Improved Model Predictive Current Control of NPC Three-Level Converter Fed PMSM System for Neutral Point Potential Imbalance Suppression

Abstract

For the neutral point clamped (NPC) three-level converter fed permanent magnet synchronous motor (PMSM) system, the performance of the conventional model predictive current control (MPCC) algorithm will be deteriorated if the amplitude of the neutral point potential (NPP) is large. Additionally, the adjustment process of the weighted coefficients of the conventional MPCC algorithm is complex because of numerous control terms in the cost function. To solve the above issues, an improved MPCC algorithm is proposed in this paper. Firstly, Newtonian iteration is used to transfer the stator current into stator voltage in the cost function. Then, the NPP term in the conventional cost function can be eliminated by introducing the partition control of the NP potential, which also eliminates the whole adjustment process of weighting coefficients. Finally, based on the amplitude of the NPP, the amplitude and phase angle of medium and small vectors are modified to improve the control performance of the torque and flux. Experimental results show that the fluctuation of the neutral point potential can be suppressed rapidly. Meanwhile, the performance of the torque, flux and current are also improved compared with the conventional MPCC.

1. Introduction

PMSMs have the advantages of a simple structure, high power density, low operating noise and high efficiency [1,2,3]. Compared with the conventional two-level converter, the NPC three-level converter possesses advantages such as low voltage stress, high output waveform quality and low switching loss [4,5,6]. Thus, the NPC three-level converter fed PMSM system is widely used in high-power medium-voltage motor drive applications such as electric traction and ship propulsion.

For MPCC, the discrete switching characteristic of the converter is considered, and the switching state minimizing the cost function is selected as the input for the next control period. The multi-objective optimization can be easily obtained by MPCC. Moreover, MPCC can be easily implemented and system constraints can be easily handled [7]. Therefore, it has been widely researched in industry and academia.

When MPCC is applied to NPC three-level inverter fed PMSM systems, the NPP term has to be added into the cost function with d-axis and q-axis stator current terms, which can achieve multi-objective optimal control of the current and NPP. In the cost function of the conventional MPCC algorithm, current and NPP terms have different dimensions, so it is necessary to adjust various weighting coefficients, which means the adjustment process is complicated. The weighting coefficient of the stator current term needs to be increased if the stator current ripple is considered as the primary control objective. However, the NPP ripple will rise if the weighting coefficient of the stator current term is much larger than that of the NPP term, which will also affect the control performance of the stator current.

The stator current terms can be transferred into either a torque/flux term [8,9,10,11] or a stator voltage term [12,13,14,15] for simplification. Alternatively, the stator current term can also be unified as the duration of the voltage vector [16]. Although the d-axis and q-axis stator current terms can be unified as one, the neutral point potential term still remains in the cost function. Thus, weighted coefficients still need to be adjusted.

In [17], a two-stage MPCC algorithm is proposed to reduce the NPP ripple of an NPC three-level inverter fed PMSM system. For the first stage, six medium vectors are used to construct the finite control set. For the second stage, large and zero vectors that do not affect the NPP and small vectors that reduce the NPP ripple are adopted to establish the finite control set. Thus, the NPP ripple can be effectively restrained. In [18], the finite control set is composed of basic voltage vectors and virtual voltage vectors that do not affect the NPP, such that an inherent DC-link voltage balancing can be achieved. In [19,20,21,22], the voltage offset is added to the reference voltage in the cost function. Medium vectors and redundant states of small vectors are selected appropriately to construct the finite control set according to the power factor. Then, the NPP ripple can be suppressed. Although the NPP ripple can be reduced by the above algorithms, the amplitude and phase angle of small and medium vectors will be changed when the imbalance of the NPP is large (DC voltage is imbalanced or the capacitance of the upper and lower DC-link capacitor is not equal). If the finite control set is still constructed according to the original basic voltage vectors, the control performance will be deteriorated.

An improved MPCC algorithm is proposed in this paper. The adjusting process of the weighted coefficients is eliminated. Meanwhile, the neutral point potential imbalance can be rapidly suppressed. For the proposed MPCC, the Newton iteration method is used to obtain the predictive model. Then, the neutral point potential partition control is introduced to eliminate the adjusting process of the weighted coefficients in the cost function. The amplitude and phase angle of basic vectors are modified when the amplitude of the neutral point potential is large. Furthermore, the FCS is reconstructed and the alternative vectors in the FCS are brought into the cost function. Then, the optimal vector for the next control period can be obtained.

2. Conventional MPCC

The topology of the NPC three-level converter fed PMSM system is shown in Figure 1. V_dc is the voltage of the DC source. C₁ and C₂ are DC-link capacitors. Each phase consists of power devices S_x1~S_x4 and clamping diodes D_x1 and D_x2, where x ϵ {A, B, C}.

The switching states and output voltages of each phase are shown in

Table 1. Output voltage corresponding to each switching state.

State	Sx1	Sx2	Sx3	Sx4	Output Voltage
P	1	1	0	0	Vdc/2
O	0	1	1	0	0
N	0	0	1	1	−Vdc/2

.

For three-level converters, there are three switching states for each phase. Thus, a total of 3³ = 27 switching states can be output for three phases, corresponding to 19 basic voltage vectors in the space vector diagram, as shown in Figure 2. According to the amplitude, they can be divided into: large vectors (V₁, V₃, V₅, V₇, V₉, V₁₁), medium vectors (V₂, V₄, V₆, V₈, V₁₀, V₁₂), small vectors (V₁₃~V₁₈) and zero vectors (V₁₉).

When the neutral point of the DC-link is directly connected to the load, the charging and discharging of the DC-link capacitor by the load current will cause the fluctuation of the neutral point potential v_o.

The relationship between v_o and neutral point current i_o can be expressed as follows:

$$v_o=-\frac{1}{2C}{\displaystyle \int i_o\mathrm{d}\mathrm{t}}$$

(1)

where C is the value of the DC-link capacitor. The neutral point current can be represented by the three-phase switching state and the load current as

$$i_o=(1-\left|S_A\right|)i_A+(1-\left|S_B\right|)i_B+(1-\left|S_C\right|)i_C$$

(2)

where S_x denotes the switching state of each phase, S_xϵ{1, 0, −1}. Substituting (2) into (1) and discretizing by the forward Euler method, the predictive value of v_o at (k + 1)T_s is obtained as

$$v_o^{k+1}=v_o^k+\frac{T_s}{2C}{\left|{\mathit{S}}_{\mathrm{A}\mathrm{B}\mathrm{C}}^k\right|}^{\mathrm{T}}{\mathit{i}}_{\mathrm{A}\mathrm{B}\mathrm{C}}^k$$

(3)

where ${\mathit{S}}_{\mathrm{A}\mathrm{B}\mathrm{C}}^k$= [|$S_{\mathrm{A}}^k$| |$S_{\mathrm{B}}^k$| |$S_{\mathrm{C}}^k$|], ${\mathit{i}}_{\mathrm{A}\mathrm{B}\mathrm{C}}^k$= [|$i_{\mathrm{A}}^k$| |$i_{\mathrm{B}}^k$| |$i_{\mathrm{C}}^k$|].

The current predictive model of PMSM in the d-q axis coordinate system is obtained by the forward Euler discretization method as

$$\begin{array}{l}{i}_{\mathrm{d}}^{k+1}=\left(1-\frac{T_{\mathrm{s}}{R}_{\mathrm{s}}}{L_{\mathrm{d}}}\right)i_{\mathrm{d}}^k+\frac{L_{\mathrm{q}}{T}_{\mathrm{s}}{\omega }_{\mathrm{r}}}{L_{\mathrm{d}}}{i}_{\mathrm{q}}^k+\frac{T_{\mathrm{s}}}{L_{\mathrm{d}}}{u}_{\mathrm{d}}^k\\ i_{\mathrm{q}}^{k+1}=\frac{-L_{\mathrm{d}}{T}_{\mathrm{s}}{\omega }_{\mathrm{r}}}{L_{\mathrm{q}}}{i}_{\mathrm{d}}^k+\left(1-\frac{T_{\mathrm{s}}{R}_{\mathrm{s}}}{L_{\mathrm{q}}}\right)i_{\mathrm{q}}^k+\frac{T_{\mathrm{s}}}{L_{\mathrm{q}}}{u}_{\mathrm{q}}^k-\frac{{\psi }_{\mathrm{f}}{T}_{\mathrm{s}}{\omega }_{\mathrm{r}}}{L_{\mathrm{q}}}\end{array}$$

(4)

where T_s is the sample period; k and k + 1 represent the kT_s and (k + 1)T_s sample period, respectively; u_d and u_q are d-axis and q-axis components of the stator voltage, respectively; i_d and i_q are d-axis and q-axis components of the stator current, respectively; L_d and L_q are d-axis and q-axis components of the stator inductance, respectively; R_s is the stator resistance; ψ_f is the stator flux; and ω_r is the rotor electricity angular speed.

For the NPC three-level converter fed PMSM system, the neutral point potential balance needs to be taken into account when designing the cost function. Therefore, the cost function of MPCC generally includes a stator current term and a neutral point potential term, shown as follows

$$g={\lambda }_i\left[{(i_{\mathrm{d}}^{k+2}-i_{\mathrm{d}}^{*})}^2+{(i_{\mathrm{q}}^{k+2}-i_{\mathrm{q}}^{*})}^2\right]+{\lambda }_{\mathrm{v}}\left|v_{\mathrm{o}}^{k+2}\right|$$

(5)

where $i_d^{*}$ and $i_q^{*}$ are the reference values of i_d and i_q, respectively, and λ_i and λ_v denote the weighted coefficients for the stator current term and the neutral point potential term.

The block diagram of MPCC is shown in Figure 3 and the algorithm is executed in the following steps:

(1)

Stator currents and neutral point potentials are sampled at kT_s;

(2)

From (3) and (4), $i_{\mathrm{d}}^{k+1}$, $i_{\mathrm{q}}^{k+1}$ and $v_{\mathrm{o}}^{k+1}$ at (k + 1)T_s are obtained and can be used as the initial values of the algorithm;

(3)

The sector in which the optimal vector of the last control period was located is determined, and the alternative vector set can be established as

Table 2. Alternative voltage vectors of conventional MPCC.

Vectors Used in the Last Control Period	Alternative Voltage Vectors
V1	V1	V2	V12	V13	V19
V2	V1	V2	V3	V13	V14	V19
V3	V2	V3	V4	V14	V19
V4	V3	V4	V5	V14	V15	V19
V5	V4	V5	V6	V15	V19
V6	V5	V6	V7	V15	V16	V19
V7	V6	V7	V8	V16	V19
V8	V7	V8	V9	V16	V17	V19
V9	V8	V9	V10	V17	V19
V10	V9	V10	V11	V17	V18	V19
V11	V10	V11	V12	V18	V19
V12	V1	V11	V12	V13	V18	V19
V13	V1	V2	V12	V13	V19
V14	V2	V3	V4	V14	V19
V15	V4	V5	V6	V15	V19
V16	V6	V7	V8	V16	V19
V17	V8	V9	V10	V17	V19
V18	V10	V11	V12	V18	V19
V19	V13	V14	V15	V16	V17	V18	V19

. The alternative vectors in Table 2 are substituted into (3) and (4) to obtain $i_{\mathrm{d}}^{k+1}$(n), $i_{\mathrm{q}}^{k+1}$(n) and $v_{\mathrm{o}}^{k+1}$(n) at (k + 2)T_s, n = 1, 2,…, m, where m denotes the number of alternative vectors;

(4)

According to (5), the cost function corresponding to each alternative voltage vector in the FCS is calculated, and the voltage vector corresponding to the minimum value of the cost function is selected to act on the converter.

3. Proposed MPCC

3.1. Newton’s Iterative Algorithm

Conventional MPCC uses a forward Eulerian formula to discretize the predictive model. Due to the truncation errors, the predictive value of the motor at the next control period is inaccurate. In this paper, the predictive accuracy of the algorithm is improved by using Newton’s iteration method to normalize the stator current term in (5) as

$$\begin{array}{l}{(i_{\mathrm{d}}^{k+2}-i_{\mathrm{d}}^{*})}^2+{(i_{\mathrm{q}}^{k+2}-i_{\mathrm{q}}^{*})}^2=g(u_{\mathrm{d}\mathrm{q}}^{\mathrm{T}})\\ =u_{\mathrm{d}\mathrm{q}}^{\mathrm{T}}{\lambda }^2{u}_{\mathrm{d}\mathrm{q}}+{\mu }_{\mathrm{k}}{u}_{\mathrm{d}\mathrm{q}}\end{array}$$

(6)

where

$$\begin{array}{l}{u}_{\mathrm{d}\mathrm{q}}=\left[\begin{array}{l}{u}_{\mathrm{d}}^k\\ u_{\mathrm{q}}^k\end{array}\right]\\ \lambda =\frac{T_{\mathrm{s}}}{L_{\mathrm{d}\mathrm{q}}}\end{array}$$

$$\begin{array}{l}{\mu }_{\mathrm{k}}=\left[{\mu }_{1\mathrm{k}} {\mu }_{2\mathrm{k}}\right]\\ {\mu }_{1\mathrm{k}}=2\lambda (a_1{i}_{\mathrm{d}}^k+a_{2\mathrm{k}}{i}_{\mathrm{q}}^k)\\ {\mu }_{2\mathrm{k}}=2\lambda (-a_{2\mathrm{k}}{i}_{\mathrm{d}}^k+a_1{i}_{\mathrm{q}}^k-i_{\mathrm{q}}^{*}-{\omega }_{\mathrm{r}}\lambda {\psi }_{\mathrm{f}})\\ {\mathrm{a}}_1=1-\frac{R_{\mathrm{s}}{T}_{\mathrm{s}}}{L_{\mathrm{d}\mathrm{q}}}\\ {\mathrm{a}}_{2\mathrm{k}}={\omega }_{\mathrm{r}}{T}_{\mathrm{s}}\end{array}$$

The MPCC algorithm can be transformed into a quadratic optimization problem of (6) under the constraint (7). In this paper, based on the Hessian matrix, the stator voltage to be controlled at the next period is iterated by the Newton iteration method. The Hessian matrix can be expressed as

$${\mathrm{H}}_g(u_{{\mathrm{d}\mathrm{q} }_{\mathrm{r}}})=\frac{\partial g}{\partial u_{\mathrm{d}}\partial u_{\mathrm{q}}}$$

(7)

where the subscript r corresponds to the rth iterations, and the forward Newton iteration algorithm is

$$\begin{array}{l}{u}_{{\mathrm{d}\mathrm{q} }_{r+1}}=u_{{\mathrm{d}\mathrm{q} }_r}-\gamma \cdot {\mathrm{H}}_g{(u_{{\mathrm{d}\mathrm{q} }_r})}^{-1}\cdot \nabla g(u_{{\mathrm{d}\mathrm{q} }_r})\\ u_{{\mathrm{d}\mathrm{q}}_0}=u_{\mathrm{d}{\mathrm{q}}_{-1}}\end{array}$$

(8)

Substituting (7) into (8) yields the following:

$$u_{\mathrm{d}{\mathrm{q}}_{r+1}}=u_{{\mathrm{d}\mathrm{q}}_r}-\gamma (u_{\mathrm{d}{\mathrm{q}}_r}+\frac{1}{2{\lambda }^2}{\mu }_{\mathrm{k}}^{\mathrm{T}})$$

(9)

where γ (0 < γ ≤ 1) is the control coefficient for the number of iterations. Moreover, the iteration convergence condition ε (0 < ε ≤ 1) is introduced, i.e., the Euclidean norm of the difference between the results of two successive iterations does not exceed the error ε.

$$\left|u_{{d\mathrm{q}}_{r+1}}-u_{\mathrm{d}{\mathrm{q}}_r}\right|⩽\epsilon$$

(10)

3.2. Dynamic Division of Sectors

When the neutral point potential is shifted significantly, the amplitude and phase angle of medium vectors and small vectors in the space vector diagram will change, which in turn affects the control performance of the system. In order to quantify the neutral point potential imbalance, the imbalance coefficients d₁ and d₂ are defined as follows:

$$\begin{array}{l}{d}_1=\frac{2v_{\mathrm{c}1}}{V_{\mathrm{d}\mathrm{c}}}\begin{array}{l}\hfill \end{array}{d}_2=\frac{2v_{\mathrm{c}2}}{V_{\mathrm{d}\mathrm{c}}}\begin{array}{l}\hfill \end{array}\\ \begin{array}{l}\hfill \end{array}\begin{array}{l}\hfill \end{array}\begin{array}{l}\hfill \end{array}\begin{array}{l}\hfill \end{array}{d}_1+d_2=2\end{array}$$

(11)

where v_c1 and v_c2 correspond to the upper and lower voltage of the DC-link capacitor, respectively.

Taking sub-regions R₁ and R₂ in sector S_I as an example, the change of the basic voltage vectors under the imbalanced neutral point potential (d₁ = 0.5 and d₂ = 1.5) are shown in Figure 4.

From Figure 4, the amplitude and phase angle of the medium vector change, from V₂ to ${\mathit{V}}_2^{\prime }$. The phase angle of small vectors remains the same while the amplitude changes, from V₁₃ to ${\mathit{V}}_{13}^{\prime }$ (ONN) and $V_{13}^{″}$ (POO), respectively, and from V₁₄ to ${\mathit{V}}_{14}^{\prime }$ (ONN) and ${\mathit{V}}_{14}^{″}$ (PPO), respectively. Assuming the reference vector ${\mathit{V}}_{\mathrm{r}\mathrm{e}\mathrm{f}}^{k+1}$ is located in the position shown in Figure 4, for conventional MPCC, the distance r₄ between V₂ and ${\mathit{V}}_{\mathrm{r}\mathrm{e}\mathrm{f}}^{k+1}$ is shorter than the distance r₃ between V₁₃ and ${\mathit{V}}_{\mathrm{r}\mathrm{e}\mathrm{f}}^{k+1}$; thus, V₂ is the optimal vector. When the neutral point potential rises, V₂ and V₁₃ will become ${\mathit{V}}_2^{\prime }$ and ${\mathit{V}}_{13}^{\prime }$/${\mathit{V}}_{13}^{″}$, respectively. The distance r₂ between ${\mathit{V}}_2^{\prime }$ and ${\mathit{V}}_{\mathrm{r}\mathrm{e}\mathrm{f}}^{k+1}$ is greater than the distance r₁ between ${\mathit{V}}_{13}^{″}$ and ${\mathit{V}}_{\mathrm{r}\mathrm{e}\mathrm{f}}^{k+1}$; thus, ${\mathit{V}}_{13}^{″}$ is actually the optimal vector.

In summary, it can be seen that the amplitude and the phase angle of basic voltage vectors will be changed when the neutral point potential rises. If the basic vectors shown in Figure 2 are still used as alternative vectors to establish the FCS, it will cause the optimal vector to be selected incorrectly. In this paper, the amplitude and the phase angle of medium and small vectors are modified according to the amplitude of the neutral point potential. The basic voltage vector in the two-phase stationary coordinate system can be calculated as follows:

$$\{\begin{array}{c}{v}_{\alpha }=\frac{\left\{\begin{array}{c}{v}_{\mathrm{c}1}\times [2(S_{\mathrm{A}1}\times S_{\mathrm{A}2})-(S_{\mathrm{B}1}\times S_{\mathrm{B}2})-(S_{\mathrm{C}1}\times S_{\mathrm{C}2})]\\ -v_{\mathrm{c}2}\times [2(S_{\mathrm{A}3}\times S_{\mathrm{A}4})-(S_{\mathrm{B}3}\times S_{\mathrm{B}4})-(S_{\mathrm{C}3}\times S_{\mathrm{C}4})]\end{array}\right\}}{\sqrt{6}}\\ v_{\mathsf{\beta }}=\frac{\left\{\begin{array}{c}{v}_{\mathrm{c}1}\times [(S_{\mathrm{B}1}\times S_{\mathrm{B}2})-(S_{\mathrm{C}1}\times S_{\mathrm{C}2})]\\ -v_{\mathrm{c}2}\times [(S_{\mathrm{B}3}\times S_{\mathrm{B}4})-(S_{\mathrm{C}3}\times S_{\mathrm{C}4})]\end{array}\right\}}{\sqrt{2}}\end{array}$$

(12)

The amplitude of the small vectors as well as the amplitude and phase angle of the medium vectors can be calculated by (12). After the actual basic vectors of the three-level converter have been calculated, the space vector diagram can be dynamically divided. Figure 5 shows the space vector diagram for the imbalance coefficients d₁ = 0.5 and d₂ = 1.5.

3.3. The Partition Control of Neutral Point Potential Imbalance

When the amplitude of the neutral point potential is small, the change of the amplitude and phase angle of basic vectors is not significant. The neutral point potential term in the cost function can be omitted. Only the stator voltage is considered as the control target. When the amplitude of the neutral point potential is large, the change of the amplitude and phase angle of basic vectors cannot be ignored. In such a case, the neutral point potential is taken as the control target, and the stator voltage term is omitted to ensure the rapid balance of the neutral point potential.

From the above analysis, it can be seen that the amplitude of the neutral point potential can be divided into two regions for control. By reasonably designing the threshold, the weighted coefficients are eliminated while ensuring the control performance of the system.

(1) Control strategy of region I

When the neutral point potential amplitude is less than the threshold, the neutral point potential term in (5) is eliminated and the cost function becomes

$$g=u_{\mathrm{d}\mathrm{q}}^{\mathrm{T}}{\lambda }^2{u}_{\mathrm{d}\mathrm{q}}+{\mu }_{\mathrm{k}}{u}_{\mathrm{d}\mathrm{q}}$$

(13)

The amplitude of the neutral point potential is small, and the effect on the amplitude and phase angle of basic voltage vectors can be ignored. The sector is still divided according to Figure 2 and the corresponding alternative vector set is shown in

Table 3. Alternative voltage vector in region I.

Sectors	Alternative Voltage Vectors
R1	V1	V2	V13	V19
R2	V2	V3	V14	V19
R3	V3	V4	V14	V19
R4	V4	V5	V15	V19
R5	V5	V6	V15	V19
R6	V6	V7	V16	V19
R7	V7	V8	V16	V19
R8	V8	V9	V17	V19
R9	V9	V10	V17	V19
R10	V10	V11	V18	V19
R11	V11	V12	V18	V19
R12	V12	V1	V13	V19

.

(2) Control strategy of region II

When the neutral point potential amplitude is greater than the threshold, the stator current term in (5) is eliminated and the cost function becomes

$$g=\left|v_{\mathrm{o}}^{k+2}\right|$$

(14)

In such a case, the amplitude of the neutral point potential is large and the amplitude and phase angle of basic vectors need to be modified according to (13). Only medium vectors as well as small vectors are adopted to establish the FCS, as shown in

Table 4. Alternative voltage vector at region II.

Sectors	Alternative Voltage Vectors
R1	${\mathit{V}}_{13}^{\prime }$	${\mathit{V}}_{13}^{″}$	${\mathit{V}}_2^{\prime }$
R2	${\mathit{V}}_{14}^{\prime }$	${\mathit{V}}_{14}^{″}$	${\mathit{V}}_2^{\prime }$
R3	${\mathit{V}}_{14}^{\prime }$	${\mathit{V}}_{14}^{″}$	${\mathit{V}}_4^{\prime }$
R4	${\mathit{V}}_{15}^{\prime }$	${\mathit{V}}_{15}^{″}$	${\mathit{V}}_4^{\prime }$
R5	${\mathit{V}}_{15}^{\prime }$	${\mathit{V}}_{15}^{″}$	${\mathit{V}}_6^{\prime }$
R6	${\mathit{V}}_{16}^{\prime }$	${\mathit{V}}_{16}^{″}$	${\mathit{V}}_6^{\prime }$
R7	${\mathit{V}}_{16}^{\prime }$	${\mathit{V}}_{16}^{″}$	${\mathit{V}}_8^{\prime }$
R8	${\mathit{V}}_{17}^{\prime }$	${\mathit{V}}_{17}^{″}$	${\mathit{V}}_8^{\prime }$
R9	${\mathit{V}}_{17}^{\prime }$	${\mathit{V}}_{17}^{″}$	${\mathit{V}}_{10}^{\prime }$
R10	${\mathit{V}}_{18}^{\prime }$	${\mathit{V}}_{18}^{″}$	${\mathit{V}}_{10}^{\prime }$
R11	${\mathit{V}}_{18}^{\prime }$	${\mathit{V}}_{18}^{″}$	${\mathit{V}}_{12}^{\prime }$
R12	${\mathit{V}}_{13}^{\prime }$	${\mathit{V}}_{13}^{″}$	${\mathit{V}}_{12}^{\prime }$

.

The block diagram of the proposed MPCC algorithm is shown in Figure 6. The implementation process of the improved MPCC algorithm is as follows:

(1)

The stator current and the neutral point potential are sampled at kT_s;

(2)

From (3) and (4), $i_{\mathrm{d}}^{k+1}$, $i_{\mathrm{q}}^{k+1}$ and $v_{\mathrm{o}}^{k+1}$ at (k + 1)T_s are obtained and can be used as the initial values of the algorithm;

(3)

The reference voltage vector is calculated by (6) and the FCS is selected according to the amplitude of the neutral point potential;

(4)

The alternative vector with the minimum value of the cost function is selected as the optimal vector and applied to the converter.

4. Experimental Results

4.1. Experimental Platform

In this paper, a three-level converter fed PMSM system is set up and the dSPACE rapid control prototyping simulator is used as the controller. The performance of the conventional model predictive current control (MPCC1, with (5) as the cost function and Table 2 as the finite control set), MPCC2 [17] and the proposed improved model predictive current control (MPCC3, with (14) and (15) as the cost function, Table 3 and Table 4 as the finite control set and the neutral point potential threshold set to 20 V) are compared. The experimental prototype is shown in Figure 7, and the experimental parameters are shown in

Table 5. Rated parameters of PMSM.

Parameters	Symbol	Value	Unit
Poles	p	4	-
Permanent magnet flux	ψf	0.45	Wb
Stator resistance	Rs	0.635	Ω
d-axis inductance	Ld	4.25	mH
q-axis inductance	Lq	4.25	mH
Rated speed	nr	1500	r/min
Rated torque	TN	10	N·m
Rated voltage	VN	220	V

.

4.2. Experimental Analysis

4.2.1. Control Performance of the Neutral Point Potential

The amplitude of the neutral point potential is set to 40 V (v_C1 = 140 V, v_C2 = 180 V) to verify the neutral point potential balance capability of MPCC1 and MPCC3. Figure 8 shows the variation of the neutral point potential (v_o = v_C1 − v_C2). It can be seen that the amplitude of the neutral point potential returns to zero for both MPCC1 and MPCC3. The settling times for MPCC1 and MPCC3 are 0.96 s and 0.61 s, respectively. The settling time of MPCC3 is shorter than that of MPCC1. The control performance of the neutral point potential is improved by the proposed MPCC.

4.2.2. Control Performance under Steady State

For different load torque T_L and motor speed n_r operating conditions (case 1: T_L = 5 N·m, n_r = 500 r/min, v_c1 = 140 V, v_c2 = 180 V; case 2: T_L =10 N·m, n_r = 500 r/min, v_c1 = 140 V, v_c2 = 180 V), the experimental results of the stator current i_A, electromagnetic torque T_e (The experimental result of torque is estimated by the stator current, flux linkage and rotator position), amplitude of stator flux |ψ_s| and neutral point potential v_o are shown in Figure 9 and Figure 10. It can be seen that when the amplitude of the neutral point potential is large, the current and torque fluctuation of MPCC3 is lower than that of MPCC1 and MPCC2. As for MPCC3, the change of the amplitude and the phase angle of basic vectors are fully considered and the actual optimal vector is selected.

The torque fluctuation rate D_T and flux fluctuation rate D_ψ are used as torque performance and flux performance indicators and are defined as follows:

$$\{\begin{array}{l}{D}_{\mathrm{T}}=\frac{T_{\mathrm{e}\_\mathrm{m}\mathrm{a}\mathrm{x}}-T_{\mathrm{e}\_\mathrm{m}\mathrm{i}\mathrm{n}}}{T_{\mathrm{e}\_\mathrm{m}\mathrm{a}\mathrm{x}}+T_{\mathrm{e}\_\mathrm{m}\mathrm{i}\mathrm{n}}}\times 100\%\\ D_{\mathsf{\psi }}=\frac{{\left|{\mathit{\psi }}_{\mathbf{s}}\right|}_{\_\mathrm{m}\mathrm{a}\mathrm{x}}-{\left|{\mathit{\psi }}_{\mathbf{s}}\right|}_{\_\mathrm{m}\mathrm{i}\mathrm{n}}}{{\left|{\mathit{\psi }}_{\mathbf{s}}\right|}_{\_\mathrm{m}\mathrm{a}\mathrm{x}}+{\left|{\mathit{\psi }}_{\mathbf{s}}\right|}_{\_\mathrm{m}\mathrm{i}\mathrm{n}}}\times 100\%\end{array}$$

(15)

where T_e_max and T_e_min represent the maximum and minimum values of electromagnetic torque and |ψ_s|_max and |ψ_s|_min represent the maximum and minimum values of stator flux.

The total harmonic distortion of the output current I_THD is used as the current performance evaluation index and is defined as follows.

$$I_{\mathrm{T}\mathrm{H}\mathrm{D}}=\frac{\sqrt{{\displaystyle \sum _{n=2}^{\infty }{I}_n^2}}}{I_1}$$

(16)

where I₁ denotes the rms value of the fundamental component of the output current and I_n denotes the rms value of the nth harmonic component. Figure 11 shows the D_T, D_ψ and I_THD for case 1 and case 2.

From Figure 11, under large neutral point potential conditions, the D_T, D_ψ and I_THD of MPCC3 are all lower than those of MPCC1 and MPCC2, which verifies that the steady state performance of the improved MPCC is superior to that of conventional MPCC.

4.2.3. Control Performance under Dynamic State

Under sudden change conditions (case 3: T_L = 0 N·m→5 N·m, n_r = 500 r/min), the experimental results of the stator current i_A, electromagnetic torque T_e, amplitude of stator flux |ψ_s| and neutral point potential v_o are shown in Figure 12.

It can be seen from Figure 12 that the dynamic processes of MPCC1, MPCC2 and MPCC3 are 3.25 ms, 2.50 ms and 1.90 ms, respectively, for case 3. The dynamic performance of the proposed MPCC is better than that of the conventional MPCC.

Compared with MPCC1 and MPCC2, MPCC3 transfers the stator current term in the cost function into a stator voltage term, and completely eliminates the weight coefficients in the cost function by neutral point potential partition control. The neutral point potential can be balanced rapidly by the modification of the amplitude and the phase angle of basic vectors and reconstruction of the finite control set when the amplitude of the neutral point potential is large. The improved algorithm can effectively improve the dynamic and steady state performance.

5. Conclusions

For the conventional MPCC of the NPC three-level converter fed PMSM system, the adjustment process of the weighted coefficients is complicated. Moreover, the flux and torque performance are deteriorated when there is a large deviation of the neutral point potential. This paper proposes an improved MPCC algorithm to solve the above issue. The improved algorithm has the following advantages:

(1) The proposed MPCC transfers the stator current term in the cost function into a stator voltage term, and completely eliminates the weight coefficients in the cost function by neutral point potential partition control.

(2) The neutral point potential can be balanced rapidly by the modification of the amplitude and the phase angle of basic vectors and reconstruction of the finite control set when the amplitude of neutral point potential is large.

The experimental results show that the improved algorithm can effectively improve the dynamic and steady state performance under the neutral point potential imbalance conditions.