (Mcal/h) or cost of fuel used per hour (Rs/h) as a function of generator output. It is a concave curve.

Incremental cost:-It is the ratio of small change in input to small change in output. Mathematically it is defined as below &Expressed as (Rs/Kwh)

Incremental cost= Δ input / Δ output i.e.dCi / dPGi

         

         

1

2

N

Scheduling without Tr. Losses

F1

F2

FN

P1

P2

P3

Pr

Scheduling Without Tr Losses

We assume that the inequality constraints is not effective, and

k k

PGi = PD or ( PGi ) – PD = 0 i=1 i=1

Using this method we define an augmented cost function (lagrangian) as

k C = C - ( PGi – PD )

i=1

( ∂C / ∂PGi ) = 0; For minimization purpose

Or dCi / dPGi = , i =1,2,3,……k

Where dCi / dPGi is the incremental cost of the ith

generator (Rs / Mwh )

dC1 / dPG1 = dC2 / dPG2 = ---- = dCn / dCGk =

This is called coordination Equation

This equation tells that optimum loading of units are so happened when ever the units face equal incremental cost

         

         

1

2

N

Scheduling With Tr. Losses

F1

F2

FN

P1

P2

P3

Pr

TR

NETWORK

LOSSES

Scheduling with Tr. Losses

)/( )(11

hRsPPPPCCm

iLDGi

m

iGii

m

iLDGi PPP

1

0

m

i

n

iLDiGi PPP

1 1

0

,0/// GiLGiiGi PPdPdCPC

(i=1, 2, . . ., m

Coordination Equation

,

ii

GiL

Gii LICorPP

dPdC)(

/1

/

GiLi PPL /1/1is called the penalty factor of the ith plant

The Lagrangian multiplier has the units of rupees

per megawatt-hour.

Minimum fuel cost is obtained, when the incremental fuel cost of each plant multiplied by its penalty factor is the same for all the plants

The partial derivative PL/PGi is referred to as the

incremental transmission loss (ITL)i, associated with

the ith generating plant.

;)(1)( ii ITLIC

This equation is referred to as the exact coordination equation

SEQUENCE OF ADDING UNITS

If there are four units A, B, C, and D feeding the power system, each having capacity 150 MW such that B has the lowest heat rate, D has next higher heat rate, A has still higher and C has the highest heat rate. The loads at which the units are to be added are given by the points of intersection as shown in fig .

TOTAL STATION LOADS, (Mw)

0 100 200 300 400 500 600

ADD D ADD A ADD C

UNITS B, D, A & C

UNITS B, D & AUNITS

B&D

UNIT B

HE

AT

RA

TE

K-C

AL

/KW

H

2600

2550

2500

2450

Constraints of Scheduling 

Power balance constraints

Spinning reserve constraints

N

∑ ( U n P G max , n ) >= ( PD, t +R t ) .

n=1

N

∑ U n PG n , t =P D, t .

n=1

Thermal Constraints

(down state)

(up state)

0

1

t1(up) t2(up) t3(up)

(up) time

t1(down) t2(down) t3(down)

Repair

Failure

DownTime

Time

Fig: Random Unit performance record neglecting scheduled outages

Security Constraints

PGi min< PGi < PGi max

1.Hydro constraints:Unit commitment cannot be completely separated from the scheduling of hydro units

2. Must Run: Some units are given a must-run status during certain times of the year for reason of voltage support on the transmission network or for such purposes as supply of steam for uses outside the steam plant itself.

:

Unit’s generation capacity constraints

Other Constraints

Fuel Constraints:A system in which some units have limited fuel, or else have constraints that require them to burn a specified amount of fuel in a given

RESULT OBTAINED BY DYNAMIC PROGRAMMING

load op-cost L.D -1 L.D –2 L.D -

3400 41625.60 160 127 113405 42557.85 162 128 115410 43500.00 164 130 116415 44452.85 166 132 117420 45415.60 168 133 119425 46388.75 170 135 120430 47372.40 172 136 122435 48365.85 174 138 123440 49370.00 176 140 124445 50384.15 178 141 126450 51408.60 180 143 127455 52443.40 183 144 128

460 53488.40 184 146 130465 54543.60 187 147 131470 55609.15 189 149 132475 56685.00 191 150 134480 57770.85 193 152 135485 58867.40 195 154 136490 59973.75 197 155 138495 61090.60 199 157 139500 62217.85 201 158 141505 63355.00 203 160 142510 64502.85 205 162 143515 65660.60 207 163 145520 66828.75 209 165 146525 68007.40 211 166 148

 

 

 

 

 

 

 

845 164773.59 343 267 2351050 248855.84 427 332 291

850 166619.16 345 269 2361055 251122.41 429 334 292

855 168475.00 347 270 2381060 253398.75 431 335 294

860 170340.84 349 272 2391065 255685.59 433 337 295

865 172217.41 351 274 2401070 257982.84 435 338 297

870 174103.75 353 275 2421075 260290.00 437 340 298

875 176000.59 355 277 2431080 262607.84 439 342 299

880 177907.84 357 278 2451080 262607.84 439 342 299

885 179825.00 359 280 2461085 264935.59 441 343 301

890 181752.84 361 282 2471090 267273.75 443 345 302

895 183690.59 363 283 2491095 269622.41 445 346 304

900 185638.75 365 285 2501100 271980.84 447 348 305

905 187597.41 367 286 2521105 274350.00 449 350 306

910 189565.84 369 288 2531110 276729.16 451 351 308

915 191545.00 371 290 2541115 279118.59 453 353 309

915 191545.00 371 290 254

1115 279118.59 453 353 309

920 193534.16 373 291 256

1120 281518.41 456 354 310

925 195533.59 375 293 257

1125 283928.41 457 356 312

930 197543.41 378 294 258

1130 286348.59 460 357 313

935 199563.41 379 296 260

1135 288779.16 462 359 314

940 201593.59 382 297 261

1140 291220.00 464 360 316

945 203634.16 384 299 262

1145 293670.84 466 362 317

950 205685.00 386 300 264

1150 296132.41 468 364 318

955 207745.84 388 302 265

1155 298603.75 470 365 320

960 209817.41 390 304 266

1160 301085.59 472 367 321

965 211898.75 392 305 268

1165 303577.84 474 368 323

970 213990.59 394 307 269

1170 306080.00 476 370 324

975 216092.84 396 308 271

1175 308592.84 478 372 325

980 218205.00 398 310 272

1180 311115.59 480 373 327

985 220327.84 400 312 273

1185 313648.75 482 375 328