# Recent Advances in Multi-Phase Electric Drives Model Predictive Control in Renewable Energy Application: A State-of-the-Art Review

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Multi-Phase Machine Model

_{th}= 10m ± 1(m = 1, 3, 5....) harmonics completely to the α-β subspace. The variables of this subspace are related to energy conversion. Since the amplitude of the higher harmonics is low, it can be considered that this subspace has only a fundamental component. The k

_{th}= 5m ± 2(m = 1, 3, 5....) harmonics are mapped on the x-y subspace, which is not related to the energy conversion but is related to the stator copper loss. The zero-sequence harmonics in the order of k

_{th}= 5m (m = 1, 3, 5....) are mapped in the zero subspace, which is always kept at zero for the five-phase machine with star winding connections [18].

_{d}

_{1}/u

_{q}

_{1}, i

_{d}

_{1}/i

_{q}

_{1}, and L

_{d}

_{1}/L

_{q}

_{1}are the d-q axes voltage, current, and inductance in the fundamental subspace, u

_{d}

_{3}/u

_{q}

_{3}, i

_{d}

_{3}/i

_{q}

_{3}, and L

_{d}

_{3}/L

_{q}

_{3}are the d-q axes voltage, current, and inductance in the third harmonic subspace, ω is the rotor angular velocity, and R is the stator resistance.

_{n}is the number of pole pairs.

## 3. Classical MPC Schemes in a Multi-Phase Machine

^{5}= 32 switching states [19], each switching state corresponds to the VVs in the α-β subspace and x-y subspace, as shown in Figure 2. All VVs can be expressed as

_{dc}is the DC-bus voltage of the VSI, and S

_{i}(i = A, B, C, D, E) ϵ {0 1} represents the switching state of different bridge arms. “1” denotes that the upper device is ON, and “0” denotes that the lower device is ON.

_{1}, medium vector group L

_{2}, and small vector group L

_{3}, with the amplitude of 0.6472V

_{dc}, 0.4V

_{dc}, and 0.2472V

_{dc}, respectively.

#### 3.1. Model Predictive Current Control

_{s}is the sampling period.

_{1}to control the machine. The vectors in this group have the maximum amplitude in the α-β subspace, and the minimum amplitude in the harmonic subspace as shown in Figure 2, so the harmonic current can be naturally reduced. To realize the simultaneous control of the fundamental and harmonic currents, the cost function (CF) is designed as

_{d}

_{1}

^{*}, i

_{q}

_{1}

^{*}, i

_{d}

_{3}

^{*}, and i

_{q}

_{3}

^{*}are the current commands in the two frames, respectively.

#### 3.2. Model Predictive Torque Control

_{e}

^{*}, ψ

_{sd}

^{*}, and ψ

_{sq}

^{*}are the torque and stator flux commands, respectively.

#### 3.3. Model Predictive Speed Control

_{m}

^{*}is the rotor mechanical angular velocity command.

## 4. Advanced Control Schemes of MPC in the Multi-Phase Machine

#### 4.1. MPC of the Multi-Phase Machine with Simplified Cost Function

_{2}. However, it is still necessary to adjust λ

_{1}to balance the control of torque and stator flux.

_{s}

^{*}is the RVV.

^{3}s)-based MPC method is proposed to eliminate the harmonic voltage. The principle of V

^{3}s is shown in Figure 6, which is synthesized from the VVs of groups L

_{1}and L

_{2}with the same direction in the α-β subspace, and the opposite direction in the x-y subspace. Therefore, the harmonic current can be eliminated by reasonably distributing the duty cycle of the two VVs in groups L

_{1}and L

_{2}. The general form of the V

^{3}s is expressed as

_{L}

_{1}and V

_{L}

_{2}are the large and medium VV, respectively; d

_{1}and d

_{2}are the duty cycles of the two VVs with the value of 0.618 and 0.382, respectively. All the synthesized V

^{3}s are shown in Figure 7.

^{3}s, a simplified CF can be obtained, which excludes the consideration of the x-y current

^{3}s method, there is no need to constrain the harmonic components in the CF [32]. Unfortunately, this method cannot compensate for the harmonic current due to the dead-time of the inverter, and the distortion of the back electromotive force of the multi-phase machine [33]. Therefore, ref. [35] proposed a modulation scheme for synthesizing the control voltage in the harmonic subspace to suppress the harmonic current. According to the deadbeat control, the dwell time range of two large VVs and two medium VVs is determined to ensure the accurate control of the fundamental subspace. Then, the harmonic voltage is optimized by the CF that only contains the x-y subspace. The CF is defined as follows

_{sx}

^{*}and u

_{sy}

^{*}are the RVVs in the x-y subspace. In this method, the modulation scheme realizes the decoupling control of fundamental and harmonic subspaces.

_{1}has the highest priority. Firstly, two large vectors that minimize g

_{1}are selected from 10 large candidate vectors. Then, the harmonic current CF g

_{3}is utilized to select the optimal vector from the two large vectors and one zero vector. In the minimum harmonic current control scheme, the harmonic current suppression is taken as the control priority. Firstly, g

_{3}is used to select one optimal large vector, one optimal medium vector, and one zero vector as the candidate vectors. Then, g

_{1}is minimized to select the optimal vector from the three candidate vectors.

#### 4.2. MPC of the Multi-Phase Machine with Harmonic Current Suppression

^{3}s-based method can not only simplify the CF but also is proven to be an effective solution for reducing harmonic currents in multi-phase drive systems [48]. Essentially, V

^{3}s is the combination of two switching states that make the average x-y voltage equal to zero [49,50]. However, since the amplitude and direction of a virtual vector are fixed in the α-β subspace, the regulation of the α-β subspace is not flexible. To mitigate this drawback, two sets of V

^{3}s with different amplitudes are employed [51], which expands the number of candidate V

^{3}s to 25. Compared with conventional V

^{3}s-based MPC, the distortion of α-β current at low speed is improved. A study by [52] proposed a V

^{3}s-based MPC method with duty cycle regulation, where the candidate VV is composed of one V

^{3}s and one zero VV. Nevertheless, the improvement of the control performance is still limited because of the constant direction of the synthesized vector. Another study [53] proposes a MPC method based on two V

^{3}s to improve the control performance in the fundamental and harmonic subspaces since the combination of two V

^{3}s can provide better voltage tracking accuracy. However, the V

^{3}s used above have static properties because they are determined off-line, that is, the duty ratio of the large vector to medium vector is fixed. Although the V

^{3}s make the average voltage in the x-y subspace equal to zero, the low-frequency current harmonics that maps to the x-y subspace will appear towing to the asymmetric windings of the machine, the non-linearity of the inverter, etc. In response to this challenge, a Bi-subspace MPC method based on V

^{3}s is proposed in [54], which uses dual V

^{3}s to achieve independent control of the fundamental and harmonic currents. Moreover, to further reduce the current tracking error, a predictive current control scheme based on space vector modulation (SVM) and V

^{3}s is proposed in [57]. In this scheme, two vectors are combined with zero vectors to enhance the control accuracy in the two subspaces. Experimental results indicate that this scheme can achieve better α-β current tracking and x-y current reduction.

#### 4.3. MPC of the Multi-Phase Machine with Computational Complexity Reduction

^{3}s-based strategy [67]. Since the average voltage in the harmonic subspace is zero, there is no need to predict the harmonic variables, and the evaluation of the harmonic variables by the CF is also eliminated. This is one of the commonly used methods to reduce the computational complexity in the MPC of the multi-phase machine [68]. It should be noted that the V

^{3}s-based MPC cannot achieve the standard PWM switching sequence, which makes the hardware implementation difficult. For easy hardware implementation, the V

^{3}s of the nonstandard switching sequence are ingeniously replaced by the corresponding equivalent virtual vector in [69], which simplifies the modulation process.

_{d}

^{*}and i

_{q}

^{*}are the reference current of the d-q axes.

_{d}

^{*}and u

_{q}

^{*}are the RVV of the d-q axes.

^{3}s for five-phase PMSM. The optimal V

^{3}s are directly selected according to the position of the RVV, which avoids the prediction of all candidate V

^{3}s. In addition, the WF is eliminated, thus reducing the calculation workload. In addition, in view of the complicated derivation of the RVV in the deadbeat-direct torque and flux control, ref. [74] proposes a simplified algorithm based on load angle control. This algorithm directly deduces the change of the load angle from the torque error, and only needs to be implemented in the stationary frame without complex coordinate transformation. Another study by [75] proposes two simplified methods for calculating RVV in the synchronous rotating frame. The first method is based on the stator flux differential. By reducing the order of the calculation equation, the calculation process of the RVV only includes basic arithmetic operations and does not involve the square root. The other simplified algorithm is based on complex power derivation, which introduces a novel reactive torque parameter to reduce the complexity.

#### 4.4. MPC of the Multi-Phase Machine with Robustness Improvement

_{s}is divided into n sub-periods, and the VV corresponding to each sub-period can be calculated as

_{c}(i) is the VV at k + (i/n)-th sampling period, i = {0,1,2…n − 1}, c ∈ {α, β, x, y}.

_{sum}(k + 1) represents the compensation part, which can be calculated as

_{p}and k

_{i}are the proportional and integral coefficients, which will affect the prediction accuracy and system stability. The stator current prediction model based on the closed-loop structure is shown in Figure 12.

_{ψ-error}(k) is the compensation part, which can be calculated as

_{dqs}(k + 1) and ψ

_{dqc}(k + 1) represent the stator flux prediction values of the voltage model and current model, respectively.

#### 4.5. MPC of Multi-Phase Machines with Fault-Tolerant Operation

_{ls}is leakage inductance.

_{y}(k) u

_{y}(k) are the stator current and voltage of the y-axis at instant k.

_{sx}= −i

_{sα}. Therefore, the x-axis current is removed from the prediction model, and the cost function under OCF is expressed as

_{sy}

^{*}= 0.2631i

_{sβ}

^{*}, and the minimum copper loss control can be achieved by controlling i

_{sy}

^{*}= 0.

^{3}s is proposed in [143], which not only simplifies the prediction model but also eliminates the turning of the WF. In [144], an improved V

^{3}s- based fault-tolerant MPTC with a deadbeat solution is proposed to quickly select the optimal vectors. In addition, the steady-state error is reduced by inserting the zero vector, and the dwell time of each vector is determined by a simple geometric principle. Moreover, ref. [145] proposes an MPCC method with continued modulation technology, which reconstructed the distribution of the post-fault VVs and compensated the back-EMF of the fault phase. A study by [146] proposes an improved fault-tolerant MPC with SVPWM solution, in which the reconstructed V

^{3}s are utilized to synthesize the reference vector, and the candidate vectors can be selected quickly by the principle of deadbeat control, thus reducing the computational burden.

^{3}s-based MPC to achieve fault-tolerant operation without changing the controller topology after OCF. The experimental results show that the V

^{3}s-based MPC shows good behavior after faults. However, the spatial position of the V

^{3}s has been shifted without proper compensation. In [148], an improved MPC method based on a health transformation matrix is proposed to deal with the single-phase OCF. To compensate for the shift of the VVs under OCF, a disturbance term is added to the candidate vector to accurately predict the machine states. Although the prediction times are reduced by selecting the outermost vectors as the control set, the harmonic variables still need to be predicted. Therefore, an improved MPCC compensation method based on V

^{3}s is proposed in [149]. By compensating for the basic VVs, 24 new V

^{3}s are constructed, avoiding the turning of WF.

#### 4.6. MPC of Multi-Phase Machines with CMV Reduction

_{dc}, ±0.3V

_{dc}, and ±0.1V

_{dc}, respectively. The common CMV suppression methods are shown in Figure 18. Among them, hardware-based methods are more costly because of the requirement of additional hardware equipment, whereas software-based methods are more economical and promising. Generally, MPC-based CMV suppression methods can be divided into two categories, namely, the CF-based optimization methods, and the control set optimization-based methods [151,152,153].

_{CML}and V

_{CMM}are the CMV of the large and medium VVs, respectively.

_{CM}is the value of CMV, and C

_{SF}is the switching frequency.

^{3}s-based MPCC method is proposed in [155], which uses only the vectors that generate small CMV to synthesize the new V

^{3}s, and the control block diagram is shown in Figure 19. According to the above analysis, the large VVs generate small CMV; therefore, three adjacent large VVs are selected in this method to construct the new V

^{3}s, and the principle is expressed as follows

_{1}, d

_{2}, and d

_{3}are the duty cycles of the three adjacent large VVs, which are 0.382, 0.236, and 0.382, respectively.

^{3}s are shown in Figure 20. Therefore, the proposed MPCC scheme enables inherent CMV suppression and harmonic current reduction. Similarly, to suppress the CMV and current harmonics of a seven-phase VSI, a simple MPCC scheme is proposed in [156], which takes 14 V

^{3}s as the control set. In this method, the V

^{3}s are synthesized by the VVs with the largest amplitude in the fundamental subspace with the lowest CMV, and the voltage components in the two harmonic subspaces are zero. Therefore, multiple WFs are eliminated in the CF. In [157], a MPCC method with inherent rejection characteristics of CMV is proposed for a five-phase VSI, in which four adjacent large VVs are utilized to synthesize V

^{3}s. The dwell time of the V

^{3}s is optimized by introducing the duty cycle optimization technique to reduce the harmonic current. In addition, the zero vector is replaced by two opposite large VVs, thus further reducing the CMV. However, the V

^{3}s-based MPC methods for suppressing the CMV have the drawback of low DC-link utilization. To address this problem, an improved MPCC method is proposed in the literature [158], which improves the utilization of the DC-link while reducing CMV. As mentioned in [159], the existence of dead-time may lead to a large CMV. Therefore, the effect of dead-time on CMV is considered in [160]. However, it requires a field programmable gate array, which increases the hardware complexity. For the sake of reducing CMV with low hardware cost, a novel MPC method for suppressing CMV during dead-time is proposed in [161]. In this method, the combination of VVs that may generate large CMV in dead-time is pre-excluded. However, it does not make full use of all possible VVs, resulting in large current ripples. According to the analysis in [160], not all non-adjacent and non-opposite VV combinations can generate large CMV, so there is no need to exclude them all. Therefore, ref. [162] proposes an improved MPC with hybrid VV pre-selection, which expanded the number of candidate VVs compared with [160]. However, since only one VV is applied during the whole sampling period, the steady-state performance is still unsatisfactory. Therefore, ref. [163] proposes a multi-vector-based MPC for CMV suppression to further improve the steady-state performance.

#### 4.7. MPC of Multi-Phase Machines with ZSC Reduction

_{0}

^{*}is set to 0 for the suppression of the ZSC.

_{eff}(n) is the sum of the CMV in the dead-time interval when the different switching states are applied on the dual inverters. This CF converts the control of current into the control of equivalent voltage, thus eliminating the prediction of stator current. In addition, the complete elimination of CMV during dead-time at low speed is achieved by adding an additional term to the CF.

^{3}s-based MPC scheme is proposed in [181] to reduce CMV and harmonic current. Firstly, the V

^{3}s of two VSI are utilized to design the control set. Secondly, only 21 vectors with ZSV are selected as the candidate control set to eliminate CMV. Finally, the optimal vector is selected with the principle of deadbeat control. Similarly, ref. [182] proposes a simplified MPCC based on V3s with zero-sequence voltage for nine-phase OEWM to eliminate the ZSC.

## 5. Future Trends of MPC for Multi-Phase Machines

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 16.**The control diagram of fault-tolerant MPCC scheme for a five-phase machine under OCF operation.

**Figure 17.**Basic VVs under single-phase OCF of the five-phase machine. (

**a**) α-β subspace. (

**b**) x-y subspace.

Control Methods | Advantages | Disadvantages |
---|---|---|

V^{3}s-based method [67,68,69] | Simplified CF and prediction model | Non-standard switching sequence |

Deadbeat solution [70,71,72,73,74,75] | Good performance and accurate vector selection | Poor robustness |

Torque and flux error-based methods [76,77,78,79,80] | Simple structure and good robustness | Rough vector selection |

Control Methods | Complexity | Disadvantages |
---|---|---|

Improved prediction model [81,82,83,84,85,86,87] | Low | Model dependency |

Parameter identification [88,89,90,91,92,93,94,95,96,97,98,99,100,101,102] | Medium | Complex tuning work |

Disturbance observer [103,104,105,106,107,108,109,110,111,112] | High | Model dependency Complex tuning work |

Model-free predictive control [113,114,115,116,117,118,119,120,121,122] | Slight high | Sensor accuracy dependency Sampling frequency dependency |

Change the structure of the controller [123,124,125,126,127,128,129,130,131] | Low | Model dependency |

Control Methods | Advantages | Disadvantages |
---|---|---|

Reduced-order transformation-based methods [137,138,139,140,141,142,143,144,145,146] | Accurate modeling and good performance | Complex structure |

Without changing structure-based methods [147,148,149] | Easy to implement and natural transition before and after the fault | Poor performance |

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## Share and Cite

**MDPI and ACS Style**

Xue, Z.; Niu, S.; Chau, A.M.H.; Luo, Y.; Lin, H.; Li, X.
Recent Advances in Multi-Phase Electric Drives Model Predictive Control in Renewable Energy Application: A State-of-the-Art Review. *World Electr. Veh. J.* **2023**, *14*, 44.
https://doi.org/10.3390/wevj14020044

**AMA Style**

Xue Z, Niu S, Chau AMH, Luo Y, Lin H, Li X.
Recent Advances in Multi-Phase Electric Drives Model Predictive Control in Renewable Energy Application: A State-of-the-Art Review. *World Electric Vehicle Journal*. 2023; 14(2):44.
https://doi.org/10.3390/wevj14020044

**Chicago/Turabian Style**

Xue, Zhiwei, Shuangxia Niu, Aten Man Ho Chau, Yixiao Luo, Hongjian Lin, and Xianglin Li.
2023. "Recent Advances in Multi-Phase Electric Drives Model Predictive Control in Renewable Energy Application: A State-of-the-Art Review" *World Electric Vehicle Journal* 14, no. 2: 44.
https://doi.org/10.3390/wevj14020044