A Hierarchical Control Strategy Based on Dual-Vector Model Predictive Current Control for Railway Energy Router

: The multiport and multidirectional energy ﬂow of railway energy routers (RERs) poses a signiﬁcant challenge when integrating photovoltaic (PV) systems and energy storage systems (ESSs). To address this issue, this paper proposes an improved hierarchical control strategy for RERs with a reference signal generation layer and an inverter control layer. In the reference signal generation layer, a time-segmentation energy allocation strategy based on a state machine is proposed to manage the multidirectional energy ﬂow in RERs resulting from PV systems and ESSs while minimizing peak power demand. In the inverter control layer, a dual-vector model predictive current control (MPCC) strategy is designed for back-to-back inverters. The dual-vector MPCC strategy eliminates the need for individual PWM blocks, thereby enhancing RER current-tracking accuracy and efﬁciency. The prominent advantage of the dual-vector MPCC strategy is its ability to achieve high current-tracking accuracy while minimizing active power losses. Simulations and hardware-in-the-loop experiments are conducted to validate the feasibility and effectiveness of the proposed method.


Introduction
Railways have become a crucial component of the national economy, with China's railways operating over 150,000 km, including 40,000 km of high-speed railways.Electrified railways commonly utilize traction transformers to convert three-phase grid AC voltage into a single-phase AC-27.5 kV traction power supply system.However, this unique configuration leads to issues such as negative-sequence current and power factor.The energy demand for electrified railways is large.For instance, the total electricity consumption in 2022 alone was as high as 787 billion kWh [1].To achieve the vision of "carbon neutrality", it becomes pragmatic to optimize the energy structure of electrified railways and seamlessly integrate new energy sources.On the other hand, electrified railways generate massive regenerative braking energy (RBE) during operation [2].Unrecycled RBE directly flows to the upstream power grid, which not only leads to a large amount of energy waste but also causes power quality deterioration in the upstream power grid [3,4].Consequently, enhancing the recovery and reuse rate of RBE emerges as a potent strategy for achieving energy conservation, emission reduction, and the optimization of the energy consumption structure.
To handle these issues, railway energy routers (RERs), which integrate both photovoltaic (PV) systems and energy storage systems (ESSs) into the traction power supply system via back-to-back converters, have been proposed [5,6].Most of the existing RER research mainly focuses on RER topology [7][8][9], compensation principle [10][11][12], or energy allocation strategy [13][14][15], while fewer studies have delved into the realm of control strategies.Most of these studies have traditionally focused on conventional RERs that lack integration with PV systems and ESSs.However, the multiport and multidirectional energy flow of RERs necessitates a fresh perspective when integrating PV systems and ESSs.Traditional control methods can no longer satisfy the RER requirements.Consequently, the adoption of a layered approach involving energy allocation strategies and device control layers becomes imperative.
For energy allocation, the scheme proposed in [5,13] integrates PV systems with the DC link of back-to-back converters, offering the capability to absorb new energy locally and enhance the flexibility and reliability of the traction power supply system.But these research studies only discuss the architecture design and lack in-depth research on energy management.Deng et al. [6,10,11] propose methods to recycle RBE by integrating ESSs into the DC link of back-to-back converters.These methods solve the problem that RBE returns to the upstream grid when surplus energy exists.But these research studies only discuss the case of integrating ESSs alone.These methods are not suitable for the case of integrating PV systems and ESSs together.The purposes of an energy allocation strategy are peak shaving and valley filling, which in turn reduce the electricity bill.Many of the aforementioned studies employ a traditional peak-shaving energy allocation strategy, capable of reducing energy charges but not both energy charges and power demand charges.
For device control, as a core component of RERs, the control of back-to-back converters determines the quality of RERs.The control methods for converters mainly include the following: (1) Hysteresis control [16][17][18][19], which has a large tracking error and large current ripple.(2) Proportional-integral (PI) control [20], which is simple and easy to use but has large overshoot during the DC-side voltage transition stage and a slow response speed.The steady-state tracking error of the AC-side compensation current is large, which causes errors between the active power/reactive power of back-to-back converter compensation and the reference command value.(3) Proportion resonance (PR) control [21,22], which requires a fundamental wave controller and multiple harmonic controllers and increases the economic cost.(4) Sliding-mode variable structure control [23], which can achieve the fast and accurate tracking of the compensation current but is prone to chattering.The model predictive control technology has been widely used in grid-connected converter control in recent years because of its simple principle, no modulation unit, high robustness, good current-tracking effect, and strong applicability in various nonlinear systems [24,25].Therefore, this paper designs a dual-vector model predictive current control (MPCC) strategy for back-to-back converters.
To solve the problem of the multiport and multidirectional energy flow of RERs when PV systems and ESSs are integrated, this paper proposes an improved hierarchical control strategy with a reference signal generation layer and a converter control layer.
In the reference signal generation layer, a time-segmentation energy allocation strategy is proposed to solve the multidirectional energy flow brought by PV systems and ESSs, as well as to reduce the maximum power demand.
In the converter control layer, a dual-vector MPCC strategy for back-to-back converters that has flexible control and does not need a separate PWM block is presented.The method improves the current-tracking accuracy and efficiency of RERs.
The remainder of this paper is organized as follows: The structure of RERs is introduced in Section 2. The four operation modes and the time-segmentation energy allocation strategy are analyzed in Section 3.An improved hierarchical control strategy for RERs with a reference signal generation layer and a converter control layer is constructed in Section 4. Simulation verification and HIL experiments are presented in Section 5. Finally, the conclusion is presented in Section 6.

System Structure of RERs
The topology of RERs is illustrated in Figure 1.Two-phase traction buses with 27.5 kV voltage are connected to a three-phase grid through a V/V traction transformer.The backto-back inverter is composed of two single-phase voltage source inverters (VSC α /VSC β ), whose AC sides are connected to the traction buses via step-down transformers T α and T β , respectively.The inverters are interconnected by a buffer capacitor on the DC side.The PV system and ESS are integrated into the traction power supply system through the back-to-back inverter's DC bus.
back-to-back inverter is composed of two single-phase voltage source in (VSCα/VSCβ), whose AC sides are connected to the traction buses via step-down formers Tα and Tβ, respectively.The inverters are interconnected by a buffer capac the DC side.The PV system and ESS are integrated into the traction power supply through the back-to-back inverter's DC bus.

Four Operating Modes of RERs
As shown in Figure 1, many uncertain factors lead to the complexity of energ e.g., whether the two feeder loads are in traction state or in regenerative braking whether the PV system is in working state or not, whether the ESS is charging or dis ing.For ease of analysis, four typical modes are summarized in Figure 2.
Pg represents the active power of the upstream utility grid; PPV represents the generated by the PV system; PESS represents the power of the ESS regardless of ch or discharging; PLα, PLβ denote the active power of α/β traction loads, respectively; P Pcβ (Qcα and Qcβ) denote the active (reactive) power compensated by VSCα and VS spectively; SOCmin, SOCmax denote the upper and lower state-of-charge (SOC) limit ESS, respectively.PH represents the value of peak-clipping power.Plow represe value of valley-filling adjustment power.
1. Mode 1: peak-clipping discharge state.In this situation, the loads satisfy (PLα PPV) ≥ PH, and the ESS satisfies SOC > SOCmin.Then, the ESS is in discharge sta power generated by the PV system or ESS is distributed to both traction through the RER.Then, a certain amount of active power is transferred fro traction bus to the other to achieve power balance.Meanwhile, the reactive po each bus is independently compensated.

Four Operating Modes of RERs
As shown in Figure 1, many uncertain factors lead to the complexity of energy flow, e.g., whether the two feeder loads are in traction state or in regenerative braking state, whether the PV system is in working state or not, whether the ESS is charging or discharging.For ease of analysis, four typical modes are summarized in Figure 2.
P g represents the active power of the upstream utility grid; P PV represents the power generated by the PV system; P ESS represents the power of the ESS regardless of charging or discharging; P Lα , P Lβ denote the active power of α/β traction loads, respectively; P cα and P cβ (Q cα and Q cβ ) denote the active (reactive) power compensated by VSC α and VSC β , respectively; SOC min , SOC max denote the upper and lower state-of-charge (SOC) limits of the ESS, respectively.P H represents the value of peak-clipping power.P low represents the value of valley-filling adjustment power.

Time-Segmentation Energy Allocation Strategy
The energy allocation strategy is the optimal approach to solve the control problem of the multiport and multidirectional energy flow when the PV system and the ESS are integrated in an RER.In China, railways have a two-part tariff policy.To reduce the electricity tariff, this subsection proposes a time-segmentation energy allocation strategy, which divides the 24 h into peak-shaving-valley-filling mode, and maximum demand mode.The peak-shaving-valley-filling mode uses lower peak-valley values to minimize the energy charge.The maximum demand mode uses higher peak-valley values to reduce the demand as much as possible, thus reducing the power demand charge.As shown in Figure 3, the time-segmentation energy management strategy is divided into the following three steps: 1. Data processing: This step involves initializing variables, detecting data, and calculating the traction load.The initialized variables include the threshold values of the SOC (SOCmin, SOCmax), the initial value of the SOC (SOC0), the rated capacity of the inverter (Smax), and the maximum discharge power of the ESS (PBM).2. Judge mode: First, based on the historical measured data, the time period, t ∈ (t1, t2), in which the maximum demand is probable to appear is estimated and set as the

Time-Segmentation Energy Allocation Strategy
The energy allocation strategy is the optimal approach to solve the control problem of the multiport and multidirectional energy flow when the PV system and the ESS are integrated in an RER.In China, railways have a two-part tariff policy.To reduce the electricity tariff, this subsection proposes a time-segmentation energy allocation strategy, which divides the 24 h into peak-shaving-valley-filling mode, and maximum demand mode.The peak-shaving-valley-filling mode uses lower peak-valley values to minimize the energy charge.The maximum demand mode uses higher peak-valley values to reduce the demand as much as possible, thus reducing the power demand charge.As shown in Figure 3, the time-segmentation energy management strategy is divided into the following three steps: 1.
Data processing: This step involves initializing variables, detecting data, and calculating the traction load.The initialized variables include the threshold values of the SOC (SOC min , SOC max ), the initial value of the SOC (SOC 0 ), the rated capacity of the inverter (S max ), and the maximum discharge power of the ESS (P BM ).

2.
Judge mode: First, based on the historical measured data, the time period, t ∈ (t 1 , t 2 ), in which the maximum demand is probable to appear is estimated and set as the maximum demand mode.Then, different peak-valley values are set for the maximum demand mode, and the peak-shaving-valley-filling mode, respectively.

3.
Judge state: Based on the instantaneous power of the traction load and PV system, and the instantaneous SOC(t) of the ESS, as well as the peak and valley values obtained in the previous step, the system operation status is judged in real time.Specifically, it is divided into peak-clipping discharge state, recovery regenerative braking state, power transmission state, and valley-filling adjustment state.The specific judgment process is shown in Figure 3, and the transition relationships among different operation states are shown in Figure 4.
lectronics 2023, 12, x FOR PEER REVIEW maximum demand mode.Then, different peak-valley values are set for demand mode, and the peak-shaving-valley-filling mode, respectively 3. Judge state: Based on the instantaneous power of the traction load an and the instantaneous SOC(t) of the ESS, as well as the peak and val tained in the previous step, the system operation status is judged in rea ically, it is divided into peak-clipping discharge state, recovery regene state, power transmission state, and valley-filling adjustment state.The ment process is shown in Figure 3, and the transition relationships am operation states are shown in Figure 4.

Hierarchical Control Strategy
The dual intermittency and volatility of PV sources and traction load cause difficulties in system control.Employing an energy allocation strategy is an effective way to solve this problem.This paper constructs a hierarchical control strategy for RERs, as shown in Figure 5, which is divided into reference signal generation layer and converter control layer.To achieve the stable operation of the three-port system of an RER, one port needs to work in master mode to stabilize the DC bus voltage and maintain the energy balance of the RER, while the remaining ports work in slave mode to manage their own output power and complete the corresponding power flow scheduling tasks.In this paper, the ESS works in master mode, and the PV system and the converters on both sides work in slave mode.

Hierarchical Control Strategy
The dual intermittency and volatility of PV sources and traction load cause difficulties in system control.Employing an energy allocation strategy is an effective way to solve this problem.This paper constructs a hierarchical control strategy for RERs, as shown in Figure 5, which is divided into reference signal generation layer and converter control layer.To achieve the stable operation of the three-port system of an RER, one port needs to work in master mode to stabilize the DC bus voltage and maintain the energy balance of the RER, while the remaining ports work in slave mode to manage their own output power and complete the corresponding power flow scheduling tasks.In this paper, the ESS works in master mode, and the PV system and the converters on both sides work in slave mode.

Reference Signal Generation
At the reference signal generation layer, the energy storage output is first calculated based on the time-segmentation energy allocation strategy shown in Figure 3.Then, the reference power of the RER is calculated based on the compensation principle [5].
Finally, according to second-order generalized integrator (SOGI), the αβ0 stationary reference frame is fabricated on the single-phase inverter.After coordinate transformation, the active power and reactive power of the single-phase voltage source converter in the dq0 rotating reference frame can be expressed as

Reference Signal Generation
At the reference signal generation layer, the energy storage output is first calculated based on the time-segmentation energy allocation strategy shown in Figure 3.Then, the reference power of the RER is calculated based on the compensation principle [5].
Finally, according to second-order generalized integrator (SOGI), the αβ0 stationary reference frame is fabricated on the single-phase inverter.After coordinate transformation, the active power and reactive power of the single-phase voltage source converter in the dq0 rotating reference frame can be expressed as Therefore, the reference-current instructions of the back-to-back inverter can be obtained and are where U α and U β denote the root-mean-square values of U α and U β , respectively.

Modeling and Control of Back-to-Back Inverter
Due to the buffer effect of the intermediate DC capacitor, VSC α /VSC β of the back-toback inverter can be controlled independently, even when active power exchange is present.This paper takes VSC α as an example to design a current controller, because VSC α and VSC β have the same structure and function.A dual-vector MPCC strategy is proposed for back-to-back converters to accurately track the reference current and acquire stable and rapid control capability.The basic principle is to seek the voltage vector that minimizes the current-tracking error as the output voltage of the voltage source converter.
Figure 6 shows an equivalent circuit of the single-phase voltage source converter.L (R) denotes the inductor (resistor) of the circuit; u and i denote the output voltage and current of VSC α , respectively; e denotes the output voltage of the grid.

Modeling and Control of Back-to-Back Inverter
Due to the buffer effect of the intermediate DC capacitor, VSCα/VSCβ of the back-toback inverter can be controlled independently, even when active power exchange is present.This paper takes VSCα as an example to design a current controller, because VSCα and VSCβ have the same structure and function.A dual-vector MPCC strategy is proposed for back-to-back converters to accurately track the reference current and acquire stable and rapid control capability.The basic principle is to seek the voltage vector that minimizes the current-tracking error as the output voltage of the voltage source converter.
Figure 6 shows an equivalent circuit of the single-phase voltage source converter.L (R) denotes the inductor (resistor) of the circuit; u and i denote the output voltage and current of VSCα, respectively; e denotes the output voltage of the grid.
Eq.( 16) Eq. (15) u ref (k+1) Eq.( 14) Referring to Figure 6, the following equation can be obtained as The current derivative in the continuous-time domain is described in Equation (4).Correspondingly, the current derivative in the discrete domain is replaced by the firstorder forward Euler approximation method with a sampling period Ts, which can be expressed as follows: Referring to Figure 6, the following equation can be obtained as The current derivative in the continuous-time domain is described in Equation (4).Correspondingly, the current derivative in the discrete domain is replaced by the first-order forward Euler approximation method with a sampling period T s , which can be expressed as follows: Then, in the discrete-time domain, the output current dynamics can be described as In MPCC, according to (6), the three different output voltages produced by VSC α can predict three different future output currents.At each sampling time point, the voltage that minimizes the cost function is selected from three possible voltages as the output voltage.And the cost function is defined as the absolute error between the present current and the future reference current.Due to the cost function, the actual output current generated by the optimal output voltage is forced to approach the future reference current in the next step.The one-step future reference current (i ref (k + 1)) can be calculated using the Lagrange extrapolation formula.
where i ref (k) denotes the present reference current; i ref (k − 1) denotes the one-step past reference current; i ref (k − 2) denotes the two-step past reference current.Due to the large amount of calculation for model predictive control, the control program has an unavoidable calculation delay in practice.Therefore, the output current of the one-step delay is obtained by dynamically moving the output current one step forward in (6), which can be described as shown in (8).
where i(k + 2) denotes the two-step future current; u(k + 1) denotes three different possible voltage vectors output by VSC α ; considering that the grid voltage changes slowly, e(k + 1) is approximately equal to e(k), i.e., e(k + 1) ≈ e(k).
At every sampling instant, the cost function that evaluates the absolute errors between the two-step future reference current (i ref (k + 2)) and the predicted output currents is described as Also, i ref (k + 2) can be calculated using the Lagrange extrapolation formula.
Due to the fact that VSC α can only produce three different output voltages (i.e., u out1 = 0; u out2 = U dc ; u out3 = −U dc ), a large current error is generated.And the delay compensation increases the current error and distortion.Although current with high-quality waveforms and smaller errors can be achieved by increasing the sampling frequency, a greater cost is needed in practice.Therefore, this paper proposes a dual-vector MPCC strategy that considers all possible future combinations produced by two voltages of VSC α (i.e., (u out1 , u out1 ), (u out1 , u out2 ), (u out1 , u out3 ), (u out2 , u out1 ), (u out2 , u out2 ), (u out2 , u out3 ), (u out3 , u out1 ), (u out3 , u out2 ), (u out2 , u out2 )).
The sampling period is assigned to two variable time intervals, corresponding to the first and second voltages in a voltage combination.At every sampling instant, the error between the actual current and the reference current is reduced by changing the variable time interval.The virtual voltage can be expressed as where u outm , u oun (m, n = 1, 2, 3) denote three different possible output voltages, respectively; t m and t n correspond to the working time of the first voltage (u outm ) and the second voltage (u outn ) during the kth sampling period, respectively, satisfying t m + t n = T s .Therefore, another way of expressing the two-step future output current is where f m (k) and f n (k) denote the derivatives of the output current produced by the first voltage (u outm ) and the second voltage (u outn ) during the kth sampling period, respectively.They can be expressed as To simplify the algorithm, modulation model predictive control is used to calculate the vector application time.The basic idea is that the larger the objective function corresponding to the corresponding vector, the larger the control error.Therefore, its action time should be as short as possible.The distribution of t m and t n satisfies where G m and G n denote the voltage cost function of the first voltage (u outm ) and the second voltage (u outn ) during the kth sampling period, respectively, which are defined as where u ref (k + 1) denotes the one-step future reference voltage value, which can be calculated using the deadbeat algorithm as According to (11) and ( 14)~( 16), the virtual vector that minimizes Equation ( 9) can be solved; then, dual-vector MPCC can be realized.
Figure 6 describes the flow diagram of the dual-vector MPC method.
Comparing the conventional MPCC strategy and the dual-vector MPC method, Figure 7 shows the output current waveforms of VSC α with the same sampling frequency.It can be seen that steady-state current error and current ripple can be reduced by utilizing the dual-vector MPC method without increasing the sampling frequency.
Electronics 2023, 12, x FOR PEER REVIEW 9 of 18 corresponding to the corresponding vector, the larger the control error.Therefore, its action time should be as short as possible.The distribution of tm and tn satisfies where Gm and Gn denote the voltage cost function of the first voltage (uoutm) and the second voltage (uoutn) during the kth sampling period, respectively, which are defined as ( 1) , ( , ) where uref(k + 1) denotes the one-step future reference voltage value, which can be calculated using the deadbeat algorithm as According to (11) and ( 14)~( 16), the virtual vector that minimizes Equation ( 9) can be solved; then, dual-vector MPCC can be realized.
Figure 6 describes the flow diagram of the dual-vector MPC method.
Comparing the conventional MPCC strategy and the dual-vector MPC method, Figure 7 shows the output current waveforms of VSCα with the same sampling frequency.It can be seen that steady-state current error and current ripple can be reduced by utilizing the dual-vector MPC method without increasing the sampling frequency.

Control of DC/DC Converter
ESSs play an important role in balancing power and adjusting the DC bus voltage for RERs.In this paper, the voltage outer loop is used to generate a reference value for the current inner loop to maintain DC bus voltage stability and manage the charge and discharge of the ESS, as shown in Figure 5b.

Control of DC/DC Converter
ESSs play an important role in balancing power and adjusting the DC bus voltage for RERs.In this paper, the voltage outer loop is used to generate a reference value for the current inner loop to maintain DC bus voltage stability and manage the charge and discharge of the ESS, as shown in Figure 5b.
Because the output characteristics are influenced by the environment and climate, a PV system should have two modes: maximum power point tracking (MPPT) mode and non-working mode.To effectively use renewable energy, it generally works in MPPT mode.However, when the environmental factors change so that the PV system cannot continuously supply power, the working mode is turned to non-working mode.The DC/DC converter control structure of the PV system is shown in Figure 5c.

HIL Experimental Verification
In this section, to verify the correctness and effectiveness of the proposed method, an HIL simulation platform consisting of a real controller (TMS320F28335 DSP) and a virtual controlled object is built, as depicted in Figure 8.An RER model and a traction power supply system model are both implemented in the RT-BOX simulator using PLECS software 4.7.5.The voltage and current signals of the converter are transmitted to the DSP controller through the A/D conversion unit.Then, the hierarchical control strategy presented in Section 4 is implemented using the DSP controller.Finally, the control signals are transmitted to the built RER from the D/A conversion unit to the control converters.

HIL Experimental Verification
In this section, to verify the correctness and effectiveness of the proposed method, an HIL simulation platform consisting of a real controller (TMS320F28335 DSP) and a virtual controlled object is built, as depicted in Figure 8.An RER model and a traction power supply system model are both implemented in the RT-BOX simulator using PLECS software 4.7.5.The voltage and current signals of the converter are transmitted to the DSP controller through the A/D conversion unit.Then, the hierarchical control strategy presented in Section 4 is implemented using the DSP controller.Finally, the control signals are transmitted to the built RER from the D/A conversion unit to the control converters.In this subsection, the data of four typical moments are selected to constitute two dynamic operating conditions to verify the correctness of the proposed dual-vector MPCC strategy.The HIL platform parameters are specified in Table 1.In this subsection, the data of four typical moments are selected to constitute two dynamic operating conditions to verify the correctness of the proposed dual-vector MPCC strategy.The HIL platform parameters are specified in Table 1.
Power(2MW/div) β-side locomotive braking t(0.04s/div)Before β-side locomotive braking, the loads are −1 MW and 7 MW, res the system is in power transmission state.The 1 MW active power genera system is absorbed; the ESS does not work; and the consumption of the upp is 5 MW.After β-side locomotive braking, the load changes to −1 MW, and in recovery regenerative braking state.The ESS is in charging state, absorb erative braking energy of the two traction buses.The 1 MW active power gen PV system is directly absorbed by the ESS, and no energy is fed back to th The DC bus voltage of the RER introduced in Figure 9c remains around its re which ensures the normal operation of the RER.The result of converter α reference current shows that the dual-vector MPC method can ensure the a ing of current.As shown in Figure 9d, when the RER works in power trans the grid-side currents of the traction transformer are almost symmetric; w works in recovery regenerative braking mode, the primary-side currents o transformer almost become 0.
2. Peak-clipping discharge state and valley-filling adjustment state: The waveforms are described in Figure 10.Before β-side locomotive braking, the loads are −1 MW and 7 MW, respectively, and the system is in power transmission state.The 1 MW active power generated by the PV system is absorbed; the ESS does not work; and the consumption of the upper-grid power is 5 MW.After β-side locomotive braking, the load changes to −1 MW, and the system is in recovery regenerative braking state.The ESS is in charging state, absorbing the regenerative braking energy of the two traction buses.The 1 MW active power generated by the PV system is directly absorbed by the ESS, and no energy is fed back to the upper grid.The DC bus voltage of the RER introduced in Figure 9c remains around its reference value, which ensures the normal operation of the RER.The result of converter α tracking the reference current shows that the dual-vector MPC method can ensure the accurate tracking of current.As shown in Figure 9d, when the RER works in power transmission state, the grid-side currents of the traction transformer are almost symmetric; when the RER works in recovery regenerative braking mode, the primary-side currents of the traction transformer almost become 0.

2.
Peak-clipping discharge state and valley-filling adjustment state: The experimental waveforms are described in Figure 10.
Power(3MW/div) β-side locomotive braking t(0.04s/div)Before β-side locomotive braking, the loads are 2 MW and 10 MW, respectively, and the system is in peak-clipping discharge state.The ESS is in discharging state, providing 1 MW active power to the traction bus; the PV system does not work; and the consumption of the upper-grid power is equal to the preset peak value (11 MW).After β-side locomotive braking, the load changes to −1 MW, and the system is in valley-filling adjustment state.The ESS is in charging state, acting as a load and consuming 1 MW active power, absorbing the regenerative braking energy of the two traction buses.The whole system consumes uppergrid power equal to the preset valley value (2 MW).The DC bus voltage of the RER introduced in Figure 10c remains around its reference value, which ensures the normal operation of the RER.The result of converter α tracking the reference current shows that the dualvector MPC method can ensure the accurate tracking of current.As shown in Figure 10d, the grid-side currents of the traction transformer are almost symmetric.

Typical Daily Measured Load Data Verification
Considering the daily periodicity of train operation, this section selects the typical daily measured load data of a traction substation in a high-speed railway section (sampling interval: 1 s/time).Correspondingly, the PV system selects the measured power generation data of the PV power station near the traction substation during a typical summer day (sampling interval: 1 min/time) to further evaluate the engineering capability of the RER.
Figure 11 shows typical daily field data of a traction substation.The traction load of the two arms is quite different, and 16.27 MWh RBE is returned to the upstream grid every day, accounting for 19.82% of the traction energy.It means that the traction substation is suitable for installing an RER system.Before β-side locomotive braking, the loads are 2 MW and 10 MW, respectively, and the system is in peak-clipping discharge state.The ESS is in discharging state, providing 1 MW active power to the traction bus; the PV system does not work; and the consumption of the upper-grid power is equal to the preset peak value (11 MW).After β-side locomotive braking, the load changes to −1 MW, and the system is in valley-filling adjustment state.The ESS is in charging state, acting as a load and consuming 1 MW active power, absorbing the regenerative braking energy of the two traction buses.The whole system consumes upper-grid power equal to the preset valley value (2 MW).The DC bus voltage of the RER introduced in Figure 10c remains around its reference value, which ensures the normal operation of the RER.The result of converter α tracking the reference current shows that the dual-vector MPC method can ensure the accurate tracking of current.As shown in Figure 10d, the grid-side currents of the traction transformer are almost symmetric.

Typical Daily Measured Load Data Verification
Considering the daily periodicity of train operation, this section selects the typical daily measured load data of a traction substation in a high-speed railway section (sampling interval: 1 s/time).Correspondingly, the PV system selects the measured power generation data of the PV power station near the traction substation during a typical summer day (sampling interval: 1 min/time) to further evaluate the engineering capability of the RER.
Figure 11 shows typical daily field data of a traction substation.The traction load of the two arms is quite different, and 16.27 MWh RBE is returned to the upstream grid every day, accounting for 19.82% of the traction energy.It means that the traction substation is suitable for installing an RER system.Based on a professional economic analysis, the specific parameters of the RER system are as follows: the rate capacity of the RER is 11 × 2 MVA; the capacity of the ESS battery is 1000 kWh; and the PV installed capacity is 1 MWp.
Figure 12 shows the changes in total active power on the traction side without and with RER deployment.Thanks to the addition of a PV system and an ESS, the power consumed by the traction is obviously decreased after RER deployment.The total traction power is reduced from 85.1 MWh to 71.8 MWh, and its rate of decrease is 15.63%.The energy recovery efficiency of RBE is 86.81%.The consumption of the PV system is 5.45 MWh, and the consumption rate is 68.76%.Figure 13 shows the instantaneous SOC of the ESS.The SOC does not exceed the limit and can be restored to SOC0 during the alternation of the day cycle, ensuring ESS state consistency.In Figure 14, it is easily inferred that the voltage imbalance degree shows an overall downward trend after RER deployment, and its 95% probability value decreases from 2.83% to 1.24%, which meets the 2% international standard.On the other hand, without RER deployment, the probability of the instantaneous power factor being equal to 1 is 33.89%, while with RER deployment, it is 75%.The power factor shows an overall increasing trend, and the average value rises from 0.61 to 0.91.Based on a professional economic analysis, the specific parameters of the RER system are as follows: the rate capacity of the RER is 11 × 2 MVA; the capacity of the ESS battery is 1000 kWh; and the PV installed capacity is 1 MWp.
Figure 12 shows the changes in total active power on the traction side without and with RER deployment.Thanks to the addition of a PV system and an ESS, the power consumed by the traction is obviously decreased after RER deployment.The total traction power is reduced from 85.1 MWh to 71.8 MWh, and its rate of decrease is 15.63%.The energy recovery efficiency of RBE is 86.81%.The consumption of the PV system is 5.45 MWh, and the consumption rate is 68.76%.Based on a professional economic analysis, the specific parameters of the RER system are as follows: the rate capacity of the RER is 11 × 2 MVA; the capacity of the ESS battery is 1000 kWh; and the PV installed capacity is 1 MWp.
Figure 12 shows the changes in total active power on the traction side without and with RER deployment.Thanks to the addition of a PV system and an ESS, the power consumed by the traction is obviously decreased after RER deployment.The total traction power is reduced from 85.1 MWh to 71.8 MWh, and its rate of decrease is 15.63%.The energy recovery efficiency of RBE is 86.81%.The consumption of the PV system is 5.45 MWh, and the consumption rate is 68.76%.Figure 13 shows the instantaneous SOC of the ESS.The SOC does not exceed the limit and can be restored to SOC0 during the alternation of the day cycle, ensuring ESS state consistency.In Figure 14, it is easily inferred that the voltage imbalance degree shows an overall downward trend after RER deployment, and its 95% probability value decreases from 2.83% to 1.24%, which meets the 2% international standard.On the other hand, without RER deployment, the probability of the instantaneous power factor being equal to 1 is 33.89%, while with RER deployment, it is 75%.The power factor shows an overall increasing trend, and the average value rises from 0.61 to 0.91. Figure 13 shows the instantaneous SOC of the ESS.The SOC does not exceed the limit and can be restored to SOC 0 during the alternation of the day cycle, ensuring ESS state consistency.Based on a professional economic analysis, the specific parameters of the RER system are as follows: the rate capacity of the RER is 11 × 2 MVA; the capacity of the ESS battery is 1000 kWh; and the PV installed capacity is 1 MWp.
Figure 12 shows the changes in total active power on the traction side without and with RER deployment.Thanks to the addition of a PV system and an ESS, the power consumed by the traction is obviously decreased after RER deployment.The total traction power is reduced from 85.1 MWh to 71.8 MWh, and its rate of decrease is 15.63%.The energy recovery efficiency of RBE is 86.81%.The consumption of the PV system is 5.45 MWh, and the consumption rate is 68.76%.Figure 13 shows the instantaneous SOC of the ESS.The SOC does not exceed the limit and can be restored to SOC0 during the alternation of the day cycle, ensuring ESS state consistency.In Figure 14, it is easily inferred that the voltage imbalance degree shows an overall downward trend after RER deployment, and its 95% probability value decreases from 2.83% to 1.24%, which meets the 2% international standard.On the other hand, without RER deployment, the probability of the instantaneous power factor being equal to 1 is 33.89%, while with RER deployment, it is 75%.The power factor shows an overall increasing trend, and the average value rises from 0.61 to 0.91.In Figure 14, it is easily inferred that the voltage imbalance degree shows an overall downward trend after RER deployment, and its 95% probability value decreases from 2.83% to 1.24%, which meets the 2% international standard.On the other hand, without RER deployment, the probability of the instantaneous power factor being equal to 1 is 33.89%, while with RER deployment, it is 75%.The power factor shows an overall increasing trend, and the average value rises from 0.61 to 0.91.

Comparison with Previous Works
1. Comparison of different control strategies: hysteresis control [16][17][18][19], PI control [20], and the proposed dual-vector MPCC strategy.The specific operating conditions involve transitioning from recovery regenerative braking state to peak-clipping discharge state.Figure 15 illustrates the current of inverter α following the application of three different control strategies.It can be observed that during transient processes, when employing the proposed dual-vector MPCC strategy, there is high precision in tracking the reference current with minimal current deviation.Figure 16 shows the current-tracking error when three different control strategies are adopted for converter α.It can be observed that the current error is minimized when employing the dual-vector MPCC strategy.Therefore, the current-tracking error can be employed effectively, and current-tracking accuracy can be improved by using the proposed dual-vector MPCC strategy.Current error(50A/div) Current error(50A/div)

Comparison with Previous Works
1.
Comparison of different control strategies: hysteresis control [16][17][18][19], PI control [20], and the proposed dual-vector MPCC strategy.The specific operating conditions involve transitioning from recovery regenerative braking state to peak-clipping discharge state.Figure 15 illustrates the current of inverter α following the application of three different control strategies.It can be observed that during transient processes, when employing the proposed dual-vector MPCC strategy, there is high precision in tracking the reference current with minimal current deviation.Figure 16 shows the current-tracking error when three different control strategies are adopted for converter α.It can be observed that the current error is minimized when employing the dual-vector MPCC strategy.Therefore, the current-tracking error can be employed effectively, and current-tracking accuracy can be improved by using the proposed dual-vector MPCC strategy.

Comparison with Previous Works
1. Comparison of different control strategies: hysteresis control [16][17][18][19], PI control [20], and the proposed dual-vector MPCC strategy.The specific operating conditions involve transitioning from recovery regenerative braking state to peak-clipping discharge state.Figure 15 illustrates the current of inverter α following the application of three different control strategies.It can be observed that during transient processes, when employing the proposed dual-vector MPCC strategy, there is high precision in tracking the reference current with minimal current deviation.Figure 16 shows the current-tracking error when three different control strategies are adopted for converter α.It can be observed that the current error is minimized when employing the dual-vector MPCC strategy.Therefore, the current-tracking error can be employed effectively, and current-tracking accuracy can be improved by using the proposed dual-vector MPCC strategy.Current error(50A/div) Current error(50A/div)

low and SOC＜SOC 0 Figure 4 .
Figure 4.The transition relationships among different operation states.Figure 4. The transition relationships among different operation states.

Figure 4 .
Figure 4.The transition relationships among different operation states.Figure 4. The transition relationships among different operation states.

Figure 5 .
Figure 5.The proposed architecture of the hierarchical control strategy.

Figure 5 .
Figure 5.The proposed architecture of the hierarchical control strategy.

Figure 6 .
Figure 6.The proposed dual-vector model predictive current control.

Figure 6 .
Figure 6.The proposed dual-vector model predictive current control.

Figure 7 .
Figure 7.The output current waveforms of VSCα with different control.(a) Conventional model predictive current control; (b) proposed dual-vector model predictive current control.

Figure 7 .
Figure 7.The output current waveforms of VSC α with different control.(a) Conventional model predictive current control; (b) proposed dual-vector model predictive current control.

Figure 8 .
Figure 8.The structure of the established HIL experimental platform.

Figure 8 .
Figure 8.The structure of the established HIL experimental platform.

Figure 9 .
Figure 9. Experimental results of power transmission state and recovery regenerative braking state.(a,b) Power balance.(c) DC bus voltage (V dc ) and inverter current (i cα /i cαref ).(d) Three-phase grid-side current (i ABC ).

Figure 10 .
Figure 10.Experimental results of peak-clipping discharge state and valley-filling adjustment state.(a,b) Power balance.(c) DC bus voltage (V dc ) and inverter current (i cα /i cαref ).(d) Three-phase grid-side current (i ABC ).

Figure 12 .
Figure 12.Changes in total active power on traction side without and with RER deployment.

Figure 13 .
Figure 13.Instantaneous SOC of the ESS.

Figure 11 .
Figure 11.Daily power profiles of traction load and PV unit.

Figure 11 .
Figure 11.Daily power profiles of traction load and PV unit.

Figure 12 .
Figure 12.Changes in total active power on traction side without and with RER deployment.

Figure 13 .
Figure 13.Instantaneous SOC of the ESS.

Figure 12 .
Figure 12.Changes in total active power on traction side without and with RER deployment.

Figure 11 .
Figure 11.Daily power profiles of traction load and PV unit.

Figure 12 .
Figure 12.Changes in total active power on traction side without and with RER deployment.

Figure 13 .
Figure 13.Instantaneous SOC of the ESS.

Figure 13 .
Figure 13.Instantaneous SOC of the ESS.

Figure 14 .
Figure 14.Improved results of power quality without and with RER deployment.(a) The 95% probability value of the voltage imbalance degree; (b) power factor.

Figure 15 .
Figure 15.Inverter α current following application of three control strategies.(a) hysteresis control; (b) PI control; (c) the proposed dual-vector MPCC strategy.

Figure 14 .
Figure 14.Improved results of power quality without and with RER deployment.(a) The 95% probability value of the voltage imbalance degree; (b) power factor.

Electronics 2023, 12 ,Figure 14 .
Figure 14.Improved results of power quality without and with RER deployment.(a) The 95% probability value of the voltage imbalance degree; (b) power factor.

Figure 15 .
Figure 15.Inverter α current following application of three control strategies.(a) hysteresis control; (b) PI control; (c) the proposed dual-vector MPCC strategy.Figure 15.Inverter α current following application of three control strategies.(a) hysteresis control; (b) PI control; (c) the proposed dual-vector MPCC strategy.

Figure 15 .
Figure 15.Inverter α current following application of three control strategies.(a) hysteresis control; (b) PI control; (c) the proposed dual-vector MPCC strategy.Figure 15.Inverter α current following application of three control strategies.(a) hysteresis control; (b) PI control; (c) the proposed dual-vector MPCC strategy.
2. Mode 2: recovery regenerative braking state.As shown in Figure2, the load (PLα + PLβ -PPV) < 0, and the ESS satisfies SOC < SOCmax.The ESS enters the ch state.RBE is first absorbed by the load on the same traction bus, and the rem part is transferred to the other traction bus to be absorbed by the RER.Then, consumed RBE is absorbed by the ESS, and the part that cannot be absorbed back to the upstream utility grid.The energy generated by the PV system is fi sorbed by the load, and the excess energy is absorbed by the ESS (if the ESS capable), otherwise it is discarded.3. Mode 3: power transmission state.In this scenario, the ESS does not work.Th of the ESS exceeds the effective threshold, or the loads meet Plow ≤ (PLα + PLβ - In this scenario, the ESS does not work.The SOC of the ESS exceeds the effective threshold, or the loads meet P low ≤ (P Lα + P Lβ − P PV ) < P H , whether in traction state or in regenerative braking state.4.Mode 4: valley-filling adjustment state.In this mode, the loads meet 0 ≤ (P Lα + P Lβ − P PV ) < P low .Specifically, if the instantaneous SOC(t) is less than the initial SOCSOC (SOCSOC 0 ), the ESS enters the charging state, otherwise it enters the standby state.The SOC values should be the same during the alternation of the daily cycle to keep the ESS continuously working.Mode 4: valley-filling adjustment state.In this mode, the loads meet 0 ≤ (PLα + PLβ -PPV) < Plow.Specifically, if the instantaneous SOC(t) is less than the initial SOCSOC (SOCSOC0), the ESS enters the charging state, otherwise it enters the standby state.The SOC values should be the same during the alternation of the daily cycle to keep the ESS continuously working.

data process
The architecture of the hierarchical control strategy.ESS =0 P ESS =P−P H P ESS =P−P low P ESS =−P BM P ESS =P BM The architecture of the hierarchical control strategy.
H ; P low ② Judge mode End Figure 3. P low ≤P L -P PV ≤P H P L -P PV ≤0 and SOC≤SOC max P L -P PV ＜P low and SOC＜SOC 0 Figure 3.The architecture of the hierarchical control strategy.P End Figure 3. L -P PV ＜P
1. Power transmission state and recovery regenerative braking state: The experimental

Table 2 .
Simulation results with different energy allocation strategies.
* The 95% probability value of voltage imbalance degree.