Power Quality Management Strategy for High-Speed Railway Traction Power Supply System Based on MMC-RPC

Abstract

This paper adopts the Modular Multilevel Converter Type Railway Power Conditioner (MMC-RPC) equipment to effectively manage the power quality of the high-speed railway traction power supply system including the reactive power and negative sequence component. Firstly, the single-phase model of the MMC was established to deduce the working characteristics of the MMC-RPC and its compensation principle for the traction power supply system with the v/v wiring transformer. Secondly, the adaptive VSG control strategy was adopted for the inverter of the MMC-RPC to provide dynamic inertial and damping support for the traction power supply system based on the virtual synchronous generator (VSG) control. Compared with the traditional double closed-loop (DCL) and VSG controls, it has better anti-disturbance and dynamic performance. The root locus analysis of control parameters based on a small signal model shows that VSG control can provide more stability margin. Furthermore, Differential Flatness Control (DFC) was used in the inner-loop controller to ensure the stable control of the inverter and the stability was verified by the Lyapunov stability analysis. For the rectifier of the MMC-RPC, a hierarchical three-level control strategy with system-level control, cluster-group voltage control, and inter-cluster voltage control for keeping the voltage balance was adopted. Finally, simulation results on the Matlab/Simulink platform verified the effectiveness and stability of the joint control applied in the MMC-RPC.

1. Introduction

At present, China’s high-speed railway is developing rapidly and its operating mileage is increasing, which is currently close to forty thousand kilometers [1]. However, power quality management is still necessary for high-speed railways. For example, traction power supply systems with low loads are still affected by the large reactive power. With the increase in traffic volume, the negative sequence problems also become more and more serious. Furthermore, voltage unbalance perturbation in the power grid is mainly caused by the electric high-speed railway loads. In addition, the access of a large number of locomotive power electronic converters has caused new problems, such as the low-frequency oscillation related to the converters with weak inertia and damping characteristic. These problems make challenges for the safe and stable operation of the railway power supply system [2,3,4,5,6].

It is difficult for traditional power quality management equipment to realize all functions of harmonic suppression, power factor, inertia improvement, and negative sequence components elimination [7,8]. One study [9] used a special connected transformer and reactive devices to realize dynamic reactive power compensation; however, its effect of negative sequence compensation is limited. In [10] the authors adopted an active power quality compensator, which can realize the comprehensive compensation; however, it is only suitable for traction stations with balance transformers to maintain the effective working voltage, and there are certain limitations in the application. In the 1980s, Japanese scholars put forward the Railway Power Conditioner (RPC), which can realize comprehensive improvement of the power quality. At present, RPC has been studied extensively in China [11]; however, the traditional RPC equipment is connected back-to-back by means of a two-level voltage source converter, and the voltage at both ends is low. It needs to be connected to the traction power supply system through step-down transformer [12], which increases the cost and volume of the whole equipment. Therefore, some researchers have proposed the MMC type RPC by replacing the step-down transformers to modular converters with better economy and efficiency. Single-phase MMC is connected back-to-back to form the MMC-RPC system [13]. While managing qualities of harmonics, negative sequence currents, and the reactive power, it also increases voltage levels to further reduce harmonics produced by the equipment itself.

At present, the difference of MMC-RPC topologies mainly lie in the number of bridge arms corresponding to different DC voltage grades and control complexity. Common MMC-RPC systems include 2, 3, and 4 bridge arms [14]. This paper takes the 4arm MMC-RPC system as the study objective due to the low requirement to the DC voltage grades and moderate control complexity.

In order to realize a better power quality management effect through RPC equipment, scholars have carried out research on different control strategies. The authors of [15] adopted a comprehensive control strategy by combining voltage-current double closed-loop and hysteresis comparison controls for traditional RPC. However, the control is not applicable to MMC-RPC even if the reference quantity can be tracked in real time. In [16], a direct power control strategy without a phase-lock loop for MMC-RPC was adopted, which has fast power response and good stability. The authors of [17,18] used current compensation and power balance controls for the MMC-RPC; however, it cannot provide inertial and damping support for the traction power supply system. Maintenance of frequency stability is limited when the traction load changes abruptly. In [19], the virtual synchronous generator (VSG) control was established, which enhances inertial and damping support; however, its fixed damping and inertia parameters have certain limitations in suppressing power oscillation.

In this study, the adaptive VSG control for the MMC-RPC was used to simulate the external characteristics of synchronous generator dynamically under different working conditions. It provides optimal inertial and damping support to improve the stability of the joint system with the traction power supply system and the MMC-RPC. The Lyapunov stability analysis has shown that the control is stable when the differential flatness theory is satisfied. Therefore, Differential Flatness Control (DFC) was applied in the inner-loop controller to ensure the stable control of the MMC-RPC. Furthermore, three-level control was adopted for the DC voltage of the rectifier to ensure the voltage balance of submodules and the voltage stability of the DC link.

Taking the v/v wiring traction power supply system as an example, this paper adopted back-to-back connection of two single-phase MMCs, which are connected to the traction network to realize the management of power quality. This paper firstly analyzes the active and reactive power compensation of the traction power supply system, establishes the mathematical model of the MMC-RPC, and studies its operating characteristics. Then, the capacitive voltage equalization control [20,21] was used in the bottom control to ensure the stability of MMC-RPC. Furthermore, the joint control of the adaptive VSG control and the DFC are adopted in the upper control to provide inertial and damping support and make the MMC-RPC more stable. A small signal model of the VSG shows the system is stable when control parameters are in a certain range; however, it is hard to realize due to the variation in the working conditions. In addition, the Lyapunov stability analysis of the DFC is established to verify the control stability. Finally, simulation verification is carried out based on Matlab/Simulink platform to verify the effectiveness of the joint control.

2. Principle of MMC-RPC

2.1. MMC-RPC Model

The power quality management scheme for high-speed railway traction power supply system based on MMC-RPC is shown in Figure 1, which is mainly composed of traction power supply system and back-to-back MMCs. The 220 kV three-phase voltage is transformed into 27.5 kV through a v/v wiring traction transformer, supplying power to the left and right traction power supply arms, respectively. The MMC-RPC is connected to the left and right traction power supply arms so as to realize energy transition on both sides and harmonic management through new control strategies. Furthermore, power control at the upper layer is applied on the inverter end, constant DC voltage control is applied on the rectifier end to ensure the stability of DC side voltage.

The topology of MMC-RPC can be obtained from Figure 1, and the left and right power supply arms are defined as α, β phase, and a and b are defined as two bridge arms in one phase, respectively. Structures and parameters of each phase and each bridge arm of the MMC-RPC are the same.

Where:

u_sk (k = α, β, which represents different phase) is the AC voltage of each phase;

i_sk is the AC current of each phase;

u_kjp and u_kjn (j = a or b, a and b represent different arms, and p and n represent the upper and lower bridge arms, respectively) are the voltage of upper and lower bridge arms of each phase;

i_kjp and i_kjn are the currents of upper and lower bridge arms of each phase;

e_kj is the ground reference voltage of the AC port of the bridge arm;

R₀ and L₀ are the bridge arm resistance and inductance, respectively;

i_dc and u_dc are the DC current and voltage, respectively.

Figure 2 shows the topology of single-phase MMC (SP-MMC) on one side of the back-to-back MMC-RPC. The lower right-hand part of Figure 2 shows the MMC submodule structure, which is composed of switching devices and capacitors. The working mode of the submodule is controlled by the switching state of the power switching devices. Furthermore, the main working states of submodules can be divided into three types: input, removal, and locking. Among them, the submodule shows the locking mode only when the MMC system fails. According to the single-phase MMC topology, the AC side port equation of the single-phase MMC is:

$$u_{\mathrm{cq}}=e_{k\mathrm{a}}-e_{k\mathrm{b}}=u_{\mathrm{sq}}-R{i}_{\mathrm{sq}}-L\frac{d{i}_{\mathrm{sq}}}{dt}-2\omega L{i}_{\mathrm{sd}},$$

(1)

The DC side voltage equation of the MMC-RPC is:

$$0.5u_{\mathrm{dc}}=u_{kj\mathrm{p}}+L_{\mathrm{S}}\frac{d{i}_{kj\mathrm{p}}}{dt}+R_{\mathrm{S}}{i}_{kj\mathrm{p}}+e_j,$$

(2)

$$0.5u_{\mathrm{dc}}=u_{kj\mathrm{n}}+L_{\mathrm{S}}\frac{d{i}_{kj\mathrm{n}}}{dt}+R_{\mathrm{S}}{i}_{kj\mathrm{n}}-e_j,$$

(3)

By using simultaneous solution for Equations (2) and (3), the MMC-RPC voltage equations on both AC side and DC side can be obtained [22]:

$$u_{\mathrm{cq}}=0.5\left(u_{kj\mathrm{n}}-u_{kj\mathrm{p}}\right)+0.5L_{\mathrm{S}}\frac{d\left(i_{kj\mathrm{n}}-i_{kj\mathrm{p}}\right)}{dt}+0.5R_{\mathrm{S}}\left(i_{kj\mathrm{n}}-i_{kj\mathrm{p}}\right),$$

(4)

$$u_{\mathrm{dc}}=\left(u_{kj\mathrm{n}}+u_{kj\mathrm{p}}\right)+L_{\mathrm{S}}\frac{d\left(i_{kj\mathrm{n}}+i_{kj\mathrm{p}}\right)}{dt}+0.5R_{\mathrm{S}}\left(i_{kj\mathrm{n}}-i_{kj\mathrm{p}}\right),$$

(5)

According to Kirchhoff’s theorem, we have:

$$i_{k\mathrm{ap}}=-i_{\mathrm{dc}}/2 - i_{\mathrm{S}},i_{k\mathrm{an}}=-i_{\mathrm{dc}}/2 + i_{\mathrm{S}},$$

(6)

$$i_{k\mathrm{bp}}=-i_{\mathrm{dc}}/2 + i_{\mathrm{S}},i_{k\mathrm{bn}}=-i_{\mathrm{dc}}/2 - i_{\mathrm{S}},$$

(7)

According to the above equations, the current and voltage relationship among upper and lower bridge arms corresponding to each phase can be obtained: i_ap = i_bn, i_an = i_bp, u_ap = u_bn, and u_an = u_bp.

2.2. Compensation Principle

In order to realize the active power balance, the reactive power compensation, the elimination of the negative sequence current at the grid side, and the suppression of the harmonic components in the traction power supply system, it is necessary to provide compensation power references for the MMC-RPC. Figure 3 shows the power compensation phasor diagram of the system under the working condition with heavy load at the left supply arm and light load at the right supply arm of the traction substation, corresponding to different phases.

Where:

U_A, U_B, and U_B are the three-phase voltages of the power grid;

P_L, P_R, Q_L, and Q_R are the active power and reactive power of the left and right supply arms, respectively;

P_Lref, P_Rref, Q_Lref, and Q_Rref are the reference compensation values of the active power and reactive power of each supply arm, respectively.

To achieve active power balance, MMC-RPC needs to transfer 0.5|P_L − P_R| power from the light load side to the heavy load side, so that the power of the supply arms on both sides is P_L′ = P_R′ after the transfer. According to Figure 3, the left and right supply arms compensate P_L′tan30° + Q_L and P_R′tan30° − Q_R reactive power, respectively, to eliminate the negative sequence current, and realize three-phase current balance on the primary side of the v/v traction transformer. Finally, the power reference components P_LH, P_RH, Q_LH and Q_RH are superimposed for the harmonic suppression, so as to obtain the MMC-RPC reference compensation values:

$$P_{\mathrm{Rref}}=\frac{P_{\mathrm{L}}-P_{\mathrm{R}}}{2}-P_{\mathrm{RH}},$$

(8)

$$P_{\mathrm{Lref}}=\frac{P_{\mathrm{R}}-P_{\mathrm{L}}}{2}-P_{\mathrm{LH}},$$

(9)

$$Q_{\mathrm{Rref}}=-Q_{\mathrm{R}}-P_{\mathrm{R}}^{\prime }\tan \frac{\pi }{6}-Q_{\mathrm{LH}},$$

(10)

$$Q_{\mathrm{Lref}}=Q_{\mathrm{L}}-P_{\mathrm{L}}^{\prime }\tan \frac{\pi }{6}-Q_{\mathrm{LH}},$$

(11)

In the actual power extraction link, the single-phase MMC-RPC has only a single degree of freedom since it is applied in the single-phase power supply system, which cannot separate the instantaneous active and reactive power through dq transformation such as the three-phase MMC. Therefore, this study constructed a virtual component orthogonal to the actual quantity with the help of an all-pass filter, which realizes the synchronous rotation coordinate transformation. The transfer function of the all-pass filter is:

$$\frac{f_{\beta }\left(\mathrm{s}\right)}{f_{\alpha }\left(\mathrm{s}\right)}=\frac{\omega -\mathrm{s}}{\omega -\mathrm{s}},$$

(12)

After constructing the virtual component, the active and reactive power of the single-phase system can be extracted:

$$\left[\begin{array}{l}p\\ q\end{array}\right]=\frac{1}{2}\left[\begin{array}{cc}{u}_{\mathrm{d}}& u_{\mathrm{q}}\\ u_{\mathrm{q}}& -u_{\mathrm{d}}\end{array}\right]\left[\begin{array}{l}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right],$$

(13)

As shown in Figure 3 and Figure 4, the phasor diagram and calculation block diagram of the overall expected compensation power can be obtained. However, the low-pass filter should be firstly used to filter harmonics in the voltage and current before they are input to the power calculation block.

3. VSG Control

3.1. VSG Model

It is necessary to establish the mathematical model of the VSG control to apply it effectively in the MMC control. At present, the rotor motion equation and the stator electrical equation of the synchronous generator are used as the mechanical and electromagnetic links of the VSG, respectively. The excitation controller and turbine regulator of the synchronous generator are also introduced to regulate the frequency and voltage, respectively, according to the active and reactive power [23].

The mathematical model of VSG is:

$$\{\begin{array}{l}\ddot{\theta }=\frac{1}{J}\left(T_{\mathrm{m}}-T_{\mathrm{e}}-D\Delta \omega \right)=\frac{1}{J}\left(\frac{P_{\mathrm{m}}}{\omega }-\frac{P}{\omega }-D\left(\omega -{\omega }_{\mathrm{g}}\right)\right)\\ \frac{\mathrm{d}\theta }{\mathrm{d}t}={\omega }_0+\Delta \omega \end{array},$$

(14)

where:

J is the inertia;

D is the damping coefficient;

T_m and T_e are the mechanical and electromagnetic torques, respectively;

ω, ω_g, and ω₀ are the angular velocity, network voltage angular velocity, and rated angular velocity respectively;

θ is the output angle.

The electrical equation of the VSG is:

$$\{\begin{array}{l}{u}_{\mathrm{d}}=E_{\mathrm{d}}-R_{\mathrm{s}}{i}_{\mathrm{d}}+\omega L_{\mathrm{s}}{i}_{\mathrm{q}}\\ u_{\mathrm{q}}=E_{\mathrm{q}}-R_{\mathrm{s}}{i}_{\mathrm{q}}-\omega L_{\mathrm{s}}{i}_{\mathrm{d}}\end{array},$$

(15)

where:

u_d, u_q, i_d, i_q, E_d, and E_q are the d-axis and q-axis components of VSG terminal voltage, stator current, and three-phase internal potential, respectively;

R_S and L_S are the VSG stator resistance and inductance, respectively.

With the help of the relationship between torque ω and power P, the turbine regulator link of the synchronous generator has been simulated, and it is adjusted as follows:

$$T_{\mathrm{m}}=\frac{P_{\mathrm{ref}}-k_{\mathrm{f}}\left(\omega -{\omega }_0\right)}{\omega },$$

(16)

where:

P_ref is the reference value of active power;

k_f is the frequency modulation coefficient.

VSG excitation controller is:

$$\begin{array}{ll}{E}_{\mathrm{d}}=& E_0+\left(k_{\mathrm{P}}+\frac{k_I}{\mathrm{s}}\right)\left(Q_{\mathrm{ref}}-Q\right)-\\ & k_{\mathrm{u}}\left(U_{\mathrm{ref}}-U_{\mathrm{m}}\right)\end{array},$$

(17)

where:

E₀ is the internal potential without load;

k_P and k_I are the reactive power proportional integral regulation parameters;

Q_ref is the reference value of reactive power;

Q is the reactive power;

k_u is the voltage regulation coefficient;

U_ref and U_m are the rated terminal voltage and terminal voltage, respectively.

3.2. Design of VSG Controller

3.2.1. Power Loop Design

According to the VSG model in Section 3.1, the operating principle of synchronous generator can be applied in the MMC power loop as shown in Figure 5:

As shown in Figure 5, the active power controller adjusts the active power through adjusting the torque of the VSG turbine regulator according to the value difference between the actual power and the expected power. Moreover, the reactive power controller adjusts the VSG excitation and bridge arm voltages to adjust the reactive power.

The active power is fed back to the control link in the form of torque. Then the synchronous angular velocity ω is obtained so the output synchronous angle is θ after the integration. Inertia and damping parameters are introduced to simulate the characteristic of the synchronous generator to keep the stability of the frequency and power under the disturbance.

3.2.2. Inner-Current Loop Design

According to the Equation (15), the VSG stator internal potential E can be established according to reactive power and terminal voltage, which can be used to calculate the expected output current of the MMC-RPC, as shown in Figure 6:

The d-axis and q-axis components of the output current can be further decoupled to establish the voltage of each bridge arm of the MMC-RPC so that the switch order of each submodule is obtained. Since the β-axis component obtained by dq-αβ inverse transformation is virtual quantity, it can be ignored. Then the expected voltages of the upper and lower bridge arms corresponding to each phase are related with:

$$\{\begin{array}{l}{u}_{\mathrm{kap}}=u_{\mathrm{kbn}}\\ u_{\mathrm{kan}}=u_{\mathrm{kbp}}\\ u_{\mathrm{kap}}=-u_{\mathrm{kbp}}\end{array},$$

(18)

3.3. Adaptive VSG Control

In traditional VSG control, the virtual inertia and damping coefficient of J and D are fixed values, which cannot reach optimal control with varied working conditions. Therefore, the adaptive VSG is applied in the MMC-RPC control. By using different values of the two coefficients, the regulation effect of the system is different, as follows:

$$\Delta \omega =\frac{T_{\mathrm{m}}-T_{\mathrm{e}}-Jd\omega /dt}{D},$$

(19)

$$\frac{d\omega }{dt}=\frac{T_{\mathrm{m}}-T_{\mathrm{e}}-D\Delta \omega }{J},$$

(20)

According to the above equations, when J is constant, the larger the D parameter is, the smaller the angular frequency deviation Δω is; when the D parameter is constant, the larger the J parameter is, the smaller the change rate dω/dt of angular velocity ω is. By adjusting J and D parameters, the change in Δω and dω/dt can be suppressed. However, the parameter values of J and D are inappropriate to affect the control stability under traditional VSG control when the system load changes.

As shown in Figure 7 and Figure 8, the power angle curve and transient angular frequency curve of VSG under power oscillation are shown, respectively. Take [t₁, t₂] time period as an example: at time t₁, the load change leads to power oscillation, Δω remains unchanged at the initial time, resulting in a sudden increase in Jdω/dt. Since J is a fixed value, dω/dt increases, which leads to an increase in Δω.

Therefore, in this time period, it is necessary to increase J to suppress the frequency deviation and increase D to suppress overshoot of the angular frequency ω. Similarly, in other time periods, J and D parameters also need to be changed accordingly. Otherwise, too large J increases the overshoot of active power and power oscillation, resulting in slow response speed and poor stability of the system. Furthermore, if the value of J is too small, the sudden change in the angular frequency in the transient process cannot be effectively suppressed. Similarly, the value of D also affects the suppression effect of VSG on active power oscillation.

By analyzing Figure 7 and Figure 8, we can obtain the selection rules of inertia J and damping coefficient D in different intervals, as shown in

Table 1. Selection rules of J and D.

Section	Δω	dω/dt	Δω (dω/dt)	J	D
[t1, t2]	>0	>0	>0	enlarge	enlarge
[t2, t3]	>0	<0	<0	reduce	enlarge
[t3, t4]	<0	<0	>0	enlarge	enlarge
[t4, t5]	<0	>0	<0	reduce	enlarge

:

Combined with the selection rules shown in Table 1, according to the deviation of power, adaptive VSG control based on dynamical regulation of J and D parameters is designed, which is as follows [24]:

$$J=J_0+K_1\left|\Delta P\right|,$$

(21)

$$D=D_0+K_2\left|\Delta P\right|,$$

(22)

where:

J₀ is the initial inertia;

D₀ is the initial damping coefficient;

ΔP is the power deviation;

K₁ and K₂ are the regulating coefficients of J and D, respectively.

The power change curves under traditional VSG and adaptive VSG control are shown in Figure 9. J and D parameters are set as 3000 and 10, respectively, for both traditional and adaptive VSG controls in the beginning. The difference is that J and D of the traditional VSG control are unchanged during the whole period of time, and they are varied for the adaptive VSG control according to values derived from Equations (21) and (22). It can be seen from the figure that the power reaches the expected values quickly and smoothly under two controls when the MMC-RPC is put into operation. However, at 0.6 s, there is a power overshoot of about 3 MW under the traditional VSG control when the load power increases to around 15 MW. However, the power changes smoothly under adaptive VSG control with the dynamical regulation of two parameters of J and D. Furthermore, the response speed is faster, and the response time is reduced by 60% under the adaptive VSG control. It can be seen that J and D changed to 5000 and 20, respectively, at 0.6 s for the adaptive VSG.

3.4. Small Signal Analysis of VSG

Taking the small signal disturbance of the variables in Equation (14) and adding them to the steady-state variables shown in Figure 5, the VSG small signal analysis model can be obtained in Figure 10.

For each variable of the VSG, small signal disturbance is taken near the steady-state value, and the following equations can be obtained:

$$\{\begin{array}{l}P=P_0+\widehat{p}\\ P_{\mathrm{m}}=P_{\mathrm{m}0}+{\widehat{p}}_{\mathrm{m}}\\ Q=Q_0+\widehat{q}\\ \omega ={\omega }_0+\widehat{\omega }\\ \delta ={\delta }_0+\widehat{\delta }\\ E=E_0+\widehat{e}\\ T_{\mathrm{m}}=T_{\mathrm{m}0}+{\widehat{T}}_{\mathrm{m}}\\ T_{\mathrm{e}}=T_{\mathrm{e}0}+{\widehat{T}}_{\mathrm{e}}\end{array},$$

(23)

where P₀, P_m0, Q₀, ω₀, δ₀, E₀, T_m0, and T_e0 are the steady-state variables and superscript ^ indicates the disturbance variables. Substituting the equation into the VSG mathematical model, ignoring the steady-state component and the disturbance components above the second order on both sides of the equation, and then performing the Laplace transform, the following equations can be derived:

$$\{\begin{array}{l}J\omega s\Delta \omega ={\widehat{p}}_{\mathrm{m}}-\widehat{p}-D\widehat{\omega }\\ \Delta E=-k_{PI}\widehat{q}\\ \widehat{\delta }=\widehat{\omega }/s\\ \widehat{p}={\widehat{T}}_e{\omega }_0\end{array},$$

(24)

Figure 11 shows the VSG model under small signal disturbance:

In combination with Equation (24), the transfer function of the VSG small signal analysis model can be obtained when the active power of VSG acts alone, in which ΔP_m is the input, and ΔP is the output [25]. Note here, ω_{g =} ω_n in steady state.

$$\begin{array}{l}G\left(\mathrm{s}\right)=\frac{\Delta P}{\Delta P_{\mathrm{m}}}=\\ \frac{E{U}_{\mathrm{s}}}{E{U}_{\mathrm{s}} + J\left(\omega L\right){\omega }_{\mathrm{n}}{\mathrm{s}}^2 + D\left(\omega L\right){\omega }_{\mathrm{n}}\mathrm{s}}\end{array},$$

(25)

According to the transfer function, the root locus diagrams with different damping coefficients D and inertia J can be obtained, which is shown in Figure 12.

According to the stability criterion, all eigenvalues in the left half plane of the imaginary axis indicate that the system is stable. It can be seen from Figure 12a that with the continuous increase in J parameter the characteristic root approaches the virtual axis and the conjugate poles gradually deviate from the real axis, resulting in the increase of system oscillation. Although the power regulation is accelerated, the system stability is reduced, so the value of J cannot be too large. It can be seen from Figure 12b that with the gradual increase in D, the system characteristic root is far away from the virtual axis and close to the real axis, the damping increases, the system oscillation decreases, and the system is in a stable operation state. Therefore, the increase in D can improve the system stability to a certain extent.

4. Inner-Loop Control Based on DFC

4.1. Theory of DFC

Differential Flatness Control (DFC) is a kind of nonlinear control. Since the system under this control has the advantages of small output fluctuation and good dynamic performance, DFC has often been used in the control of power electronic equipment in recent years. The following equation can be satisfied for a certain nonlinear system for the DFC:

$$\dot{\mathrm{x}}=f\left(\mathrm{x},\mathrm{u}\right),$$

(26)

It is assumed that the output of the system can be expressed as:

$$\mathrm{z}\left(t\right)=\mathrm{h}\left(\mathrm{x},\mathrm{u},\frac{d\mathrm{u}}{dt},\cdot \cdot \cdot ,\frac{d^{\mathrm{k}}\mathrm{u}}{d{t}^{\mathrm{k}}}\right),$$

(27)

The input u, state x, output z, and their finite derivative variables compose a vector space. If the output vector can express all input and state vectors of the system, which is shown in Equation (28), then the basis vectors composed of these vectors can be unfolded into several hyperplanes, respectively. The hyperplane is linear, and there is no warping and bending caused by nonlinearity. Therefore, the system is a flat system; the system is differential flat for the output, the output vector is called a flat output vector.

$$\{\begin{array}{l}\mathrm{x}\left(t\right)={\mathrm{h}}_{\mathrm{x}}\left(\mathrm{z},\frac{d\mathrm{z}}{dt},\cdot \cdot \cdot ,\frac{d^{\mathrm{k}}\mathrm{z}}{d{t}^{\mathrm{k}}}\right)\\ \mathrm{u}\left(t\right)={\mathrm{h}}_{\mathrm{u}}\left(\mathrm{z},\frac{d\mathrm{z}}{dt},\cdot \cdot \cdot ,\frac{d^{\mathrm{k}}\mathrm{z}}{d{t}^{\mathrm{k}}}\right)\end{array},$$

(28)

According to the differential flatness theory, a DFC system is designed for the MMC-RPC, and it can be obtained by combining Equations of (1) and (28):

$$\{\begin{array}{l}{u}_{\mathrm{cd}}=e_{k\mathrm{a}}-e_{k\mathrm{b}}=u_{\mathrm{sd}}-R_0{i}_{\mathrm{sd}}-L_0\frac{d{i}_{\mathrm{sd}}}{dt}+2\omega L_0{i}_{\mathrm{sq}}\\ u_{\mathrm{cq}}=e_{k\mathrm{a}}-e_{k\mathrm{b}}=u_{\mathrm{sq}}-R_0{i}_{\mathrm{sq}}-L_0\frac{d{i}_{\mathrm{sq}}}{dt}-2\omega L_0{i}_{\mathrm{sd}}\end{array},$$

(29)

The output vector of the system is:

$$y={\left[y_1,y_2\right]}^{\mathrm{T}}={\left[i_{\mathrm{sd}},i_{\mathrm{sq}}\right]}^{\mathrm{T}},$$

(30)

The input vector of the system is:

$$u={\left[u_1,u_2\right]}^{\mathrm{T}}={\left[u_{\mathrm{cd}},u_{\mathrm{cq}}\right]}^{\mathrm{T}},$$

(31)

The system state variables are:

$$x={\left[x_1,x_2,x_3\right]}^{\mathrm{T}}={\left[i_{\mathrm{sd}},i_{\mathrm{sq}},u_{\mathrm{dc}}\right]}^{\mathrm{T}},$$

(32)

Assuming that the DC side equivalent capacitance is C_eq, without considering the converter loss and energy exchange within the inductor, the DC side dynamic equation is:

$$C_{\mathrm{eq}}{u}_{\mathrm{dc}}\frac{d{u}_{\mathrm{dc}}}{dt}=0.5u_{\mathrm{d}}{i}_{\mathrm{sq}}-\frac{u_{\mathrm{dc}}^2}{R},$$

(33)

The solution of the Equation (33) can be expressed as:

$$u_{\mathrm{dc}}\left(t\right)=k{e}^{-t/\left(R{C}_{\mathrm{eq}}\right)}+\sqrt{0.5R{u}_{\mathrm{sd}}{i}_{\mathrm{sd}}},$$

(34)

where k is related to the initial state and stable state of the DC voltage.

According to the differential flattening theory, the input and the state variables can be expressed as:

$$\{\begin{array}{l}{x}_1=i_{\mathrm{sd}}=y_1\\ x_2=i_{\mathrm{sq}}=y_2\\ x_3=u_{\mathrm{dc}}=k{e}^{-t/\left(R{C}_{\mathrm{eq}}\right)}+\sqrt{0.5R{u}_{\mathrm{sd}}{i}_{\mathrm{sd}}}\end{array},$$

(35)

$$E\left(e_{11},e_{12},e_{21},e_{22}\right)=0.5\left(e_{11}^2+e_{12}^2+e_{21}^2+e_{22}^2\right),$$

(36)

From Equations (35) and (36), it can be seen that the output variables and its finite order derivatives can represent all input variables and state variables of the MMC-RPC. Therefore, the MMC-RPC control corresponds with the differential flattening theory, and the output variables are also flat for the flat system.

4.2. Stability Analysis of DFC

The following equation can be obtained according to Equation (29):

$$\{\begin{array}{l}{L}_0{\dot{y}}_1=-u_1+u_{\mathrm{sd}}-R_0{y}_1+2\omega L_0{y}_2\\ L_0{\dot{y}}_2=-u_2+u_{\mathrm{sq}}-R_0{y}_2-2\omega L_0{y}_1\end{array},$$

(37)

It can be seen from Equation (37) that when the output variables y₁ and y₂ reach the expected value, the expected trajectory of u₁ and u₂ can be gradually tracked.

Construct Proportional Integral (PI) error of output variables y₁ and y₂:

$$\{\begin{array}{l}{e}_{k1}=y_k^{*}-y_k\\ e_{k2}={\displaystyle {\int }_0^t\left(y_k^{*}\left(x\right)-y_k\left(x\right)\right)}dx\end{array},$$

(38)

Aiming at the global asymptotic stability of the system, the positive definite energy equation of the MMC-RPC is designed as follows:

$$2E\left(e_{11},e_{12},e_{21},e_{22}\right)=e_{11}^2+e_{12}^2+e_{21}^2+e_{22}^2,$$

(39)

When the above equation satisfies the initial conditions (when e = 0, E(0) = 0; when e ≠ 0, E > 0), the derivative of (39) can be obtained:

$$\dot{E}\left(e_{11},e_{12},e_{21},e_{22}\right)=e_{11}{\dot{e}}_{11}+e_{12}{e}_{11}+e_{21}{\dot{e}}_{21}+e_{22}{e}_{21},$$

(40)

Equation (37) is substituted into Equation (40) to obtain the following equation:

$$\begin{array}{l}\dot{E}\left(e_{11},e_{12},e_{21},e_{22}\right)=-e_{21}\left(-{\dot{y}}_2^{\ast }-\frac{u_2}{L_0}+\frac{u_{\mathrm{sq}}}{L_0}-\frac{R_0}{L_0}{y}_2-2\omega y_1-e_{22}\right)+\\ e_{11}\left({\dot{y}}_1^{\ast }+\frac{u_1}{L_0}-\frac{u_{\mathrm{sd}}}{L_0}+\frac{R_0}{L_0}{y}_1-2\omega y_2+e_{12}\right)\end{array},$$

(41)

For the differential flat system of the MMC-RPC, when e ≠ 0 and E > 0, the first derivative of $\dot{E}$ needs to be <0, so that when ||e|| tends to 0, and E tends to 0. Moreover, it can ensure that the total energy of the system is reduced to global stability along the desired trajectory. In order to ensure $\dot{E}$ < 0, it can be deduced that:

$$\{\begin{array}{l}{u}_1=L_0\left(-{\dot{y}}_1^{\ast }-\frac{u_{\mathrm{sd}}}{L_0}-\frac{R_0}{L_0}{y}_1+2\omega y_2-e_{12}-k_1{e}_{11}\right)\\ u_2=L_0\left(-{\dot{y}}_2^{\ast }-\frac{u_{\mathrm{sq}}}{L_0}-\frac{R_0}{L_0}{y}_2-2\omega y_1-e_{22}-k_2{e}_{21}\right)\end{array},$$

(42)

By introducing Equation (42) into Equation (41), it can be deduced that:

$$\dot{E}\left(e_{11},e_{12},e_{21},e_{22}\right)=-k_1{e}_{{}_{11}}^2-k_2{e}_{{}_{21}}^2\le 0,$$

(43)

5. System Control Strategy Design of MMC-RPC

5.1. Adaptive VSG-DFC Control on Inverter Side

In this study, the differential flatness theory is introduced into the MMC-RPC closed-loop control. By proving the stability of the DFC system, the Lyapunov Equation (41) is established, and the inner-loop controller Equation (42) is derived. At the same time, the DFC inner loop is combined with the adaptive VSG control. As for the outer loop, the VSG generates the output components required by the DFC. Then, through the inner-loop control based on the differential flattening theory, the voltage of each bridge arm of the MMC-RPC is established to obtain the switch order of each submodule, as shown in Figure 13.

5.2. DC Voltage Three-Level Control on Rectifier Side

Since each MMC submodule has an independent capacitor, it is necessary to balance and stabilize the capacitor voltage of the submodule, which is the prerequisite for the reliable operation of the MMC system. According to the characteristics of the MMC rectifier, a hierarchical control strategy including the system-level control, cluster-group voltage control, and inter-cluster voltage control was applied, and it was combined with carrier Phase-Shift Modulation (PSM). The capacitor voltage of the MMC submodule can be balanced and the DC link voltage can be stabilized.

5.2.1. System-Level Control on Rectifier Side

In order to ensure the reliable operation of the MMC-RPC, the constant DC bus voltage control is adopted on the rectifier side, and the DC link voltage is stabilized through PI adjustment. The specific control is shown in Figure 14:

5.2.2. Cluster-Group Voltage Control

The voltage sharing control takes the individual voltage of each phase as the control objective. For the total capacitance voltage of the upper and lower bridge arms of each phase, its average value is adjusted through the outer voltage loop, and the current is adjusted by the inner current loop. The capacitor voltage of each submodule can accurately track the expected value. The specific control is shown in Figure 15:

5.2.3. Inter-Cluster Voltage Control

This voltage stabilizing control takes the capacitor voltage of single submodule as the control objective. According to the difference between the actual value and the expected value of the capacitor voltage of each submodule, the capacitor voltage is tracked by proportional control. Moreover, the correction component is superimposed in the modulation waveform so as to realize the stable control of the submodule voltage between bridge arms. The specific control is shown in Figure 16:

6. Simulation Analysis

In this study, the MMC-RPC based on improved VSG-DFC control is established on the Matlab/Simulink platform to verify the stability and effectiveness of this management scheme.

6.1. Verification of the Adaptive VSG-DFC

It is assumed that there is 10 MW traction load on the left power supply arm of the traction network, which suddenly changes to 20 MW at 0.5 s. The adaptive VSG-DFC control and the traditional direct power control are respectively adopted on the inverter side of the MMC-RPC.

As shown in Figure 17, under the condition of sudden change in traction load, the output frequency of MMC-RPC system under the traditional control fluctuates largely, which is caused by the lack of damping and inertia support. At 0.5 s, the load changes suddenly, and then the maximum frequency fluctuation amplitude reaches 2 Hz for nearly 0.1 s, thus affecting the stability of traction power supply system. Under the adaptive VSG-DFC control, the frequency response is more stable and faster when the MMC-RPC system is put into operation, and the regulation time is reduced by 25%. In addition, the output frequency is nearly unaffected by the sudden change in the traction load. Both cases show the adaptive VSG-DFC has better stability and dynamic performance.

As shown in Figure 18, traction load changes suddenly at 0.5 s. There is a power overshoot power of 1 MW when the load is changed at 0.5 s under the traditional double closed-loop control. Furthermore, there is the power fluctuation of about 0.1 MW under the traditional control, when reaching the desired power at about 0.46 s. Moreover, the adaptive VSG-DFC control has no sudden change in the process of power transfer, which is smoother and more stable. Moreover, the power fluctuation is smaller. The VSG-DFC strategy avoids the reduction in power quality, has better power regulation effect and higher stability.

Figure 19 and Figure 20 show the voltage and current waveform under VSG-DFC control when traction load changes suddenly at 0.5 s. The inverter side current remains stable when the traction load changes suddenly. Similarly, the capacitor voltage of submodules on one bridge arm can be stabilized at the expected value of 5500 V. The voltage of each submodule remains the same, and the voltage ripple is less than 1%.

Figure 21 shows the capacitor voltages of submodule under two controls. The capacitor voltage fluctuation of submodule is smaller under the adaptive VSG-DFC control. By comparison, the joint control of VSG with current decoupling inner loop has more voltage fluctuation.

6.2. Verification of Management Effect under Adaptive VSG-DFC Strategy

On the simulation platform, under the traction load of 15 MW on the left arm and 1 MW on the right arm in the traction network, MMC-RPC is put into operation at 0.8 s to verify the power quality control effect.

Figure 22 shows the management capability under VSG-DFC, which illustrates the power quality management capability of the MMC-RPC in the traction power supply system. The MMC-RPC is put into use at 0.8 s in the traction power supply system. It can be seen from Figure 22 that the three-phase current at the power grid side is unbalanced. After the MMC-RPC system transferred active power and compensated reactive power, the three-phase current is gradually balanced. Figure 23 shows the current of traction power supply arm. Due to the imbalance of locomotive load, the current imbalance on both sides is large. Then, the current balance on both sides is realized through MMC-RPC, and the negative sequence current is well-managed.

7. Conclusions

Aiming at the power quality problems of the traction power supply system with a v/v wiring transformer in a high-speed railway, this paper proposes a new power quality management scheme based on the MMC-RPC. This paper firstly analyzed the mathematical model and control strategy of the MMC-RPC to compensate the active and reactive power in the system. The adaptive VSG control is more stable compared with traditional VSG control. Then this paper proposed a joint control by combining the adaptive VSG and the DFC. Small signal analysis and simulation on Matlab/Simulink platform were carried out and results verify the effectiveness of the control. Based on the analysis results, the following conclusions were obtained:

(1)

According to the frequency deviation and frequency derivative of the response, the J and D parameters were adaptively adjusted in the improved VSG control. In the case of sudden change in traction load, the MMC-RPC under the adaptive VSG control has better stability. The small signal analysis verified the stability of VSG control. Adaptive VSG can make the parameters change dynamically, stay away from the unstable region, and maintain the control with more margin. Besides the improved stability, the response speeds of both frequency and power are faster compared with traditional VSG and double closed-loop control, when the MMC-RPC is put into operation or the traction load is suddenly changed;

(2)

The DFC has good response speed and stability, which ensures the stable and effective operation of the MMC-RPC. Furthermore, the DFC control on the rectifier side is used to stabilize the DC side voltage. The stability of DFC control was confirmed by stability analysis. By applying the DFC into the inner loop of the adaptive control, both the AC and DC dynamic responses were improved;

(3)

Under the adaptive VSG-DFC, the MMC-RPC can realize the bi-directional power control for power supply arms of different phases. Overall, the adaptive VSG-DFC can realize the active power balance, reactive power compensation, and negative sequence current suppression. In addition, it can ensure the current and voltage balance, and realize the comprehensive management of power quality.

In the subsequent research, the hardware in the loop simulation of low-power MMC-RPC will be carried out through dsPace platform to verify the effectiveness of the system.