A. M. Elaiwa, X. Xiaa and A. M. Shehatab

aDepartment of Electrical, Electronic and Computer Engineering, University of Pretoria, South Africa.

bDepartment of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt.

The problem of allocating the customers' load demands among the available thermal power generating units in an economic, secure and reliable way has received considerable attention since 1920 or even earlier

Static Economic Dispatch (SED)

n

iii PCC

1

)(min

n

ii DP

1

,

(i) Load-generation balance

.,...,2,1,maxmin niPPP iii (ii) Generation capacity

Significant cost savingsA Small improvement in the SED

Drawbacks • It may fail to deal with the large variations of the load demand due to the ramp rate limits of the generators• It does not have the look-ahead capability

Dynamic Economic Dispatch (DED)

N

t

n

i

tii PCPC

1 1

)()(min

,,...,2,1,1

n

i

ttti NtLossDP

Subject to

.,...,2,1,,...,2,1,maxmin niNtPPP itii

,,...,2,1,1,...,2,1,.. 1 niNtTURPPTDR iti

tii

(i) Load-generation balance

(ii) Generation capacity

(iii) Ramp rate limits

P1 P2 PN

0 T 2T NT

OptimalDynamicDispatch

Constraints

Objective function

Method of Solution

Minimize Cost

Minimize Emission

Maximize Profit

Math. Programming

AI Techniques

Hybrid Metheds

Equality

Inequality

Dynamic

,)()( 2tii

tiii

tii PcPbaPC

,))(sin()()( min2 tiiii

tii

tiii

tii PPfePcPbaPC

Smooth

Non smooth

The cost function

Periodic Implementation of DED

Technical Deficiencies

If the solutions are implemented repeatedly and periodically due to the cyclic consumption behavior and seasonal changes of the demand.

,,...,2,1,.. 1 niTURPPTDR iNiii

N

t

n

i

tii PCC

1 1P

)()P(min

,,...,2,1,1

n

i

tti NtDP

Subject to

.,...,2,1,,...,2,1,maxmin niNtPPP itii

,,...,2,1,1,...,2,1,.. 1 niNtTURPPTDR iti

tii

(i) Load-generation balance

(ii) Generation capacity

(iii) Ramp rate limits

Problem DED-(P)

},...,2,1,,...,2,1,{P NjniP ji

,...,2,1,1...,2,1,1 niNtTuPP ti

ti

ti

.,...,2,1

1

1 NtTuPPt

j

jii

ti

DED in control system framework

,)(),(min1 1

1

0

11

},{ 1

N

t

n

i

t

j

jiii

UPTuPCUPC

(i)

(ii)

(iii)

,1,...,2,1,)(1

1

1

1

n

i

tt

j

jii NtDTuP

.,...,2,1,,...,2,1,max1

1

1min niNtPTuPP i

t

j

jiii

,,...,2,1,1,...,2,1, niNtURuDR itii

.,...,2,1,,1

1

niURTuDR i

N

j

jii

Problem DED-(P1,U)

}1,...,2,1,,...,2,1,{ and ),...,,(Let 112

11

1 NjniuUPPPP jin

The solution of DED is an open-loop

Modeling uncertainties External disturbancesUnexpected reaction of some of the power system components

A closed-loop solution is needed

Model predictive control method

The idea of MPC

)(),...,1(),0(

))(),((),,(min

)0()),(),(()1(

****

00

0

NuuuU

kukxlNUxJ

xxkukxfkxN

kU

MPC Algorithm

}1,...,2,1,,...,2,1,{U Njniu mjim

),...,,( 112

11

1nPPPP Input the initial status

(1) Compute the open-loop optimal solution of DED-(P1,U)

(2) The (closed-loop) MPC controller

},...,2,1,{ 1 niu mi

is applied to the system in the sampling interval [m+1, m+2) to obtain the closed loop MPC solution

112 mi

mi

mi TuPP

over the period [m+1, m+2)

and let m=0

(3) Let m:=m+1 and go to step (1)

Model Predictive Control Approach to DED

Theorem 1.

Suppose that problem DED-(P1,U) is solvable, P* is the globally optimal solution

of the DED-(P) problem, then MPC Algorithm converges to P* if 1

1

1 DPn

ii

Convergence

Theorem 2.

1112 mi

mi

mi

mi TwTuPP

Robustness

Suppose that 1- problem DED-(P1,U) is solvable, 2- P* is the globally optimal solution of DED-(P) 3- 4- the following

,LC

,ewki

is executed in step (2) of MPC Algorithm

5- the disturbance is bounded

Then MPC Algorithm converges to the set

}:{ * PPP

DED with emission limitations

The emission of gaseous pollutants from fossil-fueled thermal generator plants including 22 ,,, COandCONOSO x

,

• Installation of pollutant cleaning • Switching to low emission fuels • Replacement of the aged fuel burners with cleaner ones; • Emission/economic dispatch

(I) Emission Constrained Dynamic Economic Dispatch

(II) Dynamic Economic Emission Dispatch

Dynamic Economic Emission Dispatch (DEED)

N

t

N

t

n

i

tii

n

i

tii PEPCEC

1 1 11

)()1()(),(min

,)()( 2tii

tiii

tii PcPbaPC

,,...,2,1,1

n

i

ttti NtLossDP

.,...,2,1,,...,2,1,maxmin niNtPPP itii

,,...,2,1,1,...,2,1,.. 1 niNtTURPPTDR iti

tii

(i)

(ii)

(iii)

2)()( tii

tiii

tii PPPE

n

i

tj

n

jij

ti

t PBPLoss1 1

,,...,2,1,.. 1 niTURPPTDR iNiii

Simulation Results

Ten units system

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