# Hierarchical Sensor Placement Using Joint Entropy and the Effect of Modeling Error

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## Abstract

**:**

## 1. Introduction

## 2. Sensor Placement Strategy

#### 2.1. Errors in Wind-Speed and Wind-Direction

_{s}, where n

_{s}is a predetermined number of possible locations. The width of the histogram intervals is computed such that the frequency count in each interval is the number of model predictions that lie within the error threshold if the measured value is at the midpoint of the interval. This is done by dividing the maximum range of prediction values, max{(Y

_{max,j}− y

_{min,j}): j = 1, ..., n

_{s}} of an output variable y into intervals i = 1, ..., N

_{I}(N

_{I}the maximum number of intervals at the j

^{th}location) of width W equal to the sum of measurement errors e

_{measur}and modeling errors e

_{mod}(Figure 2). The intervals create subsets of model predictions that are then used to compute probabilities and entropy. Each subset represents model predictions that, given a potential measurement, will not be possible to separate further.

#### 2.2. Hierarchical Sensor Placement

1: | Create a list locationList containing all possible locations. |

2: | Create a set sensorOptimum to store all possible sensors. The set is empty to start with. |

3: | Create a set modelSubsets to store subsets of models that cannot be separated using the current sensor configuration. To start with this set contains a single element, which is the initialModelSet. |

4: | Add the first sensor location that corresponds to maximum entropy to sensorOptimum. |

5: | Create a list of subsets of models that cannot be separated by the first sensor location and add these subsets to modelSubsets. Remove the initialModelSet from modelSubsets. |

6: | Repeat while locationList is not empty |

{ | |

7: | Select a sensor location from locationList, let it be currentLocation. |

8: | Repeat for each set in modelSubsets |

{ | |

9: | Divide and distribute models in the current set into intervals of the currentLocation. |

} | |

10: | Calculate the entropy of the distribution of the currentLocation. |

} | |

11: | Select the sensor location with maximum entropy. Add to sensorOptimum and remove from locationList. |

#### 2.3. Joint Entropy as a Design Criterion

_{j}is the entropy of a random variable y at a measurement location j, p(y

_{i})

_{j}is the probability of the i

^{th}interval of a variable’s distribution with i = 1, ..., N

_{I}and N

_{I}the maximum number of intervals at the j

^{th}location. In order to compute the entropy, the number of models that lie within each interval m

_{i}is calculated and the probability of the interval is calculated as (m

_{i}/N).

_{j,j}

_{+1}is defined as:

_{k}and N

_{k}the maximum number of intervals at the location (j + 1).

## 3. Results

^{5}and 10.4 × 10

^{5}mesh elements.

_{*}is the atmospheric-boundary-layer friction (or shear) velocity, z

_{0}the surface roughness and κ ≅ 0.41 the von Kármán constant:

_{μ}a model constant:

_{0}, which is modified using the equivalent sand-grain roughness, k

_{s,ABL}:

_{s}is the roughness constant, set to satisfy the constraint k

_{s,ABL}≤ z

_{p}, and z

_{p}is the grid resolution (the distance of the centroid of the wall-adjacent cell to the wall).

_{d}is the drag coefficient, varying from 0.1 to 0.5, and d

_{SA}is the local leaf-area density, with range 1 to 7 [39].

_{k,j}, y

_{k,j}are the ranks of the input parameters and output variables respectively at each location j ∈ {1, …, 63}, with k = 1, …, n the size of the sample and x̄

_{j}, ȳ

_{j}the mean values.

_{mod,speed}, and wind direction, e

_{mod,dir}, following [33]:

_{j}(z) is the wind speed at height z at possible sensor locations j ∈ {1, …, 63}. Wind direction with $\frac{{u}_{j}(z)}{U(z)}<0.33$ were not considered since modeling errors were high (around ±180 deg).

#### 3.1. Effect of Modeling Error

#### 3.2. Optimum Sensor Configurations

#### 3.3. Performance Evaluation

#### 3.3.1. Wind-Flow Predictions

#### 3.3.2. Sequential vs. Hierarchical Sensor Placement

## 4. Discussion

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Constructing subsets of model predictions of width W for measurement location j using modeling and measurement errors.

**Figure 4.**Example of a two-dimensional regular grid created using the intervals of two sensor locations j and (j + 1).

**Figure 6.**CutCell Cartesian meshing for the computational domain; bottom view (left) and the domain of interest magnified (right).

**Figure 7.**Possible measurement locations displayed in the simulation environment: 3D view on the left and plan view on the right.

**Figure 8.**Comparison of the joint entropy in wind-speed predictions calculated during sensor placement; errors in predictions were considered either spatially uniform (±0.4 and ±1 m/s) or varying (only the first 15 optimum locations are displayed in the graph).

**Figure 9.**Comparison of the joint entropy in wind-direction predictions calculated during sensor placement; errors in predictions are taken to be either spatially uniform (±30 and ±180 deg) or varying (only the first 15 optimum locations are displayed in the graph).

**Figure 10.**A comparison of the joint entropy in wind-speed and wind-direction predictions calculated during sensor placement; errors in predictions vary spatially (only the first 15 optimum locations are displayed in the graph).

**Figure 11.**A comparison of the maximum number of candidate models of wind-speed and wind-direction that is expected during sensor placement; errors in predictions vary spatially (only the first 15 optimum locations are displayed in the graph).

**Figure 12.**The optimum configurations of four sensors for wind speed (left) and wind direction (right) displayed in the simulation environment; the markers represent the selected sensor locations.

**Figure 13.**Comparison of the wind-speed prediction ranges at an unseen location obtained with the optimum configuration of four sensors and the simulated measurements at this location; 15 min are taken from the 2 h measurement period.

**Figure 14.**Comparison of the wind-direction prediction ranges at an unseen location obtained with the optimum configuration of four sensors and the simulated measurements at this location; 15 min are taken from the 2 h measurement period.

**Figure 15.**Comparison of the wind-direction prediction ranges at an unseen location obtained using the optimum configuration of (

**a**) four sensors and (

**b**) six sensors and the simulated measurements at this location; 15 min are taken from the 2 h measurement period.

**Figure 16.**Comparison of the wind-speed prediction ranges at an unseen location obtained using (

**a**) hierarchical and (

**b**) sequential optimum configurations of four sensors and the simulated measurements at this location; 15 min are taken from the 2 h measurement period.

Parameter in CFD simulations | Lower bound | Upper bound | Comments |
---|---|---|---|

Height of computational domain [m] | 40 | 88 | The lower and upper bounds were set according to ([11,12,35]) |

Diagonal distance 1 from inlet boundary [m] | 83 | 117 | |

Diagonal distance 2 from inlet boundary [m] | 83 | 117 | |

Mesh growth rate | 1.05 | 1.1 | |

Terrain roughness of computational domain [m] | 0.5 | 1 | The lower and upper bounds were set according to ([36]), for suburbs (lower bound) to regularly-build large towns (upper bound). |

Terrain roughness of area of interest [m] | 3 × 10^{−3} | 0.3 | The lower and upper bounds were set according to ([36]), for concrete surfaces (lower bound) to long grass (upper bound). |

Surface roughness of BubbleZERO [m] | 0.03 | 1 | A double and inflatable membrane from PTFE was installed on the outside of the BubbleZERO. The lower bound corresponded to the typical PTFE value and the upper bound to the height of the inflated PTFE. |

BubbleZERO doors [m] | 2 × 10^{−5} | 0.16 | The lower and upper bounds were typical thickness of glazing and wooden frames. |

Inertial resistance of trees | 0.1 | 3.5 | The lower and upper bounds were set according to ([30]) |

Inertial resistance of bushes | 0.1 | 5.2 | |

Wind direction at inlet boundary [deg] | 1 | 360 | The wind direction varied from 1 to 360 degrees in order to account for possible direction values. |

Wind speed at inlet boundary [m/s] | 0 | 7.2 | The lower and upper bounds were set according to meteorological data obtained from the weather station Changi WMO in Singapore. |

TKE at inlet boundary [J/kg] | 0 | 7.2 | The lower and upper bounds were set according to ([11,12]). |

TDE at inlet boundary [m^{2}/s^{3}] | 0 | 1.3 |

**Table 2.**The selection order of the optimum configurations of four sensors for predicting wind speed and wind direction.

Selection order | Sensor location | |
---|---|---|

Wind speed | Wind direction | |

1st | L45 | L12 |

2nd | L16 | L16 |

3rd | L63 | L19 |

4th | L39 | L37 |

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**MDPI and ACS Style**

Papadopoulou, M.; Raphael, B.; Smith, I.F.C.; Sekhar, C.
Hierarchical Sensor Placement Using Joint Entropy and the Effect of Modeling Error. *Entropy* **2014**, *16*, 5078-5101.
https://doi.org/10.3390/e16095078

**AMA Style**

Papadopoulou M, Raphael B, Smith IFC, Sekhar C.
Hierarchical Sensor Placement Using Joint Entropy and the Effect of Modeling Error. *Entropy*. 2014; 16(9):5078-5101.
https://doi.org/10.3390/e16095078

**Chicago/Turabian Style**

Papadopoulou, Maria, Benny Raphael, Ian F.C. Smith, and Chandra Sekhar.
2014. "Hierarchical Sensor Placement Using Joint Entropy and the Effect of Modeling Error" *Entropy* 16, no. 9: 5078-5101.
https://doi.org/10.3390/e16095078