# Balancing the Active Power of a Railway Traction Power Substation with an sp-RPC

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## Abstract

**:**

## 1. Introduction

## 2. sp-RPC Concept

## 3. sp-RPC Proposed and Operation Principle

#### 3.1. sp-RPC Topology

_{dc}, and a voltage of 7 kV on the dedicated dc-link of each submodule, v

_{dc_xk}, and v

_{dc_yk}, k being the number of the submodule of the x-side or y-side. The value of 1 kV was defined in order to obtain a voltage value on the low-voltage common dc-link. In this way, it would be easier to integrate a solar photovoltaic system and an energy storage system with a lower voltage interface.

_{spRPCx}and L

_{spRPCy}, of the sp-RPC with the catenary [26]. Determining L

_{spRPCk}, with k being equal to x or y depending on the x or y side, requires taking into account the voltage on the dc-link of each submodule. Being a cascaded converter with 13 levels, the inductor current ripple comes from the transition between adjacent levels with 7000 V (V

_{dc_k}) on the V

_{dc}. Additionally, considering six submodules with a switching frequency of 1 kHz with a unipolar modulation at the output of each full-bridge, the resulting output frequency of the sp-RPC is 12 kHz (f

_{sw}). Finally, a 3% ripple in the inductor (∆i

_{spRPCk}) was considered, obtaining a minimum value in the coil of 48.6 mH. This value was rounded off to 50 mH

_{spRPCk}(25,000 V) the rated voltage of the sp-RPC at the point of connection to the catenary, and i

_{spRPCk}(100 A) the maximum rated current of the sp-RPC. In turn, ω (2 π 50 rad/s) relates the frequency of the catenary voltage, N (6) is the number of submodules, V

_{dc}(7000 V) is the voltage at the dc-link of each submodule, and the desired voltage ripple is defined by ∆v

_{dc}(20 V peak to peak). The value obtained was 9.5 mF and was adjusted to 10 mF. The capacitor C

_{1}was adjusted with the aid of the simulation results until the best response was obtained.

#### 3.2. Sp-RPC Control Algorithm

_{S_PG}= 5 Hz). Due to the inherent concept, the sp-RPC will have to conciliate the instantaneous monitoring of the operating variables of the power electronic converters, with periodic monitoring, caused by the time delay of sending information, of the active powers of each TPS. In this way, throughout this topic, all the implemented algorithms are presented in order to provide a continuous operation of the sp-RPC. This topic starts by analyzing the different variables to be controlled, the interaction between different control algorithms, the determination of the reference values for each power electronics converter, and the control signals for the power semiconductors.

_{PGx}and P

_{PGy}, the next step involves determining the reference active power P

_{spRPCx}* and P

_{spRPCy}*. This determination is based on Equations (6) and (7), explained in the next topic. Next, these values are used to determine the i

_{spRPCx}* and i

_{spRPCy}* operating currents (based on Equations (8) and (9), also explained in the following topic). Once this sequence is complete, it is checked if an updated value has been received in the TPS operating power. If so, the average values of the TPS are updated, repeating the entire process. Otherwise, the values of P

_{PGx}and P

_{PGy}are maintained, updating P

_{spRPCx}* and P

_{sRPCy}* only based on the instantaneous operation values of sp-RPC.

#### 3.2.1. Dynamic sp-RPC Control

_{PGx}, and the y-side, P

_{PGy}. Due to the distance from the TPS to the sp-RPC, these parameters are updated periodically. In this way, the determination of the average value of the operating power between the two TPS, P

_{avg}, represented in Equation (3), is affected by the time of sending this information (200 ms).

_{av}g value has been determined, it is necessary to determine the desired energy transfer between TPS and P

_{trans}. That is, if it is intended to absorb energy from the TPS on the x-side and inject it into the y-side or vice versa. For this, Equation (4) is necessary to determine the active power of the spRPC on the y-side, P

_{spRPCy}. By adding P

_{PGy}to P

_{spRPCy}and subtracting the value of P

_{avg}, it is possible to verify the intended energy flow. If a positive result is obtained, the power flow is carried out from the x-side TPS to the y-side TPS. Otherwise, the energy flow is carried out in the opposite way.

_{trans}value is determined, it remains to define the average operating powers for spRPCx and spRPCy, P

_{spRPCx}* and P

_{spRPCy}*, respectively. Equation (5) determines the value of P

_{spRPCx}*, while Equation (6) is used for the P

_{spRPCy}*.

_{reg}represents the regulating power to keep the common dc-link regulated. In turn, P

_{reg_spRPCx}represents the regulating power to keep all the dedicated dc-links of the spRPCx, while P

_{reg_spRPCy}represents the regulating power to keep all the dedicated dc-links of the spRPCy. The determination of these regulating powers is analyzed in the following topic. Once limited, the portion of the regulating powers used to regulate the dc-links that constitute the full sp-RPC system, as represented in Equation (7), is subsequently added.

_{spRPCx}* and P

_{spRPCy}* are set, it is necessary to determine the reference current values i

_{spRPCx}* and i

_{spRPCy}*. Equation (8) is used to determine i

_{spRPCx}*, using the values determined by the PLL control algorithm (Alg_PLL): ${\widehat{V}}_{PLLx}$ (the peak value) and v

_{PLL}(sinusoidal waveform with amplitude equal to 1). The factor of 2 arises from the ratio of the peak and rms value of the voltage and current values. An analogous approach is taken in Equation (9) to determine i

_{spRPCy}*.

#### 3.2.2. DC-Link Regulation

_{dc}*, and the average value obtained, V

_{dc}. Finally, the V_dc_error portion is used with a PI controller in order to determine the final regulating power Preg. The representative block diagram of the algorithm implemented for regulating the dc-link voltage is shown in Figure 6.

_{reg_spRPCx}, that the spRPCx needs to keep all the voltage on all the dc-link regulated. The same methodology is replicated for spRPCy.

_{spRPCx}* and P

_{spRPCy}* based on the previous Equations (7) and (6). Once P

_{spRPCx}* has been determined, it is necessary to determine the average operating power for each submodule in order to maintain not only the regulation of the dc-link but essentially impose an equal power distribution between submodules. In this way, the power value is defined as a function of the number of existing submodules, dividing P

_{spRPCx}by N. The resulting portion is then added to the representative regulating power of each submodule, thus creating a reference power. The average operating power of each submodule is then compared with the respective average power, resulting in an error value. Finally, this value is used in a proportional-integral (PI) controller in order to generate a phase angle capable of minimizing the error obtained, as illustrated in Figure 8.

#### 3.2.3. PLL Control Algorithm

_{pll}and V

_{pll}. The v

_{pll}signal corresponds to a unitary signal and is synchronized with the fundamental component of the catenary. In turn, V

_{pll}represents the amplitude of the input signal. The v

_{pll}signal can be used as a reference sinusoidal signal for current control. V

_{pll}will allow the amplitude of the reference current to be determined, knowing the desired operating power.

#### 3.2.4. Predictive Current Control Algorithm

_{out}, is the error current, i

_{error}, Equation (11) was deduced by introducing the characteristic equation of the inductor v

_{L}.

_{a}= 1/T

_{a}, it is possible to obtain the equation:

_{error}[k] defines the difference between i*[k] and i

_{out}[k], Equation (13) can be simplified to Equation (14).

#### 3.2.5. PWM Techniques for Multilevel Converter

## 4. sp-RPC Simulation Results

#### 4.1. Case Analysis Scenarios

- In Figure 12a, the load on the x-side presents an average active power P
_{Loadx}= 0.998 MW, while the load on the y-side presents P_{Loady}= 6.12 MW. Regarding the currents, it is possible to verify that i_{PGx}presents an amplitude of 56.54 A, while i_{PGy}presents an amplitude of 357.2 A. This case represents the event (i); - In Figure 12b, the load on the x-side presents an average active power P
_{Loadx}= 0.998 MW, while the load on the y-side presents P_{Loady}= −3.88 MW. Regarding the currents, it is possible to verify that i_{PGx}presents an amplitude of 56.54 A, while i_{PGy}presents an amplitude of 228.4 A. This case represents the event (ii); - In Figure 12c, the load on the x-side presents an average active power P
_{Loadx}= 0.998 MW, while the load on the y-side presents P_{Loady}= 4.12 MW. Regarding the currents, it is possible to verify that i_{PGx}presents an amplitude of 56.54 A, while i_{PGy}presents an amplitude of 243 A. This case represents the event (iii).

#### 4.2. Traction Power Substation Active Power Analyses

_{PGx}and P

_{PGy}initially present different values, and at 0.4 s, the proposed control algorithm is activated, being able to verify that P

_{PGx}and P

_{PGy}converge to an average value of 3.6 MW. At instant 1 s, the locomotive on the y-side starts braking (P

_{Loady}< 0), generating a new unbalance between P

_{PGx}and P

_{PGy}. When the control algorithm updates the reference values for the sp-RPC, it is possible to verify that P

_{PGx}and P

_{PGy}converge to similar values, presenting an average value of −1.4 MW. Finally, at the instant 2 s, the locomotive on the y-side starts to accelerate, P

_{Loady}varies again, and as soon as the reference values are updated, it is possible to verify that the average values of P

_{PGx}and P

_{PGy}converge again, presenting an average value of 2.6 MW. In Figure 13b, it is possible to see that the powers P

_{spRPCx}and P

_{spRPCy}always follow the reference values P

_{spRPCx}* and P

_{spRPCy}*. Figure 13c shows the variation of the dc voltage on the different dc-links over time. Regarding the common dc-link, it is possible to verify that the v

_{dc}remains regulated to an average value of 1 kV. In turn, the remaining dc voltages that compose each submodule, from v

_{dc_x}

_{1}to v

_{dc_x}

_{6}and from v

_{dc_y}

_{1}to v

_{dc_y}

_{6}, are also regulated to an average value of 7 kV.

#### 4.3. Detailed Analysis of the sp-RPC

_{PGx}, and v

_{PGy}, as well as the voltages on each side of the sp-RPC, v

_{spRPCx}, and v

_{spRPCy}. Moreover, it is possible to check the dc-link voltages from v

_{dc_x}

_{1}to v

_{dc_x}

_{6}and from v

_{dc_y}

_{1}to v

_{dc_y}

_{6}, showing an initial disturbance caused by the start of the operation but maintaining a constant desired average value of 7 kV. In Figure 14b, the multilevel voltages v

_{outx}and v

_{outy}are presented, being able to verify the 13 voltage levels generated by the modular power converter based on SST. Nevertheless, it is also possible to see that despite the initial perturbation, v

_{dc}remains regulated at a value of 1 kV. Finally, by analyzing Figure 14c, it is possible to verify that as soon as the sp-RPC starts operating, synthesizing i

_{spRPCx}and i

_{spRPCy}, it was possible to equalize the amplitudes of the i

_{PGx}and i

_{PGy}.

_{PGx}and v

_{PGy}, as well as the voltages on each side of the sp-RPC, v

_{spRPCx}and v

_{spRPCy}. Moreover, it is possible to see that v

_{dc_x}

_{6}has an average value of 7.02 kV, while v

_{dc_y}

_{6}has an average value of 6.99 kV during this interval. The remaining voltages have similar values of the dc-link on the same side. In Figure 15b, the multilevel voltages v

_{outx}and v

_{outy}are displayed, as well as it is also possible to see that v

_{dc}shows an average value of 1.0 kV during this interval. Finally, by analyzing Figure 15c, it is possible to verify that until the instant 1 s, the amplitudes of i

_{PGx}and i

_{PGy}were almost similar, obtaining a maximum amplitude value of 205.5 A and 213.0 A. At this time, i

_{spRPCx}has a maximum value of 148.4 A, while i

_{spRPCy}has a maximum value of 142.6 A. However, with the locomotive on the y-side starting regenerative braking (P

_{Loady}< 0), the amplitude of i

_{PGy}increases to a maximum value of 364.8 A, and a new unbalance appears. At instant 1.2 s, the proposed control algorithm updates the operation reference values, as can be seen with the amplitude change of the i

_{spRPCx}and i

_{spRPCy}, being able to verify that i

_{PGx}and i

_{PGy}converge again to similar amplitude values.

_{PGx}and v

_{PGy}, as well as the voltages on each side of the sp-RPC, v

_{spRPCx}and v

_{spRPCy}. Moreover, it is possible to see that v

_{dc_x}

_{6}has an average value of 7.06 kV, while v

_{dc_y}

_{6}has an average value of 6.94 kV during this interval. The remaining voltages have similar values of the dc-link on the same side. In Figure 16b, the multilevel voltages v

_{outx}and v

_{outy}are displayed, and it is also possible to see that v

_{dc}shows an average value of 0.99 kV during this interval. Finally, by analyzing Figure 16c, it is possible to verify that until the instant 2 s, the amplitudes of i

_{PGx}and i

_{PGy}were similar, obtaining a maximum value of 81.15 A and 85.51 A, respectively. At this time, i

_{spRPCx}has a maximum value of 137 A, while i

_{spRPCy}has a maximum value of 68.61 A. However, with the locomotive on the y-side starting to accelerate, varying the P

_{Loadx}> 0, i

_{PGx}increases to a maximum amplitude value of 381 A, and a new unbalance appears. At the time instant 2.1 s, the proposed control algorithm updates the operation reference values, as can be seen with the amplitude change of the i

_{spRPCx}and i

_{spRPCy}, being able to verify that i

_{PGx}and i

_{PGy}converge again to similar amplitude values.

_{PGx}and v

_{PGy}, as well as the voltages on each side of the sp-RPC, v

_{spRPCx}and v

_{spRPCy}. Moreover, it is possible to see that v

_{dc_x}

_{6}has an average value of 6.95 kV, while v

_{dc_y}

_{6}has an average value of 7.07 kV during this interval. The remaining voltages have similar values of the dc-link on the same side. In Figure 17b the voltages v

_{outx}and v

_{outy}are displayed, and it is also possible to see that v

_{dc}shows an average value of 1.02 kV during this interval. Finally, by analyzing Figure 17c, it is possible to verify that i

_{PGx}and i

_{PGy}have a similar amplitude value, obtaining a maximum amplitude of 146.3 A and 155 A, respectively. At this time, i

_{spRPCx}has a maximum value of 89.74 A, while i

_{spRPCy}has a maximum value of 87.46 A.

## 5. Performance Analyses of the sp-RPC

_{PG}and P

_{avg}is determined, where P

_{PG}represents the operating power of one side of the TPS, either side x-side or y-side, and P

_{avg}represents the average value of the operating power of the two TPS. The maximum value determined is then divided by P

_{avg}in order to determine the unbalance. Finally, by multiplying the result by 100, it is possible to obtain the percentage value of the unbalance.

_{PG_off}variable represents the average active power between the two TPS when the sp-RPC is off, and P

_{PG_on}is used for the same measurement when the sp-RPC is on.

_{spRPCx}and v

_{spRPCy}. Additionally, to highlight the existing unbalance in the current values of the i

_{PGx}and i

_{PGy}. On the bottom side of Table 3, the average values of the operating powers of each TPS for different initial operating load conditions (with the sp-RPC disconnected) can be seen. Analyzing the data, it is possible to verify that the greater the difference in the value of the connected load, the greater the unbalance caused by each TPS, as expected. Nevertheless, to mention that the system unbalance varies from values of 33.3% to 169.3%.

_{PGx}and i

_{PGy}rms values. Analyzing the active power values, it is possible to conclude that the proposed algorithm mitigates the existing unbalance, decreasing to values below 1.86% in the existing operating conditions.

_{Loadx}= 999 kW and P

_{Loady}= 6119 kW, initially, the average value of the two TPS with the sp-RPC disabled, P

_{PG_off}, was 3559 kW, changing to 3619 kW when the sp-RPC was enabled, P

_{PG_on}. The start-up of the sp-RPC reflected an increase of 1.7%, P

_{PG_%inc}, in the average operating power of the two TPS. The same phenomenon occurs in the remaining operating conditions.

## 6. Discussion

_{spRPCx}and v

_{spRPCy}. In turn, with regenerative braking, one encounters an increase in the rms value of the overhead contact line voltages, being most evident, once again, at the end of the overhead contact line, v

_{spRPCy}. With the sp-RPC on, it is possible to impose a similar average operating power in each power TPS. This characteristic was quantified by presenting the results obtained in the table. Analyzing the first case presented, with P

_{Loadx}= 999 kW and P

_{Loady}= 6119 kW, it is possible to verify that with the sp-RPC, it was possible to decrease an unbalance from 71.9% to 0.19%. Analyzing the initial data with sp-RPC disabled, it is possible to verify an active power between the two TPS of P

_{pg_off}= 3559 kW. When the sp-RPC starts operating, this value rises to P

_{PG_on}= 3619 kW. This increase, P

_{PG_inc%}, is due essentially to the energy losses presented by the sp-RPC.

_{PG_on}is due to the decrease in the average absolute value of the active powers of the two TPS due to the sp-RPC losses.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Simplified schematic of the proposed topology for the sp-RPC, highlighting the measurement points for the variables involved in the control algorithms.

**Figure 6.**Block diagram of the control algorithm responsible for regulating the voltage on the common dc-link.

**Figure 7.**Block diagram of the control algorithm responsible for regulating the different dc-link voltages on the spRPCx.

**Figure 8.**Block diagram of the control algorithm responsible for balancing the power between the different submodules of the spRPCx.

**Figure 9.**Block diagram of the PLL control algorithm (based on [28]).

**Figure 11.**Illustration of the carrier and modulation waveforms for the phase-shift carrier technique.

**Figure 12.**Simulation results of the power in each TPS, P

_{PGx}and P

_{PGy}, the voltages, v

_{PGx}, v

_{PGy}, v

_{spRPCx}and v

_{spRPCy}, and currents, i

_{PGx}and i

_{PGy}, for different load conditions: (

**a**) P

_{Loadx}= 0.998 MW and P

_{Loady}= 6.12 MW; (

**b**) P

_{Loadx}= 0.998 MW and P

_{Loady}= −3.88 MW; (

**c**) P

_{Loadx}= 0.998 MW and P

_{Loady}= 4.12 MW.

**Figure 13.**Evolution of active powers and voltages: (

**a**) variation of the load power, P

_{Loadx}and P

_{Loady}, and the active power on each TPS, P

_{PGx}and P

_{PGy}; (

**b**) variation of sp-RPC active power, P

_{spRPCx}and P

_{spRPCy}, as well as the reference values, P

_{spRPCx}* and P

_{spRPCy}*; (

**c**) variation of the common dc-link voltage, v

_{dc}, variation of the voltage of the dc-link on each submodule, v

_{dc_x}

_{1}to v

_{dc_x}

_{6}, and v

_{dc_y}

_{1}to v

_{dc_y}

_{6}.

**Figure 14.**Detailed real-scale simulation results during the time interval from 0.3 s to 0.6 s: (

**a**) the voltages on each TPS, v

_{PGx}and v

_{PGy}, the output voltages of the sp

_{RPCx}and sp

_{RPCy}, v

_{spRPCx}and v

_{spRPCy}, and voltages on the dc-link of each submodule, v

_{dc_x}

_{1}to v

_{dc_x}

_{6}, and v

_{dc_y}

_{1}to v

_{dc_y}

_{6}; (

**b**) the multilevel voltage of the spRPCx and spRPCy, v

_{outx}and v

_{outy}, and the voltage of the common dc-link voltage, v

_{dc}; (

**c**) the currents on each TPS, i

_{PGx}and i

_{PGy}, and the output currents of the sp-RPC, i

_{spRPCx}and i

_{spRPCy}.

**Figure 15.**Detailed real-scale simulation results during the time interval from 0.95 s to 1.25 s: (

**a**) the voltages on each TPS, v

_{PGx}and v

_{PGy}, the output voltages of the sp

_{RPCx}and sp

_{RPCy}, v

_{spRPCx}and v

_{spRPCy}, and voltages on the dc-link of each submodule, v

_{dc_x}

_{1}to v

_{dc_x}

_{6}, and v

_{dc_y}

_{1}to v

_{dc_y}

_{6}; (

**b**) the multilevel voltage of the spRPCx and spRPCy, v

_{outx}and v

_{outy}, and the voltage of the common dc-link voltage, v

_{dc}; (

**c**) the currents on each TPS, i

_{PGx}and i

_{PGy}, and the output currents of the sp-RPC, i

_{spRPCx}and i

_{spRPCy}.

**Figure 16.**Detailed real-scale simulation results during the time interval from 1.95 s to 2.25 s: (

**a**) the voltages on each TPS, v

_{PGx}and v

_{PGy}, the output voltages of the sp

_{RPCx}and sp

_{RPCy}, v

_{spRPCx}and v

_{spRPCy}, and voltages on the dc-link of each submodule, v

_{dc_x}

_{1}to v

_{dc_x}

_{6}, and v

_{dc_y}

_{1}to v

_{dc_y}

_{6}; (

**b**) the multilevel voltage of the spRPCx and spRPCy, v

_{outx}and v

_{outy}, and the voltage of the common dc-link voltage, v

_{dc}; (

**c**) the currents on each TPS, i

_{PGx}and i

_{PGy}, and the output currents of the sp-RPC, i

_{spRPCx}and i

_{spRPCy}.

**Figure 17.**Detailed real-scale simulation results during the time interval from 2.9 s to 3 s: (

**a**) the voltages on each TPS, v

_{PGx}and v

_{PGy}, the output voltages of the sp

_{RPCx}and sp

_{RPCy}, v

_{spRPCx}and v

_{spRPCy}, and voltages on the dc-link of each submodule, v

_{dc_x}

_{1}to v

_{dc_x}

_{6}, and v

_{dc_y}

_{1}to v

_{dc_y}

_{6}; (

**b**) the multilevel voltage of the spRPCx and spRPCy, v

_{outx}and v

_{outy}, and the voltage of the common dc-link voltage, v

_{dc}; (

**c**) the currents on each TPS, i

_{PGx}and i

_{PGy}, and the output currents of the sp-RPC, i

_{spRPCx}and i

_{spRPCy}.

Variable | Nominal | Unit | |
---|---|---|---|

Nominal active power for the sp-RPC | P_{sp_RPCx}, P_{sp_RPCy} | 2.5 | MW |

Common dc-Link nominal voltage | V_{dc} | 1 | kV |

Nominal voltage on the dedicated dc-links of the sp-RPC | V_{dc}__{x}_{1}, V_{dc}__{x}_{2}, V_{dc}__{x}_{3}, V_{dc}__{x}_{4}, V_{dc}__{x}_{5}, V_{dc}__{x}_{6},V _{dc}__{y}_{1}, V_{dc}__{y}_{2}, V_{dc}__{y}_{3}, V_{dc}__{y}_{4}, V_{dc}__{y}_{5}, V_{dc}__{y}_{6} | 7 | kV |

Catenary nominal voltage | v_{PGx}, v_{PGy} | 25 | kV |

sp-RPC output nominal current | i_{spRPCx}, i_{spRPCy} | 100 | A |

Communication frequency between TPS and the sp-RPC | f_{s_PG} | 5 | Hz |

Sampling frequency for the sp-RPC | f_{s} | 50 | kHz |

Switching frequency for the ac side | f_{sw_spRPC} | 1 | kHz |

Switching frequency for the DAB | f_{sw_spRPC_DAB} | 1 | kHz |

Variable | Value | Unit | |
---|---|---|---|

Line impedance inductors | L_{PGzx}, L_{PGzy}, L_{spRPCzx}, L_{spRPCzx}, | 2.5 (250 ^{#1}) | mH (mΩ) |

Coupling inductor | L_{spRPCx}, L_{spRPCy} | 50 (10 ^{#1}) | mH (mΩ) |

Capacitance on the dc-link | C_{1} | 200 (10 ^{#1}) | mF (mΩ) |

Capacitance on the dc-link | C_{x}_{1}, C_{x}_{2}, C_{x}_{3}, C_{x}_{4}, C_{x}_{5}, C_{x}_{6},C _{y}_{1}, C_{y}_{2}, C_{y}_{3}, C_{y}_{4}, C_{y}_{5}, C_{y}_{6} | 10 (10 ^{#1}) | mF (mΩ) |

^{#1}Considered as internal series resistance for each component.

**Table 3.**Currents and voltages rms values, and active power and unbalance value for different locomotive consumption before and after the sp-RPC being enabled.

sp-RPC Disabled | sp-RPC Enabled | |||||||
---|---|---|---|---|---|---|---|---|

Event | (i) | (ii) | (iii) | (i) | (ii) | (iii) | ||

Parameters | ||||||||

i_{PGx} | 39.97 | 39.97 | 39.97 | 144 | 56.9 | 102 | A | |

i_{PGy} | 245.8 | 155 | 165.4 | 143 | 63.5 | 104 | A | |

i_{spRPCx} | - | - | - | 105 | 97 | 62.4 | A | |

i_{spRPCy} | - | - | - | 100 | 100 | 61.4 | A | |

v_{PGx} | 24,992 | 24,992 | 24,994 | 24,983 | 25,004 | 24,987 | V | |

v_{PGy} | 24,982 | 25,018 | 24,985 | 24,990 | 25,005 | 24,992 | V | |

v_{spRPCx} | 24,982 | 24,982 | 24,984 | 24,923 | 25,043 | 24,950 | V | |

v_{spRPCy} | 24,940 | 25,081 | 24,968 | 25,013 | 25,014 | 25,000 | V | |

P_{Loadx} | 999 | 999 | 999 | 999 | 999 | 999 | kW | |

P_{Loady} | 6119 | −3884 | 4122 | 6119 | −3884 | 4122 | kW | |

P_{PGx} | 999 | 999 | 999 | 3612 | −1419 | 2565 | kW | |

P_{PGy} | 6119 | −3884 | 4122 | 3626 | −1367 | 2565 | kW | |

P_{spRPCx} | - | - | - | −2608 | 2424 | −1564 | kW | |

P_{spRPCy} | - | - | - | 2507 | 2507 | 1534 | kW | |

P_{PG_off} | 3559 | −1442 | 2560 | - | - | - | kW | |

P_{PG_on} | - | - | - | 3619 | −1393 | 2565 | kW | |

P_{PG_%inc} | - | - | - | 1.7 | −3.4 | 0.2 | % | |

Unbalance | 71.9 | 169.3 | 61.0 | 0.19 | 1.86 | 0 | % |

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## Share and Cite

**MDPI and ACS Style**

Barros, L.A.M.; Martins, A.P.; Pinto, J.G. Balancing the Active Power of a Railway Traction Power Substation with an sp-RPC. *Energies* **2023**, *16*, 3074.
https://doi.org/10.3390/en16073074

**AMA Style**

Barros LAM, Martins AP, Pinto JG. Balancing the Active Power of a Railway Traction Power Substation with an sp-RPC. *Energies*. 2023; 16(7):3074.
https://doi.org/10.3390/en16073074

**Chicago/Turabian Style**

Barros, Luis A. M., António P. Martins, and José Gabriel Pinto. 2023. "Balancing the Active Power of a Railway Traction Power Substation with an sp-RPC" *Energies* 16, no. 7: 3074.
https://doi.org/10.3390/en16073074