A Method for Online Locomotive Positioning and Voltage Stability Monitoring in AT Traction Network Based on PMU Measurements

Abstract

This invention discloses a method for online positioning and voltage stability monitoring of locomotives in an AT traction network based on PMU measurements. Specifically, it involves: acquiring raw current and voltage data for each phase of the AT station; calculating the locomotive's current position using the measurement data; iteratively correcting the locomotive position and phase measurement data using state estimation; calculating the current and voltage on the locomotive; and then obtaining the locomotive's active and reactive power. The locomotive is treated as a constant power load, and the main power grid and traction network at the locomotive's front end are considered as a black box. Thevenin equivalents are performed to obtain the equivalent voltage and equivalent impedance. Using the obtained equivalent voltage and equivalent impedance, the limiting power and the corresponding limiting voltage at the locomotive's current position can be calculated, thus obtaining the power margin. This invention features rapid and accurate calculations, good resistance to abnormal data, and provides accurate real-time power margin; it also achieves effective online real-time monitoring of traction network voltage stability.

Description

Technical Field

[0001] This invention belongs to the field of traction power grid monitoring, and in particular relates to a method for online positioning and voltage stability monitoring of AT traction locomotives based on PMU measurements. Background Technology

[0002] Voltage instability in power systems can lead to large-scale blackouts, causing huge economic losses and severely impacting social life. Voltage instability in the traction grid can cause locomotive shutdowns, traffic paralysis, and even serious traffic accidents. The traction grid voltage stability monitoring method based on PMU measurements directly calculates power margin and assesses voltage stability using only relevant measurement information, eliminating the need for power flow calculations, thus reducing network computational load. It can be applied online to traction grid voltage stability monitoring, which is of great significance for ensuring the safe and stable operation of the power system.

[0003] Online monitoring methods for grid voltage stability based on local voltage stability indices typically involve first determining the equivalent system parameters of a single power source's power transmission, then calculating the system's local voltage stability indices to determine the grid's voltage stability. In this process, the accuracy of the calculated equivalent system parameters determines the effectiveness of the local voltage stability indices used to determine system stability. These methods are categorized into two types based on whether the information required for calculating the equivalent system parameters is from the entire grid or a specific region. The method using partial grid region information to calculate the equivalent system parameters can be further divided into two types: first, represented by the direct line model, which generally cannot accurately handle the time-varying characteristics of actual loads and cannot track real-time load changes; second, the Thevenin equivalence method, which requires voltage and current phasors from two or more system states and uses parameter identification methods to estimate the equivalent system parameters of the single power source's power transmission. Theoretically, the Thevenin equivalence method can accurately handle the time-varying characteristics of actual loads. However, when continuous excitation conditions are not met, the identified equivalent system parameters of the single power source's power transmission may have significant errors, easily leading to misjudgments.

[0004] The voltage stability of the traction power grid, especially in mountainous areas with weak power grids, has always threatened the safe operation of locomotives. In the past, railway personnel relied on experience to judge when to start trains, without a specific calculation method to quantify it. The voltage operation requirement of the traction power grid is 19KV to 29KV. However, in some special cases, although the voltage meets the minimum operating requirements, the maximum power that the line can withstand has reached the critical value. Fluctuations in load power can easily cause voltage collapse. Summary of the Invention

[0005] To address the aforementioned problems, this invention provides a method for online positioning and voltage stability monitoring of locomotives in AT traction networks based on PMU measurements.

[0006] The present invention provides a method for online positioning and voltage stability monitoring of locomotives in an AT traction network based on PMU measurement, comprising the following steps:

[0007] Step 1: Obtain the topology and related component parameters of the traction power supply system, including: the impedance matrix per unit length of the traction network line, and the distances between traction substations, AT substations, and section substations.

[0008] Step 2, PMU measurement configuration: Set up PMUs at the beginning and end of each AT segment and at each cross line; obtain the current and voltage measurement data of each phase at the PMU configuration location.

[0009] Step 3: Based on the topology and component parameters described in Step 1, and the measurement data described in Step 2, taking into account measurement uncertainties, perform state estimation to estimate the locomotive's position, voltage, and current.

[0010] Step 4: Based on the locomotive position and the locomotive's voltage and current described in Step 3, identify the Thevenin equivalent circuit of the traction network seen from the locomotive port, including the equivalent potential E, equivalent reactance X, and equivalent resistance R.

[0011] Step 5: Determine the locomotive's maximum power and corresponding critical voltage based on the Thevenin equivalent circuit described in Step 4.

[0012] Step 6: Based on the limit power described in Step 5 and the current power of the locomotive, obtain the power margin of the locomotive at each position in the traction network and the power margin of the traction network.

[0013] Furthermore, step 3, state estimation, specifically involves:

[0014] (1) Determine the section where the locomotive is located.

[0015] In a fully parallel traction power grid with m AT stations in each direction, the locomotive's location is determined by the phase of the T-phase current in the contact network between the AT stations. Within this section, the direction of the current flowing from the AT station to the contact network is defined as positive. When the T-phase current of the contact network measured by the k-th AT station and the (k+1)-th AT station are both positive, the locomotive is located between the k-th AT station and the (k+1)-th AT station. The same applies to multi-vehicle operation.

[0016] (2) Preliminary estimate of locomotive position.

[0017] Given that the distance between two AT stations is L km, and assuming the train is located d km from the beginning of the AT segment, the voltage and current equations within this AT segment are as follows:

[0018]

[0019] In the formula, Z T Z R Z represents the self-impedance on phase T and phase R, respectively. RT Z TF Z RF These represent the mutual impedances of R relative to phase T, T relative to phase F, and R relative to phase F, respectively. This indicates the current in the contact wire, rails, and return line at the beginning of section AT. This indicates the current in the end contact wire, rail, and return line. These represent the voltage between the contact wire and the rail at the beginning and end of section AT, respectively. This indicates the current flowing through the car. This indicates the locomotive voltage.

[0020] Except for d, which is unknown, the rest are component parameters and measurements.

[0021] The initial estimate of the locomotive's position, d, can be obtained by solving the problem. (0) :

[0022]

[0023] (3) The locomotive position, locomotive voltage, current, active power and reactive power are obtained through state estimation.

[0024] Using the locomotive position described in (2) as the initial value, iterative state estimation is performed to obtain the locomotive position and state variables such as locomotive voltage and current.

[0025] The state estimation measurement equation is:

[0026]

[0027] In the formula, z, x, and ε represent the measurement, state vector, and error vector, respectively. ε T It is [ε1ε2ε3ε4ε5ε6ε7ε8].

[0028] The system's observation matrix H is:

[0029]

[0030] The state vector estimate is obtained using the least squares method:

[0031]

[0032] Using the state vector estimate, the locomotive position d is calculated using the same locomotive position calculation formula. (1) .

[0033]

[0034] Locomotive position update formula:

[0035] diff (i) =d (i) -d (i-1)

[0036] If diff (i) If the convergence condition is met, the iteration ends, and the locomotive position is obtained as d. (i) Based on the state vector estimate after the i-th iteration, the computer calculates the vehicle's voltage, current, active power, and reactive power.

[0037] Voltage at both ends of the vehicle:

[0038]

[0039] The current flowing through the car is:

[0040]

[0041] The locomotive's active power and reactive power are respectively:

[0042]

[0043] Otherwise, update the locomotive position and perform state estimation iteration:

[0044]

[0045] Furthermore, the specific steps for step 4, Thevenin equivalence, are as follows:

[0046] Thevenin equivalent system equations are:

[0047]

[0048] (1) In the traction power supply system, the equivalent impedance Z th The corresponding R th and X th Take a series of inductance values ​​X in the vicinity thi and a series of inductance values ​​R thi .

[0049] (2) For R thi X thi For each selected value, the equivalent voltage E at each data sampling point is calculated using the Thevenin equivalent system equation. thi (k).

[0050] (3) For each E thi Calculate their average value:

[0051]

[0052] Calculate the root mean square error:

[0053]

[0054] Choose ε i smallest and its two adjacent points and The equivalent impedance R is obtained by performing a quadratic fitting. th +jX th The optimal solution is R+jX.

[0055] (4) Calculate E using R+jX th (k), to obtain the equivalent voltage:

[0056]

[0057] When the equivalent impedance of the load and the equivalent impedance of the system are the same, if the critical voltage is greater than 19kV, the locomotive's power limit is:

[0058]

[0059] The corresponding critical voltage:

[0060]

[0061] In the formula, θ represents the phase angle of the locomotive port voltage. This indicates the phase angle of the locomotive port current.

[0062] If the critical voltage is less than 19kV, then the locomotive's power limit is:

[0063]

[0064] Furthermore, step 6 requires determining the power margin at each location of the traction power grid based on the limit power calculated in step 5, including the locomotive power margin P. m The calculation method is as follows:

[0065] P m =P max -P

[0066] Once the power margin is obtained, it can be determined whether the train can start at each position, and if so, what the maximum power of the train will be.

[0067] The beneficial technical effects of this invention compared to the prior art are as follows:

[0068] (1) The present invention uses the current and voltage data obtained by AT station measurement, and after filtering, the position of the locomotive can be calculated. The calculation process is fast and accurate, and has good anti-abnormal data capability.

[0069] (2) The Thevenin equivalent of this invention takes into account the high resistance characteristics of the traction power grid, which further improves the accuracy of Thevenin equivalent.

[0070] (3) The present invention utilizes power margin to monitor voltage stability online in traction power grids, providing accurate real-time power margin. Attached Figure Description

[0071] Figure 1 This is a topology diagram of the model built for the simulation of this invention.

[0072] Figure 2 This is a topology diagram of the line between two AT stations.

[0073] Figure 3 This is a schematic diagram of Thevenin's isometry.

[0074] Figure 4 The graph shows the variation of the limiting power with distance in Example 1.

[0075] Figure 5 The diagram shows the variation of the locomotive's maximum power with distance for Example 2, where one locomotive is fixed at the end of the uphill section.

[0076] Figure 6 This is a comparison diagram of the locomotive's position before and after state estimation and its actual position in Example 3. Detailed Implementation

[0077] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0078] This invention discloses an online locomotive positioning and voltage stability monitoring method for AT traction networks based on PMU measurements. First, it acquires the original current and voltage data of each phase at the AT station. Using the measured data, it calculates the locomotive's current position. State estimation iteratively corrects the locomotive position and the phase measurement data, and calculates the current and voltage on the locomotive, thereby obtaining the locomotive's active and reactive power. The locomotive is treated as a constant power load, and the main power grid and traction network at the locomotive's front end are considered as a black box. Thevenin equivalents are performed to obtain the equivalent voltage and equivalent impedance. Using the obtained equivalent voltage and equivalent impedance, the limiting power and the corresponding limiting voltage at the locomotive's current position can be calculated, thus obtaining the power margin. Specifically:

[0079] Step 1: Obtain the topology and related component parameters of the traction power supply system, including: the impedance matrix per unit length of the traction network line, and the distances between traction substations, AT substations, and section substations.

[0080] Step 2, PMU configuration: Install PMUs at the beginning and end of each AT segment and at each cross-connection; obtain the current and voltage measurement data of each phase at the PMU configuration location. The line topology diagram between the two AT stations is as follows. Figure 2 As shown.

[0081] Step 3: Based on the topology and component parameters described in Step 1, and the measurement data described in Step 2, taking into account measurement uncertainties, perform state estimation to estimate the locomotive's position, voltage, and current.

[0082] The state estimation is as follows:

[0083] (1) Determine the section where the locomotive is located.

[0084] In a fully parallel traction power grid with m AT stations in each direction, the locomotive's location is determined by the phase of the T-phase current in the contact network between the AT stations. Within this section, the direction of the current flowing from the AT station to the contact network is defined as positive. When the T-phase current of the contact network measured by the k-th AT station and the (k+1)-th AT station are both positive, the locomotive is located between the k-th AT station and the (k+1)-th AT station. The same applies to multi-vehicle operation.

[0085] (2) Preliminary estimate of locomotive position.

[0086] Given that the distance between two AT stations is L km, and assuming the train is located d km from the beginning of the AT segment, the voltage and current equations within this AT segment are as follows:

[0087]

[0088] In the formula, Z T Z R Z represents the self-impedance on phase T and phase R, respectively. RT Z TF Z RF These represent the mutual impedances of R relative to phase T, T relative to phase F, and R relative to phase F, respectively. This indicates the current in the contact wire, rails, and return line at the beginning of section AT. This indicates the current in the end contact wire, rail, and return line. These represent the voltage between the contact wire and the rail at the beginning and end of section AT, respectively. This indicates the current flowing through the car. This indicates the locomotive voltage.

[0089] Except for d, which is unknown, the rest are component parameters and measurements.

[0090] The initial estimate of the locomotive's position, d, can be obtained by solving the problem. (0) :

[0091]

[0092] (3) The locomotive position, locomotive voltage, current, active power and reactive power are obtained through state estimation.

[0093] Using the locomotive position described in (2) as the initial value, iterative state estimation is performed to obtain the locomotive position and state variables such as locomotive voltage and current.

[0094] The state estimation measurement equation is:

[0095]

[0096] In the formula, z, x, and ε represent the measurement, state vector, and error vector, respectively. ε T It is [ε1ε2ε3ε4ε5ε6ε7ε8].

[0097] The system's observation matrix H is:

[0098]

[0099] The state vector estimate is obtained using the least squares method:

[0100]

[0101] Using the state vector estimate, the locomotive position d is calculated using the same locomotive position calculation formula. (1) .

[0102]

[0103] Locomotive position update formula:

[0104] diff (i) =d (i) -d (i-1)

[0105] If diff (i) If the convergence condition is met, the iteration ends, and the locomotive position is obtained as d. (i) Based on the state vector estimate after the i-th iteration, the computer calculates the vehicle's voltage, current, active power, and reactive power.

[0106] Voltage at both ends of the vehicle:

[0107]

[0108] The current flowing through the car is:

[0109]

[0110] The locomotive's active power and reactive power are respectively:

[0111]

[0112] Otherwise, update the locomotive position and perform state estimation iteration:

[0113]

[0114] Step 4: Based on the locomotive position and its voltage and current as described in Step 3, identify the Thevenin equivalent circuit of the traction network seen from the locomotive port, including the equivalent potential E, equivalent reactance X, and equivalent resistance R. A Thevenin equivalent circuit diagram is shown below. Figure 3 As shown.

[0115] The specific method for Thevenin equivalence is as follows:

[0116] Thevenin equivalent system equations are:

[0117]

[0118] (1) In the traction power supply system, the equivalent impedance Z th The corresponding R th and X th Take a series of inductance values ​​X in the vicinity thi and a series of inductance values ​​R thi .

[0119] (2) For R thi X thi For each selected value, the equivalent voltage E at each data sampling point is calculated using the Thevenin equivalent system equation. thi (k).

[0120] (3) For each E thi Calculate their average value:

[0121]

[0122] Calculate the root mean square error:

[0123]

[0124] Choose ε i smallest and its two adjacent points and The equivalent impedance R is obtained by performing a quadratic fitting. th +jX th The optimal solution is R+jX.

[0125] (4) Calculate E using R+jX th (k), to obtain the equivalent voltage:

[0126]

[0127] When the equivalent impedance of the load and the equivalent impedance of the system are the same, if the critical voltage is greater than 19kV, the locomotive's power limit is:

[0128]

[0129] The corresponding critical voltage:

[0130]

[0131] In the formula, θ represents the phase angle of the locomotive port voltage. This indicates the phase angle of the locomotive port current.

[0132] If the critical voltage is less than 19kV, then the locomotive's power limit is:

[0133]

[0134] Step 5: Determine the locomotive's maximum power and corresponding critical voltage based on the Thevenin equivalent circuit described in Step 4.

[0135] Step 6: Based on the limit power described in Step 5 and the current power of the locomotive, obtain the power margin of the locomotive at each position in the traction network and the power margin of the traction network.

[0136] Locomotive power margin P m The calculation method is as follows:

[0137] P m =P max -P

[0138] Once the power margin is obtained, it can be determined whether the train can start at each position, and if so, what the maximum power of the train will be.

[0139] Example:

[0140] An experimental example using an AT traction station built with MATLAB / SIMULINK to collect experimental data. The topology diagram of the AT traction station is shown below. Figure 1 As shown in the diagram. The 220kV side represents an infinite power supply. The impedance matrix per unit length of the traction network line, autotransformer parameters, and distances between traction substations, AT substations, and section substations were obtained from field experiments. In the single-car example, the locomotive model represents a continuously increasing power load. In the multi-car example, the fixed locomotive model represents a constant power load with a power of 10MW, while the moving locomotive model represents a continuously increasing power load. PMU measuring devices are installed at both ends of each AT substation and on each parallel line to measure the current and voltage information of each phase. In the following examples, Pm uses 100MW as the baseline value.

[0141] Calculation example 1:

[0142] like Figure 4 As shown, to verify the accuracy of the equivalent values ​​and the effectiveness of the power margin calculation in this invention, the appendix... Figure 1 The load is continuously increased every 1 km between the two phases of the uplink TR to determine the limiting power, and the data measured by the PMU is stored for use in the method of this invention. Table 1 shows a comparison between the limiting power obtained from the simulation and the limiting power obtained by the method of this invention.

[0143] Table 1 Comparison of Limiting Power in Example 1

[0144] Location (km) Pm (simulation) Pm (This invention) error Location (km) Pm (simulation) Pm (This invention) error 1 0.5834 0.5754 0.008 16 0.4944 0.4839 0.0105 2 0.5660 0.5569 0.0091 17 0.4951 0.4851 0.01 3 0.5539 0.5441 0.0098 18 0.5003 0.4909 0.0094 4 0.5467 0.5366 0.0101 19 0.5101 0.5041 0.006 5 0.5440 0.5337 0.0103 20 0.5251 0.5200 0.0051 6 0.5455 0.5353 0.0102 21 0.4983 0.4912 0.0071 7 0.5514 0.5414 0.01 22 0.4780 0.4672 0.0108 8 0.5618 0.5522 0.0096 23 0.4632 0.4518 0.0114 9 0.5774 0.5686 0.0088 24 0.4527 0.4411 0.0116 10 0.5990 0.5917 0.0073 25 0.4462 0.4346 0.0116 11 0.5640 0.5546 0.0094 26 0.4433 0.4318 0.0115 12 0.5381 0.5290 0.0091 27 0.4439 0.4328 0.0111 13 0.5192 0.5105 0.0087 28 0.4480 0.4373 0.0107 14 0.5061 0.4958 0.0103 29 0.4558 0.4458 0.01 15 0.4980 0.4874 0.0106 30 0.4679 0.4591 0.0088

[0145] Calculation example 2:

[0146] like Figure 5 As shown, to verify the accuracy of the equivalent values ​​and the effectiveness of the power margin calculation in this invention, the appendix... Figure 1 A locomotive with a constant power of 10MW was added at a distance of 30km between the two phases of the uplink TR. The load was continuously increased every 1km in the downlink to determine the ultimate power, and the data measured by the PMU was stored for use in the method of this invention. Table 2 shows a comparison between the ultimate power obtained from the simulation and the ultimate power obtained by the method of this invention.

[0147] Table 2 Comparison of Limiting Power in Example 2

[0148]

[0149]

[0150] Calculation example 3:

[0151] like Figure 6 As shown in Table 3, in order to verify the locomotive position correction proposed in this invention through state estimation iteration, the locomotive position before and after state estimation was compared with the actual position.

[0152] Table 3 Comparison of Locomotive Position Before and After State Estimation with Actual Position

[0153]

[0154] It can be seen that the average error before state estimation is 0.3073 km, and the maximum error is 1.2521 km. After state estimation, the average error is 0.0777 km, and the maximum error is 0.2321 km.

[0155] In summary, this invention can determine the locomotive's location based on measurement data. The determination process is rapid and accurate, with strong anti-interference capabilities. It solves the problem of judging voltage stability in the traction network based on experience, quantifies voltage stability from the perspective of power margin, and provides accurate limit power and the limit voltage corresponding to the limit power, thereby achieving effective online real-time monitoring of traction network voltage stability.

Claims

1. A method for online positioning and voltage stability monitoring of locomotives in an AT traction network based on PMU measurements, characterized in that, Includes the following steps: Step 1: Obtain the topology and related component parameters of the traction power supply system, including: the impedance matrix per unit length of the traction network line, and the distances between traction substations, AT substations, and section substations; Step 2, PMU measurement configuration: Set up PMUs at the beginning and end of each AT segment and at each cross line; obtain the current and voltage measurement data of each phase at the PMU configuration location; Step 3: Based on the topology and component parameters described in Step 1, and the measurement data described in Step 2, taking into account measurement uncertainties, perform state estimation to estimate the locomotive's position, voltage, and current. The state estimation is specifically as follows: (1) Determine the section where the locomotive is located; In a fully parallel traction power grid with m AT stations in each direction, the locomotive's location is determined by the phase of the T-phase current in the contact network between the AT stations. Within this section, the direction of the current flowing from the AT station to the contact network is defined as positive. When the T-phase current of the contact network measured by the k-th AT station and the (k+1)-th AT station are both positive, the locomotive is located between the k-th AT station and the (k+1)-th AT station. The same applies to the case of multiple locomotives operating. (2) Preliminary estimation of locomotive position; Given that the distance between two AT stations is L km, and assuming the train is located d km from the beginning of the AT segment, the voltage and current equations within this AT segment are as follows: In the formula, Z T Z R Z represents the self-impedance on phase T and phase R, respectively. TR Z TF Z RF These represent the mutual impedances of phase T relative to phase R, phase T relative to phase F, and phase R relative to phase F, respectively. This indicates the current in the contact wire, rails, and return line at the beginning of section AT. This indicates the current in the end contact wire, rail, and return line. These represent the voltage between the contact wire and the rail at the beginning and end of section AT, respectively. This indicates the current flowing through the car. Indicates the locomotive voltage; Except for d, which is unknown, all others are component parameters and measurements; The initial estimate of the locomotive's position, d, can be obtained by solving the problem. (0) : (3) Locomotive position, locomotive voltage, current, active power and reactive power are obtained through state estimation; Using the locomotive position described in (2) as the initial value, iterative state estimation is performed to obtain the locomotive position and locomotive voltage, current and other state variables; The state estimation measurement equation is: In the formula, z, x, and ε represent the measurement, state vector, and error vector, respectively. ε T is [ε1ε2ε3ε4ε5ε6ε7ε8]; The system's observation matrix H is: The state vector estimate is obtained using the least squares method: Using the state vector estimate, the locomotive position d is calculated using the same locomotive position calculation formula. (1) ; Locomotive position update formula: diff (i) =d (i) -d (i-1) If diff (i) If the convergence condition is met, the iteration ends, and the locomotive position is obtained as d. (i) Based on the estimated state vector values ​​after the i-th iteration, the computer calculates the vehicle's voltage, current, active power, and reactive power. Voltage at both ends of the vehicle: The current flowing through the car is: The locomotive's active power and reactive power are respectively: Otherwise, update the locomotive position and perform state estimation iteration: Step 4: Based on the locomotive position and the locomotive's voltage and current described in Step 3, identify the Thevenin equivalent circuit of the traction network seen from the locomotive port, including the equivalent potential E, equivalent reactance X, and equivalent resistance R. Step 5: Determine the locomotive's maximum power and corresponding critical voltage based on the Thevenin equivalent circuit described in Step 4; Step 6: Based on the limit power described in Step 5 and the current power of the locomotive, obtain the power margin of the locomotive at each position in the traction network and the power margin of the traction network.

2. The method for online positioning and voltage stability monitoring of locomotives in AT traction network based on PMU measurement according to claim 1, characterized in that, The specific procedure for step 4, Thevenin equivalence, is as follows: Thevenin equivalent system equations are: (1) In the traction power supply system, the equivalent impedance Z th The corresponding R th and X th Take a series of inductance values ​​X nearby thi and a series of resistance values ​​R thi ; (2) For R thi X thi For each selected value, the equivalent voltage E at each data sampling point is calculated using the Thevenin equivalent system equation. thi (k); (3) For each E thi Calculate their average value: Calculate the root mean square error: Choose ε i smallest and its two adjacent points and The equivalent impedance R is obtained by performing a quadratic fitting. th +jX th The optimal solution is R+jX; (4) Calculate E using R+jX th (k), to obtain the equivalent voltage: When the equivalent impedance of the load and the equivalent impedance of the system are the same, if the critical voltage is greater than 19kV, the locomotive's power limit is: The corresponding critical voltage: In the formula, θ represents the phase angle of the locomotive port voltage. The phase angle representing the locomotive port current; If the critical voltage is less than 19kV, then the locomotive's power limit is:

3. The method for online positioning and voltage stability monitoring of locomotives in an AT traction network based on PMU measurement according to claim 2, characterized in that, Step 6 obtains the power margin at each location of the traction power grid based on the limit power calculated in step 5, and the locomotive power margin P. m The calculation method is as follows: P m =P max -P Once the power margin is obtained, it can be determined whether the train can start at each position, and if so, what the maximum power of the train will be.

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