A%20Simple%20Unit%20Commitment%20Problem

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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.

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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 youre in.

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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 x2115)U1 (0.255 x2215)U2
• subject to
• W 5 x1U1 - x2U2
• 0 lt x1 lt 10
• 0 lt x2 lt 10
• U may be only 0 or 1

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Lagrange Relaxation (2)

• L (0.25 x2115)U1 (0.255 x2215)U2
  l(5 x1U1 - x2U2)
• Pick a value for l and keep it fixed
• Minimize for U1 and U2 separately
• 0 d/dx1(0.25x21 15 - x1l1)
• 0 d/dx2(0.255x22 15 - x2l1)

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Lagrange Relaxation (3)

• 0 d/dx1(0.25x21 15 - x1l1)
• if the value of x1 satisfying the above falls
  outside the 0 lt x1 lt 10, we force x1 to the
  limit.
• If the term in the brackets is gt 0, set U1 to 0,
  otherwise keep it 1
• 0 d/dx2(0.255x22 15 - x2l1)
• 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 l1 around
• dL/dl (5 x1U1 - x2U2)
• l2 l1 dL/dl (a)
• if dL/dl gt 0, a 0.2
• if dL/dl lt 0, a 0.005
• After we found l2, repeat the whole
  processstarting at step 1