energies

Article Evaluation of Rail Potential Based on Power Distribution in DC Traction Power Systems

Guifu Du, Dongliang Zhang *, Guoxin Li, Chonglin Wang and Jianhua Liu

School of Information and Electrical Engineering, China University of Mining and Technology, Xuzhou 221008, China; [email protected] (G.D.); [email protected] (G.L.); [email protected] (C.W.); [email protected] (J.L.) * Correspondence: [email protected]; Tel.: +86-516-8388-5605

Academic Editor: Paul Stewart Received: 8 July 2016; Accepted: 5 September 2016; Published: 9 September 2016

Abstract: Running rails used as a return conductor and ungrounded scheme have been widely adopted in DC traction power systems. Due to the longitudinal resistance of running rails and insulation resistance of rail-to-ground, there will be a potential rise between running rails and the ground when return current flows through the running rails, which is known as rail potential. At present, abnormal rise of rail potential exists widely in DC traction power systems. The present rail potential model still cannot simulate and explain the abnormal rail potential in the system. Based on the analysis of power distribution with multiple trains running in multiple sections, a dynamic simulation model of rail potential in the whole line is proposed. The dynamic distribution of rail potential and stray current in DC traction power systems when multiple trains run in multiple sections is analyzed, and the impact of traction current distribution on rail potential is evaluated. Simulation results show that the abnormal rise of rail potential during the dynamic operation of the system can be evaluated effectively.

Keywords: DC traction power systems; rail potential; power distribution; stray current

1. Introduction Direct current (DC) traction power systems with an ungrounded scheme are widely used in urban rail transit lines; the running rails are usually used as return conductor. Due to the longitudinal resistance of running rails, there will be a potential between running rails and the ground when traction current flows back to the traction substation, which is called rail potential. Since the running rails cannot be completely insulated to the ground, some of the return current will leak from running rails into the ground and buried metallic structures, which is known as stray current [1]. At present, abnormal rise of rail potential and stray current exists in many urban railway lines, which may exceed the limit value stipulated in relative standards [2,3]. This will cause damage to human safety and electrochemical corrosion of reinforcing bars and buried metal pipelines [4,5]. In order to guarantee human safety and normal operation of equipment, the regulations to control the rail potential in DC traction power systems should be designed [6,7]. Maximum permissible body voltages as a function of time duration in DC traction systems are provided in IEC 62128-1 [3]. For long-term conditions, the accessible rail potential shall not exceed 120 V. Voltage-limiting devices are usually used to reduce the rail potential by connecting the running rails to the substation ground, in which case the current leakage from voltage-limiting devices can reach 800 A, and the stray current increases significantly. Currently, abnormal rise of rail potential cannot be simulated and evaluated by existing models when multiple trains run in multiple sections. Many anomalies in the line cannot be clarified. For example, rail potential in the first and last sections is much higher than that in the middle sections during the operation of the line. In many lines, rail potential in the first and last sections frequently

Energies 2016, 9, 729; doi:10.3390/en9090729 www.mdpi.com/journal/energies Energies 2016, 9, 729 2 of 20 surpasses the stipulated limits [8]. Although the longitudinal resistance of running rails meets the design requirements, rail potential is still high and inconsistent with the theoretical calculation results. Therefore, the dynamic distribution of rail potential needs further studying when multiple trains run in multiple sections. There are some studies on controlling the rail potential in DC traction power systems. Relevant studies show that [9,10] reducing the resistance of running rails, improving the voltage level of power supply systems and shortening the distance between traction substations can drop the rail potential effectively. In addition, different grounding schemes of the system—such as solid grounding, diode grounding and being ungrounded—have an impact on rail potential distribution [10]. A number of simulations and experimental results [9] show that the connecting cables between different railway lines are the main cause of high rail potential. What is more, running modes of trains [11], cross-track regeneration supply [12] and other factors also influence the distribution of rail potential and stray current. However, the impact of power distribution on rail potential when multiple trains run in multiple sections has broadly been ignored, and the abnormal rise of rail potential in the actual line has not yet been clarified. In the last few decades, the simulation model of rail potential and stray current have been studied. The distribution of rail potential is analyzed through finite-element methods and distributed parameter model of return circuit [12–14]. Charalambous [12] uses CDEGS software from Safe Engineering Services & Technologies, Ltd. (SES) in Canada to analyze and assess the change of stray current when the cross-track regeneration supply exists. Meanwhile, a topologically accurate model in cut-and-cover sections of DC traction power systems is established to simulate the dynamic distribution of stray current under the influence of train characteristics, time tabling, headway, etc. [13]. In a lumped parameter model of rail potential and stray current [9,14], a segment of 100 m is sufficient and accurate for the simulation, but the conductance matrixes of the whole line will be very large, and speed of the calculation will be slow. In a distributed parameter model of rail potential and stray current [11,15], the impact of power distribution between source nodes and load nodes on rail potential is always ignored. Traction current transmitting over sections exists widely when multiple trains run in multiple sections in DC traction power systems. As stated by Lin [16] and Falvo [17], in DC traction power systems, all traction substations and trains are in parallel operation for the connectivity of the catenary and running rails in the whole line. Traction current of an accelerating train comes not only from the traction substations of the section, but also from traction substations in other sections or regenerative braking trains with a long distance. The current that regenerative braking trains feed back to the catenary can supply power for accelerating trains in other sections. However, little attention has been paid to the power distribution between source nodes and load nodes. Traction current transmitting with a long distance will have an impact on rail potential. The number of nodes in the network of DC traction power systems is large, and the running mode of the traction substation and train can change, becoming very complicated. Subsequently, current flow has many paths. The current distribution between power source nodes and load nodes cannot be drawn exactly by power flow calculation. When multiple trains run in multiple sections, traction current distribution is affected by the parameters of rectifier units, the train’s running modes, as well as the coincidence degree of the trains, parameters of regenerative braking energy absorbing device (RBEAD), line parameters, etc. At present, related l studies [18,19] generally focus on the method of power flow calculation, whereas little attention has been paid to the power flow tracing in a DC traction power system with multiple nodes. Traction current distribution relations between source nodes and load nodes need to be calculated. The impact of traction current distribution on an abnormal rise of rail potential when multiple trains run dynamically in multiple sections is lack of accurate modeling and analysis. For the realization of fast and accurate calculation of rail potential, a new simulation model is proposed in this paper. The simulation model of rail potential consists of the train movement calculation module, power flow calculation module, power flow tracing calculation module and rail potential calculation module. Energies 2016, 9, 729 3 of 19

Energies 2016, 9, 729 3 of 20 established, thereby clarifying the abnormal rise of rail potential in the actual line. Field tests and simulations show that traction current transferring over sections widely exists in DC traction power systems,This and paper the illustrates abnormal the rise impact of rail of potential power distribution during the on dynamic rail potential. operation A dynamic of the system model can of rail be potentialevaluated Energies basedeffectively. 2016 on, 9 power, 729 distribution when multiple trains run in multiple sections is3 established,of 19 thereby clarifyingestablished, the thereby abnormal clarifying rise the of railabnormal potential rise of in rail the potential actual line. in the Field actual tests line. and Field simulations tests and show that2. System tractionsimulations Description current show transferring that traction over current sections transferring widely over exists sections in widely DC traction exists in powerDC traction systems, power and the systems, and the abnormal rise of rail potential during the dynamic operation of the system can be abnormalA DC rise traction of rail power potential system during mainly the dynamic consists operation of rectifier of units, the system catenaries, can be trains, evaluated running effectively. rails, evaluated effectively. buried conductor, RBEAD, etc. A schematic of a DC traction power system is shown in Figure 1. For 2. System Description the connectivity2. System ofDescription catenaries and running rails in the whole line, all the trains and traction substationsA DC traction areA DCin parallel traction power power operation. system system mainly mainly consists consists ofof rectifier rectifier units, units, catenaries, catenaries, trains, trains, running running rails, rails, buried conductor,buried conductor, RBEAD, RBEAD, etc. A etc. schematic A schematic of a of DC a DC traction traction power power system is is shown shown in Figure in Figure 1. For1. For the connectivitythe ofconnectivity catenaries of and catenaries running and rails running in the rails whole in the line, whole all the line, trains all the and trains traction and traction substations are in parallelsubstations operation. are in parallel operation.

FigureFigure 1. Schematic 1. Schematic of of DC DC tractiontraction power power system. system. Figure 1. Schematic of DC traction power system. The 24‐pulseThe 24 diode‐pulse dioderectifier rectifier units units are are usually usually set inin traction traction substations, substations, which which can provide can provide unidirectional current from the alternating current (AC) side to DC side. In this paper, assuming that unidirectionalThe 24-pulse current diode from rectifier the alternating units are current usually (AC) set side in traction to DC side. substations, In this paper, which assuming can provide that unidirectionalinternal current resistance from of the rectifier alternating units currentis changeless, (AC) output side tocharacteristic DC side. Inof thisthe rectifier paper, units assuming is that internal resistanceshown as lineof Ithe in Figurerectifier 2. Aunits train’s is runningchangeless, mode outputcan actually characteristic be divided intoof the three rectifier stages: units is internalshown as resistanceaccelerating, line I in of Figurecoasting the rectifier 2.and A braking. unitstrain’s is Regenerative changeless,running mode braking output can is characteristic theactually main braking be of divided the method, rectifier intowhich units three will is stages: shown asaccelerating, line I infeed Figure coastingthe current2. A train’sand back braking.to runningthe catenary Regenerative mode and rise can catenary actually braking voltage. be is divided the When main the into catenarybraking three voltage stages:method, rises accelerating, which to will no‐load voltage, U0, of the rectifier units, rectifier units will be out of operation. The output coastingfeed the andcurrent braking. back Regenerativeto the catenary braking and rise is thecatenary main brakingvoltage. method, When the which catenary will feedvoltage the currentrises to back to thecharacteristic catenary is and shown rise as catenary line II in Figure voltage. 2. If the When current the that catenary regenerative voltage braking rises trains to no-loadfeed back voltage, no‐load voltage,to the catenary U0, of cannot the berectifier fully absorbed units, by rectifier surrounding units accelerating will be trains, out thenof operation. catenary voltage The output U , of the rectifier units, rectifier units will be out of operation. The output characteristic is shown characteristic0 will riseis shown continuously. as line In IIorder in Figure to prevent 2. If the the catenary current voltage that fromregenerative exceeding brakingthe safety trainslimit, an feed back asto linethe catenary II inRBEAD Figure cannotis 2usually. If thebe set currentfully in trains absorbed thator traction regenerative by substations. surrounding braking When accelerating the trains catenary feed voltage trains, back at to thenRBEAD the catenary exceeds cannotvoltage bewill fully rise absorbedcontinuously.its trigger voltage, by surrounding In U ordermax, the RBEADto prevent accelerating will start the tocatenary trains, absorb the then voltage remaining catenary from energy voltage exceeding so as to will ensure the rise safetythat continuously. the limit, an In order tocatenary prevent voltage the catenarydoes not exceed voltage Umax. from In the exceeding urban railway the lines safety of China, limit, the anRBEAD RBEAD is usually is usually set at set in RBEAD istraction usually substations, set in trains and its or output traction characteristic substations. is shown When as line IIIthe in catenaryFigure 2. The voltage output characteristicat RBEAD exceeds trains or traction substations. When the catenary voltage at RBEAD exceeds its trigger voltage, Umax, its triggerof voltage, traction substation Umax, the with RBEAD RBEAD will is shown start in to Figure absorb 2. the remaining energy so as to ensure that the the RBEAD will start to absorb the remaining energy so as to ensure that the catenary voltage does not catenary voltage does not exceed Umax. In the urban railway lines of China, the RBEAD is usually set at exceedtractionU substations,max. In the urbanand its railway output characteristic lines of China, is theshown RBEAD as line is III usually in Figure set at 2. tractionThe output substations, characteristic and itsof traction output characteristicsubstation with is RBEAD shown as is lineshown III inin FigureFigure2 2.. The output characteristic of traction substation with RBEAD is shown in Figure2.

Figure 2. Output characteristic of traction substation with regenerative braking energy absorbing device (RBEAD).

Figure 2.2. Output characteristic of traction substation with regenerative braking energy absorbing device (RBEAD).

Energies 2016, 9, 729 4 of 20 Energies 2016, 9, 729 4 of 19

The train’s traction and regenerative braking curve are constrained by the traction network voltage. With the dropdrop ofof tractiontraction networknetwork voltage,voltage, train tractivetractive and regenerative braking capacity will be limited [[16].16]. In this paper, the models are simplifiedsimplified without considering the effect of traction network voltagevoltage on on train train traction traction and and braking braking performance, performance, and itand is assumed it is assumed that the that current the networkcurrent voltagenetwork can voltage meet can the needsmeet the of theneeds train of traction the train and traction braking and performance. braking performance. In the dynamic In the operation dynamic ofoperation DC traction of DC power traction systems, power thesystems, voltage the and voltage current and at current each node at each of tractionnode of substationtraction substation or train canor train be obtained can be obtained by power by flow power calculation flow calculation at each moment. at each moment. At present, At therepresent, are there many are pieces many of powerpieces of flow power calculation flow calculation software forsoftware DC traction for DC power traction systems power to systems obtain the to obtain current the and current voltage and at eachvoltage node at each [20,21 node]. Power [20,21]. flow Power calculation flow calculation software for software DC traction for DC power traction systems power mainly systems consists mainly of aconsists train movement of a train calculationmovement modelcalculation and powermodel flowand power calculation flow model.calculation With model. the route With parameters, the route trainparameters, characteristics, train characteristics, and scheduled and timetables, scheduled etc., timetables, the relation etc., between the relation power between and time, power as well and as thetime, relation as well between as the positionrelation andbetween time ofposition the trains, and can time be of obtained the trains, by the can train be movementobtained by calculation the train model.movement Then, calculation parameters model. such Then, as the current,parameters voltage such and as the power current, of traction voltage substations and power and of trainstraction at eachsubstations moment and can trains be attained at each throughmoment the can power be attained flow calculation through the model. power flow calculation model. As shown in in Figure Figure 1,1, the the return return circuit circuit consists consists of of running running rails, rails, the the buried buried conductor conductor and and the theground. ground. Up Upline line rail railand and down down line line rail rail are are connected connected by by cross cross-bonding‐bonding cables cables at at intervals of 400 m∼500 m. m. The buried conductor is set in the ballast bed to collect stray current.current. The distributed parameters ofof the the return return circuit circuit must must be convertedbe converted to lumped to lumped parameters parameters for the for calculation the calculation of power of flow.power For flow. the For realization the realization of fast and of fast accurate and accurate calculation calculation of rail potential of rail potential in this paper, in this the paper, segment the ofsegment return of circuit return between circuit tractionbetween substation traction substation and train and is converted train is converted to double πtocircuit double in π thecircuit power in flowthe power calculation. flow calculation. In order toIn ensureorder to the ensure accuracy the accuracy of the lumped of the lumped parameters parameters in double in doubleπ circuit, π thecircuit, parameters the parameterscan be calculated can be calculated by the distributed by the distributed parameter parameter model [22]. model The model [22]. inThe power model flow in calculationpower flow is calculation shown in Figureis shown3. in Figure 3.

Figure 3. Lumped parameter model in power flowflow calculation.

As shownshown inin Figure Figure3 ,3, the the train train is equivalentis equivalent to ato constant a constant power power model model in the in upthe line up orline down or down line, suchline, assuchP2 andas PP2 4and. P2 andP4. P24 andare the P4 powerare the of power trains atof this trains moment. at thisy cNmoment.is the equivalent ycN is the resistance equivalent of theresistance traction of substation. the tractionzr1 substation.and zs1 are thezr1 and equivalent zs1 are resistancesthe equivalent of the resistances rail and the of buried the rail conductor. and the yburiedg1 is the conductor. equivalent yg1 conductance is the equivalent of rail conductance to buried conductor. of rail to buriedyp1 is conductor. the equivalent yp1 is conductance the equivalent of buriedconductance conductor of buried to the conductor ground. Basedto the onground. the lumped Based parameteron the lumped model parameter at each moment model at shown each inmoment Figure shown3, the nodal in Figure voltage 3, the equations nodal voltage can be equations obtained, can and be the obtained, power and flow the can power be calculated flow can by be iterativecalculated methods. by iterative methods. For thethe realization realization of of fast fast and and accurate accurate calculation calculation of the of rail the potential, rail potential, the return the circuit return between circuit tractionbetween substation traction andsubstation train is convertedand train tois lumped converted parameters to lumped in the parameters power flow in calculation. the power In orderflow tocalculation. obtain the In rail order potential to obtain and the stray rail potential current at and each stray position current and at each evaluate position the impactand evaluate of power the distributionimpact of power on rail distribution potential, the on power rail potential, flow tracing the ofpower the system flow tracing should beof calculated,the system and should the rail be potentialcalculated, should and the be calculatedrail potential by the should distributed be calculated parameter by modelthe distributed of the return parameter circuit. model of the return circuit.

Energies 2016, 9, 729 5 of 20 Energies 2016, 9, 729 5 of 19 3. Power Flow Tracing of DC Traction Power Systems 3. Power Flow Tracing of DC Traction Power Systems When multiple trains run in multiple sections, the current of each node and branch can be When multiple trains run in multiple sections, the current of each node and branch can be drawn based on power flow calculation at a certain moment. However, it is difficult to obtain the drawn based on power flow calculation at a certain moment. However, it is difficult to obtain the traction current distribution between source nodes and load nodes due to the complexity of the current traction current distribution between source nodes and load nodes due to the complexity of the flow path. The source nodes and load nodes on the catenary at one moment are shown in Figure4. current flow path. The source nodes and load nodes on the catenary at one moment are shown in The current that the regenerative braking train at node 14 feeds back to the catenary can flow to node 7. Figure 4. The current that the regenerative braking train at node 14 feeds back to the catenary can flow to However, due to the complexity of the path and nodes, it is difficult to calculate the current distribution node 7. However, due to the complexity of the path and nodes, it is difficult to calculate the current between nodes 14 and 7. Therefore, it is necessary to trace the power flow in order to obtain the traction distribution between nodes 14 and 7. Therefore, it is necessary to trace the power flow in order to obtain current distribution coefficients between the source nodes and the load nodes. the traction current distribution coefficients between the source nodes and the load nodes. Assuming that there are N nodes on the line at a moment, the traction current distribution Assuming that there are N nodes on the line at a moment, the traction current distribution coefficient kij from the current of source node i, IGi, to current of load node j, ILj, is shown in Equation (1): coefficient kij from the current of source node i, IGi, to current of load node j, ILj, is shown in Equation (1):

IkIijIij = ij Ljkij ILj (1)(1)

IIijij isis the the current from source node i to load node j. The traction current distribution coefficientcoefficient K of allall source source nodes nodes and and all all load load nodes nodes is calculated is calculated based based on the on power the power flow tracing flow tracing of the DC of tractionthe DC powertraction system. power system.

Figure 4. Source nodes and load nodes on the catenary atat one moment.

An analytic model of currentcurrent distributiondistribution in a DCDC tractiontraction powerpower systemsystem is establishedestablished in thisthis paper.paper. InIn thisthis way,way, thethe currentcurrent distributiondistribution relationrelation cancan bebe accuratelyaccurately calculatedcalculated betweenbetween the power sources andand thethe loadsloads inin orderorder toto obtainobtain thethe currentcurrent andand powerpower supplysupply areas.areas. AA=() (aa ) N Assuming thatthat ijij NN N× Nisis the the downstream downstream distribution distribution matrix matrix of of a a power power network with N nodes and I is the current supply from node i to node j by i-j branch, the downstream distribution nodes and Iijij is the current supply from node i to node j by i‐j branch, the downstream distribution matrix cancan bebe expressedexpressed asas follows:follows: II, i jI , 0  ij−I Tj/I , i 6= ij j, I > 0   ij Tj ij aij1, aij ij = 1, i = j (2)(2)   0, else   0, else

where, i, j=1, 2, 3, …, N. ITj is the total injected current of node j. where, i, j = 1, 2, 3, . . . , N. I is the total injected current of node j. As shown in EquationTj (2), the downstream distribution matrix includes the connection As shown in Equation (2), the downstream distribution matrix includes the connection relationship relationship between nodes, current flows in branches and the injected current values. The between nodes, current flows in branches and the injected current values. The downstream distribution downstream distribution matrix is an unsymmetrical matrix. matrix is an unsymmetricalT matrix. T T I [II , ,..., I ] , I T [,II ,...,] I and I T[,II ,...,] I respectively Trepresent the ILLLLLN= [12IL1, IL2,..., ILN]GGGGN, IG 12= [IG1, IG2,..., IGNTTTTN] and12IT = [IT1, IT2,..., ITN] respectively representN‐dimension the N load-dimension current loadcolumn current vector, column source vector, current source column current vector column and vector total and injected total injectedcurrent column vector. Based on Kirchhoff’s current law, the input current of each node is equal to the

output current, so the vector IL and AIT can be expressed as:

NNI aI Iij I ij Tj TiI Tj jjjiI11,,0ij Tj (3) N IIITi ij  Li jjiI1, ,ij  0

Energies 2016, 9, 729 6 of 20 current column vector. Based on Kirchhoff’s current law, the input current of each node is equal to the output current, so the vector IL and AIT can be expressed as:

N N  −I  ∑ a I = I + ∑ ij I ij Tj Ti ITj Tj j=1 j=1,j6=i,Iij>0 N (3) = ITi − ∑ Iij = ILi j=1,j6=i,Iij>0 where, i = 1, 2, . . . , N. It can be expressed as: AIT = IL (4)

ILL = diag(IL1, IL2, ... , ILN), IGG = diag(IG1, IG2, ... , IGN) and ITT = diag(IT1, IT2, ... , ITN) are the N × N diagonal matrix, E = [1, 1, . . . , 1]T is the N-dimension unit column vector. The relation of IT and ITT can be expressed as Equation (5), and the relation of IG and IGG can be expressed as Equation (6). T IT = [IT1, IT2,..., ITN] = ITT E (5) T IG = [IG1, IG2,..., IGN] = IGGE (6)

The analytic relation between IG and IL can be expressed as Equation (7).

−1 −1 −1 IG = IGGE = IGG (ITT) IT = IGG (ITT) A IL (7)

That is, −1 −1 IG = IGG(ITT) A IL (8) The current distribution between power sources and loads is presented in Equation (8). −1 −1 IGG(ITT) A is the traction current distribution coefficients. K = (kij), then,

−1 −1 K = IGG(ITT) A (9)

So, N IGi = ∑ kij ILj (10) j=1 where, Iij = kij ILj is the current that the power source at node i supplies for the load at node j. Based on the current distribution principle, the sum of current distribution coefficients of each power source to all loads should be 1. The matrix of traction current distribution coefficients meet the relation presented in Equation (11):

T T −1 −1 T −1 E K = E IGG ITT A = (IG) (AITT) (11)

T In order to obtain the property of K, the relation between (IG) and AITT should be solved. Assuming that B = AITT, the element at i-th row and j-th column is:   (ITT)1j    (I )   TT 2j  Bij = ai1 ai2 ... aiN   (12)  ...  (ITT)Nj Energies 2016, 9, 729 7 of 20

ITT is a diagonal matrix. Based on ITT and IT, Bij = aijBjj = aij ITj. Based on Equation (2), we can deduce that:  −Iij, i 6= j, Iij > 0  Bij = ITj, i = j (13)  0, else Energies 2016, 9, 729 7 of 19 Therefore, the sum of elements in j-th column of matrix B is:

NNj 1j−1 NB BB BN Bij= sjB + jjB + sj B (14) is∑11ij ∑ sj jj sj  1∑ sj (14) i=1 s=1 s=j+ 1 By the definition of Bij and ITj, we can know that: By the definition of Bij and ITj, we can know that: NNj 1 BBIIij sj Gj  ij  BI sj  Gj (15) N j−1 N is11 iIsj ,01ij  ∑ Bij = ∑ Bsj + IGj + ∑ Iij + ∑ Bsj = IGj (15) That is: i=1 s=1 i,Iij>0 s=j+1

T I  EAIT (16) That is: GTT ( )T = T TT11 T IG 1 E TAITT (16) Then, E KEIIAGG TT I G AI TT   E, and the sum of each column element is 1, T = T −1 −1 = ( )T ( )−1 = T whichThen, compliesE K withE ItheGG IcurrentTT A distributionIG AI principle.TT E , and the sum of each column element is 1, whichBased complies on the with above the method, current distribution the current distribution principle. of all power sources and loads at a moment can beBased obtained on the in above DC traction method, power the current systems. distribution of all power sources and loads at a moment can be obtained in DC traction power systems. 4. Modeling of Rail Potential 4. Modeling of Rail Potential In DC traction power systems, multiple trains run in multiple sections. The current at load node In DC traction power systems, multiple trains run in multiple sections. The current at load comes from many power sources, and each power source supplies power to many load nodes node comes from many power sources, and each power source supplies power to many load nodes simultaneously. On the basis of traction current distribution coefficients, the line consists of multiple simultaneously. On the basis of traction current distribution coefficients, the line consists of multiple power supply sections. The return circuit is a linear model, and it follows linear superposition power supply sections. The return circuit is a linear model, and it follows linear superposition theorem. theorem. The rail potential with multiple trains running in multiple sections can be obtained by The rail potential with multiple trains running in multiple sections can be obtained by superimposing superimposing the rail potential of each power supply section in the line, assuming that the total the rail potential of each power supply section in the line, assuming that the total length of the line is L length of the line is L and there are N nodes in the line. The direction of 0 to L is the positive value of and there are N nodes in the line. The direction of 0 to L is the positive value of current in running rails. current in running rails. The positive current of the traction substation and train indicates that the The positive current of the traction substation and train indicates that the current is injected to the rail. current is injected to the rail. The power supply section from j‐th source node to i‐th load node is The power supply section from j-th source node to i-th load node is shown in Figure5a. The traction shown in Figure 5a. The traction current that IGj supplies for ILi is IkIjijiGj , the position of j‐th current that IGj supplies for ILi is Iji = kji IGj, the position of j-th source node is xj and the position of source node is xj and the position of i‐th load node is xi. i-th load node is xi. Assuming that a single power supply section exists in xi to xj, and the supply current is kjiIGj, Assuming that a single power supply section exists in xi to xj, and the supply current is kjiIGj, thenthen thethe railrail potentialpotential inin thethe lineline isis divideddivided intointo threethree continuous continuous changechange sections.sections. TheThe distributeddistributed parameterparameter modelmodel ofof thethe returnreturn circuitcircuit atat anyany point point between betweenx xto to ( (xx+ + dxdx)) isis shownshown inin FigureFigure5 5b.b.

(a) (b)

FigureFigure 5.5. ReturnReturn circuitcircuit modelmodel withwith aa powerpower supplysupply section,section, ((aa)) AA powerpower supplysupply sectionsection existsexists inin thethe line;line; ((bb)) TheThe modelmodel ofof thethe returnreturn circuitcircuit inin dxdx..

As shown in Figure 5b, urij(x), usij(x), irij(x) and isij(x) respectively represent rail potential, potential of buried conductor to the ground, current in rail and stray current in buried conductor at position x. Based on Kirchhoff’s law, these parameters meet the following equation:

Energies 2016, 9, 729 8 of 20

As shown in Figure5b, urij(x), usij(x), irij(x) and isij(x) respectively represent rail potential, potential of buried conductor to the ground, current in rail and stray current in buried conductor at position x. Based on Kirchhoff’s law, these parameters meet the following equation:  durij(x)  dx = −Rr · irij(x)  du (x)  sij = −R · i (x) dx s sij (17) dirij(x)  dx = −Gs · urij(x) + Gs · usij(x)  ( )  disij x dx = Gs · urij(x) − (Gp + Gs) · usij(x) where, Rr is the longitudinal resistance of the running rail, Ω/km; Rs is the longitudinal resistance of the buried conductor, Ω/km; Gs is the longitudinal insulation conductance of rails to the buried conductor, S/km; Gp is the longitudinal insulation conductance of the buried conductor to the ground, S/km. Due to the different boundary conditions of three sections, the undetermined coefficient of rail potential is different in the three sections. The parameters in section 0 to xi can be expressed as follows:

   −αx  urij1(x) Cij1e  u (x)   C eαx   sij1  = ×  ij2  ≤ <   H  −βx  , 0 x xi (18)  irij1(x)   Cij3e  βx isij1(x) Cij4e

The parameters in section xi to xj can be expressed as follows:

   −αx  urij2(x) Cij5e  u (x)   C eαx   sij2  = ×  ij6  ≤ <   H  −βx  , xi x xj (19)  irij2(x)   Cij7e  βx isij2(x) Cij8e

The parameters in section xj to L can be expressed as follows:

   −αx  urij3(x) Cij9e  u (x)   C eαx   sij3  = ×  ij10  ≤ ≤   H  −βx  , xj x L (20)  irij3(x)   Cij11e  βx isij3(x) Cij12e   1 1 1 1  α2 α2 β2 β2   1 − G R 1 − G R 1 − G R 1 − G R  Gp α Gpα Gp β Gp β where, H =  s r s r s r s r , rα = − − , rβ = − − ,  α − α β − β  α Rr GsRr β Rr GsRr  Rr Rr Rr Rr  rα −rα rβ −rβ r r q 2 q 2 m+n (m−n) m+n (m−n) α = 2 + 4 + mk, β = 2 − 4 + mk, m = GsRr, n = Rs(Gs + Gp), and k = RsGs. Cij1~Cij12 are the undetermined coefficients that can be obtained by the boundary conditions in each section. Energies 2016, 9, 729 9 of 20

Cij1~Cij12 can be obtained by Equation (21) as follows:  i (0) = 0  rij1  − ( ) + ( ) =  irij1 xi irij2 xi kji IGj   urij1(xi) = urij2(xi)   irij2(xj) − irij3(xj) = kji IGj   u (x ) = u (x )  rij2 j rij3 j  i (L) = 0 rij3 (21)  isij1(0) = 0   i (x ) = i (x )  sij1 i sij2 i  u (x ) = u (x )  sij1 i sij2 i   isij2(xj) = isij3(xj)   usij2(xj) = usij3(xj)   isij3(L) = 0

Then, the urij(x), usij(x), irij(x) and isij(x) can be obtained when a power supply section from xi to xj exists in the line. There may be up to N × N power supply sections in the line, and the return circuit is a linear model, which meets the linear superposition theorem. At a certain time, the rail potential with multiple trains running in multiple sections can be obtained by Equation (22), and other parameters can be attained by Equations (23)–(25). N N ur(x) = ∑ ∑ urij(x) (22) i=1 j=1

N N up(x) = ∑ ∑ upij(x) (23) i=1 j=1

N N ir(x) = ∑ ∑ irij(x) (24) i=1 j=1

N N is(x) = ∑ ∑ isij(x) (25) i=1 j=1 Then, the distribution of rail potential and stray current can be simulated with multiple trains running in multiple sections at each moment. The flow chart of simulation on rail potential dynamic distribution in this paper is shown in Figure6. It mainly consists of four modules: the train movement calculation module, the power flow calculation module, the power flow tracing module and the rail potential module. Based on the route parameters, train characteristics, and scheduled timetables, the power and position of each train changing with time can be obtained by the train movement calculation module. At each time point, the segment of return circuit between traction substation and train is converted to lumped parameters. The voltage and current of each node in the network can be calculated by the power flow calculation module. For the nonlinear output characteristics of traction substations with RBEAD, the iterative method is usually used in power flow calculations. In each iteration, the operation state of the traction substation may change based on the catenary voltage. Based on the actual characteristic of the line, if more than one traction substation needs to change the operation state, only one traction substation is permitted to change the operation state in this iteration. The iteration will continue until the i-th step when the power-balance equations ∆Pn < ε1 and mismatch voltage of each node ∆Un < ε2 are all fulfilled. Then, distribution of rail potential at this time point can be drawn by the power flow tracing module and rail potential module. The simulation will continue until the time exceeds the limit tend. By this model, the dynamic distribution of the rail potential when multiple trains run in multiple sections with different train diagrams and line parameters can be simulated. Energies 2016, 9, 729 10 of 20 Energies 2016, 9, 729 10 of 19

Pn 1 ?

U n  2 ?

FigureFigure 6. 6.FlowFlow chart chart of the of the simulation. simulation.

5. Simulation Results 5. Simulation Results Take the first‐stage project of Guangzhou Metro Line 2 as an example to simulate the dynamic Take the first-stage project of Guangzhou Metro Line 2 as an example to simulate the dynamic distribution of rail potential based on the method proposed in this paper. The parameters in the distributionsimulation are of shown rail potential in Table based1. on the method proposed in this paper. The parameters in the simulation are shown in Table1. Table 1. Simulation Parameters. Table 1. Simulation Parameters. Parameter Value Equivalent resistance of Parameterthe traction substation ycN/(Ω) 0.008 Value No‐load voltage of the traction substation U0/(V) 1593 Equivalent resistance of the traction substation ycN/(Ω) 0.008 max Trigger voltageNo-load of regenerative voltage of braking the traction energy substation absorbingU0 /(V)device U /(V) 1800 1593 Trigger voltageLongitudinal of regenerative resistance braking of the energy catenary absorbing Rw/(Ω/km) device Umax/(V)0.02 1800 LongitudinalLongitudinal resistance resistance of the of the running catenary railR wR/(r/(ΩΩ/km)/km) 0.02 0.02 R LongitudinalLongitudinal resistance resistance of the of buried the running conductor rail r /(RsΩ/(Ω/km)/km) 0.02 0.02 Longitudinal resistance of the buried conductor Rs/(Ω/km) 0.02 Longitudinal insulation conductance of rails to buried conductor Gs/(S/km) 1/15 Longitudinal insulation conductance of rails to buried conductor Gs/(S/km) 1/15 Longitudinal insulation conductance of the buried conductor to the ground Gp/(S/km) 1/3 Longitudinal insulation conductance of the buried conductor to the ground Gp/(S/km) 1/3 Mismatch power in the iteration ξ1/(kW) 10−−4 4 Mismatch power in the iteration ξ1/(kW) 10 −−4 4 MismatchMismatch voltage voltage of of each each node node in in the the iteration iteration ξξ22/(V)/(V) 1010

In the first‐stage project of Guangzhou Metro Line 2, cross‐bonding cables exist between the In the first-stage project of Guangzhou Metro Line 2, cross-bonding cables exist between the running rails of up line and down line. The catenaries of up line and down line are connected by running rails of up line and down line. The catenaries of up line and down line are connected by positive buses at the position of traction substation, and they are independent in the interval. There positiveare ten stations buses at in the the position first‐stage of project traction of substation, Guangzhou and Metro they Line are independent2. The locations in theof the interval. stations There are are tenshown stations in Table in the2. first-stage project of Guangzhou Metro Line 2. The locations of the stations are shown in Table2.

Energies 2016, 9, 729 11 of 19

Table 2. Locations of the Stations. Energies 2016, 9, 729 11 of 20 Huijiang Station Guangzhou Nan(S1) Shibi Nanpu Luoxi (S3) (S2) Table 2. Position/m 0 Locations1020 of the Stations.3367 5800 6994 Station Nanzhou Dongxiao Nan(S4) Jiangtailu Changgang Jiangnan Xi(S5) Station Guangzhou Nan(S ) Shibi Huijiang (S ) Nanpu Luoxi (S ) Position/m 9387 1 10,291 12,2652 13,171 3 14,042 Position/m 0 1020 3367 5800 6994 Station Nanzhou Dongxiao Nan(S4) Jiangtailu Changgang Jiangnan Xi(S5) FivePosition/m traction substations, 9387 S1 to S5, 10,291are respectively 12,265 set in Guangzhou 13,171 Nan Station, 14,042 Huijiang Station, Luoxi Station, Dongxiao Nan Station and Jiangnan Xi Station. Other stations are train stationsFive without traction rectifier substations, unitsS1 andto S 5RBEAD., are respectively The train set headway in Guangzhou is 180 Nan s in Station, the up Huijiangline and Station,down line, andLuoxi the Station, dwell Dongxiaotime of each Nan train Station at each and Jiangnan station is Xi 30 Station. s. The Othertime step stations in the are simulation train stations is without0.5 s. In the simulation,rectifier units the and parameters RBEAD. The of trainall trains headway in line is 180 are s inassumed the up line to andbe the down same. line, According and the dwell to the operationtime of each characteristics train at each stationof metro is 30 systems, s. The time the steptraffic in thewill simulation be regular is from 0.5 s. Inthe the time simulation, when the the train diagramparameters is stable. of all trains In the in simulation, line are assumed the train to be headway the same. is According 180 seconds to the in operationthe up line characteristics and down line, andof metro a train systems, needs 1078 the traffic s to run will from be regular the departure from the station time when to the the terminal train diagram station. is stable.There will In the be six trainssimulation, at most the in train the headwaywhole line. is 180Therefore, seconds from in the the up time line andof seventh down line, train and starting a train from needs the 1078 departure s to station,run from each the departure180 s demonstrate station to t the cyclic terminal behavior. station. Thus, There the will simulation be six trains can at be most carried in the out whole from line. 1080 s toTherefore, 1260 s. In from order the to time compare of seventh the traincurrent starting distribution from the in departure different station, situations, each 180the ssimulation demonstrate time t in thiscyclic paper behavior. is from Thus, 0 s to the 1260 simulation s. can be carried out from 1080 s to 1260 s. In order to compare the currentThe distribution position‐time in different curve of situations, each train the in simulationthe course timeof simulation in this paper is shown is from in 0 Figure s to 1260 7. s. The position-time curve of each train in the course of simulation is shown in Figure7.

FigureFigure 7. 7.Train Train diagram diagram in in the the simulation. simulation.

5.1.5.1. Model Model Validation ForFor the modelmodel validation, validation, field field tests tests of the of railthe potential rail potential were carried were carried out on 16 out August on 16 2016, August when 2016, whentwo trains two weretrains tested were in tested the line. in Inthe the line. field In tests, the field two diagrams tests, two of diagrams the trains wereof the used. trains In diagramwere used. 1, In diagramtrain 1 accelerated 1, train 1 from accelerated Dongxiao from Nan Dongxiao Station in theNan up Station line at 26in s,the while up trainline at 2 accelerated26 s, while from train 2 acceleratedNanpu Station from in theNanpu down Station line at in the the same down time. line In diagramat the same 2, train time. 1 accelerated In diagram from 2, train Dongxiao 1 accelerated Nan fromStation Dongxiao in the up Nan line Station at 0 s, and in itthe began up line to brake at 0 s, at and 38 s;it trainbegan 2 acceleratedto brake at from38 s; Nanputrain 2 Stationaccelerated in the from down line at 26 s. The regenerative braking train was in high coincidence with the accelerating train Nanpu Station in the down line at 26 s. The regenerative braking train was in high coincidence with in diagram 2. The simulation results comparing with field test results in two diagrams are shown in the accelerating train in diagram 2. The simulation results comparing with field test results in two Figure8. The field test point is set at Nanpu Station. diagrams are shown in Figure 8. The field test point is set at Nanpu Station.

EnergiesEnergies 20162016,, 99,, 729729 1212 of 2019

50 100 Simulation result Simulation result Energies 2016, 9, 729 Field test result Field test12 result of 19 40 80

50 100 Simulation result Simulation result 30 Field test result 60 Field test result 40 80

20 40 30 60 Rail potential(V) Rail potential(V) 10 20 20 40

0 Rail potential(V)

Rail potential(V) 0 10 20

-10 0 20 40 60 80 100 120 -20 0 00 20 40 60 80 100 120 Time(s) Time(s)

-10 0 20 40 60 80 100 120 -20 (a) 0 20 40 (60b) 80 100 120 Time(s) Time(s) Figure 8. Simulation model validation, (a) Rail potential in diagram 1; (b) Rail potential in diagram 2. Figure 8. Simulation model(a) validation, (a) Rail potential in diagram 1; (b) Rail(b potential) in diagram 2.

As shownFigure in 8. Figure Simulation 8, the model simulation validation, results(a) Rail potential show thein diagram same trend1; (b) Rail with potential the field in diagram test results. 2. The maximumAs shown value in of Figure the rail8, potential the simulation in field results tests is show higher the than same the trendsimulation with theresults. field This test situation results. As shown in Figure 8, the simulation results show the same trend with the field test results. The Themay maximum be causedvalue by the of increase the rail potential of rail longitudinal in field tests resistance is higher thanduring the the simulation operation. results. In diagramThis situation 1, two maymaximum be caused value by the of the increase rail potential of rail in longitudinal field tests is higher resistance than duringthe simulation the operation. results. This In situation diagram 1, trainsmay accelerated be caused atby thethe increasesame time. of rail There longitudinal is no regenerative resistance during braking the operation. current inIn thediagram line, 1, and two the two trains accelerated at the same time. There is no regenerative braking current in the line, tractiontrains current accelerated of the at trainthe same was time.supplied There by is nothe regenerative traction substations braking current with ain short the line, distance. and the The and the traction current of the train was supplied by the traction substations with a short distance. maximumtraction rail current potential of the at Nanputrain was Station supplied in the by fieldthe traction test was substations 42.9 V. When with the a short train distance.in regenerative The Thebraking maximummaximum was in railrail high potentialpotential coincidence at Nanpu with Station Station the accelerating in in the the field field test train test was was in 42.9 diagram 42.9 V. V. When When 2, thethe the railtrain train potential in inregenerative regenerative changed brakingsignificantly.braking was was in As high inshown high coincidence coincidencein Fig. 8b, with trainwith the the2 acceleratingaccelerated accelerating from train train inNanpuin diagram Station 2, 2, the the in rail railthe potential potentialdown linechanged changed at 26 s. significantly.Trainsignificantly. 1 in the As up shown Asline shown began in Fig.in to Fig. 8b,brake 8b, train train at 38 2 2 accelerated s,accelerated and fed the fromfrom regenerative Nanpu Station Station energy in in the theback down down to line the line atcatenary. 26 at s. 26 s. TrainThe maximumTrain 1 in the1 in up thevalue lineup ofline began rail began potential to braketo brake in at theat 38 38 field s, s, and and test fed fed was thethe 94.2 regenerativeregenerative V, occurring energy energy at 42 back s back when to tothe train the catenary. catenary. 1 was in Theregenerative maximumThe maximum braking. value value of In railof diagram rail potential potential 2, the in in the railthe field fieldpotential test was was rose 94.2 94.2 starkly V, V, occurring occurring when at the 42 at saccelerating 42 when s when train train 1 trainwas 1 in wasand regenerative braking. In diagram 2, the rail potential rose starkly when the accelerating train and inregenerative regenerative braking braking. train In were diagram in high 2, the coincidence. rail potential From rose the starkly comparison when the of acceleratingthe simulation train results and regenerative braking train were in high coincidence. From the comparison of the simulation results regenerativeand field test braking results, trainwe can were see inthat high the coincidence. simulation model From thecan comparisonevaluate the of rail the potential simulation effectively results and field test results, we can see that the simulation model can evaluate the rail potential effectively andin the field actual test line. results, we can see that the simulation model can evaluate the rail potential effectively in the actualin the line. actual line. 5.2. Effect of Power Distribution on Rail Potential 5.2. Effect5.2. Effect of Power of Power Distribution Distribution on on Rail Rail Potential Potential DuringDuring the simulationthe simulation time, time, the the dynamic dynamic distribution distribution of the railrail potential potential in in the the line line is asis asshown shown During the simulation time, the dynamic distribution of the rail potential in the line is as shown in Figurein Figure 9. 9. in Figure9.

(a) (b) (a) (b) Figure 9. Cont.

Energies 2016, 9, 729 13 of 20 Energies 2016, 9, 729 13 of 19

Energies 2016, 9, 729 13 of 19

(c) (d)

Figure 9. Dynamic distribution of rail potential, (a) Dynamic distribution of rail potential; (b) FigureDistribution 9. Dynamic of rail potential distribution (time‐position); of rail potential, (c) Distribution (a) Dynamic of rail potential distribution (Rail potential of rail‐position); potential; (b) Distribution(d) Distribution of rail of rail potential potential (time-position); (rail potential (‐ctime).) Distribution of rail potential (Rail potential-position); (d) Distribution of rail potential(c) (rail potential-time). (d) As shown in Figure 9, a significant difference between rail potential exists in different times and Figure 9. Dynamic distribution of rail potential, (a) Dynamic distribution of rail potential; (b) positions. The maximum rail potential in the line is 130.8 V, occurring in 1230 m at 1231 s. As shown As shownDistribution in Figure of rail9, apotential significant (time‐position); difference (c) Distribution between of rail rail potential potential (Rail exists potential in different‐position); times and in Figure 9c, the simulation result of rail potential is consistent with the distribution in the actual positions. The(d) maximumDistribution of rail rail potential potential (rail in potential the line‐time). is 130.8 V, occurring in 1230 m at 1231 s. As shown in Figureline.9c, The the rail simulation potential resultin the offirst rail and potential last sections is consistent is much higher with the than distribution that in the middle in the actualsections line. during the operation of the line. The maximum and minimum values of the rail potential in each The rail potentialAs shown in the in Figure first and 9, a significant last sections difference is much between higher rail than potential that exists in the in middledifferent sectionstimes and during sectionpositions. and traction The maximum substation rail arepotential shown in inthe Table line is 3. 130.8 V, occurring in 1230 m at 1231 s. As shown the operationin Figure of the9c, the line. simulation The maximum result of andrail potential minimum is consistent values of with the railthe distribution potential in in each the actual section and tractionline. substation TheTable rail 3. arepotential Maximum shown in and inthe Table firstMinimum and3. last Rail sections Potential is muchin Each higher Section than and that Traction in the Substation. middle sections during the operation of the line. The maximum and minimum values of the rail potential in each Table 3. Maximum andSection1 Minimum Rail PotentialSection2 in Each SectionSection3 and Traction Substation.Section4 Positionsection and tractionS1 substation are shownS2 in Table 3. S3 S4 S5 (3367 m) (3627 m) (3297 m) (3751 m) Maximum (V) 103.3 Section1130.8 83.7 Section288.6 80.6 Section398.0 72.3 Section4119.3 107.3 Position TableS 3. Maximum and MinimumS Rail Potential in EachS Section and TractionS Substation. S Minimum (V) −1 102.8 −(3367 m)135.2 −2 85.9 −(362796.7 m) −92.03 −(329792.0 m) −81.84 −(3751127.9 m) −116.55 Section1 Section2 Section3 Section4 MaximumPosition (V) 103.3 S1 130.8 83.7S2 88.6 80.6S3 98.0S4 72.3 119.3S5 107.3 (3367 m) (3627 m) (3297 m) (3751 m) MinimumAs (V) clearly−102.8 shown in−135.2 Table 3, the−85.9 rail potential−96.7 in the−92.0 first and− 92.0last sections−81.8 is higher−127.9 than that−116.5 in Maximum (V) 103.3 130.8 83.7 88.6 80.6 98.0 72.3 119.3 107.3 the middleMinimum sections. (V) − 102.8With − the 135.2same − parameters85.9 − of96.7 the − return92.0 − circuit,92.0 the − length81.8 − of section127.9 − 1 is116.5 3367 m, and the maximum rail potential in section 1 is 130.8 V. By comparison, the length of section 2 is 3627 As clearly shown in Table3, the rail potential in the first and last sections is higher than that in m, whereasAs clearlythe maximum shown in rail Table potential 3, the rail is potential88.6 V. From in the 1080 first sand to last1260 sections s, the maximumis higher than rail that potential in the middle sections. With the same parameters of the return circuit, the length of section 1 is 3367 m, in sectionthe middle 1 is 130.8sections. V, occurringWith the same in 1230 parameters m at 1231 of the s, returnand the circuit, maximum the length rail ofpotential section 1 in is section3367 m, 2 is and the maximum rail potential in section 1 is 130.8 V. By comparison, the length of section 2 is 3627 m, 88.6 andV, occurring the maximum in 6756 rail mpotential at 1178 in s. section Variation 1 is 130.8of the V. rail By potentialcomparison, at typicalthe length moments of section and 2 is positions 3627 whereasare shownm, the whereas maximum in Figure the maximum 10. rail potential rail potential is 88.6 is 88.6 V. From V. From 1080 1080 s tos to 1260 1260 s,s, the maximum maximum rail rail potential potential in section 1in is section 130.8 1 V, is occurring 130.8 V, occurring in 1230 in m 1230 at 1231 m at s, 1231 and s, the and maximum the maximum rail rail potential potential in in section section 2 is is 88.6 V, 88.6 V, occurring in 6756 m at 1178 s. Variation of the rail potential at typical moments and positions occurring150 in 6756 m at 1178 s. Variation of the rail potential150 at typical moments and positions are are shownRail potential in Figure at 1230m 10. T=826s shown in FigureRail potential 10. at 6756m T=1178s 100 T=1231s 100 150 150 Rail potential at 1230m T=826s 50 Rail potential at 6756m T=1178s T=1231s 100 50 100 0 50

Rail potential(V) Rail 500 -50 potential(V) Rail 0

-100 potential(V) Rail -500 -50 potential(V) Rail

-150 -100 -50 1080 1100 1120 1140 1160 1180 1200 1220 1240 1260 -100 Time(s) 0 2000 4000 6000 8000 10000 12000 14000 Position(m) -150 1080 1100 1120 1140 1160 1180 1200 1220 1240 1260 -100 (a) Time(s) 0 2000 4000 6000 (b8000) 10000 12000 14000 Position(m) (a) (b)

Figure 10. Variation of the rail potential, (a) Variation of the rail potential at 1230 m and 6756 m; (b) Distribution of rail potential at three moments. Energies 2016, 9, 729 14 of 20

As shown in Figure 10a, at 1230s, the rail potential at the position of 1230 m is 89.3 V. When a train in section 2 begins regenerative braking at 1231 s, the rail potential reaches 130.8 V rapidly with the increase of the traction current transferring over the section. At 1177 s, the rail potential at 6756 m is 49.3 V. When a train in section 4 begins regenerative braking at 1178 s, the rail potential rises to 88.6 V with the increase of the traction current transferring over the section. During the period of the train in regenerative braking, the rail potential remains at a high level, and changes with the traction current transfers over the sections. It can be seen that rail potential is highly affected by the power distribution. As shown in Figure 10b, although the rail potentials at 1231 s and 1178 s are influenced by traction current transferring over sections, the maximum value in section 1 is much higher than that in section 2. Therefore, it is necessary to analyze the power distribution and rail potential at the two typical moments and positions. Meanwhile, the power distribution and rail potential at the time of 826 s is analyzed as a comparison. At 826 s, the total traction current is large, but there are no trains in regenerative braking in the line, and the traction current transferring over sections is low. The maximum rail potential in the line at 826 s is 35.3 V. Table4 shows the positions and current of all the trains and traction substations at 1231 s in the line. As shown in Table4, there are six trains, Tu1~Tu6, in the up line and there are six trains, Td1~Td6, in the down line at 1231 s. At this moment, Tu2 and Td1 stop respectively at Huijiang Station and Changgang Station with a traction current of 0 A. Traction substations S2, S4 and S5 are out of operation. The current distribution between the power sources and the loads are shown in Table5. As shown in Table5, in the traction current of accelerating train Td6 at 1230 m, 1414.8 A comes from traction substation S1 with the distance of 1230 m; 266.3 A comes from the regenerative braking train Tu3 with the distance of 4571 m; 900.0 A comes from the regenerative braking train Td4 with the distance of 5598 m. Table6 shows the positions and current of all the trains and traction substations at 1178 s in the line. Table6 illustrates the current and positions of all trains and traction substations at 1178 s. At this moment, there are six trains, Tu1~Tu6, in the up line and there are six trains, Td1~Td6, in the down line. Td5 and Tu6 stop respectively at Huijiang Station and Changgang Station; traction substations S2 and S4 are out of operation. The RBEAD at substations S1 and S5 are activated to absorb the remaining braking energy. The traction current distribution between the power sources and the loads are shown in Table7. As shown in Table7, in the traction current of accelerating train Tu3 at 6756 m, 147.7 A comes from the regenerative braking train Td6 with the distance of 5740 m, 538.9A comes from the regenerative braking train Tu1 with the distance of 5592 m, 1073.2 A comes from the traction substation S3 with the distance of 238 m, 609.3 A comes from the regenerative braking train Td1 with the distance of 6149 m. Table8 shows the positions and current of all the trains and traction substations at 826 s in the line. Energies 2016, 9, 729 15 of 20

Table 4. Positions and Current of the Trains and Traction Substations at 1231 Seconds.

Symbol S1 Tu1 Td6 Tu2 S2 Td5 Tu3 Td4 S3 Tu4 Td3 S4 Tu5 Td2 Tu6 Td1 S5 Current(A) −2022.5 1187.2 2581.1 0 0 180.3 −398.6 −2349 444.2 167.4 −95.0 0 −35.4 169.8 170.5 0 0 Position(m) 0 1009 1230 3367 3367 4034 5801 6828 6994 8391 9383 10,291 10,293 11,274 12,821 13,171 14,042

Table 5. Current Distribution at 1231s.

Load(A) Source Tu1 Td6 Td5 S3 Tu4 Td2 Tu6 Total(A)

S1 607.7 1414.8 0 0 0 0 0 2022.5 Tu3 132.3 266.3 0 0 0 0 0 398.6 Td4 447.2 900.0 180.3 444.2 167.4 116.9 93 2349 Td3 0 0 0 0 0 52.9 42.1 95 Tu5 0 0 0 0 0 0 35.4 35.4 Total(A) 1187.2 2581.1 180.3 444.2 167.4 169.8 170.5

Table 6. Positions and Current of the Trains and Traction Substations at 1178 s.

Symbol S1 Td6 Tu1 Td5 S2 Tu2 Td4 Tu3 S3 Td3 Tu4 S4 Td2 Tu5 Td1 Tu6 S5 Current(A) 721.3 −387 -2133.6 0 0 178.2 1145.8 2369.1 −1214.5 184.8 615.5 0 359.9 173.7 −2879.7 0 866.5 Position(m) 0 1016 1164.0 3367 3367 3949 5808 6756 6994 8475 9390 10,291 10,294 11,189 12,905 13,171 14,042

Table 7. Current Distribution at 1178s.

Load(A) Source S1 Tu2 Td4 Tu3 Td3 Tu4 Td2 Tu5 S5 Total(A)

Td6 0 38.3 201 147.7 0 0 0 0 0 387 Tu1 721.3 139.9 733.5 538.9 0 0 0 0 0 2133.6 S3 0 0 141.3 1073.2 0 0 0 0 0 1214.5 Td1 0 0 70 609.3 184.8 615.5 359.9 173.7 866.5 2879.7 Total(A) 721.3 178.2 1145.8 2369.1 184.8 615.5 359.9 173.7 866.5

Table 8. Positions and Current of the Trains and Traction Substations at 826 s.

Symbol S1 Td5 Td4 S2 Tu1 Td3 Tu2 S3 Td2 Tu3 S4 Td1 Tu4 Tu5 S5 Current (A) −82.2 0 0 −1105.7 189.7 2810.8 192.3 −2985.4 194.7 2696.8 −4207.4 2451.3 191.4 0 −346.3 Position (m) 0 1020 3367 3367 3771 5886 6581 6994 8653 9330 10,291 10,345 11,011 13,168 14,042 Energies 2016, 9, 729 16 of 20

Table8 illustrates the current and positions of all trains and traction substations at 826 s. At this moment, the maximum rail potential in the line is 35.3 V. There are five trains Tu1~Tu5 in the up line and five trains, Td1~Td5, exist in the down line. Td5, Td4 and Tu5 stop respectively in Shibi Station, Huijiang Station, and Changgang Station. The current distribution between the power sources and the loads are shown in Table9.

Table 9. Current Distribution at 826 s.

Load (A) Source Tu1 Td3 Tu2 Td2 Tu3 Td1 Tu4 Total (A)

S1 13.1 64.2 4.9 0 0 0 0 82.2 S2 176.6 863.8 65.3 0 0 0 0 1105.7 S3 0 1882.8 122.1 145.8 834.7 0 0 2985.4 S4 0 0 0 48.9 1862.1 2279 17.4 4207.4 S5 0 0 0 0 0 172.3 174 346.3 Total(A) 189.7 2810.8 192.3 194.7 2696.8 2451.3 191.4

As shown in Table9, the total traction current of all the accelerating trains in line is 8727.2 A, whereas the total traction current is respectively 4456.3 A at 1231 s and 5027 A at 1178 s. However, the maximum rail potential at 826s is much lower than that at 1231 s and 1178 s. Although the total traction current is high at 826 s, there are no regenerative braking trains in the line at this moment. The traction current of accelerating trains comes from the traction substations with a short distance. In the traction current of the accelerating train Td3 at 5886 m, 1882.8 A comes from the traction substation S3 with the distance of 1108 m; 863.8 A comes from the traction substation S2 with the distance of 2519 m; and 64.2 A comes from S1 with the distance of 5886 m. In the traction current of the accelerating train Tu3 at 5933 m, 834.7 A comes from the traction substation S3 with the distance of 2336 m; and 1862.1 A comes from the traction substation S4 with the distance of 961 m. In the traction current of the accelerating train Td1 at 10,345 m, 2279 A comes from the traction substation S4 with the distance of 54 m, and 172.3 A comes from the traction substation S5 with the distance of 3697 m. Comparing the distribution of traction current and rail potential at 826 s, 1178 s and 1231 s, we know that rail potential will increase abnormally when regenerative braking trains feed current back to the catenary and the current is absorbed by the accelerating trains at long distance. Based on the distribution of traction current and rail potential, the mechanism that rail potential in the first and last sections is much higher than that in the middle sections is analyzed. Maximum rail potential in section 1 is 130.8 V, occurring at 1231 s at the position of Td6, 42.2 V higher than that in section 2, occurring in 1178 s at the position of Tu3. In this paper, transmission distance of traction current (TDTC) is proposed to describe the transmission distance of the load’s traction current at N node j. TDTC = ∑ Iij × xi − xj . The TDTC of Td6 is 7996.6 A·km at 1231 s, and the TDTC of Tu3 is i=1 7863.3 A·km at 1178 s. There is little difference between the TDTC of the two trains, but rail potential in the position of Td6 is much higher than that in the position of Tu3. Comparing the TDTCs, we find that unilateral nature of the Td6 is more obvious: 1740.2 A·km of the TDTC comes from the power sources in the section of 0 m to Td6, whereas 6256.4 A·km of the TDTC comes from the power sources in the section of Td6 to L. The TDTC of Tu3 is relatively bilateral at 1178 s; 3861.3 A·km of the TDTC comes from the power sources in the section of 0 m to Tu3; 4002 A·km) of the TDTC comes from the power sources in the section of Tu3 to L. Therefore, the rail potential with multiple trains running in multiple sections is influenced by the TDTC and the unilateral nature or bilateral nature of the train’s traction current. When the system is in dynamic operation, there are multiple trains in the line. If the current that the regenerative braking train feed backs to the catenary is absorbed by the accelerating trains at long distance, then the rail potential may rise abnormally. The traditional methods such as shortening Energies 2016, 9, 729 17 of 20 the distance between traction substations or ensuring the longitudinal resistance within the standard EnergiesEnergies 2016 2016, ,9 9, ,729 729 1717 of of 19 19 values cannot control the rail potential effectively. 5.3.5.3. An An Analysis Analysis of of Stray Stray Current Current 5.3. An Analysis of Stray Current AnAn analysis analysis of of stray stray current current when when multiple multiple trains trains run run in in multiple multiple sections sections is is carried carried out out in in this this An analysis of stray current when multiple trains run in multiple sections is carried out in this paper.paper. TheThe simulationsimulation resultresult ofof thethe buriedburied conductorconductor potentialpotential isis shownshown inin FigureFigure 11,11, andand thethe paper. The simulation result of the buried conductor potential is shown in Figure 11, and the simulation simulationsimulation result result of of the the stray stray current current in in the the buried buried conductor conductor is is shown shown in in Figure Figure 12. 12. result of the stray current in the buried conductor is shown in Figure 12.

(a(a) ) (b(b)) Figure 11. Distribution of buried conductor potential, (a) Distribution of buried conductor potential FigureFigure 11. 11.Distribution Distribution of of buried buried conductor conductor potential, potential, ( a(a)) Distribution Distribution of of buried buried conductor conductor potential potential (time‐position); (b) Distribution of buried conductor potential (potential‐position). (time-position);(time‐position); ( b(b)) Distribution Distribution of of buried buried conductor conductor potential potential (potential-position). (potential‐position).

(a(a) ) (b(b)) Figure 12. Distribution of stray current, (a) Distribution of stray current (time‐position); (b) FigureFigure 12. 12.Distribution Distribution of stray of stray current, current, (a) Distribution (a) Distribution of stray currentof stray (time-position); current (time (b‐)position); Distribution (b) Distribution of stray current (stray current‐position). ofDistribution stray current of (straystray current current-position). (stray current‐position).

AsAs shownshown inin FigureFigure 11,11, influencedinfluenced byby thethe distributiondistribution ofof railrail potential,potential, thethe buriedburied conductorconductor As shown in Figure 11, influenced by the distribution of rail potential, the buried conductor potentialpotential in in the the first first and and last last sections sections is is much much higher higher than than that that in in the the middle middle sections. sections. The The maximum maximum potential in the first and last sections is much higher than that in the middle sections. The maximum buriedburied conductor conductor potential potential is is 2.36 2.36 V, V, occurring occurring in in 238 238 s s at at 14,042 14,042 m. m. The The minimum minimum buried buried conductor conductor buried conductor potential is 2.36 V, occurring in 238 s at 14,042 m. The minimum buried conductor potentialpotential is is − −2.422.42 V, V, occurring occurring at at 1084 1084 s s in in 14,042 14,042 m. m. The The maximum maximum and and minimum minimum buried buried conductor conductor potential is −2.42 V, occurring at 1084 s in 14,042 m. The maximum and minimum buried conductor potentialspotentials have have exceeded exceeded the the standard standard limits. limits. Figure Figure 13a 13a illustrates illustrates the the buried buried conductor conductor potential potential at at potentials have exceeded the standard limits. Figure 13a illustrates the buried conductor potential at 826826 s, s, 1178 1178 s s and and 1231 1231 s. s. Although Although the the total total traction traction current current at at 826 826 s s is is much much higher higher than than that that at at 1178 1178 s s 826 s, 1178 s and 1231 s. Although the total traction current at 826 s is much higher than that at 1178 s andand 12311231 s,s, thethe maximummaximum buriedburied conductorconductor potentialpotential isis 0.180.18 VV atat 826826 s,s, andand thethe maximummaximum buriedburied and 1231 s, the maximum buried conductor potential is 0.18 V at 826 s, and the maximum buried conductorconductor potentials potentials at at 1178 1178 s s and and 1231 1231 s s are are 0.56 0.56 V V and and 2.06 2.06 V, V, respectively. respectively. The The buried buried conductor conductor conductor potentials at 1178 s and 1231 s are 0.56 V and 2.06 V, respectively. The buried conductor potentialpotential is is highly highly affected affected by by the the power power distribution. distribution. potential is highly affected by the power distribution. FigureFigure 12 12 illustrates illustrates the the distribution distribution of of stray stray current current in in buried buried conductor. conductor. Stray Stray current current is is low low at at bothboth ends ends of of the the line, line, and and is is high high in in the the middle middle sections. sections. The The maximum maximum positive positive stray stray current current in in the the buriedburied conductor conductor is is 29.39 29.39 A, A, occurring occurring in in 904 904 s s at at 7760 7760 m. m. The The maximum maximum negative negative stray stray current current in in the the buriedburied conductor conductor is is 28.59 28.59 A, A, occurring occurring in in 238 238 s s at at 7230 7230 m. m. The The distribution distribution of of stray stray current current at at 826 826 s, s, 1178 1178 s s

Energies 2016, 9, 729 18 of 20

Figure 12 illustrates the distribution of stray current in buried conductor. Stray current is low at both ends of the line, and is high in the middle sections. The maximum positive stray current in the buried conductor is 29.39 A, occurring in 904 s at 7760 m. The maximum negative stray current Energies 2016, 9, 729 18 of 19 in the buried conductor is 28.59 A, occurring in 238 s at 7230 m. The distribution of stray current at 826 s, 1178 s and 1231 s is shown in Figure 13b. Maximum stray current in buried conductor at 826 s, and 1231 s is shown in Figure 13b. Maximum stray current in buried conductor at 826 s, 1178 s and 1231 s 1178 s and 1231 s are 4.63 A, 12.13 A, and 25.41 A respectively. Stray current is highly affected by the are 4.63 A, 12.13 A, and 25.41 A respectively. Stray current is highly affected by the power distribution. power distribution.

2.5 30 T=826s T=826s T=1178s T=1178s 2 25 T=1231s T=1231s

20 1.5

15 1 10 0.5 5

0 Stray current(A) 0

-0.5 Potential of buried conductor(V) -5

-1 -10

-1.5 -15 0 2000 4000 6000 8000 10000 12000 14000 0 2000 4000 6000 8000 10000 12000 14000 Position(m) Position(m) (a) (b)

Figure 13. Comparison of buried conductor potential and stray current at three moments, (a) Distribution Figure 13. Comparison of buried conductor potential and stray current at three moments, of buried conductor potential at three moments; (b) Distribution of stray current at three moments. (a) Distribution of buried conductor potential at three moments; (b) Distribution of stray current at three moments. 6. Summary and Conclusion 6. SummaryIn this paper, and Conclusion a dynamic simulation model for evaluation of the rail potential when multiple trains run in multiple sections is presented, and the impact of power distribution on abnormal rise of In this paper, a dynamic simulation model for evaluation of the rail potential when multiple trains rail potential is analyzed. With the simulation model in this paper, the distribution of rail potential in run in multiple sections is presented, and the impact of power distribution on abnormal rise of rail the dynamic operation of the line can be simulated rapidly and accurately. Conclusions can be potential is analyzed. With the simulation model in this paper, the distribution of rail potential in the drawn as follows: dynamic operation of the line can be simulated rapidly and accurately. Conclusions can be drawn (1) A dynamic simulation model for evaluation of the rail potential is proposed in this paper. The as follows: model incorporates train movement calculation, power flow calculation, power flow tracing and rail (1)potentialA dynamic calculation. simulation For realization model for of fast evaluation and accurate of the calculation, rail potential return is proposed circuit is in converted this paper. to lumpedThe parameters model incorporates in power train flow movement calculation calculation, and is modeled power flowwith calculation,distributed powerparameters flow tracing in rail potentialand railcalculation. potential Based calculation. on this For model, realization the impact of fast of and power accurate distribution calculation, on rail return potential circuit in is dynamicconverted operation to lumped of the system parameters can be in analyzed power floweffectively. calculation and is modeled with distributed (2) Tractionparameters current in rail transferring potential calculation. over sections Based widely on this exists model, in theDC impact traction of power power distributionsystems when on multiplerail potentialtrains run in in dynamic multiple operation sections. ofThe the power system flow can tracing be analyzed method effectively. proposed in this paper can (2)obtainTraction the power current distribution transferring between over power sections sources widely and exists loads in effectively. DC traction power systems when (3) Railmultiple potential trains and run stray in multiple current is sections. highly Theaffected power by flow the power tracing distribution. method proposed When in the this traction paper currentcan transferring obtain the power over sections distribution increases, between rail power potential sources and stray and loads current effectively. are relatively high. Rail (3)potentialRail with potential multiple and straytrains current running is in highly multiple affected sections by the is also power influenced distribution. by the When unilateral the traction nature or bilateralcurrent nature transferring of the train’s over sections traction increases, current. rail potential and stray current are relatively high. Rail potential with multiple trains running in multiple sections is also influenced by the unilateral Acknowledgments: This work is supported by the National Natural Science Foundation of China (Grant nature or bilateral nature of the train’s traction current. no. 51147011), the Jiangsu Provincial Natural Science Foundation of China (Grant no. BK20130187), and the Research and Innovation Program of Postgraduates of Jiangsu Province (KYLX 1383). Acknowledgments: This work is supported by the National Natural Science Foundation of China (AuthorGrant No. Contributions: 51147011), the JiangsuGuifu Du Provincial wrote Naturalthe paper Science and designed Foundation the of simulation China (Grant model; No. BK20130187), Dongliang and the Research and Innovation Program of Postgraduates of Jiangsu Province (KYLX 1383). Zhang contributed to the conception of the study and provided the line and vehicle data; Guoxin Li analyzed the power distribution of DC traction power systems; Chonglin Wang analyzed the simulation data; Jianhua Liu helped to perform the analysis with constructive discussions. Conflicts of Interest: The authors declare no conflict of interest

References

Energies 2016, 9, 729 19 of 20

Author Contributions: Guifu Du wrote the paper and designed the simulation model; Dongliang Zhang contributed to the conception of the study and provided the line and vehicle data; Guoxin Li analyzed the power distribution of DC traction power systems; Chonglin Wang analyzed the simulation data; Jianhua Liu helped to perform the analysis with constructive discussions. Conflicts of Interest: The authors declare no conflict of interest

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