Yu Peng ,Jin-fang Cheng ,Run-xiang Jiang

a Military Key Laboratory for Military Target Characteristics,Naval University of Engineering,Wuhan 430033,PR China

b College of Weaponry Engineering,Naval University of Engineering,Wuhan 430033,PR China

Keywords:UEP PSO Inversion Point-electrode

ABSTRACT To model the underwater electric potential(UEP)of ships,a multiple point-electrodes method is commonly used.However,it is difficult to determine the total number of point-electrodes,their respective positions and current values.In this paper,the particle swarm optimization(PSO)method is applied to solve the number,positions and current values of these point-electrodes according to the UEP distribution at a known depth below the keel of a ship.A scaled ship model experiment is carried out to determine the effectiveness of the method.The results show that for hulls in a natural corrosion or cathodic protection state,the UEP inversion accuracy can reach 85%at different depths below the keel.This method is suitable for software implementation and can help the simulation and prediction of UEP signatures.?2020 China Ordnance Society.Production and hosting by Elsevier B.V.on behalf of KeAi Communications Co.This is an open access article under the CCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

The underwater electric potential(UEP)signature of a ship is a measure of the static electric field arising from corrosion and anticorrosion currents around the hull.It can be applied to warship detection,tracking and positioning[1,2].

There are mainly two methods to model the UEP signatures of ships.The first one is to use the boundary element method(BEM)or finite element method(FEM)software to simulate the UEP of ships,such as BEASY Corrosion&Corrosion related magnetic software[17]and COMSOL Multiphysics software[18].These software need much information about the ship including hull lines and material polarization curves.Although the modeling accuracy is very high,the computation is huge and the real-time performance is poor[3].The other method is to use the point-electrode method to simulate the UEP signature of ships.This method has much less computation which is less than 1%of the former one.Although its modeling accuracy(about 90%)is lower than the former one(more than 99%),the accuracy is sufficient to meet our requirements in most cases[4].

At present,the research of UEP modeling mainly focuses on determining the number,locations and current values of pointelectrodes based on the know n hull information.CHENG and LIU apply a pair of electric dipoles to model the ship's static electric field[5,6],which is similar to using two point-electrodes for electric field modeling.But this method is only an idealized approximation.In order to improve the accuracy of modeling,JANG adopts the least square method to calculate the number,positions and current values of point-electrodes[7].The effectiveness of this method is demonstrated by BEASY Corrosion&Corrosion related magnetic software.On the basis of his former work,JIANG proposes a fast prediction algorithm of the electric field[8].How ever,both methods require much hull information,such as the coating damage rate,hull material characteristics,areas of each part of hull surface,etc.The difficulty is to determine the positions and current values of point-electrodes,and this method contains many empirical elements.

In this paper,the PSO method[9]is applied to directly calculate the number,positions and current values of point-electrodes by using the UEP distribution at a know n depth below the keel of a ship,without knowing other information about the ship hull.We call this process the inversion method.

After the inversion,we can get the total number,positions and current values of the point-electrodes.Then we are able to calculate the UEP at other depths based on the information about the pointelectrodes.We call this process the forward calculation.

Then the inversion accuracy can be determined by comparing the UEP calculated with the measured data.

Finally,a scaled ship model experiment is carried out to examine the inversion performance when the ship is in a natural corrosion state or cathodic protection state.And the results show a good performance in both ship states.The method proposed in this paper does not need to get the information about hull lines,material characteristics and so on.It only needs to obtain the UEP distribution at a certain depth below the keel.As the UEP measured closer to the hull would contain more information about the hull,a shallower measure depth can achieve a better inversion performance.

2.Problem formulation

Ships have complex structures made of various materials,including carbon steel,low alloy steel,cast steel,copper alloy,aluminium alloy,stainless steel and titanium alloy.Due to the different electrode potentials of metal materials in seawater,a corrosive galvanic cell(electrolytic couple circuit)is formed when different metals are electrically connected by metal wires,which will generate currents in the circuit[10].The hull can be equivalent to multiple point-electrodes,and the UEP is the result of the joint action of these point-electrodes.

The coordinate definition is shown in Fig.1 where the y axis points starboard.We define the horizontal distance as the distance between the measuring point and the keel along the y axis and the vertical distance is the distance between the measuring point and the keel along the z axis in this paper.

The hull is equivalent to m point-electrodes,as shown in Fig.1.Their coordinates and current values are（xi，yi，zi）and Ii（i=1，··，m）,respectively.Then the UEP at a point P（x，y，z）on the measuring line is

Where K（Ii，P）represents a distance function between the measuring point P（x，y，z）and the point-electrode i.Under the condition of three homogeneous medium layers“air-sea-bed”,we can get[11,12].

And

Fig.1.UEP modeling based on point-electrodes.

Where h is the depth of seawater;k is the reflection coefficient of the seabed;σ1，σ2are the conductivities of seawater and the seabed,respectively;n is the number of reflective layers.According to JIANG[7],it is proper to set n=20 in engineering calculation.

If Eq.(1)is used to calculate the UEP,the current density and the area of each part of the hull surface need to be obtained beforehand.The workload is large and the feasibility is poor.

How ever,these formulae can be used for an inversion calculation,which means the number,positions and current values of equivalent point-electrodes can be automatically solved based on the know n UEP φj（j=1，…，N）of N points below the hull.As shown in Eq.(4),the parameters to be solved are current I and distance function K,where λ=（4πσ1）-1.

To make a briefer expression,Eq.(4)is abbreviated as Eq.(5).Because there are too many unknowns,it is difficult to get I and K by solving the equation directly.Therefore,a numerical approximation method can be used to approach the optimal value,and the objective function of the optimization is Eq.(6).

In addition,some scholars believe that the current values of all point-electrodes should follow the principle of electrical neutrality,which means I1+I1+…+Im=0.How ever,this principle is mainly useful in the case that a large number of point-electrodes exist.Because there are many minor damages on the hull,there will be many equivalent point-electrodes with relatively small current.How ever,the purpose of this paper is to use as few pointelectrodes as possible to achieve a higher inversion performance of the UEP signature,so the principle of electrical neutrality is not follow ed in this paper.

The analysis shows that the PSO algorithm is suitable for solving the above problem,so the PSO method is utilized to calculate the total number,locations and current values of the point-electrodes.At present,the research of PSO is relatively mature and so will not be described here in detail.It is worth noting that the inversion accuracy of the UEP will be improved but the calculation will be doubled when the number of point-electrodes increases.Therefore,the number of point-electrodes will be set first in the inversion process,and the number of point-electrodes will continue to increase till the inversion accuracy meets the requirement.The formula for calculating inversion accuracy is Eq.(7),where the measured data is φ1jand the inversion result is φj.

In order to achieve better inversion performance,the UEP should be measured close to the hull which contains more information about the hull.

3.Calculation steps

We measure the UEP at depth of 1.0B,where B represents the molded breadth of a ship,as shown in Fig.2.

Then start the inversion calculation.After the inversion process,the inversion results are used to calculate the UEP at other depths.To evaluate the inversion accuracy,the UEP of other depths are also measured.Finally the inversion accuracy is determined by comparing the calculated UEP with the measured data using Eq.(7).

Step 1.Data initialization and boundary restriction:set the number of point-electrodes as m;limit the electrodes position and current values within a reasonable range;limit the maximum number of iterations.

Step 2.PSO calculation:follow the rule of“particle initializationupdating individual optimal pbest and global optimal gbestparticle search iteration updating”.Firstly,the coordinates and current values of the point-electrodes are initialized,and the UEP on the measuring line are calculated according to Eq.(4),then the fitness of each particle is calculated by Eq.(6).At last,the particle is updated and optimized again.

Step 3.The inversion accuracy is calculated according to Eq.(7)when the number of PSO iterations exceeds the maximum number of iterations.If the inversion accuracy meets the requirement,the optimization process stops.Or if the accuracy does not meet the requirement,the number of point-electrodes m becomes m+1 and the optimization is carried out again as in step 2.

It is noteworthy that the initial positions of the point electrodes have a significant effect on the PSO efficiency.The hull can be regarded as symmetrical along the keel,thus yi=0（i=1，…，m）.At the same time,the depth of the point electrode is set to the draft depth of the ship,which can greatly reduce the computational complexity while maintaining a good inversion performance.

Placing the point-electrodes on the keel has little effect on the calculation of the UEP directly below the keel,but has some influence on the calculation of the UEP where the horizontal distance is non-zero.Therefore,when calculating the UEP where the horizontal distance is non-zero,each point electrode should be divided into two point-electrodes symmetrical along the keel.The y coordinates of the two point-electrodes correspond to the y coordinates of the centers of the two propellers respectively,and their current values are half of the current value of the previous pointelectrodes respectively.

Fig.2.Sketch of measuring the UEP.

4.Experiments

In order to determine the effectiveness of the proposed inversion method,a scaled ship model experiment is carried out.PSM(Physical Scale Modeling)has been widely used in the study of corrosion and anti-corrosion of warships in various countries[13,14].According to the scaled model theory,the UEP of the scaled ship is proportional to that of the actual ship.In this paper,the UEP at different vertical distances and horizontal distances are obtained when the ship model is in a natural corrosion state and cathodic protection state.Then the effectiveness of the inversion method proposed is determined.

4.1.Experimental conditions

4.1.1.Experimental facilities

The ship model used is scaled in equal proportion,2.8 m in length,0.34 m in width and 0.09 m in draft.The hull is made of 907A steel;the propeller is made of nickel-aluminum bronze(NAB);the stern shaft adopts electrical insulation measures;the hull is coated with marine paint(thickness is about 0.5 mm),as shown in Fig.3.

The experiment pool is 12 m long,6 m wide and 1.5 m deep.The bottom and all sides of the pool are painted with epoxy resin.The conductivity of seawater is 2.61 S/m.

In the experiment,the UEP at different depths and different horizontal distances are measured while the ship model moves in the pool.The towing speed of the ship model is set to 4.2 cm/s.

The measuring sensor comprises low noise Ag/AgCl electrodes with a 4×3 array arranged underwater.The first vertical measuring electrodes on the left side of the electrode rack are aligned with the keel as shown in Fig.3 and the scaled ship model moves along the x-axis.The measuring electrode depth dvis defined as the vertical distance between the measuring electrodes and the water surface;dhis defined as the horizontal distance between measuring electrodes and the keel.As the UEP has strong regional characteristics,we measure the UEP signals at four depths(0.5B,1B,1.5B,2B)and three horizontal distances(0B,0.5B,1B).A reference electrode is arranged at the bottom of the middle edge of the pool.The measurement circuit adopts a low noise electric field amplifier.The signal bandwidth of the amplifier is direct current(DC)to 0.5 Hz and the gain of the preamplifier is 60 d B.In the range of 0.1-10 Hz,the noise peak-peak value is less than 1μV.

4.1.2.Ship states simulation

Ships are generally in a cathodic protection state[15,16].Commonly,cathodic protection includes impressed current cathodic protection (ICCP) and sacrificial anodes cathodic protection(SACP).

Fig.3.Relative position between the ship model and measuring electrodes.

In order to make the ship state similar to the real cathodic protection state,both ICCP and SACP methods are adopted in the experiment.Two pairs of auxiliary anodes(ICCP1 and ICCP2)are installed in the middle of the hull and near the propeller respectively.Each pair of auxiliary anodes is symmetrical along the keel.The reference electrodes are installed in the center of the hull surface.At the same time,the fin stabilizer and rudder plate are installed with sacrificial zinc anodes which are equally proportionately scaled from real zinc blocks,see Fig.4.The ICCP workstation adopts the constant potential working mode,which means the potential near the reference electrode is controlled near the set value by adjusting the output current of auxiliary anodes automatically.In the experiment,the reference potential is set to-900 m V.

In the experiment,the UEP is measured firstly when the ship is in a natural corrosion state.Then the ship state is changed to a cathodic protection state to avoid the influence on the UEP in the natural corrosion state.

4.2.Experimental results and analysis

To eliminate the influence of static UEP difference between the measuring electrodes,the experiment data needs to be de-DC preprocessed,so we consider the UEP value as zero when the ship model is at the farthest distance from the measuring electrodes.

4.2.1.Natural corrosion state

Fig.5 shows the passing characteristics of UEP when the measuring electrodes are at different depths or different horizontal distances from the ship model in a natural corrosion state.The x coordinate is the pool position coordinate,and the position of ship's propeller corresponds to the minimum potential point(see Fig.5).It is clear that in a natural corrosion state,the UEP characteristics of the ship model are relatively simple,similar to the UEP signature of an electric dipole.

In the inversion process,we begin with two point-electrodes.The results show that the maximum inversion accuracy of UEP with two point-electrodes is no more than 75%at(dv=1.0B,dh=0B),but the inversion accuracy can reach about 88%when the number of point-electrodes is increased to three.The specific results are shown in Table 1.Note P_E number represents the number of point-electrodes;(1.0B,0B)represents(dv=1.0B,dh=0B).

Table 1 shows the inversion accuracy can reach more than 88%at all three depths when the number of point-electrodes is no less than three.But the inversion accuracy of dh=1.0B is lower than that of dh=0B at depth of 2.0B.The main reason is that the hull is equivalent to being symmetrical along the keel.In fact,the states of the coating on the port and starboard sides are different,which leads to a little lower inversion accuracy when the horizontal distance is not zero.

Fig.4.The location of ICCP anodes and SACP anodes.

Fig.5.UEP in a natural corrosion state.

Table 1 UEP inversion accuracy according to different numbers of point-electrodes.

Fig.6 is the UEP inversion results with three point-electrodes in a natural corrosion state.The y coordinates of three pointelectrodes are 380.7 cm,416.2 cm and 517.8 cm respectively,where y=522 cm corresponds to the position of the propeller.It can be seen that the positions of three point-electrodes correspond basically to the middle,rear parts of hull and the vicinity of the propeller.Their currents are 3.2 m A,2.4 m A and-3.9 m A,respectively.Note that the negative current means absorbing current.From the characteristics of these point-electrodes,we can find the propeller acts as the main field source.

Fig.6.UEP inversion performance with 3 point-electrodes.

It can be seen that three point-electrodes can get good inversion results at different depths and different horizontal distances.Therefore,in order to simulate a ship's static electric field or UEP in a natural corrosion state,three point-electrodes can be used.If the accuracy is controlled above 90%,four point-electrodes are enough.

4.2.2.Cathodic protection state

Compared with the natural corrosion state,most ships are in a cathodic protection state.The UEP signature of a ship in a cathodic protection state is obviously more complex than that in a natural corrosion state,as shown in Fig.7.It can be seen that the UEP in a cathodic protection state has several peaks,which correspond basically to the positions of two pairs of auxiliary anodes,as indicated in Fig.7.The main reason is that the output current of auxiliary anode is obviously larger than that of sacrificial anodes under a cathodic protection state,so the peak values of UEP correspond to the positions of auxiliary anodes.If the UEP measured is closer to the hull,smaller peaks will appear near the sacrificial anodes.

Considering the multi-peaks characteristics of the UEP in a cathodic protection state,the inversion process is carried out directly from three point-electrodes.The inversion accuracy corresponding to the number of point-electrodes is shown in Table 2.

By comparing Tables 1 and 2,we will find that the inversion accuracy is higher in a natural corrosion state with the same number of point electrodes.That is because the UEP signature is more easily affected by ICCP anodes which have relatively larger output currents.Accordingly,the number of equivalent pointelectrodes is affected by the number of ICCP anodes.How ever,most of the ships are protected by two pairs of ICCP anodes,so the results calculated have a certain universality.

The inversion performance with four point-electrodes in a cathodic protection state is shown in Fig.8.The y coordinates of the four point-electrodes are 340.5 cm,427.7 cm,482.1 cm and 532.2 cm.Their positions correspond to the hull bow,the ICCP2 auxiliary anode,the hull middle and the ICCP1 auxiliary anode respectively.Their currents are 1.0 m A,6.8 m A,-11.2 m A and 11.6 m A.It can be seen that the current output mainly consists of two pairs of auxiliary anodes of ICCP,and only a small current output exists at the bow of the ship.

In Fig.8,it can be seen that the UEP inversion accuracy is higher at the depth of 1.0B,but the accuracy is low near auxiliary anodes.It is analyzed that the output current of the auxiliary anode is relatively large,which leads to the strong non-linearity of the UEP distribution near the auxiliary anode.When calculating the UEP,there will be larger errors at the peak position,but it still has good inversion accuracy at other locations.

Fig.7.UEP in a cathodic protection state.

Table 2 UEP inversion accuracy according to different numbers of point-electrodes.

Fig.8.UEP inversion performance with 4 point-electrodes.

Fig.9 shows the UEP inversion results at dv=2.0Band dh=1.0B are less accurate compared with the results at dh=0B.The main reason is that the ship's sides would have some impact on the UEP distribution when the horizontal distance is between 0.5Band 1.0B.If the horizontal distance increases to 2.0B,the effect of the ship's side on the UEP distribution will be reduced.

The inversion results at dv=2.0B,dh=1.0B in a natural corrosion state is better than that in a cathodic protection state.There are two main reasons.(1)As the sacrificial anodes and auxiliary anodes are manually installed in the experiment,these anodes are not strictly symmetrical along the keel.In this case,some errors exist when we regard the hull as symmetrical along the keel.(2)The underwater current density is higher and its distribution is more complex in a cathodic protection state.As a result,the inversion results are better in a natural corrosion state.

Fig.9.UEP inversion performance when the horizontal distance is 1.0B with 4 pointelectrodes.

5.Conclusions

By measuring the UEP at a depth of 1.0B where B is the molded breadth of a ship,the number,positions and current values of the equivalent point-electrodes are solved by using the PSO method.Then the unknown UEP at depths of 1.5B and 2.0B below the keel,as well as a horizontal distance of 1.0B,are calculated.

The results show that for the ship in a natural corrosion state,three point-electrodes can reach more than 88%inversion accuracy,and the inversion accuracy is slightly lower for a horizontal distance of 1.0B.For the ship in a cathodic protection state,the UEP signature is more complex.And the inversion accuracy of all three depths can reach 85%by using four point-electrodes.

Therefore,in order to simulate the UEP signature around the ship,at least three point-electrodes are needed for ships in a natural corrosion state,and at least four point-electrodes are needed for ships in a cathodic protection state.

Acknowledgement

This work is based on the support by the Research project“Deep Sea of Electromagnetic Detection System”,Initiated by Pilot National Laboratory for Marine Science and Technology(Qingdao,China).