A SimpleUnit Commitment

Problem

Valentín Petrov, James Nicolaisen

18 / Oct / 1999

NSF meeting

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

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

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

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

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

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

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!

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

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)

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

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