A Capacity Allocation Method for Long-Endurance Hydrogen-Powered Hybrid UAVs Based on Two-Stage Optimization
Abstract
:1. Introduction
2. Mathematical Model
2.1. Hydrogen-Powered Hybrid UAV System Model
2.1.1. Proton Exchange Membrane Fuel Cell
2.1.2. Lithium Battery
2.1.3. Hydrogen Storage Tank
2.1.4. Electric Motor
2.2. Construction of Uncertainty Model
3. Two-Stage Stochastic Optimization Problem
- Step 1: Initial Particle Swarm Generation:
- Step 2: Fitness Calculation:
- Step 3: Update Global Best Particle and Individual Best Particle:
- Step 4: Boundary Check:
- Step 5: Repeat Steps 2–4 until the iteration conditions are met, exit the loop, and obtain the optimal solution.
3.1. Energy Balance Constraint
3.2. Energy Storage Constraint
- (1)
- Before the hydrogen hybrid UAV takes off, the hydrogen storage tank has already stored a certain amount of hydrogen energy.
- (2)
- Due to the continuity of multiple flight missions by the hydrogen hybrid UAV, when the hydrogen fuel is at no less than 20% of its maximum capacity, refueling is not required. Therefore, the amount of hydrogen in the storage tank at the end of the previous flight is assumed to be the same as the starting amount for the next flight.
- (3)
- The storage and withdrawal of hydrogen from the tank do not occur simultaneously. Since the refueling process does not affect the capacity configuration of the hydrogen hybrid UAV, the focus is more on the withdrawal rate, which should not exceed the maximum withdrawal power of the hydrogen storage tank.
4. Results
5. Conclusions
- (1)
- The two-stage stochastic optimization method proposed in this paper effectively addresses the impact of uncertain factors such as wind speed and wind direction. In the first stage, PSO is used to optimize the capacity configuration of the equipment, while the second stage evaluates the expected optimal cost under different scenarios by solving the MILP problem, ultimately achieving the system’s full lifecycle optimization.
- (2)
- This paper constructs a comprehensive evaluation metric that includes both short-term and long-term costs. The short-term costs focus on hydrogen consumption and regular maintenance expenses, while the long-term costs consider equipment investment and amortization of service life. By using a weighted sum, the short-term and long-term objectives are balanced, ensuring that the system meets the immediate operational needs while maintaining long-term sustainability.
- (3)
- To address the uncertainty of wind speed and wind direction, the Monte Carlo method is used to generate random scenarios for these variables. By combining this with the PSO algorithm, the system configuration is continuously optimized. Through iterative optimization, the PSO algorithm effectively finds the optimal solution, thereby improving the overall system performance.
- (4)
- A sensitivity analysis of the number of particles in the PSO algorithm was conducted, and the results showed that the optimization effect was best when the number of particles was set to 50. This provides valuable guidance for future optimization algorithm configurations, ensuring both computational efficiency and convergence.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
i | |||
---|---|---|---|
1 | 0.0588460 | 1.325 | 1.0 |
2 | −0.06136111 | 1.87 | 1.0 |
3 | −0.002650473 | 2.5 | 2.0 |
4 | 0.002731125 | 2.8 | 2.0 |
5 | 0.001802374 | 2.938 | 2.42 |
6 | −0.0012150707 | 3.14 | 2.63 |
7 | 0.958842 × 10−4 | 3.37 | 3.0 |
8 | −0.1109040 × 10−6 | 3.75 | 4.0 |
9 | 0.1264403 × 10−9 | 4.0 | 5.0 |
Time (min) | Wind Speed (m/s) | Wind Speed Uncertainty (±m/s) | Wind Direction (°) |
---|---|---|---|
0 | 3.5 | ±0.3 | 45 |
10 | 3.2 | ±0.3 | 42 |
20 | 3.8 | ±0.4 | 47 |
30 | 3.6 | ±0.4 | 50 |
40 | 4 | ±0.3 | 48 |
50 | 3.9 | ±0.5 | 53 |
60 | 4.3 | ±0.5 | 57 |
70 | 4.5 | ±0.4 | 55 |
80 | 4.1 | ±0.5 | 59 |
90 | 4.6 | ±0.5 | 63 |
100 | 4.8 | ±0.5 | 60 |
110 | 4.5 | ±0.5 | 65 |
120 | 5 | ±0.4 | 68 |
130 | 4.7 | ±0.6 | 66 |
140 | 5.2 | ±0.6 | 70 |
150 | 5.4 | ±0.6 | 72 |
160 | 5.1 | ±0.6 | 75 |
170 | 5.3 | ±0.7 | 73 |
180 | 5.6 | ±0.7 | 77 |
190 | 5.8 | ±0.6 | 75 |
200 | 5.4 | ±0.7 | 78 |
210 | 5.9 | ±0.7 | 82 |
220 | 6.1 | ±0.7 | 80 |
230 | 6 | ±0.8 | 85 |
240 | 6.2 | ±0.8 | 83 |
250 | 6.3 | ±0.8 | 87 |
260 | 6.5 | ±0.5 | 89 |
270 | 6.6 | ±0.8 | 86 |
280 | 6.4 | ±0.9 | 91 |
290 | 6.8 | ±0.3 | 88 |
300 | 6.9 | ±0.8 | 92 |
310 | 6.7 | ±0.4 | 90 |
320 | 7 | ±0.9 | 95 |
330 | 6.8 | ±1.2 | 93 |
340 | 7.2 | ±1.0 | 98 |
350 | 7.1 | ±1.0 | 100 |
360 | 7.3 | ±0.3 | 97 |
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Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|
ω1 | 0.4 | K1 | 0.2 | P1 | 500 |
ω2 | 0.6 | K2 | 0.4 | P2 | 2000 |
k1 | 0.1 | K3 | 0.1 | P3 | 300 |
k2 | 0.9 | K4 | 0.2 | P4 | 100 |
k3 | 0.5 | C1 | 500 | P1,ev | 50 |
k4 | 0.5 | C2 | 800 | P2,ev | 250 |
Pf | 0.0025 | C3 | 20 | P3,ev | 50 |
Nm | 50 | C4 | 400 | P4,ev | 20 |
Nmn | 4 | Ne | 4 | Nu | 2000 |
Parameter | Value |
---|---|
N | 50 |
Max_iteration | 1000 |
dim | 4 |
lb | [0, 0, 0, 0] |
ub | [5, 10, 100, 12] |
Device | Parameter |
---|---|
Fuel cell (Kw) | Pmax = 3.367, Pr = 2.525 |
Lithium battery (kWh) | Battery Capacity 7.694 |
Cooling system (W) | Thermal Power 53.128 |
Hydrogen storage tank (L) | V = 9.328 |
Indicators | PSO | GA |
---|---|---|
The optimal lifecycle cost (CNY ten thousand) | 30.15 | 31.20 |
Number of iterations | 143 | 256 |
Computation time (s) | 63 | 102 |
Post-iteration error (%) | 0.5 | 1.2 |
Number of feasible solutions (%) | 98 | 92 |
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Li, H.; Wang, C.; Yuan, S.; Zhu, H.; Sun, L. A Capacity Allocation Method for Long-Endurance Hydrogen-Powered Hybrid UAVs Based on Two-Stage Optimization. Algorithms 2025, 18, 196. https://doi.org/10.3390/a18040196
Li H, Wang C, Yuan S, Zhu H, Sun L. A Capacity Allocation Method for Long-Endurance Hydrogen-Powered Hybrid UAVs Based on Two-Stage Optimization. Algorithms. 2025; 18(4):196. https://doi.org/10.3390/a18040196
Chicago/Turabian StyleLi, Haitao, Chenyu Wang, Shufu Yuan, Hui Zhu, and Li Sun. 2025. "A Capacity Allocation Method for Long-Endurance Hydrogen-Powered Hybrid UAVs Based on Two-Stage Optimization" Algorithms 18, no. 4: 196. https://doi.org/10.3390/a18040196
APA StyleLi, H., Wang, C., Yuan, S., Zhu, H., & Sun, L. (2025). A Capacity Allocation Method for Long-Endurance Hydrogen-Powered Hybrid UAVs Based on Two-Stage Optimization. Algorithms, 18(4), 196. https://doi.org/10.3390/a18040196