IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006 703

Calculation of DC Current Distribution in AC PowerSystem Near HVDC System by Using Moment

Method Coupled to Circuit EquationsBo Zhang1, Xiang Cui2, Rong Zeng1, and Jinliang He1

State Key Lab of Power Systems, Department of Electrical Engineering, Tsinghua University, Beijing 100084, ChinaDepartment of Electrical Engineering, North China Electric Power University, Hebei 071003, China

When dc current is injected into the earth through the grounding electrodes of a high-voltage direct current (HVDC) system, trans-formers in ac system may be under dc bias if the dc currents flowing through the transformers are large enough. In this paper, a nu-merical method coupling the method of moments (MoM) to circuit equations is presented to calculate the dc current distribution in acpower system caused by an HVDC system. The MoM is used to calculate the electric fields in complex earth structure caused by all thegrounding systems including the dc grounding electrodes, the ac substation grounding systems and the long metal pipe lines. The circuitequations are coupled to the moment method to take account of the effects of the transmission lines. By using the method, the dc currentdistribution in an ac power system caused by an HVDC system is analyzed. Some useful conclusions are drawn from the analyzed results.

Index TermsCurrent distribution, grounding electrodes, high-voltage direct current (HVDC) transmission, method of moments(MoM).

I. INTRODUCTION

THE high-voltage direct current (HVDC) power transmis-sion constitutes an important technology in the develop-ment of large interconnected power networks. However, whenHVDC system uses earth as its current return path, great dccurrent will flow in the earth, which will bring great groundpotential differences in a large area around the dc groundingelectrodes [1]. Thus, dc currents will flow through the trans-formers in the ac substations if their neutral points are grounded,and the transformers may be under dc bias. The dc bias cancause acute vibration, great noise, high temperature of the trans-former, even make the protection miswork. In addition, a lot ofharmonic waves will be produced which can lower the powerquality [2]. Although the transformers in the substations nearthe dc grounding electrodes have more possibility to be underdc bias, dc bias does not always take place on them because thetransmission lines connected to the substations can also greatlyaffect the currents flowing through the transformers. In order toavoid the transformers under dc bias, it is not only necessary toinvestigate the tolerance level of transformer but also necessaryto analyze the dc current distribution in the ac system to esti-mate where measures should be taken.

Many papers have investigated the transformers under dc bias[3][5]. Because their dc biases are mainly caused by geomag-netism, few papers paid attention to the dc current distributionin ac power system caused by HVDC system. The current distri-bution is determined by many factors among which the groundpotential rise at each substation is an important one. This groundpotential rise is not only affected by the substations position,but also the ground resistance of the substation. Because the dis-tance among the two grounding electrodes of an HVDC system

Digital Object Identifier 10.1109/TMAG.2006.871460

and the ac substations is so far that much current maybe flow inthe deep layer of the earth [1], the ground resistance of each sub-station is determined not only by the structure of its groundingsystem but also by the deep layer of the earth. In order to an-alyze the dc current distribution in ac power system, both thestructures of the grounding systems and the deep layer of theearth should be considered. In this paper, a method coupling themethod of moments (MoM) to circuit equations is presented tocalculate the dc current distribution. The method can take ac-count of the structure of the earth, the structure of ac powersystem, the dc resistances of the transmission lines, the rela-tive positions and the structures of the grounding systems alltogether. The effect of the buried long metal pipe lines can alsobe considered.

II. CALCULATION MODEL

The whole system is just like an electric field under theground combined with a resistance network in the air. The elec-tric field is generated by the distribution of the leakage currentsfrom the dc grounding electrodes, the substation groundingsystems, and the buried long metal pipe lines which can alsobe regarded as grounding systems. The resistance networkconsists of the transmission lines. The key task to analyzethe dc current distribution in ac power system is to obtain thedistribution of the leakage currents, from which the potentialof each substation can be obtained, and the current flowingthrough the substation can be calculated.

Because the areas of the substations are very large and theirgrounding systems are made of steel, the grounding materialsresistance can not be neglected and the potential on eachgrounding system is unequally distributed. In order to considerthe effect of grounding materials resistance especially thoseof the buried long metal pipe lines, MoM coupled to circuitequations is used to complete this mission.

0018-9464/$20.00 2006 IEEE

704 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006

Due to the complex current distribution on each groundingsystem, MoM is used. Based on the idea of the MoM, for thecomplex conductor networks like the substation grounding sys-tems, the dc grounding electrodes, and the buried long metalpipe lines, it is necessary to divide them into conductor segments[6]. Let us assume that the longitudinal current of each seg-ment is centralized on the axis, and the leakage current flowsout from the central point of the segment. (This means that thereare two longitudinal currents in each segment, one flows fromthe segments start point to its central point, the other flows fromits central point to its end point.) Fig. 1 shows these currents onthe th segment.

Suppose that there be segments and nodes. The columnmatrix of the potentials at the central points of segments andthe column matrix of the leakage currents have followingrelation:

(1)where is a matrix with order of whose entry is equalto the potential at the central point of segment caused by aunit current leaking from segment . Based on the boundarycondition that the potential difference on the inner surface of theconductor must be equal to that on the outer surface, the entriesof can be regarded as voltage sources and a circuit modelcan be set up. Fig. 2 shows the equivalent circuit at nodesand . Applying the nodal analysis approach to the equivalentcircuit, following equation can be obtained:

(2)where is a column matrix of potentials at the nodes, is acolumn matrix of the injected currents at the nodes whose en-tries are usually zero except those of current injected nodes onthe dc grounding electrodes, is a relational matrix reflectingthe connection relationship between nodes and segments, whoseentry is 1 if node is connected to segment , otherwise

is zero, is a diagonal matrix with order of whoseentries at the diagonal are two times of the self-conductancesof corresponding segments, and is a nodal conduc-tance matrix of the nodal potential equations whose entry istwo times of the sum of all self- conductances of the segmentsconnected to node , if there is a transmission line connected tothe node should also plus the transmission lines conduc-tance, and should be as in the equation at the bottomof the page. Substituting (1) into (2) yields

(3)

Fig. 1. Two grounding systems connected with a transmission line.

At the same time, the currents flowing through the segmentstwo ends and can be obtained from and , shown in(4) at the bottom of the page, where is a relational matrix re-flecting the connection relationship between the nodes and thestart points of the segments, whose entry is 1 if the startpoint of segment is connected to node , otherwise iszero; and is also a relational matrix reflecting the connec-tion relationship between the nodes and the end points of thesegments, whose entry is 1 if the end point of segment isconnected to node , otherwise is zero.

The leakage currents and the currents flowing through the twoends of the segments have following relation:

(5)Substituting (4) into (5) gives the following equation:

(6)where is an identity matrix. By solving (6), the leakage cur-rents can be obtained. From the leakage currents, the potentialof each grounding system can be calculated and the currentsflowing through the transmission lines and transformers can bedetermined.

From this section it can be seen that the structure of the earth,the structure of ac power system, the dc resistances of the trans-mission lines, the relative positions and the structures of thegrounding systems are all embedded in the method.

III. PRACTICAL APPLICATION

In this section, the dc current distribution in an ac powersystem caused by an HVDC system will be analyzed, fromwhich the validity of above method is also testified.

A. Effect of Ground Wires on dc Current DistributionBecause the ground wires connect the substation grounding

systems with the tower grounding systems, currents will not

(4)

ZHANG et al.: CALCULATION OF DC CURRENT DISTRIBUTION IN AC POWER SYSTEM NEAR HVDC SYSTEM 705

Fig. 2. Equivalent circuit of Fig. 1.

Fig. 3. Analyzed transmission line.

only flow in the phase conductors but also flow in the groundwires and the tower grounding systems. Although the currentsin the ground wires do not flow through the transformers, theyflow through the substation grounding systems, which may af-fect the ground potential rises of the substations and then affectthe currents flowing through the transformers. Thus, before thedc current distribution in ac power system is analyzed, the cur-rent distribution along a transmission line shown in Fig. 3 is cal-culated to find the effect of the ground wires on the dc currentsflowing through transformers. If the effect is great, both theground wires and the tower grounding systems should be addedin the calculation of dc current distribution in ac power system,or else the ground wires and the tower grounding systems canbe neglected and the calculation will be greatly simplified.

In Fig. 3, the phase conductors connect the two substationgrounding systems via the transformers neutral points. Twoground wires connect the substation grounding systems withthe tower grounding systems. The areas of the two substationgrounding systems with depth 0.8 m are all 150 150 m . Thegrounding system of each tower with depth of 1 m is a criss-cross with length of 7.5 m in each direction. The dc resistanceof the three phase conductors in parallel with unit length is 0.02

/km. The dc currents through the dc grounding electrodes are1500 A. The earth structure shown in Table I is determined ac-cording to the composition of gross layering of the earth andthe practical situation in China [8]. Fig. 4 shows the potentialdistribution along the transmission line. Table II shows the dccurrent distribution on the transmission line under different dcresistance of ground wire.

From Fig. 4 it can be seen that although the transmission lineis 50 km away from the dc grounding electrode, the potentialdifference between the substations is still tens of volts due to theexistence of the layer with very high resistivity in the earth. The

Fig. 4. Potential distribution along the transmission line. Note that the dcresistance of each ground wire with unit length is 2.50 /km.

TABLE IPARAMETERS OF THE EARTH STRUCTURE

TABLE IIDC CURRENT DISTRIBUTION

potential with transmission line decreases slowly compared withthat without transmission line. Also, the potential differencewith transmission line is smaller than that without transmissionline. However, this potential difference is still high enough togenerate great currents flowing through the transformers.

From Table II it can be seen that the dc currents at the trans-formers neutral points hardly vary with the dc resistance ofground wires, even the ground wires do not exist, which showsthat the effect of the ground wires on the dc currents flowingthrough transformers is very small. Thus, the ground wires andthe tower grounding systems can be neglected and the calcula-tion will be greatly simplified.

B. dc Current Distribution in an ac Power SystemRecently, a kV HVDC system has been put into ser-

vice in China. However, when the system uses earth as currentreturn path, transformers in some ac substations are in abnormalstate, especially the transformer in substation A shown in Fig. 5.When the dc grounding electrode leaks current, engineers findthat the transformer in substation A makes great noise. Mea-surement shows that the current flowing through the transformerneutral point is 34.5 A when 1500 A current is leaked from the

706 IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 4, APRIL 2006

Fig. 5. Substations near a dc grounding electrode.

Fig. 6. dc current distribution in the substations and transmission lines.

dc grounding electrode while the currents of the other two sub-stations are not so great. In this section, the dc current distribu-tion near substation A is analyzed. The relative location of thesubstations to the dc grounding electrode, the relationship of thesubstations, and the dc resistances of the transmission lines areshown in Fig. 5. The earth structure is the same with that shownin Table I. The areas of the substation grounding systems withdepth 0.8 m are all 150 150 m . The dc grounding electrodeconsists of two concentric rings with radii 400 m and 300 m,respectively. Substation B and substation C are also connectedwith other substations which are not shown in Fig. 5 but havebeen taken into account in the calculation. The calculated dccurrent distribution is shown in Fig. 6.

From Fig. 6 it can be seen that when 1500 A current is leakedfrom the dc grounding electrode, the current flowing through thetransformer neutral point of substation A is much greater thanthose of the other two substations and reaches 31.2 A, which re-flects the same result with actual one and verifies the validationof the method presented in this paper.

Although substation B is more close to the dc grounding elec-trode than substation A, which makes the ground potential ofsubstation B high, the current flowing through its transformeris much smaller than that of substation A. This indicates thatthe distance to the dc grounding electrode does not directly de-

termine the current value. The current flowing through a sub-stations transformer is determined by the currents in the trans-mission lines connected to the substation which are determinedby the potential differences among the substations. The poten-tial differences have some relations with the distances to the dcgrounding electrode. Because the current in the two transmis-sion lines connected to substation B is small, the current flowingthrough its transformer is small. Although the current in thetransmission line connecting substation C and substation A isgreat due to the long distance, most of it flows to other substa-tions from the corresponding transmission lines and the currentflowing through the transformer of substation C is not as greatas that of substation A. This result will be useful for the futuredesign and site selection of new dc grounding electrode and acsubstations.

IV. CONCLUSIONA numerical method coupling the MoM to circuit equations

is presented to calculate the dc current distribution in ac powersystem caused by an HVDC system. The effect of the groundwires on the dc currents flowing through the transformers is an-alyzed. The dc current distribution in ac power system causedby an HVDC system is calculated. Results show that the effectof the ground wires on the dc currents flowing through trans-formers is very small. The current flowing through a transformeris determined by many factors such as the locations of the sub-stations and the dc grounding electrode, the interconnection re-lationship of the substations, and the dc resistances of the trans-mission lines. The method is useful to estimate the effect ofthe dc current from the dc grounding electrode on the ac powersystem.

REFERENCES[1] J. Eduardo, T. Villas, and C. M. Portela, Calculation of electric field

and potential distributions into soil and air media for a ground electrodeof a HVDC system, IEEE Trans. Power Del., vol. 18, pp. 867873, Jul.2003.

[2] R. J. Ringlee and J. R. Stewart, Geomagnetic effects on power systems,IEEE Power Eng. Rev., vol. 9, pp. 69, Jul. 1989.

[3] Y. Yao, C. S. Koh, and G. Ni, 3-D Nonlinear transient eddy current cal-culation of online power transformer under dc bias, IEEE Trans. Magn.,vol. 41, pp. 18401843, May 2005.

[4] W. C. Viana, R. J. Micaleff, S. Young, F. P. Dawson, and E. P. Dick,Transformer design considerations for mitigating geomagnetic inducedsaturation, IEEE Trans. Magn., vol. 35, pp. 35323534, Sep. 1999.

[5] E. F. Fuchs, Y. You, and D. J. Roesler, Modeling and simulation, andtheir validation of three-phase transformers with three legs under dcbias, IEEE Trans. Power Del., vol. 14, pp. 443449, Apr. 1999.

[6] R. F. Harrington, Field Computation by Moment Methods. Hampshire,U.K.: MacMillan, 1968.

[7] B. Zhang, Z. Zhao, X. Cui, and L. Li, Diagnosis of breaks in substa-tions grounding grid by using electromagnetic method, IEEE Trans.Magn., vol. 38, pp. 473476, Mar. 2002.

[8] HVDC Ground Electrode Design, IEC, San Francisco, CA, 1981.EPRI EL-2020, Project 1467-1.

Manuscript received June 20, 2005 (e-mail: shizbcn@tsinghua.edu.cn).

tocCalculation of DC Current Distribution in AC Power System Near HBo Zhang ${}^{1}$, Xiang Cui ${}^{2}$, Rong Zeng ${}^{1}$, and J${}^1$ State Key Lab of Power Systems, Department of Electrical I. I NTRODUCTIONII. C ALCULATION M ODEL

Fig.1. Two grounding systems connected with a transmission lineIII. P RACTICAL A PPLICATIONA. Effect of Ground Wires on dc Current Distribution

Fig.2. Equivalent circuit of Fig.1 .Fig.3. Analyzed transmission line.Fig.4. Potential distribution along the transmission line. NoteTABLE I P ARAMETERS OF THE E ARTH S TRUCTURETABLE II DC C URRENT D ISTRIBUTIONB. dc Current Distribution in an ac Power System

Fig.5. Substations near a dc grounding electrode.Fig.6. dc current distribution in the substations and transmissIV. C ONCLUSIONJ. Eduardo, T. Villas, and C. M. Portela, Calculation of electriR. J. Ringlee and J. R. Stewart, Geomagnetic effects on power syY. Yao, C. S. Koh, and G. Ni, 3-D Nonlinear transient eddy curreW. C. Viana, R. J. Micaleff, S. Young, F. P. Dawson, and E. P. DE. F. Fuchs, Y. You, and D. J. Roesler, Modeling and simulation,R. F. Harrington, Field Computation by Moment Methods . HampshirB. Zhang, Z. Zhao, X. Cui, and L. Li, Diagnosis of breaks in sub

HVDC Ground Electrode Design, IEC, San Francisco, CA, 1981. EPRI