1 Introduction

Nowadays, the development of railway infrastructure shows a rebirth in many countries including Ethiopia after a long period of recession. This revival of the railway is mainly due to the new trend of renewable energy production and consumption. As a consequence, the expansion of a traction power supply system is growing massively –without it, only the weaker and less energy effcient steam and diesel locomotives could be used [1]. An accurate power supply system analysis offers important information for planning, operation and design. Almost in all traction power supply system analyses and studies, the widely adopted trend and cost effective mechanism is simulation. In[2], widespread applications which deal with many railway power feeding system simulators were discussed.

1.2.3 发展了新业态和培育了新动能。全市立项“创投”项目中，种植水稻、柑橘、蔬菜、茶叶等81个，其他特色种养15个；其中有15个项目同步发展乡村旅游业，12个项目配套茶叶加工，有1 666.7 hm2推行稻田综合种养模式。这些新业态培育了产业发展新动能，项目建成投产后，年产值12亿元以上。

In this study, a different approach, using MATLAB,is developed. Several alternative analysis methods are used for designing the electrical scheme of the power supply system for electrified railways [3]. Selection of an appropriate supply system is always very dependent on the railway system objectives. Many studies show that direct linking of the feeding transformer to the overhead catenary system and the rails at each substation is relatively simple and economical. Nevertheless, there are some drawbacks to this arrangement such as high impedance of feeders with high losses, high rail-to-earth voltage and the interference to neighboring communication circuits [2] [4]. Moreover,the autotransformer feeding configuration has many advantages and solves many disadvantages of the direct feeding system. The addition of autotransformer at every 8-15 km intervals improves the voltage preformance along the traction line and increases the substation coverage to 50-100 km [4]. The electromagnetic interference in an AT system is normally much lower compared with a direct feeding (1 × 25 kV) system [5]. In addition, for high power locomotives and high speed trains, direct feeding systems cannot be applied because most countries are replacing the existing (1 × 25 kV) systems with autotransformer fed supply systems (2 × 25 kV) [6].In this paper, analysis, modeling and simulation of an autotransformer traction power supply system are presented and a comparative analysis with a direct power supply system in terms of voltage profile along the traction network is performed.

在区分高、低自尊者的基础上，研究者进行了两方面的比较，一方面是进行了三组中高自尊者及低自尊者职业认同的组间比较； 另一方面是分别对三组中同一组的高自尊和低自尊者职业认同进行了组内比较。

2 System configuration

In this system, the traction transformers are supplied from state grid, at 132 kV voltage level. This voltage is further stepped-down to 55 kV at traction substations by using 132/55 kV transformers with center tap on the secondary side to have ±27.5 kV between the center-tap and the respective terminals. Each traction substation has two 132 kV independent power lines. The secondary terminals of the traction power transformers are selected to give a voltage of ±27.5 kV in order to compensate for any voltage drop caused by power supply line prior to the catenary system. The nominal voltage of the catenary system is considered to be 25 kV.

where v is the speed of the train; Ft-max is the maximum tractive force of the locomotive;ξ is the slippage ratio.In order to obtain more realistic results, some losses and auxiliary power consumptions must be taken into account.These losses can be modeled with the parameter ηloco which is the locomotive’s efficiency, as:

In addition, the system consists of center-tapped autotransformers located every 15 km where the outer terminals are connected between the catenary and feeder wires. The autotransformer-fed system enables power to be distributed along the system at higher than the train utilization voltage. As a nominal value, power is distributed at 55 kV (line-to-line) while the trains operate at 25 kV(line-to-ground). The system voltages for the proposed system conform to European standards EN 50163: 2004[10] and their values given by the above standards are as follows.

The auxiliary power consumption Paux can be considered as well in the calculation which includes the cooling systems,train heating and the power for travelers (for the case of passenger train). The electrical active power demand will be:

2. The maximum permanent voltage allowed in the supply line shall be 27.5 kV.

3. The maximum non-permanent voltage that should be allowed for a short period of time shall be 29 kV.

4. The minimum permanent voltage shall be 19.0 kV.

5. The minimum non-permanent voltage that should be allowed for a short period of time shall be 17.5kV.

3 Longitudinal force acting on the train

The train, as a load, is on the move and considered to be one of the main problems of longitudinal rail dynamics and is governed by the Fundamental Law of Dynamics applied in the longitudinal direction of the train’s forward motion [7].

The left side of the equal sign as shown in (1) is the sum of all the forces acting in the longitudinal direction of the train, where Ft is the tractive or braking effort; Fex is the force that opposes the forward motion of the train; M*is the total mass (train mass + passenger or freight mass)of the train, but due to rotational inertia effect, the effective linear mass of the train increases and this value varies from 5% to 15% depending on the number of motored axles,the gear ratio and the type of car construction; a is the longitudinal acceleration experienced by the train. Various literatures [8] [9] [11] show that different countries use different starting acceleration ranging from 0.08m/s2 to 0.25m/s2 for freight trains.

3.1 Forces against the train

The total force acting on a train against its direction of motion (Fex) can be expressed mathematically as follows:

where Fr is mechanical and aerodynamics resistance; Fgr is gradient resistance; Fc is curves resistance.

3.1.1 Mechanical and aerodynamic resistance

The force created due to mechanical and aerodynamic resistance Fr is given by:

Generally, A+Bv is rolling resistance and cv2 is aerodynamic resistance. The value of A can be approximately computed as [12]:

Naxle is the number of trailing car axles and B can be expressed as a function of total train length rather than train mass.

where, LT is the total length of the train. The aerodynamics drag, the part depending upon the speed squared is usually written for no wind condition as:

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where Af is the projected cross sectional area; CD is the air drag area; ρ is the air density which is equal to 1.3 kg/m3.

3.1.2 Force due to gradient

Gradient resistance is also a component of the train load against the direction of travel. It is positive for uphill gradients and negative for downhill gradients (i.e.pushes the train forward). Thus the gradient resistance is determined by:

where i is the percentage gradient and g is the gravitational acceleration. For freight and passenger lines, the location with maximum gradient needs to be considered in design. Therefore, for the profile design of railways, we need to lower the maximum gradient to ensure that the freight train passes through this section at no less than calculated speed.

2.1.3 Force due to curve

Additional curving resistance Fc mainly corresponds to the increased energy dissipation that occurs in the wheel rail interface, due to sliding motions (creep) and Friction phenomena. It is dependent on wheel rail friction as well as the stiffness and character of the wheel. The resistive force produced by the curve is modeled by the following equation [13]:

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3.2 Maximum tractive effort of the locomotive

The tractive effort can be increased by increasing the motor torque but only up to a certain point. Beyond this point any increase in the motor torque does not increase the tractive effort but merely causes the driving wheels to slip. The transmitted force is limited by adhesion and the maximum force that can be transmitted can be written as[12]:

where M and µa are mass of the train in ton and coefficient of adhesion, respectively. The adhesion coefficient µacan be found based on Curtius and Kniffler derived adhesion curve in [14]:

3.3 Power demand of the train

The maximum tractive force of the locomotive multiplied by the train velocity gives the maximum power that the locomotive consumes. The mechanical tractive power of the motor is computed by [15]:

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1. The nominal voltage shall be 25 kV.

In order to simplify the analysis, a constant power model is used, in which the train power and power factor are assumed constant. The locomotive has a maximum speed of 100 km/h [5]. It has six axle loads each weighing 25 tons. The length of a single locomotive and trailing train is 15.5 m and 14.27 m respectively and the total train weight is 5000T.

4 Modeling of traction power supply system

Electrical power system analysis always depends on a mathematical model which mainly comprises mathematical equations that define the relationships between various electrical power quantities with required precision.Therefore, based on the objective of the electrical power system analysis, various models for a given system may be applicable. In this section, the traction power supply system mathematical model is presented.

4.1 Train or locomotive model

Fig. 1 shows MATLAB simulation results of tractive effort and resistance force with respect to the speed of the train.

where STR, VR, VC are the power demand of the train, rail voltage and contact line voltage respectively.

4.2 Model of substation power transformer

Today, there are many transformers which are used in railway power supply systems such as open delta or Vv,Scott, YNd11, Wood-Bridge, single phase transformers,etc. These different transformers are used in various feeding configurations. For example, in direct feeding systems, the Vv transformers are the best choice whereas in autotransformer feeding arrangements, the three winding specially built single phase traction power transformers are used. In this type of transformer arrangement, the primary terminal of the transformer has a voltage rate of 132, 275 or 400 kV and the secondary terminal voltage is 55 kV.For the line being studied, the single phase three-winding transformer is used with the voltage designated as 132 kV/27.5 kV-0-27.5 kV. Fig. 2 illustrates such connections.

For the substation power transformer, the Norton equivalent circuit in the multi-conductor model is formed as Fig. 2, defining Z A =ZO +Z1 +6Z2and ZB =6 Z2+24Ze,where ZO is short circuit impedance of the high voltage grid; Z1 and Z2 are impedance of the primary and secondary windings respectively; Ze is impedance connecting the center tap of the 2nd winding and the rails; a is the transformer’s turn ratio.

I1 is the primary current of the transformer caused by the contact line current IC and the negative feeder current IF as:

Consider the primary-side circuit, using Kirchhoff’s law,and replacing I1 in (19), we have:

Use Kirchhoff’s law on the secondary-side contact to rail line (C-R) circuit,

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The quantity Dad is a function of both the earth resistivity and the frequency (f) and is defined by the relation:

多面体和旋转体(29)：棱柱、棱锥、棱台的概念、性质、画法、面积，圆柱、圆锥、圆台、球的概念、性质、画法、面积，球冠及其面积，体积的概念与公理，柱、锥、台的体积，拟柱体的体积，球、球缺的体积.

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Based on the assumption that all currents are supplied by the substation and eventually returned to the substation, we have:

Equations (19)-(24) shown above can be written in another form of the Norton equivalent circuit as shown in equation (25)

where IC, IR and IF are the contact line current, the rail line current and the negative feeder current, respectively; VO,VC, VR and VF are the nominal voltage, contact line voltage,rail line voltage and negative feeder voltage, respectively.In addition, JSS is the substation current injected, VSS is the substation voltage and YSS is the substation admittance.The short circuit impedance of the grid is assumed zero,which is equivalent to assuming the primary voltage of the substation transformer 132 kV. The no load secondary voltage between the overhead catenary system and feeder is 55 kV with a grounded center tap. The nameplate of the substation transformer shows 132/55 kV, 60 MVA.Calculated based on annual transportation demands, X/R ratio equals to 10 based on ANSI/IEEE C37.010-1979 and the impedance of the traction transformer is 15% based on IEC 60076 standards.

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where ke(m) is the track gauge coefficient and r is the radius of the curve.

4.3 Model of autotransformer

The autotransformer has a single winding connection between the catenary and the feeder wires. The rail system (rails and grounded wires) is connected to a center point on the winding. The usual supply voltage is 50 kV between the catenary and feeder with a transformation ratio of 2∶1 to obtain a 25 kV from catenary to rail and from rail to the return feeder. The station where the autotransformer is located is also called paralleling station because the two tracks (C wire and F wire) are connected in parallel. Thus, the admittance model of the autotransformer becomes:

Z1 and Z2 are primary and secondary leakage impedances;Zm is magnetizing impedance; Im is magnetizing current; I1 and I2 are primary and secondary currents; E1 and E2 are electromotive forces windings; N1 and N2 are turns of the two windings. Assuming N1 = N2 and Z1 = Z2 = Zg, we have

Substituting equation (26) to (27), we can easily find the following equations:

The conductor a - a1 shown in Fig. 4 carries a current Ia with a return through circuit d - d1 beneath the surface of the earth. The earth is considered to have a uniform resistivity and to be semi-infinite. The current Ia in the ground spreads out over a large area. Using the Fig. 4 and ignoring skin effect, it is possible to write the self and mutual impedances of the catenary system. Carson found that the earth resistance (rd) is a function of frequency and he derived the empirical formula as:

The self-impedance (Zaa) of wire with earth return can be expressed as:

采用延时合闸策略时，A相首合后，该相磁链工作曲线与合闸角度及剩磁大小、方向密切相关，也是决定励磁涌流大小的关键。以A相电压正半周为例(负半周的关系同理)，A相开关的理想合闸相角为90°，考虑到开关动作离散性和预击穿特性的影响，其可能的合闸角将落在90°±90°的区间上，对应为正半周，存在一个上升沿和下降沿。

where ro is the radius of the conductor (m) and Dad is the equivalent conductor at depth. Also, k = 2.10-7, ω = 2πf Finally, the mutual impedance (Zad1) is stated as:

Define

If no actual earth resistivity data is available, it is common to assume ρe to be 100 ohm-meter. The earth resistivity ρe depends on the nature of the soil and is commonly found in the literature.

发现木马和僵尸网络程序控制端共计294个IP，环比减少32.10%。近四个季度我省木马和僵尸网络程序控制端数量发展趋势如图2所示：

4.4.1 Conductor configuration and standards

1. Contact wire (C): Part of the overhead contact line system which establishes contact with the current collector.To avoid errors, the impedances of the messenger and the contact wire are calculated independently.

2. Messenger wire (M): Part of the overhead contact line system used to support the contact wire.

3. Rails (R): The two rails in the same track are also treated as independent conductor parts. They are connected to the autotransformer at the center tap.

4. Negative feeder (F): Besides the catenary, another outer tap of the autotransformer is connected to the feeder wire.

5. Static wire (S): It is used for safety reasons and included in the calculation of the matrix impedance for greater accuracy.

The conductor configuration or arrangement of the overhead line is based on the industry standards of conductor clearances. According to IEC, the minimum electrical clearances of the conductor must be maintained under all line loading and environmental conditions. Since

Autotransformer components are modeled by their equivalent circuits in terms of inductance and resistance.The magnetizing impedance Zm of the autotransformer is taken as infinite and also the impedance Z1 = Z2 because the two windings are similar. The earthing resistance Ze is assumed to form the resistance between center-tap and remote earth. The calculated results of the AT ratings are:50/25 kV, 10 MVA, with 7.5% impedance and an X/R ratio of 10.

4.4 Modeling of the catenary system

In 1923, Carson published an impressive paper which discussed the impedance of the overhead conductor with earth return [6]. This paper has been used in many researches for the calculation of the impedance of the overhead power supply line where current flows through the earth, especially in the railway system where significant amount of current flows via the earth to the traction substation[6]. In this paper, Carson’s line model has been used for impedance calculation of the traction network.The following Carson equation is used to compute mutual impedance (Zad1) and self-impedance (Zaa) of the catenary system.the actual sag clearance of conductors on overhead contact line is seldom monitored, sufficient allowance for this clearance (safety buffer) must be considered in the process of the initial design.

Minimum horizontal and vertical distances from energized conductor to ground, other conductors, vehicles,and objects such as buildings (“electrical clearances”),are defined based on three parameters. Clearances are defined based on the transmission line to ground voltage,the use of ground fault relaying, and the type of object or vehicle expected within proximity of the line. The IEC 270 Rules cover both vertical and horizontal clearances to the energized conductors. The electric static clearance, which is the minimum distance required between the live parts of the overhead wire equipment and structure or the earthed parts of the overhead wire equipment under 25 kV must be at least 320 mm as per IEC 270. The minimum electrical clearance to earth or another conductor is 150 mm under adverse condition and the minimum clearance between two parallel wires in open overlaps is 250 mm but may be reduced to 150 mm as absolute minimum under the worst case.

The international standards covering most conductor types are IEC 61089 (which supersedes IEC 207, 208, 209 and 210) , EN 50182 and EN 50183. In this paper, for all negative feeder wires, earth wires and messenger wires,aluminum conductor steel reinforced (ACSR) is used.

5 Simulation results and discussion

The purpose of this simulation is to evaluate the autotransformer-fed power supply system for a typical railway line and make a comparative analysis with direct fed return conductor configuration. Since voltage profiles and voltage regulations along the line are the most important parameters to evaluate the system performance,a computer-aided steady state load-flow simulation in terms of voltage was performed. The analysis is done for both configurations. Performance of the traction power system has been investigated by applying different headway distances on the trains and by increasing the number of trains along the feeding section. Since traction power supply system is always designed by considering two (minimum) or more locomotives fed from the same section, this simulation is performed based on two and three locomotives which are moving along the 55 km long feeding section and results are taken one at a time at points 15 km, 30 km, 45 km and 55 km from the traction substation. Fig. 8 - Fig. 12 show simulation results of the direct fed configuration and Fig. 13 - Fig. 17 show results of the autotransformer fed system. Table 2 - Table 6 show comparison of voltage profiles of the train at various positions and for a number of trains.

5.1 Simulation results of direct fed with return system

The simulation results (Fig. 8 - Fig. 12) are found based on a model developed using MATLAB/Simulink as shown in Fig. 7.

5.2 Simulation results of the autotransformer fed system

The simulation results (Fig. 13 - Fig. 17) are found based on a model developed using MATLAB/Simulink as shown in Fig. 6.

5.3 Comparative analysis of both configurations

In this section, based on the above simulation results,a comparative analysis between the direct power supply mode and the autotransformer power supply network has been done. The comparison is performed based on the voltage profiles of the train at various positions and for a number of trains.

Table 2 - Table 6 show that the train voltages of both 1 × 25 kV and 2 × 25 kV traction network change with distance and the number of trains. However, 2 × 25 kV traction network voltage profiles are always greater than 1 × 25 kV traction power supply mode. The 2 × 25 kV traction supply system shows less voltage variations over distance from the substation, which in turn decreases the number of traction substations.

6 Conclusion

In this paper, first, the mathematical modeling and analysis of major components of the autotransformer traction power supply system have been done. Second,simulation models of both direct power supply mode and autotransformer supply system have been performed based on MATLAB/Simulink. Third, simulation of the supply system with variation of distance of electric locomotives and the number of trains across the line has been conducted.Finally, a brief comparison of both traction power supply systems in terms of voltage profile has been made. From the results obtained, it can be concluded that the train voltage peak values for both configurations decrease with the increase of distance from locomotive to the traction substation and the number of trains along the feeding section. In addition, simulation results also indicate that the autotransformer supply network is far more superior than direct power supply mode in terms of train voltage profile.In the end, the results obtained from the autotransformer model confirms with the industry standards and this clearly indicates the successful use of developed mathematical model for simulating traction power supply systems.

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