3.1. Path Loss Model Design
The transmission line environment in a mountainous area has the characteristics of a deep valley, curved line, and rich surrounding vegetation. Therefore, the interference of radio signals in a mountainous area is stronger than that in a plain area, and the path loss is higher [
36]. The transmission of a LoRa signal in a mountainous environment is easily affected by electromagnetic interference [
37], mountain shelter, vegetation reflection and absorption, and other factors. It can be seen from [
38] that in the mountain monitoring environment, mountain occlusion had the greatest interference on the LoRa signal. Therefore, the path loss model was established according to the influence of mountain shelter on the LoRa signal.
The signal strength of the LoRa transceiver module at a certain point is the difference between the field strength excited by the base station at the antenna and the field strength loss when the signal propagates to the point:
where
ET is the field strength excited by the LoRa transceiver module at the antenna and
Lps is the median path loss caused by signal propagation. When the signal has no direct propagation path, the diffraction effect will occur, as shown in
Figure 3. The distance of the diffraction path is calculated as follows:
where
dD is the diffraction distance;
d1 is the distance between the LoRa transceiver module antenna and the mountain;
d2 is the distance between the receiving antenna and the mountain;
hm is the altitude of the mountain;
hT is the altitude of the LoRa transceiver module antenna; and
hr is the altitude of the receiver antenna. In order to display the diffraction principle of a wireless signal in a mountainous area more intuitively, the schematic diagram, as shown in
Figure 3, was constructed by simulating the topography of the southwest mountainous area.
Figure 3 shows the LoRa signal transmission diffraction model established according to the western mountain environment. The LoRa signal passes through the diffraction path ① from point
a to point
b, and its diffraction law and signal field strength can be analyzed by the Egli model. The Egli propagation model is a simplified wireless propagation model on irregular terrain, which is a statistical model based on a large number of test data, which can reflect the attenuation law and signal trend in an irregular environment. Compared with other terrains, the Egli model can be used to evaluate hilly and mountainous areas as the field strength of the shape is more accurate. The empirical equation of the Egli wireless signal transmission loss is as follows:
where
f is the radio frequency;
ht is the height of transmitting antenna of LoRa transceiver module;
hr is the height of receiving antenna;
d is the distance between receiving and transmitting antennas; and
Kh is terrain correction factor. When the average relief height
Kh of the terrain around the test point was equal to 15 m,
Kh was taken as 0; when the terrain relief height
H around the test point was greater than or less than 15 m, the terrain correction factor should be added. For the 150 MHz frequency band, the terrain correction factor
Kh can be obtained by the following equation:
where
H is the average relief height of the terrain around the test point.
3.2. Network Model Design
In this section, the mathematical modeling of the transmission line architecture design is carried out, and a priority multi-objective optimization model of LoRa dynamic grouping is proposed. The optimal installation location and usage quantity of the cellular wireless transmission module under the specified delay conditions were found through the algorithm, and different networking modes can be realized under the constraint of different system energy consumption and delay transformation.
In this paper, the network architecture diagram can be modeled as a data directed graph. The transmission tower near the substation can realize end-to-end data transmission directly through the LoRa transmission module. Therefore, the directional graph mainly considers the transmission tower that needs to transmit data through the cellular wireless transmission module. As shown in
Figure 4, suppose that a transmission line contains
N transmission line towers, where
i represents the end-to-end data transmission link between transmission poles and towers, and
j represents the data transmission link of towers through cellular wireless communication. Therefore, any data transmission link can be expressed as a vector (
i,j), and
P is the set of all wireless transmission communication links, that is, all communication link vectors (
i,j) are included in
P,
. The purpose of this model is to find a feasible communication link for each communication path used, minimize the delay caused by multiple links, and consider the energy constraint of each link.
First, it is necessary to limit the delay generated in any communication link to be less than or equal to the maximum communication delay required, as shown in Equation (5):
where
Di,j,k denote that the communication node
k uses the data link (
i,j) as the generated delay;
Mi,j,k are binary variables; if node
k uses the data link (
i,j) as the communication link,
Mi,j,k = 1; otherwise, it is 0; and Dmax represents the maximum transmission delay required by the system administrator.
Equation (6) ensures that there are towers using cellular wireless transmission modules regardless of transmission mode. Where
Gi and
Li are binary variables, if the
i tower uses thee cellular wireless transmission module, then
Gi = 1, otherwise it is 0; if tower
i uses the LoRa transmission module, then
Li = 1, otherwise it is 0.
Equation (7) ensures that each communication link is reused and the link cost is calculated at most once, where
Oi,j are binary variables. If the data transmission link (
i,
j) is used,
Oi,j = 1, otherwise it is 0.
Equation (8) determines that the decision variables Mi,j,k, Gi, Li, and Oi,j are binary variables.
The delay function of the network is composed of Equations (5)–(8). The binary variables in the equation are decision variables.
Oi,j indicates that the link is used by at least one node. If node
k chooses link (
i,
j) as its transmission path,
Mi,j,k = 1, and when there is a delay constraint, the frequency of the cellular wireless transmission access path
j of transmission tower
i is significantly higher than that of other transmission towers, then the cellular wireless transmission module must be installed on the tower. Therefore, the network delay function is shown in Equation (9):
where
is the network energy consumption function;
d and
b represent the energy consumption of a single cellular wireless transmission module and LoRa transmission module, respectively; and
Ci,j represents the communication energy consumption generated on the data link (
i,j). The energy consumption function mainly includes transmission module energy consumption and wireless communication energy consumption, and the path loss function
Lps must be included in the energy consumption of wireless communication. As shown in Equation (10), system energy consumption is the sum of all communication energy consumption used for data transmission during the whole operation period and the energy consumption and path loss of installing the LoRa transmission module or cellular wireless transmission module on the selected tower.
According to the above equation, the LoRa dynamic networking model of transmission lines in mountainous areas is shown in Equation (11):
3.3. Delay Model and Energy Consumption Model Design
According to [
39], if the spread spectrum factor (SF), coding rate (CR), and signal bandwidth (BW) are known, the calculation formula issued by the Semtech company can calculate the air transmission time of a single LoRa node packet. By understanding the key parameters that users can control and according to the definition of the symbol rate, the LoRa symbol rate
Rs and symbol period
Ts can be calculated by the following formula:
The LoRa data transmission time is equal to the sum of preamble time and packet transmission time. The length of the preamble can be calculated by the following formula:
where
np represents the set preamble length, and its value is determined by two registers in the transmission chip.
The payload transfer time can be calculated by the following formula:
where
ɛ is the number of payload symbols, which can be calculated by the following formula:
where
PL is the number of bytes of payload; when the header is used,
H = 0, otherwise
H = 1;
DE is determined by the chip register; and
CR is the coding rate, and the value range is 1–4.
Finally, the data transmission time of LoRa is equal to the preamble transmission time plus the payload transmission time. The formula is as follows:
The energy generation process of LoRa data transmission is shown in
Figure 5. In this paper, the energy consumption model was established based on the calculation method of communication equipment.
If
ETX is the transmission energy consumption,
ERX is the receiving energy consumption,
n is the data size, and
d is the data transmission distance, then:
Among them, ETelec and ERelec, respectively, represent the energy consumption of LoRa communication equipment for sending and receiving data; εamp is the energy consumption of power amplifier transmitting data per unit distance; k is the propagation attenuation index of transmission environment with the value range of 2 ≤ k ≤ 5; k should be taken as 4 in a mountain environment; and Ecc(n) refers to the energy consumption of cellular wireless data transmission module when transmitting n byte data.
3.4. Multi Objective Particle Swarm Optimization Algorithm
Through the modeling of delay and energy consumption, it can be seen that the delay model is mainly related to the parameters of LoRa transmission equipment and the amount of data transmitted. The energy consumption model is mainly related to the receiving and sending power of the LoRa module, the wireless transmission power of the cellular, the amount of data transmitted, and the transmission distance. Low delay and low energy consumption are contradictory goals. In order to meet the design requirements of the transmission line online monitoring system, particle swarm optimization (PSO) [
40] can be used to optimize the delay
Tall and energy consumption
Eall.
The model of the multi-objective optimization algorithm with d-dimension decision variables and m-objective is defined as follows:
where
x is the decision vector;
X is the decision space;
y is the target vector; and
Y is the target space.
In the PSO algorithm, each particle is a solution in the solution space. It adjusts its flight according to its own flight experience and the flight experience of its peers. The best position each particle has experienced in the flight process is the optimal solution found by the particle itself. The best position that the whole group has experienced is the optimal solution found by the whole group at present.
Let be the dimensional position of particle a(a = 1,2,...,s), be the flight speed of particle a, be the individual optimal solution of particle a, and be the global optimal solution of the whole population.
In each iteration, the velocity and position of each particle are updated with Equations (23) and (24):
where
a = 1, 2,...,
s,
s is the number of particles in the population,
b = 1, 2,...,
D,
hab ∈ [
Lb,
Ub],
Lb and
Ub represent the lower bound and last term of the search space,
i ∈ [
vmin,
D,
vmax,
D],
vmin and
vmax respectively represent the minimum and maximum speed of particle flight;
c1 and
c2 are learning factors;
r1 and
r2 are random numbers between 0 and 1; and
θ is inertia weight.