# Automated Floodway Determination Using Particle Swarm Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Mathematical Representation of the Problem

_{fw}in this section. In a similar way, the maximum and minimum floodway areas (A

_{fw,max}and A

_{fw,min}, respectively) can be calculated by connecting the 100-year floodplain extents (diamonds) and channel banks (dots), respectively. Our interest is to minimize the floodway area A

_{fw}while satisfying hydraulic requirements.

_{fw}is the floodway surface area, A

_{fw,min}and A

_{fw,max}are the minimum and maximum surface areas of the floodway, respectively as shown in Figure 4, $\mathbf{s}$ and $\mathbf{Fr}$ are the surcharge and Froude number vectors, respectively, for all cross sections, ${\mathbf{s}}_{i}$ and ${\mathbf{Fr}}_{i}$ are the surcharge and Froude number, respectively, for cross section i, and N is the total number of cross sections in the model including n cross sections to optimize and those used as boundary conditions.

_{fw}and A

_{fw,min}, respectively) to the surface area difference between the maximum and minimum floodways (A

_{fw,max}and A

_{fw,min}, respectively). Provided that all the hydraulic conditions are satisfied, the values of this ratio are 0 and 1 when the floodway is at its minimum and maximum widths, respectively. If one or more of the hydraulic conditions are violated, the objective function is set to unity and extra penalties are added based on how severely the surcharge and Froude number are deviating from their respective allowable limits. The penalty functions are designed so that the magnitude of deviation from the intended surcharge range or Froude number range is directly correlated to the magnitude of penalty added to the objective function. Any penalty will force the objective function to take on a larger value, which is an undesirable trait in a minimization problem. Now, floodway optimization is defined as a mathematical equation that can be evaluated objectively by computer code. The main goal of the proposed approach is to minimize the objective function $f\left(\mathbf{M}\left(\mathbf{g}\left(\mathbf{x}\right)\right)\right)$ by optimizing the variable vector $\mathbf{x}$ that represents encroachment limits $\mathbf{e}$, using a heuristic algorithm.

^{2}and 100 km

^{2}, respectively. Trial 1 evaluates the minimum floodway area, but the surcharges and Froude numbers for both cross sections violated hydraulic conditions (i.e., ${\mathbf{s}}_{i}>0.305\text{}\mathrm{m}$ and ${\mathbf{Fr}}_{i}\ge 1$ for $i=1,2$). Since there are hydraulic violations, the penalty case of Equation (1) is used to calculate the objective function value in the Equation (1) column. That is, f (

**M**(

**g**(

**x**))) = 1 + max(0 − 0.366/0.305, 0) $+max(0.366/0.305-1,0)+max(1.1-1,0)+max(0-0.397/0.305,0)+max(0.397/0.305-1,0)+max(1.2-1,0)=1.8$. Similarly, trials 2–3 have some violations and use the same equation to evaluate the objective function. Trial 4 does not violate any hydraulic conditions, so the acceptable case of Equation (1) is used to calculate the objective function value. In this case, $f\left(\mathbf{M}\left(\mathbf{g}\left(\mathbf{x}\right)\right)\right)=\frac{80-50}{100-50}=0.6$. Similarly, trials 5–6 do not have any violations and use the same floodway area ratio as their objective function values. Assuming that there are no more trial simulations, trial 4 would be the best floodway model because it minimizes the floodway area while satisfying all hydraulic requirements. In actual optimization runs, trials will evolve based on a heuristic algorithm introduced in Section 2.3.

#### 2.3. Isolated-Speciation-Based Particle Swarm Optimization

#### 2.4. Automated Floodway Optimizer for HEC-RAS

_{max}tells ISPSO the total number of iterations to perform for optimization. The maximum number of HEC-RAS model runs is defined by iter

_{max}times the swarm size S (iter

_{max}× S). The HEC-RAS model is represented by

**M**(·) and requires that two plans be defined: (1) 100-year floodplain and (2) floodway. The boundary conditions specify how the downstream or upstream end of the floodway should tie into adjacent existing floodways. There are four possible boundary conditions: (1) No existing floodways at the upstream and downstream ends of the study reach (BC = None), (2) floodway only at the downstream end (BC = DS), (3) floodway only at the upstream end (BC = US), and (4) floodways at both ends (BC = Both). The boundary conditions fix the encroachment limits at the most upstream, downstream or both cross sections, and therefore the problem dimension can be determined based on the number of cross sections and the number of boundary conditions. For example, when there are no upstream or downstream floodways to tie into (BC = None), all cross sections should be optimized, whereas the number of cross sections to optimize reduces by either 1 or 2 if one boundary condition (either BC = DS or BC = US) or two boundary conditions (BC = Both) are specified, respectively. The number of cross sections to optimize is indicated by n and the problem dimension is the total number of left and right encroachment limits on those n cross sections, which is $D=2\times n$. The recommended swarm size of $S=10+2\sqrt{D}$ [50] was used.

Algorithm 1 Pseudocode for automated floodway optimization for HEC-RAS. | |

Require: iter_{max} | ▹ Maximum number of iterations |

Require:M(·) | ▹ HEC-RAS model with 100-year and floodway plans and profiles |

Require: BC ∈ {None, DS, US, Both} | ▹ Boundary conditions for the encroachment limits |

Extract cross section information from M(·) | |

$N\leftarrow $ Number of cross sections | |

$n\leftarrow $ $N-$ Number of boundary conditions | |

$D\leftarrow 2\times n$ | ▹ Problem dimension |

$S\leftarrow 10+\u23a32\sqrt{D}\u23a6$ | ▹ Swarm size |

A_{fw,min}, A_{fw,max} ← Minimum and maximum possible areas of the floodway | |

X $\in {[0,1]}^{S\times D}\leftarrow S$ number of D-tuples randomly sampled from ${[0,1]}^{D}$ | ▹ Initial population |

Let $\mathbf{g}:{[0,1]}^{D}\to {\mathbb{R}}^{D}$ that maps particles to encroachment limits | |

iter $\leftarrow 1$ | |

repeat | ▹ ISPSO loop |

for $i\leftarrow 1,\dots ,S$ do | |

X_{i} ← Row i from X | ▹ ${i}^{\mathrm{th}}$ trial encroachment limits or particle i in ISPSO |

Simulate M(g(X_{i})) using CLIRAS | ▹ Execute the HEC-RAS program |

Evaluate f (M(g(X_{i}))) | ▹Equation (1) |

if $i=1$ or f (M(g(X_{i}))) < f (M(g(X_{best}))) then | ▹ If X_{i} is better than X_{best} |

X_{best} ← X_{i} | ▹ Store the best encroachment limits from the current iteration |

end if | |

end for | |

if iter = 1 or f (M(g(X_{best}))) < f (M(g(x_{best}))) then | ▹ If X_{best} is better than x_{best} |

x_{best} ← X_{best} | ▹ Store the best encroachment limits so far |

end if | |

Evolve X using ISPSO | ▹ Evolution of the swarm in ISPSO |

iter ← iter + 1 | |

until iter = iter_{max} or other conditions are satisfied | |

Optimized encroachment limits $\leftarrow \mathbf{g}$(x_{best}) | ▹ Found the best encroachment limits |

_{fw,min}and A

_{fw,max}, respectively, are calculated by straightening the reach and calculating the area of the polygon defined by the bank stations for A

_{fw,min}or the 100-year floodplain extents for A

_{fw,max}. These minimum and maximum floodway areas A

_{fw,min}and A

_{fw,max}, respectively, are used in the objective function in Equation (1) to assess the fitness of the floodway defined by trial encroachment limits

**g**(

**X**

_{i}). Once the problem is defined, ISPSO initializes the swarm and starts evolving the particles by evaluating trial encroachment limits. The final solution ${\mathbf{e}}_{\mathrm{best}}=\mathbf{g}$(

**x**

_{best}) is a set of optimized left and right encroachment limits, which then becomes input for the HEC-RAS floodway model.

#### 2.5. Numerical Experiments

## 3. Results and Discussion

#### 3.1. Comparison of Different Approaches

_{fw}− A

_{fw,min}, not the total floodway area (A

_{fw}). On average for all eight cases (i.e., manual and HEC-RAS cases), the reduction of the floodway area outside the channel was 20%, which can significantly increase encroachment areas for development.

#### 3.2. Sensitivity of Encroachment Limits to the Boundary Condition

#### 3.3. Optimization Performance

#### 3.4. AFORAS as a Tool for Floodway Optimization

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**River and cross section geometries for Beaver Creek near Kentwood, Louisiana, in the FLODENCR model. The arrow indicates the direction of flow. A bridge structure is located at 8.69 km.

**Figure 4.**Plan view of a floodway along with a river, cross sections, channel banks, and 100-year floodplain extents.

**Figure 5.**Graphical views of the left and right encroachment limits for the four different boundary conditions. Vertical lines at encroachment station 0 represents the river line. Negative and positive stations represent the left and right encroachment limits, respectively. Straightened river, Automated Floodway Optimizer for HEC-RAS (AFORAS) floodway, Manual floodway, and Hydrologic Engineering Center’s River Analysis System (HEC-RAS) floodway.

**Figure 6.**Two-dimensional projections of all particles

**X**

_{i}$\in {[0,1]}^{D}$ for $1\le i\le S$ satisfying the three hydraulic criteria from all 30 AFORAS runs for BC = Both. Since the total number of particles meeting hydraulic requirements was excessively large—130,395 out of 570,000 ($30\phantom{\rule{4.pt}{0ex}}\mathrm{AFORAS}\phantom{\rule{4.pt}{0ex}}\mathrm{runs}\times 19,000\phantom{\rule{4.pt}{0ex}}\mathrm{model}\phantom{\rule{4.pt}{0ex}}\mathrm{runs}/\mathrm{AFORAS}\phantom{\rule{4.pt}{0ex}}\mathrm{run}$)—10,000 particles were sampled to construct each subplot, which represents one cross section (XS). Particles that perform better are plotted darker in front of those that perform worse and are in a lighter gray.

**Figure 7.**Two-dimensional projections of all particles

**X**

_{i}$\in {[0,1]}^{D}$ for $1\le i\le S$ satisfying the three hydraulic criteria from all 30 AFORAS runs for BC = None. Since the total number of particles meeting hydraulic requirements was excessively large—142,937 out of 540,000 ($30\phantom{\rule{4.pt}{0ex}}\mathrm{AFORAS}\phantom{\rule{4.pt}{0ex}}\mathrm{runs}\times 18,000\phantom{\rule{4.pt}{0ex}}\mathrm{model}\phantom{\rule{4.pt}{0ex}}\mathrm{runs}/\mathrm{AFORAS}\phantom{\rule{4.pt}{0ex}}\mathrm{run}$)—10,000 particles were sampled to construct each subplot, which represents one cross section (XS). Particles that perform better are plotted darker in front of those that perform worse and are in a lighter gray.

**Figure 8.**Cumulative minimum values of the objective function vs. the number of model runs. The gray lines and black line represent 30 runs of AFORAS and the mean of those runs, respectively.

**Table 1.**Example model outputs and their objective function values. For all trials, $N=2$, A

_{fw,min}= 50 km

^{2}, and A

_{fw,max}= 100 km

^{2}.

Trial | ${\mathbf{A}}_{\mathbf{f}\mathbf{w}}$ | ${\mathbf{s}}_{1}$ (m) | ${\mathbf{s}}_{2}$ (m) | ${\mathbf{F}\mathbf{r}}_{1}$ | ${\mathbf{F}\mathbf{r}}_{2}$ | Violations | Equation (1) | Equation (2) |
---|---|---|---|---|---|---|---|---|

1 | 50 | 0.366 | 0.397 | 1.1 | 1.2 | 4 | 1.80 | 0.50 |

2 | 60 | 0.336 | 0.366 | 1.0 | 0.8 | 3 | 1.30 | 0.30 |

3 | 70 | 0.310 | 0.320 | 1.1 | 0.9 | 1 | 1.17 | 0.07 |

4 | 80 | 0.295 | 0.295 | 0.9 | 0.7 | 0 | 0.60 | 0.07 |

5 | 90 | 0.244 | 0.214 | 0.8 | 0.6 | 0 | 0.80 | 0.50 |

6 | 100 | 0.305 | 0.305 | 0.9 | 0.8 | 0 | 1.00 | 0.00 |

**Table 2.**Objective function values for the test cases with different boundary conditions. Since Automated Floodway Optimizer for HEC-RAS (AFORAS) solves a minimization problem, lower values represent better models. The numbers inside parentheses indicate what percent of the objective function value could be improved by running AFORAS.

BC | Problem Dimension | AFORAS | Manual | HEC-RAS |
---|---|---|---|---|

None | 24 | 0.270 | 0.348 (29%) | 0.379 (40%) |

DS | 22 | 0.278 | 0.345 (24%) | 0.379 (36%) |

US | 22 | 0.333 | 0.347 (4%) | 0.379 (14%) |

Both | 20 | 0.338 | 0.342 (1%) | 0.379 (12%) |

BC | XS 8.05 km | XS 8.15 km | XS 8.26 km–9.27 km | XS 9.45 km | XS 9.64 km |
---|---|---|---|---|---|

None | 5a | 5b | 5c–5j | 5k | 5l |

DS | 6a | ∼6b | ∼5c–5j | ∼5k | ∼5l |

US | ∼5a | ∼5b | ∼5c–5j | ∼6k | 6l |

Both | 6a | 6b | 6c–6j | 6k | 6l |

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**MDPI and ACS Style**

Cho, H.; Yee, T.M.; Heo, J. Automated Floodway Determination Using Particle Swarm Optimization. *Water* **2018**, *10*, 1420.
https://doi.org/10.3390/w10101420

**AMA Style**

Cho H, Yee TM, Heo J. Automated Floodway Determination Using Particle Swarm Optimization. *Water*. 2018; 10(10):1420.
https://doi.org/10.3390/w10101420

**Chicago/Turabian Style**

Cho, Huidae, Tien M. Yee, and Joonghyeok Heo. 2018. "Automated Floodway Determination Using Particle Swarm Optimization" *Water* 10, no. 10: 1420.
https://doi.org/10.3390/w10101420