*Article* **Wildfire Risk Assessment of Transmission-Line Corridors Based on Naïve Bayes Network and Remote Sensing Data**

**Weijie Chen 1, You Zhou 1,\*, Enze Zhou 2, Zhun Xiang 2, Wentao Zhou 1 and Junhan Lu 1**

	- 2 Electric Power Research Institute, Guangdong Power Grid, Guangzhou 510080, China<EMAIL_ADDRESS>(E.Z<EMAIL_ADDRESS>(Z.X.)

**Abstract:** Considering the complexity of the physical model of wildfire occurrence, this paper develops a method to evaluate the wildfire risk of transmission-line corridors based on Naïve Bayes Network (NBN). First, the data of 14 wildfire-related factors including anthropogenic, physiographic, and meteorologic factors, were collected and analyzed. Then, the relief algorithm is used to rank the importance of factors according to their impacts on wildfire occurrence. After eliminating the least important factors in turn, an optimal wildfire risk assessment model for transmission-line corridors was constructed based on the NBN. Finally, this model was carried out and visualized in Guangxi province in southern China. Then a cost function was proposed to further verify the applicability of the wildfire risk distribution map. The fire events monitored by satellites during the first season in 2020 shows that 81.8% of fires fall in high- and very-high-risk regions.

**Keywords:** wildfire; risk assessment; Naïve bayes; transmission-line corridors

#### **1. Introduction**

Due to the inhomogeneous distribution of energy resources and power loads in China, a large number of overhead transmission lines pass through forests and mountains to achieve an optimal allocation of power resources [1,2]. Wildfires are prone to occur in transmission-line corridors during the high-incidence periods. If a wildfire occurs under an overhead transmission line, the high temperature and smoke in wildfire would reduce the insulating strength of air gap drastically and induce a trip fault of transmission line. In addition, the reclosing usually fails under wildfire conditions with continuous high temperature, which seriously endangers the reliable operation of power grid [3–5].

To improve the wildfire prevention capability of transmission lines, scholars carried out much research, including distribution analysis of widfire occurrence [6,7], transmissionline trip mechanisms caused by wildfire [8,9], fire-spot monitoring algorithms [10–13] and wildfire risk assessment methods [14–16]. Remote sensing satellite is an economical and effective way to monitor wildfires in transmission-line corridors continuously. After identifying the flame strength of wildfire, the tripping risk would be evaluated by comparing the height of fire and transmission line [17,18]. However, this method needs numerous land-surface and environmental parameters to estimate the possible height of wildfire. Once a wildfire occurs near the transmission lines, it may cause a trip within tens of minutes. Furthermore, multiple wildfires usually occur simultaneously during the period of the Spring Festival, Qingming Festival, and autumn harvest in China. It is difficult for operation and maintenance personnel to rush the field and put out the fire in time. Therefore, it is necessary to assess occurrence probability of wildfire to propose differentiated wildfire prevention strategies.

**Citation:** Chen, W.; Zhou, Y.; Zhou, E.; Xiang, Z.; Zhou, W.; Lu, J. Wildfire Risk Assessment of Transmission-Line Corridors Based on Naïve Bayes Network and Remote Sensing Data. *Sensors* **2021**, *21*, 634. https://doi.org/10.3390/s21020634

Received: 16 December 2020 Accepted: 14 January 2021 Published: 18 January 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

The wildfire risk assessment was initially proposed by forest and fire departments to supporting decisions or policies for general fire, forest management, and fire suppression. Forestry departments in the US and Canada evaluated the large-scale wildfire risk by combining meteorological factors [19,20]. Nevertheless, these meteorology-based assessment methods cannot meet the requirement of spatial accuracy for transmission-line corridors due to the geographical differences. In 2016, State Grid Corporation of China issued a standard of drawing guidelines (DG) for region distribution map of wildfires near overhead transmission lines. In this standard, fire-spot densities and vegetation burning hazard grades are introduced to construct a risk assessment matrix [21]. However, the wildfire occurrence risk is affected by multi-dimension factors [22]. Besides meteorology and vegetation, physiographic and human-related factors are believed to have an important role in affecting wildfire occurrences [23]. Unfortunately, due to the complexity of wildfire occurrence there is still no physical model that could assess wildfire risk with specific variables.

Bayesian network is an effective approach to estimate uncertainty in risk evaluation in terms of the likelihood of risks and hazards [24]. In this paper, we aim to propose a wildfire risk assessment model based on Naïve Bayes Network (NBN) and remote sensing data. The region of Guangxi province which locates in southern China, is selected as the study area. A total of 14 sub-categories of wildfire-related factors including anthropogenic, physiographic, and meteorologic factors are collected. Then the spatial data are divided into grids of 1 km × 1 km to meet the spatial accuracy requirement of power grid. Considering the historical wildfire occurrences, the grids are divided based on whether there have been wildfires. After the importance evaluation by the relief algorithm, an NBN-based model is built with the optimal factor subset to map the wildfire risk distribution of Guangxi province. The wildfire risks are then divided into four levels based on the wildfire occurrence probability. In addition, a cost function is proposed to evaluate the applicability of wildfire risk map.

#### **2. Study Area and Data Collection**

#### *2.1. Study Area*

Guangxi province is in South China with the latitude of 20◦54 –26◦24 and the longitude of 104◦28 –112◦04 . The total area is 237,600 square kilometers. It is located on the southeastern edge of the Yunnan–Guizhou Plateau. In addition, the terrain is high in the northwest and low in the southeast. The region of Guangxi province is dominated by subtropical and tropical monsoon climate, in which the precipitation is synchronous with high temperature. With a large forest area, Guangxi is vulnerable to wildfire disasters due to its changeable climate, complex topography, and various vegetation.

#### *2.2. Wildfire-Related Factors*

#### 2.2.1. Anthropogenic Factors

Based on survey, more than 90% of wildfires are linked directly or indirectly to intentional and unintentional human activities [25]. It mainly includes productive fires such as wasteland and slash burning, and non-productive fires such as smoking in the wild and incensing on the grave [26]. To represent the influence of human activities, five kinds of anthropogenic sub-factors, which are Distance to Roads (DR), Distance to Settlements (DS), Population Density, Gross Domestic Product (GDP) and Historical Fire-Spot Density, are selected.

Human settlements and roads are the main areas of human activities, in where there are more human-caused fires. Population density reflects the regional population aggregation. A high population density generally corresponds to more human activities and consequently high probability of wildfire occurrence [27]. However, this law may be only suitable to explain the fire events in forests and rural areas. In large urbans, the higher population density may cause the less fire occurrence due to lack of fuels [28]. GDP represents the economic status of regions, which affects the human's fire habits. The data of

roads, settlements, population density were downloaded from the website of the Resource and Environment Science and Data Center (RESDE) with a resolution of 1 km × 1 km. In addition, the DR and DS of grids were calculated by using ArcGIS 10.4.

Historical Fire-Spot Density represents the spatial distribution of wildfires in the past few years. It is related to not only human activities but also other meteorological and physiographic factors. The database of historical fire-spots was monitored by polar orbiting meteorological satellites from 2010 to 2019, which is provided by the National Meteorological Center. The fire-spot data during 2010–2014 is used to calculate the Historical Fire-Spot Density of grids, whereas the remains (from 2015 to 2019) are used to train and test the NBN model. To calculate the Historical Fire-Spot Density, the study area is divided into grids of 2.5 km × 2.5 km first. Then the fire-spots from 2010–2014 were allocate into grids based on their longitude and latitude. In addition, the final Historical Fire-Spot Density of grids was obtained by using Kriging interpolation method from the resolution of 2.5 km × 2.5 km into 1 km × 1 km.

#### 2.2.2. Physiographic Factors

The physiographic factors include the land cover and landscape parameters of grids. The land cover factors are consisted of the Land-Usage Type, Vegetation Type, Fuel Load and Normalized Difference Vegetation Index (NDVI), whereas the landscape parameters are average Elevation, Slope, and Aspect of underlying surface.

The vegetations provide the fuel basis for wildfires' development. Different types of vegetations differ from their burning capacity in fire ignition and spread. Wildfires prone to ignite and spread rapidly at woodlands, shrubberies and meadows [29]. Therefore, the Land-Usage Type and Vegetation Type are taken to describe the flammability of the underlying surface. To construct the NBN model, the Land-Usage Type and Vegetation Type of study area are classified into four levels according to their flammability, which are shown in Tables 1 and 2, respectively. The Fuel Loading is represented by the drying weight of fuel per unit area, which affects the spread velocity and flame intensity of fires. The NDVI represents the coverage of surface vegetation. It is another representative parameter about fuel contents on the underlying surface.

**Table 1.** Classification of Vegetation Types.

**Table 2.** Classification of Land-Usage Types.

Complex topography influences not only the distribution of vegetations, but also the spreading behavior of fires directly. The elevation brings differences of temperature and humidity to affect the composition of vegetations. Moreover, human population tends to cluster at the region with low elevation and gentle slopes, which increases the fire activities. In addition, the slope also has a direct impact on the spreading speed of wildfires [30]. The increase of slope leads to the faster surface runoff, which is beneficial to the drying of

vegetations. Slope aspect determines the amount of solar radiation, therefore the humidity of atmosphere and vegetations.

The data of Fuel Loading was obtained from the National Meteorological Center. In addition, other land cover factors and landscape parameters were downloaded from RESDE. All the data resolution are 1 km × 1 km.

#### 2.2.3. Meteorologic Factors

Annual Precipitation and Temperature which have great influences on vegetation growth are selected as the meteorologic factors. In the high precipitation regions, the growth of vegetations is flourishing, which provides fuel conditions for wildfires to burn. However, the reduced transpiration in these areas increases the humidity of vegetation and reduces the flammability. The water-holding capacity of soils and air humidity are also increased. The fires' ignition and spread are therefore restrained. The higher temperature in forests generally benefits plant growth. In addition, the increase of temperature also accelerates the transpiration of vegetations, which promoting the rapid drying of vegetations. The data of Annual Precipitation and Annual Temperature were obtained from the RESDE, with a resolution of 1 km × 1 km.

#### *2.3. Sample Preparation and Pre-Processing*

The influencing degree of wildfires in transmission-line corridors varies with the distance. Generally, the fire that is 1 km away from the transmission-line is regarded as the highest risk, whereas the fire that is 3 km away is assumed to be no impact on the operation of transmission-line [31]. To meet the spatial accuracy requirement, the study area is divided into 1 km × 1 km grids. Then all the wildfire-related factors are allocated into grids. Specifically, the grids which have monitored fire-spots from 2015–2019 are taken as the fire samples. Considering the ignition, spread, and extinction of wildfire last for several hours, those monitoring fire-spots within 4 h and 3 km are regarded as a same fire-spot. Meanwhile, a same number of the non-fire samples are sampled randomly from the rest grids of 3 km away from the grids of the fire samples.

Compared to processing continuous factors, the Bayesian model has a higher efficiency and better robustness for discrete factors [24]. Therefore, the equal frequency method integrating empirical knowledge was used to discretize the factors. The standards of factor discretization are shown in Table 3.

**Table 3.** Factor discretization standards.

#### *2.4. Spatial Distribution of Factors*

The factor distributions of the study area are visualized by using the ArcGIS software, as shown in Figure 1. It can be seen from the spatial distribution map that the historical wildfire high-incidence areas mainly distribute in the northwest, east, and central parts of Guangxi province. The distribution of GDP is similar to that of population density. The population concentrates in the vicinity of large municipal districts such as Nanning City and Liuzhou City in the central part, and Wuzhou City in the east. The elevation, slope, and NDVI in Guangxi province are relatively similarly distributed. Guangxi province has a wide distribution of karst landforms with mountains and hills. The west and north of Guangxi province are adjacent to the Yunnan–Guizhou Plateau, in where the elevation and slope values are relatively large. As the elevation increases, the climate, hydrothermal conditions change, leading to the plant no longer flourish in these regions. In addition, the higher elevation and steeper slope at the northern of Guangxi province make the less human settlements and higher vegetation coverages and higher NDVI values in this region. For the meteorological distribution, the annual temperature in the northern plateau is slightly lower. That is caused by the increase of latitude and elevation in plateau. The precipitation distribution shows that a circular decreasing trend from the east to the west. In addition, the annual precipitation in the northeast of Guangxi province is the largest. In Guangxi province, sparse forests and shrub forests are the main vegetation and land-Usage Type. In addition, coniferous forests and theropencedrymions mainly locates in the eastern region of Guangxi province, whereas the western region is dominated by broad-leaved forests and shrubs.

**Figure 1.** *Cont.*

**Figure 1.** *Cont.*

(**m**) Annual precipitation (**n**) Annual temperature

**Figure 1.** Spatial factor distribution of Guangxi province.

#### **3. Importance of Wildfire-Related Factors**

Wildfire occurrences are influenced by many feature factors. However, the relationship among these factors and wildfire is complex and differs from the location of regions [32]. Some of wildfire-related factors may play an important role in some regions but contribute little on the occurrence probability of wildfires in other regions. In addition, they even introduce noisy information for risk analysis in this region. Moreover, the redundancy of data increases the complexity of model and reduces the evaluation performance. A feature selection method, the relief algorithm, was used to appraise the contribution importance of wildfire-related factors on the wildfire occurrences before model building.

The relief algorithm was first proposed by Kira and Rendell [33], which is a factor weighting algorithm for binary classification based on the correlation between factors and the sample classification. Figure 2 gives the basic idea about how the relief algorithm appraises the importance of two factors. For a random sample *Si* and its factor *xk*,*D*(*xk*, *Si*, *SNM i* ) represent the distance between and *Si* and its closest different-class sample *SNM i* . In addition, *D*(*xk*, *Si*, *SNH i* ) represent the distance between *Si* and its closest same-class sample *SNH i* . If the same-class distance is greater than the different-class distance, the factor is more useful to classify the sample and should be given a higher weight. In Figure 2, if the random *Si* is a fire sample, then the factor *x*1 has a smaller distance from its same-class samples than that of the factor *x*2, indicating the factor *x*1 is more important than the factor *x*2.

**Figure 2.** Basic idea of Relief algorithm.

The basic steps of the relief algorithm are as follows:

$$D(\mathbf{x}\_{\circ}, \mathbf{S}\_{i\prime}, \mathbf{S}\_{i}^{NM}) = \begin{cases} \begin{array}{ll} 0 & \mathbf{x}\_{\circ} \neq \mathbf{x}\_{\circ}^{NM} \\ 1 & \mathbf{x}\_{\circ} = \mathbf{x}\_{\circ}^{NM} \end{array} \tag{1} \\ \end{cases}$$

else,

$$D(\mathbf{x}\_{\circ}, \mathbf{S}\_{i\prime}, \mathbf{S}\_{i\prime}^{NM}) = \left| \frac{\mathbf{x}\_{\circ} - \mathbf{x}\_{j}^{NM}}{\max(\mathbf{x}\_{j}) - \min(\mathbf{x}\_{j})} \right| \tag{2}$$

In this study, the most of factors are continuous variables, except the Vegetation Type and Land-Usage Type.

(4) Update the weight of factor *xj* after multiple sampling

$$
\omega\_j^\* = \omega\_j + \sum \left( \frac{D(\mathbf{x}\_{j'} S\_{i'} S\_i^{NM})}{m} - \frac{D(\mathbf{x}\_{j'} S\_{i'} S\_i^{NM})}{m} \right) \tag{3}
$$

where *ωj* and *ω*∗ *j* are the initial and updated weight, respectively. *m* is the number of random sampling. The calculated weights of the 14 wildfire-related factors are listed in Table 4.

Based on relief algorithm, the DS, Vegetation Type, and DR shows the most three important impacts on distinguishing the fire and non-fire samples. This is mainly because of the plentiful of fire activities near the living and transportation regions of populations. In addition, the type of vegetation affects the ignition and spread of wildfires. Population Density, Annual Temperature, Slope, and GDP are the least four important factors. The population in Guangxi province is concentrated in the regions of cities, whereas the rest large parts of region are subtropical forests. In urban area, it is hard to inflame due to the lack of fuels and timely fire-fighting behavior. On the other hand, in sparsely populated forests, the happen probability of wildfire is also low due to the low human activities. The wildfire is likely to be happened at the interfacial region of forests and human settlements, which leads the population and GDP to fade into insignificance in this study. The study area is the provincial power grid in southern of China with a small temperature difference. Thus, the average temperature has little influence on the wildfire occurrence.

#### **4. Naïve Bayes Network-Based Wildfire Risk Assessment**

*4.1. Bayes Theorem and Independence Assumption*

Bayes' theorem was initially proposed by Thomas Bayes in the 18th century. It expresses the relationship between the conditional probabilities of two events statistically [34]. It has been widely used in uncertain fields such as disaster prediction, medical diagnosis, speech recognition, and so on. The Bayes' theorem is as follows:

$$P(X\_i|Y) = \frac{P(Y|X\_i) \cdot P(X\_i)}{\sum\_{j} P(Y|X\_j) \cdot P(X\_j)} \tag{4}$$

where *P*(*Xi*) is the priori probability which are obtained from the past experience or data distribution. *P*(*Y*|*Xi*) is the probability of event *Y* occurring under the condition of known event *Xi*. *P*(*Xi*|*Y*) is the probability of *Xi* when the result *Y* is known. Based on this theorem, the probabilities of wildfire-related factors and then the wildfire occurrence probability under certain conditions could be estimated statistically. However, the estimation of joint probability *P*(*Y*|*Xi*) is difficult due to the limited number of samples. Therefore, the Naïve Bayes Network (NBN), in which the factors are assumed to be independent with each other, is used to model the risk of wildfire occurrences. Even through it sacrifices the interaction of factors but still gets an acceptable performance of model in many applications.

#### *4.2. Model Construction*

The construction of a Bayes network includes the structure learning and the parameter learning. Due to the independence assumption, the structure of NBN is simplified into a directed acyclic graph with factor nodes connecting to a class node. In addition, the parameter learning process is as follows:

$$P(\mathbf{x}\_{i\bar{j}}|Y) = \frac{n\_{i\bar{j}}}{\sum\_{k=1}^{4} n\_{ik}} \tag{5}$$

where *xij* is the *i*th wildfire-related factor fall in *j*th level; and *nij* is the number of *xij*.*k* represented the discretization level of factors. *P*(*xij Y*= 0) and *P*(*xij Y*= 1) are the probabilities of *xij* for the girds with non-fires and fires, separately.

(3) The conditional probability *P*(*Y* = 1|*x*1, *x*2,..., *xn*) and *P*(*Y* = 0|*x*1, *x*2,..., *xn*) of samples in the testing subset were calculated based on the Bayes' theorem. In addition, the final wildfire occurrence probability *P(Y)* was obtained after normalization.

$$P(Y) = \frac{P(Y = 1 | \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n)}{P(Y = 1 | \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) + P(Y = 0 | \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n)} \tag{6}$$

(4) To test and optimize model's performance, the value of 0.5 was selected as the threshold to divide the samples into two classes: "Prone to fire" and "Prone to nonfire". The least important factors were then eliminated in turn to study the influence of factors composition on the model's performance.

**Figure 3.** Samples of testing subset in the Guangxi province.

#### *4.3. Model Assessment*

The confusion matrix is used to evaluate the assessment performance of NBN model. It compares the predicted fire tendency of the testing subset with the actual wildfire events to measure the following four indexes, as shown in Table 5.

**Table 5.** Definition of confusion matrix.

*TP*: True Positive, represents the number of fire events correctly predicted as "Prone to fire".

*TN*: True Negative, indicates the grids where are non-fire and that were identified as "Prone to non-fire", correctly.

*FN*: False Negative, reflects the fire events that were identified as "Prone to non-fire", mistakenly.

*FP*: False Positive, recounts the grids where are non-fire but that are predicted to be "Prone to fire".

It is obviously that larger shares of *TP* and *TN* indicate a better predicted performance of model. Therefore, the indexes of *Accuracy Pa*, *Recall Pr*, *Precision Pp* and a more balanced index *F*-score are introduced on the basis of the confusion matrix.

$$P\_a = \frac{TP + TN}{TP + TN + FP + FN} \tag{7}$$

$$P\_I = \frac{TP}{TP + FN} \tag{8}$$

$$P\_p = \frac{TP}{TP + FP} \tag{9}$$

$$F = \frac{(1+\beta^2)P\_pP\_r}{\beta^2P\_p + P\_r} \tag{10}$$

where *β* reflect the attention degree of power grids on fault tolerance. Considering the wildfire would induce outage of transmission lines, bring huge economic losses, and even casualties, this study takes *β* as 3.

As mentioned in Table 4, the weights of wildfire-related factors have been evaluated by using relief algorithm. Then the NBN models were re-trained and re-tested by eliminating the least important wildfire-related factor one by one. The assessment performances of models are shown in Figure 4.

**Figure 4.** Variation of assessment indexes with wildfire-related factors.

With the reduced number of wildfire-related factors, the impact of noise, which is bring by unimportant factors, on the model is gradually reduced, which leading to the improvement of *Accuracy Pa* and *Precision Pp*. The *Accuracy Pa* is only 70.14% when all 14 wildfire-related factors are used. It reaches the maximum of 75.93% when the number of factors is reduced to six. The *Recall Pr* and *F*-score remain about 81% until the used factor number less than six. However, the performance of NBN model deteriorates significantly when the number of wildfire-related factors reduces from six to five, indicating that the remaining factors have greatest impacts on the wildfire occurrence.

The *F*-score reaches the highest value of 81.23% when eight wildfire-related factors are used. The results of confusion matrix are listed in Table 6. 81.92% of actual fire events in the testing subset are predicted correctly. The eight wildfire-related factors include three anthropogenic factors (DS, DR, Elevation and Fire-spot density), three physiographic factors (Vegetation Type, land-Usage Type, and NDVI) and only one meteorological factor (Annual Precipitation). The composition of important wildfire-related factors indicates an important role of human activities in increasing the risk of wildfires. Based on survey, more than 90% of wildfires in Guangxi province are human-caused, deliberately and unintentionally.

**Table 6.** Results of confusion matrix when 8 wildfire-related factors are used

#### **5. Visualization of Wildfire Risk and Discussion**

To guide the wildfire prevention of transmission corridors, the wildfire occurrence probability is calculated in all 1 km × 1 km grids of the study area. The wildfire risks of grids are then divided into four levels based on the probability: Low-(0% ≤ *p* < 25%), Medium- (25% ≤ *p* < 50%), High-(50% ≤ *p* < 75%) and Very-high-(*p* ≥ 75%) risk. The conditional probability distribution of wildfire-related factors and an example of probability inference for grids are shown in Figure 5. The remarkable difference of conditional probability

distribution gives a visualized explanation of why and how these wildfire-related factors affect wildfire occurrence.

**Figure 5.** Conditional probability distribution of wildfire-related factors and an example of probability inference.

The NBN-based wildfire risk distribution of Guangxi province is mapped by using the ArcGIS software (Figure 6a). For comparison, another wildfire risk distribution map is drawn according to the DG of State Grid Corporation of China (Figure 6b) [21]. In addition, a total of 527 fire events, which were monitored by remoting satellite during the first season in 2020, allocate on the maps.

**Figure 6.** Wildfire risk distribution in Guangxi province.

It can be observed in the NBN-based map that the very-high-risk regions mainly locate at the northwest, south, and east of Guangxi province, which has a high spatial consistency with the historical fire-spot density distribution. In addition, the northwest of Guangxi province is in the border of Yunnan–Guizhou Plateau. The higher elevation and less annual precipitation result in low moisture in air and vegetation, which is the flammable condition for wildfires. In the eastern and southern regions, the population density is higher, and the settlements are widely distributed, therefore more fire-using activities. In addition, lowrisk regions distribute at the middle of Guangxi province, which are generally municipal districts with few fuels and the deep forests without human activities.

The distribution of very-high-risk regions in DG-based map are almost the same as that in NBN-based map, but few low-risk areas are observed in the DG-based map. The area proportions of risk levels, as well as the location proportions of fire events, are summarized in Table 7. Due to the huge ratio of forest in Guangxi province, more areas are assessed as high-and very-high-risk level by both NBN-based map and DG-based map. Compared to NBN-based map, the area proportions of risk levels in DG-based map are more inhomogeneous. More than half of regions in Guangxi province are assessed to be high risk.

**Table 7.** Comparison of NBN-based map and DG-based map

Table 7 also gives the proportions of fire events in risk level regions. 45.35% of fire events distribute in the very-high-risk regions of NBN-based map, whereas 32.45% in that of DG-based map, indicating a higher prediction precision of NBN-based map. However, it should be noticed that the higher predicted precision of NBN-based map may be caused by the bigger proportion of very-high-risk region, which are 31.67% and 23.00% for the NBN-based map and DG-based map, respectively. A larger area of high-risk region indicates a greater probability of fire event happening in the region, which leads to the better prediction of model. On the other hand, the larger area of high-risk region needs

more costs for wildfire rescue and management. To balance the contradiction of prediction precision and management cost, a cost index -score is proposed.

$$R = \sum\_{i=1}^{4} \left( k\_i \times \mathbb{S}\_i + f\_i \times N\_i \right) \tag{11}$$

where *ki* are the maintenance cost for the region with the *i*th risk level; The maintenance costs of different risk levels need further studies in power grid cases, and are simplified to 1, 2, 4, 8 for the low-risk, medium-risk, high-risk, and very-high-risk regions, respectively. *Si* is the area proportion of the ith risk region. In addition, *fi* represents the misjudgment cost when the wildfires are happened in the *i*th risk level. If a fire happened in the low-risk and medium-risk region, it is likely to be enlarged and bring disaster cost due to negligence of management. Thus, *fi* are set as 8, 8, 2, and 0 for simplicity. The *R*-score of NBN-based map is 6.15, which is lower than that of DG-based map (6.62). The reduced cost also indicates the applicability of NBN-based map.

#### **6. Conclusions**

This study develops a spatial framework to assess and map wildfire risk of transmissionline corridors by integrating remote sensing data. The proposed NBN-based wildfire risk assessment combines empirical knowledge and machine learning into the discretization of input factors, construction of conditional probabilities of wildfire-related factors, and mapping of wildfire risk distribution. NBN as a core algorithm is used to infer the probabilities of wildfire occurrence. The remote sensing data including a total of 14 sub-categories of wildfire-related factors is assembled to construct NBN model. Based on the relief algorithm, the number of key wildfire-related factors is reduced into 8, indicating human activities and fuels at underlying surface play more important roles in wildfires occurrence. This spatial framework was implemented in a case study of Guangxi province in the south of China. In addition, a cost index, R-score, is proposed to reflect the maintenance costs and misjudgment costs. The results show that the NBN-based wildfire risk map has a higher prediction precision and lower costs for power grids than the traditional method. 45.35% of new monitored fire events distribute at the very-high-risk regions. The visual wildfire risk distribution can assist decision maker of power grids to optimize both supplies and staff resources and make strategies for responding damage control in the future.

**Author Contributions:** Funding acquisition, Z.X.; Investigation, E.Z.; Methodology, W.Z.; Project administration, Y.Z.; Software, W.Z.; Supervision, Z.X.; Writing—original draft, W.C.; Writing—review & editing, Y.Z. and J.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research was funded by the Science and technology projects funded by China Southern Power Grid Corporation for supporting this research under Contract No. GDKJXM20198386, by the Changsha University of Science and Technology Professional Master's Innovation Project SJCX202045, by the Changsha University of Science and Technology Academic Master's Research and Innovation Project CX2020SS56, by the Hunan Province Natural Science Foundation of China under Grant No. 2019JJ50658, and by the Scientific Research Foundation of Hunan Education Department under Grant No.20B007 for scientific research.

**Institutional Review Board Statement:** No statement.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** No statement.

**Acknowledgments:** The authors would like to thank Sanyi Sui from the Changsha University of Science and Technology for his helpful suggestions. The authors also would like to thank the anonymous reviewers for their constructive suggestions and comments.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


2.1. Heat pipe construction

Additionally, the single-phase systems need to working fluid circulation mechanical pumps or fans, while in the capillary-driven two-phase systems the working fluid circulate without any additional mechanical devices, which make such systems more reliable and free of electric power. The best known capillary-driven two-phase system is heat pipe and its schema is shown in Figure 1. The concept of the heat pipe was first suggested by Gaugler in 1944 [1] and by Trefethen [2] but was not widely publicized until serious development work by Grover and his co-worker in Los Alamos laboratory [3]. Heat pipes are passive heat transfer devices that transport heat from one point (heat source) to another (heat sink) with extremely high thermal conductance due the latent heat of vaporization of a working fluid. As shown in Figure 1, it consists of container, working fluid, wick structure and has three sections (an

Since one of the most important parts of the heat pipe HP and loop heat pipe LHP is the porous wick structure, this work is focused on experiments influencing porous wick structure on the

The heat pipe is heat transfer device using the phase-change of working fluid to transfer heat from a heat source to heat sink and capillary forces generated in the wick structure to working fluid circulation. The heat pipe consists of hermetically closed container with wick structure on the inner surface and working fluid near its saturation temperature. Heat absorbed through the container to the liquid in the evaporator causes liquid evaporation and vapor flow through the open core of the heat pipe evaporator. The vapor flows out of the evaporator through the adiabatic section into the condenser. There the vapor condenses and released heat is transfer through wick structure and container wall into condenser ambient. Condensed liquid saturates the wick structure and creates capillary pressure; thus the liquid is pumped back into the evaporator. The operation of the heat pipe depends mainly on the parameters of container, working fluid and wick structure. Proper selection and design of the main heat pipe parts influence its operational characteristics defined by the heat transfer limitations, effective

evaporator section, an adiabatic section, and a condenser section) [4].

heat transport ability of the heat pipe and loop heat pipe.

2. Heat pipe

Figure 1. Schema of heat pipe.

142 Porosity - Process, Technologies and Applications

The heat pipe may have several basic parts depending on its type. During the heat pipe development, the main components and materials remained the same. The simplest type of heat pipe consists of two basic parts, the body (container) and the working medium. A capillary structure (wick) can be placed inside the heat pipe body to allow the condensed liquid phase of the working fluid wicking against the vapor flow due the capillary action. Such a heat pipe is called a wick heat pipe. The heat pipe without capillary structure, is called gravitational heat pipe because it returns the liquid phase from the condenser part to the evaporator part which is due to gravity [5].


Meet the editor

Ezgi Günay has been an associate professor at the Mechanical Engineering Department at Gazi University in Turkey since 2010. She graduated from the Engineering Sciences Department (Applied Mechanics Department) of the Engineering Faculty at the Middle East Technical University (METU, Ankara, Turkey) in 1985. She received her Master of Science degree in 1989 from the same department. The title of the thesis was "Development

of a Preprocessor and Modification of a Finite Element Procedure for the Analysis of Metal Forming Processes" (December 1989, METU). She graduated with her PhD degree at Gazi University from the Mechanical Engineering Department in 1996. The title of the thesis was "A Nonlocking Finite Element Model for Nonlinear Analysis of Thin and Thick Composite Plates." She worked as a research assistant between 1993 and 1999 and studied academically by giving basic courses as an assistant professor between 1999 and 2010 at the same department. During these years she has given courses in the following subjects: Technical Drawing, FOR-TRAN-Computer Programming Languages, Applied Mathematics for Mechanical Engineers, Differential Equations, Statics, Dynamics, Strength of Materials, Introduction to Numerical Analysis, Introduction to Composite Materials, Introduction to Finite Element Analysis (FEA), Finite Element Method (FEM), Plate and Shell Theories, and Elasticity. She has authored about 40 papers published in both national and international proceedings and journals. She has three chapters published in international books. She has written two programming e-books in FORTRAN language (co-author) for engineering students.

Ezgi Günay is presently working on the following subjects: finite element linear and geometrically nonlinear analyses of fiber composite-stiffened shell and plate structures, linear and nonlinear FEA buckling analysis of fiber composite-stiffened plates, numerical and experimental studies on micromechanical investigation of layered fiber composites and natural fiber composite wood, FEA and analytical studies on single-walled carbon nano-tubes (SWCN), FEA of stress transfer mechanisms from matrix to fiber in SWCN-reinforced nanocomposites, and experimental and FEA studies on natural fiber composite wood (transversely isotropic) materials.

**Preface III**

General Theorems in Elasticity **1**

**Chapter 1 3**

**Chapter 2 11**

Engineering Applications in Theory of Elasticity **31**

**Chapter 3 33**

**Chapter 4 53**

**Chapter 5 73**

Repair Inspection Technique Based on Elastic-Wave Tomography Applied

*by Katsufumi Hashimoto, Tomoki Shiotani, Takahiro Nishida and Nobuhiro Okude*

FEA and Experimentally Determination of Applied Elasticity Problem

Introductory Chapter: Analytical and Numerical Approaches in

An Overview of Stress-Strain Analysis for Elasticity Equations *by Pulkit Kumar, Moumita Mahanty and Amares Chattopadhyay*

Concept of Phase Transition Based on Elastic Systematics

**Section 1**

Contents

Engineering Elasticity

for Fabricating Aspheric Surfaces

*by Paul S. Nnamchi and Camillus S. Obayi*

for Deteriorated Concrete Structures

*by Duc-Nam Nguyen*

*by Ezgi Günay*

**Section 2**

## Contents

Preface

During the nineteenth century, mechanics was mainly considered to be a part of applied mathematics. At the beginning of the twentieth century, all topics under the subject of mechanics, "solid mechanics" and "fluid mechanics," were defined with an index notation, and by re-explanation of these issues according to their coordinate systems, continuum mechanics has been created. With the gathering of all subjects in a single title, a new area has been opened in macroscopic (largescale) mechanics. The exact solutions to linear/nonlinear elasticity problems have been discovered by applying analytical, numerical, and experimental methods and techniques to the new generation of computers and experimental equipment. Numerical solutions such as finite differences and finite elements methods have

The purpose of writing this book has been summarized in three main concepts. The first objective was to provide the basic information and principles about the history and theory of elasticity. The second was to explain the fundamental equations. The third was to present the recent engineering application studies that have been collected under the following basic headings: "FEA and Experimental Determination of Applied Elasticity Problems for Fabricating Aspheric Surfaces," "Repair Inspection Technique based on Elastic-Wave Tomography Applied for Deteriorated Concrete Structures," and "Concept of Phase Transition Based on

The main subject of this book is engineering elasticity and consists of five chapters in two sections. The title of the first section is "General Theorems in Elasticity" and the first chapter of this section belongs to the editor and is entitled "Analytical and Numerical Approaches in Engineering Elasticity." In this chapter, the historical development of "elasticity theory" is presented briefly, and recent studies performed regarding the elasticity concept are categorized and listed according to their basic engineering problem groups. A literature survey has been performed and categorized between the years 2014 and 2018 and represented in a statistical plot.

The second chapter in the first section is entitled "A General Overview of Stress-Strain Analysis for the Elasticity Equations" and explains the results of elasticity equations and the analysis of stress, strain, and stress–strain relationships through particular sections. In this section the concept of normal and shear stresses, principal stresses, plane stress, Mohr's circle, stress invariants, stress equilibrium equations, linear elasticity, generalized Hooke's law, and stress–strain relationships for triclinic, monoclinic, orthotropic, transversely isotropic, fibre-reinforced, and isotropic materials are discussed by researchers P. Kumar, M. Mahanty, and A.

The title of the second section is "Engineering Applications in Theory of Elasticity"

Determination of Applied Elasticity Problems for Fabricating Aspheric Surfaces"

and consists of three chapters on engineering applications on elasticity.

The first chapter of the second section is entitled "FEA and Experimental

been extensively used to approximate the exact results.

Elastic Systematics."

Chattopadhyay.


**Contents**

## **About the Special Issue Editor**

**Luis Martinez-Salamero** taught circuit theory, analog electronics, and power processing from 1978 to 1992 at the Escuela Tecnica Superior de Ingenieros de Telecomunicaci ´ on de Barcelona, Polytechnic ´ University of Catalonia, Barcelona, Spain. From 1992 to 1993, he was visiting scholar at the Center for Solid State Power Conditioning and Control, Department of Electrical Engineering, Duke University, Durham, NC. He continued his cooperation with Duke University for three months per year in 1995 and 1996. From 2003 to 2004, 2010 to 2011, and March–August of 2018, he was visiting scholar at the Laboratory of Architecture and Systems Analysis (LAAS), National Agency for Scientific Research (CNRS), Toulouse, France. Since 1995 he has been a full professor with the Department of Electrical Electronic and Automatic Control Engineering ,School of Electrical and Computer Engineering, Rovira i Virgili University (URV), Tarragona, Spain, where he managed the Research Group in Industrial Electronics and Automatic Control (GAEI) in the period 1998–2018 (http://deeea.urv.cat/gaei/index.php). His research interests include the structure and control of power conditioning systems, namely, the electrical architecture of satellites, power distribution in hybrid and electric vehicles, as well as the nonlinear control of converters and drives, and power conditioning for renewable energy. He has published a large number of papers in scientific journals and conference proceedings in the fields of modelling, simulation, and control of power converters, and holds a U.S. patent on dual-voltage electrical distribution in vehicles. He was Guest Editor of the IEEE Transactions on Circuits and Systems Special Issue on Simulation, Theory and Design of Switched-Analog Networks (Aug. 1997). In cooperation with the European Space Agency (ESA), he organized the 5th European Space Power Conference (ESPC-98) in Tarragona and served during two terms (1996–2002) as a Dean of the School of Electrical and Computer Engineering in URV. He has supervised 15 doctoral theses and given lectures in the universities of Toulouse, Bordeaux, and Tel-Aviv. He has conducted numerous seminars on the use of non-linear control in power converters in both Spanish and foreign universities, and has been invited speaker in four plenary sessions in international conferences. He was president of the Spanish Joint Chapter of the IEEE Power Electronics and Industrial Electronics Societies from 2005 to 2008, and distinguished lecturer of the IEEE Circuits and Systems Society in the period 2001–2002. He served as a president of the committee for Communications, Electronics and Computer Science for the Research Activity Evaluation (CNEAI) of the Spanish Ministry of Education in 2012, and as a national coordinator in the area of Electrical, Electronic and Automatic Control Engineering of the National Agency of Prospect and Evaluation (ANEP) of the Spanish Ministry of Science and Innovation in the period 2008–2011. He is currently a distinguished professor of Rovira i Virgili University.

## **Preface to "Sliding Mode Control of Power Converters in Renewable Energy Systems"**

The first paper on dc-to-dc switching converters published in a scientific journal had only one page and did not mention the term power converter in the description of that emergent family of circuits (1). Almost fifty years later, around 7000 patents have been issued on dc–dc conversion in the USA alone, making us consider whether the interdisciplinary field known as Power Electronics reached maturity long time ago, and if there is still space for innovation.

Some data can help us to answer affirmatively to the second question. They are supported by the concern for the global warming, which has renewed the interest in renewable energy technologies. Underpinning a sustainable environment implies the use of more efficient devices and better control strategies in the generation, transport, and conversion of electric energy. Two vectors catalyze the new research: advances in technology and the new paradigm in power distribution. In the first case, the increasing use of wide-gap power devices and digital processors is contributing to reduce the size, weight, and internal losses of power converters in parallel with better dynamic performances. In the second case, electric vehicles and smart grids are the main industrial actors of the new research, in which the notion of a single converter has been substituted by the concept of multi-converters (i.e., an important number of converters interacting with the energy sources or the loads, using intermediate elements called buses for ac or dc power distribution).

In this context, sliding-mode control continues to become established in the control of power converters since it constitutes a reliable and efficient solution to many open problems. It offers robustness in the face of parametric uncertainty, fast response, and systematic approach in the control design, which feeds on a rigorous general theory that has been successfully applied in other engineering fields like process control and electromechanical systems. This Special Issue of Energies reflects the state of the art of the sliding-mode control of power converters, and the selected papers are representative of different applications in the field of renewable energies.

The first paper illustrates the control design of a three-phase voltage source supplying dc loads without a rectifier. In this work, Alsmadi, Chairez, and Utkin develop switching commands for the power devices of a three-phase PWM AC/DC voltage source converter in such a way that the output voltage can track a desired positive time-varying function.

In the second paper, Yang and Tan exhaustingly cover the recent development of sliding-mode control applications for renewable energy systems and examine the current trends to improve their efficiency and load protections in the face of large-signal variations. A comparative study between sliding-mode control and proportional–integral control is presented in three cases (i.e., a low-power wind energy conversion system, a series–series compensated wireless power transfer system, and a multiple energy storage system in a DC microgrid).

The next paper addresses the design of the input filter of a power converter supplying a constant power load (CPL). Anderson, More, and Puleston present a Lyapunov analysis to tackle the ´ nonlinear internal dynamics of the sliding-mode controlled converter with CPL. Using a Lienard-type description, the authors establish the stability conditions and provide a secure operation region, which is eventually translated into filter design guidelines.

Tuning the parameters of a second-order sliding-mode control of the grid-side converter of a doubly-fed induction generator (DFIG) under unbalanced and harmonically distorted grid voltage is the subject of the fourth paper by Susperregui, Herrero, Martinez, Tapia-Otaegui, and Blasco. A multi-objective optimization is applied for tuning, and two versions of the control algorithm are compared. A Pareto front is eventually derived to help the designer to understand the trade-off among objectives and select the final solution.

A novel adaptive-gain second-order sliding mode direct power control strategy for a wind-turbine-driven DFIG is the subject of the fifth paper. Han and Ma present the control scheme in detail, derive the adaptive gains from a Lyapunov stability approach, and show the resulting reduction of the rotor voltage chattering. Active and reactive power regulation is attained under a two-phase stationary reference frame for both balanced and unbalanced grid voltage.

The dual-stator winding generator (DWIG) is a promising electrical machine for wind energy systems in the low/mid power range. In the sixth paper, Talpone, Puleston, Cendoya, and Barrado-Rodrigo propose a super-twisting algorithm sliding-mode control with moderate real-time computation burden to maximize power extraction during low wind regimes. The results show an extended range of operation of the machine, fast finite convergence for maximum power tracking, reduction of both chattering and mechanical stress, and robustness against parameter uncertainties.

Efficient lighting has become a must for either domestic or industrial applications, the discharge lamps being a representative example of the devices involved in the conversion of electricity into light. In the seventh paper, Valderrama-Blavi, Leon-Masich , Olalla and Cid-Pastor present a versatile ballast for discharge lamps of different type. The ballast has two stages. The first stage is a boost converter exhibiting loss-free resistor characteristics imposed by a sliding-mode control. The second stage is a resonant inverter supplying the discharge lamp at high frequencies. Successful ignition, warm-up and both nominal and dimming operations are illustrated in the paper.

An example of an inductive device with potential use in sea-wave energy harvesting is introduced in the next paper by Garriga-Castillo, Valderrama-Blavi, Barrado-Rodrigo, and Cid-Pastor. The energy obtained by a magnetic pick-up is stored in a battery and used to supply a dc load. An interface circuit for impedance matching between the magnetic pick-up and the battery is used.  Two candidates for the role of interface, i.e., a SEPIC converter and the cascade connection of a buck and a boost converters show similar performances after being compared on equal experimental basis.

The control of both output and circulating currents in a modular multilevel converter by means of sliding-mode control is described in the ninth paper. Uddin, Zeb, M.A. Khan, Ishfaq, I. Khan, S.ul Islam, Kim, Park , and Lee use a first-order switching strategy to control the output current, and a second-order switching law-based super-twisting algorithm to control the circulating current and suppress its second harmonic. The proposed control shows better performances than the conventional proportional-resonant regulation scheme.

Digital implementation of sliding-mode control is an important research area today because it has recently opened the way to implement sliding-mode controllers resulting in constant switching frequency. One of these ways is analyzed in detail in the tenth paper by Vidal-Idiarte, Restrepo, El Aroudi, Calvente, and Giral. They interpret a digital input-output linearization strategy in a buck converter under the optics of discrete-time sliding-mode control theory, and their theoretical approach is verified by means of simulations and experiments.

Designing boost inverters is a well-known problem for engineers due to the difficulty of the analytic understanding of how these circuits operate in either autonomous or grid-connected mode. Lopez-Caiza, Flores-Bahamonde, Kouro, Santana, Muller, and Chub present in the eleventh ¨ paper in a dual boost inverter a combination of a resonant control and a sliding-mode control to regulate the current injected to the grid and the balance between the inductor currents respectively. The experimental results validate the theoretical predictions and meet the requirements of power quality standards.

Micro-inverters development is an example of recent directions in power conversion for photovoltaic systems. A design based on the cascade connection of two boost converters and a full-bridge circuit is analyzed in the twelfth paper by Valderrama-Blavi, Rodriguez-Ramos, Olalla, and Genaro-Munoz. Two sliding-mode control alternatives are analyzed and their performances ˜ measured in an experimental prototype. The first one regulates the system energy through the control of the input current while the second one enforces a self-oscillating transformer behavior with variable transformer ratio.

Rivera, Ortega-Cisneros, and Chavira address the control of a boost converter for output tracking of a DC biased sinusoidal signal in the thirteenth paper. They apply discontinuous output regulation based on the use of a sliding function made of a linear combination of tracking errors and an integral term. Experimental results show good tracking of the output voltage with THD less than the 5% standard limit, and excellent rejection of input voltage disturbances.

Multiphase converters have become a practical solution in industry to improve efficiency while reducing size of passive components. In the fourteenth paper, Pajer, Chowdhury, and Rodic propose a multiphase buck converter for battery emulation, which is regulated by a cascade control scheme. The inner loop uses a sliding-mode control for phase currents while the outer loop applies a proportional controller with output current feedforward. Disturbance observers are used in both loops for mismatch compensation. The theoretical analysis is corroborated by experimental results in a 4-phase synchronous prototype.

Power converters supplying a constant power load (CPL) are open-loop unstable, so they can only operate in closed-loop with an appropriate control scheme. El Aroudi, Martinez-Trevino, ˜ Vidal-Idiarte, and Cid-Pastor approach the problem in the fifteenth paper by proposing a digital sliding mode -based control with PWM that results in inrush current limitation. The paper covers exhaustingly the design of a cascade control that eventually yields excellent output voltage regulation and the suppression of inrush current in a boost converter experimental prototype feeding a CPL of 1 kW.

Lopez-Santos, Cabeza-Cabeza, Garcia, and Martinez-Salamero illustrate in the last paper the use of sliding-mode control to obtain high power factor in the bridgeless isolated version of the SEPIC converter, which is used as unidirectional isolated interface between an AC source and a low voltage DC distribution bus. Zero-crossing points are considered here as an additional mode, which is analyzed in detail to demonstrate how the switching surface is reached and the sliding motions ensured. The simplicity of the implementation and the low level of resulting THD show that the proposed control is comparable to the best strategies reported in the technical literature.

Finally, I want to express my gratitude to the large number of reviewers who carried out manuscript reviews in record time. The quality of this special issue is also due to their deep and thorough work.

> **Luis Martinez-Salamero** *Special IssueEditor*

## *Article* **Design of a Continuous Signal Generator Based on Sliding Mode Control of Three-Phase AC-DC Power Converters**

#### **Yazan M. Alsmadi 1,\*, Isaac Chairez 2 and Vadim Utkin 3**

Received: 11 October 2019; Accepted: 18 November 2019; Published: 23 November 2019

**Abstract:** In recent years, hundreds of technical papers have been published which describe the use of sliding mode control (SMC) techniques for power electronic equipment and electrical drives. SMC with discontinuous control actions has the potential to circumvent parameter variation e ffects with low implementation complexity. The problem of controlling time-varying DC loads has been studied in literature if three-phase input voltage sources are available. The conventional approach implies the design of a three-phase AC/DC converter with a constant output voltage. Then, an additional DC/DC converter is utilized as an additional stage in the output of the converter to generate the required voltage for the load. A controllable AC/DC converter is always used to have a high quality of the consumed power. The aim of this study is to design a controlled continuous signal generator based on the sliding mode control of a three-phase AC-DC power converter, which yields the production of continuous variations of the output DC voltage. A sliding mode current tracking system is designed with reference phase currents proportional to the source voltage. The proportionality time-varying gain is selected such that the output voltage is equal to the desired time function. The proposed new topology also o ffers the capability to get rid of the additional DC/DC power converter and produces the desired time-varying control function in the output of AC/DC power converter. The e ffectiveness of the proposed control design is demonstrated through a wide range of MATLAB/Simulink simulations.

**Keywords:** sliding mode control (SMC); power converter; continuous signal generator; equivalent control; AC-DC power converter

## **1. Introduction**

Sliding mode control (SMC) has high order reduction property, good dynamic performance, low sensitivity to disturbances, and plant parameter variations, allowing SMC to handle nonlinear systems with uncertain dynamics and disturbances. Additionally, SMC is decoupled into independent lower dimensional subsystems, simplifying feedback control design. These properties allow SMC to be used in a wide range of applications such as automotive control, robotics, aviation, power systems, power electronics, and electric motors [1–5].

Power electronic converters are controlled by switching electrical components, which can produce two dissimilar values at the gating terminals [6,7]. Their controlled variables may take values from a two valued discrete set. Moreover, linearization is not required [1,2,8,9]. Hence, SMC is a preferred method to realize the control of power converter devices.

Within the wide diversity of available power electronic devices, the well-known three-phase AC/DC is commonly applied in energy conversion plants. Nevertheless, inherent complications appear with regards to reactive power generation, as well as the higher harmonic content in the input current. These characteristics appear as practical disadvantages that have become more relevant as the AC/DC converter capacity turn out to be larger and larger [10–12].

The idealized AC/DC converter shows up as a constant DC voltage as controlled output (or current) and a sinusoidal input set of currents at unity power factor at the AC line. Nevertheless, the current technology of thyristor phase-controlled converters has two intrinsic disadvantages: First, the larger firing angle, the smaller power factor; and second, the line current has moderately large harmonics components [13,14]. As AC/DC converters are more and more controlled using PWM switching patterns, the input as well as the output performances improve. These PWM AC/DC converters offer numerous advantages compared to some traditional rectifiers [15,16]: Unity power factor, low harmonic components in input current, bidirectional power flow, and low ripple in output voltage. These characteristics make simpler the filtering processes on both AC and DC sides of the proposed converter [1,2,10,16].

Conventional control design techniques for this type of power converter device usually solve the maintaining of the DC output voltage at a given reference level firstly, and at second place, try to seek for the minimization of high order harmonics and reactive power at the input. SMC of AC/DC power converters, presented in [1], o ffers the inverse sequence of actions. First, a current tracking system was designed with sinusoidal current references that are proportional to the AC input voltages with a constant gain of proportionality. This automatically sets the reactive power to zero. Second, it was proven that the output voltage will be constant and only depends on the amplitude of the reference input. However, the proposed control method requires an additional DC/DC converter to control the DC load. This introduces the following control design challenge: Is it possible to avoid using an additional DC/DC converter and generate any arbitrary desired time varying function at the output of the AC/DC converter such that the DC load can be directly controlled?

The main contributions of this study are:

(a) Sliding mode control is an appropriate tool for application for wide range of power converters. The first publications on DC/DC converters [1–4,7,8] demonstrated its e fficiency. The methods of minimization heat losses and of chattering amplitude based on harmonic cancellation principle, switching frequency control for DC/DC converters can be found in [2–4]. Multidimensional sliding modes were utilized in power converters to control AC load with DC energy source [5,6]. The design methodology to control output constant voltage and power factor simultaneously for AC-DC converter was developed in [7]. We are not aware of publications with our problem statement—to have an arbitrary time function (not constant) in the output of AC/DC converter. The attempts to find time of the varying gain as an algebraic state function of state failed, because it should satisfy the algebraic equation.

(b) A SMC has been proposed to generate continuous waveforms based on a controlled switched sequence of a three-phase AC-DC converter. This achievement was a consequence of solving a trajectory tracking of estimated reference currents. The realization of such tracking enforces the production of bounded derivative DC output voltage. The tracking controller implemented a time-varying relationship between the currents and voltages on the AC side of the converter. Such given positive relation between voltage and current justified the positive power efficiency of the controlled power converter.

## **2. Problem Statement**

The design problem considered in this study is the generation of switching commands for the power electronics-switching elements of the AC/DC converter (shown in Figure 1) in such a way that the output voltage can track the desired output of the converter *f*(*t*), which should be positive. This condition agrees with the classical realization of AC/DC, buck, and boost power converters. Since the input AC voltage is bounded, the output capacitor *C* can be charged at a limited velocity, which means that the time derivative *f*(*t*) should be bounded. Therefore, the problem can be described as fixing the switching sequence such that:

.

$$\lim\_{t \to \infty} \left| f(t) - \upsilon\_{dc} \right| = 0 \tag{1}$$

**Figure 1.** Scheme of the three-phase PWM AC/DC voltage source converter.

The power efficiency can be maximized if each input phase current of the power converter is proportional to the corresponding phase voltage with a positive gain. Therefore, the objective of this paper is to design a new control algorithm such that the input phase currents track preselected reference inputs and the positive gain of the proportionality is selected as a time varying function. Accordingly, the output voltage is equal to the desired function *f*(*t*).

#### **3. Circuit Model of the Three-Phase PWM AC**/**DC Voltage Source Converter Scheme**

Figure 1 shows the three-phase PWM AC/DC voltage source converter scheme. *ea*, *eb*,*ec* are the balanced three-phase AC input voltages; *idc* is dc-link current; *RL* is a resistive load connected to the DC side; *iL* is the load current; *ia*, *ib*, *ic* are the three-phase AC input currents; *Cdc* is the dc-link capacitance *vdc* is dc-link voltage; *R g* and *Lg* represent the grid-side resistance and inductance, respectively.

The balanced three-phase AC input currents are given by:

$$L\_{\mathcal{S}} \frac{d\dot{i}\_a}{dt} = \varepsilon\_a - R\_{\mathcal{S}} \dot{i}\_a - \upsilon\_{an} \tag{2}$$

$$L\_{\S} \frac{di\_b}{dt} = \mathcal{e}\_b - R\_{\S} i\_a - \mathcal{v}\_{bn} \tag{3}$$

$$L\_{\mathcal{S}} \frac{di\_{\mathcal{C}}}{dt} = \varepsilon\_{\mathcal{C}} - R\_{\mathcal{S}} i\_{\mathcal{C}} - \upsilon\_{\mathcal{C}n} \tag{4}$$

where *van*, *vbn*, *vcn* are the AC side phase voltages of the converter. The balanced three-phase AC voltages are given by:

$$
\varepsilon\_a = E\_0 \sin(\omega t) \tag{5}
$$

$$x\_b = E\_0 \sin\left(\omega t - \frac{2\pi}{3}\right) \tag{6}$$

$$e\_c = E\_0 \sin\left(\omega t + \frac{2\pi}{3}\right) \tag{7}$$

Here, ω is the AC power source angular frequency and *E*0 is the amplitude of the phase voltages. Assume that *iabc* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *iaibic* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, *eabc* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *eaebec* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, *vs* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *van vbn vcn* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, then Equations (2)–(4) can be re-written in a compact form:

$$L\_{\circ} \frac{d\dot{l}\_{\text{abc}}}{dt} = \varepsilon\_{\text{abc}} - R\_{\circ}\dot{\iota} - \upsilon\_{\text{s}} \tag{8}$$

Define the switching function *S* of each switch as:

$$S\_j = \begin{cases} \begin{array}{ll} 1, & S\_j \quad \text{is close} \\ -1, & S\_j \quad \text{is open} \end{array} & j = a, b, c \end{cases} \tag{9}$$

As a result, the voltage vector *vs* can be given in terms of the switching functions *S* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *Sa Sb Sc* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ as:

$$v\_s = \frac{1}{3} v\_{dc} \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix} \mathbf{S} \tag{10}$$

By substituting Equation (10) into (8), the AC input current equations can be given by:

$$\mathcal{L}\_{\mathcal{S}} \frac{d\dot{q}\_{\rm abc}}{dt} = \varepsilon\_{\rm abc} - R\_{\mathcal{S}} i\_{\rm abc} - \frac{1}{3} \upsilon\_{\rm dc} \left| \begin{array}{cc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array} \right| \mathcal{S} \tag{11}$$

In conclusion, the output voltage can be given by:

$$\mathbf{C}\frac{d\mathbf{v}\_{\rm dc}}{dt} = -\mathbf{i}\_{L} + \mathbf{i}^{T}\mathbf{S} \tag{12}$$

#### **4. Sliding Mode Current-Tracking Control**

As previously indicated, a sliding mode-based current tracking system is designed such that sinusoidal reference inputs are tracked by phase currents proportional to input AC voltages. Rewriting Equation (11) as:

$$L\_{\%} \frac{di\_{\rm abc}}{dt} = \varepsilon\_{\rm abc} - R\_{\rm g} i\_{\rm abc} - \frac{1}{3} v\_{\rm dc} \Gamma\_{\rm O} S \tag{13}$$

where, matrix Γ0 is given by:

$$
\Gamma\_0 = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}, \det \Gamma\_0 = 0 \tag{14}
$$

Since the sum of the three-phase currents is zero, only three state variables should be controlled in a system with a three-dimensional control vector S: Two phase currents and output voltage. However, because matrix Г0 is singular, the conventional sliding mode approach cannot be directly applied. Therefore, a tracking system for two phase currents only is first designed. As a result, the sliding mode should be enforced on the intersection of two surfaces σ*a* = *Lg iare f* − *ia* and σ*b* = *Lg ibre f* − *ib* , or in a vector form:

$$
\sigma\_{ab} = \mathcal{L}\_{\mathfrak{J}} (i\_{abref} - i\_{ab}) \tag{15}
$$

Excluding the phase current *ic* = −*ia* − *ib* yields:

$$L\_{\mathcal{K}} \frac{di\_{ab}}{dt} = \varepsilon\_{ab} - R\_{\mathcal{K}} i\_{ab} - \frac{1}{3} \upsilon\_{dc} \Gamma \mathcal{S} \tag{16}$$

*Energies* **2019**, *12*, 4468

.

$$C\frac{dv\_{dc}}{dt} = -\frac{v\_{dc}}{R\_L} + (i\_{a\prime}i\_{b\prime} - i\_a - i\_b)S\tag{17}$$

where Γ is a 2 × 3 matrix given by:

$$
\Gamma = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix} \tag{18}
$$

The ideal tracking system is based on the Lyapanov function:

$$V = \frac{1}{2} \,\,\sigma\_{ab}\,^T \,\sigma\_{ab} \tag{19}$$

 *V* has to be negative definite and calculated on the system trajectory when selecting discontinuous control:

$$\dot{V} = \sigma\_{ab}\,^T F(.) - \, (\mathfrak{u}\_{dc}/L) (\mathfrak{a}\_\prime \mathfrak{z}\_\prime \gamma) S \tag{20}$$

where *F*(.) is state function, which does not depend on control. α, β, and γ are given by:

$$\begin{cases} \alpha = (2\sigma\_a + \sigma\_b) \\ \beta = (\sigma\_a + 2\sigma\_b) \\ \gamma = (\sigma\_a + \sigma\_b) \end{cases} \tag{21}$$

If *vdc*/*L* is large enough, *F*(.) can be suppressed with control *S* given by:

$$S = \begin{cases} S\_{\mathfrak{a}} = \operatorname{sign}\left(\alpha\right) \\ S\_{\mathfrak{b}} = \operatorname{sign}\left(\beta\right) \\ S\_{\mathfrak{c}} = \operatorname{sign}\left(\gamma\right) \end{cases} \tag{22}$$

such that . *V* = σ*abTF*(.) − (*udc*/*L*)|α| + --β--- + --γ--- < 0. As a result,σ*ab* tends to zero and σ*ab* becomes zero after a finite time interval [17–19]. Consequently, sliding mode occurs with *iab* = *iabre f* . Therefore, the current tracking system is developed with sinusoidal current references proportional to the input AC voltages as:

$$i\_{ab} = K(t) \times \varepsilon\_{ab} \tag{23}$$

Calculate the equivalent control [2]:

$$(\Gamma S)\_{\alpha\eta} = \left(-L\_{\mathcal{g}}\frac{d\dot{i}\_{abref}}{dt} + \mathfrak{e}\_{ab} - R\_{\mathcal{g}}\dot{i}\_{ab}\right)\frac{\mathfrak{Z}}{\mathfrak{u}\_{\mathfrak{c}}}\tag{24}$$

The three phase input currents can also be expressed as:

$$
\begin{pmatrix} i\_a & i\_{b\_\succ} \ -i\_a \ -i\_b \end{pmatrix} = \frac{1}{3} (\dot{i}\_a - \ i\_{c\succ} \ \dot{i}\_b - \ i\_c) \ \Gamma \tag{25}
$$

The sliding mode equation can be obtained by substituting Equations (24) and (25) into (17):

$$\mathcal{L}\left(\frac{d\upsilon\_{dc}}{dt}\right) = \left(-\frac{\upsilon\_{dc}}{R}\right) + \frac{1}{\upsilon\_{dc}}\left(i\_a - i\_{c\prime}i\_b - i\_c\right)\left(-L\_{\text{g}}\frac{d}{dt}i\_{ab\upsilon\prime f} + c\_{ab} - R\_{\text{g}}i\_{ab}\right) \tag{26}$$

After sliding mode occurs (*iab* = *K*(*t*)*eab*):

$$i\_{ab} = i\_{ab,ref} \\\frac{d}{dt}i\_{ab} = \frac{d}{dt}i\_{ab,ref}$$

Therefore,

$$\frac{d}{dt}\dot{\iota}\_{ab} = K(t)\frac{d}{dt}\varepsilon\_{ab} + \varepsilon\_{ab}\frac{d}{dt}k(t) \tag{27}$$

where, *ia* − *ic* = *K*(*t*)(*ea* − *ec*,*eb* − *ec*).

> It can be shown that the output voltage *vdc* is given by:

$$\mathbf{C}\frac{d\upsilon\_{dc}}{dt} = \frac{-\upsilon\_{dc}}{R\_L} + \frac{1}{\upsilon\_{dc}} \Big( -\frac{3}{2} L\_\text{\space{R}} K E\_0^2 \frac{d}{dt} \mathbf{K} + \frac{3}{2} K E\_0^2 - \frac{3}{2} R\_\text{\space{R}} K^2 E\_0^2 \Big) \tag{28}$$

If *y* = *v*2*dc*, then the gain *K* should satisfy the differential equation

$$\left(\frac{1}{2}\mathbb{C}\frac{dy}{dt} + \frac{1}{R\_L}y\right)\frac{2}{3KE\_0^2} = -L\_\% \frac{dK}{dt} + 1 - R\_\% K\tag{29}$$

Assume that *f* is the voltage reference and *y*ˆ = *f* 2, then the equation

$$\left(\frac{1}{2}\mathbb{C}\frac{d\boldsymbol{\mathcal{Y}}}{dt} + \frac{1}{R\_L}\boldsymbol{\mathcal{Y}}\right)\frac{2}{3\mathcal{K}\mathcal{E}\_0^2} = -L\frac{d\mathcal{K}}{dt} + 1 - R\_{\mathcal{K}}\boldsymbol{\mathcal{K}}\tag{30}$$

can be simulated in the controller,

$$
\Delta y = \hat{y} - y
$$

$$\frac{1}{2}\mathbb{C}\frac{d\Delta y}{dt} + \frac{1}{R\_L}\Delta y = 0\tag{32}$$

Equation (32) shows that Δ*y* → 0 and the output voltage tends to be the reference input *vdc* → *f* . Assume that

$$\begin{aligned} \left| \frac{3}{2} \left( \frac{1}{2} \mathbb{C} \frac{d}{dt} f^2 + \frac{1}{R\_L} f^2 \right) \right| &\le M\_\prime \\\\ L\dot{K} &\ge 1 - R\_{\mathcal{S}} K - \frac{M}{K E\_0^2} \end{aligned} \tag{33}$$

If *RgK* = ε < 1, then

$$\lim\_{K\_0^2 \to \infty} \left( 1 - R\_\% K - \frac{M}{KE\_0^2} \right) < 1\tag{34}$$

It means that

then:

$$L\frac{dK}{dt}\_{\mathcal{R}\_\delta K=\varepsilon} > 0\tag{35}$$

and *K* is always positive if *RgK*(0) > ε. This corresponds to a power factor equal to 1 (*iabc* = *Keabc*) for a high enough amplitude of the input source voltage. It is important that *L* . *K* in (29) will be negative for high enough values of *K*. Hence, the gain *K* is bounded.

Remark: The implementation of the proposed controller in embedded systems requires the online measurement of the dc voltage at the capacitor and the possibility of realizing fast enough oscillations on the switching electronic elements. This part of the problem can be solved using available fast dedicated microcontrollers devices. Notice that the accuracy of the produced signal is a function of the relative relationship between the switching devices operation frequency and the main frequency components of the desired signals. Evidently, the exact reconstruction of the desired signal cannot be acquired. Nevertheless, the studies regarding discrete implementation of sliding mode has shown that the accuracy of the sliding mode realization is proportional to the square power of the sampling period if the explicit discretization is considered. For more details on the implementation issues, please see references [20–23]. Remark: The proposed control strategy offers an alternative to some other sliding mode controllers designs considering adaptive pulse width modulation [24], sub-optimal regulation [25], and multitype restrictive [26] approaches which have been applied on DC-DC power converters to obtain arbitrary signals. However, not one of them has been tested on the AC-DC device.

## **5. Simulation Results**

In order to evaluate the proposed sliding mode control design procedure several computer simulations have been conducted using MATLAB/Simulink software. The control algorithm is represented in the following flow diagram (Figure 2):

**Figure 2.** Flow diagram describing the sliding mode control realization.

Different generated signals have confirmed the abilities of the proposed continuous waveform generators.

The simulation was performed for power converter governed by Equation (10) with control (21) and different desired functions *f(t)* in the converter output. The differential equation for time varying gain *K(t)* in Equation (29).

#### *5.1. Sinusoidal Waveform Generation*

The first signal is a pure sinusoidal waveform, which corresponds to a traditional signal used in diverse signal generators. The selected reference waveform is:

$$f(t) = 250(\sin(\omega t) + 1) \ge 0 \text{ (35)}\tag{36}$$

The results of the simulation are shown in Figure 3.

**Figure 3.** *Cont*.

**Figure 3.** Comparison of the reference and the generated sinusoidal continuous waveform for a fixed period: (**a**) (0,10.0) and (**b**) (0,0.1) s.

The input currents obtained by the application of the suggested first sliding-mode controller show a modulated sinusoidal shape in all three branches (Figure 4).

**Figure 4.** *Cont*.

**Figure 4.** Time variation of the input currents at the three branches of the AC source on the period (2.0–2.5) s (closer view) for the three branches: (**a**) *ia*, (**b**) *ib*, and (**c**) *ic* with the a sinusoidal reference.

The time dependence of the current ˆ *is* also shows the expected modulation with the frequency of the desired output current, which is 15 Hz (Figure 5).

**Figure 5.** Time variation of the controller gain K.

The phase relations between the input currents hold both in the transient and the steady state periods. In the period between 0.0 and 0.1 s, the current decreases exponentially to the steady state, which is detected after 0.1 s. A phase shift of 2/3 is evidenced, which also confirms the efficiency of the suggested controller. Notice that the simultaneous dependence of *K* with respect to the reference waveform as well as its derivative does not relate it to the reference voltage form. The gain variation is continuous but not necessarily differentiable, considering the gain structure estimated in this study.

#### 5.1.1. Variable Frequency Sinusoidal Waveform Generation

The second proposed reference signal is a composite sinusoidal waveform, which corresponds to a class of simplified chirp signal. Such waveforms can be used for testing the spectral response of diverse systems for calibration purposes. The selected composite sinusoidal reference waveform is:

$$f\_2(t) = \begin{cases} 200(\sin(5t) + 1) & if \quad 0 \le t < 2\\ 100(\sin(t) + 1) & if \quad 2 \le t < 8\\ 200(\sin(5t) + 1) & if \quad 8 \le t \le 10 \end{cases} \tag{37}$$

Once more, the selected bias constants mean a positive waveform. The selected transition times between the sinusoidal forms can design a continuous composite waveform with a bounded derivative. The simulation results are shown in Figure 6.

**Figure 6.** Comparison of the reference and the generated sinusoidal continuous waveform for a fixed period: (**a**) (0,10.0) and (**b**) (0,0.1) s.

The input currents agree with the variation of the sinusoidal frequency by the application of the suggested first sliding mode controller with the corresponding modulated sinusoidal shape in all three branches (Figure 7).

When looking at the time variation of the gain *K* for the controller, notice that the simultaneous dependence of *K* with respect to the reference waveform, as well as its derivative does not relate it to the reference voltage form. The second waveform considered in this study produces a smoother variation of the gain *K*. The gain variation is continuous considering the gain structure estimated in this study (Figure 8).

**Figure 7.** Time variation of the input currents at the three branches of the AC source on the period (2.0–2.5) s (closer view) for the three branches: (**a**) *ia*, (**b**) *ib*, and (**c**) *ic* with the reference sinusoidal.

**Figure 8.** Time variation of the controller gain *K*.

#### 5.1.2. Triangular Waveform Generation

The third suggested reference signal is a triangular signal, which is also a common signal used in the calibration of diverse devices. Notice that this signal has a bounded but not continuous derivative. Consequently, the suggested controllers are applicable (Figure 9).

**Figure 9.** Comparison of the reference and the generated triangular waveform for a fixed period: (**a**) (0,10.0) and (**b**) (0,0.1) s.

For the class of triangular signal, the controller succeeded at reconstructing the suggested reference signal as shown in Figure 9a. Consequentially, the tracking error of the reference voltage is reduced to less than 0.05% over a period of 0.08 s (Figure 9b).

The time variation of the gain *K* for the controller with the reference triangular signal appears in Figure 10. The gain variation function is continuous.

**Figure 10.** Time variation of the controller gain *K*.

Even if the exact sliding motion can be acquired if and only if the switches in the power converter oscillate at the infinite frequency, the current available technology allows to oscillate at such high frequencies ensuring the existence of the practical sliding motion. On the other hand, the required high frequency oscillations of the sliding mode may produce heath losses which could damage the switching circuit. In all the presented cases, chattering phenomenon should mentioned always when applying sliding mode control. The set of chattering suppression methods has been developed in the framework of sliding mode control theory. They are surveyed in [27]. The harmonics cancellation principle is the most efficient for power converters and can be applied for our case. The design idea consists in using several parallel converters with controlled phases such that high order harmonics can be cancelled.

## **6. Conclusions**

This paper has presented the control design procedure to directly control DC loads using a three-phase voltage source without a rectifier. It consists of two steps. First, the current tracking problem is solved with reference currents proportional to phase voltages. Then, the time varying proportionality coefficient is selected such that the output voltage is equal to the desired time function. It has been shown that the proportionality gain should satisfy the first order differential equation, which is implemented in the controller. The behavior of the system with a positive coefficient is equivalent to having a unity power factor. Stability of the complete system (the power converter and controller dynamics) was also proved. A wide range of computer simulations were provided to demonstrate efficiency of the proposed control design for different types of sinusoidal and triangular functions as voltage reference inputs.

**Author Contributions:** Formal analysis, Y.M.A., I.C., and V.U.; Investigation, Y.M.A., I.C., and V.U.; Methodology, Y.M.A., I.C., and V.U.; Supervision, V.U.; Validation, V.U.; Writing—Original draft, Y.M.A., I.C., and V.U.

**Funding:** This research work was partially funded by the [Consejo Nacional de Ciencia y Tecnología] grant number [CB-20181457] and the [Instituto Politécnico Nacional] grant number [SIP-20191724].

**Conflicts of Interest:** The authors declare no conflict of interest.


**7. The process of socialization in organization**

Parents impart societal standards to their children, but socialization is not a oneway process. Students are active participants in the socialization process. They do not receive from the socializer in a passive manner. They are physically or culturally predisposed to be socialized more or less easily in various aspects of their lives. They process socialization-related information, accepting or rejecting it as appropriate. They are more accepting of some forms of control than others, in part because some forms of misbehavior, such as bodily and psychological injury to others, are fundamentally more problematic than others, such as social convention violations [61]. Some steps are required in the socializing process. Stages of socialization are a term used to describe these steps [62].

The initial stage of socialization is investigation. During this stage of the socialization process, an individual assesses a group to see if it is a good fit for him or her. The group follows suit. The end of this stage occurs when the group extends an invitation to the newcomer and the newcomer accepts. The second phase is known as socialization. The new member adopts the culture of the group, whether it is correct or wrong, and good or bad. They blend in with the group, adopting its norms, attitudes, and beliefs. The maintenance stage is the third stage of the organization's socialization process. The new member and the group agree what the group expects of the new member, such as how they should act or another contribution, during the maintenance stage. If an individual fails to meet collective expectations, they can be kicked out, or they can renegotiate and rejoin the group. The other option is to re-socialize. Depending on the conclusion of the maintenance stage, the member will either be welcomed back into the group or taught to act appropriately, or they will be evicted and forced to learn to live outside of it. The last stage is the remembering stage. In this section, people who have left the group reflect on their experiences [17, 44].


**Acknowledgements**

Special thanks to all the medical and health workers, the public health organizations, and their staff, who strive against all odds to bring succour to Africans during the pandemic.


*2.2. Power Measuring Principle*

2.2.1. Parameter Symbol Definition

Variable symbols are defined as follows:

*k*—the transmission ratio the hydraulic pump and the engine

#### 2.2.2. Measuring Principle Analysis

According to the machine power roadmap of the maize silage harvester, the power sources of key operation parts include both mechanical drive and hydraulic drive.

For mechanically-driven working parts, the operating power can be acquired by measuring the rotational speed and rotating torque. For rotation torque measurement, the principle of torsion resistance strain gauge is used, and the strain gauge is attached to the rotating elastic shaft to form a strain bridge detection circuit, which outputs a frequency signal proportional to the torsion value, and thus the working torque is obtained. For rotation speed measurement, a speed sensor is mounted on the fixed part of the torque sensor, and an encoder with multiple detection positions is installed on the rotating part of the torque sensor. When the encoder rotates with the working shaft, the sensor outputs a series of speed pulse signals, and the working speed is measured. According to the definition of mechanical torque, the torque can be calculated by Formula (1).

$$T\_r = FR \tag{1}$$

The relationship between linear speed and rotational speed is shown as follows:

$$
v\_r = \frac{n\_r}{60} 2\pi R\tag{2}$$

The power consumption of the mechanical rotating parts is then given as follows:

$$P\_r = \frac{1}{1000} Fv\_r = \frac{1}{1000} \times \frac{T\_\mathbf{r}}{R} \times \frac{n\_\mathbf{r}}{60} \times 2\pi R = \frac{n\_\mathbf{r} T\_\mathbf{r}}{9549.3} \tag{3}$$

For hydraulic-driven working parts, the operating power can be obtained by the differential pressure between the outlet and inlet and the hydraulic flow of the hydraulic pump, and the power of the hydraulic parts is calculated as follows.

$$P\_h = \frac{(p\_o - p\_i)}{60} q = \frac{n\_0 k V\_0 \eta\_v}{60} (p\_o - p\_i) \tag{4}$$