## 1. Introduction

Mine ventilation plays a very important role in underground mining. It bear significant responsibility for supplying enough fresh air to designated areas, diluting methane, and maintaining appropriate climatic conditions and removing underground contaminants effectively [

1,

2,

3,

4,

5]. Ventilation is often considered one of the biggest restraining factors in mine production [

6].

To ensure the safe and stable operation of the ventilation system, mine ventilation network (MVN) models have been widely applied in simulating and optimizing ventilation systems. This is an effective tool for ventilation planning prior to mining, understanding the current ventilation condition, and predicting the condition for further ventilation development [

7,

8]. To avoid air disorder or air inversion, mine ventilation simulation needs to be done before reforming the ventilation systems. Through applying a number of changes (adding/removing branches and nodes, changing resistance of roadways, revising parameters of fans, etc.) to the original model, simulation experiments are conducted to obtain the air distribution situation after new development or ventilation upgrade. These numerical model experiments can provide quick feedback, diagnose existing or potential problems, and help the ventilation engineer to evaluate the proposed ventilation plan [

9].

For building the MVN models, several commercial computer programs are available, such as Ventsim, VnetPC, and VentGraph. As these software have graphical representations of an MVN and are easy to operate, they have become widespread in mining [

10]. Feng [

8] used Ventsim to simulate the MVN for a mine that is more than 1 km deep with high ventilation pressure and a complex ventilation system. The model identified several areas that can reduce the resistance. A new model was built to simulate the ventilation condition after resistance reduction, and results showed that the airflow rate increased with reduced fan pressure and power cost. This helped the implementation of the proposed modification to the current ventilation setup. These software were also applied in ventilation and fire simulations for various underground constructions [

11]. To evaluate underground storage facilities, alternative configurations of the underground space was simulated with VnetPC [

12]. The study considered both ventilation system characteristics and economic factors. Based on the MVN model results, the most appropriate design, which is both secure and economically acceptable, was proposed.

To accurately simulate the airflow rate of roadways, good calibration of the ventilation network model is required [

13,

14]. In MVN modeling, one of the key parameters inputted to the model is the air way resistance, according to which airflow created by the fan is distributed. If resistances are supposed to be erroneous, it is impossible to accurately simulate the airflow performance in ventilation modeling software [

15]. However, due to various sources of errors, it is challenging for the measured airway resistances to reflect true values. Some of the measurement errors include: instrumental and operational errors; partial airway measurement results that are applied to the entire airway or similar airways; and empirical resistance that is applied to airways that are difficult to access. Due to such reasons, once the measured resistance data is input into the MVN model, the simulated airflow distribution results usually do not match the measured data. However, it is believed that the airflow quantity measurement is more accurate than that of the resistance. This is because the resistance is calculated based on measured pressure drop and airflow quantity over a section of the airway, or computed based on empirical values. Conversely, the airflow is directly measured with minimum procedure and operational errors. Thus, for effective ventilation planning, it is vital to ensure that the simulated airflow distribution agrees with that of the actual measured airflow distribution.

The aim of this paper is to apply the non-linear optimization algorithm to calibrate the MVN model to match the actual measured airflow data. The proposed method can calibrate the MVN model to agree with the measured airflow data, and control the errors of all other parameters within minimum range. After calibration, the underground airflow simulation result will be in accordance with the on-site measured data. When the calibrated model is used for further ventilation development simulation, it can provide results nearest to the actual situation, and therefore improve the accuracy for underground mine ventilation planning. Beyond that, as the ventilation model was widely used in scientific research related to mine ventilation, mine fire, and mine dust simulations, the method proposed in this paper will be a very helpful tool for these models to achieve more accurate results.

## 2. Methods

Assuming there are

b branches and

j junctions in the ventilation network, the measured and simulated airflow quantities in the branches can be denoted as

${Q}_{i}$ and

${q}_{i}\text{}\left(1\le i\le b\right)$, and the measured and simulated airway resistances for the branches can be denoted as

${R}_{i}$ and

${r}_{i}\text{}\left(1\le i\le b\right)$. The general processes for the proposed calibration method is demonstrated in the flow chart shown in

Figure 1. The first step is to obtain the original airflow quantities and ventilation resistances through consulting the ventilation survey report. Then, based on Kirchhoff’s second law, the actual measured airflow quantities (

${Q}_{i}$) and the airway resistances (

${r}_{i}$) are used in the calibrated MVN model to establish (

b – j + 1) sets of independent equations. The constraint conditions used for

${r}_{i}$ are within the range of plus and minus 10% of the measured resistance values. The airway resistances (

${r}_{i}$) used in the calibrated MVN model can be obtained by solving the objective function, which minimizes the difference between

${R}_{i}$ and

${r}_{i}$.

Ventilation survey is a systematic procedure of obtaining data of pressures and air quantities in roadways, and the distribution of airflow in ventilation systems. It is a common practice to conduct the ventilation survey periodically, and record and analyze the data in a ventilation survey report. Therefore, the most up-to-date data required for building an MVN model can be found in that report. As the objective for the calibration is to calibrate the MVN model so that it achieves the same airflow quantities as the actual measured data, the numerical expression is: ${q}_{i}={Q}_{i}$.

In an MVN with

b branches and

j junctions, there are

$\left(b-j+1\right)$ independent circuits [

16]. The most common way to obtain the number of independent circuits in an MVN is through calculating the independent circuit matrix. The process is shown as below. Firstly, all of the the branches and nodes in the MVN are numbered. Secondly, a minimum spanning tree weighted by ventilation resistance is selected. Then, the fundamental incidence matrix

B is obtained through combining the cotree’s branches with the tree’s branches, as shown in Equation (1).

where

B_{c} and

B_{t} represent the submatrix of the cotree’s branches and the submatrix of the tree’s branches, respectively.

Finally, the independent circuit matrix

C can be calculated through Equation (2). [

17,

18].

where

I is a unit matrix.

Using

$\left(b-j+1\right)$ independent circuits, the same number of independent equations can be established based on Kirchhoff’s second law, as shown in Equation (3).

As we only have

$\left(b-j+1\right)$ independent equations to solve

b number of

${r}_{i}\text{}(\left(b-j+1\right)b)$, there will be unlimited groups of solutions of

${r}_{i}$. It is assumed that although the measured resistance data has errors, it should be reasonable and close to the true resistance. Therefore, the resistance values used in the model should be constrained within a small range of the measured values. The constraint conditions for

${r}_{i}$ are set according to Equation (4).

To calibrate the MVN model, the modified resistances

${r}_{i}$ are used. One of the assumptions is, although errors exist, the measured resistances

${R}_{i}$ should be close to the true values. Thus, the objective function used in the algorithm is to minimize the difference between

${R}_{i}$ and

${r}_{i}$. There are many numerical ways to achieve this objective function, we have choosen to minimize the root mean square deviation (RMSD) between

${R}_{i}$ and

${r}_{i}$, as shown in Equation (5).

If an optimal solution is found that satisfies Equations (3) and (4), the ${r}_{i}$ results can be used in the MVN model. This guarantees that the simulated airflow quantities will match the measured data, whereas the airway resistances used are within a reasonable range of the measured or assumed values. Sometimes, an optimal solution may not be found due to large ventilation survey error or inaccurately recorded data. Under such circumstances, the ventilation survey data need to be validated, or the constraint condition can be loosen until the optimal solution is found.

## 4. Discussion and Conclusions

In recent years, several optimization analyses for mine ventilation have been conducted with different algorithms [

21,

22,

23]. However, none of them were applied to calibrate data for more accurate simulation. In this paper, a simple and effective way to calibrate a mine ventilation network model is proposed. The method uses a non-linear optimization algorithm to find resistance solutions nearby the measured data and achieves the simulated airflow rates that are very close to the measured ones. This method was then applied to a case study to calibrate roadways’ resistance. Finally, the results were discussed and the errors were analyzed. The results show that the simulated airflows in the calibrated model have much smaller errors, which proves the proposed calibrating method to be effective.

Such a calibrated MVN model ensures that the utilization and further modeling of such a network can provide accurate and instructional guidance for ventilation planning. Compared to existing methods, this method has the following advantages. First of all, this calibrating method is significantly effective with very high simulation precisions after calibration. Secondly, the proposed method is operational and cost-effective. Compared to the traditional calibration method of using repeated and time-consuming ventilation surveys, this mathematical calibration method can calibrate the ventilation network model quickly through solving a nonlinear optimization problem. Finally, the error of this method is controllable. A certain percent of error is allowed initially; if no optimal solutions can be found under such conditions, it means either incorrect data has been recorded or the ventilation survey accuracy needs to be improved.

Even though this calibrating method has many strengths, it still need to be improved in the following aspects. At first, as the same constraint conditions are applied to all of the roadways, some erroneous measured data may result in no solution for the objective function or the increase of the global error. It would be better if a further improved model could identify larger measured data. In addition, to achieve the minimum root mean square deviation (RMSD), some roadways’ error percentage may magnify. This can be improved by using the RMSD for relative resistance differences. Finally, it takes a great deal of time to write code for nonlinear optimization and input data. It would be more helpful if it were embedded in a commercial mine ventilation network simulation software, such as Ventsim or VNetPC. In conclusion, the calibration method proposed in this paper has great practical and scientific significance as it can calibrate ventilation models effectively to achieve much more accurate results.