A Robustness Temperature Inversion Method for Cable Straight Joints Based on Improved Sparrow Search Algorithm Optimized BPNN

The temperature of cable conductor is of great significance to improve the current carrying capacity, asset utilization and safe operation of cable lines. Aiming at the problems of slow calculation speed, low accuracy and weak anti-interference ability of the current temperature calculation methods, this paper proposes an inversion method based on improved sparrow search algorithm (ISSA) optimized back propagation neural network (BPNN). Tent mapping was used to increase the initial population diversity of sparrow. Modified sparrow optimization formula to improve convergence speed. Chaotic perturbation is applied to the optimal individual to improve the global and local search ability of SSA. The multi-physics simulation model of 110kv straight connector was established, and the temperature distribution data under three different working conditions were obtained. According to the simulation data and CEC2017 standard test experiments, the optimization ability of the improved model is compared with particle swarm optimization (PSO), whale optimization algorithm (WOA), SSA and MSWOA. To verify the generalization performance and migration ability of the proposed method, the thermal cycle test and inversion calculation of the 10kV cable straight-through joint were carried out. The results show that ISSA-BPNN has high accuracy, fast convergence speed, good robustness, and is less affected by cable joint type, load current and cable environment conditions. It has good engineering practicability.


I. INTRODUCTION
In response to the green development policy, economic and 19 environmental protection has become the goal of modern 20 power system to transmit electric energy. With the accelera- 21 tion of urbanization, electricity demand surges. This increase 22 has placed a greater burden on existing lines. Therefore, it is 23 necessary to fully tap the hidden capacity of lines to improve 24 the actual transmission capacity of transmission lines [1], [2]. 25 The capacity of the line depends on the maximum long-term 26 allowable operating temperature of the insulating material. 27 Therefore, the cable core temperature monitoring and diag-  The associate editor coordinating the review of this manuscript and approving it for publication was Alireza Sadeghian.
Cable joints are more prone to overheating due to their 31 physical structure and material properties. There are two 32 main ways to obtain the temperature of the conductor core. 33 One is the implantable temperature measurement, that is, 34 the temperature sensor or optical fiber is buried inside 35 the cable joint when the cable joint is made; the other is 36 non-implantable temperature measurement, such as infrared 37 imaging temperature measurement technology [3], [7], [8], 38 [9]. The built-in temperature measurement unit is at high 39 potential and withstands long-term high temperature, and its 40 safety and stability cannot be guaranteed. When transmit-41 ting signals to the external receiving unit module, it will 42 also be interfered by strong electromagnetic fields. Infrared 43 temperature measurement is also susceptible to meteoro-44 logical factors such as ambient temperature, humidity and 45 wind speed, and the farther the test distance, the lower the 46 obtain a hybrid algorithm with shorter time, and performed 102 well in fluid computing and other fields. Li et al. [21] intro-103 duced Gaussian distribution and adaptive weight to construct 104 the algorithm variant of GDS-WOA, and realized the opti-105 mization of constraint problems. Hsu et al. [28] used GWO 106 and denoising convolutional neural network (QnCNN) to 107 refine the recognition effect of quaternion discrete cosine 108 transform (QDCT) on image watermark. Zhang et al. [29] 109 combined mayfly algorithm (MA) and SSA, introduced 110 levy flight and nonlinear weight to balance the relationship 111 between global search and local search. Tuerxun et al. [30] 112 mixed SSA and SVM to improve the accuracy of wind tur-113 bines fault diagnosis. There are more other improved models 114 [31], [34], [35], [36]. Although earlier research has increased 115 the algorithm's accuracy and speed of convergence, the global 116 search and local development capabilities of SSA, a recently 117 developed swarm intelligence algorithm, remain uneven and 118 the system is still prone to falling into the local optimum. 119 Additionally, we need to make improvements to it in order to 120 strengthen its robustness and optimize its optimization effect. 121 To deal with complicated and variable operating conditions, 122 the upgraded hybrid algorithm and variant of the swarm 123 intelligence algorithm must be used to the joint temperature 124 inversion.

125
In conclusion, we propose a hybrid approach to enhance 126 SSA and use the improved SSA to optimize the weights and 127 thresholds of BPNN. The common test function CEC2017 128 assesses the upgraded algorithm's optimization performance. 129 Using modeling and experimental data, the algorithm's 130 impact on the inversion of the cable joint temperature is exam-131 ined. We assess the performance of the algorithm using δ MAE , 132 δ MAPE , δ RMSE , R 2 four indicators. The outcomes demonstrate 133 that the revised algorithm performs significantly better than 134 the original algorithm.

135
The main contributions of this study are summarized as 136 follows.
where t represents the temperature; q v is the heat generated 188 by the heat source in the unit time and area; λ is the thermal 189 conductivity.

190
The solution of the heat conduction equation in (1) also 191 required the initial conditions and boundary conditions as 192 the solution conditions for the partial equations. In the finite 193 element method for partial equation solution, there are three 194 main boundary conditions for the thermal field.

195
The first boundary conditions: the temperature of the 196 boundary was specified as a constant.
The second boundary conditions: the heat flux density on 199 the boundary was specified as a fixed value. The third boundary condition: the surface heat transfer 202 coefficient h between the object boundary and the surround-203 ing fluid, of which temperature was specified as where 1 , 2 , 3 is the boundary, n is the normal unit vector 206 of the boundary, h is the convective heat dissipation coeffi-207 cient, q n is the heat flux density, t f is the fluid temperature, 208 and f (x, y) is a constant.

210
The material parameters of cable joint is shown in Table. 1. 211 The actual operating environment of the cable joint was 212 complex, and the boundary conditions were difficult to be 213 acquired and determined. When the air velocity around the 214 cable is less than 0.15 m/s, it is a kind of natural convection. 215 The surface of the joint adopted the third type of boundary 216 conditions, the natural convection heat transfer coefficient 217 was set to 8 W/(m 2 · • C), and the ambient temperature was 218 set to 20 • C. When the cable conductor at both ends of the 219 joint is more then 5 m, the axial heat conduction is assumed 220 to be in equilibrium state. The simulation results indicated 221 that the axial heat transfer distance of the joint was 2-3 m 222 away from the crimping position of the joint. The total length 223 of the model was 8 m, so both ends can be set to the second 224

231
The heat source of a cable joint mainly included the Joule 232 heat of the conductor core, the loss of the insulating medium 233 and the circulation loss of the aluminum sheath. Since the 234 outer sheath of high-voltage cable was cross-connected or 235 single-ended grounded, the circuit current losses were negli-236 gible. The Joule heat of the conductor core included the Joule 237 heat of the core itself and the heat generated by the contact 238 resistance of the crimping position.
where R 20 is the resistance value of the wire core at 20 • C, α is  algorithm simulated the process of individual sparrows 279 avoiding predators and constantly approaching the food 280 location. The population was consisted of three roles: pro-281 ducer, follower and early warning. Producers had a high 282 energy reserve and a larger foraging area, which provided 283 foraging area and direction information for the population. 284 Followers approached producers and grabbed food resources. 285 The early warning gave warning signal when danger was 286 appearing, and if necessary, gives up food to avoid dan-287 ger. The producer location was updated in the following 288 way: where, t represents current iterations, Xt i,j denotes posi-291 tion information of the j th dimension of the i th sparrow in 292 t iterations, α is a random number in the range of [0, 1], 293 iter max is the maximum number of iterations; R 2 takes the 294 value in [0, 1], which represents the warning value; ST takes 295 the value in [0.5, 1], which represents the safety thresh-296 old; Q is a random number obeying the standard normal 297 distribution; L is a 1 × d matrix with all elements being 298 1. When R 2 < ST, the population is not in danger and 299 the foraging range of sparrows will increase; when R 2 ≥ 300 ST, natural enemies appear and the sparrows move to safe 301 areas. 302 VOLUME 10, 2022 where D is the population dimension and r is a random Generally, swarm intelligence algorithms were randomly ini-340 tialized populations, but the uniformity of population was 341 stochastic distribution in space. The initial population distri-342 bution affected the convergence speed and accuracy of the 343 algorithm.
[33], [34]. At the later period of iteration, SSA still 344 had the common problem of swarm intelligence algorithm. 345 The population approached to the food location, the foraging 346 space shrinks, the population diversity decreased, and the 347 algorithm was easy to fall into the local optimal solution. 348 The randomness of chaotic mapping could enrich population 349 diversity and improve the ability of the algorithm to jump 350 out of local optimum.
The chaotic values obtain from equation (13) were mapped 365 to the sparrow population as follow where, x n is the individual after perturbation; x is the individ-375 ual to be perturbed.

377
The basic topological structure of BPNN, shown in Fig.4, 378 consists of an input layer, a hidden layer, and an output layer. 379 BPNN is a multi-layer feed-forward network trained 380 according to the error back propagation algorithm, which 381 has the advantages of strong nonlinear mapping capability 382 and flexible network structure. The forward propagation of 383 the information of the input feature quantity is processed 384 by the hidden layer to obtain the actual output. If the error 385 between the output value and the expected value does not 386 meet the set operation termination condition, the error is 387 propagated from the output layer forward layer by layer, 388 and the gradient descent method is adopted. Through the 389 adjustment of the weights and thresholds of the hidden layer 390 neurons and the connected neurons, the network training is 391 where η is learning rate, which is between 0 and 1. The cable joint core temperature depended on the dynamic 426 balance relationship between heat generation and heat dis-427 sipation in a cable joint. Therefore, the load currents and 428 the external surface temperatures of the cable joint end 429 (T B ), which were easy to measure and can reflect the rela-430 tionship between the heat generation and heat dissipation, 431 were selected as input characteristics, and the temperature 432 of the joint core (T C ) was the inversion results. To verify 433 the performance of the temperature inversion method based 434 on ISSA-BPNN, The simulation and test were carried out 435 to obtain the joint temperature data of the following four 436 operating conditions for inversion.

437
(1) Simulation load currents were applied in the form of a 438 single step to simulate the actual cable line load stabi-439 lization period. Temperature rises of the joints were not 440 large when the load current was small, and there was no 441 over-temperature danger. Therefore, load currents were 442 set from 800 A to 1300 A, with 100 A interval for a total 443 of five groups of simulations.

444
(2) The current was applied in the form of multiple steps 445 with no fixed time interval and large changes. In sim-446 ulation such as winter and summer periods with large 447 fluctuations in daily electricity consumption, The relation 448 between joint temperatures and currents applied were as 449 shown in Fig.5.

450
(3) To better simulating the actual operating currents, the 451 actual daily load curve of a residential area was simulated 452 by a segmentation function with one hour interval, and 453 Wave-forms of straight joint temperatures of joints and 454 equivalent current were as shown in Fig.6.

455
(4) To verify the generalization capability of the algorithm, a 456 single-core cold-shrink straight joint of 10 kV 185 mm 2 457 was subjected to a thermal cycling test and surface tem-458 perature inversion to verify the current and the measured 459 temperature of each layer of the joint were as shown 460 in Fig.7.

461
Under each of the above operating conditions, the ambient 462 temperature range was set to 18∼23 • C. 80% of the sim-463 ulation test data were randomly selected as the training set 464 and 20% as the test set, and the specific number of samples 465 collected was shown in Table 2. is large, the autonomous learning time consuming will be 471 greatly increased. Therefore, in this paper, ISSA was used to 472 optimize the weights and thresholds of BPNN to reduce the 473 VOLUME 10, 2022     (2) Initialization of sparrow population parameters: input 479 population size, the maximum number of iterations, 480 number of discoverers, number of early warning, safety 481 threshold, alarm value.

483
(4) Calculation and ranking of fitness values for each 484 sparrow.

485
(5) Update the positions of producers, followers and early 486 warnings.

487
(6) Determine whether the fitness value reaches the con-488 vergence condition, and perform a chaotic perturbation 489 update if it was not reached.

490
(7) If the maximum number of iterations was reached, the 491 position information of the global optimal sparrow was 492 output, and vice versa, returned to the fifth step to con-493 tinue the cycle.

494
(8) Assigning the optimal parameters obtained by SSA to 495 the weights and thresholds of the neural network for 496 temperature inversion.

498
Based on the results of temperature inversion, the five models 499 were compared using δ MAE (mean absolute error), δ MAPE 500 (mean absolute percentage error), δ RMSE (root mean square 501 error), and R 2 (goodness of fit) as model evaluation indexes 502 with the following equations:     Table 3.  Table 4. To compare the con-526 vergence speed and accuracy of each algorithm model more 527 intuitively, this paper also gives the test function convergence 528 curve shown in Fig.9.

529
On F1-F4, the optimization effect of ISSA is better than 530 that of PSO, WOA, SSA and MSWOA. For F1, SSA, 531 MSWOA and ISSA can find their optimal values, but the 532 average and standard deviation of ISSA are 0, indicating 533 that it has strong stability. The optimization effect of ISSA 534 on F2 is also tens of orders of magnitude higher than 535 other algorithms. According to the overall performance of 536 the model on the single-extremum test function, it can be 537 seen that it has certain advantages over other swarm intelli-538 gence optimization algorithms in its local development abil-539 ity. On the F5 function, although the algorithm failed to 540 find the ideal optimal solution, but the convergence curve 541 shows that the convergence rate of ISSA is still faster. 542 This is because chaos perturbation enriches the population 543 individuals and improves the global search efficiency. The 544 variants of the swarm intelligence algorithm on F6-F9 can 545 find the optimal value, indicating that the ISSA and MSWOA 546 algorithms have better optimization capabilities under low-547 dimensional conditions. The convergence speed of ISSA 548 in Fig.9 (a) (b) (c) (e) (f) is faster. In addition, the conver-549 gence speed of ISSA and MSWOA in the convergence graph 550 is not much different, but the convergence speed of the 551 improved swarm intelligence algorithm model is significantly 552 faster than that of the original swarm intelligence algorithm. 553 VOLUME 10, 2022 F8 function for each algorithm is easy to find its optimal 554 solution in the beginning of the iteration directly converges 555 to the optimal solution, so the graph is almost a straight 556 line. In summary, it can be proved that the optimization 557 performance of ISSA has been significantly improved, and 558 the stability is also strong.  Table 5. BPNN uses relu as the hidden layer 570 activation function. The input layer and output layer nodes 571 are 2 and 1 respectively, and the number of hidden layers is 572 1. The number of neurons is determined to be 9 according to 573 the empirical formula (26) and the mean square error on the 574 training set. The average mean square error on the training set 575 and the test set is selected as the fitness function, as shown in 576 Equation (27).  where n is the number of neurons in the input layer, m is the and MSWOA converge after 21 iterations, but the former 591 converges to 6.7 × 10 −2 , and the convergence accuracy is 592 higher. The minimum fitness value of ISSA means the highest 593 convergence accuracy. The first inflection point indicates that 594 the ISSA curve converges fastest. 595 Fig.11 shows the inversion error of single-step load. 596 The BPNN models optimized by the five optimization 597 algorithms can accurately reflect the joint mandrel temper-598 ature. Among them, the inversion error of ISAA-BPNN and 599 MSWOA-BPNN is the smallest, and the maximum error does 600 not exceed 0.2 • C. The PSO-BPNN error is the largest, but the 601 maximum error does not exceed 0.7 • C. When the load does 602 not fluctuate greatly, the accuracy of each algorithm is high, 603 and the advantage of ISSA is not easy to reflect.

604
Compared with the single step working condition, 605 as shown in Fig.12, the joint temperature inversion error 606 under multi-step load is generally increased, but the 607 ISSA-BPNN still has excellent performance, and the error is 608 stably distributed within 0.5 • C. The inversion error of other 609 VOLUME 10, 2022    Taking multi-step loading as an example, the inversion 616 effects of five models are evaluated. As shown in Table 6, 617 FIGURE 13. Error comparison of temperature inversion results for 110kV cable joints under equivalent actual load current test. the R 2 of ISSA-BPNN and MSWOA-BPNN are more than 618 99%, but the former reaches 99.84%, which is closer to 1, 619 and the error is closer to 0. Combined with Fig.12 and Fig.13, 620 it can be seen that under dynamic load, with the acceleration 621 of load change time, the inversion effect of the neural network 622 model optimized by the original heuristic algorithm becomes 623 worse. However, the performance of ISSA-BPNN is stable, 624 the fluctuation of inversion error is small, and the change of 625 load current size and change rate has little effect on the model 626 inversion effect. It shows that the improved model has strong 627 generalization ability. To further verify the generalization capability of the model, 631 a straight joint for 10 kV 185 mm 2 cable was replaced for a 632 three-day thermal cycling test. The inversion validation was 633 performed based on the data obtained from the tests. As can 634 be seen from Fig.7, the environmental conditions of the test 635