analysis of rail potential and stray currents in a direct-current transit system

1516 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 25, NO. 3, JULY 2010

Analysis of Rail Potential and Stray Currentsin a Direct-Current Transit System

Yii-Shen Tzeng and Chien-Hsing Lee, Senior Member, IEEE

Abstract—The diode-grounded scheme for stray-current col-lection in some systems, such as the Taipei rapid transit systems(TRTS), has been constructed to gather the stray current leakingfrom the running rails and avoid corrosion damage to the systemas well as the surrounding metallic objects. During operationof the TRTS, a high potential between the negative return busand system earth bus at traction substations, referred to as railpotential, has been observed on the Blue line between BL13 andBL16. Since the Blue and Red-Green lines have their running railsand stray-current collector mats in junction at the G11 station,the TRTS suspects that the impedance bond at G11 is the causeof rail potential rise. This paper presents the results of field testsfor studying whether the impedance bond at G11 of the tie linehas an impact on rail potential and stray currents in TRTS. Theresults show the rail potential can be reduced by disconnecting theimpedance bond at G11 of the tie line so that the negative returncurrent of the Blue line cannot flow to the rails of the Red-GreenLine, and vice-versa. In addition, rail potential and stray currentsoccurring at a station of the Blue line are numerically simulatedby using a distributed two-layer ladder circuit model. The simula-tion results are compared with the field-test results and they areconsistent with each other.

Index Terms—DC electrified railways, diode-grounded, directgrounded, rail potential, stray currents, ungrounded.

I. INTRODUCTION

I N SOME systems, such as Taipei rapid transit systems(TRTS), the diode-grounded scheme for stray-current

collection is designed to drain the stray current (a portionof the negative return current leaking from the running railsdue to the resistances of running rails and rail to ground) intothe insulated traction earth bus (TEB) at a traction substation(TSS). When the stray current is found on the collector matand returns to the TEB through the collector cable, it is calledprimary stray current as shown in Fig. 1. This current leavingfrom the running rails may cause corrosion to the rails/railfasteners themselves and further leak into the surrounding soiland into any metallic conductors (such as the reinforcementbar in concrete) due to the limited conductivity of the collector

Manuscript received August 22, 2009; revised November 25, 2009. Currentversion published June 23, 2010. This work was supported by the NationalScience Council, Taiwan, under Grants NSC 98-2221-E-006-245 and NSC-97-2221-E-161-009-MY3. Paper no. TPWRD-00636-2009.

Y.-S. Tzeng is with the Department of Electrical Engineering, Oriental In-stitute of Technology, Panchiao, Taipei 220, Taiwan (e-mail: [email protected]).

C.-H. Lee is with the Department of Systems and Naval MechatronicEngineering, National Cheng Kung University, Tainan 701, Taiwan (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRD.2010.2040631

Fig. 1. Schematic representation of a stray-current return system.

mat. The sum of further leakage of the primary stray currentand the uncollected primary stray current may flow back tothe system earth bus (SEB) through the ground; this is calledsecondary stray current. This current is particularly harmful tothe external infrastructure, tunnels, etc. Nevertheless, both straycurrents eventually flow back to the negative return bus (NRB)if the drainage diode is at the turn-on state. The sum of thesestray currents is then called the aggregate stray current. As canbe observed in Fig. 1, there are three permanently installedresistors, including (6000 A/30 mV), (4000 A/60mV), and (500 A/60 mV) which are used for measuring thenegative return current, aggregate stray current, and secondarystray currents, respectively.

Generally, to minimize the influence by stray currents, astray-current control, such as cross-bonding of the running railsand stray-current collector cables (traction earth conductors) isrequired to balance negative traction return currents, to reduceresistances of the negative return circuit, as well as to decreasethe rail potential [1]–[17]. Historically, stray-current collectormats either with or without drainage diodes have been installedon a dc transit system for assessing, investigating, recording,and isolating the offending source of increased stray-current ef-fects [8]–[13]. They are constructed from an assembly of drivenrods and bare copper conductor. All joints are exothermicallywelded. Typically, the mats are located at a minimum of 1 mbelow the finished grade and the cross-sectional area of thecollector cable is 70- or 120-mm CU [7]–[9]. For the code ofpractice in TRTS [15], it requires a collector mat of six 12-mmdiameter steel-reinforcing bars to be bonded to a 120-mm Custray-current collector cable at 200-m intervals.

0885-8977/$26.00 © 2010 IEEE

TZENG AND LEE: ANALYSIS OF RAIL POTENTIAL AND STRAY CURRENTS 1517

Fig. 2. Network of the completed and planned rapid transit system in Taipei.

II. SYSTEM DESCRIPTION

The completed and planned network of the TRTS is illus-trated in Fig. 2. The electric power for TRTS is supplied bythe Taiwan power company (TPC) through 161-kV incomingfeeder units. The bulk supply substations (BSSs) connect theMetro power supply system to the public power grid, step the161 kV down to 22.8 kV (referred to as 22 kV). Then, it dis-tributes the 22-kV supply to the rectifier substations [or trac-tion supply substation (TSS)], to the station supply substations(SSS), and to the depots. TSS provides the energy for the trac-tion supply and the depots. The average distance between thetwo TSS is about 1.5 km. Each TSS comprises medium high-voltage switchgear units, rectifier transformers, rectifiers, anddc-voltage switchgear units. The rectifier transformers are de-signed to step down the voltage from 22.8 kV to 589 V for thesupply of the rectifiers, which convert the 589-V ac to the trac-tion voltage of 750-V dc. The SSS provides the energy to operateauxiliary installations, such as escalators, workshops, illumina-tion, etc. within the stations and depots. It is fed directly fromthe related medium high-voltage switchgear units of the allo-cated BSS. The power transformers step down the voltage from22.8 kV to 380 V feeding the low-voltage switchgear. As forsome important data of TRTS, such as the rating of the tractionsubstations and location of impedance bonds, please refer to Ta-bles III–VI .

During operation of the TRTS, two features have been ob-served from the supervisory control and data acquisition at theoperation control center: 1) The rail potential at stations from

BL13 to BL16 was relatively high and close to the IEC 62128-1limit [14], [16]; 2) the number of substations with diode-on atthe Red line is always more than that of the Blue line. Moreover,there were no more than three TSSs with their drainage diodesturned on at any moment in the Red line based on the results ofthe stray-current monitoring system. This means that stray cur-rents outflowing from the rail sections of a total of 21 stations(R12–R33) will flow back collectively to only one to three TSSs.However, the location of these TSSs may vary randomly whichdepends on the position and status (i.e., acceleration, coasting,or deceleration) of the multiple trains through the entire Redline. As a result, the TRTS has postulated that stray currentsoutflowing from the rail of the Blue line could flow back to thethird rail at TSSs with diode on at the Red line. Due to the longdistance of stray current flowing to the diode-on substation, therail potential was thus relatively high, and close to the IEC limit.Since the Blue and Red-Green lines have their running rails andstray-current collector mats in junction at the G11 station, thefield tests of disconnecting the tie line were performed to eval-uate whether the negative return current of the Blue line willflow to the rails of the Red-Green Line, and vice-versa.

III. FIELD TESTS

During operation of the TRTS, it has been observed that therail potential at the Red-Green line increases by more than about20–30 V after the Blue line entered commercial operation inDecember 1999. Since the Blue and Red-Green lines have theirrunning rails and stray-current collector mats in junction at theG11 station, the TRTS suspects that the impedance bond at G11is the cause of rail potential rise. As a result, the field tests havebeen performed for studying whether the impedance bond atG11 of the tie line has an impact on rail potential and straycurrents in TRTS.

Since the Blue line has not yet fully operated at the time ofthe field test (i.e., only operated from BL01 to BL13), the se-lected field test site was at BL13 TSS, which was conductedon May 4, 2001 [14]. The impedance bond at G11 was discon-nected between 11:00 A.M. and 16:00 P.M.. As seen in Fig. 3, railpotentials had been lowered by almost half after disconnectingthe impedance bond (i.e., the highest rail potential was about112.5 V). This improvement was because the negative returncurrent of the Red and Green lines cannot flow to the rails ofthe Blue line and vice-versa. As a result, the disconnection ofthe impedance bond at G11 has been implemented permanentlysince 2003.

Moreover, the stray currents recorded at BL09 TSS as shownin Fig. 4 is utilized for assessing the performance of stray-cur-rent collection systems installed in TRTS since its measuredstray currents was the highest among other stations. The stray-current collection system efficiency is defined as the ratio of asum of the primary stray current divided by a sum of the ag-gregate stray current, and the stray-current leakage percentageis defined as the ratio of a sum of the secondary stray cur-rent divided by a sum of the aggregate stray current. As a re-sult, the stray-current collection system efficiency and stray-current leakage percentage were obtained to be 40.29% and59.71%, respectively. To increase the stray-current collection


Fig. 3. Measurement results of the potential at BL13 TSS during the field teston May 4, 2001, by disconnecting the impedance bond at G11.

Fig. 4. Measurement results of stray currents at BL09 TSS during the field test.

efficiency, the collection system generally has to offer a signif-icantly lower resistance path than the segment reinforcement ina tunnel, buried services, and the surrounding soil itself [11].The code of practice in TRTS [15] requires a minimum of 150

-km be maintained between the running rails and the earth.It also requires that the collector mat of six 12-mm diametersteel-reinforcing bars be bonded to a 120-mm Cu stray-currentcollector cable at 200-m intervals. Thus, for a uniform leakagealong the track, the model for current leaking from the track intothe stray-current collector cable is that of using resistances on a1-m basis.

Nevertheless, an effective way suggested by Dekker [8] hasbeen utilized for reducing the resistance of the collection systemin TRTS which alters the welding point of the collection systemand the connection between the collection mat and the collectorcable. Currently, the code of practice in TRTS requires that thelongitudinal and latitudinal steel bars in the collection mat undereach plinth should not form into a loop configuration as shown

Fig. 5. Comparison of different welding of the collection system. (a) The struc-ture installed by TRTS. (b) The structure suggested by Dekker.

in Fig. 5(a) in order to avoid the loop current occurring withinthe collection system. Thus, only one of the latitudinal steel barsin the collection mat is used to connect with the collector cable.However, this design obviously may not be able to decrease theself-resistance of the collection system. As a result, TRTS hasmodified the welding point of the collection system to be that asshown in Fig. 5(b), which has been suggested by Dekker [8].

IV. MODELING OF RAIL POTENTIAL AND STRAY CURRENTS

Most of the computer simulations on rail potential and straycurrents in a dc transit system were reported by researchersapproximately ten years ago [2]–[4], [6], [17]–[20]. To pre-cisely model the negative return circuit and the stray-currentcollection system, a distributed two-layer ladder circuit modelis used instead of a single-layer transmission-line model. Thedistributed two-layer ladder model consists of the stray-currentcollector mat and stray-current collector cable and provides anadvantage of studying the efficiency of a stray-current collectionsystem. However, the single-layer transmission-line model doesnot have the advantage in evaluating the stray-current collectionsystem efficiency. Moreover, a train model based on the detailedtrain movement simulation and dc load-flow calculation for thewhole system is used instead of assuming a typical profile of thetraction power as a train running between two stations [18]. Im-portantly, a model for determining operation states (i.e., on oroff) of drainage diodes as shown in the Appendix is developedand used instead of assuming only one turn-on drainage diodefor the whole system [14], [17].

Generally, the power supply of a dc transit system is classifiedinto positive and negative circuit networks. The former consistsof a positive terminal from the rectifier of a TSS as well as theconductor rail, whereas the latter consists of a negative terminalfrom the rectifier of a TSS, the running rails, and the stray-cur-rent collection system. As a result, the rail potential and straycurrents are behaviors that belong to the negative circuit net-work. Fig. 6 shows an equivalent circuit of the dc negative net-work in a transit system with the ladder circuit model. The pos-itive circuit network is assumed as a super node to simplify thisstudy.

A. Equivalent Circuit of the Negative Circuit Network

The train can be modeled as a current source as shownin Fig. 6. As a train runs, its position and dissipated/regener-ated power are all functions of time. Meanwhile, the output


Fig. 6. Equivalent circuit of the dc negative network in a transit system.

power of the TSS rectifier will change with time. Thus, the rec-tifier and the train can be considered as stationary and nonsta-tionary time-variant current sources, respectively. The currents

and at any instant of time are dependent on the in-stantaneous power and voltage obtained from the train move-ment simulation and the dc load-flow calculation.

Moreover, the negative return circuit is used to describe selfand mutual resistances among the running rails, the stray-cur-rent collection system, and the ground. For example, Fig. 7shows one of the negative return circuit segments including fiveresistors , , , , and . Resistances of the mutualresistors , , and are dependent on the degree of in-sulation. A segment of 100 m is sufficient and accurate for thestray current and rail potential analysis [19]. Cross-bonding ca-bles are used at a certain distance to balance the negative returncurrent between up and downtrack running rails, to decrease re-sistances of the negative return circuit, and to decrease rail po-tential. As seen in Fig. 6, resistors and are used to rep-resent these cross-bonding cables. Similarly, stray-current col-lector cables located up and downtrack are crossly bonded todecrease resistances of the stray-current return path. In addition,three shunt resistors , , and are used for measuringnegative return currents, aggregate stray currents, and secondarystray currents, respectively. The resistor connected betweenthe SEB and the ground represents the ground resistance of theTSS. For the drainage diode, it can be equivalent with a cur-rent source parallel with a turn-on resistor if the diode is on.Moreover, resistors and are used to represent theresistances of cables connected up and downtrack of the NRB,respectively. Likewise, resistors and are used torepresent the resistances of cables located up and downtrack ofthe TEB, respectively.

B. Flowchart of Computing Rail Potential and Stray Currents

A flowchart of computer simulation on rail potential and straycurrents is shown in Fig. 7, which consists of three main mod-ules (i.e., train movement, dc load flow, and stray-current mod-ules [21].

Fig. 7. Flowchart of computer simulation on rail potential and stray currents.

The kernel of the train movement module is the numerical in-tegration of the motion equation. Based on the train characteris-tics, route data, operation parameters, and scheduled timetable,trains can be simulated sequentially and then be put on, or beremoved from, the route. Each train on the route is accord-ingly represented as a moving bus in the dc positive networkwith a constant power load, which may be positive or nega-tive depending upon its powering or braking mode. After thetrain movement module has been performed at each time point,the dc load-flow module calculated by using Newton-type loadflow equations in the dc positive network can be executed to ob-tain equivalent current sources of trains and TSSs for each timepoint. Based on the status of TSS as well as the instantaneouspower and voltage of the train at each time step, the bus currentmatrix and admittance matrix, of the negative cir-cuit network can be obtained by using sparse programming tech-niques [22]. Then, the matrix equation issolved by using the LU decomposition method. The tolerableerror of bus voltages is set to be less than 0.0001 V. For each it-eration, the algorithm for determining the operation states of thedrainage diodes at TSSs is referred to in the Appendix. Once thebus voltages have been found from the matrix equation, currentsflowing through each branch in the dc negative circuit networkcan be computed.

V. SIMULATION RESULTS

The Blue line in TRTS, as shown in Fig. 2, is used as a testsystem here and their data are summarized in Tables III–VI.The station substation located in the passenger station is notincluded in the analysis since it belongs to the ac system. Pa-rameters of the equivalent circuit of the dc negative network


used for the simulation are , m ,, m , /km,

/km, -km, -km,-km, V (turn-on voltage of the drainage

diode), m ,m , m , m . These data

are system design parameters provided by the TRTS. The trainheadway is about 5 min and 40 s, which is obtained from thecontrol center of TRTS.

For evaluating stray currents, the following are defined:1) Total stray current (TSC): the total current leaked to the

ground through the running rails and the stray-current col-lection system at time t.

2) One-hour’s equivalent gross leakage charge (GLC): this isobtained by integrating the TSC against time for duration

and then normalizing to one-hour’s equivalent [4].Moreover, the maximum positive and negative potentials over

the period of simulation time are used for assessing the railpotential.

A. Effects of Grounding Strategies on Rail Potential and StrayCurrents

Several cases, as listed in Table I, are studied for the effectsof grounding strategies on rail potential and stray currents. Thescenario of Case is currently used by the Blue line. However,the grounding scheme of the extended Blue line has been mod-ified to the diode-grounded scheme with a series of normallyopen switches. In other words, its grounding scheme of TSSs isungrounded during normal operations. Fig. 8 shows the simu-lation results of rail potential and stray currents for Case . Asseen in Fig. 8(a), a maximum aggregate stray current of 13.18A flows back to the BL04 TSS and 94% of this current con-sists of primary stray currents. On the contrary, about 60% ofTSCs leaked to the ground directly from the running rails asseen in Fig. 8(b), and the remaining 40% will leak to the groundfrom the stray-current collection system. As seen in Fig. 8(c),the maximum positive and negative potentials of the runningrails are about 134.6 V and 23.4 V, respectively.

To verify the proposed model of stray-current calculation, thestray currents and NRB voltage at BL04 TSS have been mea-sured from 4:00 A.M. to 9:00 A.M. on April 16, 2007. As a re-sult, the maximum stray current occurred between 07:01:36 to07:02:36 as shown in Fig. 9. As seen in Fig. 9, the drainagediode was turned on for three times as the NRB voltage becomesslightly smaller than the zero voltage. Moreover, the aggregateand secondary stray currents at time 11.69 s are 57.7 A and7.0 A, respectively. This means that the primary stray currentis equal to 50.7 A at this point of time. Thus, the ratios of theprimary and secondary stray currents to the aggregate stray cur-rent are 88% and 12%, respectively. Comparing the measuredresults as shown in Fig. 9 with the simulated values as shown inFig. 8(a), the measured stray currents are several times greaterthan the simulated values. This situation may result from thedeteriorated rail insulation because the Blue line has been op-erating commercially for more than ten years. Nevertheless, theratios of the primary and secondary stray currents to the aggre-gate stray current are close to the results obtained from Fig. 8(a),

Fig. 8. Simulation results of the blue line for case a. (a) Stray currents occurredat BL04 TSS. b) Total stray current occurred at the blue line. (c) Maximumpositive and negative rail potentials occurred at the blue line.

TABLE ISUMMARY OF SIMULATION CASES FOR THE TEST LINE

which are 94% and 6%, respectively. Simulation results for theother cases are summarized in Table II.


Fig. 9. Measurement results on stray currents and NRB voltage at BL04 TSS.

TABLE IISUMMARY OF SIMULATION RESULTS FOR ALL CASES

B. Effects of Cross-Bonding on Rail Potential and StrayCurrents

The Blue line of TRTS has 12 cross-bonding locations be-tween the up and downtrack as well as stray-current collectorcables. The cables used for cross-bonding of running rails andstray-current collector cables are 4 250 mm Cu EPR and1 120 mm Cu EPR, respectively. Detailed cross-bonding lo-cations are listed in the Appendix. Fig. 10 shows the maximumpositive rail potential and TSC as considering whether the run-ning rails and stray-current collector cables are cross-bondedor not. As seen from Fig. 10(a), the rail potential can be re-duced from 167.6 V to 134.6 V when the running rails arecross-bonded. However, the TSC will not be effectively reducedas shown in Fig. 10(b) after cross-bonding of running rails andstray-current collector cables.

C. Effects of a Short-Circuited Drainage Diode on StrayCurrents

Some of the drainage diodes at TSSs at the Blue line had beenburned out during the system integration test. One of the reasonsis the signaling worker forgetting to connect the negative return

Fig. 10. Maximum positive rail potential and total stray current with andwithout cross-bonding of running rails and stray-current collector cables. (a)Maximum positive potential occurred at the blue line. (b) Total stray currentoccurred at the blue line.

cables to the impedance bond after the track-circuit test. As a re-sult, the return current flows into the rails, the stray-current col-lection system, and the ground. Eventually, it flows back to theNRBs via the drainage diodes and the drainage diodes are over-load and damaged consequently. After the capacities of diodeswere changed from 1100 A to 5800 A, they were not burned outanymore. To illustrate the about event and explain possible rea-sons, a virtual case is presented here.

If the drainage diode at BL04 TSS is assumed to be short-cir-cuited, the grounding scheme at BL04 TSS will become di-rect grounded but the grounding scheme of other TSSs at theBlue line is still diode-grounded. Simulation results as shown inFig. 11 present the effects of a short-circuited drainage diode onstray currents. As seen from Fig. 11(a), the aggregate stray cur-rent flowing from the NRB to the TEB is about 636.8 A after thedrainage diode has been shorted. However, the aggregate straycurrent flowing through the BL04 TSS under normal operationsis 13.2 A. Thus, if a drainage diode is shorted, the aggregatestray current flowing through the TSS will increase more thanseveral ten times. Moreover, the large aggregate stray currentwill not occur only at the shorted TSS but it may flow throughother normally operated drainage diodes. As seen from Fig.11(b), the aggregate stray current flowing back to the BL05 TSShas been increased from 8.1 A to 360.2 A for a short-circuiteddrainage diode at the BL04 TSS. In addition, the total stray cur-rent will be affected in this case. As seen from Fig. 11(c), thetotal stray current occurring at the Blue line is about 12.9 A


Fig. 11. Effects of a short-circuited drainage diode at BL04 TSS on stray cur-rents. (a) Aggregate stray current flowing through the BL04 TSS. (b) Aggregatestray current flowing back to the BL05 TSS. (c) Total stray current occurred atthe Blue line.

under normal operations. After a drainage diode is shorted atthe BL04 TSS, the total stray current will increase from 12.9 Ato 570.2 A.

D. Effects of Rail Insulation on Stray Currents

This subsection investigates the effects of three rail insula-tion cases on stray currents. Parameters of the test system forCase are the rail-to-ground resistance to be -km,the rail-to-collector-mat resistance to be -km, andthe collector-mat-to-ground resistance to be -km.The aforementioned values are the minimum-designed criteriafor the track-bed without ballast in TRTS [15]. Resistances ,

Fig. 12. Effects of rail insulation on maximum aggregate stray currents at theblue line.

, and of Case are 40, 40, and - , respec-tively. These values are also the minimum-designed criteria forthe track-bed with ballast in TRTS. For Case , the uptrackrail insulation between the BL03 and BL04 passenger stationsis assumed to significantly deteriorate at the distance of 100 to200 m from the BL04 passenger station. This will result inbeing - and the resistances and being the sameas Case (i.e., -km and -km).

Fig. 12 shows the maximum aggregate stray current flowingthrough TSSs under the influence of deteriorated rail insulationfor Case compared with Cases and . As seen from Fig. 12,the maximum aggregate stray current will increase about six toseven times at all TSSs compared to Case with Case whenthe rail insulation deteriorates. The situation assumed in Case

means the distance of 100 m away from the BL04 passengerstation for the uptrack running rails has the ground resistance of

. Compared to Case with Case , the maximum aggregatestray current flowing through the BL04 TSS will decrease from13.2 A to 8.3 A when the rail insulation deteriorates around theBL04 passenger station. Nevertheless, the maximum aggregatestray currents flowing through other TSSs will increase about6.23 to 7.74 times. The highest increase of the maximum aggre-gate stray currents flows through the BL05 TSS and the secondhighest increase of the maximum aggregate stray currents flowsthrough the BL09 TSS (about 7.34 times). In conclusions, thelowest value of the maximum aggregate stray currents occurs atthe BL04 TSS, which is the closest to the deteriorated point ofthe rail insulation and the largest increase of the maximum ag-gregate stray currents occurs at the BL05 TSS, which is the nextclosest to the deteriorated point of the rail insulation. This meansthat the stray current leaking from the rails to the ground will beobviously increased when the rail-to-ground insulation deterio-rates somewhere close to a TSS. However, this increasing straycurrent will flow back to the further TSS, not the closer TSS.

VI. SUMMARY AND CONCLUSION

This paper presents the results of field tests and simulationsof rail potential and stray currents at several stations of the Blueline of TRTS. Based on the results, concluding remarks havebeen drawn as follows.

1) Although cross-bonding running rails can effectivelyimprove the problem of rail potential, the problem of


TABLE IIIDESIGNED ELECTRICAL DATA

TABLE IVSTATION MILEPOST OF THE BLUE LINE SYSTEM

stray currents still cannot be significantly mitigated.As for the Blue line of TRTS, the rail potential canbe reduced to about 20% from 167.4 to 134.6 V forcross-bonding running rails per 700–900 m.

2) As for all TSSs using the diode-grounded scheme, theTSC and maximum rail potential will both reduce toabout 10% for one of the TSSs using the direct-groundedscheme and others using ungrounded.

3) If the insulation of running rails deteriorates, the ag-gregate stray current flowing through the TSS, whichis the closest to the deteriorated point, will slightly de-crease. Nevertheless, the aggregate stray current flowingthrough other TSSs will increase. This will help main-tenance workers determine the section with the deterio-rated insulation of running rails.

4) On the issues of rail potential and stray currents, thediode-grounded scheme has the highest rail potentialand its maximum value is about 50% higher than the railpotential obtained by the ungrounded scheme. Thus, theungrounded scheme has less of a problem with stray cur-rents and rail potential than the diode-grounded scheme.

APPENDIX

ALGORITHM FOR DETERMINING THE OPERATION

STATES OF THE DRAINAGE DIODES

For a diode-grounded system, each drainage diode installedat the negative network can be considered as a two-state (i.e., onand off) digital switch. Thus, a system consisting of n TSSs will

TABLE VCROSS-BONDING LOCATIONS OF RUNNING RAILS

TABLE VICROSS-BONDING LOCATIONS OF STRAY CURRENT COLLECTOR CABLES

have operation states of drainage diodes. Nevertheless, onlyone state at an instant of time can satisfy either an on or off oper-ation state. This paper adopts a method which is widely used inpower-electronics simulation programs to distinguish discontin-uous points between on and off states. As a result, for nonlinearoperation characteristics of drainage diodes, one can accuratelycalculate the switching time between on and off states as wellas obtain the operation states of drainage diodes correctly.

Fig. 13 shows a flowchart for finding the switching time ofdrainage diodes. To simplify the process, the simulation step

is assumed to be small enough (e.g., 0.1 s) and the cur-rent changes at TSSs and trains within can be linearly ap-proximated. In addition, the train position is assumed linearlychanged. Based on these assumptions, the equivalent currentsources of trains and TSSs as well as the train positions at aninstant of time within can be obtained by using the interpo-lation method.

As seen from Fig. 13, the states of drainage diodes atcan be assumed to be the same as the states of drainage

diodes at time when it is simulated from t to . Ifthe calculated result shows that the operation states of drainagediodes have not been changed (i.e., no reverse current appears atthe turn-on diode and the forward-biased voltage is less than thethreshold turn-on voltage for the turn-off diodes), the switchinginstant search of drainage diodes is completed and goes backto the main program to do the simulation at the next instant of


Fig. 13. Flowchart for determining the operation states of drainage diodes.

time. However, if the calculated result shows that the operationstates of one or more drainage diodes are not satisfied (i.e., areverse current appears at the turn-on diode or a forward-bi-ased voltage is greater than the threshold turn-on voltage for theturn-off diodes), this means at least one of the drainage diodeshas changed its operation state from t to . Generally, abisection method is used to search the switching time (i.e.,

) for the earliest changing state of the drainagediode.

When a system consisted of many drainage diodes, it is pos-sible that more than one drainage diode can have its opera-tion states change within a simulation step . Since the cur-rent sources and train position are assumed to be changed lin-early, the diode with the largest mismatch of the forward voltage

among all drainage diodes (i.e., the largest error of the for-ward-bias voltage and the threshold turn-on voltage) will firstchange its operating state.

If the time difference during the bisection search isless than the time tolerance (e.g., s) and the largest mis-match of the forward voltage of the drainage diodeis less than the voltage tolerance (e.g., 10 V), the firstchanging time point will then be obtained. As a result, the equiv-alent current sources of the train and TSS should be recalculatedsince the operation states of the drainage diode and the admit-tance matrix of the negative network have been changed. More-over, the voltage at each bus of the negative network should beresolved to verify whether the operation states of other drainagediodes will vary at the switching instant as the circuit topologychanges. The aforementioned calculation and searching pro-cesses will be continuous at the instant of changing states untilall drainage diodes satisfy the operation condition.

Note that when the first switching instant is found, the cir-cuit will be calculated based on the operation states of drainagediodes at that instant. The searching and calculation processeswill be repeated until the operation states of all drainage diodeshave not been changed at the time of and or the timedifference between and is less than the time tolerance

.

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Yii-Shen Tzeng was born in Taiwan in 1966. Hereceived the B.S.E.E. and Ph.D. degrees in electricalengineering from the National Taiwan Institute ofTechnology in 1991 and 1995, respectively.

Currently, he is an Associate Professor in the De-partment of Electrical Engineering, Oriental Instituteof Technology, Panchiao, Taiwan. His research inter-ests include the converter modeling and the planningand operation of rapid-transit power systems.

Chien-Hsing Lee (S’93-M’98–SM’06) received theB.S. degree in electrical engineering from ArizonaState University, Tempe, in 1993 and the M.S.E.E.and Ph.D. degrees from the Georgia Institute of Tech-nology, Atlanta, in 1995 and 1998, respectively.

Currently, he is an Associate Professor at NationalCheng Kung University, Taiwan. His research in-terests are power system grounding analysis, powersystem transient modeling, and applications ofwavelet theory in power systems.

analysis of rail potential and stray currents in a direct-current transit system

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