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Scaling Up Pareto Optimization for Tree Structures with Affine Transformations: Evaluating Hybrid Floating Solar-Hydropower Systems in the Amazon
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Extends the tree-structured Pareto dynamic programming solver with affine-transformation dominance pruning and dynamic child ordering, and applies it to hydropower-plus-floating-solar portfolio planning across the Amazon basin.
Problem
Computing Pareto frontiers for dam portfolio selection on large tree-structured river networks is computationally intractable as the number of objectives and dam configurations (with and without floating PV) grows.
Approach
Joining child sub-frontiers is recast as affine transformations (a_i z + b_i) so that dominated transformations can be pruned before enumerating solutions, combined with a ranking heuristic that chooses which children to join first, while preserving exact/FPTAS optimality guarantees; a multi-tree formulation handles multiple dam configurations.
Key results
- Affine transform dominance consistently reduced running time versus no-transform, and combining it with child ordering gave order-of-magnitude speedups in most cases (e.g., ECG on Amazon: 25567s baseline to 369s with node-size order and transforms).
- Node-size and frontier-size child ordering both greatly outperformed the static baseline order; many baseline runs exceeded the 24-hour limit.
- Adding floating PV shifts Pareto frontier energy values higher and widens their distribution, and increases the frequency of eastern lowland dams in optimal portfolios.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add affine-transform dominance to node-size child ordering (ablation) | Pareto optimization algorithm · Amazon River basin, ECG, ϵ=0.025 | running time (seconds): 449 -> 369 | Helped |
| Add affine-transform dominance to node-size child ordering (ablation) | Pareto optimization algorithm · Marañón River basin, All criteria, ϵ=0.05 | running time (seconds): 53372 -> 50387 | Helped |
| Use node-size child ordering instead of baseline input order, with affine-transform dominance (ablation) | Pareto optimization algorithm · Amazon River basin, ECG, ϵ=0.025 | running time (seconds): >86400 -> 369 | Helped |
| Use node-size child ordering instead of baseline input order, with affine-transform dominance (ablation) | Pareto optimization algorithm · Marañón River basin, All criteria, ϵ=0.05 | running time (seconds): >86400 -> 50387 | Helped |
| Use frontier-size rather than node-size child ordering (ablation) | Pareto optimization algorithm · Amazon River basin, ECG, ϵ=0.025 | running time (seconds): 369 -> 421 | Hurt |
| Use frontier-size rather than node-size child ordering (ablation) | Pareto optimization algorithm · Marañón River basin, All criteria, ϵ=0.05 | running time (seconds): 50387 -> 43236 | Helped |
| Pair hydropower dams with floating photovoltaic panels | Amazon hydropower system · Amazon River basin Pareto frontier | energy | Helped |
Surprising findings
- Joining the children with the largest sub-trees or frontiers first, rather than in input order, reduces the number of solutions considered.
Limitations
- Experiments limited to the Amazon and Marañón basins with six specific criteria
- Floating PV assumption of up to 5% of reservoir or 30 km2 per dam site is a modeling assumption
Experimental setup
Datasets: Amazon River basin data (Flecker et al. 2022) Marañón sub-basin data (Flecker et al. 2022)
Benchmarks: Amazon River basin Marañón sub-basin
Metrics: running time (seconds) epsilon-approximate Pareto frontier size energy distribution of Pareto solutions
Baselines: original tree Pareto DP with static child order (Gomes-Selman et al. 2018) no-transform baseline Compression and Expansion (Bai et al. 2023)
Builds on: Gomes-Selman et al. 2018 tree Pareto DP Wu et al. 2018 FPTAS Compression and Expansion (Bai et al. 2023)
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