Appendix A
Select the actual locations of the 75 cities in remote areas to normalize the coordinates, and the values are: (0.48,0.21), (0.52,0.26), (0.55,0.50), (0.50,0.50), (0.41,0.46), (0.51,0.42), (0.55,0.45), (0.38,0.33), (0.33,0.34), (0.45,0.35), (0.40,0.37), (0.50,0.30), (0.55,0.34), (0.54,0.38), (0.26,0.13), (0.15,0.05), (0.21,0.48), (0.29,0.39), (0.33,0.44), (0.15,0.14), (0.16,0.19), (0.12,0.17), (0.50,0.40), (0.22,0.53), (0.21,0.36), (0.20,0.30), (0.26,0.29), (0.40,0.20), (0.36,0.26), (0.62,0.48), (0.67,0.41), (0.62,0.35), (0.65,0.27), (0.62,0.24), (0.55,0.20), (0.35,0.51), (0.30,0.50), (0.45,0.42), (0.21,0.45), (0.36,0.06), (0.06,0.25), (0.11,0.28), (0.26,0.59), (0.30,0.60), (0.22,0.22), (0.27,0.24), (0.30,0.20), (0.35,0.16), (0.54,0.10), (0.50,0.15), (0.44,0.13), (0.35,0.60), (0.40,0.60), (0.40,0.66), (0.31,0.76), (0.47,0.66), (0.50,0.70), (0.57,0.72), (0.55,0.65), (0.02,0.38), (0.07,0.43), (0.09,0.56), (0.15,0.56), (0.10,0.70), (0.17,0.64), (0.55,0.57), (0.62,0.57), (0.70,0.64), (0.64,0.04), (0.59,0.05), (0.50,0.04), (0.60,0.15), (0.66,0.14), (0.66,0.08), (0.43,0.26).
Two additional sets of normalized coordinates for 75 cities will be uploaded as attachments for spatial reasons.
Appendix B
Select the actual locations of the 50 cities in remote areas to normalize the coordinates, and the values are: (0.41,0.94), (0.37,0.84), (0.54,0.67), (0.25,0.62), (0.07,0.64), (0.02,0.99), (0.68,0.58), (0.71,0.44), (0.54,0.62), (0.83,0.69), (0.64,0.60), (0.18,0.54), (0.22,0.60), (0.83,0.46), (0.91,0.38), (0.25,0.38), (0.24,0.42), (0.58,0.69), (0.71,0.71), (0.74,0.78), (0.87,0.76), (0.18,0.40), (0.13,0.40), (0.82,0.07), (0.62,0.32) (0.58,0.35), (0.45,0.21), (0.41,0.26), (0.44,0.35), (0.04,0.50), (0.48,0.21), (0.52,0.26), (0.55,0.50), (0.50,0.50), (0.41,0.46), (0.51,0.42), (0.55,0.45), (0.38,0.33), (0.33,0.34), (0.45,0.35),(0.40,0.37), (0.50,0.30), (0.55,0.34), (0.54,0.38), (0.26,0.13), (0.15,0.05), (0.21,0.48), (0.29,0.39), (0.33,0.44), (0.15,0.14).
Two additional sets of normalized coordinates for 50 cities will be uploaded as attachments for spatial reasons.
Appendix C
Select the actual locations of the 100 cities in remote areas to normalize the coordinates, and the values are: (0.48, 0.21), (0.52, 0.26), (0.55, 0.50), (0.50, 0.50), (0.41, 0.46), (0.51, 0.42), (0.55, 0.45), (0.38, 0.33), (0.33, 0.34), (0.45, 0.35), (0.40, 0.37), (0.50, 0.30), (0.55, 0.34), (0.54, 0.38), (0.26, 0.13), (0.15, 0.05), (0.21, 0.48). (0.29, 0.39), (0.33, 0.44), (0.15, 0.14), (0.16, 0.19), (0.12, 0.17), (0.50, 0.40), (0.22, 0.53), (0.21, 0.36), (0.20, 0.30). (0.26, 0.29), (0.40, 0.20), (0.36, 0.26), (0.62, 0.48), (0.67, 0.41), (0.62, 0.35), (0.65, 0.27), (0.62, 0.24), (0.55, 0.20). (0.35, 0.51), (0.30, 0.50), (0.45, 0.42), (0.21, 0.45), (0.36, 0.06), (0.06, 0.25), (0.11, 0.28), (0.26, 0.59), (0.30, 0.60). (0.22, 0.22), (0.27, 0.24), (0.30, 0.20), (0.35, 0.16), (0.54, 0.10), (0.50, 0.15), (0.44, 0.13), (0.35, 0.60), (0.40, 0.60). (0.40, 0.66), (0.31, 0.76), (0.47, 0.66), (0.50, 0.70), (0.57, 0.72), (0.55, 0.65), (0.02, 0.38), (0.07, 0.43), (0.09, 0.56), (0.15, 0.56), (0.10, 0.70), (0.17, 0.64), (0.55, 0.57), (0.62, 0.57), (0.70, 0.64), (0.64, 0.04), (0.59, 0.05), (0.50, 0.04). (0.60, 0.15), (0.66, 0.14), (0.66, 0.08), (0.43, 0.26), (0.10,0.10), (0.90,0.50), (0.90,0.10), (0.45,0.90), (0.90,0.80), (0.70,0.90), (0.10,0.45), (0.45,0.10), (0.40,0.44), (0.24,0.15), (0.17,0.23), (0.23,0.71), (0.51,0.94), (0.87,0.65), (0.68,0.52), (0.84,0.36), (0.66,0.25), (0.61,0.26), (0.91,0.45), (0.83,0.72), (0.16,0.82), (0.66,0.10), (0.79,0.79), (0.82,0.70), (0.22,0.98).
Two additional sets of normalized coordinates for 100 cities will be uploaded as attachments for spatial reasons.
Appendix D
Select the actual locations of the 1000 cities in remote areas to normalize the coordinates. Due to space reasons, the coordinates will be uploaded as an attachment.
Figure 1.
Inverted bifurcation diagram.
Figure 1.
Inverted bifurcation diagram.
Figure 2.
Lyapunov exponent diagram.
Figure 2.
Lyapunov exponent diagram.
Figure 3.
Algorithm flow chart.
Figure 3.
Algorithm flow chart.
Figure 4.
Optimal path of 75 cities (Note: yellow five-pointed star 65 indicates the starting city, green five-pointed star 63 indicates the second city).
Figure 4.
Optimal path of 75 cities (Note: yellow five-pointed star 65 indicates the starting city, green five-pointed star 63 indicates the second city).
Figure 5.
Friedman test figure.
Figure 5.
Friedman test figure.
Figure 6.
The path-planning diagram.
Figure 6.
The path-planning diagram.
Figure 7.
Environmental model represented by raster method.
Figure 7.
Environmental model represented by raster method.
Figure 8.
The correspondence between raster coordinates and serial numbers.
Figure 8.
The correspondence between raster coordinates and serial numbers.
Figure 10.
The path image generated by the proposed algorithm.
Figure 10.
The path image generated by the proposed algorithm.
Figure 11.
Location of remote wind farms. The green line represents mountains and the blue line represents rivers.
Figure 11.
Location of remote wind farms. The green line represents mountains and the blue line represents rivers.
Figure 12.
Solution of path planning.
Figure 12.
Solution of path planning.
Figure 13.
The results of path planning (Note: The yellow star represents the starting point).
Figure 13.
The results of path planning (Note: The yellow star represents the starting point).
Table 1.
Classification of literature review on path planning.
Table 1.
Classification of literature review on path planning.
Whether It Is a Hybrid Optimization Algorithm | Time | Main Research Work | Algorithm cATEGORY |
---|
No | 2019 | Liu et al. [5] used an ant colony algorithm to find the optimal path for inspection robots. Through simulation analysis, the ant colony algorithm reduces the time of shortest path searching, and increases the success rate of finding the optimal path. | Metaheuristic methods based on swarm |
No | 2021 | Dong et al. [6] proposed a path planning method for an ultra-high voltage substation based on an ant colony optimization algorithm. Experimental results show that the algorithm could significantly improve the number of iterations. | Metaheuristic methods based on swarm |
No | 2022 | In 2022, based on the improved biologically inspired neural network algorithm, Chen et al. [7] proposed a method of multi-mobile robot cooperative full-area coverage inspection. Simulation experiments verify the feasibility of the proposed multi-robot cooperative inspection scheme. | Metaheuristic methods based on biology |
Yes | 2018 | He et al. [8] proposed an improved genetic simulated annealing algorithm. Compared with the optimization results of other path optimization algorithms, the proposed algorithm can obtain better travel path. | Metaheuristic methods based on biology and metaheuristic methods based on physics |
Yes | 2019 | Yuan et al. [9] proposed a robot path-planning method based on simulated an annealing ant colony algorithm. Simulation results show that the algorithm can quickly plan the shortest and optimal inspection path. | Metaheuristic methods based on swarm and metaheuristic methods based on physics |
Table 2.
The number of legal paths varies with α.
Table 2.
The number of legal paths varies with α.
The Value of α | Number of Valid Paths n |
---|
0 < α ≤ 0.03 | 0 < n ≤ 25 |
0.03 < α ≤ 0.05 | 30 < n ≤ 60 |
0.06 < α ≤ 0.08 | 80 < n < 100 |
0.09 < α ≤ 0.1 | 30 < n ≤ 50 |
0.11 < α ≤ 0.13 | 0 < n ≤ 10 |
α > 0.13 | n = 0 |
Table 3.
The number of legal paths varies with β.
Table 3.
The number of legal paths varies with β.
The Value of β | Number of Valid Paths n |
---|
0.001 < β ≤ 0.004 | 0 < n ≤ 15 |
0.005 < β ≤ 0.007 | 30 < n ≤ 70 |
0.008 < β ≤ 0.009 | 80 < n < 100 |
0.01 < β ≤ 0.02 | 30 < n ≤ 50 |
0.025 < β ≤ 0.04 | 0 < n ≤ 10 |
0.4 < β < 1 | n = 0 |
Table 4.
The number of legal paths varies with I0.
Table 4.
The number of legal paths varies with I0.
The Value of I0 | Number of Valid Paths n |
---|
0 < I0 ≤ 0.1 | 0 < n ≤ 15 |
0.1 < I0 ≤ 0.45 | 30 < n ≤ 75 |
0.45 < I0 ≤ 0.65 | 80 < n < 100 |
0.65 < I0 ≤ 0.75 | 30 < n ≤ 40 |
0.75 < I0 ≤ 0.8 | 0 < n ≤ 10 |
0.8 < I0 < 1 | n = 0 |
Table 5.
The number of legal paths varies with z (0).
Table 5.
The number of legal paths varies with z (0).
The Value of z (0) | Number of Valid Paths n |
---|
0 < z (0) ≤ 0.3 | 0 < n ≤ 20 |
0.3 < z (0) ≤ 0.6 | 30 < n ≤ 50 |
0.6 < z (0) ≤ 0.8 | 80 < n < 100 |
0.8 < z (0) ≤ 0.9 | 45 < n ≤ 60 |
0.9 < z (0) ≤ 1 | 10 < n ≤ 30 |
z (0) > 1 | n = 0 |
Table 6.
The number of legal paths varies with ε.
Table 6.
The number of legal paths varies with ε.
The Value of ε | Number of Valid Paths n |
---|
0 < ε ≤ 0.02 | 0 < n ≤ 30 |
0.02 < ε ≤ 0.04 | 40 < n ≤ 60 |
0.04 < ε ≤ 0.1 | 80 < n < 100 |
0.1 < ε ≤ 0.3 | 20 < n ≤ 30 |
0.3 < ε ≤ 0.5 | 10 < n ≤ 30 |
ε > 0.5 | n = 0 |
Table 7.
The number of legal paths varies with Pc.
Table 7.
The number of legal paths varies with Pc.
The Value of Pc | Algorithm Optimization Performance |
---|
Pc < 0.4 | The speed of producing new individuals is slow |
0.4 ≤ Pc ≤ 0.99 | Obtain the optimal solution |
Pc > 0.99 | The excellent pattern of population is easily destroyed |
Table 8.
The number of legal paths varies with Pm.
Table 8.
The number of legal paths varies with Pm.
The Value of Pm | Algorithm Optimization Performance |
---|
Pm < 0.01 | The ability to generate new individuals and inhibit premature phenomenon will be poor, which will affect the optimization performance of the algorithm. |
0.01 ≤ Pm ≤ 0.1 | Obtain the optimal solution |
Pm > 0.1 | More new individuals can be generated, and many good patterns may be destroyed. The performance of the genetic algorithm is similar to that of a random search algorithm. |
Table 9.
Comparison of TSP City Travel Business Problem (75 cities).
Table 9.
Comparison of TSP City Travel Business Problem (75 cities).
| Operation Results | Best Results | Worst Result | Average Time | Average Optimization Rate |
---|
Algorithm | |
---|
Grey Wolf Optimizer (GWO) | 9.5523 | 15.0516 | 56.619 s | 75.77% |
Firefly Algorithm (FA) | 7.1349 | 7.0772 | 159.6186 s | 31.29% |
Ant Colony Algorithm (ACA) | 5.6281 | 6.5804 | 107.784 s | 3.56% |
Genetic algorithm (GA) | 6.0926 | 6.9553 | 77.56 s | 12.11% |
TCNN | 6.124 | 8.9322 | 26.545 s | 12.69% |
proposed algorithm | 5.4345 | 6.2969 | 33.94 s | ≈0 |
Table 10.
Comparison of TSP City Travel Business Problem (50 cities).
Table 10.
Comparison of TSP City Travel Business Problem (50 cities).
| Operation Results | Best Results | Worst Result | Average Time | Average Optimization Rate |
---|
Algorithm | |
---|
Grey Wolf Optimizer (GWO) | 7.4258 | 8.4002 | 34.87 s | 35.99% |
Firefly Algorithm (FA) | 7.30 | 11.38 | 120.1204 s | 33.69% |
Ant Colony Algorithm (ACA) | 5.58 | 7.204 | 90.435 s | 2.19% |
Genetic algorithm (GA) | 5.7495 | 7.6771 | 64.608 s | 5.29% |
TCNN | 6.3398 | 7.153 | 21.7 s | 16.11% |
proposed algorithm | 5.5551 | 6.6146 | 28.94 s | 1.73% |
Table 11.
Comparison of TSP City Travel Business Problem (100 cities).
Table 11.
Comparison of TSP City Travel Business Problem (100 cities).
| Operation Results | Best Results | Worst Result | Average Time | Average Optimization Rate |
---|
Algorithm | |
---|
Grey Wolf Optimizer (GWO) | 17.6394 | 20.1142 | 76.809 s | 94.45% |
Firefly Algorithm (FA) | 14.65 | 27.03 | 2905.2109 s | 58.56% |
Ant Colony Algorithm (ACA) | 8.1728 | 16.0399 | 115.314 s | 2.44% |
Genetic algorithm (GA) | 8.6926 | 13.4465 | 99.79 s | 8.96% |
TCNN | 16.7864 | 19.0772 | 58.545 s | 72.8% |
proposed algorithm | 8.0345 | 10.2969 | 66.94 s | 0.71% |
Table 12.
Comparison of TSP City Travel Business Problem (1000 cities).
Table 12.
Comparison of TSP City Travel Business Problem (1000 cities).
| Operation Results | Best Results | Worst Result | Average Time |
---|
Algorithm | |
---|
Grey Wolf Optimizer (GWO) | 421.7637 | 482.7657 | 690.728 s |
Firefly Algorithm (FA) | 328.65 | 437.03 | 11,322.9213 s |
Ant Colony Algorithm (ACA) | 265.5747 | 328.1901 | 915.314 s |
Genetic algorithm (GA) | 287.3142 | 326.639 | 444.598 s |
TCNN | 331.8651 | 349.1945 | 271.265 s |
proposed algorithm | 251.0795 | 279.5731 | 300.807 s |
Table 13.
“Improvement” of different instances for 75 cities (The running time is 110s).
Table 13.
“Improvement” of different instances for 75 cities (The running time is 110s).
| % Improvement | Instance 1 | Instance 2 | Instance 3 |
---|
Algorithm | |
---|
Grey Wolf Optimizer (GWO) | 52.696% | 54.47% | 57.64% |
Firefly Algorithm (FA) | 17.42% | 18.74% | 18.97% |
Ant Colony Algorithm (ACA) | 18.41% | 18.92% | 19.62% |
Genetic algorithm (GA) | 76.79% | 79.67% | 79.51% |
TCNN | 45.24% | 47.31% | 48.54% |
proposed algorithm | 79.65% | 80.99% | 82.35% |
Table 14.
“Improvement” of different instances for 50 cities (The running time is 100s).
Table 14.
“Improvement” of different instances for 50 cities (The running time is 100s).
| % Improvement | Instance 1 | Instance 2 | Instance 3 |
---|
Algorithm | |
---|
Grey Wolf Optimizer (GWO) | 55.40% | 57.89% | 47.71% |
Firefly Algorithm (FA) | 20.20% | 21.38% | 19.42% |
Ant Colony Algorithm (ACA) | 16.74% | 17.66% | 12.91% |
Genetic algorithm (GA) | 70.26% | 77.34% | 69.22% |
TCNN | 49.10% | 53.47% | 47.81% |
proposed algorithm | 77.09% | 79.03% | 76.45% |
Table 15.
“Improvement” of different instances for 100 cities (The running time is 150 s).
Table 15.
“Improvement” of different instances for 100 cities (The running time is 150 s).
| % Improvement | Instance 1 | Instance 2 | Instance 3 |
---|
Algorithm | |
---|
Grey Wolf Optimizer (GWO) | 54.63% | 52.31% | 53.36% |
Firefly Algorithm (FA) | 19.58% | 16.52% | 17.34% |
Ant Colony Algorithm (ACA) | 20.16% | 17.99% | 18.67% |
Genetic algorithm (GA) | 81.48% | 78.75% | 79.26% |
TCNN | 53.45% | 50.78% | 51.48% |
proposed algorithm | 84.52% | 81.96% | 82.49% |
Table 16.
The obtained algorithm comparison sequence values.
Table 16.
The obtained algorithm comparison sequence values.
Data Set | Proposed Algorithm | TCNN | GA | ACA | FA | GWO |
---|
D1 | 1 | 4 | 3 | 2 | 5 | 6 |
D2 | 1 | 4 | 3 | 2 | 5 | 6 |
D3 | 1 | 5 | 3 | 2 | 4 | 6 |
D4 | 1 | 5 | 3 | 2 | 4 | 6 |
The average sequence value | 1 | 4.5 | 3 | 2 | 4.5 | 6 |
Table 17.
Commonly used critical values for F-test.
Table 17.
Commonly used critical values for F-test.
α = 0.1 | |
---|
Number of Data Sets N | Number of Algorithms k |
---|
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|
4 | 5538 | 3.463 | 2.813 | 2.480 | 2.273 | 2.130 | 2.023 | 1.940 | 1.874 |
5 | 4.545 | 3.113 | 2.606 | 2.333 | 2.158 | 2.035 | 1.943 | 1.870 | 1.811 |
8 | 3.589 | 2.726 | 2.365 | 2.157 | 2.019 | 1.919 | 1.843 | 1.782 | 1.733 |
10 | 3.360 | 2.624 | 2.299 | 2.108 | 1.980 | 1.886 | 1.814 | 1.757 | 1.710 |
15 | 3.102 | 2.503 | 2.219 | 2.048 | 1.931 | 1.845 | 1.779 | 1.726 | 1.682 |
20 | 2.990 | 2.448 | 2.182 | 2.020 | 1.909 | 1.826 | 1.762 | 1.711 | 1.668 |
Table 18.
qα values commonly used in Nemenyi test.
Table 18.
qα values commonly used in Nemenyi test.
α | Number of Algorithms k |
---|
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|
0.1 | 1.645 | 2.052 | 2.291 | 2.459 | 2.589 | 2.693 | 2.780 | 2.855 | 2.920 |
Table 19.
Comparison of wind farm path inspection results with detour obstacles.
Table 19.
Comparison of wind farm path inspection results with detour obstacles.
| Operation Results | Best Results | Worst Result | Average Time |
---|
Algorithm | |
---|
Grey Wolf Optimizer (GWO) | 7.1136 | 9.8808 | 18.219 s |
Firefly Algorithm (FA) | 5.2821 | 7.6176 | 86.3244 s |
Ant Colony Algorithm (ACA) | 4.7551 | 6.7835 | 30.939 s |
Genetic algorithm (GA) | 4.8965 | 5.0656 | 46.053 s |
TCNN | 4.9696 | 5.8965 | 23.354 s |
proposed algorithm | 4.5366 | 4.7446 | 22.09 s |
Table 20.
Comparison of wind farm path inspection results with non-detour obstacles.
Table 20.
Comparison of wind farm path inspection results with non-detour obstacles.
| Operation Results | Best Results | Worst Result | Average Time |
---|
Algorithm | |
---|
Grey Wolf Optimizer (GWO) | 7.1525 | 8.9197 | 10.42 s |
Firefly Algorithm (FA) | 5.0085 | 7.6565 | 65.2261 s |
Ant Colony Algorithm (ACA) | 4.793 | 6.8224 | 22.939 s |
Genetic algorithm (GA) | 4.9354 | 5.1454 | 39.77 s |
TCNN | 5.321 | 6.4954 | 17.616 s |
proposed algorithm | 4.5755 | 4.7835 | 21.768 s |