Finding Multiple Optimal Solutions to Optimal Load Distribution Problem in Hydropower Plant
Abstract
:1. Introduction
2. Problem Description
3. Solution Approach for Finding Multiple Solutions
3.1. Review of General Dynamic Programming (DP) Framework
- (1)
- Setting the decision variable as the load allocated to the generator.
- (2)
- Setting the stage variable as the accumulative load of the first i generators: .
- (3)
- Establishing the state equation as:
- (4)
- Define as the minimum total water discharge when the accumulative electric load from the first generator to the one is , i.e.,
3.2. Discretized OLD as a Shortest Path Problem
3.3. Finding MOS
- For :We set for as follows:
- For to n:We set for as follows:
- For :We set for as follows:
- For to n:We set for as follows:
3.4. Multiple Solutions Space
4. Case Study
4.1. Geheyan Hydropower Plant
4.2. Global Optimal Solution
Load L (MW) | Output of units (MW) | Total discharge (m/s) | Total discharge (m/s) | |||
---|---|---|---|---|---|---|
Unit 1 | Unit 2 | Unit 3 | Unit 4 | |||
500 | 255 | 245 | \ | \ | 518 | 517.1 |
550 | 295 | 255 | \ | \ | 562 | 562.0 |
600 | 300 | 300 | \ | \ | 608 | 608.0 |
650 | 295 | 285 | 70 | \ | 688 | 686.3 |
700 | 265 | 255 | 180 | \ | 734 | 731.0 |
750 | 250 | 250 | 250 | \ | 777 | 775.6 |
800 | 295 | 255 | 250 | \ | 821 | 820.5 |
850 | 300 | 295 | 255 | \ | 866 | 865.5 |
900 | 300 | 300 | 300 | \ | 912 | 912.0 |
950 | 295 | 295 | 295 | 65 | 991 | 989.5 |
1,000 | 255 | 255 | 255 | 235 | 1,036 | 1,034.2 |
1,050 | 275 | 265 | 255 | 255 | 1,079 | 1,079.0 |
1,100 | 275 | 275 | 275 | 275 | 1,124 | 1,124.0 |
1,150 | 295 | 295 | 295 | 265 | 1,169 | 1,169.0 |
1,200 | 300 | 300 | 300 | 300 | 1,216 | 1,216.0 |
4.3. Multiple Solutions
- (1)
- In the case that total electric load is equal to 500 MW (), the optimal solution consumes water discharge of 518 m/s and distributes them to unit 1 and unit 2 (Table 1).Table 2 shows single or multiple solutions under different discrete intervals. When the discrete interval of DP is 100 MW, a single solution is found but it is not optimal compared to the optimal water discharge found from MILP (518 m/s). When the discrete interval is 10 MW, a single optimal solution is located. When the discrete interval decreases to 1 MW, multiple solutions are found and they all reach the optimal discharge. With a smaller interval (0.1 MW), even more optimal solutions are identified.
Discrete interval | Unit 1 | Unit 2 | Total discharge |
---|---|---|---|
Δ (MW) | (MW) | (MW) | (m/s) |
100 | 300 | 200 | 521 |
10 | 250 | 250 | 518 |
1 | 255 | 245 | 518 |
254 | 246 | ||
⋮ | ⋮ | ||
251 | 249 | ||
250 | 250 | ||
0.1 | 255 | 245 | 518 |
254.9 | 245.1 | ||
⋮ | ⋮ | ||
250.1 | 249.9 | ||
250 | 250 |
- (2)
- When total load is set to 650 MW, four multiple solutions spaces are identified as follows.
- (a)
- , and . These ranges, if satisfying OLD constraints, can be verified by the records of the I/O curve, e.g., . Furthermore, we can obtain multiple solutions from , and given any values of and , and .
- (b)
- , and .
- (c)
- , and .
- (d)
- , and .
- (3)
- When the total load is set to 1100 MW, four generation units are used. By using the multiple solution approach, each unit’s output is either 295 MW, 285 MW, 275 MW, 265 MW or 255 MW. That is, if a solution sampled from above values satisfies the OLD constraints, it is an optimal solution. It is shown that the multiple solutions exist in the form of a few scattered points.
4.4. Non-Linear I/O Function
- (1)
- In the case that total electric load is equal to 500 MW, the optimal solution consumes water discharge of 518 m/s and distributes them to unit 1 and unit 2. Indeed, the multiple optimal solutions are the same as that from the piece-wise linear approximation (Figure 6) because the nonlinear and linear approximation are the same within the range of [240 MW, 255 MW].
- (2)
- In the case that total electric load is equal to 650 MW, (296.7 MW, 286.7 MW, 66.6 MW) is the unique optimal solution.
- (3)
- In the case that total electric load is equal to 1100 MW, a few scattered points, the same as that of piece-wise linear I/O curve, are the multiple optimal solutions.
4.5. Application
- (1)
- Improving unit stability: Multiple solutions provide choices to reduce the readjustment efforts when certain conditions change. For example, the total load L often changes from time to time as shown in Table 3, which lists historical data from a practical operation of the Geheyan hydropower plant. If the decision maker has only a single solution, the solutions with different values of L could be very different from each other and hence the readjustment cost is often high when the plant switches from one total load to another. However, if the decision maker has MOS for each different L, he/she can choose the solutions that require the least adjustment. For example, the decision maker could change only the load for unit 2 and keep the load of unit 3 constant when the total load L changes (because the solutions in the space [75 MW, 85 MW] have the same cost). This choice can improve the stability of the overall system.
Date | Output of Units (MW) | Load (MW) | |||
---|---|---|---|---|---|
Unit 1 | Unit 2 | Unit 3 | Unit 4 | ||
2006-3-23 10:30 | 0 | 76.5 | 78.9 | 0 | 155.4 |
2006-3-23 10:31 | 0 | 80.9 | 78.9 | 0 | 159.8 |
2006-3-23 10:32 | 0 | 81.9 | 78.9 | 0 | 160.8 |
2006-3-23 10:33 | 0 | 80.9 | 78.9 | 0 | 159.8 |
- (2)
- Avoiding unit adjustment over the vibration area: For example, the current total load is MW and the I/O function is piece-wise linear. We can apply our algorithm to find multiple solutions for this case. The MOS include the following two solutions:Solution 1: MW, MW, MW and MW.Solution 2: MW, MW, MW and MW.
5. Conclusions
Appendix
Finding an Optimal Solution Using Lagrangian Relaxation
5.1. Finding a Single Solution Using Mixed Integer Linear Programming
5.2. EXAMPLE 3D in [4]
Value | Unit 1 | Unit 2 | Unit 3 |
---|---|---|---|
A | 749.55 | 1,285.0 | 1,531.0 |
B | 6.95 | 7.051 | 6.531 |
C | |||
D | |||
Minimum (MW) | 320 | 300 | 275 |
Maximum (MW) | 800 | 1,200 | 1,100 |
Acknowledgements
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Liu, P.; Nguyen, T.-D.; Cai, X.; Jiang, X. Finding Multiple Optimal Solutions to Optimal Load Distribution Problem in Hydropower Plant. Energies 2012, 5, 1413-1432. https://doi.org/10.3390/en5051413
Liu P, Nguyen T-D, Cai X, Jiang X. Finding Multiple Optimal Solutions to Optimal Load Distribution Problem in Hydropower Plant. Energies. 2012; 5(5):1413-1432. https://doi.org/10.3390/en5051413
Chicago/Turabian StyleLiu, Pan, Tri-Dung Nguyen, Ximing Cai, and Xinhao Jiang. 2012. "Finding Multiple Optimal Solutions to Optimal Load Distribution Problem in Hydropower Plant" Energies 5, no. 5: 1413-1432. https://doi.org/10.3390/en5051413