1. Introduction

2. The Proposed Method

2.2. The Method for Improving D-S Evidence Theory

In the processing of data from similar sensors, the feature-level data fusion method can effectively eliminate ambiguous data and enhance the monitoring accuracy of environmental characteristics by similar sensors. For data from multiple types of sensors, decision-level fusion enables the complementary integration of multi-source data obtained from multiple sensors, thereby improving the accuracy of the monitoring results.

In tunnel fire detection, the raw data from temperature, CO concentration, and smoke concentration sensors are first subjected to feature extraction and correlation analysis of similar data. These features are then fused to form a comprehensive decision based on multi-source monitoring data, including temperature, CO concentration, and smoke concentration. On this basis, a decision-level fusion of multi-source data (temperature, CO concentration, smoke concentration) is performed to obtain the BPA of the evidence theory. The data fusion algorithm is then used to integrate these decision results, yielding the final fire probability. The proposed tunnel multi-sensor data fusion algorithm based on the improved DS evidence theory is illustrated in Figure 2.

Step 1: Primary fusion of data from sensors of the same type to obtain BPA.

The key aspect of data fusion using Dempster-Shafer (DS) evidence theory is obtaining the BPA. Following the primary fusion process for similar sensor data, multi-sensor monitoring data are first collected, and invalid data are removed to obtain feature data. Then, a primary fusion of similar sensor data is performed. Finally, the BPA is preliminarily calculated based on the feature intervals.

Step 2: Secondary fusion of data from multiple types of sensors to obtain tunnel condition information.

The improved DS evidence theory addresses evidence conflict issues and yields the final fusion results of multi-sensor data. According to the secondary fusion process for multi-sensor data, the basic probability numbers are first derived from the BPA. Then, an evidence distance matrix is constructed to calculate the degree of conflict between pieces of evidence, and normalization is performed to obtain the relative conflict degree. Subsequently, the trust coefficients between pieces of evidence are calculated to further optimize the BPA. Finally, classical DS evidence theory is used to fuse the evidence and determine the tunnel conditions.

2.2.1. Primary Fusion of Data from Sensors of the Same Type

The process for obtaining the improved DS evidence theory BPA function for the fusion of data from sensors of the same type involves the following steps:

(1) Similar data screening based on Euclidean distance

To exclude anomalous data when calculating the BPA, this paper proposes a similar data screening algorithm based on Euclidean distance. The primary method involves measuring the similarity between data points by calculating the Euclidean distance between similar data. A smaller distance indicates a higher degree of similarity and greater data authenticity, while a larger distance indicates a lower degree of similarity and lesser data authenticity. Therefore, by calculating the pairwise distances between all similar data points and setting an appropriate threshold, anomalies can be identified and excluded [29].

Let the number of sensors be n, and the distance between the collected like data$s_i=\left\{s_i|i=1,2,\mathrm{...},n\right\}$, The distance $d\left(s_i\right)$ [28] between $s_i$ and other similar data excluding $s_i$ can be expressed as:

$$\Delta d(i)=d\left(s_i\right)-d_m$$

(8)

$$d\left(s_i\right)={\left[{\displaystyle \frac{1}{n}}\sum _{j=1}^n{\left(s_i-s_j\right)}^2\right]}^{{\displaystyle \frac{1}{2}}},i\ne j,i=1,2,\cdots ,n$$

(9)

The distance $d_y$ of anomalous data is greater than the distance $d_z$ of normal data. Therefore, the difference $\Delta d(i)$ between the distances of anomalous and normal data can be used to identify anomalous data. In the equation, $d_m$ represents the median of $d\left(s_i\right)$.

Eliminating the magnitude of $\Delta d(i)$ by the following equation.

$$dw\left(s_i\right)={\displaystyle \frac{\Delta d(i)}{s_m}}$$

(10)

where $s_m$ represents the median of the set of sensor data of the same type $s_i$.

when $dw\left(s_i\right)\ge 0.02$, the sensor is considered to have a large error and should be rejected.

(2) Primary fusion of homogeneous sensor data using a weighted averaging method

If the weights are the same between sensors of the same type, then it is known that

$$\overline{x}={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n{x}_i}$$

(11)

We defined $\overline{x}$ as the fused value of data from the same type of sensors and $L(\overline{x})$ as the distance from $\overline{x}$ to other sensors. When $L(\overline{x})$ reached its minimum value, $\overline{x}$ was closest to the local multi-sensor monitoring result.

$$L(\overline{x})={{\displaystyle {\sum }_{i=1}^n(\overline{x}-x_i)}}^2$$

(12)

derived that.

$$L(\overline{x})=n{\overline{x}}^2+{\displaystyle {\sum }_{i=1}^n{x}_i^2}-2\overline{x}{\displaystyle {\sum }_{i=1}^n{x}_i}$$

(13)

derivation on the left and right sides yields that

$${\displaystyle \frac{dL(\overline{x})}{d\overline{x}}}=2n\overline{x}-2{\displaystyle {\sum }_{i=1}^n{x}_i}$$

(14)

when $\overline{x}={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n{x}_i}$ , The minimum value of $L(\overline{x})$ is obtained, which corresponds to the average of the data from all sensors.

(3) Acquisition of BPA functions based on feature intervals

To monitor tunnel fire conditions in real-time, it is essential to collect three key types of data: carbon monoxide (CO) concentration, smoke concentration, and temperature. The identification framework is $\Theta =\{A_1,A_2,\mathrm{...},A_{n-1}{A}_n\}$, categorizes the tunnel fire status into three levels: {Normal Conditions, Warning Conditions, and Fire Conditions}.

Interval T represents the range within which the identification framework $\Theta$ is situated. This interval divides the n objects within the identification framework into n characteristic intervals $T_i$, $i=1,2,\cdots ,n$, which describe the range $[T_{iL},T_{iR}]$ where the identification object $A_i$ is located. The midpoint $T_{iM}$ of each interval is set as the characteristic value for that feature interval.

$$T_{iM}={\displaystyle \frac{T_{iL}-T_{iR}}{2}}i=1,2,\cdots ,n$$

(15)

where $T_{iL}$ represents the left boundary of the characteristic interval $T_i$, while $T_{iR}$ represents the right boundary of the characteristic interval $T_i$.

Let $l{(j)}_i$ represent the distance between the sensor's measured value $a(j)$ and each characteristic value:

$$l{(j)}_i=\left|a(j)-T_{iM}\right|i=1,2,\cdots ,n$$

(16)

Dividing $l{(j)}_i$ by the length of each characteristic interval yields the dimensionless distance $lw{(j)}_i$ between the sensor's measured value and each interval's characteristic value:

$$lw{(j)}_i={\displaystyle \frac{1}{T_{iL}-T_{iR}}}l{(j)}_{i}i=1,2,\cdots ,n$$

(17)

Taking the reciprocal of $lw{(j)}_i$ and normalizing it yields the probability assignment function $mass(j)$ and the basic probability number $mass{(j)}_i$ for the measured value $a(j)$.

$$mass{(j)}_i={\displaystyle \frac{1/(lw{(j)}_i+C_0)}{{\displaystyle \sum _{k=1}^n[1/(lw{(j)}_k+C_0)]}}}i=1,2,\cdots ,n$$

(18)

where $C_0$ is a constant.

According to equation (18) , the closer the sensor's measured value is to the characteristic value $T_{iM}$ of the interval, the larger $mass{(j)}_i$ becomes, indicating that $a(j)$ is closer to the identification object $A_i$. However, during a tunnel fire, some sensor measurements may exceed the right boundary $T_{nR}$ of the characteristic interval $T_n$, causing $lw{(j)}_i$ to become too large and, consequently, $mass{(j)}_i$ to decrease, which moves $a(j)$ further away from the identification object $A_i$. Therefore, when calculating $mass{(j)}_i$, if $a(j)$ is too large, it should be replaced with $a{(j)}^{\text{'}}$. Based on the characteristic interval $T_i$, this adjustment ensures that $a(j)$ remains close to the corresponding characteristic value $T_{iM}$.

$$a{(j)}^{\text{'}}=T_{iM}+C_1$$

(19)

where $C_1$ is a constant.

2.2.2. Secondary Fusion of Multi-Type Sensor Data

The improved DS evidence theory for multi-type sensor data fusion is applied to a secondary fusion process.

First, let the identification framework be $\Theta =\{A_1,A_2,A_3,\mathrm{...},A_n\}$, where $A_n$ represents the n-Th condition of tunnel fire, and $E=\{E_1,E_2,E_3,\mathrm{...},E_m\}$ denotes the evidence set, with $E_m$ being the m-Th piece of evidence related to the tunnel fire. The definition of the distance $d_{ij}$ between evidence $mas{s}_j$ and $mas{s}_i$ is given by:

$$d_{ij}={\mathrm{e}}^{1-{\text{<}\mathrm{E}}_i,{\mathrm{E}}_j\text{>}}i,j=1,2,\cdots ,m$$

(20)

where

$${\text{<}\mathrm{E}}_i,{\mathrm{E}}_j\text{>}={\displaystyle \frac{{\displaystyle \sum _{s=1}^m}mas{s}_i\left(A_s\right)mas{s}_j\left(A_s\right)}{\sqrt{{\displaystyle \sum _{s=1}^m}mas{s}_i^2\left(A_s\right)}\sqrt{\sum _{s=1}^{m}mas{s}_j^2\left(A_s\right)}}}$$

(21)

As $d_{ij}$ approaches 1, it indicates a higher degree of mutual support between the pieces of evidence. Conversely, as $d_{ij}$ approaches e, it signifies a higher level of conflict between the pieces of evidence. Based on this, an evidence distance matrix D can be constructed:

$$D=\left[\begin{array}{cccc}1& d_{12}& \cdots & d_{1m}\\ d_{21}& 1& \cdots & d_{2m}\\ ⋮& ⋮& \ddots & ⋮\\ d_{m1}& d_{m2}& \cdots & d_{mm}\end{array}\right]$$

(22)

Defined $con(E_i)$ as the level of conflict between evidence.

$$con\left(E_i\right)=\sum _{i=1,i\ne j}^m{d}_{ij}$$

(23)

Normalize the evidence conflict degree to obtain the relative conflict degree of the evidence $in(E_i)$ [30].

$$in\left(E_i\right)={\displaystyle \frac{con\left(E_i\right)}{{\displaystyle \sum _{j=1}^m}con\left(E_j\right)}}$$

(24)

Let $\lambda (E_i)$ be the trust coefficient of the evidence, representing the importance of evidence $E_i$ and its influence on the fusion result. The definition of $\lambda (E_i)$ is:

(1)

Let $m_i$ be the original probability assignment function. The optimized probability assignment function $mas{s}_i^{*}$ is:

$$mas{s}_i^{*}(A_j)=\lambda \left(E_i\right)\cdot mas{s}_i(A_j)$$

(26)

$$mas{s}_i^{*}(X)=1-\lambda \left(E_i\right)$$

(27)

where $mas{s}_i^{*}(X)$ denotes uncertainty.

In summary, the specific steps for improving the DS evidence theory for the secondary fusion of multi-sensor data in tunnels are as follows:

Firstly, an evidence distance matrix D is constructed to calculate the degree of conflict between the evidence and normalized to obtain the relative degree of conflict.

Secondly, calculates the trust coefficients between the evidence to optimize the BPA.

Finally, the evidence is fused using the classical DS theory of evidence.

3. Numerical Examples and Simulation Verification

3.2. The Proposed Holistic Approach

To further demonstrate the feasibility and effectiveness of the data fusion method, this subsection applies the proposed improved DS evidence theory data fusion method to tunnel fire detection.

In tunnel fire detection, temperature sensors, CO sensors, and smoke sensors are commonly used to monitor fires [34]. The PyroSim software, based on Fire Dynamics Simulator (FDS) and Computational Fluid Dynamics [35], can simulate large-scale slow-moving vortices and accurately obtain critical parameters such as fire heat release, fire smoke, CO concentration, and temperature.

The fire simulation process using PyroSim is illustrated in Figure 4. First, a comprehensive tunnel geometric model is created, including elements such as lining, vehicles, and ventilation systems, to accurately describe the internal structure of the tunnel. Next, multiple sensors, including temperature, CO, and smoke sensors, are installed to enable real-time monitoring of the fire scene. The heat release rate is determined based on the actual fire conditions. Subsequently, a tunnel fire simulation model is established, and simulation parameters such as temperature and velocity are set to realistically simulate the fire situation. Finally, FDS is run to perform the fire simulation and obtain multi-sensor monitoring data.

3.2.1. Simulation Model

(1) Establishing the tunnel geometric model

The tunnel geometric model is constructed with the following specifications: a height of 8.5 meters, a width of 14.5 meters, and a length of 600 meters. The tunnel is lined with 31 concrete walls forming the lining and surrounding rock structure. The wall material has a density of 2280.0 kg/m³, a specific heat capacity of 1.04 kJ/ (kg·K) , and a thermal conductivity of 1.8 W/ (m·K) . In consideration of practical scenarios, the dimensions of the vehicles are set as follows: width 2.4 meters, height 2.5 meters, and length 7 meters. The fire materials carried by large trucks are wood and plastic. In the tunnel model, a blue square on the top represents the jet fan, which has an outlet area of 1.21 m² and a wind speed of 27.9 m/s. These parameters are used to establish the tunnel geometric model, with the specific structure illustrated in Figure 5.

(2) Installation of fire monitoring equipment

A 3x3 sensor matrix is arranged on the tunnel ceiling, with each matrix containing 3 temperature sensors, 3 CO sensors, and 3 smoke sensors. The distance between adjacent sensor matrices is 10 meters. Additionally, pairs of smoke sensors are installed on the right side of the tunnel at heights of 2 meters and 3 meters from the ground, with a distance of 100 meters between adjacent pairs. The specific arrangement of the sensors is illustrated in Figure 6.

(3) Establishing the fire model

The tunnel accommodates various vehicle types, including large trucks and small cars, so the corresponding heat release rates are set to 20 MW for large trucks and 5 MW for small cars. During a fire, higher wind speeds result in faster smoke spread and greater disruption to the smoke layer, increasing danger. Based on traffic volume survey data, the simulation wind speed is set to 2.5 m/s, with 3 m/s and zero wind speed used as control conditions. The tunnel is set with a pressure of 94.5 kPa, an initial temperature of 20°C, an initial CO concentration of 42 cm³/m³, and an initial smoke concentration K of 0.004 m^-1.

When a fire occurs at the midpoint of the tunnel, the distance for personnel evacuation to the exits is maximized, which is unfavorable for evacuation; simultaneously, the tunnel fan is located farthest from the fire source, making smoke control less effective. In Figure 7. ignition point 1 is located at the midpoint of the tunnel, representing the main fire condition, while ignition point 2 is near the entrance fan, serving as the control condition. The farther the sensor is from the ignition point, the more challenging it is to monitor the tunnel fire. Therefore, the distance between the ignition point and the downstream sensor matrix is set to 7.5 meters.

Based on the established fire conditions, with other fire conditions remaining consistent, four types of tunnel fire conditions are set as shown in Table 3.

Table 3. Tunnel fire conditions.

Table 3. Tunnel fire conditions.

Condition Number	Ignition Point L	Heat Release Rate (MW)	Wind Speed (m/s)
LZ-R5-S2.5	Tunnel Midpoint-Z	5	2.5
LZ-R20-S0	Tunnel Midpoint-Z	20	0
LZ-R20-S2.5	Tunnel Midpoint-Z	20	2.5
LEN-R20-S3	Tunnel Entrance 200m-EN	20	3

3.2.2. Tunnel Fire Simulation

Based on the tunnel fire conditions set in Section 3.2.1, simulations are conducted in PyroSim. Using the condition LZ-R5-S2.5 as an example, the multi-sensor monitoring data for tunnel fires are obtained. The monitoring data for temperature sensors, smoke sensors, and CO sensors in the fire monitoring nodes are shown in Table 4.

3.2.3. Tunnel Fire Simulation Data Fusion

Based on the "Highway Tunnel Ventilation Design Specifications" and relevant design standards for tunnel engineering, the characteristic intervals for temperature, smoke concentration, and CO concentration are categorized as shown in Table 5.

The primary fusion data of sensor readings under four different operating conditions, combined with Table 5, were used to calculate the probability assignment functions as shown in Table 6, according to Equations (18) and (19) .

The probability assignment function is optimized using Equations (26) and (27) , and the probability assignment function for unknown conditions is calculated. Combining the optimized probability assignment function with Equation (6) , the probabilities of normal operation, fire warning, fire state, and unknown state under various tunnel conditions are determined, as shown in Table 7.

For the different conditions mentioned above, the simulation results using the same method are shown in Figure 8.

The fire occurrence probabilities under various conditions are analyzed, and the resulting curves for the four types of fire conditions are shown in Figure 9.

Based on the multi-sensor monitoring data and fusion results for tunnel fires under different conditions, the following conclusions can be drawn:

A higher heat release rate and wind speed make it easier to detect tunnel fires. The fire condition LZ-R5-S2.5, with a lower heat release rate, presents the greatest challenge for fire detection. The improved DS evidence theory multi-sensor data fusion algorithm reached a conclusion of fire occurrence at 7.2 seconds based on changes in tunnel temperature and smoke concentration. Thus, the improved DS evidence theory multi-sensor data fusion algorithm demonstrates good fusion performance under zero wind speed and low heat release rate conditions.

The probabilities of tunnel fire occurrence under the four conditions are 67.5%, 82.8%, 83.5%, and 71%, respectively. This indicates that the proposed method has a high monitoring accuracy.

The comparison results of the improved DS evidence theory multi-sensor fusion algorithm with those of Sun and the original methods are shown in Figure 10.

4. Conclusion

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