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