Three-Level Inverter-PMSM Model Predictive Current Control Based on the Extended Control Set

Abstract

In the neutral point clamped (NPC) three-level inverter-permanent magnet synchronous motor system, traditional model predictive current control (MPCC) uses the system predictive model to traverse the 27 basic voltage vectors, to achieve the d-q axis current component and neutral point voltage of the multi-objective optimal control. Finite control set model predictive control predicts the state change of the control target at future moments based on a finite number of switching states of the inverter. The control principle of this method is simple and easy to implement, but the control effectiveness of this control strategy is limited because only one basic vector can be selected as the optimum output per control period. In this paper, a model predictive current control strategy based on an extended control set (ECS-MPCC) is proposed, which can improve the control performance of the system by extending the control set to select multiple vectors in a single control period compared to the traditional strategy. In addition, to address the disadvantage of extending virtual space vectors leading to an increase in computation, this paper proposes a fast search method for optimal vector based on region reduction. The proposed method avoids the optimization process traversing all virtual space vectors, thus enabling a fast search for the optimal vector. The experimental results show that the proposed control strategy has good steady-state and dynamic performance.

1. Introduction

Permanent magnet synchronous motors (PMSM) are used extensively in servo drives, energy extraction, and rail transportation owing to their advantages of compact size, light weight, simple structure, high efficiency, and power factor [1,2].

The neutral-point clamped (NPC) inverter is characterized by a moderate number of voltage levels and a simpler structure than other multilevel inverters. Since there are more power devices than two-level inverter and the neutral point of the DC side is connected to the load, the number of output level is extended, which produces a higher sinusoidal output waveform and raises the inverter’s overall efficiency [3]. Therefore, the NPC inverter-permanent magnet synchronous motor system is widely used in medium and high-voltage high-power and high-performance motor drive applications. However, the particular topology of the NPC inverter causes neutral point voltage fluctuation when it is operating. To ensure the performance of the control strategy, the basic vector should be reasonably selected to achieve balanced control of the neutral point voltage [4].

Model predictive control (MPC) strategies were born out of industrial practice in the 1960s and have become an important research direction in the field of motor control as the computing performance of microprocessors has increased [5]. The NPC inverter-PMSM system has the characteristics of multi-input and multi-output. The principle of finite set model predictive control (FCS-MPC) is simple and has great practical value in motor drive systems [6]. However, in FCS-MPC, only one optimal vector is selected in a local time per control period, and the basic vector of NPC inverter cannot be fully utilized, which limits the control effect. In addition, if the control effect is improved only by shortening the control period, the average switching frequency of the inverter will increase. Therefore, it is of great significance to improve the control effect of the MPC strategy at the same switching frequency [7].

At present, there are two ways to improve the control effect of FCS-MPC: multi-step prediction and multi-vector.

Multi-step prediction improves the accuracy of the optimization result by estimating the change of the control target in the long period range. However, the number of predictions also increases exponentially with the increase of the step size, which increases the computational burden of the controller [8]. In the literature [9], the ability of extending the prediction step to improve the control effect of the strategy is proved, and the calculation amount of the strategy is reduced by spherical decoding, so that the strategy can be implemented on low-cost FPGA. The literature [10] centers on the reference flux vector, sets a bandwidth that satisfies the flux and torque error requirements, and reduces the computational effort by filtering out sequences that do not meet the error requirements.

To address the drawback that the FCS-MPC strategy only uses one single vector as the optimal output, researchers have proposed a multi-vector-based strategy. In the literature [11,12], the zero vector is introduced and the amplitude of the basic space vector is adjusted by duty ratio control as a way to extend the control set of the FCS-MPC to achieve the suppression of torque and flux linkage fluctuations. This strategy is improved in the literature [13] to address the issue of increasing computational effort after the introduction of the zero vector by first solving the optimal duty cycle in the three directions chosen by the strategy, and then re-establishing the control set to carry out the optimization search in order to achieve streamlined computation.

In the above strategy, because the zero vector does not affect the neutral point voltage, it can be directly extended to the NPC inverter system. However, the addition of the zero vector can only change the amplitude of the basic space vector and has limited ability to extend the control set. In the literature [14,15], two adjacent basic vectors and zero vectors are used in each control period to continue extending the control set in order to achieve error-free control of the target, which improves the control effect of the strategy, but the computational process is relatively complex. In this regard, the literature [16] simplifies the selection process of the optimal basic vector based on the torque error and calculates the duty cycle based on the slope of the vector affecting the torque and flux linkage. In the literature [17], the selection of three vectors is simplified in the drive system with the NPC inverter, but the least squares optimization method is still needed to calculate the duty cycle. In [18,19], some additional virtual space vectors are introduced in the two-level space voltage vector diagram to extend the control set. In order to reduce the operation burden caused by too many candidate vectors, the strategy selects the candidate vectors near the optimal vector by region partition, which reduces unnecessary prediction process. Nevertheless, as the number of virtual vectors continues to increase, the strategy still needs a lot of computation. At present, when applying multi-vector model predictive control in the NPC inverter system, due to the increase in the number of basic vectors and the need to consider the balance of midpoint voltage, the calculation method of the vector duty cycle and the sorting of output vectors remain difficult. Therefore, it is necessary to conduct in-depth research on this.

In this paper, a model predictive current control strategy based on an extended control set (ECS-MPCC) is proposed. The main improvements are:

(1)

Improve motor control effect. It extends the control set of traditional model predictive current control (MPCC) strategy by adding virtual space vector.

(2)

Simplify the optimization process. The proposed strategy analyzes the relationship between the predicted values of the virtual vector and the basic vector, omits the unnecessary prediction process, and then uses the region reduction to quickly calculate the duty ratio of the vector.

(3)

Output vector online sorting. Considering the switching frequency, the motor control effect, and the neutral point voltage fluctuation of the inverter, an online sorting output vector method is proposed.

2. Traditional MPCC

Neglecting core material saturation, eddy current losses and hysteresis losses, the mathematical model of PMSM in the d-q coordinate system can be obtained by coordinate transformation as follows:

$$\left\{\begin{array}{l}{u}_{\mathrm{d}}=R_{\mathrm{s}}{i}_{\mathrm{d}}+{\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{d}}}{\mathrm{d}t}}-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{q}}\\ u_{\mathrm{q}}=R_{\mathrm{s}}{i}_{\mathrm{q}}+{\displaystyle \frac{\mathrm{d}{\psi }_{\mathrm{q}}}{\mathrm{d}t}}+{\omega }_{\mathrm{e}}{\psi }_{\mathrm{d}}\end{array}\right.$$

(1)

$$\left\{\begin{array}{l}{\psi }_{\mathrm{d}}=L_{\mathrm{d}}{i}_{\mathrm{d}}+{\psi }_{\mathrm{f}}\\ {\psi }_{\mathrm{q}}=L_{\mathrm{q}}{i}_{\mathrm{q}}\end{array}\right.$$

(2)

where R_s is the stator resistance; i_d, i_q, ψ_d, and ψ_q are the components of the current and the stator flux in a two-phase rotating d-q coordinate system respectively; and ω_e is the electrical angular speed of the rotor.

Assuming the system sampling time is T_s, discretizing the continuous mathematical model of PMSM shown in Equations (1) and (2), the traditional current prediction model can be expressed as follows:

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

(3)

where the superscript k indicates that the variable is at moment k and the variable is the sampled value; the superscript k + 1 indicates that the variable is at moment k + 1 and the variable is the predicted value; v_d and v_q indicate the voltage components in the d-q coordinate system.

The topology of the NPC inverter is shown in Figure 1, with per phase bridge consisting of four power switching devices and two clamping diodes.

The switching states of the NPC inverter are defined as 1, 0, and −1, corresponding to three output levels of V_dc/2, 0, and −V_dc/2 for each phase, so there are 3³ = 27 switching states for the three phases, corresponding to 27 basic vectors in the space vector plane. As shown in Figure 2, the basic vector can be divided into large vectors V₁~V₆, medium vectors V₇~V₁₂, small vectors V_13U~V_18U and V_13L~V_18L, and zero vector V₁₉.

As shown in Figure 1, in some switching states, the load is directly connected to the neutral point of the upper and lower capacitors on the DC side to charge and discharge the capacitors, generating neutral point voltage fluctuations. The larger neutral point voltage fluctuation will make the amplitude and phase of the basic vector change obviously, which will seriously affect the prediction results. The neutral point voltage v_o is related to the working state and output current of the inverter, which can be defined as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{o}}=(1-|S_{\mathrm{A}}|)i_{\mathrm{A}}+(1-|S_{\mathrm{B}}|)i_{\mathrm{B}}+(1-|S_{\mathrm{C}}|)i_{\mathrm{C}}\\ v_{\mathrm{o}}=-{\displaystyle \frac{1}{2\mathrm{C}}}\int i_{\mathrm{o}}\mathrm{d}t\end{array}\right.$$

(4)

where v_o is the neutral point voltage; S_A, S_B, and S_C correspond to the switching states of each phase of the inverter; i_A, i_B, and i_C are the three-phase output current; C is the capacitance of the DC side capacitor.

By discretizing (4), the neutral point voltage prediction model can be obtained as follows:

$$v_{\mathrm{o}}^{k+1}=v_{\mathrm{o}}^k+\frac{T_{\mathrm{s}}}{2\mathrm{C}}|S_{\mathrm{ABC}}^k{|}^{\mathrm{T}}{i}_{\mathrm{ABC}}^k$$

(5)

where i^k_ABC = [i^k_A i^k_B i^k_C], S^k_ABC = [|S^k_A| |S^k_B| |S^k_C|].

Considering neutral point voltage control and delay compensation, the cost function for model predictive current control (MPCC) can be expressed as follows:

$$g=|{i_{\mathrm{d}}}^{*}-{i_{\mathrm{d}}}^{k+2}|+|{i_{\mathrm{q}}}^{*}-{i_{\mathrm{q}}}^{k+2}|+{\lambda }_{v_{\mathrm{o}}}|v_{\mathrm{o}}^{k+2}|$$

(6)

where the superscript “*” means the reference value of the variable.

The structure block diagram of traditional MPCC is shown in Figure 3. The basic control flow is as follows:

(1)

Sampling to obtain the three-phase current, electrical angular speed, rotor position angle, and upper and lower capacitance voltages on the DC side of the inverter at moment k.

(2)

Applying the optimal voltage vector at moment k obtained from the search to predict the current and neutral point voltage at moment k + 1;

(3)

Using (3) to predict the current component i_d and i_q at moment k + 2 in the rotating coordinate system;

(4)

Using (5) to predict the neutral point voltage v_o at k + 2 time;

(5)

According to (6), the optimal voltage vector at k + 1 time is obtained by substituting the predicted value of each variable into the cost function.

3. ECS-MPCC

To improve the control performance of the system, a model predictive current control based on an extended control set (ECS-MPCC) is proposed in this paper. The strategy improves the control effect by introducing the virtual space vector to the traditional MPCC and reduces the computational load by region reduction. In this section, the neutral point voltage control, virtual vector extension, prediction, and optimization simplification process and virtual vector equivalent sequence online determination are introduced respectively.

3.1. Neutral Point Voltage Control

According to (4), when the medium and small vector are selected, the current will flow through the DC side neutral point of the inverter, resulting in capacitor voltage fluctuation. The corresponding neutral point currents for these vectors are shown in

Table 1. Output vector and neutral point current.

V13U~V18U	io	V13L~V18L	io	V7~V12	io
100	−iA	0−1−1	iA	10−1	iB
110	iC	00−1	−iC	01−1	iA
010	−iB	−10−1	iB	−110	iC
011	iA	−100	−iA	−101	iB
001	−iC	−1−10	iC	0−11	iA
101	iB	0−10	−iB	1−10	iC

.

According to Table 1, each pair of small vectors with the same amplitude and angle has an opposite effect on the neutral point voltage. The proposed strategy uses this feature to filter the small vectors in the control set in order to suppress the current neutral point voltage fluctuation and eliminate the corresponding weight coefficient. After filtering the small vectors, the cost function can be simplified as follows:

$$g=|{i_{\mathrm{d}}}^{*}-{i_{\mathrm{d}}}^{k+2}|+|{i_{\mathrm{q}}}^{*}-{i_{\mathrm{q}}}^{k+2}|$$

(7)

3.2. Virtual Vector Extensions

For the virtual space vector at any position in the vector diagram, it can be represented by several basic vectors and corresponding duty ratio using the volt-second balance principle. This is expressed as follows:

$$\left\{\begin{array}{l}{\mathit{V}}_{\mathrm{vir}n}{T}_{\mathrm{s}}=d_x{T}_{\mathrm{s}}{\mathit{V}}_x+d_y{T}_{\mathrm{s}}{\mathit{V}}_y+d_z{T}_{\mathrm{s}}{\mathit{V}}_z\\ d_x+d_y+d_z=1\end{array}\right.$$

(8)

where V_virn is the extended virtual space vector, n = 1, 2, 3…; V_x,y,z and d_x,y,z represent the basic space vector and corresponding duty ratio required to synthesize this vector, x, y, z ∈ [1, 2, …, 19].

For the convenience of analysis, when synthesizing virtual vectors with V_x,y,z, all virtual vectors and basic vectors can be expressed by duty ratio combination as follows:

$$\left\{\begin{array}{l}{\mathit{V}}_x=D_{x}V , D_x=[1 0 0]\\ {\mathit{V}}_y=D_{y}V , D_y=[0 1 0]\\ {\mathit{V}}_z=D_{z}V , D_z=[0 0 1]\\ {\mathit{V}}_{\mathrm{vir}n}=D_{\mathrm{vir}n}V , D_{\mathrm{vir}n}=[d_x d_y d_z]\\ D_{\mathrm{vir}n}=d_x{D}_x+d_y{D}_y+d_z{D}_z\end{array}\right.$$

(9)

where V = [V_x V_y V_z]^T; the uppercase symbol D denotes the array, representing the combination of duty ratios of the three fundamental vectors required to synthesize the vector. The lowercase symbol d denotes the value, representing the duty ratio of the basic vectors.

The method of extending the virtual vector is shown in Figure 4. The proposed strategy is based on the volt-second balance principle and makes two adjacent vectors act on T_s/2 in each control period to synthesize the virtual space vector. The virtual space vector after the single-time extension is marked with blue dots in the figure. Compared with Figure 2, the number of vectors in the space vector plane in Figure 4 increases, which means that the finite control set is extended.

From Figure 4, it can be seen that the virtual space vectors V_vir1, V_vir2, and V_vir3 taken are related to the basic vectors as follow:

$$\left\{\begin{array}{l}{\mathit{V}}_{\mathrm{vir}1}=({\mathit{V}}_1+{\mathit{V}}_7)/2\\ {\mathit{V}}_{\mathrm{vir}2}=({\mathit{V}}_7+{\mathit{V}}_{13})/2\\ {\mathit{V}}_{\mathrm{vir}3}=({\mathit{V}}_1+{\mathit{V}}_{13})/2\end{array}\right.$$

(10)

On the basis of Figure 4, the existing adjacent voltage vectors operating T_s/2 are chosen in order to enhance the performance of FCS-MPC, and additional virtual space vectors are added to extend the control set. By repeating the above steps, the intersections in the vector diagram will become more intensive, and the number of virtual vectors per extension can be described as follows:

$$N_{j+1}=N_j(N_j+1)/2$$

(11)

where N_j and N_j+1 represent the total number of vectors after the jth and (j + 1)th extension of virtual vectors, respectively.

According to (11), each time the virtual space vector is added, the control set is extended, and at the same time, the storage space requirements and the computational burden on the control set increase accordingly. To address this drawback, this paper proposes a fast optimization method for the proposed virtual vector expansion approach.

From (8), any virtual vector synthesized using the method described in this paper may be expressed as follows:

$$\left\{\begin{array}{l}{\mathit{V}}_{\mathrm{AB}\_\mathsf{\alpha }}=({\mathit{V}}_{\mathrm{A}\_\mathsf{\alpha }}+{\mathit{V}}_{\mathrm{B}\_\mathsf{\alpha }})/2\\ {\mathit{V}}_{\mathrm{AB}\_\mathsf{\beta }}=({\mathit{V}}_{\mathrm{A}\_\mathsf{\beta }}+{\mathit{V}}_{\mathrm{B}\_\mathsf{\beta }})/2\end{array}\right.$$

(12)

$$D_{\mathrm{AB}}=(D_{\mathrm{A}}+D_{\mathrm{B}})/2$$

(13)

where V_A and V_B can represent any adjacent basic vector and V_AB is the synthetic virtual vector. The subscripts α, β indicate the components in a stationary coordinate system.

After Park transformation, the voltage component of virtual space vector in rotating d-q coordinate system can be obtained as follows:

$$\left\{\begin{array}{l}{\mathit{V}}_{\mathrm{AB}\_\mathrm{d}}={\mathit{V}}_{\mathrm{AB}\_\mathsf{\alpha }}\cos \theta +{\mathit{V}}_{\mathrm{AB}\_\mathsf{\beta }}\sin \theta \\ {\mathit{V}}_{\mathrm{AB}\_\mathrm{q}}=-{\mathit{V}}_{\mathrm{AB}\_\mathsf{\alpha }}\sin \theta +{\mathit{V}}_{\mathrm{AB}\_\mathsf{\beta }}\cos \theta \end{array}\right.$$

(14)

Substituting (12) and (14) into (3) can obtain the equation as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{AB}\_\mathrm{d}}^{k+1}=(i_{\mathrm{A}\_\mathrm{d}}^{k+1}+i_{\mathrm{B}\_\mathrm{d}}^{k+1})/2\\ i_{\mathrm{AB}\_\mathrm{q}}^{k+1}=(i_{\mathrm{A}\_\mathrm{q}}^{k+1}+i_{\mathrm{B}\_\mathrm{q}}^{k+1})/2\end{array}\right.$$

(15)

where $i_{\mathrm{AB}\_\mathrm{d}}^{k+1}$ and $i_{\mathrm{AB}\_\mathrm{q}}^{k+1}$ are the predicted values of the currents corresponding to the virtual vectors.

Equation (15) indicates that the current prediction corresponding to the virtual voltage vector synthesized in MPCC in the manner described in this paper can be calculated from the current prediction corresponding to the voltage vector required for its synthesis. Based on this conclusion, the proposed strategy can omit the prediction process after the extension of the virtual space vector, thus reducing the computational load.

The MPCC cost function after eliminating the neutral point voltage term weighting factor is Equation (7), where only the currents i_d, i_q need to be optimized. Therefore, disregarding the control of neutral point voltage, the proposed strategy can quickly search for the optimal vector by region reduction. The process of fast search for the optimal virtual vector is represented in Figure 5. If the three basic vectors corresponding to the minimum value of the value function in the traditional MPCC are V₁, V₇, and V₁₃, then the optimal vector should be located in the triangular region formed by the vertices of the above three vectors. The current prediction values corresponding to each virtual vector can be calculated directly from the current prediction values corresponding to V₁, V₇, and V₁₃ by using (15). By adding the calculation results to the optimization search process, the vectors closer to the optimal vector can be found as V_vir1, V_vir2, and V_vir3. If the virtual vectors are continuously synthesized in the above way and added to the calculation process, the three optimal vectors can also be updated. By repeating the whole process M times, the minimum value function can be updated and the region where the optimal voltage vector is located can be reduced. Figure 6 is the flow chart for searching the optimal vector rapidly after screening small vectors.

3.3. Output Vector Sorting

NPC inverter has four power devices on each bridge, and the switching times of power devices are different with different vector switching sequences. After adding the virtual space vector, 1~3 basic vectors may be output in each control period, so it is difficult to determine the switching sequence by look-up table.

For the optimal vector synthesis method (including the optimal basic three vectors and duty ratio set) obtained in the previous section, this paper proposes a method to determine the vector sequence online in order to take into account the motor control effect, inverter switching frequency and neutral point voltage fluctuation control. As shown in Figure 7, in order to reduce the switching frequency of the inverter, the switching sequence used in this method needs to satisfy the following conditions:

(1)

The switching sequence of the inverter is symmetrical pulse.

(2)

During the control period, only one phase switching state of the inverter is changed when switching vectors.

(3)

Only one power device in the upper bridge is actuated when switching vectors.

According to the proposed strategy, the duty ratio set may be used to determine the optimal vector type, and the basic principles for selecting the switching sequence are as follows:

(1)

When the optimal vector is a single vector, it can be output directly.

(2)

When the optimal vector is synthesized from two or three vectors, if the switching principle shown in Figure 7 is satisfied, the basic vector can be output directly after sorting.

(3)

When the optimal vector is composed of two or three vectors and does not satisfy the switching principle shown in Figure 7, any small vector in the optimal basic vector set should be replaced by a redundant small vector, and then output after sorting.

As shown in Figure 8a, when the optimal vector is synthesized of two vectors: 110 and 100/0−1−1, if the small vector is 100, then the switching principle of this paper is satisfied and the output vector can be sorted directly. If the small vector is 0−1−1, the control of neutral point voltage should be abandoned in this control period, and the small vector should be replaced by 100 before the output vector sorting. Similarly, the three-vector sorting method is shown in Figure 8b. When the basic vector direct sorting does not meet the switching principle of this paper, any small vector still needs to be replaced with the corresponding redundant small vector.

The structure block diagram of the proposed strategy is shown in Figure 9. The basic control flow is as follows:

(1)

Sampling to obtain the three-phase current, electrical angular speed, rotor position angle, and upper and lower capacitance voltages on the DC side of the inverter at moment k.

(2)

Applying the optimal voltage vector at moment k obtained from the search to predict the current and neutral point voltage at moment k + 1;

(3)

According to Table 1 and the information on the current neutral point voltage and three-phase current, the small vectors which can restrain the fluctuation of neutral point voltage in the control set can be selected.

(4)

In the process shown in Figure 6, MPCC is used to determine the basic three vectors needed to synthesize the optimal vector. Subsequently, the duty ratio and prediction value corresponding to the synthesized virtual vector are calculated and added to the optimization search, and the optimal three vectors are continuously updated to achieve triangle area reduction and fast duty ratio calculation.

(5)

Using the obtained basic vector and duty ratio set, the output voltage vector sequence is sorted to generate the PWM pulses operating on the inverter.

4. Experimental Results and Analysis

To verify the performance of the ECS-MPCC strategy proposed in this paper, the experimental system for the permanent magnet synchronous motor fed by NPC three-level inverter was established. The experimental system uses the rapid control prototyping (RCP) module DS1007 from dSPACE^® (Paderborn, Germany) as the controller, the Infineon^® I-Type three-level module F3L100R07W2-E3_B11 (Infineon Technologies, Neubiberg, Germany) in parallel to form an inverter, and the load feed and sudden change using SIEMENS^® S120 frequency convertor (Munich, Germany) driving load motors as dynamometers. The parameters of the PMSM are shown in

Table 2. Parameters of PMSM.

Parameters	Symbol	Value	Unit
pole pairs	p	2	-
PM 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

, and the experimental system is shown in Figure 10.

Through experimental analysis, the number of area reduction M for the ECS-MPCC strategy was set to 3, and then the performance of traditional MPCC and the proposed strategy is compared.

4.1. Steady-State Performance Analysis

First, in the steady-state experiment, the reference speed of the motor is set to 1000 r/min, and the load torque is set to 5 N·m. The effects of different region reduction times M on average switching frequency f_sw, the strategy computational complexity and the current fluctuation of d and q axis in ECS-MPCC strategy are analyzed. In this experiment, the control period of the proposed ECS-MPCC strategy is 100 μs.

Table 3. The impact of M on strategy.

M	Tta	fsw	σ id	σ iq
0	9.95 μs	1242 Hz	0.424	0.361
1	12.58 μs	1997 Hz	0.248	0.242
2	13.12 μs	3082 Hz	0.142	0.101
3	13.68 μs	3412 Hz	0.0644	0.0743
4	14.25 μs	3494 Hz	0.0622	0.0662
5	14.93 μs	3498 Hz	0.0576	0.0660
6	15.55 μs	3500 Hz	0.0549	0.0647
7	16.17 μs	3510 Hz	0.0548	0.0638

presents the experimental results.

The average switching frequency f_sw is calculated from the number of switching times N_T of all power devices within a fixed time T (T = 1 s in this paper), which can be expressed as:

$$f_{sw}=\frac{N_T}{12T}$$

(16)

The current fluctuations of the motor are evaluated using standard deviations σ_id and σ_iq, which are calculated as follows:

$${\sigma }_u=\sqrt{\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{(u_i-\overline{u})}^2}}, \overline{u}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{u}_i}$$

(17)

where n represents the number of sampling points, and this paper sets n = 1 × 10⁴.

In the experimental system of this paper, the DS5202 board based on dSPACE is used to drive the motor. The turnaround time T_ta representing the computational complexity of the strategy can be seen in the experimental software (Control Desk 6.2) on the PC.

According to Table 3, when the number of area reduction M is less than 3, with the increase of M, the average switching frequency f_sw of the inverter increases obviously, the standard deviation σ_id and σ_iq decrease, and the d and q axis current fluctuations of the motor decrease significantly, indicating that the motor control effect has been greatly improved. When M exceeds 3, the improvement of switching frequency and motor control effect is not obvious, indicating that enough virtual vectors have been added to the control set. In addition, it can be seen that when the number of area reductions changes from M = 0 to M = 1, the calculation amount of the strategy increases significantly. This is because the introduction of virtual space vector not only increases the calculation process of duty cycle, but also increases the calculation steps of online calculation of output sequence and delay compensation. After adding the virtual space vector, the computational complexity of the strategy increases slightly with the increase of M. At this time, the computational burden is increased only because of the increase of region reduction and optimization times. Therefore, considering the control effect of the strategy and the computational burden, M = 3 is given in this paper. On this basis, continuing to reduce the region and increase the virtual vector, it is no longer meaningful to improve the performance of the strategy.

In order to verify the steady-state performance of the strategy, the control period of the proposed ECS-MPCC strategy is further given to be 290 μs, and the traditional MPCC control period is given to be 100 μs. The control effects of the two strategies on the output voltage V_AB, d-q axis current i_d and i_q, and neutral point voltage v_o are compared at the same switching frequency. Figure 11 and Figure 12 illustrate the experimental results under different operation conditions.

In Figure 11, the given motor reference speed is 500 r/min, and the given load torque is 2 N·m. In Figure 12, the given motor reference speed is 1000 r/min and the given load torque is 5 N·m. In the steady-state experiment, it can be seen that the value of q-axis current i_q increases with the increase of reference speed and load torque. According to Figure 11 and Figure 12, when applying the proposed ECS-MPCC strategy, the d-axis and q-axis current fluctuations are smaller. As the motor speed increases, the frequency of the output voltage of the NPC inverter becomes larger. The experimental results under both conditions show that compared with the traditional MPCC strategy, the output voltage has a higher sine degree when the proposed strategy is applied. From the midpoint voltage experiment results, both strategies can suppress the midpoint voltage below 0.5 V, far less than the DC side voltage.

The steady-state experimental results indicate that the proposed ECS-MPCC strategy makes full use of the basic voltage vector of the three-level inverter after extending the control set, and its steady-state control effect is better than the traditional MPCC strategy at the same switching frequency.

4.2. Transient-State Performance Analysis

Transient experiments verify the control performance of the ECS-MPCC control strategy in the case of motor speed mutation and load torque mutation. In the speed mutation experiment, the load torque of the motor is given as 2 N·m, and the reference speed is stepped from 500 r/min to 1000 r/min to verify the response of the motor to the reference speed step under the proposed strategy. In the torque mutation experiment, the reference speed of the motor is set to 1000 r/min, and the load torque is stepped from 0 N·m to 5 N·m to verify the response of the motor to the load step under the proposed strategy.

The transient experiment observes the tracking of motor speed and the changes in current and neutral point voltage during the whole process. The results of the experiment are displayed in Figure 13.

As shown in Figure 13a, when the reference speed is stepped from 500 r/min to 1000 r/min, the i_q of the motor will increase rapidly to achieve rapid tracking of the target speed. After the dynamic process, the i_q is reduced to the original value. During the whole process, the motor reaches the target speed within about 0.06 s, and the overshoot is small. As shown in Figure 13b, when the motor encounters a sudden change in load torque during operation, the speed of the motor decreases, and the error with the reference speed increases, resulting in an increase in i_q, ensuring that the motor can return to a given speed. During the whole dynamic process, the motor only has a small speed fluctuation, and can recover to the given speed within 0.04 s. After the dynamic process, the i_q of the motor increases with the increase of the load torque. In the transient process, due to the sudden increase of current, the fluctuation of v_o will increase, but still less than 1 V, indicating that the proposed ECS-MPC strategy can effectively control the neutral voltage fluctuation of NPC inverter. From the above analysis, the transient experimental results show that the FCS-MPC strategy has good dynamic control performance.

5. Conclusions

This paper presents an ECS-MPCC strategy for the NPC three-level inverter-fed permanent magnet synchronous motor system. In the strategy, the weight coefficient is eliminated and the cost function is simplified using the characteristic that the paired small vectors have opposite effects on the neutral point voltage. Then, the strategy uses the volt-second balance principle to extend the virtual vector in the space vector diagram. On this basis, a fast calculation method for the vector duty ratio is designed. In the strategy, after determining the basic voltage vector required for synthesis by minimizing the value function, the duty ratio of the optimal virtual vector is calculated using region reduction. The ECS-MPCC strategy avoids traversing all virtual space vectors and directly calculates the predicted value corresponding to the virtual space vector using the existing predicted value, which reduces the calculation burden. The experimental results show that the proposed ECS-MPCC strategy has better steady-state performance and good dynamic performance compared to the traditional MPCC.