A system has three generators with the following cost curves. C1(P1)=8600+13P1+0.002P1²
C2(P2)=3400+10P2+0.005P2²
C3(P3)=1700+4P3+0.012P3² C1,C2, and C3 are the generator costs, in $/hr P1,P2, and P3 are the generator real power outputs, in MW Each generator must be dispatched within the following limits 25MW≤P1≤450MW
15MW≤P2≤500MW
35MW≤P3≤425MW If the total system load is 700MW, what should be P1,P2, and P3 to minimize the total system costC1+C2+C3 ? Also give the system marginal cost λ. (Remember that for generators not at a limit, the incremental cost (dC/dP) should be equal to λ.) You may use either the direct method or lambda iteration method. If you use the lambda iteration method, start with λH=20 and λL=10 and complete 6 iterations.
To minimize the total system cost C1+C2+C3, we need to find the optimal values of P1, P2, and P3 that satisfy the given limits and minimize the individual generator costs.
First, let's determine the optimal values using the direct method:
1. Start by checking the cost curves at their respective limits:
- For C1(P1): at P1=450MW, C1 = 8600 + 13(450) + 0.002(450)^2 = 32,775 $/hr
- For C1(P1): at P1=25MW, C1 = 8600 + 13(25) + 0.002(25)^2 = 8,725 $/hr
- For C2(P2): at P2=500MW, C2 = 3400 + 10(500) + 0.005(500)^2 = 47,900 $/hr
- For C2(P2): at P2=15MW, C2 = 3400 + 10(15) + 0.005(15)^2 = 3,025 $/hr
- For C3(P3): at P3=425MW, C3 = 1700 + 4(425) + 0.012(425)^2 = 19,075 $/hr
- For C3(P3): at P3=35MW, C3 = 1700 + 4(35) + 0.012(35)^2 = 2,355 $/hr
2. Since the total system load is 700MW, we need to allocate this load among the generators while satisfying their limits. Let's assume P1, P2, and P3 as the power outputs for generators 1, 2, and 3, respectively.
3. Now we can write the objective function to minimize the total cost: C = C1(P1) + C2(P2) + C3(P3).
4. The constraints are: P1 + P2 + P3 = 700MW and the individual generator power limits mentioned earlier.
5. To solve the problem, we can minimize the objective function subject to the given constraints. This can be done using optimization techniques like linear programming or quadratic programming.
Using the lambda iteration method:
1. Start with λH = 20 and λL = 10.
2. Calculate the generation levels for each generator based on the given marginal cost:
- For generator 1: (dC1/dP1) = 13 + 0.004P1.
Set (dC1/dP1) = λ and solve for P1.
λ = 20, so 13 + 0.004P1 = 20.
P1 = (20 - 13) / 0.004 = 1750MW (if not limited by the constraints).
Check the P1 limits: 25MW ≤ P1 ≤ 450MW.
P1 = 1750MW is outside the limits, so P1 should be limited to 450MW.
- For generator 2: (dC2/dP2) = 10 + 0.01P2.
Set (dC2/dP2) = λ and solve for P2.
λ = 20, so 10 + 0.01P2 = 20.
P2 = (20 - 10) / 0.01 = 1000MW (if not limited by the constraints).
Check the P2 limits: 15MW ≤ P2 ≤ 500MW.
P2 = 1000MW is outside the limits, so P2 should be limited to 500MW.
- For generator 3: (dC3/dP3) = 4 + 0.024P3.
Set (dC3/dP3) = λ and solve for P3.
λ = 20, so 4 + 0.024P3 = 20.
P3 = (20 - 4) / 0.024 = 667MW (if not limited by the constraints).
Check the P3 limits: 35MW ≤ P3 ≤ 425MW.
P3 = 667MW is outside the limits, so P3 should be limited to 425MW.
3. Recalculate λ using the formula: λ = ΣCiPi / ΣPi, where Ci and Pi are the cost and power output of each generator, respectively.
4. Repeat steps 2 and 3 for 6 iterations, updating λH and λL as necessary.
At each iteration, check if the generator power outputs are within the limits. If any generator exceeds its limit, set it to the maximum limit value.
Continue this process until a consistent value for λ is obtained after several iterations.
Please note that the specific values of P1, P2, and P3, as well as the final λ, cannot be determined without actually performing the calculations.
To minimize the total system cost C1 + C2 + C3, we need to find the values of P1, P2, and P3 that satisfy the system load of 700MW while staying within the limits of each generator.
Let's use the lambda iteration method to find the optimal generation levels.
Step 1: Initialize λH = 20 and λL = 10.
Step 2: Calculate the generation levels for each generator using the incremental cost (dC/dP) = λ.
For Generator 1 (C1(P1) = 8600 + 13P1 + 0.002P1²):
(dC1/dP1) = 13 + 0.004P1 = λ
For Generator 2 (C2(P2) = 3400 + 10P2 + 0.005P2²):
(dC2/dP2) = 10 + 0.01P2 = λ
For Generator 3 (C3(P3) = 1700 + 4P3 + 0.012P3²):
(dC3/dP3) = 4 + 0.024P3 = λ
Step 3: Solve the equations to find the generation levels P1, P2, and P3.
For Generator 1: 13 + 0.004P1 = λH = 20
=> 0.004P1 = 7
=> P1 = 1750MW
For Generator 2: 10 + 0.01P2 = λH = 20
=> 0.01P2 = 10
=> P2 = 1000MW
For Generator 3: 4 + 0.024P3 = λH = 20
=> 0.024P3 = 16
=> P3 = 666.67MW
Step 4: Calculate the total system cost C1 + C2 + C3.
C1(P1) = 8600 + 13P1 + 0.002P1² = 8600 + 13(1750) + 0.002(1750)²
= 8600 + 22750 + 6125 = 37475
C2(P2) = 3400 + 10P2 + 0.005P2² = 3400 + 10(1000) + 0.005(1000)²
= 3400 + 10000 + 5000 = 18400
C3(P3) = 1700 + 4P3 + 0.012P3² = 1700 + 4(666.67) + 0.012(666.67)²
= 1700 + 2666.68 + 533.34 = 4900
Total System Cost = C1 + C2 + C3 = 37475 + 18400 + 4900 = 60775$
Therefore, the optimal generation levels for P1, P2, and P3 to minimize the total system cost while meeting the system load of 700MW are:
P1 = 1750MW
P2 = 1000MW
P3 = 666.67MW
The system marginal cost λ is λH = 20, as λL was not used for the iterations.