Inter national J our nal of P o wer Electr onics and Dri v e System (IJPEDS) V ol. 16, No. 1, March 2025, pp. 185 ∼ 194 ISSN: 2088-8694, DOI: 10.11591/ijpeds.v16.i1.pp185-194 ❒ 185 Backstepping multiphase induction machine contr ol impact in pr esence of open phases fault Chak er Berrahal, Abderrahim El F adili LSIB Laboratory , FST Mohammedia, Hassan II Uni v ersity of Casablanca, Casablanca, Morocco Article Inf o Article history: Recei v ed month dd, yyyy Re vised month dd, yyyy Accepted month dd, yyyy K eyw ords: Backstepping control DC/A C In v erter L yapunouv stability Multiphase induction machine Open phases f ault ABSTRA CT As po wer requirements increase, multiphase induction machines (MPIMs) present a promising alternati v e to con v entional three-phase induction machines. These machines help reduce the current switched by the in v erter and circulat- ing through the windings , which in turn mitig ates torque ripple. Moreo v er , in- corporating more than three phases enhances system reliability , allo wing the machine to maintain operation e v en in the e v ent of one or more phase f ailures. This mak es MPIMs particularly suitable for high-reliability applications, such as electric v ehicles. While most pre vious studies ha v e concentrated on speed and ux control of MPIMs, less attention has been gi v en to handling open-phase f aults. This paper e xplores the rob ustness of the backstepping control method applied to MPIMs, particularly in scenarios in v olving open-phase f a ults. The proposed multi-loop nonlinear controller is de v eloped to achie v e tw o main ob- jecti v es: precise speed re gulation across a wide range of speed references, and ef fecti v e rotor ux control. The con v er gence of the feedback control system is rigorously analyzed using L yapuno v’ s stability theory . Simulation results sho w that, although the control objecti v es are met, stator current demands increase as more phases e xperience f aults. This observ ation highlights the need for further de v elopment of MPIM models that tak e phase f aults into consideration. This is an open access article under the CC BY -SA license . Corresponding A uthor: Chak er Berrahal LSIB Laboratory , FST Mohammedia, Hassan II Uni v ersity of Casablanca Casablanca, Morocco Email: chak er .berrahal@gmail.com 1. INTR ODUCTION Multiphase induction machines are increas ingly popular in industrial applications due to their reli a- bility and high operational a v ailability [1]. The distrib ution of po wer across multiple phases results in lo wer per -phase con v erter currents, reducing stress on the machine windings and po wer electronics semiconductors. The y of fer numerous adv antages, including higher torque density , impro v ed torque quality , and increased o v er - all ef cienc y [2]. This paper aims to pro vide an o v ervie w of the k e y de v elopments in multiphase induction machines, emphasizing their ability to operate under f ault conditions. Extensi v e research has been conducted on this topic [3], [4], focusing on operational analysis, modeling, control, and f ault diagnosis algorithms. As pre viously mentioned, these machines of fer se v eral benets o v er their three-phase counterparts [5]. The dis- trib ution of po wer o v er a greater number of phases reduces electrical and thermal stress per phase, enhancing reliability and po wer density [6]. Moreo v er , the increased number of phases pro vides better operational redun- danc y , which is v aluable in critical applications such as aerospace and electric traction [2], [4]. In the e v ent of a phase f ailure, the machine can continue operat ing, albeit with reduced performance, b ut remains functional [7]. J ournal homepage: http://ijpeds.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
186 ❒ ISSN: 2088-8694 These machines also perform well at lo w speeds and high torque [7], making them ideal for electric v ehicles and rene w able ener gy systems [4], [8]. The rene wed interest in multiphase induct ion machines is supported by adv ancements in po wer electronics and adv anced control techniques. Control techniques are crucial to fully e x- ploit the characteristics of multiphase induction machines, especially in the presence of f aults. These machines dif fer from con v entional three-phase machines in the number of stator phases and are typically po wered by multiphase con v erters, allo wing better control and greater operational e xibility . A k e y feature of multiphase induction machines is their ability to operate in a de graded mode under f ault conditions. Thanks to their redun- danc y and po wer distrib ution across multiple phases, t hese machines can continue functioning, albeit slightly reduced, in the e v ent of one or more phase f ailures [9]. Thi s is a major adv antage for critical applications such as aerospace, marine, and electric traction, where reliability and continuity of service are essential. Se v eral control techniques ha v e been de v eloped for multiphase induction machines, enabling them to operate under normal conditions and in the presence of f aults [6], [9]. These include: − Multi v ariable v ector control [10], [9]: This allo ws independent control of the machine’ s torque and ux. This approach is particularly suited to multiphase induction machines due to the comple xity associated with the high number of phases. − F ault-tolerant control strate gies [11] : These aim to maintain machine performance e v en in the e v ent of phase f ailures. These techniques typically in v olv e reconguring the control system to adapt to the ne w operating conditions . − Nonlinear control techniques [12]–[14]: Such as backstepping and sliding modes, which of fer better performance and greater Impact to disturbances and model uncertainties. These control approaches ha v e been e xtensi v ely studied in the literature , demonstrating the pot ential of multiphase induction machines for critical applications requiring reliability and f ault tolerance [15], [16]. The objecti v e of this paper is to present the modeling and control of the inte gration between a DC/A C con v erter and a multiphase induction machine connected to a battery . The Impact of the re gulator , designed and analyzed using the backstepping technique [17], will be e v aluated by introducing f aults in one or more arms of the DC/A C con v erter supplying the multiphase machine. The controller’ s Impact will be tested through simulations conducted in the MA TLAB/Simulink en vironment [18], [19]. The paper is or g anized as follo ws: Section 2 introduces the model of the MultiPhase Induction Ma- chine (MPIM) with n phases and its association with a DC/A C in v erter , e xpressed in the x ed frame ( α , β ), used to dri v e a battery-po wered electric v ehicle. Section 3 is dedicated to the synthesis of a multi-loop nonlin- ear controller using the backstepping technique and L yapuno v stability . Section 4 presents simulation results to illustrate the control Impact in the e v ent of an open-phase f ault. 2. MODELING OF THE SYSTEM F or our study , we consider an n-phase induction machine (MPIM) po wered by an n-le g v oltage in v erter (DC/A C in v erter), used to dri v e a battery-po wered electric v ehicle. The schematic diagram is sho wn in Figure 1. 2.1. MPIM model The model of a MultiPhase induction machine (MPIM) with n phases, e xpressed in the x ed frame ( α , β ), w as deri v ed using the P ark transformation. Using the stator current components ( i sα and i sβ ) and the rotor ux components ( ϕ r α and ϕ r β ) as state v ariables, the tw o-phase model of the MPIM is described by (1). dω dt = p M sr J L r ( ϕ r α i sβ − ϕ r β i sα ) − T L J − f v J ω di sα dt = − γ i sα + M sr R r σ L s L 2 r ϕ r α + M sr σ L s L r pω ϕ r β + 1 σ L s v sα di sβ dt = − γ i sβ + M sr R r σ L s L 2 r ϕ r β − M sr σ L s L r pω ϕ r α + 1 σ L s v sβ dϕ r α dt = − R r L r ϕ r α + pω ϕ r β + R r M sr L r i sα dϕ r β dt = − R r L r ϕ r β − pω ϕ r α + M sr R r L r i sβ (1) The follo wing notations are used: ( i sα , i sβ are stator current α β components), ( ϕ r α , ϕ r β are rotor ux α β components),( v sα , v sβ are stator v oltage α β components),( ω is rotor speed, ( R s , R r are stator and rotor resistances),( L s , L r are stator and rotor self-inductances), ( M sr is Mutual inductance between the stator and Int J Po w Elec & Dri Syst, V ol. 16, No. 1, March 2025: 185–194 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 187 the rotor), ( f v is friction coef cient,( J is rotor inertia), ( T L is load torque) and ( p is the number of pole pairs). with the parameters γ and σ are dened as: γ = L 2 r R s + M 2 sr R r σ L s L 2 r and σ = 1 − M 2 sr L s L r . 2.2. DC/A C in v erter model The modeli n g of the con v erter in the multiphase reference frame, sho wn in Figure 2, in v olv es e xpress- ing the v oltages at nodes 1, 2, ..., t o n of the in v erter with respect to the midpoint M. These v oltages can be e xpressed in terms of the switch connection functions and the input v oltage as in (2). v S 1 v S 2 . v S i . v S n = V dc n × ( n − 1) − 1 − 1 . . − 1 − 1 ( n − 1) − 1 . . − 1 . . . . . . − 1 − 1 . ( n − 1) . − 1 . . . . . . − 1 − 1 − 1 . . ( n − 1) × k 1 k 2 . k i . k n (2) Where: − v S 1 , v S 2 , . . . , v S n are the v oltages at nodes 1, 2, ..., to n with respect to the midpoint M. − k 1 , k 2 , . . . , k n are the switch connection functions (binary v ariables that tak e the v alue 0 or 1). − V dc is the input v oltage (the v oltage across the battery). T o simplify thi s system of (2), we apply the P ark transformation. The ne w system of equations is then represented in the ( α , β ) reference frame. The in v erter is characterized by the independent control of the stator v oltage components v sα and v sβ . Accordingly , these v oltages are e xpressed as functions of the corresponding control inputs. v sα = V dc .u 1 v sβ = V dc .u 2 (3) Where u 1 and u 2 represent the a v eraged v alues of the α , β -components within the multiphase duty ratio system, obtained by applying the P ark transformation to the duty ratio signals ( k 1 , k 2 ,..., k n ) and a v eraging o v er PWM periods. Figure 1. Controlled system K K' K' K' K' K K K 1 2 i n 1 2 i n V S1 S2 Si Sn V V V V dc 1 2 i n Figure 2. Multiphase DC/A C-in v erter 2.3. Modeling of the entir e system No w , let us dene the a v eraged state v ariables, as in (4). x 1 = ω , x 2 = i sα , x 3 = i sβ , x 4 = ϕ r α , x 5 = ϕ r β , (4) As clear from the conte xt, the notation • refers to a v eraging o v er the PWM periods. Then, it is pro v ed in man y places that instantaneous ‘MPIM-in v erter association’ representation (1) assumes the follo wing a v eraged form, in v olving the a v eraged v ariables (3) and (4). ˙ x 1 = − f v J x 1 + p M sr L r 1 J ( x 3 x 4 − x 2 x 5 ) − T L J (5) ˙ x 2 = R r M sr σ L s L 2 r x 4 + pM sr σ L s L r x 5 x 1 − γ x 2 + V dc σ L s u 1 (6) Bac kstepping multiphase induction mac hine contr ol impact in ... (Chak er Berr ahal) Evaluation Warning : The document was created with Spire.PDF for Python.
188 ❒ ISSN: 2088-8694 ˙ x 3 = R r M sr σ L s L 2 r x 5 − pM sr σ L s L r x 4 x 1 − γ x 3 + V dc σ L s u 2 (7) ˙ x 4 = − R r L r x 4 + R r L r M sr x 2 − p x 1 x 5 (8) ˙ x 5 = − R r L r x 5 + R r L r M sr x 3 + p x 1 x 4 (9) 3. CONTR OLLER DESIGN 3.1. Contr ol objecti v es The operational control objecti v es are tw ofold: − Speed re gulation: the machine speed ω must closely follo w a gi v en reference signal ω r ef . − Re gulating the rotor ux norm ϕ r = p x 2 4 + x 2 5 should be maintained at a reference v alue ϕ r ef , ideally equal to its nominal v alue. 3.2. Motor speed and r otor norm ux contr ol design The task of controlling the rotor speed and rotor ux norm is no w considered for the multiphase induction motor described by (5)-(9). The speed reference x ∗ 1 = ω r ef is an y bounded and dif ferentiable function of time, with its rst tw o deri v ati v es being a v ailable and bounded. These conditions can al w ays be satised by ltering the original (possibly non-dif ferentiable) reference through a unit static g ain second-order linear lter . The rotor ux reference ϕ r is set to its nominal v alue. The controller design, illustrated in Figure 3 (backstepping control scheme), will be carried out in tw o steps using the backstepping technique [20]-[23], Figure 3. Backstepping control scheme 3.2.1. Step 1: Intr oducing tracking err ors Let’ s dene the tracking errors as (10) and (11). z 1 = ω r ef − x 1 (10) z 2 = ϕ 2 r ef − ( x 2 4 + x 2 5 ) (11) From (5), (8), and (9), it follo ws that the errors z 1 and z 2 are go v erned by the follo wing dif ferential equations, as (12) and (13). z 1 = ˙ ω r ef + f v x 1 J − pM sr J .L r ( x 3 x 4 − x 2 x 5 ) + T L J (12) z 2 = 2 ϕ r ef ˙ ϕ r ef − 2 R r M sr L r ( x 2 x 4 + x 3 x 5 ) + 2 R r L r ( ϕ 2 r ef − z 2 ) (13) In (12) and (13), the terms p M sr J L s ( x 3 x 4 − x 2 x 5 ) and 2 R r M sr L r ( x 2 x 4 + x 3 x 5 ) emer ge as virtual control signals. W ere these actual control signals, the error system (12) and (13) could be globally asymptotically stabilized by setting p M sr J L s ( x 3 x 4 − x 2 x 5 ) = µ 1 and 2 R r M sr L r ( x 2 x 4 + x 3 x 5 ) = ν 1 , where: Int J Po w Elec & Dri Syst, V ol. 16, No. 1, March 2025: 185–194 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 189 µ 1 def = ( c 1 z 1 + ˙ ω r ef ) + T L J + f v J ( ω r ef − z 1 ) (14) ν 1 def = c 2 z 2 + 2 ϕ r ef ˙ ϕ r ef + 2 R r L r ( ϕ 2 r ef − z 2 ) (15) Here, c 1 and c 2 represent positi v e design parameters. Ho we v er , since p M sr J L s ( x 3 x 4 − x 2 x 5 ) and 2 R r M sr L r ( x 2 x 4 + x 3 x 5 ) are not the actual control signals, the y cannot be equated to µ 1 and ν 1 respecti v ely . Nonetheless, we retain the e xpressions of µ 1 and ν 1 as initial stabilizing functions and introduce the ne w errors [23]-[25]. z 3 = µ 1 − p M sr J .L r ( x 3 x 4 − x 2 x 5 ) (16) z 4 = ν 1 − 2 R r L r M sr ( x 2 x 4 + x 3 x 5 ) (17) Then, using the notations (10)-(17), the dynamics of the errors z 1 and z 2 , initially described by (12) and (13), can be re written as (18) and (19). z 1 = − c 1 z 1 + z 3 (18) z 2 = − c 2 z 2 + z 4 (19) 3.2.2. Step 2: Deri ving contr ol signals T o ensure the con v er gence of all errors ( z 1 , z 2 , z 3 , z 4 ) to zero, we need to elucidate the dependenc y of these errors on the actual control signals ( u 1 , u 2 ). Initially focusing on z 3 , we deri v e its dynamics from (16). z 3 = ˙ µ 1 − p M sr J L r ( ˙ x 3 x 4 + x 3 ˙ x 4 − ˙ x 2 x 5 − x 2 ˙ x 5 ) (20) Utilizing (5)-(9) and (14), we simplify (20) to (21). z 3 = µ 2 − p M sr J L r V dc σ L s ( x 5 u 1 − x 4 u 2 ) (21) Where, as in (22). µ 2 = h c 1 ( − c 1 z 1 + z 3 ) + ¨ ω r ef + ˙ T L J − f v .T L J 2 i + pM sr J .L r ( f v J + R r L r + γ )( x 4 x 3 − x 2 x 5 ) + p 2 M sr x 1 J .L r ( x 3 x 5 + x 2 x 4 ) + p 2 M 2 sr σ J .L s L 2 r x 1 ( x 2 4 + x 2 5 ) (22) Similarly , for z 4 from (17), we ha v e (23). z 4 = ˙ ν 1 − 2 R r M sr L r ( ˙ x 2 x 4 + x 2 ˙ x 4 + ˙ x 3 x 5 + x 3 ˙ x 5 ) (23) Substituting (5)-(9) and (15) into (23), we get (24). z 4 = ν 2 − 2 R r L r M sr V dc σ L s ( u 1 x 4 + u 2 x 5 ) (24) Where, as in (25). ν 2 = c 2 ( − c 2 z 2 + z 4 ) + 2( ˙ ϕ 2 r ef + ϕ r ef ¨ ϕ r ef ) − (2( R r L r ) 2 + 2( R r L r ) 2 M sr σ L s L r M sr )( x 2 4 + x 2 5 ) Bac kstepping multiphase induction mac hine contr ol impact in ... (Chak er Berr ahal) Evaluation Warning : The document was created with Spire.PDF for Python.
190 ❒ ISSN: 2088-8694 − 2( R r M sr L r ) 2 ( x 2 2 + x 2 3 ) + 2 pR r M sr L r x 1 ( x 2 x 5 − x 3 x 4 ) + (4( R r L r ) 2 M sr + 2 R r L r γ M sr )( x 2 x 4 + x 3 x 5 ) (25) No w , considering the error system (18)-(19), (21), and (24), we propose an augmented L yapuno v function candidate. V = 1 2 z 2 1 + 1 2 z 2 2 + 1 2 z 2 3 + 1 2 z 2 4 (26) Its time-deri v ati v e along the trajectory of the state v ector ( z 1 , z 2 , z 3 , z 4 ) is as (27). V = ˙ z 1 z 1 + ˙ z 2 z 2 + ˙ z 3 z 3 + ˙ z 4 z 4 (27) Utilizing (18)-(19), (21), (24), and (27) e xpands as (28). V = z 1 ( − c 1 z 1 + z 3 ) + z 2 ( − c 2 z 2 + z 4 ) + z 3 ( µ 2 − pM sr V dc σ L s L r ( x 5 u 1 − x 4 u 2 )) + z 4 ( ν 2 − 2 R r M sr V dc σ L s L r ( x 4 u 1 + x 5 u 2 )) (28) Supplementing c 3 z 2 3 − c 3 z 2 3 + c 4 z 2 4 − c 4 z 2 4 to the right side of (28) and rearranging terms, we obtain (29). V = − c 1 z 2 1 − c 2 z 2 2 − c 3 z 2 3 − c 4 z 2 4 + z 3 h µ 2 + z 1 + c 3 z 3 − pM sr V dc σ L s L r ( x 5 u 1 − x 4 u 2 ) i + z 4 c 4 z 4 + z 2 + ν 2 − 2 R r M sr V dc σ L s L r ( x 4 u 1 + x 5 u 2 ) (29) Here, c 3 and c 4 represent ne w positi v e real design parameters. (29) implies that the control signals u 1 , u 2 must be selected to render the quantities within the curly brack ets on the right side of (29) to zero. Setting these quantities to zero and solving the resulting second-order linear equation system with respect to ( u 1 , u 2 ) yields the follo wing control la w , as in (30). u 1 u 2 = λ 0 λ 1 λ 2 λ 3 − 1 z 1 + c 3 z 3 + µ 2 z 2 + c 4 z 4 + ν 2 (30) W ith: λ 0 = p V dc σ L s M sr L r x 5 , λ 1 = − p V dc σ L s M sr L r x 4 λ 2 = 2 R r V dc σ L s M sr L r x 4 , λ 3 = 2 R r V dc σ L s M sr L r x 5 (31) Notably , the matrix λ 0 λ 1 λ 2 λ 3 is nonsingular , as its determinant, D = λ 0 λ 3 − λ 2 λ 4 = − 2 R r L r V dc σ L s M sr ( x 2 4 + x 2 5 ) , remains non-zero since the ux ϕ r = p x 2 4 + x 2 5 ne v er v anishes practically due to the presence of the remnant ux. Substituting the control la w (30) into ( u 1 , u 2 ) on the right side of (29), we get (32). V = − c 1 z 2 1 − c 2 z 2 2 − c 3 z 2 3 − c 4 z 2 4 (32) Since the right side of (32) is a ne g ati v e denite function of the state v ector ( z 1 , z 2 , z 3 , z 4 ), the latter globally asymptotically v anish [14]. This result is more e xplicitly formulated in the follo wing theorem: Theorem: stability and con v er gence analysis: consider the closed-loop system comprising: Int J Po w Elec & Dri Syst, V ol. 16, No. 1, March 2025: 185–194 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 191 − The MPIM-DC/A C con v erter association, described by model (5)-(9) − The nonlinear controller dened by the control la w (30) Then, the follo wing properties hold: − The closed-loop error system, e xpressed in the ( z 1 , z 2 , z 3 , z 4 ) coordinates, e v olv es according to (33)- (36). z 1 = − c 1 z 1 + z 3 (33) z 2 = − c 2 z 2 + z 4 (34) z 3 = − c 3 z 3 − z 1 (35) z 4 = − c 4 z 4 − z 2 (36) − The abo v e linear system is stable with respect to the L yapuno v function V = 1 2 z 2 1 + 1 2 z 2 2 + 1 2 z 2 3 + 1 2 z 2 4 , and the errors ( z 1 , z 2 , z 3 , z 4 ) e xponentially con v er ge to zero. Proof: (33) and (34) directly stem from (18)-(19). (35) is deri v ed by substituting the control la w (30) into ( u 1 , u 2 ) on the right side of (21). Similarly , (36) is obtained by substituting the control la w (30) into ( u 1 , u 2 ) on the right side of (24). Thus, P art 1 is established. Furthermore, from (26), it is e vident that V = 1 2 z 2 1 + 1 2 z 2 2 + 1 2 z 2 3 + 1 2 z 2 4 serv es as a (radially unbounded) L yapuno v function for the error syst em (33)-(36). Since ˙ V is a semi-ne g ati v e denite function of the state v ector ( z 1 , z 2 , z 3 , z 4 ), by L yapuno v’ s equilibrium point theorem, the entire error system is e xponentially asymptotically stable, and the errors con v er ge to zero. 4. SIMULA TION AND RESUL TS The control system, described by model (5)-(9), and the control la w (30), are e v aluated via sim ulation. The system’ s characteristics are summarized in T able 1. The Impact of the backstepping multiphase induction machine control is assessed both with and without open phase f aults. The machine load remains constant, and the MPIM operates at a high speed ( ω r ef = 100 r d/s ). Initially , the machine operates without open phase f aults o v er the interv al [ 0 , 10 s ] . At t = 10 s , an open f ault occurs in phase number one, and another f aul t occurs in phase number four at t = 14 s . The reference v alue Φ r for the rotor ux norm is set to its nominal v alue ( 1 w b ). The indicated v alues of design parameters c 1 , c 2 , c 3 , c 4 ha v e been selected using a ’ try-and-error’ search method and pro v ed to be suitable. The e xperimental setup is simulated within the MA TLAB/Simulink en vironment with a calculation step of 5 µs . This v alue is chosen considering that the in v erter frequenc y commutation is 15 kHz. T able 1. System features Feature Symbole V alue Unit Induction machine Nominal po wer P n 7 . 5 k W Nominal current I n 9 . 6 A Stator resistance R s 0 . 63 Ω Stator c yclic inductance L s 0 . 098 H Rotor resistance R r 0 . 40 Ω Rotor c yclic inductance L r 0 . 09 H Mutual inductance M sr 0 . 09 H Inertia J 0 . 22 k g .m 2 V iscous rubbing f v 0 . 001 N m/r d/s Number of pole pairs p 2 Number of phases n 5 DC-A C con v erter Continuous v oltage Bus V dc 500 V PWM frequenc y F m 15 . 00 k H z The control performance obtained is depicted in Figures 4-8, illustrating the impact of nonlinear back- stepping control ag ainst one or tw o phase f ailure f aults on one or tw o supply phases, respecti v ely , of the mul- tiphase induction machine. Initially , the MPIM operates without an y f ault until t = 10 s. At t = 10 s, the rst Bac kstepping multiphase induction mac hine contr ol impact in ... (Chak er Berr ahal) Evaluation Warning : The document was created with Spire.PDF for Python.
192 ❒ ISSN: 2088-8694 open phase f ault occurs in phase number one, follo wed by a second f ault simultaneously in phase number four , starting from t = 14 s. Figures 4- 8 demonstrate the responses of rotor speed, rotor ux, the phase stator current unaf fected by an y f ault, and electromagnetic torque. F or both controlled v ariables ( ω , Φ r ), the tracking quality is highly sati sf actory across all speed ranges (refer to Figures 4 and 5). In both scenarios, the rotor ux norm and rotor speed closely match their references and con v er ge to them after the occurrence of each f ault. Re- markably , e v en in the pre sence of a single or double f ault, the motor speed and the rotor ux con v er ge to w ards their references. Figure 7 illustrates the electromagnetic torque generated by the multiphase induction machine before and after the occurrence of open-phase f aults. The machine is loaded with a constant torque T l set at 20 N .m . Initially , without an y f aults, the electromagnetic torque con v er ges, follo wing a transient state, to w ards 20 N .m . The slight torque ripples observ ed in Figure 7 stem from the po wer supply of the MPIM by the DC/A C con v erter . After the rst open-phase f ault, the a v erage v alue of the electromagnetic torque remains unchanged, b ut the am plitude of the ripples increases by 10%. This increase further intensies with the pres ence of tw o simultaneous open-phase f aults, reaching up to 40% (as depicted in Figure 8). 0 2 4 6 8 10 12 14 16 18 20 Time (s) 0 10 20 30 40 50 60 70 80 90 100 S p e e d  ( r d / s ) w _ r e f w Figure 4. Rotor speed response 0 2 4 6 8 10 12 14 16 18 20 Time (s) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F l u x r ( w b ) P h i _ R _ r e f P h i R Figure 5. Rotor Flux response 0 2 4 6 8 10 12 14 16 18 20 Time (s) -40 -30 -20 -10 0 10 20 30 40 L 3  C u r r e n t  ( A ) I s ( L 3 ) F i r s t S e c o n d f a u l t i n L 1 f a u l t i n L 4 Figure 6. Stator current in phase number 3 response 0 2 4 6 8 10 12 14 16 18 20 Time (s) 0 5 10 15 20 25 30 35 T o r q u e ( N . m ) T l T e F i r s t f a u l t i n L 1 S e c o n d f a u l t i n L 4 Figure 7. Electromagnetic torque response Figure 8. Zoom in electromagnetic torque Int J Po w Elec & Dri Syst, V ol. 16, No. 1, March 2025: 185–194 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 193 In Figure 6, the stator current absorbed by phase number 3, the phase unaf fected by an y f aults, is depicted. At the onset of a single open phase f ault at t = 10 s, the stator current of phase number 3 sur ges by 50% compared to its v alue in the absence of a f ault. W ith the occurrence of t w o simultaneous open phase f aults starting from t = 14 s, af fecting phases number 1 and number 4, this current escalates by o v er 150%. 5. CONCLUSION This study e xplores the impact of backstepping control for multiphase induction machines in the presence of open-phase f aults. The controller w as designed and analyzed using the backstepping technique, in- te grating the multiphase induction machine and the DC/A C in v erter modeled in the x ed frame ( α , β ), without considering open-phase f aults. The proposed controller incorporates nonlinear loops aimed at ensuring precise re gulation of rotor speed and ux. The impact of backstepping control w as e v aluated in single and double open-phase f ault scenarios. In both cases, the rotor ux norm and rotor speed closely foll o wed their references, con v er ging to them after each f ault e v ent. Ev en in the presence of open-phase f aults, the motor speed and rotor ux con v er ged to their references. After the occurrence of open-phase f aults, the stator current absorbed by the non-f aulty phases increased by 50% and 150%, respecti v ely . Although the electromagnetic torque generated by the multiphase induction machine maintained the a v erage v alue of the load torque a fter open-phase f aults, the ripple amplitude increased by 40%. Hence, the design and analysis of backstepping control must consider the presence of open-phase f aults. REFERENCES [1] E. 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