Power Market Economics and Security

Problem 1: The generator data are given in Table 1 and the load and reserve data are given in
Table 2. The fuel consumption functions of the generating units are quadratic H(P) = af + bf * P

• cf * P2
  (MBtu). The fuel prices are all 1 $/MBtu. The unit shutdown costs and the system
  losses are assumed to be zero. Unit 3 has a fuel contract of 2000 MBtu. The initial Lagrangian
  multipliers for power balance and reserve requirements are given in Table 3. The initial
  multiplier  for Unit 3’s fuel constraint is zero. The adjustment steps of multipliers are given in
  Table 4. Set Γ=10,000 if the ED is infeasible based on a given commitment. Use the LR method
  to solve the UC problem. Obtain two different feasible solutions and show the corresponding
  relative duality gaps.
  Table 1: Generator data
  Unit af
  (MBtu)
  bf
  (MBtu/MW)
  cf
  (MBtu/MW2
  )
  Pmin
  (MW)
  Pmax
  (MW)
  Min
  ON
  (h)
  Min
  OFF
  (h)
  Startup
  Cost ($)
  Initial
  Status
  (h)/(MW)
  1 2000 62.3 0.06 100 400 2 2 2000 ON 4 /300
  2 2100 64 0.07 80 400 2 1 1500 ON 4 /200
  3 1900 59 0.05 40 200 1 2 0 OFF 4 /0
  Table 2: Load and reserve data
  Hour Load (MW) Reserve (MW)
  1 500 50
  2 700 70
  3 800 80
  4 400 40
  Table 3: Initial  and 
  Hour  (Power balance)  (Reserve requirements)
  1 75 0
  2 100 0
  3 110 0
  4 70 0
  Table 4: Adjustment steps of Lagrangian multipliers
  Lagrangian Multiplier k1 k2
   (Power balance) 0.02 0.01
   (Reserve requirements) 0.02 0.005
   (Fuel constraint for unit 3) 0.005 0.001

Sample Solution