Analysis of theoretical line loss in metro distribution network

1 Introduction

Conventional DC power supply systems in urban rail transit, although capable of supporting frequent start-stop operations, face persistent challenges such as stray currents causing infrastructure corrosion, low efficiency in regenerative braking energy utilization, negative sequence currents affecting power quality, and phase splits that limit train speed (Dolara et al., 2012). To overcome these issues, a novel AC traction power supply scheme has been proposed. This scheme integrates new contact supply devices, AC-DC-AC locomotives, and co-phase compensation devices, thereby eliminating phase splits and stray currents, reducing construction costs, and improving the utilization of regenerative braking energy (Steczek et al., 2023; Chen et al., 2016).

For the effective operation of this system, accurate theoretical line loss calculation is essential (Cai et al., 1996; Goodman and Kulworawanichpong, 2002). It provides the basis for optimizing operational strategies, planning network upgrades, and guiding equipment maintenance. However, in metro systems the stochastic and dynamic nature of train loads has a significant impact on line losses. As trains accelerate, decelerate, and run at constant speed across different substations, the traction load continuously shifts along the network, complicating the loss evaluation (Zhang Y. et al., 2025).

Existing research has addressed traction power supply in various forms, including direct supply, BT supply, and AT supply (Kulworawanichpong, 2015), with equivalent circuit models often employed for engineering analysis (Al-Khalidi and Kalam, 2008). Line loss calculation methods in distribution networks can be broadly categorized into: (1) traditional approaches based on substation output data, which are simple but neglect time-domain characteristics (Li et al., 2020); (2) methods using FTU/TTU measurements, which are convenient but cannot directly capture the effect of load variations (Zhang W. et al., 2025; Ni et al., 2019); (3) real-time data-driven methods, which improve timeliness but remain limited in analyzing dynamic load impacts (Gulakhmadov et al., 2022).

This paper aims to extend these studies to urban rail traction networks by explicitly considering the dynamic characteristics of AC-DC-AC locomotives. By combining traction characteristic curves with train operation diagram is established. The main contributions of this paper are summarized as follows: (1) A traction-oriented cascade equivalent model is established, covering the substation–feeder–section–train structure of metro AC traction networks. (2) An improved impedance aggregation method for multi-core cables and traction transformers is proposed, enabling position-dependent equivalent impedance evaluation. (3) A time-sequential loss calculation framework is developed, coupling dynamic train load, nodal-voltage iteration, and component-level loss decomposition along the line.

The rest of this paper is organized as follows. System structure and network circuit equivalence process are introduced in Section 2. Section 3 presents a power distribution calculation model. Under the condition that load is in traction and regenerative braking while single train and multi-train are working, power distribution in system is explored in Section 4. At last, corresponding conclusions are induced in Section 5.

2 Traction power supply system

2.1 System structure

As illustrated in Figure 1, the traction power supply system consists of a central substation (SS), traction cables, single-phase traction transformers, an overhead contact system (OCS), and a current return line (CRL). The central substation connects to the 110 kV three-phase grid and delivers 35 kV single-phase power through traction cables. Each traction substation is equipped with a single-phase transformer (T_*il*1, T_*il*2, T_*il*3), stepping down the voltage to the catenary level for train operation (Li, 2016).

The central substation employs combined co-phase power supply technology, consisting of a traction transformer and a back-to-back negative sequence compensation device (NCD). The NCD transfers active power from the grid side to the traction side while compensating harmonics and reactive power (Shu et al., 2013). In practice, 35 kV two-core cables are used for traction power, while three-core cables are reserved for auxiliary supply (Huang et al., 2023). The traction network supplies locomotives with same-phase power along the entire line, improving supply reliability and reducing the installed capacity of traction transformers.

2.2 Network equivalent circuit

2.2.1 Traction cables

Urban rail AC traction systems typically employ 35 kV cables, either two single-core or one double-core type. A single-core armored cable can be modeled as a linear passive element at the operating frequency, and a uniform line segment is equivalent to a Π-type circuit, as shown in Figure 2.

The impedance and admittance matrices of the cable are expressed as

ZL=[Z11⋯Z1m⋮⋱⋮Zm1⋯Zmm], YL2=[Y11⋯Y1m⋮⋱⋮Ym1⋯Ymm].(1)

Here, Z_*ii* and Y_*ii* denote the self-impedance and self-admittance of phase i, while Z_*ij* represents mutual impedance. Due to shielding, Y_*ij* = 0. The capacitances C_*ics*, C_*isa*, C_*iae* describe coupling between core–shield, shield–armor, and armor–ground (Ghassemi, 2019). These parameters allow calculation of impedance for arbitrary line length and configuration, enabling accurate current and voltage distribution analysis in traction networks. This equivalent model also provides the basis for subsequent power flow and line loss calculations.

2.2.2 Central substation

As shown in Figure 3, the traction port consists of the traction transformer and the co-phase compensation device (CPD), which share the load and ideally eliminate the negative-sequence current. The cable-side current satisfies

IL=IT+IC, IC=KNIL2, IT=IL(1-KN2),(2)

where K_*N*∈[0, 1] is the compensation factor.

Applying Thevenin's theorem, the traction port has open-circuit voltage equal to the rated secondary voltage, and equivalent resistance

Z0=(1-KN)ZS+ZT,(3)

which depends only on transformer and system short-circuit capacity, and is independent of cable and load.

2.3 Traction supply system equivalent model

The AC traction power network of urban rail transit is essentially a 35 kV cable-fed system. The central substation connects to the contact line through single-phase traction transformers in parallel, supplying locomotives. Since the network involves multiple voltage levels, all circuits are converted to a unified 35 kV voltage level for modeling purposes.

As shown in Figure 4, the traction network can be represented by a multi-loop equivalent circuit. The central substation is modeled by an ideal voltage source Ẽ₀ with internal impedance Z_*s*. The line is divided into two parts: - a local loop, which contains the locomotive and the nearby traction substations, and - a distal loop, which represents the remaining sections equivalently.

In the local loop, each traction substation is represented by its transformer impedance (Z_*T*1, Z_*T*3, Z_*T*4, …). The catenary and return conductors are represented by impedances Z_*OCS* and Z_*CS*. The locomotive is modeled as a current source i_*n* located at position x, with the distance between adjacent substations denoted as D and the total line length as L. The distal loop is reduced to an equivalent impedance ZT′ obtained by recursive parallel conversion of the remote sections.

This equivalent model explicitly describes the current distribution across substations (I₁, I₂, …) and forms the basis for subsequent derivation of line loss equations.

2.4 Line loss calculation method

Consider n traction substations uniformly distributed along the line. Let D denote the spacing between two adjacent substations along the track. Let l be the longitudinal position of the locomotive measured from the line origin (0 ≤ l ≤ nD). Thus the locomotive lies within the m-th supply section, where m = ⌈l/D⌉, and the relative position inside this section is x = l−(m−1)D, satisfying 0 ≤ x ≤ D.

Let [I₁, I₂, …, I_*n*] be the vector of substation currents. The current distribution coefficient is defined as

ki=IiIL, [k1,k2,…,kn],

where I_*L* is the total traction load current. The distal loop can be equivalently reduced to ZT′, obtained by recursive parallel conversion of transformer and contact-line impedances (Kim et al., 2010).

The nodal equations of the system can be expressed compactly as

A·[I1,I2,…,Im+1]T=[0,0,…,0,IL]T,(4)

where matrix A is determined by system parameters and locomotive position l. Normalizing by I_*L* yields the current distribution vector

A·[k1,k2,…,km+1]T=[0,0,…,0,1]T.(5)

Given locomotive position l, the local circuit index m and distance x within that circuit are obtained from

m=⌈l/D⌉, x=l-(m-1)D,(6)

with D the spacing between substations. Thus the equivalent impedance at the locomotive is

Z=Z0+ZCS∑h=1N∑i=h+1nki+ZTkN+1+xDZOCS∑i=1N+1ki,(7)

and the corresponding load voltage

Ul=E0-ILZ.(8)

Finally, the line loss is obtained as the sum of resistive losses of cables and transformers and reactive losses due to power factor. The procedure is summarized as:

• Obtain phasors of three-phase voltage and current from transformer measurements and convert to the HV side;
• Compute equivalent impedance Z from network parameters and locomotive position l;
• Determine current distribution {k_*i*} and calculate losses of each load;
• Sum all load losses to obtain the total network loss and the contribution of each substation.

3 Dynamic traction load modeling based on train operation diagrams

3.1 Methodology for establishing a traction load model

This section establishes a dynamic traction load model by describing the forces acting on the locomotive and converting them into power demand on the contact wire side.

3.1.1 Traction and braking forces

Locomotives are subject to traction, braking, and resistance forces (Figure 5), where traction and braking do not occur simultaneously. For the CRH2 locomotive (Huang et al., 2016), the traction force TE (kN) and regenerative braking force F_*b* are

TE={176-0.36v,v≤12516250/v,v>125,Fb={172,v≤70172-(v-70)/2,v>70.(9)

3.1.2 Resistance

The running resistance consists of a basic term and additional terms due to gradients, curves, and tunnels:

w=(A+Bv+Cv2)+wg+wc+wt,(10)

with w_*g* = 1000tan(θ), w_*c* = K_*c*/R, and w_*t* = 0.00013L_*t*. Thus the net force on the locomotive is

F=TE-Fb-w.(11)

3.2 Power calculation for locomotive operation

The wheelset output power is

P=F·v·10003.6.(12)

and the equivalent power on the contact line side, considering transmission efficiency η, auxiliary load P_*aux*, and power factor angle φ_*T*, is

{Traction:Pnet=Pη+Paux, Q=Pnet/tanφTCoasting:Pnet=Paux, Q=Pnet/tanφTBraking:Pnet=-Pη+Paux, Q=-Pnet/tanφT.(13)

3.3 Dynamic load generation procedure

Using the train operation diagram, the dynamic load profile is obtained by:

• Integrating operational and locomotive data (traction/braking curves, power factor, P_*aux*, η, running time and speed).
• Determining position X_*n* and speed v_*n* at time T_*n*, then computing force F.
• Calculating active power P_*n* and apparent power S_*n* using Equation 13, forming data rows [T_*n*, X_*n*, v_*n*, P_*n*, S_*n*].

This process yields the time-varying traction load along the line.

4 Example calculation of metro distribution network line losses

4.1 Calculation procedure of line losses

At each sampling instant T_*n*, the algorithm proceeds as follows:

• Input and preprocessing. The equivalent impedance parameters are imported, including the main substation impedance, cable unit-length impedance, transformer equivalent impedance, and contact-line impedance. Based on Section 2, the branch-coefficient matrices [X, K] and the equivalent impedance matrix [X, Z] are generated.
• Time-step initialization. The simulation clock is initialized at T₀. For the current locomotive position X_*n*, the corresponding complex traction power S_*n* is obtained from the load model.
• Equivalent-impedance extraction. Using the position X_*n* and the matrix [X, Z], the equivalent impedance Z_*n* seen from the locomotive is computed.
• Initialization of nodal voltage and current. The voltage and current are initialized as
• Iterative voltage update. For iteration index k: If k = k_max, the point is marked as not convergent and the algorithm proceeds to the next time step. Otherwise, if convergence is declared.
• Current update. Once Un(k+1) converges, the current is updated as
• Acquisition of additional nodal information. Using the converged voltage and current, the voltages, currents, and losses of all other target nodes along the line are computed. These are used for the loss-calculation steps described in Section 4.2.
• Time advancement. The simulation time is updated by

If T_*n*+1>T, the simulation terminates; otherwise, the next time step is computed.

4.2 Raw data

This section selects a line from that in Section 2 for validation. Based on the model of the traction power supply system, Figure 6 shows the traction load distribution at a specific moment on a particular substation along the line. In this system, the primary side of the traction substation is connected to a 220kV grid, with the system short-circuit capacity at the connection point being 4,000 MVA. The secondary side voltage level is 35kV. The traction transformer adopts a Vv connection, with the phase sequence connected as ABC. The transformer capacity is 80 MVA, and the percentage of short-circuit impedance is 10.5%. Each individual traction transformer has a capacity of 10 MVA, with a percentage impedance of 8.5%. In this simulation, the contact wire is selected as the power supply device, and the voltage level is set to 3.5kV. To ensure that the traction power supply system meets operational requirements, the local circuit length is set to 5 km. Considering various on-site factors, the simulation parameters are set as shown in Table 1, with traction transformers placed at intervals of 0 km, 5 km, 10 km, 15 km, and 20 km.

Table 1. Line parameters.

At present, there is a lack of actual data on the AC traction power supply systems of urban rail transit. Therefore, this paper selects the CJ6 locomotive as the operating locomotive, adopts the operation curve of the locomotive on this line calculating line losses. The speed-time curve of a single load operation on this line is shown in Figure 6. Validation is performed for both traction and braking conditions.

4.3 Calculation of losses generated by the load

When the resistance and power supply model are determined, the branch coefficients of each traction substation are obtained based on the method from Section 2. The variation of branch coefficients with train position is shown in Figure 7. By combining line information, the equivalent impedance Z between each position and the 35 kV external power supply is calculated according to (7). After obtaining the distribution of Z, the power flow is solved using the equivalent load model from Section 3, considering the train's operation along the entire line. Based on the locomotive voltage obtained from the power flow analysis, the power characteristics of the main substation and traction substations, as well as the system losses, are evaluated.

As shown in Figure 7 and Table 2, traction substations mainly supply power to their adjacent local loops during train operation. The power distribution exhibits a Gaussian-like pattern, being higher in the middle and lower at both ends, with maximum output when the train passes near a substation. Since substations 1 and 5 only feed one adjacent section, while substations 2–4 supply two neighboring sections, the average output power of substations 2–4 is nearly twice that of substations 1 and 5. Furthermore, as illustrated in Figure 8, because the reactance of the traction network equivalent impedance is much larger than its resistance, reactive power losses dominate over active power losses.

Table 2. Statistical power characteristics of each traction substation.

The loss calculation follows the definitions in Equations 14, 15. At each simulation step i, the system loss is obtained as

ΔSi=35000·I*-Si=ΔPi+jΔQi,(14)

where S_*i* is the complex power delivered to the locomotive load, and I^* is the conjugate of the source current from the equivalent circuit. The relative loss level is then expressed as

δi=|ΔSi||35000·I*|×100%.(15)

These equations are applied to compute the active loss ΔP_*i*, reactive loss ΔQ_*i*, and total loss ratio δ_*i* during train operation. The results under different train positions are presented in Figures 8, 9.

The supply capability of the traction system is mainly constrained by the permissible voltage range of locomotives. By defining the ratios of maximum and minimum line voltages to the nominal level, the acceptable range is set as [0.76, 1.16]. Simulation results show that as voltage level increases, the minimum number of traction substations required decreases. For instance, when the voltage is no less than 5 kV, two substations are sufficient to cover the entire line. Conversely, for a fixed number of substations, the minimum feasible voltage level decreases as more substations are added.

In terms of efficiency, the daily active power losses between the main substation and traction loads decrease with both higher voltage levels and more substations, as illustrated in Figure 9. Note that the calculation in this section is purely mechanism-based. The accuracy of the results depends on the correctness of cable and transformer parameters. In real systems, measurement-based or data-driven models can be used to calibrate uncertain parameters, and thus can serve as a complementary tool to enhance the accuracy of the proposed mechanism-based method.

In summary, increasing the supply voltage and adding more traction substations are both effective strategies for reducing system losses. Higher voltage levels reduce the required number of substations, lowering construction costs, while fewer substations may increase overall losses. This highlights the essential role of theoretical line loss calculations, which directly reflect system mechanisms and provide a reliable foundation for planning and optimization of metro traction networks.

5 Conclusion

This paper investigates a theoretical line loss calculation method for metro traction distribution networks, using actual metro load data as the case study. The research highlights how theoretical line loss models, rather than statistical methods, directly capture the mechanisms of losses and thus provide a solid foundation for system evaluation and planning. The main findings are as follows:

• By analyzing the operational characteristics of the traction network, the line loss calculation formula was derived using the loop current method. This clarified the composition of losses and explained their relationship with dynamic metro loads.
• Considering the network's structural features, a method was established to allocate the cross terms of line loss to individual loads, enabling more precise loss attribution across substations and circuits.

Future work can be expanded in the following directions:

• Develop optimization schemes for targeted loss reduction indicators, thereby providing actionable strategies for system operation and planning.
• Explore the long-term relationship between line losses and load curve characteristics, building line loss models expressed through curve parameters and comparing them with traditional statistical approaches.

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Footnotes

Edited by: Chengxi Liu, Wuhan University, China

Reviewed by: Lijun Yang, Yanshan University, China

Thai Nguyen, Ho Chi Minh City University of Transport, Vietnam

Dong Doan Van, Ho Chi Minh City University of Transport, Vietnam

Jian Zhang, The University of Tennessee, Knoxville, United States