economic 6

8/10/2019 Economic 6

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ECONOMIC OPERATION

OF POWER SYSTEMS

Faculty ofEngineering

Tanta University

Dr. Ahmed Mohamed AzmyDepartment of Electrical Power and Machine Engineering

Tanta University - Egypt


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Economic load dispatch

The unit commitment studyIt optimally defines the required generators tomeet the expected load and to provide aspecified margin of operating reserve over a

specified period of timeThe economic dispatchIt determines the output power of each plantthat would minimize the overall fuel cost. Thisrepresents a coordination process betweenthe unit productions in an economic manner.


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System Constraints

The technical limitations that have not to be violatedunder any conditionThe violation of these constraints affects the powerquality and the general operation of the powersystem and causesstabilityproblems

The constraints can be divided into two groups:equality constraintsandinequality constraintsInequality constraints can have a hard nature,where the variables are definite and specific like the

tap-changing transformer, or a soft nature, wherethe variables are smoothly varied within a specificrange like the nodal voltages


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System Constraints

Constraints

Equality constraints Inequality constraints

hard nature soft nature

The variables are

definite and specificlike the tap-changing

transformer

The variables are

smoothly varied withina specific range like

the nodal voltages


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System Constraints

Equality ConstraintsThe main equality constraints are the basic load flowequations that establish the flow balance equationsFor example, the equality constrains according to

Newton-Raphson Method are

)--sin(yVVQ

)--cos(yVVP

ijji

n

1jijjii

ijji

n

1jijjii


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System ConstraintsInequality Constraints

Generator constraints

The thermal stability of generators requires that thetotal VA "Sg" loading of any generator has not to

exceed a certain maximum value Sg-max:

max,g2g

2gg SQPS

where: Pgand Qgare the active and reactivegenerated power respectively


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Generator constraints

The upper limit of the active power Pmax isconstrained by the thermal consideration

The lower limit Pmin is constrained by the flameinstability of the boiler

Consequently, the generated power from any unit"Pg" has to be kept within the limits:

maxgmin PPP


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Voltage constraints

Both the magnitudes and angles of node voltageshave to be controlled in order to keep them within

acceptable limitsThe power quality necessitates that the voltagemagnitudes at load terminals are kept withinspecific limits or else the equipments will notoperate satisfactorily

The regulation of the voltage starts from thegenerators (exciters) to reduce the cost of extravoltage regulating devices


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Voltage constraints

The upper limit of phase angles is definedregarding the transient stability of power systems

On the other hand, the lower limit of the angles isdefined taking into account achieving an efficientutilization of transmission facility

Typical operating angle of transmission line liesbetween 30 - 45


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Voltage constraints

where: is the magnitude of node angle with anangle ofdnat node n

VVV

maxnmin

maxnmin


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Running spare capacity constraints

To ensure the existence of enough spinning reserveto overcome any emergency situation

The generation should guarantee a minimum sparecapacity in addition to load demand and power losses

Pg> PLoad+ Ploss

This difference, i.e. spare capacity, is definedaccording to economic issues and technical aspectslike the ramping rates of the generators


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Network security constraints

Violation of constrains can take place subsequent toabnormal conditions like a line outage, eitherscheduled or forced, which affects the security of thenetwork

Sometimes a so called (x-1 study) is performed toexamine the reliability and security of the system

Thex-1studymeans that the network is studied withoutage of one branch at a time


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Transformer tap settings

For auto-transformers, tmaxcan be unity

Sometimes, there is a possibility to change thevoltage in steps, i.e. to chose between different

values rather than varying the voltage smoothly

maxtt0


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Transformer tap settings

Phase shift-ing transformers have phase shift limits

maxmin


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Economic Dispatch Problem

Economic load dispatch concerns with theoperatingcostrather than the fixed costOnly fuel cost is considered in the study where allother costs that are depend on the generated powerwill be included in the expression of the fuel costObviously, the cost of fuel is concerned since thermalplants are assumed

An early approach of power dispatch was to supply

power from the most efficient plant till the point ofmaximum efficiency and then from the next mostefficient plant and so on


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Another approach was to load the machines so thatall units have the same incremental cost ofproductionHowever, the locations of power plants and thetrans-mission losses are not consideredGenerating power in far plants requires supplying thetransmission losses in addition to the load demandEconomic distribution of load demand amonggenerating units necessitates expressing thecostas

a function of thegenerated powerThe performance curve ofboiler-turbine-generatorset is required, which is called input-output curve



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input-output curve

Input fuel (Btu/h)

Output power (MW)

PmaxPmin


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The incremental fuel rate is given as

Incremental fuel rate =(P)d

(F)d

(output)

(input)

With Fis the input fuel and Pis the output powerThe reciprocal of the incremental heat rate is knownasthe incremental efficiencyTo operate at the maximum fuel efficiency, the point

of the minimum heat rate has to be definedMultiplying the incremental fuel rate by the fuel cost,the incremental fuel cost is obtained in($/MWh)

(Btu/MWh)



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The fuel is given as:

F=(40+4*P + 0.012*P2)*106

Also, the heat rate can be calculated from equation

(MW)poweroutput

(Btu/h)fuelinputrateHeat

The fuel cost as a function of the output power is:C=0.12*10-6 *F= 4.8 + 0.48*P + 0.0014*P2 ($/h)

The incremental fuel cost is:

Ifr= 0.48 + 0.0028*P ($/MWh)


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P (MW) 10 20 30 70 80 90 100

F*10-6 (Btu/h) 81.2 124.8 170.8 378.8 436.8 497.2 560

Heat rate *10-6

(Btu/MWh)8.12 6.24 5.69 5.41 5.46 5.52 5.6

IFR ($/MWh) 0.5 0.53 0.56 0.67 0.7 0.73 0.76


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10 40 70 1000

2

4

6 Input fuel (Btu/h) * 108

Output power (MW)

Input output curve


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Heat rate curve

10 40 70 1005

6

7

8

Output power (MW)

Heat rate(Btu/MWh) * 105


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Incremental fuel cost

10 40 70 100

0.5

0.6

0.7

0.8 Incremental fuel cost ($/MWh)

Output power (MW)


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Operating cost of thermal plants

According to the input-output curve of thermalpower plant, the cost function is simplified in aquadrate form:

C = a+ b*P + *P2

This relation gives a good approximation to theactual variation of the cost with the variation of

the output power


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Economic Dispatch Neglecting Losses

It is assumed that allgenerators are connectedat the same bus withoutlosses in transmission linesThe economic dispatch

problem neglecting alllosses in the transmissionsystem is defined as:

1

2

n

.

.

.

F1

F2

Fn

P1

P2

Pn

PD

Min FT=

n

1iiF Subject to

imax,iimin,

1D

PPP

P

n

i

iP


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Min FT=

n

1iiF Subject to

imax,iimin,

1D

PPP

P

n

i

iP

With FTis the total input fuel to the generators, Fiis

the input fuel to ith unit, PDis the total load demand,Piis the output power of the i

th unit and Pmin,iandPmax,iare the lower and upper limits of unit i

For simplicity, assume only two units connected atthe same bus

PD= P1+ P2 FT= F1+ F2


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For minimum total fuel consumption and selectingthe power of the first unit as a variable

PD

= P1

+ P2

FT

= F1

+ F2

000 1P2P

2dP2dF

1dP1dF

1P2F

1dP1dF

1PTF

Differentiating the load demand equation withrespect to the power of generating 1 we get:

1011P2P

1P2P

PD= P1+ P2


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012

2

2

1

1

P

P

dP

dF

dP

dF

101 1P2P

1P2P

ndPndF

2dP2dF

1dP1dF ... Condition of

optimal operation

iPdiFdThe term represents the incremental production

cost of the ith plant in $/MWh

All generators have to operate at the sameincremental cost of production to achieve aneconomic operation

where: is the Lagrange multiplier


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n

nDP

2...22 22

11

b

b

b

n

n

2

2

1

1

n21

D

2222

1

2

1

2

1P ......

b

n

1i i

n

1i i

iD

2

1

2P

economic 6

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