a simple unit commitment problem valentín petrov, james nicolaisen 18 / oct / 1999 nsf meeting
TRANSCRIPT
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A SimpleUnit Commitment
Problem
Valentín Petrov, James Nicolaisen
18 / Oct / 1999
NSF meeting
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Economic Dispatch (Covered last time)
• With a given set of units running, how of the load much should be generated at each to cover the load and losses? This is the question of Economic dispatch.
• The solution is for the current state of the network and does not typically consider future time periods.
G
G
G
GG
G
G
G G
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Deciding which units to “commit”
• When should the generating units (G) controlled by the GENCO be run for most economic operation?– Concern must be given to environmental effects
• How does one define “economic operation”? Profit maximizing? Cost minimizing? Depends on the market you’re in.
G
G
G
GG
G
G
G G
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Problem Setup• Last meeting we discussed the economic
dispatch problem
• Now we will see how the unit commitment fits into the general picture
• Unit commitment is bound to the economic dispatch
• Use similar optimization methods
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What is Unit Commitment (1)
• We have a few generators (units)
• Also we have some forecasted load
• Besides the cost of running the units we have additional costs and constraints– start-up cost– shut-down cost– spinning reserve– ramp-up time... and more
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What is Unit Commitment (2)
• It turns out that we cannot just flip the switch of certain units on and use them!
• We need to think ahead, and based on the forecasted load and unit constraints, determine which units to turn on (commit) and which ones to keep down
• Minimize cost, cheap units play first
• Expensive ones run only when demand is high
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How Do We Solve the Problem• If a unit is on, we designate this with 1 and
respectively, the off unit is 0
• So, somehow we decide that for the next hour we will have "0 1 1 0 1" if we have five units
• Based on that, we solve the economic dispatch problem for unit 2, 3 and 5
• We start turning on U2, U3, U5
• When the next hour comes, we have them up and running
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To Come Up With Unit Commitment• The question is, _how_ do we come up with
this unit commitment "0 1 1 0 1" ?
• One very simplistic way: if we have very few units, go over all combinations from hour to hour
• For each combination at a given hour, solve the economic dispatch
• For each hour, pick the combination giving the lowest cost!
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Lagrange Relaxation (1)
• Min f = (0.25 x21+15)U1 + (0.255 x2
2+15)U2
• subject to:– W = 5 – x1U1 - x2U2
– 0 < x1 < 10
– 0 < x2 < 10
• U may be only 0 or 1
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Lagrange Relaxation (2)
• L = (0.25 x21+15)U1 + (0.255 x2
2+15)U2 + (5 – x1U1 - x2U2)
• Pick a value for and keep it fixed
• Minimize for U1 and U2 separately
• 0 = d/dx1(0.25x21 + 15 - x11)
• 0 = d/dx2(0.255x22 + 15 - x21)
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Lagrange Relaxation (3)
• 0 = d/dx1(0.25x21 + 15 - x11)
– if the value of x1 satisfying the above falls outside the 0 < x1 < 10, we force x1 to the limit.
– If the term in the brackets is > 0, set U1 to 0, otherwise keep it 1
• 0 = d/dx2(0.255x22 + 15 - x21)
– same as above
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Lagrange Relaxation (4)• Now assume the variables x1, x2, U1, U2 fixed
• Try to maximize L by moving around
• dL/d = (5 – x1U1 - x2U2)
• dL/d– if dL/d– if dL/d
• After we found 2, repeat the whole process
starting at step 1