04 unit commitment

8/8/2019 04 Unit Commitment

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Unit Commitment

l Given load profile(e.g. values of the load for each hour of a day)

l Given set of units availablel When should each unit be started, stopped and how

much should it generate to meet the load at minimumcost?

G G GLoad Profile

? ? ?

TypicalSummer AndWinter Demands


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Why try to optimise?

l Energy traded through the Electricity Pool of England andWales: ~ 7 billion per year

l 0.1% cost reduction through better scheduling:~ 7 million

A Simple Example

l Unit 1: P Min = 250 MW, P Max = 600 MW C 1 = 510.0 + 7.9 P 1 + 0.00172 P 12 /h



l What combination of units 1, 2 and 3 will produce 550MW at minimum cost?

l How much should each unit in that combinationgenerate?


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Cost of the various combinations

1 2 3 P min P max P 1 P 2 P 3 C total Off Off Off 0 0 Infeasible

Off Off On 150 500 Infeasible

Off On Off 200 400 Infeasible

Off On On 350 900 0 400 150 5418

On Off Off 250 600 550 0 0 5389

On Off On 400 1100 400 0 150 5613On On Off 450 1000 295 255 0 5471

On On On 600 1500 Infeasible 5617

l Far too few units committed:Cant meet the demand

l Not enough units committed:Some units operate above optimum

l Too many units committed:Some units below optimum

l Far too many units committed:Minimum generation exceeds demand

l No-load cost affects choice of optimal combination

Observations on the example:


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Another Example

Load

Time

12 18 24

500

1000

l Optimal generationschedule for a loadprofile

l Decompose the profileinto a set of period

l Assume load is constantover each period

l For each time period,

which units should becommitted to generate atminimum cost during thatperiod?

Optimal combination for each hour

Load Unit 1 Unit 2 Unit 3

1100 On On On

1000 On On Off

900 On On Off

800 On On Off700 On On Off

600 On Off Off

500 On Off Off


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Matching the combinations to the load

Load

Time1260 18 24

Unit 1

Unit 2

Unit 3

Issues

l Must consider all constraintsn Unit constraintsn System constraints

l Some constraints create a link between the periodsl Starting up a generating unit costs money in addition to

the running cost considered in economic dispatchl Curse of dimensionality


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Unit Constraints

l Constraints that affect each unitindividually:

n Maximum generating capacityn Minimum stable generationn Flexibilityn Minimum up timen Minimum down time

n Ramp rate

Flexible Plants

l Power output can be adjusted (within limits)l Examples:

n Coal-firedn Oil-firedn Open cycle gas turbinesn Combined cycle gas turbinesn Hydro plants with storage

l Status and power output can be optimised

Thermal units


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Inflexible Plants

l Power output cannot be adjusted for technical or commercial reasons

l Examples:n Nuclear n Run-of-the-river hydron Renewables (wind, solar,)n

Combined heat and power (CHP, cogeneration)l Output treated as given when optimising

Notations

X i ( t ) : Status of unit i at period t

X i ( t ) = 1: Unit i is on during period t

X i ( t ) = 0 : Unit i is off during period t

Pi ( t ) : Power produced by unit i during period t


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Unit Constraints

l Minimum up timen Once a unit is running it may not be shut down

immediately:

l Minimum down timen Once a unit is shut down, it may not be started

immediately

If X i (t ) = 1 and t iup < t i

up ,min then X i (t + 1) = 1

If X i (t ) = 0 and t idown

< t idown ,min then X i (t + 1) = 0

Unit Constraints

l Maximum ramp ratesn To avoid damaging the turbine, the electrical output of a unit

cannot change by more than a certain amount over a periodof time:

Pi (t + 1) Pi (t ) Piup ,max

P i (t ) P i (t + 1) P idown ,max

Maximum ramp up rate constraint:

Maximum ramp down rate constraint:


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System Constraints

l Constraints that affect more than one unitn Load/generation balancen Reserve generation capacityn Crew constraintsn Emission constraintsn Network constraints

System Constraints: Load/generation balance

P i (t )iC ( t ) = L (t )

C (t ) = i X i (t ) = 1{ }: Set of units committed at time t


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System Constraint: Reserve Capacity

l Unanticipated loss of a generating unit or aninterconnection causes unacceptable frequency drop if not corrected

l Need to increase production from other units to keepfrequency drop within acceptable limits

l Rapid increase in production only possible if committedunits are not all operating at their maximum capacity

P imax

iC ( t ) L (t )+ R (t )

R (t ): Reserve requirement at time t

How much reserve?

l Protect the system against credible outagesl Deterministic criteria:

n Capacity of largest unit or interconnectionn Percentage of peak load

l Probabilistic criteria:n Takes into account the number and size of the

committed units as well as their outage rate


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Types of Reservel Spinning reserve

n Primary quick response for a short time

n Secondary slower response for a longer time

n High frequency ability to reduce output when frequency is high

l Scheduled or off-line reserven Unit that can start quickly (e.g. gas turbines)

l Other sources of reserven Pumped hydro plantsn Demand reduction

l Reserve must be spread around the network

Cost of Reserve

l Reserve has a cost even when it is not calledn More units scheduled than required

Units not operated at their maximum efficiency Extra start up costs

n Must build units capable of rapid responsen Cost of reserve proportionally larger in small systems

Important driver for the creation of interconnections between systems


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Crew Constraints

l It may not be possible to start more than one generatingunit at a time in a power station because of the number of people required to supervise the start-up

l Less of a problem than it use to be thanks to automation

Emission Constraints

l Amount of pollutants that generating units can emit maybe limited

l Pollutants:n SO 2, NO x

l Various forms:n Limit on each plant at each hour n Limit on plant over a year n Limit on a group of plants over a period


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Network Constraints

l Transmission network may have an effect on thecommitment of units

n Some units must run to provide voltage supportn The output of some units may be limited because their

output would exceed the transmission capacity of thenetwork

Cheap generatorsMay be constrained off

More expensive generator May be constrained on

A B

Start-up Costs

l Thermal units must be warmed up before they can bebrought on-line

l Warming up a unit costs moneyl Start-up cost depends on time unit has been off

SC i (t iOFF ) = i + i (1 e

t i

OFF

i )

t iOFF

i

i + i


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Solving the Unit Commitment Problem

l Decision variables:n Status of each unit at each period:

n Output of each unit at each period:

l Combination of integer and continuous variables

X i (t ) 0,1{ } i , t

Pi ( t )

X i (t )

Pi (t ) 0, P i

min ; P imax{ } i , t

Optimisation with integer variables

l Continuous variablesn Can follow the gradientsn Any value within the feasible set is OK

l Discrete variablesn There is no gradientn Can only take a finite number of valuesn Must try combinations of discrete values


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How many combinations are there?

111

110

101

100

011

010

001

000

l Examplesn 3 units: 8 possible statesn N units: 2 N possible states

How many solutions are there anyway?

1 2 3 4 5 6T=

l Optimisation over a timehorizon divided into intervals

l A solution is a path linking onecombination at each interval

l How many such path arethere?


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How many solutions are there anyway?

1 2 3 4 5 6T=

l Optimisation over a timehorizon divided into intervals

l A solution is a path linking onecombination at each interval

l How many such path arethere?

l Answer:

2 N ( )2 N ( )K 2 N ( )= 2 N ( )T

The Curse of Dimensionality

l Example: 5 units, 24 hours

l Processing 10 9 combinations/second, this would take 1.9

10 19 years to solvel There are over 100 units in England and Wales...l Many of these combinations do not satisfy the constraints

2 N ( )T

= 2 5( )24

= 6.2 10 35 combinations


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How do you Beat the Curse?

Brute force approach wont work!

l Need to be smartl Try only a small subset of all combinationsl Cant guarantee optimality of the solutionl Try to get as close as possible within a reasonable

amount of time

Main Solution Techniques

l Priority list / heuristic approachl Dynamic programmingl Lagrangian relaxationl Mixed Integer Programming

l Characteristics of a good techniquen Solution close to the optimumn Low computing timen Ability to model constraints

04 unit commitment

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