Citation: Li, X.; Li, C.; Bai, F.; Cao, F.

Study on the Interference

Distribution Characteristics of the

HVDC Grounding Electrode Current

with Buried Pipelines Based on MoM

and FEM. Appl. Sci. 2022, 12, 4433.

https://doi.org/10.3390/

app12094433

Academic Editor: Gilsoo Jang

Received: 4 April 2022

Accepted: 25 April 2022

Published: 27 April 2022

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applied sciences

Article

Study on the Interference Distribution Characteristics of theHVDC Grounding Electrode Current with Buried PipelinesBased on MoM and FEMXiong Li 1,*, Chengrong Li 1, Feng Bai 2 and Fangyuan Cao 2

1 State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North ChinaElectric Power University, Beijing 102206, China; crl@ncepu.edu.cn

2 China Electric Power Research Institute, Beijing 100192, China; baifeng@epri.sgcc.cn (F.B.);caofangyuan@epri.sgcc.com.cn (F.C.)

* Correspondence: leex@ncepu.edu.cn

Abstract: It is very important to calculate the interference of high voltage direct current (HVDC)grounding current with pipelines accurately and take proper protective measures to ensure energytransmission safety. There is still a lack of systematic research into related prediction methods thatconsider the nonlinear polarization on the interface between the soil and pipelines. In this paper,a methodology is proposed for calculating the coupling voltage and current on buried pipelinesinduced by grounding currents that accounts for nonlinear polarization based on the method ofmoments (MoM) and the finite element method (FEM). The validity of the proposed mathematicalmodel is verified by a scale experiment. The interference distribution characteristics under differentparameters are analyzed using the proposed method. The results show that the relationship betweenthe maximum leakage current density and the electrode grounding current satisfy the linear function.The corrosion area becomes more concentrated as the grounding current increases. The corrosionarea range increases substantially as the distance increases, whereas the influence degree decreasessubstantially. Both the maximum leakage current density and corrosion risk area are positivelycorrelated with the soil resistivity.

Keywords: HVDC grounding electrode; buried pipelines; nonlinear polarization; method of moments (MoM);finite element method (FEM); pipe-to-soil potential; leakage current density; distribution characteristics

1. Introduction

High voltage direct current (HVDC) transmission technology has been widely used inthe power system for its advantages in large transmission capacity and long transmissiondistance. The grounding electrode current can reach several kilo-amperes when the HVDCtransmission project is under monopole operation [1]. This current may cause severeinterference with buried pipelines in the vicinity of the electrode [2–4]. The currents canflow into the pipeline at one point and flow out at another point. Metal corrosion can occurat the current-in point, while hydrogen embrittlement may occur at the current-out point,which both threaten the operation safety of pipeline engineering [5,6]. Therefore, it is veryimportant to determine the interference characteristics of the electrode current with buriedpipelines for pipeline risk assessment and safety protection.

In pipeline engineering, pipe-to-soil potential (PSP) and leakage current density areoften used to indicate the DC interference degree [7]. In the past, numerous tests wereneeded to obtain these two indicators along the pipelines [8,9]. In situ measurement isoften costly and time-consuming so simulations represent an interesting option for thestudy of such phenomena. Many calculation methods have been proposed to determinethe interference under different conditions. Lagace et al. used the transmission line modelto simulate long pipelines by applying the pipeline conductor parameters and coating

Appl. Sci. 2022, 12, 4433. https://doi.org/10.3390/app12094433 https://www.mdpi.com/journal/applsci

Appl. Sci. 2022, 12, 4433 2 of 13

parameters to the transmission line equations [10]. Yu et al. extended the simulationmethodology by adding a local damage equivalent branch and cathodic protection branchfor more practical engineering applications [11]. Giao et al. first calculated the currentdistribution of the HVDC grounding electrode by the method of moments (MoM) [12]. ThenDawalibi et al. developed the multi-step method to calculate the interaction of the multi-electrodes based on MoM [13,14]. Zhang et al. proposed a field-circuit coupling model toconsider the impacts of different protective measures [15]. However, the above-mentionednumerical methods cannot consider the nonlinear electrochemical polarization effectively,which may lead to inaccurate results for density and PSP calculations [16]. The PSP deviatesfrom the equilibrium potential when polarization occurs. This deviation becomes greateras the leakage current density increases. Besides, the polarization relationship is nonlinearand it can result in improper protective measures for pipeline safety.

There are many parameters that can influence the calculation results, such as theelectrode grounding current, soil resistivity, soil structures, pipeline coating characteristics,and the distance between the electrode and pipeline. Differences in these parameterscan lead to different pipeline safety assessments and protective measures. Therefore, it isnecessary to study the interference distribution characteristics under the various above-mentioned parameters.

In this paper, a methodology based on a combination of MoM and FEM is proposedto calculate the pipeline interference caused by HVDC electrode current. The proposedmethod takes into account the pipe-soil interface polarization by introducing a nonlinearboundary condition which is difficult to consider for traditional methods. Then the interfer-ence distribution characteristics under different parameters are studied using the proposedmethod. These characteristics can be used to develop guidelines for practical engineering.

2. Interference Mechanism of the Grounding Current

The high-voltage DC grounding electrode is a grounding facility that provides a lowresistance path during the operation of the HVDC transmission system [17]. Figure 1 showsthe interference process of the DC grounding electrode current on the buried metallicpipelines. The earth currents flow into the pipeline in the area close to the groundingelectrode when the electrode is under anode operation. The currents flow out of the pipelineat the two ends which are far away from the electrode. As we can see, electrochemicalcorrosion occurs in the area currents flow out. The pipe-soil potential also has an obviouspositive shift.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 2 of 13

the interference under different conditions. Lagace et al. used the transmission line model

to simulate long pipelines by applying the pipeline conductor parameters and coating pa-

rameters to the transmission line equations [10]. Yu et al. extended the simulation meth-

odology by adding a local damage equivalent branch and cathodic protection branch for

more practical engineering applications [11]. Giao et al. first calculated the current distri-

bution of the HVDC grounding electrode by the method of moments (MoM) [12]. Then

Dawalibi et al. developed the multi-step method to calculate the interaction of the multi-

electrodes based on MoM [13,14]. Zhang et al. proposed a field-circuit coupling model to

consider the impacts of different protective measures [15]. However, the above-mentioned

numerical methods cannot consider the nonlinear electrochemical polarization effectively,

which may lead to inaccurate results for density and PSP calculations [16]. The PSP devi-

ates from the equilibrium potential when polarization occurs. This deviation becomes

greater as the leakage current density increases. Besides, the polarization relationship is

nonlinear and it can result in improper protective measures for pipeline safety.

There are many parameters that can influence the calculation results, such as the elec-

trode grounding current, soil resistivity, soil structures, pipeline coating characteristics,

and the distance between the electrode and pipeline. Differences in these parameters can

lead to different pipeline safety assessments and protective measures. Therefore, it is nec-

essary to study the interference distribution characteristics under the various above-men-

tioned parameters.

In this paper, a methodology based on a combination of MoM and FEM is proposed

to calculate the pipeline interference caused by HVDC electrode current. The proposed

method takes into account the pipe-soil interface polarization by introducing a nonlinear

boundary condition which is difficult to consider for traditional methods. Then the inter-

ference distribution characteristics under different parameters are studied using the pro-

posed method. These characteristics can be used to develop guidelines for practical engi-

neering.

2. Interference Mechanism of the Grounding Current

The high-voltage DC grounding electrode is a grounding facility that provides a low

resistance path during the operation of the HVDC transmission system [17]. Figure 1

shows the interference process of the DC grounding electrode current on the buried me-

tallic pipelines. The earth currents flow into the pipeline in the area close to the grounding

electrode when the electrode is under anode operation. The currents flow out of the pipe-

line at the two ends which are far away from the electrode. As we can see, electrochemical

corrosion occurs in the area currents flow out. The pipe-soil potential also has an obvious

positive shift.

Grounding Electrode

Pipeline

PSPAnode zone Anode zone

Cathode zone

Figure 1. Interference process of HVDC electrode currents on pipelines. Figure 1. Interference process of HVDC electrode currents on pipelines.

Different electrochemical reactions occur for different interference areas [18]. The elec-trochemical reaction for the anode interference area is usually described by:

2Fe → 2Fe2++4e

Appl. Sci. 2022, 12, 4433 3 of 13

For the cathode interference area, hydrogen evolution reaction or oxygen absorptionreactions dominate under different soil medium environments. In general, it can be shownas follows:

2H+ + 2e− → H2 acid oxygen-free conditionH2O+2e− → 2OH− + H2 neutral or alkaline oxygen-free conditionO2 + 4H+ + 4e− → 2H2O acid aerobic conditionO2 + 2H2O+4e− → 4OH− alkaline aerobic condition

The above electrochemical reactions can finally lead to a specific relationship betweenthe PSP and the leakage current which can be defined as follows:

J = fp(ϕp − ϕs) (1)

where J, ϕp and ϕs are the leakage current density, the pipe-side potential and the soil-sidepotential interface between the soil and pipe, respectively. The nonlinear function f p, whichdescribes the relationship between the pipe-to-soil potential and the leakage current density,can be measured by experiment. The three-electrode system is often used to measure thepolarization curves with the potential sweep method [19]. Figure 2 shows the typicalthree-electrode system for measuring polarization curves.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 13

Different electrochemical reactions occur for different interference areas [18]. The

electrochemical reaction for the anode interference area is usually described by:

2+e e2F 2F +4e

For the cathode interference area, hydrogen evolution reaction or oxygen absorption

reactions dominate under different soil medium environments. In general, it can be shown

as follows:

2

2 2

2 2

2 2

2H 2e H acid oxygen-free condition

H O+2e 2OH H neutral or alkaline oxygen-free condition

O 4H +4e 2H O acid aerobic condition

O 2H O+4e 4OH alkaline aerobic condition

The above electrochemical reactions can finally lead to a specific relationship be-

tween the PSP and the leakage current which can be defined as follows:

p p s( )J f (1)

where J, p and s are the leakage current density, the pipe-side potential and the soil-side

potential interface between the soil and pipe, respectively. The nonlinear function fp,

which describes the relationship between the pipe-to-soil potential and the leakage cur-

rent density, can be measured by experiment. The three-electrode system is often used to

measure the polarization curves with the potential sweep method [19]. Figure 2 shows the

typical three-electrode system for measuring polarization curves.

Luggin capillary

Electrochemical workstation

CE WE RE

Counter electrode

Workingelectrode

Reference electrode

V

A

Figure 2. Typical three-electrode system for measuring polarization curves.

The discrete measurement data can be fitted by the Butler-Volmer equation which is

one of the most fundamental equations in electrochemical kinetics. Equation (2) shows the

typical Butler-Volmer equation [20]:

P ( ) (exp( ) exp( ))corr corrcorr

a c

E E E EJ f E J

(2)

where J and E are current density and PSP, respectively, and are obtained by measure-

ment. corr corr, , ,a cE J are the metal self-corrosion potential, self-corrosion current den-

sity, Tafel slope of the anode reaction and Tafel slope of the cathode reaction, respectively.

The four parameters to be fitted can be obtained using the least square method. Then, the

fitted functions can be used in the proposed methodology. The methodology must

Figure 2. Typical three-electrode system for measuring polarization curves.

The discrete measurement data can be fitted by the Butler-Volmer equation which isone of the most fundamental equations in electrochemical kinetics. Equation (2) shows thetypical Butler-Volmer equation [20]:

J = fP(E) = Jcorr(exp(E− Ecorr

βa)− exp(−E− Ecorr

βc)) (2)

where J and E are current density and PSP, respectively, and are obtained by measurement.Ecorr, Jcorr, βa, βc are the metal self-corrosion potential, self-corrosion current density, Tafelslope of the anode reaction and Tafel slope of the cathode reaction, respectively. The fourparameters to be fitted can be obtained using the least square method. Then, the fittedfunctions can be used in the proposed methodology. The methodology must consider thenonlinear polarization to calculate the current and potential distribution accurately.

3. Mathematical Model Combining MoM with FEM

Pipeline corrosion is mainly caused by the resistive coupling of the grounding currentson the pipeline metal. The computation domain includes the soil area, the pipeline con-ductor area, as well as the infinite air-earth boundary, the grounding conductor boundary,and the interface boundary between the soil area and the pipeline conductor area. Figure 3

Appl. Sci. 2022, 12, 4433 4 of 13

shows the boundary value problem of the resistive coupling model between the groundingelectrode and pipelines.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 13

consider the nonlinear polarization to calculate the current and potential distribution ac-

curately.

3. Mathematical Model Combining MoM with FEM

Pipeline corrosion is mainly caused by the resistive coupling of the grounding cur-

rents on the pipeline metal. The computation domain includes the soil area, the pipeline

conductor area, as well as the infinite air-earth boundary, the grounding conductor

boundary, and the interface boundary between the soil area and the pipeline conductor

area. Figure 3 shows the boundary value problem of the resistive coupling model between

the grounding electrode and pipelines.

Figure 3. Schematic diagram of the boundary value problem.

The conductivities are assumed to be homogeneous. Then, the soil field Ωs and pipe

conductor field Ωp belong to the constant current field so the potential satisfies Laplace’s

equations. The air can be considered as electrical insulating medium so the current flow-

ing out of the air-earth boundary (Sg) is zero. The total leakage current of the grounding

electrode should equal the injected current in the HVDC transmission system based on

the charge conservation law. According to the interface conditions for electric field vec-

tors, the current density on the soil side should equal the current density on the pipe con-

ductor side. Besides, the leakage current density also should satisfy the polarization rela-

tionship with the pipe-to-soil potential ( s p ). The theoretical equations for the above

three-dimensional boundary value problem can be formulated as follows:

ele

s s

p p

sg

ss hvdc ele

p sp s

p

2

s p

2

p

( ) 0,

( ) 0

( )

( ),

( ) ( )

0,

( ),( ) ( )

S

r r

r r

rr S

n

rI r S

n

r r

n n

r r r S

ds

f

，

(3)

where r

is an arbitrary coordinate in the computation domain, Ihvdc is the grounding cur-

rent of the HVDC grounding electrode, and p and s are the conductivity of the pipe

conductor and soil medium, respectively. A numerical method combining MoM and FEM

is proposed to solve the above equations. The Laplace equation in the soil area is discre-

tized by MoM which is suitable for open-domain problems. The Laplace equation in pipe

conductor area is discretized by FEM which is suitable for enclosure domain problems.

By applying the nonlinear polarization boundary conditions between the two areas, the

two methods can be coupled for solving.

The potential of an arbitrary point in the soil area can be expressed as follows:

Figure 3. Schematic diagram of the boundary value problem.

The conductivities are assumed to be homogeneous. Then, the soil field Ωs and pipeconductor field Ωp belong to the constant current field so the potential satisfies Laplace’sequations. The air can be considered as electrical insulating medium so the current flowingout of the air-earth boundary (Sg) is zero. The total leakage current of the groundingelectrode should equal the injected current in the HVDC transmission system based on thecharge conservation law. According to the interface conditions for electric field vectors,the current density on the soil side should equal the current density on the pipe conductorside. Besides, the leakage current density also should satisfy the polarization relationshipwith the pipe-to-soil potential (ϕs − ϕp). The theoretical equations for the above three-dimensional boundary value problem can be formulated as follows:

∇2 ϕs(→r ) = 0,

→r ∈ Ωs

∇2 ϕp(→r ) = 0,

→r ∈ Ωp

∂ϕs(→r )

∂n = 0,→r ∈ Sg∫

Seleσs

∂ϕs(→r )

∂n ds = Ihvdc,→r ∈ Sele

σp∂ϕp(

→r )

∂n = σs∂ϕs(

→r )

∂n= − fp(ϕp(

→r )− ϕs(

→r )),

→r ∈ Sp

(3)

where→r is an arbitrary coordinate in the computation domain, Ihvdc is the grounding

current of the HVDC grounding electrode, and σp and σs are the conductivity of the pipeconductor and soil medium, respectively. A numerical method combining MoM andFEM is proposed to solve the above equations. The Laplace equation in the soil area isdiscretized by MoM which is suitable for open-domain problems. The Laplace equation inpipe conductor area is discretized by FEM which is suitable for enclosure domain problems.By applying the nonlinear polarization boundary conditions between the two areas, thetwo methods can be coupled for solving.

The potential of an arbitrary point in the soil area can be expressed as follows:

ϕs(→r ) =

x

sG(→r ,→r′) · Jleak(

→r′)ds, s ∈ Sele, Sp (4)

where→r′

is the current source coordinate in the soil area, Jleak(→r ) is the leakage current

density for the buried conductor, and G(→r ,→r′) is the Green’s function for the above

boundary problem. The electrode conductor and pipe conductor are divided into M and

Appl. Sci. 2022, 12, 4433 5 of 13

N segments, respectively. The basis function for different segment current density can beexpressed as follows:

vi =

1→r′∈ ∆i

0→r′

/∈ ∆i(5)

where ∆i is the definitional domain of the buried conductors so Jleak(→r ) can be expressed

as follows:

Jleak(→r′) =

M+N

∑i=1

civi (6)

c1, c2, · · · cM are the current density of the electrode conductor segments. cM+1, cM+2 · · ·cM+N are the current density of the pipe conductor segments. Based on the MoM and pointmatch method the following algebraic equation set can be formed.

G(r1,1) · · · G(r1,M) G(r1,M+1) · · · G(r1,M+N)...

. . ....

.... . .

...G(rM,1) · · · G(rM,M) G(rM,M+1) · · · G(rM,M+N)

G(rM+1,1) · · · G(rM+1,M) G(rM+1,M+1) · · · G(rM+1,M+N)...

. . ....

.... . .

...G(rM+N,1) · · · G(rM+N,M) G(rM+N,M+1) · · · G(rM+N,M+N)

c1...

cMcM+1

...cM+N

=

ϕ(e,0)...

ϕ(e,0)ϕ(s,1)

...ϕ(s,N)

(7)

The electrode is considered as an equipotential ϕ(e,0). The pipe potential on the soilside is ϕ(s,1), ϕ(s,2), · · · ϕ(s,N).

Using the finite element Galerkin method for the pipe conductor area, another equationset can be formed as follows:

K1,1 · · · K1,N K1,N+1 · · · K1,N+P...

. . ....

.... . .

...KN,1 · · · KN,N KN,N+1 · · · KN,N+P

KN+1,1 · · · KN+1,1 KN+1,N+1 · · · KN+1,N+P...

. . ....

.... . .

...KN+P,1 · · · KN+P,N KN+P,N+1 · · · KN+P,N+P

ϕ(p,1)...

ϕ(p,N)

ϕ(p,N+1)...

ϕ(p,N+P)

=

F(

ϕ(p,1) − ϕ(s,1)

)...

F(

ϕ(p,N) − ϕ(s,N)

)0...0

(8)

K is the Stiffness matrix element for different nodes. ϕ(p,1), ϕ(p,2), · · · ϕ(p,N) are thepipe potential on the pipe side. ϕ(p,N+1), ϕ(p,N+2), · · · ϕ(p,N+P) are the non-boundary nodepotentialin the pipe conductor area. F(ϕ(p,i) − ϕ(s,i)), i = 1, 2, · · ·N are the load elementswhich depend on the nonlinear polarization function. According to charge conservation,the electrode segment currents obey the following equation:

M

∑i=1

ci∆si = Ihvdc (9)

By combining Equations (7)–(9), the distribution of the PSP and leakage current alongthe pipeline can be solved with the Newton–Raphson Method.

4. Experimental Verification4.1. Experiment System

To verify the effectiveness of the proposed methodology, a scale problem experimentwas conducted. Figure 4 shows the experiment system layout for the scale groundingelectrode and pipeline.

Appl. Sci. 2022, 12, 4433 6 of 13

Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 13

4. Experimental Verification

4.1. Experiment System

To verify the effectiveness of the proposed methodology, a scale problem experiment

was conducted. Figure 4 shows the experiment system layout for the scale grounding elec-

trode and pipeline.

Electrode

Auxiliary grounding electrode

DC Source

pipeline

Cable

Cable28m

Figure 4. Layout of the scale experiment system.

The scale electrode was a 30 m diameter steel circle buried at a depth of 1.5 m. The

pipeline was buried 28 m away from the electrode at a depth of 1 m. The grounding cur-

rent was supplied by a current generator with 300 A maximum output current. The pipe

material was X80 steel. The soil was considered as a homogeneous medium and the soil

resistivity is r, pH = 8. The leakage current density and PSP along the pipeline were meas-

ured with the test coupons. The test points layout is shown in Figure 5.

Test Pile

200m

Insulated joint Test point

P

−100 −82.5 −65 −47.5 −30 −15 0 15 30 47.5 65 82.5 100

Figure 5. Test points layout along the pipeline.

For each test point, the test circuit is shown in Figure 6. based on the external coupon

method [21].

Figure 4. Layout of the scale experiment system.

The scale electrode was a 30 m diameter steel circle buried at a depth of 1.5 m.The pipeline was buried 28 m away from the electrode at a depth of 1 m. The groundingcurrent was supplied by a current generator with 300 A maximum output current. The pipematerial was X80 steel. The soil was considered as a homogeneous medium and the soilresistivity is r, pH = 8. The leakage current density and PSP along the pipeline weremeasured with the test coupons. The test points layout is shown in Figure 5.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 6 of 13

4. Experimental Verification

4.1. Experiment System

To verify the effectiveness of the proposed methodology, a scale problem experiment

was conducted. Figure 4 shows the experiment system layout for the scale grounding elec-

trode and pipeline.

Electrode

Auxiliary grounding electrode

DC Source

pipeline

Cable

Cable28m

Figure 4. Layout of the scale experiment system.

The scale electrode was a 30 m diameter steel circle buried at a depth of 1.5 m. The

pipeline was buried 28 m away from the electrode at a depth of 1 m. The grounding cur-

rent was supplied by a current generator with 300 A maximum output current. The pipe

material was X80 steel. The soil was considered as a homogeneous medium and the soil

resistivity is r, pH = 8. The leakage current density and PSP along the pipeline were meas-

ured with the test coupons. The test points layout is shown in Figure 5.

Test Pile

200m

Insulated joint Test point

P

−100 −82.5 −65 −47.5 −30 −15 0 15 30 47.5 65 82.5 100

Figure 5. Test points layout along the pipeline.

For each test point, the test circuit is shown in Figure 6. based on the external coupon

method [21].

Figure 5. Test points layout along the pipeline.

For each test point, the test circuit is shown in Figure 6. based on the external couponmethod [21].

Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 13

uDL2 Data Logger

Terminal

Reference electrode Coupon

Pipeline

Test pile

Soil

Figure 6. Test circuit based on external coupon method.

Figure 7 shows the polarization relationship for the above-mentioned soil and pipe

material. The current density is plotted with logarithmic coordinates for a better view.

“log J” means the common logarithm of the leakage current density.

- 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3

- 4

- 3

- 2

- 1

0

1

2

3

pola

riza

tion

pot

enti

al(V

CS

E)

current density (logJ A/m2)

Figure 7. The relationship between polarization potential and current density for the X80 pipe steel

under 100 m , pH = 8 soil condition.

4.2. Verification of Calculation Method with Expriments Results

The buried conductors was subdivided into about 3000 segments. Boundary condi-

tions were considered in the Green’s function when applying MoM. The pipeline conduc-

tor domain was subdivided into 45,800 tetrahedron elements. The nonlinear polarization

conditions can be treated as a Neumann boundary in each iterative computation when

applying FEM. The leakage current density and PSP were measured under 100 A and 150

A grounding current, respectively. The results of the experiment and calculation are

shown in Figure 8.

Figure 6. Test circuit based on external coupon method.

Appl. Sci. 2022, 12, 4433 7 of 13

Figure 7 shows the polarization relationship for the above-mentioned soil and pipematerial. The current density is plotted with logarithmic coordinates for a better view.“log J” means the common logarithm of the leakage current density.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 7 of 13

uDL2 Data Logger

Terminal

Reference electrode Coupon

Pipeline

Test pile

Soil

Figure 6. Test circuit based on external coupon method.

Figure 7 shows the polarization relationship for the above-mentioned soil and pipe

material. The current density is plotted with logarithmic coordinates for a better view.

“log J” means the common logarithm of the leakage current density.

- 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3

- 4

- 3

- 2

- 1

0

1

2

3

pola

riza

tion

pot

enti

al(V

CS

E)

current density (logJ A/m2)

Figure 7. The relationship between polarization potential and current density for the X80 pipe steel

under 100 m , pH = 8 soil condition.

4.2. Verification of Calculation Method with Expriments Results

The buried conductors was subdivided into about 3000 segments. Boundary condi-

tions were considered in the Green’s function when applying MoM. The pipeline conduc-

tor domain was subdivided into 45,800 tetrahedron elements. The nonlinear polarization

conditions can be treated as a Neumann boundary in each iterative computation when

applying FEM. The leakage current density and PSP were measured under 100 A and 150

A grounding current, respectively. The results of the experiment and calculation are

shown in Figure 8.

Figure 7. The relationship between polarization potential and current density for the X80 pipe steelunder 100 Ω ·m, pH = 8 soil condition.

4.2. Verification of Calculation Method with Expriments Results

The buried conductors was subdivided into about 3000 segments. Boundary condi-tions were considered in the Green’s function when applying MoM. The pipeline conductordomain was subdivided into 45,800 tetrahedron elements. The nonlinear polarizationconditions can be treated as a Neumann boundary in each iterative computation whenapplying FEM. The leakage current density and PSP were measured under 100 A and 150 Agrounding current, respectively. The results of the experiment and calculation are shown inFigure 8.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 13

- 100 - 50 0 50 100

- 6

- 4

- 2

0

2

4

6

8

10

leak

age

curr

ent

den

sity

(mA

/cm

2 )

position relative to the pipeline midpoint (m)

100A test 100A Calculation 150A test 150A Calculation

- 100 - 50 0 50 100

- 6

- 4

- 2

0

2

4

6

8

10

pipe

-to-

soil

pot

enti

al(V

)

position relative to the pipeline midpoint (m)

100A test

100A Calculation 150A test 150A Calculation

(a) (b)

Figure 8. Results of test and calculation. (a) leakage current density along the pipeline; (b) pipe-to-

soil potential along the pipeline.

The results show that, the current density and PSP at the center point are the largest

along the pipeline for both test results and calculation results. The maximum test leakage

current density changed from 5.987 mA/cm2 to 8.719 mA/cm2 when the grounding current

changed from 100 A to 150 A. Accordingly, the maximum calculated leakage current den-

sity changed from 5.667 mA/cm2 to 8.604 mA/cm2. The relative error between the calcu-

lated results and the measured results for leakage current is 1.32–5.35%. In the same way,

the relative error for PSP can be obtained, that is 2.63–7.72%. As the grounding current

increases, the relative error becomes smaller. Therefore, the validity of the calculation

method is verified by experiment results.

5. Interference Distribution Characteristics

Based on the proposed calculation model, the interference distribution characteristics

were analyzed under different parameters. These parameters include the electrode cur-

rents, soil resistivity, and the distance between the electrode and pipelines. Considering

the following typical interference model, the length of the pipeline is L, and the distance

between the electrode and pipeline is d. The soil is considered as a homogeneous medium

and the soil resistivity is r, pH = 8. The pipe material was X80 steel. A schematic diagram

of the pipeline and HVDC grounding electrode arrangement is shown in Figure 9.

HVDCGrouding Electrode

L

dpipeline

Figure 9. Schematic diagram of the pipeline and HVDC grounding electrode arrangement.

Figure 8. Results of test and calculation. (a) leakage current density along the pipeline; (b) pipe-to-soilpotential along the pipeline.

The results show that, the current density and PSP at the center point are the largestalong the pipeline for both test results and calculation results. The maximum test leakagecurrent density changed from 5.987 mA/cm2 to 8.719 mA/cm2 when the grounding currentchanged from 100 A to 150 A. Accordingly, the maximum calculated leakage current densitychanged from 5.667 mA/cm2 to 8.604 mA/cm2. The relative error between the calculated

Appl. Sci. 2022, 12, 4433 8 of 13

results and the measured results for leakage current is 1.32–5.35%. In the same way,the relative error for PSP can be obtained, that is 2.63–7.72%. As the grounding currentincreases, the relative error becomes smaller. Therefore, the validity of the calculationmethod is verified by experiment results.

5. Interference Distribution Characteristics

Based on the proposed calculation model, the interference distribution characteristicswere analyzed under different parameters. These parameters include the electrode currents,soil resistivity, and the distance between the electrode and pipelines. Considering thefollowing typical interference model, the length of the pipeline is L, and the distancebetween the electrode and pipeline is d. The soil is considered as a homogeneous mediumand the soil resistivity is r, pH = 8. The pipe material was X80 steel. A schematic diagramof the pipeline and HVDC grounding electrode arrangement is shown in Figure 9.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 8 of 13

- 100 - 50 0 50 100

- 6

- 4

- 2

0

2

4

6

8

10

leak

age

curr

ent

den

sity

(mA

/cm

2 )

position relative to the pipeline midpoint (m)

100A test 100A Calculation 150A test 150A Calculation

- 100 - 50 0 50 100

- 6

- 4

- 2

0

2

4

6

8

10

pipe

-to-

soil

pot

enti

al(V

)

position relative to the pipeline midpoint (m)

100A test

100A Calculation 150A test 150A Calculation

(a) (b)

Figure 8. Results of test and calculation. (a) leakage current density along the pipeline; (b) pipe-to-

soil potential along the pipeline.

The results show that, the current density and PSP at the center point are the largest

along the pipeline for both test results and calculation results. The maximum test leakage

current density changed from 5.987 mA/cm2 to 8.719 mA/cm2 when the grounding current

changed from 100 A to 150 A. Accordingly, the maximum calculated leakage current den-

sity changed from 5.667 mA/cm2 to 8.604 mA/cm2. The relative error between the calcu-

lated results and the measured results for leakage current is 1.32–5.35%. In the same way,

the relative error for PSP can be obtained, that is 2.63–7.72%. As the grounding current

increases, the relative error becomes smaller. Therefore, the validity of the calculation

method is verified by experiment results.

5. Interference Distribution Characteristics

Based on the proposed calculation model, the interference distribution characteristics

were analyzed under different parameters. These parameters include the electrode cur-

rents, soil resistivity, and the distance between the electrode and pipelines. Considering

the following typical interference model, the length of the pipeline is L, and the distance

between the electrode and pipeline is d. The soil is considered as a homogeneous medium

and the soil resistivity is r, pH = 8. The pipe material was X80 steel. A schematic diagram

of the pipeline and HVDC grounding electrode arrangement is shown in Figure 9.

HVDCGrouding Electrode

L

dpipeline

Figure 9. Schematic diagram of the pipeline and HVDC grounding electrode arrangement.

Figure 9. Schematic diagram of the pipeline and HVDC grounding electrode arrangement.

5.1. Electrode Current

The relationship between the polarization potential and current density is the same asin Section 5.1 and shown in Figure 7. The other parameters are listed in Table 1.

Table 1. Parameters for the study model.

Parameters Value

HVDC grounding current 100 A, 500 A, 1000 A, 2000 A, 5000 Apipeline length 100 km

distance between electrode and pipelines 10 kmsoil parameters 100 Ω·m, pH = 8

pipeline steel material X80pipeline coating resistivity 3.3 × 107 Ω·m

The distribution of the leakage current density and pip-to-soil potential under differentHVDC ground currents are shown in Figure 10.

Appl. Sci. 2022, 12, 4433 9 of 13

Appl. Sci. 2022, 12, x FOR PEER REVIEW 9 of 13

5.1. Electrode Current

The relationship between the polarization potential and current density is the same

as in Section 5.1 and shown in Figure 7. The other parameters are listed in Table 1.

Table 1. Parameters for the study model.

Parameters Value

HVDC grounding current 100 A, 500 A, 1000 A, 2000 A, 5000 A

pipeline length 100 km

distance between electrode and pipelines 10 km

soil parameters 100 Ω·m, pH = 8

pipeline steel material X80

pipeline coating resistivity 3.3 × 107 Ω·m

The distribution of the leakage current density and pip-to-soil potential under differ-

ent HVDC ground currents are shown in Figure 10.

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50- 2

- 1

0

1

2

3

4

5

leak

age

curr

ent

den

sity

(A/m

2)

position relative to the pipeline midpoint (km)

100A

500A 1000A

2000A 5000A

electorde current

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50

- 1.5

- 1.0

- 0.5

0.0

0.5

1.0

pip

e-to

-so

il p

ote

nti

al(V

)

position relative to the pipeline midpoint (km)

100A

500A

1000A

2000A

5000A

electorde current

(a) (b)

Figure 10. Calculation results under different HVDC grounding currents. (a) leakage current den-

sity along the pipeline; (b) PSP along the pipeline.

As can be seen in Figure 10, the leakage current density and PSP reach 3.92 A/m2 and

0.89 V, respectively, when the grounding current is 5000 A. Both maximum leakage cur-

rent density and maximum PSP increase as the electrode grounding current increases.

Regression analysis shows that the relationship between the maximum leakage current

density Jmax and the electrode grounding current Ihvdc is linear, that is Jmax = a1·Ihvdc + b1 where

a1 and b1 are fitting constant coefficients.

The maximum value is located at the midpoint of the pipeline. When currents flow

out of the pipe, the pipeline faces corrosion risk, especially for the central area of the pipe-

line. Figure 11 shows the corrosion risk area distribution under different grounding cur-

rents.

Figure 10. Calculation results under different HVDC grounding currents. (a) leakage current densityalong the pipeline; (b) PSP along the pipeline.

As can be seen in Figure 10, the leakage current density and PSP reach 3.92 A/m2

and 0.89 V, respectively, when the grounding current is 5000 A. Both maximum leakagecurrent density and maximum PSP increase as the electrode grounding current increases.Regression analysis shows that the relationship between the maximum leakage currentdensity Jmax and the electrode grounding current Ihvdc is linear, that is Jmax = a1·Ihvdc + b1where a1 and b1 are fitting constant coefficients.

The maximum value is located at the midpoint of the pipeline. When currents flow outof the pipe, the pipeline faces corrosion risk, especially for the central area of the pipeline.Figure 11 shows the corrosion risk area distribution under different grounding currents.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 13

Figure 11. Corrosion risk area distribution under different grounding currents.

The length of anode area is 34.5 km under 100 A, whereas the length of anode area is

25.7 km under a 5000 A grounding current The result shows that the length facing corro-

sion risk decreases as the HVDC grounding current increases, which means the corrosion

area is becoming more concentrated. The effect of grounding current on the anode area

range is not very significant when the current exceeds 1000 A.

5.2. Distance between Electrode and Pipelines

The interference scope of the HVDC grounding current can reach tens of kilometers.

Therefore, the potential and density distribution characteristics were studied in the range

from 2 km to 50 km. A schematic diagram for the pipeline and HVDC grounding electrode

arrangement is shown in Figure 9. Five distance types (2 km, 5 km, 10 km, 20 km, 50 km)

were chosen for study. The HVDC grounding current is 3000 A. The other parameters are

the same as those listed in Table 1. The relationship between polarization potential and

current density is the same as in Section 5.1 and shown in Figure 7.

The results are shown in Figure 12 for different distances between the electrode and

pipelines.

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50- 5

0

5

10

15

20

25

leak

age

curr

ent

den

sity

(A/m

2)

position relative to the pipeline midpoint (km)

2km 5km

10km

20km 50km

distance betweenelectrode and pipeline

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50

- 2

0

2

4

6

8

10

pip

e-to

-so

il p

ote

ntia

l(V

)

position relative to the pipeline midpoint (km)

2km 5km

10km 20km

50km

distance between

electrode and pipeline

(a) (b)

Figure 12. Calculation results under different distance between electrode and pipeline. (a) leakage

current density along the pipeline; (b) PSP along the pipeline.

The maximum leakage current density and PSP reach 25.2 A/m2 and 8.74 V, respec-

tively, when the distance between electrode and pipeline is 2 km. As the distance in-

creases, the maximum current density and PSP decrease gradually. Regression analysis

shows that the relationship between the maximum leakage current density Jmax and the

distance d satisfy negative exponential function, that is 2

max 2bJ a d (b2 < 0) where a2

Figure 11. Corrosion risk area distribution under different grounding currents.

The length of anode area is 34.5 km under 100 A, whereas the length of anode area is25.7 km under a 5000 A grounding current The result shows that the length facing corrosionrisk decreases as the HVDC grounding current increases, which means the corrosion areais becoming more concentrated. The effect of grounding current on the anode area range isnot very significant when the current exceeds 1000 A.

5.2. Distance between Electrode and Pipelines

The interference scope of the HVDC grounding current can reach tens of kilometers.Therefore, the potential and density distribution characteristics were studied in the rangefrom 2 km to 50 km. A schematic diagram for the pipeline and HVDC grounding electrodearrangement is shown in Figure 9. Five distance types (2 km, 5 km, 10 km, 20 km, 50 km)were chosen for study. The HVDC grounding current is 3000 A. The other parameters are

Appl. Sci. 2022, 12, 4433 10 of 13

the same as those listed in Table 1. The relationship between polarization potential andcurrent density is the same as in Section 5.1 and shown in Figure 7.

The results are shown in Figure 12 for different distances between the electrode and pipelines.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 13

Figure 11. Corrosion risk area distribution under different grounding currents.

The length of anode area is 34.5 km under 100 A, whereas the length of anode area is

25.7 km under a 5000 A grounding current The result shows that the length facing corro-

sion risk decreases as the HVDC grounding current increases, which means the corrosion

area is becoming more concentrated. The effect of grounding current on the anode area

range is not very significant when the current exceeds 1000 A.

5.2. Distance between Electrode and Pipelines

The interference scope of the HVDC grounding current can reach tens of kilometers.

Therefore, the potential and density distribution characteristics were studied in the range

from 2 km to 50 km. A schematic diagram for the pipeline and HVDC grounding electrode

arrangement is shown in Figure 9. Five distance types (2 km, 5 km, 10 km, 20 km, 50 km)

were chosen for study. The HVDC grounding current is 3000 A. The other parameters are

the same as those listed in Table 1. The relationship between polarization potential and

current density is the same as in Section 5.1 and shown in Figure 7.

The results are shown in Figure 12 for different distances between the electrode and

pipelines.

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50- 5

0

5

10

15

20

25

leak

age

curr

ent

den

sity

(A/m

2)

position relative to the pipeline midpoint (km)

2km 5km

10km

20km 50km

distance betweenelectrode and pipeline

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50

- 2

0

2

4

6

8

10

pip

e-to

-so

il p

ote

ntia

l(V

)

position relative to the pipeline midpoint (km)

2km 5km

10km 20km

50km

distance between

electrode and pipeline

(a) (b)

Figure 12. Calculation results under different distance between electrode and pipeline. (a) leakage

current density along the pipeline; (b) PSP along the pipeline.

The maximum leakage current density and PSP reach 25.2 A/m2 and 8.74 V, respec-

tively, when the distance between electrode and pipeline is 2 km. As the distance in-

creases, the maximum current density and PSP decrease gradually. Regression analysis

shows that the relationship between the maximum leakage current density Jmax and the

distance d satisfy negative exponential function, that is 2

max 2bJ a d (b2 < 0) where a2

Figure 12. Calculation results under different distance between electrode and pipeline. (a) leakagecurrent density along the pipeline; (b) PSP along the pipeline.

The maximum leakage current density and PSP reach 25.2 A/m2 and 8.74 V, respec-tively, when the distance between electrode and pipeline is 2 km. As the distance increases,the maximum current density and PSP decrease gradually. Regression analysis shows thatthe relationship between the maximum leakage current density Jmax and the distance dsatisfy negative exponential function, that is Jmax = a2 · db2 (b2 < 0) where a2 and b2 arefitting constant coefficients. That means that the interference current density increases aspower function when the distance between electrode and pipeline decreases.

Figure 13 shows the interference range for the anode area and cathode area along thepipeline for different distances between the electrode and pipeline.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 13

and b2 are fitting constant coefficients. That means that the interference current density

increases as power function when the distance between electrode and pipeline decreases.

Figure 13 shows the interference range for the anode area and cathode area along the

pipeline for different distances between the electrode and pipeline.

Figure 13. Corrosion risk area distribution for different distances between HVDC electrode and

pipeline.

The distance between the electrode and pipeline has a significant influence on the

anode area range. The anode area range increases as the distance increases. Although the

anode area range extends, the influence degree decreases substantially.

5.3. Soil Resistivity

Soil resistivity influences not only the grounding current flow path but also the po-

larization. Four soil resistivity types (10, 30, 100, 1300 Ω·m) were chosen for study. The

HVDC grounding current is 3000 A. The other parameters are the same as those listed in

Table 1. The polarization curves were measured under different soil conditions. Figure 14

shows the test polarization curve data.

- 5 - 4 - 3 - 2 - 1 0 1 2

- 4

- 3

- 2

- 1

0

1

2

3

pola

riza

tio

n po

tent

ial(

V)

leakage current density (logJ A/m2)

X80-10 m pH=8

X80-30 m pH=8

X80-100 m pH=8

X80-1300 m pH=8

Figure 14. Polarization curve under different soil resistivity.

The calculation results of leakage current density and PSP are shown in Figure 15.

Figure 13. Corrosion risk area distribution for different distances between HVDC electrodeand pipeline.

The distance between the electrode and pipeline has a significant influence on theanode area range. The anode area range increases as the distance increases. Although theanode area range extends, the influence degree decreases substantially.

5.3. Soil Resistivity

Soil resistivity influences not only the grounding current flow path but also the polar-ization. Four soil resistivity types (10, 30, 100, 1300 Ω·m) were chosen for study. The HVDC

Appl. Sci. 2022, 12, 4433 11 of 13

grounding current is 3000 A. The other parameters are the same as those listed in Table 1.The polarization curves were measured under different soil conditions. Figure 14 showsthe test polarization curve data.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 12 of 14

and b2 are fitting constant coefficients. That means that the interference current density

increases as power function when the distance between electrode and pipeline decreases.

Figure 13 shows the interference range for the anode area and cathode area along the

pipeline for different distances between the electrode and pipeline.

Figure 13. Corrosion risk area distribution for different distances between HVDC electrode and

pipeline.

The distance between the electrode and pipeline has a significant influence on the

anode area range. The anode area range increases as the distance increases. Although the

anode area range extends, the influence degree decreases substantially.

5.3. Soil Resistivity

Soil resistivity influences not only the grounding current flow path but also the

polarization. Four soil resistivity types (10, 30, 100, 1300 Ω·m) were chosen for study. The

HVDC grounding current is 3000 A. The other parameters are the same as those listed in

Table 1. The polarization curves were measured under different soil conditions. Figure 14

shows the test polarization curve data.

−5 −4 −3 −2 −1 0 1 2

−4

−3

−2

−1

0

1

2

3

po

lari

zati

on

po

ten

tial

(V)

leakage current density (logJ A/m2)

X80-10 m pH=8

X80-30 m pH=8

X80-100 m pH=8

X80-1300 m pH=8

Figure 14. Polarization curve under different soil resistivity.

The calculation results of leakage current density and PSP are shown in Figure 15.

Figure 14. Polarization curve under different soil resistivity.

The calculation results of leakage current density and PSP are shown in Figure 15.Appl. Sci. 2022, 12, x FOR PEER REVIEW 13 of 14

−50 −40 −30 −20 −10 0 10 20 30 40 50

−2

−1

0

1

2

3

4

5

leak

age

curr

ent

den

sity

(A/m

2)

position relative to the pipeline midpoint (km)

10 m

30 m

100 m

1300 m

soil resistivity

−50 −40 −30 −20 −10 0 10 20 30 40 50

−10

−5

0

5

10

pip

e-to

-so

il p

ote

nti

al (

V)

position relative to the pipeline midpoint (km)

10 m

30 m

100 m

1300 m

soil resistivity

(a) (b)

Figure 15. Calculation results under different soil resistivity. (a) leakage current density along the

pipeline; (b) PSP along the pipeline.

The maximum leakage current density and PSP increase as the soil resistivity

increases. Regression analysis shows that the relationship between the maximum leakage

current density Jmax and the soil resistivity ρs satisfy logarithmic function, that is

3max s3 ln( )J a b . a3 and b3 are fitting constant coefficients.

Figure 16 shows the interference range for the anode area and cathode area along the

pipeline for different distances between the electrode and pipeline.

Figure 16. Corrosion risk area distribution under different soil resistivity.

The results indicate that the anode area range extends as the soil resistivity becomes

higher. Both the maximum leakage current density and corrosion risk area are positively

correlated with the soil resistivity. Therefore, it is very important to evaluate the

interference level accurately when dealing with highly resistive soil conditions.

6. Conclusions

(1) A methodology based on a combination of MoM and FEM is proposed for calculating

the coupling voltage and current on buried pipelines induced by grounding currents

while taking into account nonlinear polarization. The validity of the calculation

method was verified by experiment.

(2) Both the maximum leakage current density and maximum PSP increase as the

grounding electrode current increases. The relationship between the maximum

leakage current density and the electrode grounding current is linear. The corrosion

area becomes more concentrated as the grounding current increases.

(3) The maximum leakage current density increases exponentially when the distance

between electrode and pipeline decreases. The distance between the electrode and

Figure 15. Calculation results under different soil resistivity. (a) leakage current density along thepipeline; (b) PSP along the pipeline.

The maximum leakage current density and PSP increase as the soil resistivity increases.Regression analysis shows that the relationship between the maximum leakage currentdensity Jmax and the soil resistivity ρs satisfy logarithmic function, that is Jmax = a3 ·ln(ρs) + b3. a3 and b3 are fitting constant coefficients.

Figure 16 shows the interference range for the anode area and cathode area along thepipeline for different distances between the electrode and pipeline.

Appl. Sci. 2022, 12, 4433 12 of 13

Appl. Sci. 2022, 12, x FOR PEER REVIEW 12 of 13

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50

- 2

- 1

0

1

2

3

4

5

leak

age

curr

ent

den

sity

(A/m

2)

position relative to the pipeline midpoint (km)

10 m

30 m

100 m

1300 m

soil resistivity

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50

- 10

- 5

0

5

10

pipe

-to-s

oil

pote

nti

al (

V)

position relative to the pipeline midpoint (km)

10 m

30 m

100 m

1300 m

soil resistivity

(a) (b)

Figure 15. Calculation results under different soil resistivity. (a) leakage current density along the

pipeline; (b) PSP along the pipeline.

The maximum leakage current density and PSP increase as the soil resistivity in-

creases. Regression analysis shows that the relationship between the maximum leakage

current density Jmax and the soil resistivity ρs satisfy logarithmic function, that is

3m ax s3 ln ( )J a b . a3 and b3 are fitting constant coefficients.

Figure 16 shows the interference range for the anode area and cathode area along the

pipeline for different distances between the electrode and pipeline.

Figure 16. Corrosion risk area distribution under different soil resistivity.

The results indicate that the anode area range extends as the soil resistivity becomes

higher. Both the maximum leakage current density and corrosion risk area are positively

correlated with the soil resistivity. Therefore, it is very important to evaluate the interfer-

ence level accurately when dealing with highly resistive soil conditions.

6. Conclusions

(1) A methodology based on a combination of MoM and FEM is proposed for calculating

the coupling voltage and current on buried pipelines induced by grounding currents

while taking into account nonlinear polarization. The validity of the calculation

method was verified by experiment.

(2) Both the maximum leakage current density and maximum PSP increase as the

grounding electrode current increases. The relationship between the maximum leak-

age current density and the electrode grounding current is linear. The corrosion area

becomes more concentrated as the grounding current increases.

(3) The maximum leakage current density increases exponentially when the distance be-

tween electrode and pipeline decreases. The distance between the electrode and

Figure 16. Corrosion risk area distribution under different soil resistivity.

The results indicate that the anode area range extends as the soil resistivity becomeshigher. Both the maximum leakage current density and corrosion risk area are positivelycorrelated with the soil resistivity. Therefore, it is very important to evaluate the interferencelevel accurately when dealing with highly resistive soil conditions.

6. Conclusions

(1) A methodology based on a combination of MoM and FEM is proposed for calculatingthe coupling voltage and current on buried pipelines induced by grounding currentswhile taking into account nonlinear polarization. The validity of the calculationmethod was verified by experiment.

(2) Both the maximum leakage current density and maximum PSP increase as the ground-ing electrode current increases. The relationship between the maximum leakagecurrent density and the electrode grounding current is linear. The corrosion areabecomes more concentrated as the grounding current increases.

(3) The maximum leakage current density increases exponentially when the distancebetween electrode and pipeline decreases. The distance between the electrode andpipeline has a significant influence on the corrosion area range. The corrosion arearange increases substantially as the distance increases, whereas the influence degreedecreases substantially.

(4) The maximum leakage current density and PSP increase as the soil resistivity increases.The relationship between the maximum leakage current density and the soil resistivitysatisfies the logarithmic function. Both the maximum leakage current density andcorrosion risk area are positively correlated with the soil resistivity.

Author Contributions: X.L. conducted theoretical research, characteristics computation, and wrotethe paper; F.B. and F.C. are responsible for the experiment design and data analysis; C.L. gave somegood advice during the experiments and paper writing. All authors have read and agreed to thepublished version of the manuscript.

Funding: This research received no external funding.

Informed Consent Statement: Not applicable.

Acknowledgments: The authors would like to thank the State Grid UHVDC Test Site of China forproviding the experiment environment.

Conflicts of Interest: The authors declare no conflict of interest.

References1. Liu, L.; Yu, Z.; Jiang, Z.; Hao, J.; Liu, W. Observation research on the effect of UHVDC grounding current on buried pipelines.

Energies 2019, 12, 1279. [CrossRef]2. Verhiel, A.L. The effects of high-voltage DC power transmission systems on buried metallic pipelines. IEEE Trans. Ind. Gen. Appl.

1971, 3, 403–415. [CrossRef]

Appl. Sci. 2022, 12, 4433 13 of 13

3. Gong, Y.; Xue, C.L.; Yuan, Z.L.; Li, Y.X.; Dawalibi, F.P. Advanced analysis of HVDC electrodes interference on neighboringpipelines. J. Power Energy Eng. 2015, 3, 332–341. [CrossRef]

4. Ying, B. The influence on safe operation of the long-distance pipeline of electrode of high voltage direct current transmissionsystem. Oil-Gasfield Surf. Eng. 2014, 33, 23–24. (In Chinese)

5. Hardie, D.; Charles, E.; Lopez, A. Hydrogen embrittlement of high strength pipeline steels. Corros. Sci. 2006, 48, 4378–4385.[CrossRef]

6. Yang, C.; Li, J.; Li, Z.; Zhang, S.; Dai, L.; Zhang, C. Study on interference and protection of pipeline due to high-voltage directcurrent electrode. Corros. Rev. 2019, 37, 273–281.

7. Lorena-de, A.-R.; Villanueva-Balsera, J.; Ortega-Fernandez, F.; Rodriguez-Perez, F. Methods to Evaluate Corrosion in Buried SteelStructures: A Review. Metals 2018, 8, 334.

8. Hopper, A.T.; Gideon, D.N.; Berry, W.E. Analysis of the Effects of High-Voltage Direct-Current Transmission Systems on Buried Pipelines;Technical Toolboxes: Houston, AK, USA, 1967.

9. Nicholson, P. Study details HVDC stray-current effects on pipelines. Oil Gas J. 2010, 108, 74–78.10. Lagace, P.J.; Houle, J.; Greiss, H.; Mukhedkar, D. Computer aided evaluation of pipeline current near toroidal HVDC ground

electrodes. IEEE Trans. Power Deliv. 1989, 4, 216–222. [CrossRef]11. Yu, Z.; Liu, L.; Wang, Z.; Li, M.; Wang, X. Evaluation of the interference effects of HVDC grounding current on a buried pipeline.

IEEE Trans. Appl. Supercond. 2019, 29, 1–5. [CrossRef]12. Giao, T.N.; Sarma, M.P. On the potential and field distribution around a ground electrode for HVDC transmission. IEEE Trans.

Power Appar. Syst. 1971, PAS-90, 2793–2801. [CrossRef]13. Dawalibi, F.; Mukhedkar, D. Multi step analysis of interconnected grounding electrodes. IEEE Trans. Power Appar. Syst. 1976, 95,

113–119. [CrossRef]14. Dawalibi, F.; Pinho, A. Computerized analysis of power systems and pipelines proximity effects. IEEE Trans. Power Deliv. 1986, 1,

40–48. [CrossRef]15. Zhang, B.; Cao, F.; Meng, X.; Liao, Y.; Li, R. Numerical approach of impressed potential on buried pipelines near high-voltage DC

grounding electrodes. IET Gener. Transm. Distrib. 2018, 12, 1177–1182. [CrossRef]16. Stern, M.; Geary, A.L. Electrochemical polarization: I. A theoretical analysis of the shape of polarization curves. J. Electrochem. Soc.

1957, 104, 56. [CrossRef]17. Marzinotto, M.; Mazzanti, G.; Nervi, M. Ground/sea return with electrode systems for HVDC transmission. Int. J. Electr. Power

Energy Syst. 2018, 100, 222–230. [CrossRef]18. Zoski, C.G. Handbook of Electrochemistry; Elsevier: Amsterdam, The Netherlands, 2006.19. Stansbury, E.E.; Buchanan, R.A. Fundamentals of Electrochemical Corrosion; ASM International: Almere, The Netherlands, 2000.20. Newman, J.; Thomas-Alyea, K.E. Electrochemical Systems; Wiley: New York, NY, USA, 2004.21. NACE-TM0497. Measurement Techniques Related to Criteria for Cathodic Protection on Underground or Submerged Metallic Piping

Systems; NACE International: Houston, TX, USA, 2002.