1.

INTRODUCTION

Most modern naval vessels are made of ferromagnetic materials. Under the influence of the earth’s magnetic field, they will generate magnetic anomalous signals in their existing areas. These signals can easily become signal sources for underwater magnetic attack weapons and anti-submarine aircraft^1,2. In order to minimize these signals and maintain the safety and concealment of the naval vessels, it is necessary to install a degaussing system which uses the magnetic field generated by the coils to offset magnetic anomaly signals³. Since the earth’s magnetic field can be represented by three orthogonal components, three-directional degaussing coils should be installed to effectively offset the influence of each component^4,5.

The design of the coils and the calculation of the optimal current are two bottlenecks in the design of modern degaussing system⁶. It is necessary to select the best winding combination from multiple windings for specific magnetic abnormal signals in order to minimize the magnetic anomaly signal. Finding the best combination of the coils is a complex optimization problem. Because that the degaussing coils have many potential installation positions⁷. Therefore, we arrange a large number of coils on the naval vessels, and then select different combinations of coils for different magnetic field signals.

In the current degaussing system design, the coils are evenly arranged on the surface of the naval vessels based on the prior knowledge of experts. Bekers et al.⁷ and Modagekar et al.⁸ proposed the application of multiple linear regression to the design of degaussing system. This method lays a large number of degaussing coils on the naval vessels, and then sorts according to the effects of the coils on reducing the magnetic anomaly signals. The algorithm runs in an iterative manner. In each iteration, a new variable (coils) is added, and the current of the coils is calculated at the same time, the residual error between the magnetic anomaly signal and the magnetic field generated by the coils is minimized. In practical applications, this method can find the most suitable installing position if the number of coils limited to a certain number.

However, this paper only carries out simulation verification in the software, and does not consider the influence of the fixed magnetic field of the actual model on the test results.

Although this method can provide the current required by the coil, the current provided by this method can only minimize the root mean square of the compensated ship’s magnetic field signal. However, the task of ship degaussing is a multiobjective optimization problem^9-12. Not only must the root mean square error of the compensated ship magnetic signal be minimized, but also the peak value of the magnetic field must be minimized.

In order to maximize the efficiency of the degaussing system, a large number of degaussing coils are firstly laid on the model in this paper, and then LARS¹³ algorithm is used to sort the coils according to the contribution of coils for the magnetic abnormal signals generated by ships under specific conditions. In practical application, a certain number of windings are selected from the multiple windings from the front to the back to obtain the windings combination mode that is most suitable for compensating the current magnetic abnormal signal. Then the multi-objective optimization model of current optimization adjustment is established and the multi-objective optimization problem is transformed into a single objective optimization problem by selecting a group of appropriate weighted coefficients by linear weighted sum method. On this basis, the current obtained by LARS is substituted into the multi-population Particle Swarm Optimization (MPSO) algorithm. In this way, the initial position and velocity of particles can be constrained, which greatly speeds up the algorithm.

The organization structure of this paper is as follows. The second section introduces the types of naval vessels degaussing system coils, including rib coils that cancel the longitudinal magnetic field, side coils that cancel the transverse magnetic field, and latitude coils that cancel the vertical magnetic field. The third section shows the multi-dimensional linear regression technology based on LARS and the multi-objective optimization technology based on MPSO. The fourth section discusses the on-site measurement results of the degaussing effect of the naval vessels based on the above methods. Finally, the fifth section summarizes the reference value of this research for practical applications.

2.

DESIGN OF DEGAUSSING COILS

The naval vessels degaussing system consists of a magnetic field measurement module, coil current control module, degaussing coils, etc.^14,15. Because naval vessels have three types of magnetism: longitudinal magnetism, transverse magnetism and vertical magnetism, the degaussing system coil used to cancel the naval vessels magnetic field also has three directions¹⁶, including the rib coil to cancel the longitudinal magnetic field, the side coil to cancel the transverse magnetic field and the latitude coil to cancel the vertical magnetic field¹⁷.

To verify the effectiveness of the automatic modeling algorithm and the multi-objective particle swarm algorithm, it is essential to lay a large number of degaussing coils in the three directions of the model, and then select the most effective part of the coils according to practical application restrictions (such as the total number of coils does not exceed a certain number, some positions cannot Install coils). Take a submarine model as an example. As shown in Figure 1, 68 coils (including 32 axial coils, 18 transverse coils and 18 vertical coils) are placed on this model. Each coil is located on the surface of the naval vessel model, and has its own center, direction and size.

Figure 1.

Three axis degaussing coils. (a): axial coil; (b): transverse coil; (c): vertical coil.

It is worth noting that the shapes of coils XQ14, XQ15, XQ16, XQ17, XQ18 are different from other axial coils. Because the special structure of this part of the naval vessels. This special-shaped coil can effectively reduce the magnetic anomaly signal generated by the submarine shall.

The ultimate goal of the degaussing system is to use the coils to generate a magnetic field equal in magnitude and opposite to the magnetic signal of the submarine¹⁸. The final target magnetic field is the linear sum of the magnetic fields generated by each sub-coil. In general, the kind of degaussing system design, which use as few coils as possible within the constraints of the coil current to compensate for magnetic abnormal signals to the greatest extent is the best.

3.

OPTIMIZATION MODEL

3.1

Automatic modeling algorithm

Assuming that the naval vessels have P independent degaussing coils, each of which is powered independently, and there are N uniaxial sensors under the naval vessels keel, then the external magnetic field of the naval vessels is

where b_i is the magnetic field measurement value at the measurement position i.

In the absence of external magnetic field excitation, a current of unit ampere is applied to each coil, the magnetic induction intensity value is measured on the same measurement line to form the degaussing efficiency matrix K of the degaussing coil

where k_ij represents the degaussing efficiency of coil j in the i-th sensor. Then we add a set of current values to the coil

The external magnetic field of naval vessels can be changed to achieve the purpose of magnetic stealth.

The set of current I can be obtained by solving the following equation:

The above problems can be solved by LARS algorithm, which can quickly get linear regression results. And the final model can be obtained after at most P steps (p is the number of regression variables).

The basic steps of the LARS algorithm are as follows:

• 1) Firstly, we find a variable K1 (cosine similarity) that is most relevant to variable B, and make the vector X=K1.

• 2) Then we extend in the direction of the vector X until it finds another variable K2, so that the correlation coefficients of the two variables are the same as those of the current residuals, at this point K and K2 have the same degree of correlation with respect to B, and record the distance from this point to the origin as |λK|.

• 3) The parameter is updated as X = B – λK.

• 4) Steps 2) and 3) are repeated until all vector directions are found.

An example of the LARS method when the above calculation process gives two predictors (K1 and K2) is shown in Figure 2.

Figure 2.

An example of minimum angle regression based on two predictor variables (K1 and K2).

Among them, the black line is the feature vector, the red line is the target residual vector of each round, the green line is the projection value of each round, and the superscript indicates the i-th round.

3.2

MPSO optimization model

The current solved by the above LARS algorithm can only minimize the root mean square of the compensated naval vessels magnetic field signal. However, in practical applications, it is necessary to ensure that the peak value of the submarine’s magnetic field is minimized, and it is also necessary to minimize the submarine’s magnetic field gradient as much as possible, so the objective function is:

At present, many scholars have conducted extensive and in-depth research on multi-objective optimization problems, and formed a variety of evolutionary multi-objective optimization algorithms^19,20. Although these algorithms can effectively solve multi-objective problems, they are complex and time-consuming. Directly select the multi-objective optimization algorithm to finally obtain a non-inferior solution set, and it is necessary to design additional processes to select individual solutions that meet the requirements from the non-inferior solution set. So they are not suitable for degaussing systems.

The linear weighted sum method is used to create penalty functions²¹ and the method of converting multi-objective problems into single-objective problems is simple, effective and less time-consuming, which is often applied in practical work.

The basic idea of linear weighted sum method is to give large weights to important objects, so the multi-object vector problem is transformed into a scalar problem of weighted sum of all objects²². According to this principle, equation (5) can be transformed into:

where ω₁ > 0, ω₂ > 0 and ω₁ + ω₂ =1. For the experimental content of Section 4 in this article, all the experimental results in the article are obtained in advance of ω₁ = ω₂ =0.5, and different weighting coefficients can be selected according to the specific conditions in actual application.

Equation (6) can be solved by MPSO. The difference between MPSO and ordinary PSO algorithm is that MPSO algorithm divides the entire particle swarm into three independent particle swarms and each particle swarm evolves according to different rules [23]. The algorithm not only maintains the independence and superiority of each subgroup and algorithm, but also does not increase the complexity of the algorithm. It has improved global search, convergence speed, accuracy and stability. The basic steps of the MPSO algorithm are as follows:

• 1) Determining the target function is the minimization problem, the size of the population is determined and the three subgroups are divided, then the maximum number of iterations and other correlation coefficients are set.

• 2) The parameters are initialized. A random position and speed at t = 0 is generated, and its adaptation value is calculated.

• 3) The value of iteration is updated as t = t + 1, when the number of times the algorithm is later reaches the set threshold, a disturbance operation is performed.

• 4) The particles are expanded and mutated that are in the global optimum of the subgroup.

• 5) Particles without subgroups are evolved according to different rules, the speed and position of each particle and the historical optimal position of the particle at the same time are updated, and then the global optimal particle of the subgroup is determined.

• 6) The global optimal particle of the total group is selected from the global optimal particles of the three subgroups.

• 7) The algorithm stops when the algorithm reaches the maximum number of iterations or when a satisfactory solution is found, if it is not, we return to Step 3), otherwise the iteration ends and the result is output.

The overall flow chart of the algorithm is shown in Figure 3.

Figure 3.

Overall flow chart.

4.

MEASURED RESULT

The actual test model is a 1:20 scaled model. The length (L) of the model is 4.5 m, the maximum diameter (B) is 0.27 m, the length of the stern of the naval vessels is 0.7m, and the minimum radius of the stern is 0.1 m. Since the actual model is secret, a schematic of the scene is shown here. The schematic diagram of the model and sensor placement is shown in Figure 4.

Figure 4.

Test schematic.

A three-axis cartesian coordinate system was established with the center of the model as the origin. At 1.2 m below the model, 10 sensors were placed from -L/2 to L/2 in the X-axis direction with an interval of 0.5 m, and 5 sensors were placed from -0.4 m to 0.4 m in the Y-axis direction with an interval of 0.2 m, so as to establish the measurement plane grid.

18 degaussing coils were placed in the axial direction, 10 in the vertical direction and 10 in the transverse direction, so a total of 38 degaussing coils were placed. In practice, however, the number of coils is determined by the number of power supplies.

The number of measurement points in the grid at this time is 50. Since the magnetic field observed at each measurement point is three-component, a total of 150 magnetic field data are measured on the entire plane grid. Assign the following numbers to the coils:

First we measure the efficiency of each degaussing, which means, passing a 1A current through the coil, and measuring the magnetic field value at each point in the grid. A total of 38 coils are set on this model, so the dimension of matrix K is 150×38. All coil currents are set to 0, and the induction field is set to the original geomagnetic field in Wuhan. The X-axis of the sensor points to the due west direction, the Y-axis points to the vertical downward direction, and the Z-axis points to the due south direction. The geomagnetic field measured in Wuhan is about (4305nT, 35825nT, 33669nt). We record the measurement data in the grid to obtain the magnetic anomaly matrix B. The dimension of the matrix B is 150×1.

The matrix K and matrix B are substituted into the LARS algorithm, and the result of the selection order of coils is obtained as the number of iterations increases, as shown in Figure 5.

Figure 5.

Selection order of coils.

The peak value and average value of the total magnetic anomaly field caused by the target decrease with the increase of the number of iterations, and the result is shown in Figure 6.

Figure 6.

(a): Peak value of total magnetic anomalies; (b) Average value decrease with the number of iterations of the LARS algorithm.

The three-component magnetic field of the magnetic anomaly is studied separately, and its peak value and average value decrease with the increase of the number of iterations. The results are shown in Figure 7.

Figure 7.

(a): Peak value of three-component of magnetic anomalies; (b): Average value decrease with the number of iterations of the LARS algorithm.

The actual test results show that the LARS algorithm can indeed effectively reduce the magnetic anomaly signal. From Figures 6 and 7, it can be seen that the total magnetic field and three components of the magnetic anomaly decrease with the gradual introduction of the coil, and the magnetic anomaly can be rapidly reduced when the coil is introduced in the early stage, and slowly reduced in the later stage. In Wuhan’s original geomagnetic field, when the number of coils is 23, the total magnetic anomaly field and the three-component magnetic field are reduced to below 100nT.

As shown in Table 1, after all the above coils are added into the system, the final peak value of magnetic anomaly is reduced to 44.93nT and the average value to 9.99nT. Taking the current value after LARS iteration as the initial value of the MPSO algorithm, the peak value of the magnetic anomaly decreases from 44.93nT to 29.43nT, and the average value increases from 9.99nT to 10.92nT. With the introduction and fusion of the MPSO algorithm, the peak value of residual magnetic anomalies has been reduced by 34.48%, and the average value has only increased by 0.93nT. Based on the two optimization goals, the degaussing effect is significantly improved.

Table 1.

Experimental result.

5.

CONCLUSIONS

The experimental results in this paper show that the LARS algorithm can provide guiding suggestions for the layout design of the degaussing coils. In this method, the measurement grid plane is first established at a certain position outside the model. Then, the magnetic field data in the grid is recorded for a specific external magnetic field. Finally, LARS sorts the coils according to their contribution to the reduction of the magnetic field in the grid. The test results show that whether it is a unidirectional external magnetic field or a three-directional external magnetic field, LARS can select coils according to a specific magnetic field, and LARS has determined that the first few coils can quickly reduce the magnetic anomaly signal in the grid. The addition of the MPSO algorithm solves the problem that the LARS algorithm does not consider the peak value of the magnetic field as the optimization target. According to the field experiment results, with the introduction of MPSO algorithm, the peak value of residual magnetic anomaly is reduced by 34.48%, so the demagnetization effect is significantly improved. In addition, constraining the initial value of particles through the LARS algorithm can also compress the search space, improve the probability of the optimization algorithm to obtain the optimal solution, and greatly reduce the running time of the MPSO program, which has certain practical guiding significance.

In practical applications, the layout of degaussing coils is based on the prior knowledge of experts, the algorithm provided in this article can provide guidance for actual engineering, in addition, the actual scale-down model test in this article includes not only the induced magnetic field but also the fixed magnetic field. This is a direction that has not been studied in many literatures. In summary, this research should have reference value in practice.