# Modelling and Post-Fault Current Control Strategy for Multi-Three-Phase Machines

Azlia Abdul Rahman, Meiqi Wang, Lorenzo Carbone, Gioele Baia, Michele Degano

► To cite this version:

Azlia Abdul Rahman, Meiqi Wang, Lorenzo Carbone, Gioele Baia, Michele Degano. Modelling and Post-Fault Current Control Strategy for Multi-Three-Phase Machines. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0306 . hal-05079941

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https://utc.hal.science/hal-05079941v1

Submitted on 22 May 2025

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# Modelling and Post-Fault Current Control Strategy for Multi-Three-Phase Machines

1st Azlia Abdul Rahman

Power Electronics, Machine and
Control (PEMC) Research Institute

University of Nottingham
Nottingham, United Kingdom
azlia.abdulrahman@nottingham.ac.uk

2nd Meiqi Wang

Power Electronics, Machine and
Control (PEMC) Research Institute

University of Nottingham
Nottingham, United Kingdom
meiqi.wang2@nottingham.ac.uk

3rd Lorenzo Carbone

Power Electronics, Machine and
Control (PEMC) Research Institute

University of Nottingham
Nottingham, United Kingdom
lorenzo.carbone1@nottingham.ac.uk

4th Gioele Baia

dept. Council of Science
Technology and Society
IUSS Advanced School of Pavia

Pavia, Italy
gioele.baia@iusspavia.it

5th Michele Degano

Power Electronics, Machine and
Control (PEMC) Research Institute

University of Nottingham
Nottingham, United Kingdom
michele.degano@nottignham.ac.uk

**Index Terms**—Multi-three-phase machine (MTPM), fault tolerance control, interior Permanent Magnet (IPM) synchronous machine control, modeling, Harmonic Inductance Variation.

**Abstract**—The growing development of Multi-Three-Phase Machines (MTPM) as competitive to conventional three-phase machines has attracted more attention to the higher reliability and fault-tolerant capability of the electrical drives system especially for safety-critical applications. In this paper, modelling technique and post-fault current control strategy for dual three-phase Interior Permanent Magnet (IPM) synchronous machines are presented, utilizing the Vector Space Decomposition (VSD) algorithm. As the multi-stator (MS) or dq modelling is an ideal option for post-fault strategies, the comparative study between VSD and MS methods are demonstrated in order to analyse their dynamic performance in nominal and faulty conditions. This take into account the equivalent harmonic inductance variation, which is then used to introduce the post-fault control strategy. A double three-phase IPM machine is modelled and operated in two different operating conditions: nominal and one three-phase set of windings in open-circuit (OC). In OC fault, the proportional and integral (PI) controllers are re-designed while adapting the inductance variation. The current dynamics responses are analysed in both healthy and faulty conditions by means of Matlab/Simulink ® simulations.

## I. INTRODUCTION

Multi-phase machines have gained significant attention in recent years, especially in applications where safety and reliability are critical. Compared to traditional three-phase machines, multi-phase machines provide additional degrees of freedom, which contribute to higher reliability, fault tolerance, efficiency, torque, and power density [1], [2]. One particularly notable subclass of these machines is the Multi-Three-Phase Machine (MTPM), characterized by isolated neutral points, which further enhances fault tolerance and simplifies control strategies [3]. MTPMs allow for the reuse of existing techniques from conventional three-phase machines, such as control strategies, fault detection, fault isolation, and winding topology design, thus bridging the gap between new technology and established methodologies [4]–[6]. In critical applications, such as aerospace, multi-three-phase drives with independently controlled and isolated three-phase windings are preferred for their superior

fault tolerance, particularly in scenarios where failure is not an option [7], [8]. Although various fault-tolerant control strategies have been proposed, Open circuit (OC) faults, which are common in stator windings, remain a significant challenge for maintaining reliable operation [9]–[12]. The complexity of OC fault scenarios, which shaped by factors such as phase arrangement, winding configuration, and fault location, demands robust and efficient control solutions [13]–[16]. Most research focuses on fault detection and mitigation strategies under open-circuit conditions, yet there remains room for innovation in optimizing control methods [9], [11], [13].

Interior Permanent Magnet (IPM) machines have emerged as a preferred solution in safety-critical applications due to their high power density, superior efficiency, reduced acoustic noise, and wider constant-power speed range [17], [18]. IPM machines benefit from advanced control algorithms, and numerous studies have explored nonlinear models that account for effects like saturation, cross-coupling, spatial harmonics, and temperature [19], [20]. Nevertheless, idealized models are often used in the simulation of IPM control systems, leaving a gap in the literature for more comprehensive models that address fault tolerance, especially under OC fault conditions.

In this paper, modelling techniques and fault-tolerant current controllers for a double three-phase IPM machine are implemented based on multi-stator (MS) and Vector Space Decomposition (VSD) algorithm. Based on the work

<figure><img src="image_2.png" /><figcaption>Fig. 1. Fully modular and redundant system implementation [21].</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2. Leakage inductances corresponding to the i-th and j-th stator of three-phase set.</figcaption></figure>

presented in [22], a comparative study between MS and VSD methods is extended by incorporating various current controller design approaches. The machine model is developed for both nominal and OC conditions, considering asymmetrical winding configurations of the MTPM. While the MS method accounts for higher harmonic contributions from the back electromotive force (BEMF), the VSD technique models only the first fundamental component which is responsible for torque production. The system is implemented with modularity and redundancy. This manuscript presents a comparison of post-fault control strategies and dynamic performances for both MS and VSD methods. Two approaches for the MS methods are explored: the first involves using the same proportional-integral (PI) gains as the VSD method, referred to as the hybrid dq model; the second employs a PI controller designed based on the dq model with different PI gains. This comparative study provides original insights into these methodologies.

The paper is arranged as follows, in section II, the mathematical modeling of the machine under both nominal and fault conditions is presented to provide a comprehensive understanding of the proposed model. Section III discussed the design of the current controller within the dq0 reference frame and the Vector Space Decomposition (VSD) technique. In section IV, an equivalent simplified model of the PI current controller is introduced, highlighting the different control approaches across various operating conditions. The case studies are described in section V, followed by simulation results, where the current responses under nominal and OC conditions are analyzed. Finally, conclusions are discussed in section VI.

## II. MULTI-STATOR (DQ0) MODEL OF MTPMS

In the early studies of the asymmetrical n-phase MTPM, a MS modelling approach has been utilized to aid the understanding of the machine's operation [3], [23]. By defining $\mathbf{R}$ and $\mathbf{L}(\theta)$ as resistance and inductance $n \times n$ matrices, $\theta$ as the mechanical position of the rotor, and $n \times 1$ vectors $\bar{v}$, $\bar{i}$, and $\bar{\psi}_0$ as the phase voltage, current, and zero-current linked flux, respectively, the voltage equation can be written as follows:

$$ \bar{v} = \mathbf{R}\bar{i} + \frac{d\theta}{dt} \frac{\partial \mathbf{L}(\theta)}{\partial \theta} \bar{i} + \mathbf{L}(\theta) \frac{d\bar{i}}{dt} + \frac{d\theta}{dt} \frac{\partial \bar{\psi}_0(\theta)}{\partial \theta} \quad (1) $$

Distributed current control is achieved within the rotor orthogonal dq0 (denoted with a subscript dq) reference frame by applying the Park's Transformation to the machine stator variables (denoted with a subscript abc). Since the machine is considered a MTPM, the global Park's Transformation matrix is defined by the following equations:

$$ \mathbf{T} = \begin{pmatrix} \mathbf{T}_1 & \cdots & \mathbf{0}_3 \\ \vdots & \ddots & \vdots \\ \mathbf{0}_3 & \cdots & \mathbf{T}_N \end{pmatrix}_{n \times n} \quad (2) $$

$$ \mathbf{T}_h = \sqrt{\frac{2}{3}} \begin{pmatrix} \cos[\theta - (h-1)\alpha] & \sin[\theta - (h-1)\alpha] & 0 \\ -\sin[\theta - (h-1)\alpha] & \cos[\theta - (h-1)\alpha] & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & -1/2 & -1/2 \\ 0 & \sqrt{3}/2 & -\sqrt{3}/2 \\ 1/\sqrt{2} & 1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix} \quad \text{with } h = 1..N \quad (3) $$

where $\mathbf{0}_3$ is a $3 \times 3$ null matrix [5].

Subsequently, this machine voltage equation in (1) could be written in new coordinate system of dq0 reference frame:

$$ \bar{v}_{dq} = \mathbf{R}_{dq} \bar{i}_{dq} + \frac{d\theta}{dt} \frac{\partial \mathbf{L}_{dq}(\theta)}{\partial \theta} \bar{i}_{dq} + \mathbf{L}_{dq}(\theta) \frac{d\bar{i}_{dq}}{dt} + \frac{d\theta}{dt} \frac{\partial \bar{\psi}_{0dq}(\theta)}{\partial \theta} \quad (4) $$

Whilst $\mathbf{R}_{dq} = \mathbf{R} = r_s \mathbf{I}_{(n \times n)}$ where $r_s$ is the stator phase resistance and $\mathbf{I}_{(n \times n)}$ is an identity matrix. The inductance matrix $\mathbf{L}_{dq}$ is obtained as follows:

$$ \mathbf{L}_{dq} = \begin{pmatrix} \mathbf{L}_{dq(1,1)} & \cdots & \mathbf{L}_{dq(1,N)} \\ \vdots & \ddots & \vdots \\ \mathbf{L}_{dq(N,1)} & \cdots & \mathbf{L}_{dq(N,N)} \end{pmatrix} $$

$$ \text{with } \mathbf{L}_{dq(i,j)} = \mathbf{L}_{dq(j,i)}^T = \mathbf{T}_h \mathbf{L}_{abc(i,j)} \mathbf{T}_h^T = \frac{3}{2} \begin{pmatrix} L_{md} & 0 & 0 \\ 0 & L_{mq} & 0 \\ 0 & 0 & 0 \end{pmatrix} + \begin{pmatrix} M_{i-j} & -X_{i-j} & 0 \\ X_{i-j} & M_{i-j} & 0 \\ 0 & 0 & H_{i-j} \end{pmatrix} \quad (5) $$

with

$$ \bar{v}_{dq} = [\bar{v}_{dq1} \cdots \bar{v}_{dqN}]^T, \quad \bar{i}_{dq} = [\bar{i}_{dq1} \cdots \bar{i}_{dqN}]^T \\ \text{and } \bar{\psi}_{0dq} = [\psi_{0dq1} \cdots \psi_{0dqN}]^T \quad (6) $$

where $L_{md}$ and $L_{mq}$ are denoted as $d$, $q$ auto inductances. Parameters $M_k$, $X_k$, $H_k$ are the stator leakage inductances expressed within the dq0 reference frame [5], [24], where $M_k$ and $X_k$ are mutual stator leakage inductances whilst $H_k$ is homopolar stator leakage inductances. Their physical meaning and relationship are explained in Fig. 2 with $i$ and $j$ are the stator identifiers 1...N and they are computed using finite element (FE) analysis or analytical method [3], [25].

### A. dq0 State Space Model

Assuming the speed loop is not affecting the inner current control one, the current control loop design based on (4) can be done in locked rotor position. However, further simplifications to design the controllers are needed. By neglecting the Back-Electro-Motive-Force (BEMF) and assuming $\mathbf{L}$ not dependant on the rotor position ($\mathbf{L}(\theta) = \mathbf{L}$) the mechanical speed $\omega$ is zero and Eq. (4) becomes:

$$ \bar{v}_{dq} = \mathbf{R}_{dq} \bar{i}_{dq} + \mathbf{L}_{dq} \frac{d\bar{i}_{dq}}{dt} \quad (7) $$
The following state space model can be derived based on (7):

$$ \dot{\bar{x}}_{dq} = \mathbf{A}_{dq}\bar{x}_{dq} + \mathbf{B}_{dq}\bar{u}_{dq} \quad \text{and} \quad \bar{y}_{dq} = \mathbf{C}\bar{x}_{dq} + \mathbf{D}\bar{u}_{dq} \quad (8) $$

where $\bar{x}_{dq}$ is the current state vector, $\bar{u}_{dq}$ is the applied voltage input vector, $\bar{y}_{dq}$ is the output current vector, while $\mathbf{A}_{dq} = -\mathbf{L}_{dq}^{-1}\mathbf{R}_{dq}$, $\mathbf{B}_{dq} = \mathbf{L}_{dq}^{-1}$, $\mathbf{C}$ and $\mathbf{D}$ are respectively identity and null $n \times n$ matrices [3], [12].

Observing the variables in Fig. 3, the two $i_q$ currents ($i_{q1}$ and $i_{q2}$) from $dq0$ reference frame variables are mapped into state space variables $x_2$ and $x_5$ for the two three-phase sets of six-phase IPM machine. For the higher phase number with $h = 1...N$, every $i_{qh}$ is mapped into its relative $x_{3h-1}$ state space variable.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Mapping of $dq0$ variables into state space model variables.</figcaption></figure>

From the above state space model, the transfer function in the $dq$ space is derived as follows:

$$ \mathbf{G}_{dq} = \mathbf{C}(s\mathbf{I} - \mathbf{A}_{dq})^{-1}\mathbf{B}_{dq} + \mathbf{D} = \mathbf{y}_{dq}/\mathbf{u}_{dq} \quad (9) $$

This $\mathbf{G}_{dq}$ is the transfer function linking the $i$-th input voltage to the $j$-th output current under balanced voltage supply (i.e.: $v_{d1} = v_{d2}$, $v_{q1} = v_{q2}$, and $v_{01} = v_{02}$ for dual three-phase set of windings), therefore the $i_q$ current of the first set is a function of all the linking input voltages.

Obviously the $\mathbf{G}_{dq}$ is not diagonal since $\mathbf{L}_{dq}$ is not diagonal matrix. Therefore it is not possible to get the decoupled transfer functions between the $i$-th input and $j$-th output from (9). For this reason, the first harmonic inductances of $d_1$ and $q_1$ are determined in $\mathbf{L}_{vsd}$ matrix which computed through diagonalization of the inductance matrix in order to design the current controllers in NC, which refers to the equivalent current controller in this paper.

## B. State Space Model in Open Circuit Condition

In Sec. II, the number of phases per set of windings and the total number of phases have been defined as $m$ and $n$, respectively. Assuming an OC fault where a three-phase set of windings (the first three-phase winding set) is disconnected from the inverter, the total number of phases in OC can be defined as follows:

$$ n_{OC} = n_{NC} - m \quad (10) $$

By deleting the last three rows and the last three columns from the NC matrices and the last three elements from the NC vectors given in Sec. II-A, the OC transfer function matrices can be derived as before:

$$ G_{vsdOC(2,2)} = \sum_{k=1}^{n_{OC}} G_{dqOC(k,2)} = G_{AdqOC} \quad (11) $$

Consequently, the circulating fault current is equal to zero under this OC ($i_{a2} = i_{b2} = i_{c2} = 0$).

## III. VSD MODEL OF MTPMS

Using the VSD algorithm, diagonalization of the inductance matrix can be achieved in order to compute the harmonic inductances analytically. In the other words, the benefits of modelling a multiphase machine based on VSD are decoupled machine model and unique mapping of all odd-order harmonics within the resulting subspaces. This is resulted by multiplying machine phase variables with the VSD matrix, as shown in Fig. 4.

Analytically, the equivalent decoupled $\mathbf{L}_{vsd}$ matrix so called harmonic inductance matrix can be computed as in the following:

$$ \mathbf{L}_{vsd} = \mathbf{T}_{vsd}\mathbf{L}_{abc}\mathbf{T}_{vsd}^{T} \quad (12) $$

where $\mathbf{T}_{vsd}$ ($\mathbf{T}_{vsd}^{T} = \mathbf{T}_{vsd}^{-1}$) represents the final real-valued orthonormal decoupling transformation as demonstrated in (14). Then, the diagonal pattern of $\mathbf{L}_{vsd}$ decoupled harmonic inductance for $6 \times 6$ matrix is presented as follows:

$$ \mathbf{L}_{vsd(6,6)} = \begin{pmatrix} d_1 & 0 & 0 & 0 & 0 & 0 \\ 0 & q_1 & 0 & 0 & 0 & 0 \\ 0 & 0 & d_3 & 0 & 0 & 0 \\ 0 & 0 & 0 & q_3 & 0 & 0 \\ 0 & 0 & 0 & 0 & d_5 & 0 \\ 0 & 0 & 0 & 0 & 0 & q_5 \end{pmatrix} \quad (13) $$

where the subscripts (1, 2...$h$) indicate a harmonic orders, and $d$ and $q$ denotes the inductance for corresponding axis of the rotor-attached orthogonal $dq0$, while (13) represents $\mathbf{L}_{vsd(6,6)}$ which the one used in this case study. In nut-shell, the highlighted $d_1$ and $q_1$ are represent the first harmonic inductance that will be used together with the stator resistance $r_s$ value to design the proposed current controller discussed in the next sections. More precisely, the design of an equivalent current control loop for nominal condition and OC fault are implemented according to the same VSD procedure on computation of equivalent harmonic inductance.

Using the VSD model, a six-phase machine can be represented using three orthogonal subspaces, i.e., the $\alpha-\beta$, $x-y$, and zero-sequence subspaces. The parameters are useful for the multiphase machine control and for development of pulse width modulation (PWM) techniques. Additionally, unlike the double-$dq$ or multi-stator model, the VSD model is applicable to any type of multiphase machines with any phase number and windings configuration. However, only six-phase or double three-phase IPM machine will be considered in this paper.

The development behind this theory led to definition of complex numbers that represented as vectors moving in different planes (generally named $x-y$, $\alpha-\beta$, or $d-q$) called as space vectors. A system of $k$ variables of the $x-th$ quantity can be fully identified by $k$ complex numbers, where the complex numbers are referred to the vector of the $h-th$ plane or $h-th$ space vector. It is important to note that the VSD process is determined by the phase variables $k$ and $n$-phases. More precisely, these parameters identify selection of the harmonics order for VSD. In Fig. 4, the VSD transformation:

$$ \mathbf{T}_{vsd}(\theta) = \mathbf{T}(\theta)\mathbf{W} \quad (14) $$
<figure><img src="image_5.png" /><figcaption>Fig. 4. Two-step VSD transformation for a generic multiphase model.</figcaption></figure>

is resulted from two cascaded procedures, the following sequential demonstrates the two-step for VSD transformation:

1) Geometrical transformation (W): this is the mapping process of the actual winding structure into conventional one; the details of operation will be clarified in the next section.

2) Decoupling transformation (T(θ)) where θ is the rotor position: this transformation deliver machine variables onto a set of mutually orthogonal subspaces.

In order to form the VSD transformation in particular multi-three-phase system, the machine variable in terms of n-winding arrangement either symmetrical or asymmetrical is initially determined.

### A. Equivalent State Space Model in Nominal Condition

After the inductance matrix has been diagonalised, VSD transfer function $G_{vsd}$ can be derived as how the equation (9) is obtained. Similar state space model in (8) is valid in both the $dq0$ and $vsd$ space, hence defining the new decoupled transfer function where it has been computed in the $vsd$ space and is resulted to the following equation:

$$G_{vsd} = C(sI - A_{vsd})^{-1}B_{vsd} + D = y_{vsd}/u_{vsd} \quad (15)$$

where the new state space matrices $A_{vsd} = -L_{vsd}^{-1}R_{vsd}$, $B_{vsd} = L_{vsd}^{-1}$, $u_{vsd}$ is the applied voltage input vectors, and $y_{vsd}$ is the output current vectors with respect to the state space output current $x_{vsd}$.

$$G_{vsdNC} =$$

$$\begin{pmatrix} \frac{1}{sd_1+r_s} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{1}{sq_1+r_s} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{sd_3+r_s} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{sq_3+r_s} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{sd_5+r_s} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{sq_5+r_s} \end{pmatrix} \quad (16)$$

It is well known that the $L_{vsd}$ and $R_{vsd}$ are diagonal ($R_{vsd} = R = R_{dq}$), therefore the matrix of $G_{vsd}$ for the dual three-phase IPM machines system in NC ($G_{vsdNC}$) is also diagonal as presented in (16). In general of multi-three-phase applications, the matrix in (16) can be demonstrated up to $2v \pm 1-th$ harmonic order on both $d$ and $q$ axes according to n-phase. The correlation between $G_{vsd}$ and $G_{dq}$ transfer function matrices will be presented in the next section.

### B. Relation of dq0-VSD state-space model

The difference between $G_{vsd}$ and $G_{dq}$ lies in the inductance matrices: $L_{vsd}$ is diagonal, $L_{dq}$ is not. As such, in the $dq0$ representation, each $j$-th output is not independent from each $i$-th input. On the contrary, within the $vsd$ space, each output current depends on the respective input voltage [3], [12]. In this

paper, the analytical approach of derived equations are mainly focused on q-axis elements, and it is important to note that the same consideration should be taking into account for derivation along d-axis.

The $G_{vsd}$ and $G_{dq}$ equivalence can be appreciated by comparing the transfer function linking the input voltages to the output currents of one set of winding. From (16), the transfer function linking the first harmonic input voltage to the first harmonic output current along the q axis is as follows:

$$G_{vsd(2,2)} = \frac{1}{sq_1 + r_s} \quad (17)$$

On the other hand, the transfer function linking the q output current of the first set of windings to the input voltages takes into account for all the $dq$ voltages from all the other sets of windings as follows:

$$\sum_{k=1}^{n} G_{dq(k,2)} = G_{Adq} \quad (18)$$

Equations (17) and (18) are equivalent when the machine is supplied in nominal condition (NC) with balanced voltages and no faults are present within the system as shown in the following:

$$G_{vsdNC(2,2)} = G_{AdqNC} \quad (19)$$

The equation (19) is obviously verified in the work presented in [22], where it is implemented on two orthonormal spaces of $dq0$ and $vsd$ with the absence of interactions among different axes of different sets of windings.

## IV. CURRENT CONTROLLERS DESIGN

Fig. 5 presents the developed simulation model used to compare $vsd$ and $dq0$ current control strategies. The PI controllers design for the $vsd$ method starts with the synthesis of the equivalent current control loop shown in Fig. 6. Firstly, the error between the measured current and its reference is calculated and fed to the PI controller which produces the $vsd$ voltage reference. In order to take into account the actuation delay naturally present on the control system, the delay transfer function $AD(s)$ is introduced in the current loop as shown in Fig. 6, and is presented in (20).

$$AD(s) = \frac{1}{s1.5T_s + 1} \quad (20)$$

This delay can be modelled as a first order transfer function with the time constant $\tau = 1.5T_s$, where $T_s$ represent the switching period [22]. Finally the delayed voltage reference thus obtained represent the plant $G_{vsd}(s)$ input variable. The transfer function on the control feedback $LPF(s)$ is used to simulate the filtering effect of the measurement devices.

$$LPF(s) = \frac{\omega_f^2}{s^2 + \sqrt{2\omega_f s + \omega_f^2}} \quad (21)$$

It can be noted that the dynamic of the plant is completely considered by considering the transfer function $G_{vsd}(s)$. In summary, to design the PI controller parameters, two equivalent control loops have been considered: one for the NC scenario and one for the OC scenario. Depending on the scenario, the PI controller transfer function $PI(s)$ and the plant transfer function $G_{vsd}(s)$ are adjusted accordingly. Lastly, the actuation delay and current filter, introduced previously, can be used to evaluate stability margin variations, while the PI gains remain constant under fault conditions [5].

### A. Nominal Condition

Under normal operating condition with two healthy three-phase sets of IPM machine windings and balanced voltage supply the PI and plant transfer function are defined as follows:

$$PI_{NC} = \frac{sK_{pINC} + K_{iINC}}{s} \quad (22)$$

$$G_{vsdNC} = G_{vsd(2,2)} = \frac{1}{sq_1 + r_s} \quad (23)$$

where $K_{pINC}$ and $K_{iINC}$ are the proportional and integral PI gains parameters in NC, respectively. The optimized PI gains parameters are then computed based on the stator resistance $r_s$,
<figure><img src="image_6.png" /><figcaption>Fig. 5. Difference control diagrams of vsd and dq0 controls.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Equivalent current control loop design for NC.</figcaption></figure>

switching period $T_s$, measurement cut-off bandwidth $\omega_f$, and first harmonic inductance $d_{1NC}$ and $q_{1NC}$ as derived in (22) for nominal condition (NC).

## B. Open-circuit Fault Condition

The PI and plant transfer function for the OC scenario are reported below:

$$PI_{OC} = \frac{sK_{pIOC} + K_{iIOC}}{s} \qquad (24)$$

$$G_{vsdOC(2,2)} = \frac{1}{s q_{1OC} + r_s} \qquad (25)$$

where $K_{pIOC}$ and $K_{iIOC}$ are the proportional and integral PI gains parameter. It is important to re-computed the first harmonic inductance ($q_{1OC}$) under OC, from presented $L_{vsdOC}$ in (13). By substituting the computed first OC harmonic inductance $q_{1OC}$ into the plant transfer function, current dynamics in OC operation can be kept constant. Similar procedure for analytical approach to obtain the PI gains in NC (Sec. IV-A) is applied to re-compute the $K_{pIOC}$ and $K_{iIOC}$ under OC fault. These PI gains are computed on the updated first harmonic inductance $q_{1OC}$ and constant stator resistance $r_s$. As presented in Fig. 7, in OC scenario it is notable that the stability margins are affected. Considering a real case implementation of MTPM windings under OC fault, the current dynamic can be restored as in nominal operation while keeping the current bandwidth $\omega_c$ constant.

As proposed in Sec. IV-A, defining $AD(s)$, $LPF(s)$, $PI(s)$ in NC and OC, respectively, the stability margins in OC fault for with and without updating the control loop are shown and compared against NC stability margins. In Fig. 7, stability margins of closed loop gain transfer functions in OC fault condition are illustrated; the $G_{clNC}$ (in green asterisks), $G_{clOC_{NUP}}$ (in red asterisks), and $G_{clOC_{UP}}$ (in magenta circle) are the closed loop transfer functions for the NC and OC fault condition, respectively, as described in the following:

$$G_{clNC}(s) = PI_{NC}(s)AD(s)G_{vsd_{NC(2,2)}}(s)LPF(s) \qquad (26)$$

$$G_{clOC_{NUP}}(s) = PI_{NC}(s)AD(s)G_{vsd_{OC(2,2)}}(s)LPF(s) \qquad (27)$$

<figure><img src="image_8.png" /><figcaption>Fig. 7. Stability margins in OC fault condition for not updating (NUP) and updating (UP) the PI compared against stability margin in NC.</figcaption></figure>

$$G_{clOC_{UP}}(s) = PI_{OC}(s)AD(s)G_{vsd_{OC(2,2)}}(s)LPF(s) \qquad (28)$$

Despite the effects of OC fault are minimal, it can be observed that the current dynamics can be reimposed by updating the PI according to the control diagram in Fig. 5. In the next Section V, further analysis on the current dynamic performance is discussed based on the simulation results in order to validate this theory. The comparison of dynamic performances for MS and VSD methods are demonstrated.

## V. SIMULATION RESULTS

Under nominal and OC conditions, the PI current controllers are designed with the constant input parameters of current filter cut-off frequency $\omega_f$, switching period $T_s$, control loop bandwidth $BW_c$, and phase margin $PM_c$ based on the case study parameters shown in Table I.

According to the proposed simplified equivalent current controller model in Sec. IV, this section presents the analysis and comparison of the current dynamic responses in between the nominal and OC conditions for both MS and VSD methods, and only q-axis currents are presented for simplicity.

<table><caption>TABLE I
SIMULATION PARAMETERS</caption><thead><tr><td>Parameter</td><td>Value</td><td>Unit</td></tr></thead><tbody><tr><td>Cut-off frequency ω<sub>f</sub></td><td>1.2566 · 10<sup>6</sup></td><td>rad/s</td></tr><tr><td>Switching Period T<sub>s</sub></td><td>0.625 · 10<sup>-6</sup></td><td>s</td></tr><tr><td>Control loop bandwidth BW<sub>c</sub></td><td>2444</td><td>rad/s</td></tr><tr><td>Phase margin PM<sub>c</sub></td><td>65.85</td><td>degree</td></tr></tbody></table>

In all operating conditions, the equivalent model denoted as EQ is assigned as a bench mark and validated against the dq model (denoted with DQ in the plots) and Hybrid dq. To recall, the EQ model is developed by using VSD control algorithm in order to obtain first harmonic inductance where linking each input voltages to output currents for the system transfer function derivation (Sec. III). In parallel to the implemented EQ models shown in Fig. 6, respective reference dq models presenting mutual couplings between different sets have been run. As aforementioned, there are two approaches for MS; one is the dq models are employed with the vsd control where the PI gains are computed based on the first harmonic inductance, by means EQ and dq models are sharing the same PI gains (named as Hybrid dq model) and the second MS approach is where the PI controller is designed based on the dq model (or multi-harmonic) and different PI gains are employed.
<figure><img src="image_9.png" /><figcaption>Fig. 8. Currents dynamic responses in OC fault condition comparing to nominal condition (desired) for MS and VSD methods.</figcaption></figure>

## A. Nominal Condition

In Fig. 8a and 8b, current dynamic responses for both three-phase set of the six-phase IPM machine in NC are demonstrated. The responses of *q* currents *i*<sub>q1</sub> and *i*<sub>q2</sub> of the two sets of healthy windings are respectively the 2-nd and 5-th element of the state space vector *x*<sub>dq</sub> in (8). For EQ and Hybrid *dq* models, the responses are identical as they are sharing similar PI gains which are designed based on the first harmonic inductance, but the difference dynamic performances of the *dq* model is appreciated. For the PI current control design of the *dq* model, the first three-phase set PI gains are calculated with respect to *L*<sub>q1</sub> inductance, and for the second three-phase set PI gains are calculated with respect to *L*<sub>q4</sub> inductance.

## B. OC Fault Condition

Under OC fault conditions, the effect of Not-Updated (NUP) PI gains and Updated (UP) PI gains on current dynamics are compared against NC for EQ, Hybrid *dq*, and *dq* models. Fig. 8c and 8d demonstrates the current dynamics for the first healthy set, where obviously *i*<sub>q2</sub> = 0 from the second three-phase set under open-circuit winding. The results show the difference between NUP PI gains in OC, against the desired NC output currents in EQ, Hybrid, and *dq* models, where the *dq* model presents slower rise time and higher overshoot. While Fig. 8e and 8f demonstrate the relative and equivalent between the updating UP PI gains in OC compared against the desired response in NC. Based on the output currents transient, the match can be appreciated between UP PI gains in OC and healthy set in their equivalent model, where the PI gains in OC must be updated to compensate the fault. The current responses with NUP PI gains is faster in its rise time without a proper dynamic performance compared to the NC and the OC fault operation with UP PI gains. The effects indicate that mutual coupling still exists in the three-phase winding sets of the machine. Even though the transient response between NC and OC with NUP the PI gains is considered minimal de-rating, it is worth noting that the dynamic current performance can be kept as in nominal operation by updating the PI gains.

## VI. CONCLUSION

In this paper, the post-fault control strategies for the multi-three-phase six-phase IPM machine under OC fault conditions in one three-phase set, compared to nominal operation, are proposed and analyzed. A fully modular and redundant control system has been developed to ensure the machine can continue delivering optimal power following an OC fault. The proposed post-fault control methods have been validated through simulations; a prototype has been built in the lab, and the experimental results will be presented in an extended version of this paper.

The results indicate that the simplified equivalent model for the current control loop, based on the VSD method, maintains consistent dynamic performance in both healthy and faulty conditions, resulting in satisfactory steady-state and dynamic current responses. For OC faults, new proportional-integral PI controllers have been designed to adapt to variations in machine parameters. Despite the reduction in torque during post-fault operation, the proposed control strategies ensure continuous operation of the remaining healthy modules, maintaining a dynamic response similar to that observed under nominal conditions. This validated control method is applicable to multi-three-phase systems where high reliability and fault tolerance are essential.

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