Using Optimisation Meta-Heuristics for the Roughness Estimation Problem in River Flow Analysis
Abstract
:1. Introduction
- the ‘physical’ parameters are those describing the properties of materials—these are usually constant values;
- the empirical parameters are those added to deal with the complexity and the variability of specific aspects characterising hydraulic properties of a channel. Amongst these, the so-called ‘roughness’ coefficient, which indicates the resistance to flood flows in both channels and flood plains, is one of the most important. Other empirical parameters are the vegetation, the alignment of the channel and its irregularities, shape and size, stage and discharge, suspended sediment load and bed sediment loads. These have to be calculated mathematically through analytical models [16].
- Section 2 introduces the designed objective function which allows formulating the coefficient estimation as an optimisation problem;
- Section 3 describes the hydrodynamic computational model which is used as a blackbox component inside the designed objective function;
- Section 4 briefly recalls the five meta-heuristic considered in this work to optimise the proposed objective function;
- Section 5 presents two case studies and describes the set-up of the conducted experiments;
- Section 6 analyses the experimental results;
- Finally, conclusions are drawn in Section 7 where future lines of research are also depicted.
2. Formulating the Optimisation Problem
- (i)
- a candidate solution , having the previously described structure, must be produced and fed to the HEC-RAS simulator;
- (ii)
- HEC-RAS is set with Manning’s coefficients from the candidate solution and a simulation is run to obtain the expected depth of the water at the gauge station deployed along the river—this value is referred here to as ;
- (iii)
- the absolute difference between , i.e., the simulated depth, and , i.e., the true depth observed at the gauge station, is calculated to obtain the loss score as shown below in Equation (1):
3. Hydrodynamic River Flows Analysis
3.1. Mathematical Formulation and Software Settings for Implementing the Model
- z indicates the bottom elevation of the river;
- h is the river’s water depth;
- v is the water’s mean velocity in the channel cross-section at hand;
- the so-called St. Venant coefficient is a correction factor that takes into consideration undesired effects due to e.g., non-uniformity of the water’s velocity in the considered profile of the river;
- g is the universal gravitational constant.
4. The Employed Optimisation Methods
4.1. (1 + 1)-ES
4.2. Differential Evolution
4.3. Covariance Matrix Adaptation Evolution Strategy
4.4. Particle Swarm Optimisation
4.5. Bayesian Optimisation
5. Experimental Set-up
5.1. Case Study 1—The River Paglia
5.2. Case Study 2—The River Aniene
6. Experimental Results
- DE and CMA-ES look, in general, to be preferable to PSO and BO;
- BO is not competitive enough in both case studies and, in particular, its effectiveness deteriorates in the larger case study, probably because BO is not good enough for large dimensionalities (indeed, it is generally used to tune the few hyper-parameters of machine learning models [30]);
- regarding robustness, (1 + 1)-ES is clearly the most robust algorithm;
- the high variance for the results of the PSO algorithm may indicate the presence of a considerable amount of local optima in the fitness landscape, given that PSO is prone to prematurely converge to local optima [56];
7. Conclusions and Future Work
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Algorithm | Final Fitness Values for the River Paglia | |||
---|---|---|---|---|
Average | Minimum | Maximum | St.Dev. | |
(1 + 1)-ES | <10 | 0 | 0.00005 | <10 |
DE | 0.00147 | 0.00004 | 0.00665 | 0.00211 |
CMA-ES | 0.00167 | 0.00013 | 0.00355 | 0.00139 |
BO | 0.01364 | 0.00538 | 0.02898 | 0.00807 |
PSO | 0.03719 | 0.00199 | 0.08304 | 0.03013 |
Std Method | 1.63000 |
Algorithm | Final Fitness Values for the River Aniene | |||
---|---|---|---|---|
Average | Minimum | Maximum | St.Dev. | |
(1 + 1)-ES | 0 | 0.00021 | ||
DE | 0.00362 | 0.00020 | 0.01190 | 0.00333 |
CMA-ES | 0.00431 | 0.00051 | 0.01572 | 0.00442 |
PSO | 0.00815 | 0.00048 | 0.01770 | 0.00532 |
BO | 0.01212 | 0.00085 | 0.02762 | 0.00914 |
Std Method | 1.78000 |
Cross Section | Std Dev. of the Manning’s Values for River Paglia | ||
---|---|---|---|
Left Channel | Main Channel | Right Channel | |
1 | 0.007826 | 0.007999 | 0.009895 |
5 | 0.008051 | 0.007276 | 0.008452 |
10 | 0.007449 | 0.008903 | 0.006296 |
15 | 0.003794 | 0.008512 | 0.008731 |
20 | 0.006102 | 0.008623 | 0.008709 |
Cross Section | Std Dev. of the Manning’s Values for River Aniene | ||
---|---|---|---|
Left Channel | Main Channel | Right Channel | |
1 | 0.006855 | 0.006383 | 0.006739 |
10 | 0.007071 | 0.006672 | 0.006709 |
20 | 0.006489 | 0.006073 | 0.006354 |
30 | 0.006764 | 0.006162 | 0.006629 |
40 | 0.008475 | 0.005951 | 0.007752 |
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Agresta, A.; Baioletti, M.; Biscarini, C.; Caraffini, F.; Milani, A.; Santucci, V. Using Optimisation Meta-Heuristics for the Roughness Estimation Problem in River Flow Analysis. Appl. Sci. 2021, 11, 10575. https://doi.org/10.3390/app112210575
Agresta A, Baioletti M, Biscarini C, Caraffini F, Milani A, Santucci V. Using Optimisation Meta-Heuristics for the Roughness Estimation Problem in River Flow Analysis. Applied Sciences. 2021; 11(22):10575. https://doi.org/10.3390/app112210575
Chicago/Turabian StyleAgresta, Antonio, Marco Baioletti, Chiara Biscarini, Fabio Caraffini, Alfredo Milani, and Valentino Santucci. 2021. "Using Optimisation Meta-Heuristics for the Roughness Estimation Problem in River Flow Analysis" Applied Sciences 11, no. 22: 10575. https://doi.org/10.3390/app112210575
APA StyleAgresta, A., Baioletti, M., Biscarini, C., Caraffini, F., Milani, A., & Santucci, V. (2021). Using Optimisation Meta-Heuristics for the Roughness Estimation Problem in River Flow Analysis. Applied Sciences, 11(22), 10575. https://doi.org/10.3390/app112210575