economic dispatch power system

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Economic Operation of Power

Systems

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Economic Operat ion of Power Systems

Economic operation is very important for a powersystem to return a profit on the capital invested. Rates fixed by regularity bodies and theimportance of conversation fuel place pressure

on power companies to achieve maximumpossible efficiency. Maximum efficiency minimizes the cost of KWhr to the consumer and the cost of the company to

deliver that KWhr in the face of constantly riseprices for fuel, labor, supplies and maintenance.

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Characterization of Power Generation Units: ! The following figure shows a typical fossil-fuel

generating unit. (Thermal Unit)

Fuel Input

Boiler Steam Turbine

Generator

To Transmission

Lines

Cooling Tower

Pm Pe

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The unit consists of a single boiler thatgenerates steam to the turbine . Fuel isburned and its chemical energy is convertedinto heat. The heat is used to convert the

water into steam that enters that turbine toprovide a mechanical power on the shaft of synchronous generator. Finally, the mechanicalenergy is converted into electrical energy

through the synchronous generator.

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Generation Characteristics: ! Thermal Efficiency:

How much fuel is needed to produce 1 MWhr ofelectrical energy.

! Load Factor: (known as Capacity Factor) It is the ratio of annual energy produced to(8760*Electrical capacity of plant)

nnual Energy Produced8760 Unit Capacity LF =

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! Maintenance Requirements: For how long out of service for maintenance?

! Reliability: Probability of power system to fulfill its function for agiven period of time.

! Capital Cost, Fuel Cost, Operation andMaintenance Cost.

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Operating Cost of a Fossil-Fuel Generating UnitOperat ing Cost (C i )

The operating cost is always function of real poweroutput from the unit ( Pi ).

The operating cost can be controlled by operating

strategy enter into the Economic Dispatchformulation.

Variable Operat ing Costs Fuel Cost Maintenance Cost

Fixed Costs Capital cost of installing

the generation unit.

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The following figure shows the typical operating costci of a fossil-fuel unit versus real output power Pi .

!"# re 18


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In practice, ci is constructed of piecewise continuous

functions. The discontinuities may be due the firingof equipment such as additional boilers orcondensers as power output is increased.

It is convenient to express ci in terms of KJ/hr orBTU/hr, which is relatively constant over the lifetimeof unit, rather than $/hr which can change monthlyor daily.

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Note: ! ci can be converted to $/hr by multiplying the fuel

input KJ/hr by the cost of fuel in $/KJ . ! BTU (British Thermal Unit): it is a unit of energy

equal to about 1055 J1BTU = 1.055 KJ

1KWhr = 3412 BTU

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Unit incremental operating cost: ! It is defined as the derivatives (slope) of the unit

operating cost C i versus the unit output Pi .

! When C i consists only of fuel costs, dC i /dP i is calledThe Heat Rate . (BTU/KWhr)

! The reciprocal of the heat rate, which is the ratio

of output energy to input energy, gives a measureof fuel efficiency for the unit.

Unit Incremental Cost = ii

dC

dP

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!"# re 212


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From the previous figure, maximum efficiency occurs atPi=600 MW.

Heat rate = C/P = ( 5.4*10^9 )/(600*10^3)= 9000 KJ/ Kwhr

= (9000 KJ/ Kw hr ) / (1.055 KJ/ BTU )= 8531 BTU/ KWhr

The efficiency at this output power;

= (1/ 9000) ( KWhr/ KJ )*(3413) ( BTU/ KWhr ) *(1.055) ( KJ/ BTU )= 40%

r $% !"# re 1

&s ' n !"# re 2

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ECONOMIC DISPATCH: ! The economic dispatch problem is to select the real outputpower of each controlled generating unit in an area tomeet a given load in such a way that the total operatingcosts in the area are minimized.

Ci : Unit Operating Cost.PLoad: Load Power.

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P1+P2++PN=PLoad S.T. minimizing C T =C 1++C N

Operating cost ($/hr) includes fuel cost ,Maintenance , capital cost of installing thegenerations .

Unit Incremental Operating Cost: ! It is defined as the derivative (slope) of unit operating

cost with respect to the unit output.

$/MWhri

i

dCUnit Incremental Cost =

dP

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Lets consider an area with N units operatingon economic dispatch as in figure 3;

! The total operating cost of these units is:

$/hr

! The total Load demand PT in the area is:

( ) ( ) ( )1 1 2 21 ... N

T i N N iC C C P C P C P == = + + +

1 2 1...

N

T N ii P P P P P == + + + =

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Mathematically the economic dispatch problem is:

The solution of economic dispatch problem can bebased on using Lagrange Funct ion .

( )1 1min subject to N N

i i i T i iC P P P = = =

function constraint

( ) ( ( ) 0) f x g x = Lagrange M ultiplier

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( )1 1 N N

i i T i iC P P = == 1 1

1 1 1

2 2

2 2 2

0

0

.

.

.

0 N N N N N

dC dC d dP dP dP

dC dC d dP dP dP

dC dC d dP dP dP

= = =

= = =

= = =18


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All units should have the same incrementaloperating cost. (Criterion for the solution ofeconomic dispatch)

1 2

1 2

= ...= N N

dC dC dC dP dP dP

= =

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Example 11.4:

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Effect of inequality Constraint If the inequality constraints are included, theeconomic dispatch should be modified. If theone or more units reach their limited value,

then these units are held constant at theirlimits and the remaining units operate atequal incremental operating cost.

The incremental operating cost of the area is !for the units that are not at their limits.

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Example 11.5:

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Effect of Transmission Losses on Economic

Dispatch Although one unit may be very efficient with alow incremental operating cost, it may also belocated far from the load centre.

In general, using generators closer to the loadresults in lower losses.

The transmission losses associated with this unitmay be so high that the economic dispatchsolution require the unit to decrease its outputwhile other units with higher incrementaloperating costs but low transmission lossesincreases their output.

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When the transmission losses are included,the constraint function becomes:

To solve the economic dispatch problem, wecan use a Lagrange Function :

1

N

i T Li P P = = +

Total Load Demand

Tot al Tr.Losses

( )1 1 N N i i T i i LC P P = == 29


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1 1

1 1 1 1

1

2 2

2 2 2 2

2

11 0

1

11 0

1

.

.

.

11 01

L

L

L

L

N N L

L N N N N

N

dC P dC d

P dP dP P dP P

dC P dC d P dP dP P dP

P

dC dC P d P dP dP P dP P

= = = = = =

= = =

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The economic dispatch problem (including thetransmission losses) is:

All of units must have the same incremental

operating cost dC i / dP i multiplied by thepenalty factor Li .

1 i=1,2,...,N

1

i i

Li ii

i

LdC dC

P dP dP P

= =

Penalty Factor

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Example 11.7:

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The penalty factor

at the slack bus isalways unity!

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Unit Commitment: !

When should each unit be started, stopped and how muchshould it generate to meet the load of minimum cost?? ! Economic dispatch is not concerned with determining

which units to turn on/off (this is the unit commitmentproblem).

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Optimal Power Flow: ! The solution of optimizing the generation while

enforcing Transmission Lines is to combine theeconomic dispatch with power flow.

! The result is known as the Optimal Power Flow(OPF).

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economic dispatch power system

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