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System planning 2013

• Lecture L9: Short term planning of thermal systems

• Chapter 5.3.1-5.3.3• Contents:

– General about thermal power– Production cost– Start up costs– Minimum up and down times– Limited generation changes

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General about thermal power

• Generates power by combustion of fuel:– Oil/petroleum– Gas– Coal– Uranium

• Large thermal plants uses the steam cycle when generating power.

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General about thermal power

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What do we want?

• Want to optimize the operation of our thermal stations.

• Want to use linear equations.

• Want to model the characteristics of the thermal stations.

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Production cost

• Depends on the amount of fuel that is used

• The efficiency is not constant, varies with the amounts of fuel that is used. The fuel input:

)()(

Gh

GGF

F(G) = fuel input [ton/h or m3/h]G = power production [MWh/h]h = heat contents [MWh/ton or MWh/m3](G) = efficiency at production G [%]

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Production cost

• Let denote the fuel price.• Production cost:

)()()(

Gh

GGFGC

• Nonlinear cost function. Often approximated by:

2)( GGGC

• Here approximated by:

GGC )(

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Operation limitations

• Changes in fuel input do not immediately result in changes in power production (large plants)

Operation limitations:– Start up times and costs– Minimum operation time and stop time– Limited generation changes

• Need to consider binary variable to state the operation status for station g during hour t:

, {0,1}g tu

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Start up costs

• If the thermal power plant is in operation a specific hour, the generation is limited by the installed capacity. If not in operation, the upper limit is zero:

gtgtg GuG ,,

g

tg

tg

G

u

G

,

, = Generation, plant g, hour t

= Operation status, plant g, hour t

= Maximum production, plant g

• If in operation, there can also exist a lower generation limit (keep the plant going):

gtgtg GuG ,, gG = Minimum production, plant g

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Start up costs

• Starting up a thermal power plant cost fuel to heat up the plant to operational temperature. Dependent on how long time the plant has been shut off.

fixedt

startcoldupstart CeCtC )1()( /

fixed

startcold

upstart

C

C

tC

)( = cost for starting after t hour stop= cost for starting a cold plant

= cooling time constant

= fixed startup cost

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Start up costs

• If the plant only will be off-line for a short period of time, beneficial to not stop the combustion in the boiler. Reduce it to minimum and do not produce power (banking)

tCC bankingupstart

bankingC = cost per hour for banking

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Start up costs

• To include start up costs in the objective of an optimization problem, binary variables must be included.

• Should be 1 the hour the plant is started, and 0 otherwise.• The start up costs depends on the time the plant has been

switched off separate variables for the different times.• Here: Only look at off-line time one hour and two hours or

more:

**,

*,

tg

tg

s

s = start after one hour stop for plant g during hour t

= start after at least two hour stop for plant g during hour t

}1,0{, **,

*, tgtg ss

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Start up costs

• The total start up cost during the planning period:

Gg Tt

tggtgg sCsC **,

***,

*

**

*

g

g

C

C = start up cost for plant g after one hour stop

= start up cost for plant g after at least two hour stop

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Start up costs

• Since the start up will cause costs, the optmization will try to set the start up variables to zero. Thus, must add constriants to make sure they take the correct values.

**,1,,

*,

2,1,,**,

tgtgtgtg

tgtgtgtg

suus

uuus

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Start up costs

**,1,,

*,

2,1,,**,

tgtgtgtg

tgtgtgtg

suus

uuus

tgu ,

**,1,, tgtgtg suu

1*, tgs

2,1,, tgtgtg uuu 0**, tgs

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Start up costs

**,1,,

*,

2,1,,**,

tgtgtgtg

tgtgtgtg

suus

uuus

tgu ,

2,1,, tgtgtg uuu

1**, tgs

**,1,, tgtgtg suu

0*, tgs

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Minimal up and down times

• Constant start up costs often used.• Introduce requirements of the down times of a

decommited unit and up times for a commited unit.

tgs , = start up variable for plant g during hour t

(1 if the plant is started before t, 0 otherwise)

}1,0{, ,, tgtg ss

tgs , = stop variable for plant g during hour t

(1 if the plant is stopped before t, 0 otherwise)

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Minimal up and down times

• The total start and stop cost:

Gg Tt

tggtgg sCsC ,,

g

g

C

C = start up cost for plant g

= stop cost for plant g

• Must couple the operation status variables and the start up and stop variables:

tgtgtgtg ssuu ,,1,,

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Minimal up and down times

• The requirements for minimal up and down times can be formulated as:

1

1

1

,,

1

,,

g

g

tt

tkkgtg

tt

tkkgtg

ss

ss

g

g

t

t = minimal up time for plant g

= minimal down time for plant g

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Limited generation changes

• Cannot change production at any rate. Must include ramping constraints that limits the generation increase/decrease between hours. Introduce the maximum generation increase and decrease:

Gg

Gg ,

• Consider increase: The first hour after that the plant has been started, the increase can be larger than the maximum increase. Maximum generation the first hour after start:

1gG

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Limited generation changes

• Decrease ramping constraints:

, 1 , ,

G Gg ggg t g t g tG G s G

• Increase ramping constraints:

1

, , 1 ,

G Gg ggg t g t g tG G s G

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Limited generation changes

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Planning problem

• Short term thermal power planning problem:

maximize income during period -production costs during

period

regarding minimal up/down timesramp rateslegal agreementsphysical limitations

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PROBLEM 15 - Symbols for Short-term Thermal Power Planning Problems

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PROBLEM 15 - Symbols for Short-term Thermal Power Planning Problems

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PROBLEM 16 - Total Operation Cost

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PROBLEM 17 - Unit Commitment