A Long-Term Evaluation on Transmission Line Expansion Planning with Multistage Stochastic Programming
Abstract
:1. Introduction
2. Transmission Line Expansion Planning Modelling
2.1. Objective Function of TLEP
2.2. Constraints of TLEP
3. Decomposition Method Formulations
3.1. Basic Form of MSSP and Nonanticipativity Constraint
- (1)
- The PH designates implementable by the projection but derives proximally.
- (2)
- The PB solves the dual problem of (17) to get a representative for and verify its degree of improvement.
3.2. Progressive Hedging Method
Algorithm 1 Progressive Hedging Algorithm |
Initiate:, , , ,
Repeat:
1–4 until certain threshold is reached such that . |
3.3. Proximal Bundle Method
Algorithm 2 Proximal Bundle Algorithm |
Initiate:, ,, , , ,
Repeat: 1–5 until certain threshold is reached such that . |
3.4. Branch and Bound in MSSP
- The node would be "fathomed" if new bound is higher than the current optimal or is infeasible.
- The node would be updated as the Best UB if new bound is lower than the current Best UB, and the NC is satisfied.
- The node would be updated as a candidate for the Best LB if new bound is lower than the current Best UB, but the NC is still not satisfied.
Algorithm 3 Branch and Bound Algorithm |
Initiate:,, , ,
Repeat:
1–5 until all leaf nodes in are fathomed or. |
4. Simulation Results
4.1. Test System Configuration
4.2. Stopping Criteria Testing
4.2.1. Progressive Hedging
4.2.2. Proximal Bundle
4.2.3. Branch and Bound
4.3. Numerical Results Analysis
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
MSSP | Multistage stochastic programming |
MILP | Mixed-integer linear programming |
TLEP | Transmission line expansion planning |
DD | Dual decomposition |
PH | Progressive hedging |
PB | Proximal bundle |
BB | Branch-and-bound |
DDSIP | Dual decomposition in stochastic integer programming |
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Nomenclature | |||
---|---|---|---|
Sets | Description | Indices | Description |
Set of indices of the stages | t | Index of stage | |
Set of indices of prospective transmission lines | l | Index of prospective transmission line | |
Set of indices of existing transmission lines | k | Index of existing transmission line | |
Set of indices of all possible transmission lines | m | Index of all possible transmission line | |
Set of indices of generating units | i | Index of generating unit | |
Set of indices of demands | j | Index of demand | |
Set of indices of buses | b | Index of bus | |
Set of lines connected to bus b | |||
Set of generators located at bus b | |||
Set of demands at bus b |
Network Information | ||||||||
---|---|---|---|---|---|---|---|---|
Transmission Line Network Side | Generation Side | |||||||
Candidates | Bus[from - to] | Capacity | Bus | Capacity | Coef1 | Coef2 | Coef3 | |
[] | [] | [] | ||||||
A | 1–2 | 0.5 | 130 | 1 | 80 | 0 | 2 | 0.02 |
B | 2–4 | 0.5 | 65 | 2 | 80 | 0 | 1.75 | 0.0175 |
C | 2–6 | 0.5 | 65 | 13 | 50 | 0 | 1 | 0.0625 |
D | 4–6 | 0.5 | 90 | 22 | 55 | 0 | 3.25 | 0.00834 |
E | 6–8 | 0.5 | 32 | 23 | 30 | 0 | 3 | 0.025 |
F | 10–22 | 0.5 | 32 | 27 | 40 | 0 | 3 | 0.025 |
G | 15–18 | 0.5 | 16 | |||||
H | 15–23 | 0.5 | 16 | |||||
I | 4–11 | 0.5 | 16 | |||||
J | 6–11 | 0.5 | 16 | |||||
K | 6–12 | 0.5 | 16 | |||||
L | 11–12 | 0.5 | 16 |
Objective | Time | Convergence Rate | Termination | |
---|---|---|---|---|
0.1 | 4014.25 | 5977.37 | 0.007 | ✗ |
1 | 4239.17 | 2836.41 | 9.98 × 10 | ✓ |
5 | 4338.48 | 3220.80 | 4.69 × 10 | ✓ |
10 | 4918.19 | 8238.30 | 1.013 | ✗ |
Case | ST | SP | Method | [A–L] | [from–to] | Time (sec) | Objective | MIP Gap | ||
---|---|---|---|---|---|---|---|---|---|---|
PH | - | 8–28, 16–17 | 14 | 2359.67 | ✓ | ✓ | 0.001 | |||
PB | - | 8–28, 16–17 | 39 | 2276.68 | ✓ | ✗ | 0.001 | |||
1 | 2 | 10 | DDSIP | - | 8–28, 21–22 | 47 | 2279.24 | ✓ | ✓ | 0.001 |
PH + DDSIP | - | 8–28, 21–22 | 42 | 2279.24 | ✓ | ✓ | 0.001 | |||
EF | - | 8–28, 21–22 | 15 | 2279.32 | ✓ | ✓ | 0.001 | |||
PH | - | 8–28, 21–22 | 2836 | 4239.17 | ✓ | ✓ | 0.001 | |||
PB | - | - | 378 | 3960.96 | ✓ | ✗ | 0.001 | |||
2 | 4 | 7 | DDSIP | J | 8–28 | 3309 | 4014.61 | ✓ | ✓ | 0.001 |
PH + DDSIP | J | 8–28 | 3104 | 4,014.66 | ✓ | ✓ | 0.001 | |||
EF | J | 8–28 | 115,590 | 4013.42 | ✓ | ✓ | 0.01 | |||
PH | - | - | - | 6460.84 | ✗ | ✗ | 0.01 | |||
PB | - | - | 6501 | 5355.88 | ✓ | ✗ | 0.01 | |||
3 | 6 | 5 | DDSIP | L | 8–28, 21–22 | 55,100 | 6254.78 | ✓ | ✓ | 0.01 |
PH + DDSIP | L | 8–28, 21–22 | 46,915 | 6239.28 | ✓ | ✓ | 0.01 | |||
EF | - | - | - | - | ✗ | ✗ | 0.01 |
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Han, S.; Kim, H.-J.; Lee, D. A Long-Term Evaluation on Transmission Line Expansion Planning with Multistage Stochastic Programming. Energies 2020, 13, 1899. https://doi.org/10.3390/en13081899
Han S, Kim H-J, Lee D. A Long-Term Evaluation on Transmission Line Expansion Planning with Multistage Stochastic Programming. Energies. 2020; 13(8):1899. https://doi.org/10.3390/en13081899
Chicago/Turabian StyleHan, Sini, Hyeon-Jin Kim, and Duehee Lee. 2020. "A Long-Term Evaluation on Transmission Line Expansion Planning with Multistage Stochastic Programming" Energies 13, no. 8: 1899. https://doi.org/10.3390/en13081899