**1. Introduction**

Pollution-induced flashover is among the most serious power accidents, which seriously threatens the safety and stability of the power system [1–5]. In the past three decades, China has suffered from air pollution due to rapid economic growth, industry-led urbanization, and a lack of environmental protection [6]. Severe air pollution aggravates the possibility of pollution flashover [7]. Therefore, much literature has focused on the issue of insulator contamination [8–21].

It will cost a lot of workforces and material resources to test the pollution degree of insulators operating in transmission lines. Therefore, some scholars have first studied insulator contamination in a wind tunnel and other pollution accumulation systems [8–12]. For example, insulator contamination characteristics were studied by wind tunnel simulation in the literature [8], and the results sugges<sup>t</sup> that insulator structures, wind speed and RH (relative humidity) have obvious impacts on the contamination degree of insulators. Research on the contamination characteristics in the winter environment [9,10] also show that the wind speed has a greater effect on insulator contamination and NSDD. Y. Liu et al. [11] set up a natural pollution accumulation system and uses it to analyze the contamination characteristics and the micro-shape features of the insulator surface. The results show that the DC electric field has a significant effect on agglomeration characteristics.

In addition, the analysis methods based on finite element, grey theory, etc. are also applied to the insulator pollution model [13–18]. For example, the coupling physics model of a three-unit XP-160 insulator string was established in the literature [13]. Moreover, the contamination deposition process was simulated using the multiphysics simulation software Comsol. Based on the grey theory, X. Qiao et al. [14] established the insulator pollution model, but the result shows that there are still some errors. The error is the error between the calculated ESDD (equivalent salt deposit density) and the actual ESDD. The error mainly comes from the defect of the algorithm and the calculation accuracy fluctuates with the sample selection. Z. Zhang et al. [15] presented the contamination results using a volume fraction which was obtained by a Eulerian two-phase flow model and further proved the feasibility of this method. What is more, little literature has studied insulator contamination in the natural environment [19–21]. However, Z. Zhang et al. [19] pointed out that it will take long time and high expense to ge<sup>t</sup> reliable results.

In addition, the pollution model used in the power system is usually static, but the environmental parameters are dynamic. There is less study on the quantitative relationship between the pollution degree and the dynamic environmental parameters. Moreover, now in the power system, the actual insulator's NSDD (non-soluble deposit density) is usually determined by measuring the reference XP-160 insulator's NSDD. However, even in the same pollution condition, the pollution levels of insulators with di fferent structures are various.

Therefore, the dynamic pollution prediction model of insulators based on atmospheric environmental parameters was built, and insulator structure coe fficients were proposed based on the model in this paper.

#### **2. Dynamic Pollution Prediction Model of Insulators**

#### *2.1. Numerical Simulation Based on Eulerian Two-Phase Model*

The practicability of the method based on the Eulerian two-phase model in engineering has been verified in the literature [15,22–24]. In the Eulerian simulation model, the di fferent phases are treated mathematically as interpenetrating continua. In the simulation model, the standard k-ε model is used to describe the e ffects of turbulent fluctuations of velocities. The basic equations of the k-ε model have been given in our previous research [15]. Ti (turbulence intensity) can be calculated according to Re (Reynolds number). More specifically, Ti = 0.16Re−1/8.

In this paper, seven-unit 3D insulator string models are established by the Solidworks. The insulators' structure and structural parameters are shown in Table 1. The seven-unit 3D insulator string models are imported into Ansys, and a wind tunnel computational domain (9 m × 2.5 m × 7 m) is created according to the studies [13,25,26], as is shown in Figure 1a. One side of the computational domain is the air and particles inlet, and the other side is the flow outlet.


**Table 1.** Profile parameters of insulators.


**Table 1.** *Cont.*

(**a**) 


**Figure 1.** Calculation model and simulation results: (**a**) the calculation model in Ansys; (**b**) thesimulationresultsofU210BP/170,U210BP/170and XP-160.

Furthermore, a size function is attached to the four regions around the insulator to control the size of the grid cells, and these regions mesh with tetrahedral cells. The outer regions are meshed with hexahedron and prism cells to reduce the number of grids. Practical experience shows [15] that this grid meshing technique improves the calculation accuracy and reduces the cost of calculation time. Then, the boundary condition setting of the calculation domain is processed. The inlet of the

domain is set to the "velocity-inlet" boundary type, and the outlet of the domain is set to the "out-flow" boundary type.

The calculated results in this paper are shown in Figure 1b. When the initial concentration settings are the same (0.06), the pollution performance of the insulator was mainly affected by the particle diameter and wind speeds. Specifically, when the initial concentration is 0.06, the simulation results relationship between the volume fraction of the three insulators and the different particle diameters and different wind speeds are shown in Figure 2. It can be seen that the wind speed (*Wi*) and particle size (*dp*) have a grea<sup>t</sup> influence on the pollution performance, but the influence on each insulator are not the same. Besides, the Euler two-phase flow simulation led to steady-state results; therefore, it is necessary to establish a connection between the simulated accumulation results and the actual accumulated pollution results. Thus, the simulation results with the same environmental parameters are compared with the wind tunnel experiments. Finally, the NSDDs of each insulator string are obtained by simulation and comparison.

**Figure 2.** Simulation results: (**a**) the volume fraction with a different particle diameter; and (**b**) the volume fraction with a different wind speed.

#### *2.2. Dynamic XP-160 Pollution Model Based on Meteorological Data*

The pollution accumulation degree depends onmeteorological conditions. For example, the number of pollution particles adhered to the surface of insulator increases with the increase in pollution concentration. According to the corresponding meteorological data, the pollution amount of the insulator surface area under the condition of the pollution concentration can be obtained. In consideration of the time-varying dynamic change of atmospheric environmental parameters, the pollution amount of insulator surface area should be superposed by the accumulated pollution amount of each period, namely:

$$\begin{aligned} \Delta \phi\_{mi} &= \int\_{0}^{d\_{\rm pM}} \frac{c\_{pi}(d\_{p})}{c\_{p0}} \cdot t\_{i} \cdot \rho\_{m}(V\_{i\nu} d\_{p}) dt\_{p} \\ \phi\_{m}(H) &= \sum\_{i=1}^{N} \Delta \phi\_{mi} H = \sum\_{i=1}^{N} t\_{i} \end{aligned} \tag{1}$$

where, *cp*0 is the reference concentration, 15 mg/m3. *cpi*(*dp*) is the concentration corresponding to the polluted particles with the particle size of *dp* in the *i* time period, mg/m3. *Vi* is the wind speed in this time period, m/s; *ti* is the time length in the *i* time period. <sup>ρ</sup>*m*(*Vi*,*dp*) is the pollution accumulation per unit time on the insulator surface. *dpM* is the maximum particle size of the atmospheric particles in the *i* time period, μm. ΔΦ*mi* is the pollution increment of the insulator surface area in the *i* time period, mg/cm2. *H* is the total time in each time period. Φ*m*(*H*) is the final accumulated pollution amount of the polluted particles on the insulator surface, mg/cm2.

The *cpi*(*dp*) cannot be directly measured, but it needs to be obtained through the relationship between the particle size and its mass concentration, but the relationship is difficult to measure and obtain. Therefore, the approximate method is adopted, and it is considered that the mass fraction and particle size of the polluted particles meet the rosin rammer distribution:

$$
\lambda\_i(d\_p) = 1 - \exp\left(-n\_2 \cdot d\_p^{n\_1}\right) \tag{2}
$$

where <sup>λ</sup>*i*(*dp*) is the mass fraction of the polluted particles, whose particle size is less than *dp* in the *i* time period. *n*1 is the distribution characteristic index. *n*2 is the distribution characteristic coefficient.

The meteorological department usually classifies the polluted particles according to the air quality index standard of real-time monitoring: PM2.5 (*dp* < 2.5 μm), PM10 (*dp* < 10 μm), TSP (total suspended particulate) (*dp* < 100 μm). The units of these three parameters are μg/m3. According to the data measured in the *i* time period, the PM2.5, PM10 and TSP can be obtained:

$$\begin{aligned} \text{PM2.5/TSP} &= 1 - \exp\left(-n\_2 \cdot 2.5^{n\_1}\right) \\ \text{PM10/TSP} &= 1 - \exp\left(-n\_2 \cdot 10^{n\_1}\right) \\ 1 &= 1 - \exp(-n\_2 \cdot 100^{n\_1}) \end{aligned} \tag{3}$$

By fitting Equation (3), *n*1, *n*2 can be obtained, and then the mass fraction particle size distribution function of the *i* time period can be obtained. Taking the air pollution monitoring data of an area as an example, the TSP, PM10 and PM2.5 measured in the period *i* are about 200 μg/m3, 120 μg/m3, 24 μg/m3, then the values of *n*1, *n*2 are 1.42 and 0.03, respectively:

$$
\lambda\_i(d\_p) = 1 - \exp\left(-0.03 \cdot d\_p^{1.42}\right) \tag{4}
$$

The fitting degree R<sup>2</sup> is 0.99, which shows that the fitting result is reasonable. After the mass fraction-particle size relationship is obtained, Equation (2) can be discretized to approximate the mass fraction size corresponding to each particle size, and then the concentration *cpi*(*dp*) can be obtained:

$$c\_{p\bar{n}}(d\_p) \approx \text{TSP} \cdot \left[\lambda \left(d\_p\right) - \lambda \left(d\_p - \Delta d\_p\right)\right] \tag{5}$$

In order to improve the calculation efficiency and take into account the accuracy, setting Δ*dp* as 1 μm. In general, the probability of an air pollution particle size less than 50 μm is 99%. Therefore, only the influence of a pollution particle size less than 50 μm needs to be considered in the prediction of pollution accumulation. Based on the discretization of Equation (5), the following results are obtained:

$$\begin{aligned} c\_{pi}(1) &= \text{TSP} \cdot \left[ 1 - \exp(-n\_2 \cdot 1^{n\_1}) \right] \\ c\_{pi}(2) &= \text{TSP} \cdot \left[ -\exp(-n\_2 \cdot 2^{n\_1}) + \exp(-n\_2 \cdot 1^{n\_1}) \right] \\ &\dots \\ c\_{pi}(50) &= \text{TSP} \cdot \left[ -\exp(-n\_2 \cdot 50^{n\_1}) + \exp(-n\_2 \cdot 49^{n\_1}) \right] \end{aligned} \tag{6}$$

Taking the particle mass fraction–particle size distribution function obtained in Equation (4) as an example, and using the method of Equation (6) to discretize, the pollution particle concentration *cpi*(*dp*) corresponding to each *dp* value can be obtained, as shown in Figure 3:

Taking Equation (6) into Equation (1) to ge<sup>t</sup> the final amount of air pollution particles on the insulator surface after *H* time of pollution accumulation:

$$\phi\_m(H) = \sum\_{i=1}^{N} \Delta \phi\_{mi} = \sum\_{i=1}^{N} \left( \sum\_{n=1}^{50} \frac{c\_{pi}(d\_p)}{c\_{p0}} \cdot t\_i \cdot \rho\_m(V\_{i\prime}, d\_p) \right) \tag{7}$$

As mentioned before, <sup>ρ</sup>*m*(*Vi*,*dp*) is the pollution accumulation per unit time on the insulator surface, which can be obtained through several ways, such as the numerical simulation of pollution deposition, wind tunnel tests and nature tests. Furthermore, the pollution degree and insulator structure pollution coefficients (the coefficients will be discussed in the discussion section) can be calculated according to the flow chart based on the proposed model in this paper, as is shown in Figure 4:

**Figure 3.** An example of the calculation result of the particle concentration–diameter relationship.

**Figure 4.** Flow chart of the pollution degree and the coefficients calculation.
