A Simple Unit Commitment Problem

1. A SimpleUnit Commitment Problem Valentín Petrov, James Nicolaisen 18 / Oct / 1999 NSF meeting

2. G G G G G G G G G 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.

3. G G G G G 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.

4. 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

5. 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

6. 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

7. 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

8. 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!

9. Lagrange Relaxation (1) • Min f = (0.25 x21+15)U1 + (0.255 x22+15)U2 • subject to: • W = 5 – x1U1 - x2U2 • 0 < x1 < 10 • 0 < x2 < 10 • U may be only 0 or 1

10. Lagrange Relaxation (2) • L = (0.25 x21+15)U1 + (0.255 x22+15)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)

11. Lagrange Relaxation (3) • 0 = d/dx1(0.25x21 + 15 - x1l1) • 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 - x2l1) • same as above

12. 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 > 0, a = 0.2 • if dL/dl < 0, a = 0.005 • After we found l2, repeat the whole processstarting at step 1