Two-Stage Multiple-Vector Model Predictive Control for Multiple-Phase Electric-Drive-Reconstructed Power Management for Solar-Powered Vehicles

Abstract

Electric-drive-reconstructed onboard chargers (EDROCs), also known as electric-drive-reconstructed power management systems, are a promising alternative to conventional onboard chargers due to their characteristics of low cost and high power density. The model predictive control offers a fast dynamic response, simple implementation, and the ability to control multiple targets simultaneously. In this paper, a two-stage multi-vector model predictive current control (MPCC) of a six-phase EDROC for solar-powered electric vehicles (EVs) is proposed. Firstly, the topology for the EDROC incorporating a six-phase symmetrical permanent magnet synchronous machine (PMSM) is introduced, and the operation principles of the DC charge mode, the drive mode, and, especially, the in-motion charge mode are analyzed in detail. After that, a two-stage multi-vector MPCC method is proposed by using the multi-vector MPC technique and designing a two-stage MPC structure to eliminate the regulation of the weighting factor of the MPC. Finally, the effectiveness of the proposed method is verified on a self-designed 2 kW EDROC platform.

1. Introduction

With the energy crisis and environmental concerns, electric vehicles (EVs) are receiving more and more attention since they use no fossil fuels and emit no greenhouse gases [1]. However, the convenience of charging and the range of EVs are still two major barriers to the promotion of EVs due to the limitations of the development of battery materials and the imperfections in infrastructure construction [2]. Depending on where they are placed, EV chargers can be divided into onboard chargers and offboard chargers (or charging piles) [3]. Although EVs can be rapidly charged through the offboard charger, it is expensive and requires valuable land resources, making large-scale construction impossible in the short term. Onboard chargers can be used to charge EVs at any time where there is a power outlet, and thus, they are an effective solution to the problem of charging EVs. Among onboard chargers, electric-drive-reconstructed onboard chargers (EDROCs), also known as electric-drive-reconstructed power management systems, charge EVs by reconstructing the machine windings and multiplexing the drive inverter [4]. Therefore, they are more economical and have higher power density compared to independent onboard chargers, and they have been studied by a large number of scholars [5].

In combination with a three-phase electric drive, a single-phase EDROC is investigated in [6] and, in particular, the torque generated by the machine rotor during charging is analyzed. In [7], a three-phase electric drive system is reconfigured as a Buck converter to regulate the battery charging current and charging voltage. Compared to three-phase machines, multiphase machines have a smoother torque output and are more fault-tolerant, making them particularly suitable for use in EVs [8]. Therefore, a large number of EDROCs based on multiphase machines have been proposed [9]. In [10], In combination with a six-phase permanent magnet synchronous machine (PMSM), a three-phase EDROC is studied. In [11], the six-phase electric drive system is reconfigured as a single-phase EDROC in order to perform a convenient single-phase charging.

The charging problem of EVs can be solved by the EDROC mentioned above. However, the range problem of EVs still exists. In addition, since EVs are charged through the grid where energy is currently derived to a large extent from fossil fuels, they are essentially indirect greenhouse gas emitters. Because solar-powered EVs can charge their traction battery through vehicle roof photovoltaic panels (PPs), it not only enhances the range of EVs but also increases the proportion of the renewable energy consumed by the EV, thus reducing indirect greenhouse gas emissions [12]. If the vehicle roof PPs have an output of 1 kW, the power generated by the vehicle roof PPs in one hour can extend the range by an average of 12 km. If the vehicle roof PP is recharged for an average of 5 h, an additional 60 km of range can be achieved per day. As a result, solar-powered EVs are researched by many scholars [13], and they are being designed by car companies like Audi, Toyota, NEDO, etc. For this type of EV, a new EDROC system is investigated in [14], where the EV can be charged by vehicle roof PPs during driving, i.e., in-motion charging. However, a DC–DC converter needs to be added in order to regulate the output current of the PPs and, thus, achieve the maximum power point (MPPT) output of the PPs. An improved EDROC based on a six-phase symmetric PMSM for solar-powered EVs is proposed in [15]. In this work, no power devices need to be added except for a few mode-switching switches. Additionally, based on [15], a multi-energy interface EDROC is designed for solar-powered EVs, where the traction battery can be charged by vehicle roof VVs, DC grid, or AC grid [16].

In the above-mentioned work, the EDROC system is mainly realized through the PI controller. However, with the increase in the number of controlled targets, the adjustment of PI controller parameters takes a large amount of effort and is quite difficult [17]. In contrast, model predictive control (MPC) offers a fast dynamic response, simple implementation, the ability to control multiple targets simultaneously, and no requirement to adjust parameters [18]. As a result, MPC is introduced to the control of the EDROC to simplify control and eliminate parameter tuning. In [19], a three-phase EDROC incorporating a six-phase induction machine is constructed, and the model protective current control (MPCC) for the EDROC is designed. Additionally, an MPC for a single-phase EDROC is proposed in [20]. However, the MPC for the EDROC for solar-powered EVs in [15] has not been developed. For the EDROC in [15], especially for in-motion charging mode, not only do the machine output speed and torque need to be controlled but also the vehicle roof PP’s output power needs to be regulated at the same time. Hence, the characteristics of the MPC are particularly well suited to the requirements of the EDROC studied in [15].

In this paper, based on the topology of the EDROC in [15], a two-stage multi-vector MPCC (that can also be known as a two-stage continuous control set MPCC) or EDROC is developed. The proposed method regulates the dq-axis currents and, hence, the machine output torque and speed by using six active voltage vectors (VVs) that are zero in the xy-plane, which eliminates the xy-axis currents’ control and, thus, simplifies the control complexity. In addition, the multi-vector MPC technique is employed to enhance the steady-state performance of the MPC. Additionally, a two-stage MPC structure is proposed, with one stage for controlling the machine dq-axis currents and another stage for controlling the 01-axis current to regulate the vehicle roof PP’s output power or the battery charging power, thus eliminating the design of the weighting factor of the MPC. Finally, the proposed method is verified on a self-designed 2 kW EDROC platform to validate the feasibility of the proposed method.

2. Operation Principle of EDROC

2.1. Topology

As illustrated in Figure 1, the EDROC system consists of a six-phase inverter, six-phase symmetrical PMSM, traction battery, vehicle roof PPs, and mode-switching switches K₁ and K₂. When K₁ and K₂ are both open, neither the vehicle roof PPs nor the DC grid is connected to the drive system. In this case, the EDROC system working in the state of the normal drive mode and the traction battery is employed to drive the EVs, as shown in Figure 2a. If the EV is idle and a DC grid socket can be acquired, the traction battery can be charged by closing the switch K₁ to link to the DC grid, and the system works in the DC charging state, as displayed in Figure 2b. In this case, the six-phase inverter as well as the windings of the six-phase symmetrical PMSM are reconstructed as a special triple-parallel boost converter, which has been analyzed in detail in [15]. Thus, the voltage of the DC grid must be lower than the voltage of the traction battery.

In addition, if the lighting is strong enough, the vehicle roof VV is plugged into the EDROC system by turning on K₂. In this scenario, if the EV is stationary, the traction battery is charged by the vehicle roof VVs, and the operation state is similar to the DC charging state in Figure 2b. Otherwise, the system operates in in-motion charging mode. When the energy generated by vehicle roof VVs is not enough to provide the energy driving the EV, the traction battery and the vehicle roof VVs together supply the energy required to travel in the EV, as shown in Figure 2c. As a result, the output power of the traction battery is reduced, and the mileage of the EV is enhanced. If not, the vehicle roof VV not only provides the energy required to drive the EV but also charges the traction battery.

2.2. Model of Six-Phase Symmetrical PMSM

The derivation of the model of the six-phase symmetrical PMSM is required before analyzing the operation principle of the EDROC. For six-phase symmetrical PMSM, its mathematic model can be expressed as

$$u_s=R{i}_s+L_s{\displaystyle \frac{d{i}_s}{dt}}+e_s$$

(1)

where the subscript s = A, B, C, U, V, and W; and u, i, R, L, and e are the stator voltage, current, resistance, inductance, and the back electromotive force, respectively. Because the mathematic model of a six-phase symmetrical PMSM in the natural coordinate frame is strongly coupled, it is generally decoupled into two mutually perpendicular 2-D coordinate systems (αβ and xy) and two zero axes (0₁ and 0₂) through employing the vector space decomposition (VSD) transformation T_VSD to streamline the analysis and control the strategy design.

$$T_{\mathrm{VSD}}={\displaystyle \frac{1}{3}}\left[\begin{array}{cccccc}1& 1/2& -1/2& -1& -1/2& 1/2\\ 0& \sqrt{3}/2& \sqrt{3}/2& 0& -\sqrt{3}/2& -\sqrt{3}/2\\ 1& -1/2& -1/2& 1& -1/2& -1/2\\ 0& \sqrt{3}/2& -\sqrt{3}/2& 0& \sqrt{3}/2& -\sqrt{3}/2\\ 1/2& -1/2& 1/2& -1/2& 1/2& -1/2\\ 1/2& 1/2& 1/2& 1/2& 1/2& 1/2\end{array}\right] \left|\begin{array}{c}\mathsf{\alpha }\\ \mathsf{\beta }\\ \mathrm{x}\\ \mathrm{y}\\ 0_1\\ 0_2\end{array}\right.$$

(2)

After using T_VSD for Equation (1) and converting αβ to dq using Park transformation, the decoupled six-phase symmetrical PMSM mathematical model is obtained as

$$\left[\begin{array}{c}{u}_{\mathrm{d}}\\ u_{\mathrm{q}}\end{array}\right]=R\left[\begin{array}{c}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right]+\left[\begin{array}{cc}{L}_{\mathrm{d}}& 0\\ 0& L_{\mathrm{q}}\end{array}\right]{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right]+{\omega }_{\mathrm{e}}\left[\begin{array}{c}-L_{\mathrm{q}}{i}_{\mathrm{q}}\\ L_{\mathrm{d}}{i}_{\mathrm{d}}+{\psi }_{\mathrm{f}}\end{array}\right],$$

(3a)

$$\left[\begin{array}{c}{u}_{\mathrm{x}}\\ u_{\mathrm{y}}\end{array}\right]=R\left[\begin{array}{c}{i}_{\mathrm{x}}\\ i_{\mathrm{y}}\end{array}\right]+L_{\sigma }{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{\mathrm{x}}\\ i_{\mathrm{y}}\end{array}\right],$$

(3b)

$$\left[\begin{array}{c}{u}_{01}\\ u_{02}\end{array}\right]=R\left[\begin{array}{c}{i}_{01}\\ i_{02}\end{array}\right]+L_{\sigma }{\displaystyle \frac{d}{dt}}\left[\begin{array}{c}{i}_{01}\\ i_{02}\end{array}\right]$$

(3c)

where ω_e represents the electrical angular speed; ψ_f stands for the permanent magnet flux; and L_σ is the stator leakage inductance. Additionally, it is worth noting that only the current in the dq subspace is involved in the conversion of electromechanical energy, and the currents in the xy subspace and 0₁ and 0₂ axes merely increase the system loss. Therefore, xy subspace currents are regulated to zero, and 0₁ and 0₂ axis currents are not controlled since they are naturally zero in the normal drive mode.

2.3. Analysis of Operation Principle

For the EDROC, it can be equated to a three-port converter: the vehicle roof VV/DC grid is the first port, and it performs as a voltage source; the traction battery is used as the second port, which can be used not only as a voltage source but also as a load; the six-phase symmetrical PMSM serves as the third port, and it is only applied as a load. Thus, according to the law of conservation of energy, the power relationship for the EDROC can be formulated as

$$\begin{array}{c}{P}_{\mathrm{S}}=P_{\mathrm{T}}-P_{\mathrm{F}}\\ P_{\mathrm{S}}=V_{\mathrm{b}}{I}_{\mathrm{b}}, P_{\mathrm{T}}=k_{\mathrm{m}}{n}_{\mathrm{m}}{T}_{\mathrm{e}}, P_{\mathrm{F}}=V_{\mathrm{PP}}{I}_{\mathrm{PP}}\end{array}$$

(4)

where P_S is the traction battery output power, and P_S > 0 (<0) when the traction battery is discharged (charged); V_b and I_b are, respectively, the traction battery output voltage and current; P_T is electromechanical power for the six-phase symmetrical PMSM, and it is always greater than or equal to zero; n_m and T_e, respectively, represent the machine speed and output electromagnetic torque; k_m is the P_m coefficient; P_F is the vehicle roof VV/DC grid output power; V_PP and I_PP are, respectively, the output voltage and current of the vehicle roof VV/DC grid. In normal drive mode, the vehicle roof VV or DC grid is not connected, and P_F equals zero. Therefore, the case is not analyzed here.

In DC charging mode, the EV is stationary so that P_T is equal to zero. Thus, the traction battery performs as a load, and the charging power is the same as the output power of the vehicle roof VV/DC grid. As a result, the 0₁-axis current is regulated to control the output power of the vehicle roof VV/DC grid, adjusting the charging power. In addition, the speed of the EV is adjusted to zero, and the xy-axis currents are controlled to zero in this mode. In in-motion charging mode, it can be found from Equation (4) that the traction battery output power can be controlled by regulating P_T and P_F. Here, P_T is indirectly controlled by regulating the dq-axis current, thus adjusting the machine speed and output electromagnetic torque. Likewise, P_F is controlled by regulating the 0₁-axis current, and the xy-axis currents are controlled to zero to reduce system losses.

3. Proposed MPCC of EDROC

3.1. Traditional MPCC

The discrete mathematical model of the six-phase symmetrical PMSM can be obtained by discretizing Equation (3) using the Euler formula as

$$\left\{\begin{array}{l}{i}_{\mathrm{d}}\left(k+1\right)=i_{\mathrm{d}}\left(k\right)+T_{\mathrm{s}}/L_{\mathrm{d}}\left(u_d(k)-R{i}_{\mathrm{d}}\left(k\right)+{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}{i}_{\mathrm{q}}\left(k\right)\right)\hfill \\ i_{\mathrm{q}}\left(k+1\right)=i_{\mathrm{q}}\left(k\right)+T_{\mathrm{s}}/L_{\mathrm{q}}\left(u_{\mathrm{q}}(k)-R{i}_{\mathrm{q}}\left(k\right)-{\omega }_{\mathrm{e}}{L}_{\mathrm{d}}{i}_{\mathrm{d}}\left(k\right)-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\right)\hfill \end{array}\right.,$$

(5a)

$$\left\{\begin{array}{l}{i}_{\mathrm{x}}\left(k+1\right)=i_{\mathrm{x}}\left(k\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{\mathrm{x}}(k)-R{i}_{\mathrm{x}}\left(k\right)\right)\hfill \\ i_{\mathrm{y}}\left(k+1\right)=i_{\mathrm{y}}\left(k\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{\mathrm{y}}(k)-R{i}_y\left(k\right)\right)\hfill \end{array}\right.,$$

(5b)

$$\left\{\begin{array}{l}{i}_{01}\left(k+1\right)=i_{01}\left(k\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{01}(k)-R{i}_{01}\left(k\right)\right)\hfill \\ i_{02}\left(k+1\right)=i_{02}\left(k\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{02}(k)-R{i}_{02}\left(k\right)\right)\hfill \end{array}\right.$$

(5c)

where T_s is the control cycle; (k) and (k + 1) denote the kth and (k + 1)th control cycles, respectively. In normal drive mode, the 0₁- and 0₂-axis current loops are not established, and thus, only the dq-axis and xy-axis currents are regulated. Due to the updating mechanism of the microcontroller, the one-step delay phenomenon exists, and the optimal voltage vector (VV) obtained in the (k − 1)th control cycle is executed in the kth cycle. Therefore, the current in the (k + 1)th control cycle can be predicted according to Equation (5)

$$\left\{\begin{array}{l}{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)=i_{\mathrm{d}}\left(k\right)+T_{\mathrm{s}}/L_{\mathrm{d}}\left(u_d^{\mathrm{ref}}(k)-R{i}_{\mathrm{d}}\left(k\right)+{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}{i}_{\mathrm{q}}\left(k\right)\right)\hfill \\ i_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)=i_{\mathrm{q}}\left(k\right)+T_{\mathrm{s}}/L_{\mathrm{q}}\left(u_{\mathrm{q}}^{\mathrm{ref}}(k)-R{i}_{\mathrm{q}}\left(k\right)-{\omega }_{\mathrm{e}}{L}_{\mathrm{d}}{i}_{\mathrm{d}}\left(k\right)-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\right)\hfill \\ i_{\mathrm{x}}^{\mathrm{p}}\left(k+1\right)=i_{\mathrm{x}}\left(k\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{\mathrm{x}}^{\mathrm{ref}}(k)-R{i}_{\mathrm{x}}\left(k\right)\right)\hfill \\ i_{\mathrm{y}}^{\mathrm{p}}\left(k+1\right)=i_{\mathrm{y}}\left(k\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{\mathrm{y}}^{\mathrm{ref}}(k)-R{i}_y\left(k\right)\right)\hfill \end{array}\right.,$$

(6)

where $u_N^{\mathrm{ref}}(k)$(N = d, q, x, and y) is the optimal VV obtained in the (k − 1)th control cycle. Then, the currents after performing VV Vⁿ can be predicted as

$$\left\{\begin{array}{l}{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+2\right)=i_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)+T_{\mathrm{s}}/L_{\mathrm{d}}\left(u_d^n-R{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)+{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}{i}_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)\right)\hfill \\ i_{\mathrm{q}}^{\mathrm{p}}\left(k+2\right)=i_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)+T_{\mathrm{s}}/L_{\mathrm{q}}\left(u_{\mathrm{q}}^n-R{i}_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)-{\omega }_{\mathrm{e}}{L}_{\mathrm{d}}{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\right)\hfill \\ i_{\mathrm{x}}^{\mathrm{p}}\left(k+2\right)=i_{\mathrm{x}}^{\mathrm{p}}\left(k+1\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{\mathrm{x}}^n-R{i}_{\mathrm{x}}^{\mathrm{p}}\left(k+1\right)\right)\hfill \\ i_{\mathrm{y}}^{\mathrm{p}}\left(k+2\right)=i_{\mathrm{y}}^{\mathrm{p}}\left(k+1\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{\mathrm{y}}^n-R{i}_{\mathrm{y}}^{\mathrm{p}}\left(k+1\right)\right)\hfill \end{array}\right.,$$

(7)

where $u_N^n$(N = d, q, x, and y) is the N-axis component of Vⁿ. For the six-phase symmetrical PMSM drive system fed by a two-level six-phase inverter, 2⁶ = 64 VVs can be obtained, as shown in Figure 3, where the VV is represented by the octal number of (S_A S_B S_C S_U S_V S_W), and S_i = 1(i = A, B, C, U, V, and W) means that the upper of the i-bridge arm is closed. Finally, the VV that minimizes the cost function Equation (8) among the 64 VVs is chosen as the optimal VV and applied in the (k + 1)th control cycle.

$$g_{\mathrm{T}}={\left(i_{\mathrm{d}}^{\mathrm{ref}}-i_{\mathrm{d}}^{\mathrm{p}}\left(k+2\right)\right)}^2+{\left(i_q^{\mathrm{ref}}-i_q^{\mathrm{p}}\left(k+2\right)\right)}^2+w_1\left(i_{\mathrm{x}}^{\mathrm{p}}{\left(k+2\right)}^2+i_{\mathrm{y}}^{\mathrm{p}}{\left(k+2\right)}^2\right)$$

(8)

Here, $i_{\mathrm{d}}^{\mathrm{ref}}$ is the d-axis current reference value, and it is usually set to zero; $i_{\mathrm{q}}^{\mathrm{ref}}$ is the q-axis current reference value, and it is the output of the speed controller; w₁ represents the xy-axis currents’ weighting factor. In a traditional MPCC, the weighting factor must be designed, and an inappropriate weighting factor inevitably deteriorates the system performance. Particularly, more weighting factors should be designed in the MPCC for the EDROC. Additionally, the steady-state performance is poor for a traditional MPCC because only one VV is used in a control cycle. In order to address these issues, a two-stage multi-vector MPCC for the EDROC is designed below.

3.2. Designer of the Proposed MPCC

In the EDROC, only the dq-axis currents and the 0₁-axis current are employed to regulate the machine speed (or output torque) and charging power (or vehicle roof VV’s output power). In Figure 3, “45, 64, 26, 32, 13, 51” are six large VVs in the αβ subspace, but they produce zero voltage in xy subspace. Consequently, only these six VVs are utilized to control the dq-axis current to eliminate the regulation of the xy-axis current weighting factor. Furthermore, “70 and 07” in Figure 3 have equal amplitudes and opposite signs on the 0₁-axis and produce zero voltage in both the xy subspace and the αβ subspace so that they are employed to control the 0₁-axis current. If the dq-axis current and 01-axis current are evaluated using the single cost function, the problem of the weighting factor remains. Additionally, when only one VV is executed during one control period, the state-steady performance is poor. In order to bypass this issue and enhance the state-steady performance, a two-stage multi-vector MPCC for the EDROC is developed in the following:

(1) First stage: dq-axis current control:

As shown in Figure 4, the synthesizable αβ voltage subspace can be divided into six triangular regions by “45, 64, 26, 32, 13, 51” that are labeled V₁, V₂, …, and V₆, and the VV within each triangular region can be synthesized by two VVs adjacent to the triangular region. Hence, two VVs during one control period are used to regulate the dq-axis currents. Then, the dq-axis currents after executing two VVs can be predicted as

$$\left\{\begin{array}{l}{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+2\right)=i_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)+T_{\mathrm{s}}/L_{\mathrm{d}}\left(\left(d_m{u}_d^m+d_n{u}_d^n\right)-R{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)+{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}{i}_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)\right)\hfill \\ i_{\mathrm{q}}^{\mathrm{p}}\left(k+2\right)=i_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)+T_{\mathrm{s}}/L_{\mathrm{q}}\left(\left(d_m{u}_{\mathrm{q}}^m+d_m{u}_{\mathrm{q}}^n\right)-R{i}_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)-{\omega }_{\mathrm{e}}{L}_{\mathrm{d}}{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\right)\hfill \end{array}\right.,$$

(9)

where $u_d^m$ and $u_{\mathrm{q}}^m$ are, respectively, dq-axis voltage components of V_m; $u_d^n$ and $u_{\mathrm{q}}^n$ are, respectively, dq-axis voltage components of V_n; and d_m and d_n denote the duty cycles of V_m and V_n, respectively. Then, the g₁ can be obtained as

$$g_1={\left(i_{\mathrm{d}}^{\mathrm{ref}}-i_{\mathrm{d}}^{\mathrm{p}}\left(k+2\right)\right)}^2+{\left(i_q^{\mathrm{ref}}-i_q^{\mathrm{p}}\left(k+2\right)\right)}^2,$$

(10)

When the derivatives of the cost function g₁ Equation (10) with respect to dm and dn are zero, dm and dn are optimal. The derivatives of the cost function g₁ Equation (10) with respect to dm and dn can be represented as

$$\left\{\begin{array}{l}-2T_{\mathrm{s}}{u}_d^m/L_{\mathrm{d}}\left(i_{\mathrm{d}}^{\mathrm{ref}}-i_{\mathrm{d}}^{\mathrm{p}}\left(k+2\right)\right)-2T_{\mathrm{s}}{u}_{\mathrm{q}}^m/L_{\mathrm{q}}\left(i_{\mathrm{q}}^{\mathrm{ref}}-i_{\mathrm{q}}^{\mathrm{p}}\left(k+2\right)\right)=0\hfill \\ -2T_{\mathrm{s}}{u}_d^n/L_{\mathrm{d}}\left(i_{\mathrm{d}}^{\mathrm{ref}}-i_{\mathrm{d}}^{\mathrm{p}}\left(k+2\right)\right)-2T_{\mathrm{s}}{u}_{\mathrm{q}}^n/L_{\mathrm{q}}\left(i_{\mathrm{q}}^{\mathrm{ref}}-i_{\mathrm{q}}^{\mathrm{p}}\left(k+2\right)\right)=0\hfill \end{array}\right.,$$

(11)

By solving for Equation (11), dm and dn can be calculated as

$$\left\{\begin{array}{c}{d}_m=\frac{\left(i_d^{ref}-i_d^p\left(k+1\right)-s_{0d}\right)\left(s_{nq}-s_{0q}\right)+\left(i_q^{ref}-i_q^p\left(k+1\right)-s_{0q}\right)\left(s_{nd}-s_{0d}\right)}{\left(s_{md}-s_{0d}\right)\left(s_{nq}-s_{0q}\right)-\left(s_{nd}-s_{0d}\right)\left(s_{mq}-s_{0q}\right)}\\ d_n=\frac{\left(i_d^{ref}-i_d^p\left(k+1\right)-s_{0d}\right)\left(s_{md}-s_{0d}\right)+\left(i_q^{ref}-i_q^p\left(k+1\right)-s_{0q}\right)\left(s_{mq}-s_{0q}\right)}{\left(s_{md}-s_{0d}\right)\left(s_{nq}-s_{0q}\right)-\left(s_{nd}-s_{0d}\right)\left(s_{mq}-s_{0q}\right)}\end{array}\right.$$

(12)

where $s_{0d}=T_{\mathrm{s}}/L_{\mathrm{d}}\left(-R{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)+{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}{i}_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)\right)$; $s_{xd}=T_{\mathrm{s}}/L_{\mathrm{d}}\left(u_d^x-R{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)+{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}{i}_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)\right)$ (x = m, n); $s_{0\mathrm{q}}=T_{\mathrm{s}}/L_{\mathrm{q}}\left(-R{i}_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)-{\omega }_{\mathrm{e}}{L}_{\mathrm{d}}{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\right)$; and $s_{\mathrm{xq}}=T_{\mathrm{s}}/L_{\mathrm{q}}\left(u_{\mathrm{q}}^x-R{i}_{\mathrm{q}}^{\mathrm{p}}\left(k+1\right)-\right.$ $\left({\omega }_{\mathrm{e}}{L}_{\mathrm{d}}{i}_{\mathrm{d}}^{\mathrm{p}}\left(k+1\right)-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\right)$. Then, the following limitations are performed in order to ensure the viability of the synthesized voltage

$$d_m=\left\{\begin{array}{cc}{d}_m& , d_m+d_n\le 1\\ {\displaystyle \frac{d_m}{d_m+d_n}}& , d_m+d_n>1\end{array}\right. d_n=\left\{\begin{array}{cc}{d}_n& ,d_m+d_n\le 1\\ {\displaystyle \frac{d_n}{d_m+d_n}}& ,d_m+d_n>1\end{array}\right.,$$

(13)

$$d_m=\left\{\begin{array}{ll}0\hfill & , d_m<0\hfill \\ d_m\hfill & , 0\le d_m\le 1\hfill \\ 1\hfill & , d_m\ge 1\hfill \end{array}\right. d_n=\left\{\begin{array}{ll}0\hfill & , d_n<0\hfill \\ d_n\hfill & , 0\le d_n\le 1\hfill \\ 1\hfill & , d_n\ge 1\hfill \end{array}\right.$$

(14)

Finally, the VV pair that minimizes the cost function g₁ Equation (10) among the six VV pairs is chosen as the optimal VV pair.

(2) Second stage: 01-axis current control:

During a control cycle, “70” and “07”, which are marked V₇ and V₈, are used to control the 0₁-axis current in the residual (1-d_m-d_n)T_s time after the optimal VV pair and the corresponding duty cycles that are applied to adjusted dq-axis currents have been obtained. The values on the 0₁-axis of V₁–V₈ are listed in

Table 1. Voltage on 0₁-axis.

Voltage Vectors	01-Axis Values	Voltage Vectors	01-Axis Values
V1 (45)	−0.167 Vb	V5 (13)	−0.167 Vb
V2 (64)	0.167 Vb	V6 (51)	0.167 Vb
V3 (26)	−0.167 Vb	V7 (70)	0.5 Vb
V4 (32)	0.167 Vb	V8 (07)	−0.5 Vb

, and the 0₁-axis current can be calculated as

$$i_{01}^{\mathrm{p}}\left(k+1\right)=i_{01}\left(k\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{01}^{\mathrm{ref}}(k)-R{i}_{01}\left(k\right)-V_{\mathrm{PP}}/2\right),$$

(15)

$$i_{01}^{\mathrm{p}}\left(k+2\right)=i_{01}^{\mathrm{p}}\left(k+1\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(d_m{u}_{01}^m+d_n{u}_{01}^n+d_{01}{u}_{01}^7+\left(T_{re}-d_{01}\right)u_{01}^8-R{i}_{01}^{\mathrm{p}}\left(k+1\right)-V_{\mathrm{pp}}/2\right),$$

(16)

where $u_{01}^{\mathrm{ref}}(k)$ is the optimal VV obtained in the (k − 1)th control cycle; $u_{01}^m$, $u_{01}^n$, $u_{01}^7$, and $u_{01}^8$ are 01-axis current parts of V_m, V_n, V₇, and V₈, respectively; d₀₁ is duty cycle of V₇; and T_re = 1-d_m-d_n. Then, the g₂ can be calculated as

$$g_2={\left(i_{01}^{\mathrm{ref}}-i_{01}^{\mathrm{p}}\left(k+2\right)\right)}^2,$$

(17)

When the derivatives of the cost function g₂ Equation (17) with respect to d₀₁ is zero, d₀₁ is optimal. The derivatives of the cost function g₂ Equation (17) with respect to d₀₁ can be represented as

$$d_{01}={\displaystyle \frac{i_{01}^{\mathrm{p}}\left(k+1\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(d_m{u}_{01}^m+d_n{u}_{01}^n+T_{re}{u}_{01}^8-R{i}_{01}^{\mathrm{p}}\left(k+1\right)-V_{\mathrm{pp}}/2\right)}{T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{01}^8-u_{01}^7\right)}},$$

(18)

By solving for Equation (18), d₀₁ can be calculated as

$$d_{01}={\displaystyle \frac{i_{01}^{\mathrm{p}}\left(k+1\right)+T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(d_m{u}_{01}^m+d_n{u}_{01}^n+T_{re}{u}_{01}^8-R{i}_{01}^{\mathrm{p}}\left(k+1\right)-V_{\mathrm{pp}}/2\right)}{T_{\mathrm{s}}/L_{\mathsf{\sigma }}\left(u_{01}^8-u_{01}^7\right)}},$$

(19)

where $i_{01}^{\mathrm{ref}}$ is the reference value of the 0₁-axis current. Finally, to ensure the validity of V₇ and V₈, the following restrictions on d₀₁ are applied:

$$d_{01}=\left\{\begin{array}{ll}0\hfill & , d_{01}<0\hfill \\ d_m\hfill & , 0\le d_{01}\le T_{re}\hfill \\ T_{re}\hfill & , d_m\ge T_{re}\hfill \end{array}\right.$$

(20)

3.3. Implementation of Proposed MPCC

The control diagram of the proposed two-stage multi-vector MPCC for the EDROC is displayed in Figure 5. The proposed control method can be implemented in a digital controller following the steps below.

Step 1: Sample the machine currents (i_A, i_B, i_C, i_U, i_V, and i_W), battery current I_b and voltage V_b, and vehicle roof VV or DC grid current I_PP and voltage V_PP, and measure the machine electrical degree θ_e and speed n. Then, convert the machine currents to the dq-axes and 01-axis by using T_VSD Equation (2) and Park transformation.

Step 2: Perform PI speed control to determine the q-axis current reference value, and set the d-axis current reference value to zero.

Step 3: Calculate the 01-axis current reference value using the maximum power point tracking (MPPT) control, which is used to achieve the maximum power output of the vehicle roof VVs and the battery charger controller [16].

Step 4: Obtain the optimal VV pairs V_m and V_n and their corresponding duty cycles d_m and d_n using Equations (10)–(13). Then, calculate d₀₁ using Equations (17) and (18).

Step 5: Generate the control signal of the six-phase inverse system using Equation (19) and apply it to the six-phase inverter.

$$d_x=d_m{S}_x^m+d_n{S}_x^n+d_{01}{S}_x^7+(1-d_m-d_n-d_{01})S_x^8$$

(21)

Here, d_x (x = A, B, C, U, V, and W) is the duty cycle corresponding to the switch of the upper bridge arm on phase x; $S_x^m$, $S_x^n$, $S_x^7$, and $S_x^8$ are the switching states of the x-phase upper bridge arm switch for V_m, V_n, V₇, and V₈. In addition, it should be noted that when the adopted voltage vectors are directly adopted sequentially in a cycle, the switching frequency of the converter is higher than the control frequency and will vary in real time. To solve this problem, the equivalent duty cycle of the converter is adopted instead of the switching state of the adopted voltage vector.

4. Experimental Results

For the sake of verifying the feasibility and effectiveness of the proposed two-stage multi-vector MPCC for an EDROC, a 2 kW EDROC experimental platform is built, as shown in Figure 6. A 2 kW six-phase symmetrical PMSM is used to simulate the traction machine for EVs, and its main parameters are listed in

Table 2. Main parameters of the six-phase symmetrical PMSM.

Parameters	Values
Rated power	2.0 kW
Rated speed	2000 rpm
Number of pole pairs	5
Direct axis inductance	5.56 mH
Quadrature axis inductance	7 mH
0-axis inductance	0.125 mH
Phase resistance	0.3 Ω
Stator-PM magnetic flux	0.042 Wb

. A magnetic power brake is employed to generate a load. A 144 V/50 AH lithium battery serves as the traction battery. Two 36 V/300 W PPs and a 3 kW DC power supply are used to simulate the vehicle roof VVs and DC grid, respectively. The six-phase inverter is constructed from six FF300R12ME4 IGBT modules, and the operation mode is switched by two MGR-1DD220D40 solid-state relays. The current and voltage are measured by HCS-LTS3-15A current sensors and HVS-AS3.3-05mA voltage sensors, respectively. The control method is executed in the TMS320F28335 DSP microcontroller. The sampling and control frequencies are both 10 kHz. The execution time of the proposed method in the TMS320F28335 DSP microcontroller is about 65 μs, which means that the proposed method can be implemented at a control frequency of more than 10 kHz in the TMS320F28335 DSP microcontroller. Additionally, it is also worth noting that the proposed method can be executed not only in DSP but also in commonly used MCUs, such as STM32F407.

Firstly, the normal drive mode is validated under 1000 rpm with a 5 Nm load, as shown in Figure 7. It can be observed from Figure 7 that the machine speed n is regulated to its set value of 1000 rpm, and the phase currents i_A and i_U are particularly smooth and sinusoidal. Additionally, the q-axis current i_q always follows its reference value ${\mathrm{i}}_{\mathrm{q}}^{\mathrm{r}\mathrm{e}\mathrm{f}}$, and the xy-axis currents are always zero. In addition, it can also be noted that the ripples of i_q, i_x, and i_y are less than 1 A. The experimental results show that the normal drive mode can be successfully performed using the proposed method.

Then, the performances of the DC charging mode are evaluated, as demonstrated in Figure 8. It is worth mentioning that in this mode, the machine speed is set to 0, and the DC power supply or the PPs are plugged in. The experimental result of the DC charging mode when the PPs are connected under the condition of 950 W/m² and 35 °C is displayed in Figure 8a. First, it can be observed that the battery output current is approximately −2 A, which means the battery is being charged. Moreover, the battery charging current and voltage remain constant during the charging process, which meets the basic requirements for battery charging. Additionally, the 01-axis current i₀₁ is constant, and thus, the output current of the PPs remains unchanged because it is equal to −3 times the 01-axis current. The phase current of the machine i_A is kept constant, and therefore, the machine does not generate a rotating electromagnetic torque, which fulfills the requirement to remain stationary during the charging of the EV. Afterward, the PPs are substituted by the power supply. The DC charging dynamic response performance is evaluated by changing the charging current in a step manner from 5 A to 2 A, as shown in Figure 8b. It can be noticed that the charge current is immediately adjusted from 5 A to 2 A after the charge current step change command is executed. Furthermore, the charge current remains constant throughout the dynamic regulation process. Additionally, when the charging current is regulated from 5 A to 2 A, the 01-axis current is controlled from −4 A to −1.6 A. The experimental results demonstrate that the proposed method has excellent dynamic performance.

Finally, the in-motion charging mode is conducted, where the PPs are under the conditions of 1000 W/m² and 34 °C. Figure 9 illustrates the steady-state experimental result of this mode under a 500 rpm speed and a load of 5 Nm. The speed is maintained at its set value of 500 rpm, and the phase currents are particularly smooth and sinusoidal. Nevertheless, due to the injection of the 0₁-axis current producing opposite effects in the two sets of three-phase machine windings, a DC current difference of about 3.2 A can be observed in the amplitude of currents i_A and i_U. Similar to the normal drive mode, the q-axis current follows its reference value, and the xy-axis currents always remain 0. Furthermore, their ripple is less than 1 A. The 01-axis current is approximately 1.6 A with 0.4 A of third-order current harmonic, which is caused by the third harmonic part of the magnetic flux generated by the permanent magnets. This problem can be solved by adding a perturbation observer, which is beyond the scope of this paper. The charging voltage is constant, and the charging current is less than 0. Thus, the battery is charged in this case. The experimental result shows that the PP provides the energy consumed by driving the EV while charging the battery. Therefore, the range of the EV can be increased.

Additionally, the dynamic response performance when mode switching from normal driving to in-motion charging modes is tested under the set speed of 500 rpm with a 5 Nm load, as shown in Figure 10. It can be observed that the whole regulation process takes about 30 ms, which means that the proposed control strategy has a good dynamic performance. Additionally, the speed and q-axis current remain constant throughout the regulation process. The xy-axis currents are always zero. In addition, the battery current changes from a positive to a negative sign, i.e., the battery state changes from discharging to charging.

The efficiency curve is presented in Figure 11. Generally, the efficiency increases with the charging power. For DC charging mode, the peak efficiency achieved at 1500 W input power is up to 84.4%. The peak efficiency is 89.5% at 1200 W input power with respect to in-motion charging mode.

5. Conclusions

The paper proposes a two-stage multi-vector MPCC for an EDROC. First, the operation principle of the EDROC studied in the paper is analyzed in detail and comprehensively. Then, the concept of MPC is introduced. After that, in conjunction with the characteristics of the EDROC, a two-stage multi-vector MPCC for the EDROC is designed. In the proposed method, the multi-vector MPC technique is employed to enhance the steady-state performance of the MPC, and a two-stage MPC structure is proposed to eliminate the design of the weighting factor of the MPC. Finally, a 2 kW EDROC platform is constructed to demonstrate the feasibility and effectiveness of the proposed method. The experimental results show that the proposed method can be successfully implemented in normal drive mode, DC charging mode, and in-motion charging mode. Additionally, the proposed method also enables fast and smooth switching between normal drive mode and in-motion charging mode.