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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
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