inter-turn short-circuits fault of induction machine operating under two control ... is introduced, based on analysis of measured three-phase stator currents and ...
ICEMIS2017, Monastir, Tunisia
Observer-based Inter-Turn Short-Circuit Fault Detection in Closed-loop Controlled Induction Machine Amal Guezmil, Hanen Berriri, Anis Sakly and Mohamed Faouzi Mimouni Research Unit of Industrial Systems Study and Renewable Energy (ESIER) National Engineering School of Monastir, Ibn Eljazzar City, 5019 University of Monastir, Tunisia. Abstract—This paper deals with detecting of stator winding inter-turn short-circuits fault of induction machine operating under two control methods namely the backstepping control and the integral sliding mode control. The fault detection method is introduced, based on analysis of measured three-phase stator currents and estimated ones given by an unknown input observer. The difference between these behaviors, namely residuals, are used to detect the inter-turn short-circuits fault. Simulation results obtained from an induction machine with inter-turn shortcircuits fault working under backstepping control and integral sliding mode control are presented.
I. I NTRODUCTION Nowadays, Induction Machines (IM) constitute a significant part of industrial systems. Thereby, their reliability and early detection of electrical and mechanical fauts are becoming more and more essential in order to minimize substantial financial losses that result from that and increase machine lifetime [1]. Induction machines are exposed to faults occurrences which causes the disruption of a normal operation of IM. According to [2], stator winding faults are one of the most common reasons of the IM breakdowns and make from 20% to 40% of all faults. These kinds of faults start as an unnoticeable Inter-Turn Short-Circuit (ITSC), which in consequence spread over the whole winding, causing the main ITSC. It leads to an emergency stop of the machine and necessity to repair it or substitute. Therefore, early detection of the ITSC fault is required. In the litterature, numerous ITSC Fault Detection (FD) techniques have been developed [3]. Among them, the observerbased methods. The iddea of these methods relies on calculating difference between estimated and measured variables [4]. This difference is namely residual. The FD is obtained by comparing residual signals with a predefined threshold that overstepping indicates the ITSC fault occurrence. Hence the majority of ITSC FD methods are applied for an IM oparating in open-loop control structures [5]–[7]. On the contrary, when the IM works under the closed-loop control, ITSC fault occurrence causes perturbations in normal operation of the control structure [11]. Uncontrolled increase of the damage can lead to unstable operation of the IM. Some previous papers have addressed the ITSC FD in closed-loop controlled IM [9]–[11].
The purpose of this paper is the design of FD method for closed-loop controlled induction machine working under ITSC fault. Simulation results for an IM operating under two nonlinear control techniques are presented: backstepping control [12] and Integral Sliding Mode (ISM) control [13]. Residual signals generated by comparing measured three-phase stator currents and estimated ones using an Unknown Input Observer (UIO) are chosen to detect the ITSC fault occurrence. This paper is organized in the following way, first, the healthy model of the IM is described. Next, the backstepping and ISM controls are synthesized. The UIO is then developed. Detailed results are presented to show the influence of the ITSC fault in the closed-loop performance. After that, the proposed FD method is described. Simulation results illustrate the proposed FD effectiveness. II. H EALTHY IM MODEL The IM nonlinear model in (𝑑 − 𝑞) reference frame can be expressed as [14]: ⎧ (𝐼 )2 𝐼˙𝑠𝑑 = 𝑎1 𝐼𝑠𝑑 + 𝑎2 𝜙𝑟 + 𝜔𝐼𝑠𝑞 + 𝑏1 𝜙𝑠𝑞𝑟 + 𝑐𝑉𝑠𝑑 𝐼 𝐼 ⎨ 𝐼˙𝑠𝑞 = 𝑎1 𝐼𝑠𝑞 − 𝑎3 𝜔𝜙𝑟 − 𝜔𝐼𝑠𝑑 − 𝑏1 𝑠𝑑𝜙𝑟𝑠𝑞 + 𝑐𝑉𝑠𝑞 ˙ 𝜙𝑟 = 𝑏2 𝜙𝑟 + 𝑏1 𝐼𝑠𝑑 ˙ = 1 (𝑚𝜙𝑟 𝐼𝑠𝑞 − 𝑇𝐿 − 𝑓 Ω) Ω 𝐽 ⎩ ˙ 𝐼 𝜃𝑠 = Ω + 𝑏1 𝜙𝑠𝑞𝑟
(1)
where 𝐼𝑠𝑑 and 𝐼𝑠𝑞 , 𝑉𝑠𝑑 and 𝑉𝑠𝑞 are respectively, the direct and quadratic stator current and voltage components. 𝜙𝑟 is the rotor flux modulus. Ω is the rotor speed. 𝑁𝑝 is the pole-pairs number, 𝑓 is the friction coefficient. 𝐽 is the Inertia moment. 𝑇𝐿 is the load torque and 𝜃𝑠 is the stator field angle. Expressions of 𝜎, 𝑎1 , 𝑎2 , 𝑎3 , 𝑏1 , 𝑏2 , 𝑐 (and 𝑚 are depend ) 2 2 𝑀𝑠𝑟 𝑀𝑠𝑟 𝑅𝑠 on IM parameters: 𝜎 = 1 − 𝐿𝑠 𝐿 ; 𝑎 = + 1 𝜎𝐿𝑠 𝜎𝐿𝑠 𝐿𝑟 𝑇𝑟 ; 𝑟
𝑀𝑠𝑟 𝐿𝑠 1 𝑠𝑟 ; 𝑎3 = 𝜎𝐿 ; 𝑐 = 𝜎𝐿 ; 𝑇𝑠 = 𝑅 ; 𝑎2 = 𝜎𝐿𝑀 𝑠 𝐿𝑟 𝑇 𝑟 𝑠 𝐿𝑟 𝑠 𝑠 𝑁𝑝 𝑀𝑠𝑟 𝐿𝑟 𝑀𝑠𝑟 1 𝑇𝑟 = 𝑅𝑟 ; 𝑏1 = 𝑇𝑟 ; 𝑏2 = − 𝑇𝑟 ; 𝑚 = 𝐿𝑟 ; Ω = 𝑁𝑝 𝜔. where 𝑅𝑠 and 𝑅𝑟 are the stator and rotor resistances, respectively. 𝐿𝑠 and 𝐿𝑟 are the stator and rotor magnetizing inductance, respectively. 𝑀𝑠𝑟 is the stator/rotor mutual inductance. The electromagnetic torque expressed, in terms of quadratic stator current 𝐼𝑠𝑞 and rotor flux modulus 𝜙𝑟 , as follow:
978-1-5090-6778-7/17/$31.00 2017 IEEE
𝑇𝑒 =
3 𝑁𝑝 𝑀𝑠𝑟 𝜙𝑟 𝐼𝑠𝑞 2 𝐿𝑟
(2)
III. BACKSTEPPING C ONTROL D ESIGN Backstepping is a constructive tool for non linear system, tries to make the rotor speed and the rotor flux track the prescribed trajectories. Based on recursive algorithm, the most interesting point of backstepping technique is to deal with nonlinearity of high-order system using virtual control [15]. To design this controller, the system is divided into simpler and smaller ones. In each one, it deal with an easier and single input single output design problem, and each step provides a reference for the next design step. A. Flux and speed regulators The goal of this step is to research the virtual control that ensures the asymptotic convergence to zero of both the flux and speed tracking error. Let’s define rotor speed and flux modulus errors as: {
𝑒Ω = Ω∗ − Ω 𝑒𝜙 = 𝜙∗𝑟 − 𝜙𝑟
(3)
By directly differentiating (3) and taking (1) into account, it yields that {
( ) ˙ ∗ − 1 (𝑚𝜙𝑟 𝐼𝑠𝑞 − 𝑇𝐿 − 𝑓 Ω) 𝑒˙ Ω = Ω 𝐽 ∗ 𝑒˙ 𝜙 = 𝜙˙ 𝑟 − 𝑏2 𝜙𝑟 − 𝑏1 𝐼𝑠𝑑
(4)
In order to check the tracking error stability, the first candidate Lyapunov function associated with the rotor speed and flux errors is defined as : 𝑒2Ω + 𝑒2𝜙 𝑄1 = 2
Stator voltages 𝑉𝑠𝑑 and 𝑉𝑠𝑞 are included in (10). These could be establishing a second candidate Lyapunov function 𝑄2 based on errors of speed, rotor flux modulus and stator current (𝑑 − 𝑞) components. 𝑄2 is given as: 𝑄2 =
𝑄˙ 2 = −𝑧1 𝑒2Ω − 𝑧2 𝑒2𝜙 − 𝑧3 𝑒2𝐼𝑠𝑞 − 𝑧4 𝑒2𝐼𝑠𝑑 + 𝑔1 𝑒𝐼𝑠𝑞 + 𝑔2 𝑒𝐼𝑠𝑑 {
∗ − 𝑎1 𝐼𝑠𝑞 + 𝑎3 𝜔𝜙𝑟 + 𝜔𝐼𝑠𝑑 + 𝑏1 𝑠𝑑𝜙𝑟𝑠𝑞 − 𝑐𝑉𝑠𝑞 𝑔1 = −𝑧3 𝑒𝐼𝑠𝑞 + 𝐼˙𝑠𝑞 (𝐼 )2 ∗ 𝑔2 = −𝑧4 𝑒𝐼𝑠𝑑 + 𝐼˙𝑠𝑑 − 𝑎2 𝜙𝑟 − 𝜔𝐼𝑠𝑞 − 𝑏1 𝜙𝑠𝑞𝑟 − 𝑐𝑉𝑠𝑑 (13) 𝐼
(7)
𝑧1 and 𝑧2 must be positive parameters, in order to ensure that tracking errors dynamics will converge exponentially to zero. In this case equivalent stator current references are given by: ∗ ⎩ 𝐼𝑠𝑑 =
𝐽 𝜙𝑟 𝑚 1 𝑏1
(
(
˙∗+ 𝑧1 𝑒Ω + Ω
𝑓Ω 𝐽
𝑧2 𝑒𝜙 + 𝜙˙ ∗𝑟 − 𝑏2 𝜙𝑟
)
+
𝑇𝐿 𝐽
)
(8)
The virtual controls in (8) are chosen to satisfy the aim of stator current regulation and are considered as references for the next step. B. Currents regulators Previous references, chosen to ensure a stable dynamic of speed and flux tracking error, can’t be imposed to the virtual controls without considering errors between its. Let 𝑒𝑑 and 𝑒𝑞 be the direct and quadratic currents errors, are defined as: ⎧ ⎨ 𝑒𝐼𝑠𝑞 =
⎩ 𝑒𝐼 = 𝑠𝑑
𝐽 𝜙𝑟 𝑚 1 𝑏1
(
(
˙∗+ 𝑧1 𝑒Ω + Ω
𝑓Ω 𝐽
+
𝑇𝐿 𝐽
)
) 𝑧2 𝑒𝜙 + 𝜙˙ ∗𝑟 − 𝑏2 𝜙𝑟 − 𝐼𝑠𝑑
− 𝐼𝑠𝑞
(9)
Using (1), differentiation of (9) yields to following errors dynamics: {
∗ − 𝑎1 𝐼𝑠𝑞 − 𝑎3 𝜔𝜙𝑟 − 𝜔𝐼𝑠𝑑 − 𝑏1 𝑠𝑑𝜙𝑟𝑠𝑞 + 𝑐𝑉𝑠𝑞 𝑒˙ 𝐼𝑠𝑞 = 𝐼˙𝑠𝑞 (𝐼 )2 ∗ 𝑒˙ 𝐼 = 𝐼˙𝑠𝑑 − 𝑎1 𝐼𝑠𝑑 + 𝑎2 𝜙𝑟 + 𝜔𝐼𝑠𝑞 + 𝑏1 𝑠𝑞 + 𝑐𝑉𝑠𝑑 𝐼
𝑠𝑑
𝜙𝑟
𝐼
To render the Lyapunov function 𝑄2 dynamic negative definite, 𝑔1 and 𝑔2 must be chosen equal to zero. Therefore, the control law is taken as: ⎧ ∗ ⎨ 𝑉𝑠𝑞 =
1 𝑐
∗ ⎩ 𝑉𝑠𝑑 =
1 𝑐
(
) 𝐼 𝐼 ∗ − 𝑎1 𝐼𝑠𝑞 + 𝑎3 𝜔𝜙𝑟 + 𝜔𝐼𝑠𝑑 + 𝑏1 𝑠𝑑𝜙𝑟𝑠𝑞 𝑧3 𝑒𝐼𝑠𝑞 + 𝐼˙𝑠𝑞 ( ) (𝐼 )2 ∗ 𝑧4 𝑒𝐼𝑠𝑑 + 𝐼˙𝑠𝑑 − 𝑎2 𝜙𝑟 − 𝜔𝐼𝑠𝑞 − 𝑏1 𝜙𝑠𝑞𝑟 (14)
To guarantee that current dynamics converge faster than those of the speed and flux, 𝑧3 and 𝑧4 must be positive parameters. (4) and (9) can be written as: ⎧ 𝑒˙ Ω = −𝑧1 𝑒Ω ⎨ 𝑒˙ = −𝑧 𝑒 2 𝜙 𝜙 𝑚 ⎩ 𝑒˙ 𝐼𝑠𝑞 = −𝑧3 𝑒𝐼𝑠𝑞 − 𝐽𝜙𝑟 𝑒Ω 𝑒˙ 𝐼𝑠𝑑 = −𝑧4 𝑒𝐼𝑠𝑑 + 𝑏2 𝜙𝑟 𝑒𝜙
(15)
Equation (15) can be also, take the following form:
(6)
Equation (6) can be also, written such as:
⎧ ∗ ⎨ 𝐼𝑠𝑞 =
(12)
where
In view of (4), the time derivative of 𝑄1 satisfies
˙ = −𝑧1 𝑒2Ω − 𝑧2 𝑒2𝜙 𝑄1
(11)
The time derivative of 𝑄2 with respect to time yields to:
(5)
( ( )) ˙ ∗ − 1 (𝑚𝜙𝑟 𝐼𝑠𝑞 − 𝑇𝐿 − 𝑓 Ω) 𝑄˙ 1 = 𝑒Ω Ω 𝐽 ( ) +𝑒𝜙 𝜙˙ ∗𝑟 − 𝑏2 𝜙𝑟 − 𝑏1 𝐼𝑠𝑑
𝑒2Ω + 𝑒2𝜙 + 𝑒2𝐼𝑠𝑞 + 𝑒2𝐼𝑠𝑑 2
𝐼
(10)
⎡
⎤ ⎤ ⎡ ⎡ −𝑧1 𝑒˙ Ω 𝑒Ω ⎢ 𝑒˙ 𝜙 ⎥ [ ] ⎢ 𝑒𝜙 ⎥ ⎢ 0 ⎣𝑒˙ ⎦ = 𝐸 ⎣𝑒 ⎦ = ⎣ 𝑚 𝐼𝑠𝑞 𝐼𝑠𝑞 𝐽𝜙𝑟 𝑒˙ 𝐼𝑠𝑑 𝑒𝐼𝑠 0
0 −𝑧2 0 𝑏 2 𝜙𝑟
0 0 −𝑧3 0
⎤ ⎡ ⎤ 0 𝑒Ω 0 ⎥ ⎢ 𝑒𝜙 ⎥ 0 ⎦ × ⎣𝑒𝐼𝑠𝑞 ⎦ 𝑒𝐼𝑠𝑑 −𝑧4
[ ] The control scheme stability is insured if the matrix 𝐸 (regression matrix) is Hurwitz what is verified by a goal choice of 𝑧𝑖 , ∀𝑖 = 1, 2, 3, 4. IV. I NTEGRAL SLIDING M ODE C ONTROL D ESIGN According to design of conventional SMC [16], the ISM control design procedure is divided into two steps: first, a sliding surface is designed, so that the controlled system yields the desired dynamic performance. The second step is to design a control law such that the trajectory of the system remains on the sliding surface from the initial time [17]. A. Integral sliding surface design Four integral sliding surface arranged in twos are defined by (16) and (17) {
∫ 𝑆1 = Ω − Ω∗ + 𝑚1 ∫ (Ω − Ω∗ ) 𝑑𝑡 𝑆2 = 𝜙𝑟 − 𝜙∗𝑟 + 𝑚2 (𝜙𝑟 − 𝜙∗𝑟 ) 𝑑𝑡 { ) ∫( ∗ ∗ 𝑆3 = 𝐼𝑠𝑞 − 𝐼𝑠𝑞 𝑑𝑡 + 𝑚3 ∫ 𝐼𝑠𝑞 − 𝐼𝑠𝑞 ∗ ∗ 𝑆4 = 𝐼𝑠𝑑 − 𝐼𝑠𝑑 + 𝑚4 (𝐼𝑠𝑑 − 𝐼𝑠𝑑 ) 𝑑𝑡
(16) (17)
∗ ∗ where Ω∗ , 𝜙∗𝑟 , 𝐼𝑠𝑞 and 𝐼𝑠𝑑 are the speed, rotor flux and stator current references respectively. 𝑚1 , 𝑚2 , 𝑚3 and 𝑚4 are positive constants.
B. Integral sliding mode control laws
C. Closed-loop stability analysis
The proposed controller concepts consists of two phases: the sliding phase with 𝑆𝑖 = 0 and 𝑆˙𝑖 = 0, 𝑖 = 1, 2, 3, 4 to perform the equivalent control and the reaching phase with 𝑆𝑖 = 0 to achieve the switching control. These can be derived separately. 1) Rotor flux and speed regulators: Let’s define the speed Ω and rotor flux modulus 𝜙𝑟 equivalent control in the following way:
Let 𝑒Ω , 𝑒𝜙 , 𝑒𝑞 and 𝑒𝑑 be the speed, the rotor flux and (𝑑−𝑞) stator current tracking errors defined by (27). Their dynamics are expressed by (28).
{
{
𝑆1 = 0 𝑆2 = 0
˙ −Ω ˙ ∗ + 𝑚1 (Ω − Ω∗ ) = 0 𝑆˙ 1 = Ω ˙ ˙ 𝑆2 = 𝜙𝑟 − 𝜙˙ ∗𝑟 + 𝑚2 (𝜙𝑟 − 𝜙∗𝑟 ) = 0
⇒
(18)
Based on (1) and (18), the equivalent reference currents are given by : ⎧ ∗ ⎨ 𝐼𝑠𝑞 = 𝑒𝑞 ∗ ⎩ 𝐼𝑠𝑑 = 𝑒𝑞
1 𝑏1
(
˙∗+ Ω
(𝑓 )
Ω − 𝑇𝐿 − 𝑚1 (Ω − Ω∗ ) ( ∗ ) 𝜙˙𝑟 − 𝑏1 𝜙𝑟 − 𝑚2 (𝜙𝑟 − 𝜙∗𝑟 )
𝐽 𝑚𝜙𝑟
)
𝐽
(19)
∗ = −𝑘1 𝑠𝑔𝑛 (𝑆1 ) 𝐼𝑠𝑞 𝑠𝑤 ∗ = −𝑘2 𝑠𝑔𝑛 (𝑆2 ) 𝐼𝑠𝑑 𝑠𝑤
(20)
with 𝑘1 and 𝑘2 are positive constant. Finally, (19) and (20) lead us to determine reference currents such as the following expression. ⎧ ∗ ⎨ 𝐼𝑠𝑞 = ∗ ⎩ 𝐼𝑠𝑑 =
(
) Ω − 𝑇𝐿 − 𝑚1 (Ω − Ω∗ ) − 𝑘1 𝑠𝑔𝑛 (𝑆1 ) ( ∗ ) 𝜙˙𝑟 − 𝑏1 𝜙𝑟 − 𝑚2 (𝜙𝑟 − 𝜙∗𝑟 ) − 𝑘2 𝑠𝑔𝑛 (𝑆2 ) (21)
𝐽 𝑚𝜙𝑟 1 𝑏1
˙∗+ Ω
where
1 𝑐
∗ ⎩ 𝑉𝑠𝑑 = 𝑒𝑞
1 𝑐
{
𝐼𝑠𝑑 𝐼𝑠𝑞 𝜙𝑟 (𝐼 )2 𝑏1 𝜙𝑠𝑞𝑟
𝑓1 = 𝑎1 𝐼𝑠𝑞 − 𝑎3 𝜔𝜙𝑟 − 𝜔𝐼𝑠𝑑 − 𝑏1 𝑓2 = 𝑎1 𝐼𝑠𝑑 + 𝑎2 𝜙𝑟 + 𝜔𝐼𝑠𝑞 +
∗ = −𝑘3 𝑠𝑔𝑛(𝑆3 ) 𝑉𝑠𝑞 𝑠𝑤 ∗ 𝑉𝑠𝑑 = −𝑘4 𝑠𝑔𝑛(𝑆4 ) 𝑠𝑤
(25)
with 𝑘3 and 𝑘4 are positive constant. To satisfy the reaching condition, the equivalent control given in (23) is augmented by a switching control term (25), to be determined. The ISM control is designed such that: ⎧ ∗ ⎨ 𝑉𝑠𝑞 =
1 𝑐
∗ ⎩ 𝑉𝑠𝑑 =
1 𝑐
(
( )) ∗ ∗ 𝐼˙𝑠𝑞 − 𝑘3 𝑠𝑔𝑛(𝑆3 ) − 𝑓1 − 𝑚3 𝐼𝑠𝑞 − 𝐼𝑠𝑞 ( ) ∗ ∗ ˙ 𝐼𝑠𝑑 − 𝑓2 − 𝑚4 (𝐼𝑠𝑑 − 𝐼𝑠𝑑 ) − 𝑘4 𝑠𝑔𝑛(𝑆4 )
𝐽
(31)
) 1( 2 𝑒Ω + 𝑒2𝜙 + 𝑒2𝑞 + 𝑒2𝑑 2
(32)
(33)
𝑚1 ≥ ∣𝑘Ω 𝑠𝑔𝑛(𝑆1 )∣ 𝑎𝑛𝑑 𝑚2 ≥ ∣𝑘𝜙 𝑠𝑔𝑛(𝑆2 )∣
{
𝑚3 ≥ ∣𝑘𝑞 𝑠𝑔𝑛(𝑆3 )∣ 𝑚4 ≥ ∣𝑘𝑑 𝑠𝑔𝑛(𝑆4 )∣
With respect to above conditions, (33) becomes:
The attractive control law are ensured by: {
(𝑓 )
To assure the stability, 𝑚𝑖 , (𝑖 = 1...4) must be chosen as follows: {
(24)
˙∗+ Ω
( 𝑟 ) 𝑉˙ = 𝑒𝜔 𝑚𝜙 𝑒𝑞 − 𝑚1 𝑒Ω − 𝑘Ω 𝑠𝑔𝑛(𝑆1 ) 𝐽 +𝑒𝜙 ((𝑏1 𝑒𝑑 − 𝑚2 𝑒𝜙 − 𝑘𝜙 𝑠𝑔𝑛(𝑆2 )) ) 𝑟 +𝑒𝑞 −𝑚3 𝑒𝑞 − 𝑚𝜙 𝑒Ω − 𝑘𝑞 𝑠𝑔𝑛(𝑆3 ) 𝐽 +𝑒𝑑 (−𝑚4 𝑒𝑑 − 𝑏1 𝑒𝜙 − 𝑘𝑑 𝑠𝑔𝑛(𝑆4 ))
(22)
(23)
(
𝑉 =
(
( )) ∗ ∗ 𝐼˙𝑠𝑞 − 𝑓1 − 𝑚3 𝐼𝑠𝑞 − 𝐼𝑠𝑞 ( ) ∗ ∗ 𝐼˙𝑠𝑑 − 𝑓2 − 𝑚4 (𝐼𝑠𝑑 − 𝐼𝑠𝑑 )
(28)
Then, a Lyapunov function and it’s derivative are given by (32) and (33).
In this case, the control laws are given by : ⎧ ∗ ⎨ 𝑉𝑠𝑞 = 𝑒𝑞
𝐽 𝑚𝜙𝑟
⎧ 𝑟 𝑒˙ = 𝑚𝜙 𝑒𝑞 − 𝑚1 𝑒Ω − 𝑘Ω 𝑠𝑔𝑛(𝑆1 ) 𝐽 ⎨ Ω 𝑒˙ 𝜙 = 𝑏1 𝑒𝑑 − 𝑚2 𝑒𝜙 − 𝑘𝜙 𝑠𝑔𝑛(𝑆2 ) 𝑟 𝑒˙ 𝑞 = −𝑚3 𝑒𝑞 − 𝑚𝜙 𝑒Ω − 𝑘𝑞 𝑠𝑔𝑛(𝑆3 ) 𝐽 ⎩ 𝑒˙ 𝑑 = −𝑚4 𝑒𝑑 − 𝑏1 𝑒𝜙 − 𝑘𝑑 𝑠𝑔𝑛(𝑆4 )
)
∗ ∗ 𝑆˙ 3 = 𝐼˙𝑠𝑞 − 𝐼˙𝑠𝑞 =0 + 𝑚3 𝐼𝑠𝑞 − 𝐼𝑠𝑞 ∗ ∗ ˙ ˙ ˙ 𝑆4 = 𝐼𝑠𝑑 − 𝐼𝑠𝑑 + 𝑚4 (𝐼𝑠𝑑 − 𝐼𝑠𝑑 )=0
𝑚𝜙𝑟 ∗ 𝐼𝑠𝑞 𝐽
) Ω − 𝑇𝐿 − 𝑚1 (Ω − Ω∗ ) − 𝑘Ω 𝑠𝑔𝑛 (𝑆1 ) ( ) ∗ ⎩ 𝐼𝑠𝑑 = 1𝑏 𝜙˙ ∗𝑟 − 𝑏1 𝜙𝑟 − 𝑚2 (𝜙𝑟 − 𝜙∗𝑟 ) − 𝑘𝜙 𝑠𝑔𝑛 (𝑆2 ) (29) ⎧ ( ) ( ) ∗ ∗ ∗ ⎨ 𝑉𝑠𝑞 − 𝑘𝑞 𝑠𝑖𝑔𝑛(𝑆3 ) = 1𝑐 𝐼˙𝑠𝑞 − 𝑓1 − 𝑚3 𝐼𝑠𝑞 − 𝐼𝑠𝑞 ( ) (30) ∗ ∗ ∗ ⎩ 𝑉𝑠𝑑 = 1𝑐 𝐼˙𝑠𝑑 − 𝑓2 − 𝑚4 (𝐼𝑠𝑑 − 𝐼𝑠𝑑 ) − 𝑘𝑑 𝑠𝑖𝑔𝑛(𝑆4 )
𝐽
(
(27)
𝑟 Using the following variable change: 𝑘Ω = 𝑘1 ( 𝑚𝜙 𝐽 ), 𝑘𝜙 = 𝑏𝑘2 , 𝑘𝑞 = 𝑐𝑘3 and 𝑘𝑑 = 𝑐𝑘4 , (𝑑−𝑞) stator current and voltage references can be expressed such as (29) and (30) and their tracking errors dynamics will be described by (31).
(𝑓 )
2) (𝑑 − 𝑞) Currents regulators: According to the current surfaces derivatives and in order to determinate equivalent reference control laws, one has: {
⎧ 1 Ω∗ − ⎨ 𝑒˙ Ω = 𝐽 (𝑚𝜙𝑟 𝑒𝑞 − 𝑇𝐿 ∗− 𝑓 Ω) − ∗ 𝑒˙ 𝜙 = 𝑏2 𝜙𝑟 + 𝑏1 𝑒𝑑 − 𝜙𝑟 + 𝑏1 𝐼𝑠𝑑 ∗ ⎩ 𝑒˙ 𝑞 = 𝑓1 + 𝑐𝑉𝑠𝑞 − 𝐼𝑠𝑞 ∗ 𝑒˙ 𝑑 = 𝑓2 + 𝑐𝑉𝑠𝑑 − 𝐼𝑠𝑑
⎧ ∗ ⎨ 𝐼𝑠𝑞 =
The control laws which ensures the attractively are: {
⎧ 𝑒 = (Ω − Ω∗ ) ⎨ Ω 𝑒𝜙 = (𝜙𝑟 − 𝜙∗𝑟 ) ∗ ) ⎩ 𝑒𝑞 = (𝐼𝑠𝑞 − 𝐼𝑠𝑞 ∗ ) 𝑒𝑑 = (𝐼𝑠𝑑 − 𝐼𝑠𝑑
(26)
( ) 𝑉˙ < −𝑚1 𝑒2Ω − 𝑚2 𝑒2𝜙 − 𝑚3 𝑒2𝑞 − 𝑚4 𝑒2𝑑
(34)
According to (34), it is shown (that the )Lyapunov function time derivative is negative definite 𝑉˙ ≤ 0 and yields the desired stability and convergence properties. This implies that the error variables 𝑒Ω , 𝑒𝜙 , 𝑒𝑞 and 𝑒𝑑 are globally uniformly bounded. Remark: To guarantee good performances, both control structures consider that the speed Ω and the rotor flux 𝜙𝑟 are accessible for closed-loop control and the IM parameters are known exactly.
V. U NKNOWN INPUT OBSERVER DESIGN
TABLE I IM PARAMETERS
In this section, to estimate IM states under faulty conditions, an UIO is designed as in [18]. This observer is known as a robust observer to load torque and parameter variations. Its design based upon the full order high gain observer theory and does not need the output differentiation. The UIO synthesis assumes that the unknown inputs dynamics are bounded without making any hypothesis on how these inputs vary. In order to obtain a better estimation, it is necessary to have dynamic representation based on the stationary (𝛼 − 𝛽) reference frame: ⎧ ⎨ 𝐼˙𝑠 = 𝑎1 𝐼2 𝐼𝑠 + 𝑎2 𝐴𝜙𝑟 + 𝐶𝑉𝑠 𝜙˙ = 𝑏 𝐼 𝐼 − 𝐴𝜙𝑟 ⎩ ˙𝑟 1 1 2 𝑠 Ω = (𝑚𝐽2 𝜙𝑟 𝐼𝑠 − 𝑇𝐿 − 𝑓 Ω)
𝑃𝑛
𝑉𝑠
𝑁𝑝
𝑅𝑠
𝑅𝑟
𝐿𝑠
11
220
2
1.2
0.8
0.16
𝐿𝑟
𝑙𝑠
𝑙𝑟
𝑀𝑠𝑟
𝐽
𝑓
0.16
0.0075
0.011
0.0.15
0.1
0.003
TABLE II BACKSTEPPING , ISM CONTROL STRUCTURES PARAMETERS
(35)
𝑧1
𝑧2
𝑧3
𝑧4
𝑚1
𝑚2
300
1200
2000
1500
2000
500
𝑚3
𝑚4
𝑘1
𝑘2
𝑘3
𝑘4
3200
1000
40
20
15
10
𝐽
]𝑇 ]𝑇 [ [ with 𝐼𝑠 = 𝐼𝑠𝛼 𝐼𝑠𝛽 , 𝜙𝑟 = 𝜙𝑟𝛼 𝜙𝑟𝛽 and 𝑉𝑠 = [ ]𝑇 𝑉𝑠𝛼 𝑉𝑠𝛽 are the stator current, the rotor flux and stator voltage vectors, respectively. Expressions of(𝐼2 , 𝐴, ) 𝐶 and 𝐽2 1
matrix are depend on IM parameters. 𝐼2 = 0 ( ) ( ) ( 𝐴=
−𝑏2 −𝜔
𝜔 ;𝐶 = −𝑏2
𝑐 0
0 ; 𝐽2 = 𝑐
0 1
0 ; 1) −1 . 0
One considers ]𝑇 [ ∈ ℝ𝑛=5 is the system state. - 𝑥 = 𝐼𝑠 𝜙 𝑟 Ω Thus, let us define the vector 𝑥 such as: 𝑥1 = [ 1 ]𝑇 𝑥1 = 𝐼𝑠 𝑥12 = Ω ∈ ℝ𝑛1 =3 , 𝑥2 = 𝜙𝑟 ∈ ℝ𝑛2 =2 for 2 ∑ 𝑘 = 1, 2 and 𝑛1 ≥ 𝑛2 , 𝑛𝑘 = 𝑛. 𝑘=1
- The measurable output vector can be portioned as fol]𝑇 [ lows: 𝑦 = 𝐼𝑠 Ω ∈ ℝ𝑝=3 . - Furthermore, the input vector 𝑒 = (𝑢, 𝑣) ∈ 𝔼 is a compact set of ℝ𝑠=3 , the subvector 𝑢 = 𝑉𝑠 ∈ ℝ𝑠−𝑚=2 is a known input of 𝑒 while the remaining subvector 𝑣 = 𝑅𝑠 ∈ ℝ𝑚=1 is unknown. Consequently, the IM model defined in (35) can be written under the following form: {
𝑥˙ = 𝑓 (𝑥, 𝑢, 𝑣) ¯ 𝑦 = 𝐶𝑥
(36)
In fact, it is easy to see that the [different]elements of the 𝑇 IM are contracted as follows: 𝑥 = 𝑥1 𝑥2 ; 𝑓 (𝑥, 𝑢, 𝑣) = [ 1 1 2 ] ] [ 𝑇 𝑓 (𝑥 , 𝑥 , 𝑢, 𝑣) 𝑓 2 (𝑥1 , 𝑥2 , 𝑢, 𝑣) and 𝐶¯ = 𝐼3 , 03×2 . The basic routine of the design algorithm is to verify certain assumptions [19]. If all these hypotheses are checked, the system observability is clearly shown. Hence, as a preliminary step, change ˜ = ( 1 ) has been introduced: 𝑥 ) ( 1 )a simple( notation 𝑥11 𝑥2 𝑥 ˜ 1 2 ;𝑥 ˜ = ;𝑥 ˜ = ; 𝐶 = 𝑑𝑖𝑎𝑔{𝐶1 , 𝐶2 }; 𝐶1 = 𝑥 ˜2 𝑣 𝑥2 ( 1 ) ) ( ( ) 𝑓˜ (˜ 𝑥, 𝑢) 𝐼2 0 ; 𝐶2 = 1 0 𝑓 (˜ 𝑥, 𝑢) = ˜2 𝑥, 𝑢) = ; 𝑓˜1 (˜ 𝑓 (˜ 𝑥 , 𝑢) ( 1 1 2 ( 1 1 2 ) ) 𝑓1 (𝑥 , 𝑥 , 𝑢, 𝑣) 𝑓2 (𝑥 , 𝑥 , 𝑢, 𝑣) 𝑥, 𝑢) = . ; 𝑓˜2 (˜ 0 𝑓 2 (𝑥, 𝑢, 𝑣) Therefore, the appropriate UIO, according to (35), given by the form developed in [19], can be designed as the following form:
⎧ ⎨ ⎩
˙ 𝐼ˆ𝑠 = 𝑎1 𝐼2 𝐼ˆ𝑠 + 𝑎2 𝐴𝜙ˆ𝑟 + 𝐶𝑉𝑠 − 2𝜃(𝐼ˆ𝑠 − 𝐼𝑠 ) [ ]+ ˆ˙ 𝑠 = −𝜃2 𝑎1 𝐼ˆ𝑠 (𝐼ˆ𝑠 − 𝐼𝑠 ) 𝑅 )⎞ ⎛ ( ) 1 ˆ𝑠𝑇 𝐽2 𝜙ˆ𝑟 ) − 𝑇𝐿 − 𝑓 Ω ˆ (𝑚 𝐼 ˙ˆ Ω ⎟ ⎜𝐽 =⎝ ⎠− ˙ ˆ 𝜙𝑟 𝑏1 𝐼2 𝐼ˆ𝑠 − 𝐴𝜙ˆ𝑟 ⎞+ ⎛ 1 ⎞−1 ⎛ ⎞ ⎛ 1 0 0 0 2 𝜃 ˆ − Ω) ⎠ .⎝ ⎠ . ⎝ ⎠ .(Ω ⎝ 1 0 𝜃12 0 𝑚𝐼ˆ𝑠𝑇 𝐽2
(
(37)
By choosing a real 𝜃 , representing the only design parameter of the UIO, the whole states and the unknown input are estimated. This parameter should be chosen as a compromise between the speed of convergence and noise convergence. VI. BACKSTEPPING AND ISM C ONTROL STRUCTURES PERFORMANCE UNDER FAULTY CONDITIONS
Simulation tests were performed using MATLAB/Simulink software using Euler method with sampling time 1𝑒−6 s. An 11 𝑘𝑊 IM is used whose parameters are cited in Table I. Fig. 1 illustrates the block diagram of the backstepping and ISM control structures under faulty condition using UIO. A faulty IM model was developed in [11]. The developed control law supposes that the three-phase stator current and voltage and the rotor speed are available. The rotor speed and flux references are set respectively to Ω∗ =157𝑟𝑑/𝑠 and 𝜙∗𝑟 =0.9𝑊 𝑏. Performance of the both control methods under faulty conditions are shown in Fig. 2. The following scenario is illustrated: start-up of the machine, then an ITSC fault in 12.5% of phase 𝑎 winding appears after 0.5𝑠 and finally a load torque appears equal to 6𝑁.𝑚 at 𝑡 = 0.75𝑠. All of the controlled variables follow their reference values precisely. Fig. 2a shows the rotor speed. It is stabilized at 0.3𝑠. The speed response is almost the same in the case of both control structures. After the ITSC appears, some small oscillations appears and the speed value still around its reference. The appearance of the load torque variation, increase these oscillations. A similar situation takes place in the case of the
ITSC fault
𝐹𝑎 = ∣𝐼𝑠𝑎 − 𝐼ˆ𝑠𝑎 ∣;
ɏ
Iˆsα β
V sabc
V sαβ
ab c
I sabc
Unknown Input Observer
I sd
I sabc abc
I sq
V sabc
V P W M
* a bc V sabc
dq
V
{
Backstepping / ISM control
dq
* sq
* sd
Quadratic current regulator
I sq*
Direct current regulator
I sd*
Speed regulator
Flux regulator
𝐹𝑐 = ∣𝐼𝑠𝑐 − 𝐼ˆ𝑠𝑐 ∣
(38)
The generated residuals in the case of fault absence and parametric variations are established in the aim to define threshold 𝜖. A good choice of this threshold is used to detect ITSC fault while avoiding false alarms. So, a simplest FD strategy could be designed as follows [11]:
ˆ Φ r
I sα β
αβ
𝐹𝑏 = ∣𝐼𝑠𝑏 − 𝐼ˆ𝑠𝑏 ∣;
ɏ ɏΎ ˆ Φ r Φ*r
U dc
Fig. 1. Block diagram of the backstepping and ISM control structures under faulty condition using UIO.
rotor flux amplitude (Fig. 2b). However, some inconsiderable oscillations in the rotor flux amplitude are introduced after the fault occurrence. Oscillations is similar for both control structures. So, one can deduce that for both control structure, the speed and the flux can track their references even under faulty condition. At 𝑡 = 0.5𝑠, the amplitudes of three-phase stator currents grow significantly and quite large asymmetry between them, it can be seen in Fig. 2c for the backstepping control and Fig. 2d for the ISM control. The faulted phase current becomes the largest one, however, the remaining ones amplitudes are close to each other. This asymmetry can be used in order to design an observer-based FD method. After the load torque is generated, the three-phase stator currents grow. While, as shown in Fig 2e, estimated three-phase stator currents are not affected by the fault and they still symmetry. After the ITSC fault occurs, large oscillations arise the electromagnetic torque for both control structures as illustrated in fig. 2f.
if 𝐹𝑖 ≤ 𝜖 if 𝐹𝑖 > 𝜖
then IM is healthy then IM is faulty
(39)
The three residual signals are presented in Fig. 3, when an ITSC fault in 12.5% of phase 𝑎 winding appears at 𝑡 = 0.5𝑠 in phase 𝑎 for IM working under both closed-loop control structures and load torque application at 𝑡 = 0.75𝑠. It can be seen in Fig. 3a for the backstepping control and Fig. 3b for the ISM control. Any thresholding technique can be utilized for unsupervised on line FD. When residuals exceed threshold (𝜖 = 1.5𝐴), the overtaking is caught by fault indicators that indicate fault presence. The highest value of the stator currents recorded by fault indicator 𝐼𝑠𝑎 , 𝐼𝑠𝑏 and 𝐼𝑠𝑐 corresponds to the stator phase affected by the ITSC. The analyze of residuals shows that all residuals are affected. However, only the first residual 𝐹𝑎 according to affected current phase 𝑎 exceed its threshold value. That allows the detection of an ITSC fault in the phase 𝑎. VIII. C ONCLUSION This paper deals with the ITSC FD of the IM operating with a closed-loop control structures. Two non linear control structures, namely the backstepping control and the ISM control, are designed to ensure the robust tracking performance and provide an accurate ITSC FD. This fault detection method is based strictly on UIO to estimate the rotor flux and generate residual signals. On the basis of the simulation tests, conducted for the IM with ITSC fault, working under both control structures, it is shown that it is possible to detect this damage type even at its initial stage.
VII. E VALUATION OF THE FAULT DETECTION METHOD
R EFERENCES
In order to detect the ITSC fault in the IM stator winding, three-phase stator currents can be used, as easy measurable signals. Analysis shown in previous section proves that threephase stator current amplitudes changes are observed. By comparing measured three-phase stator currents and estimated ones using UIO, three residual signals can be conducted. These signals are used as alarms to indicate the occurrence of the fault, and if properly designed, give information from which the source of the fault may be identified. Generally, residuals are wanted to be identified such that they are close to zero when the IM behaviour in healthy mode, and promptly become non zero when the associated ITSC fault occurs. In our case, the three-residual signals defined in (38) are expressed as an absolute value of the difference between measured variable and his estimate ones by UIO.
[1] H. Henao, et al. Trends in fault diagnosis for electrical machines: A review of diagnostic techniques., IEEE industrial electronics magazine vol. 8, no. 2, pp. 31-42, 2014. [2] M. Riera-Guasp, J. A. Antonino-Daviu and G. A. Capolino, Advances in electrical machine, power electronic, and drive condition monitoring and fault detection: state of the art., IEEE Transactions on Industrial Electronics, vol. 62, no. 3, pp. 1746-1759, 2015. [3] K. M. Siddiqui, K. Sahay andV. K. Giri, Health monitoring and fault diagnosis in induction motor-a review., International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering, vol. 3, no. 1, pp. 6549-6565, 2014. [4] J. Chen and J. Patton, Robust model-based fault diagnosis for dynamic systems. vol. 3. Springer Science & Business Media, 2012. [5] C. H. De Angelo, et al. Online model-based stator-fault detection and identification in induction motors., IEEE Transactions on Industrial Electronics, vol. 56, no. 11, pp. 4671-4680, 2009. [6] N. R, Devi, D. V. S, Sarma andP. V. R, Rao, Detection of stator incipient faults and identification of faulty phase in three-phase induction motorsimulation and experimental verification., IET Electric Power Applications, vol. 9, no. 8, pp. 540-548, 2015.
Reference
ISM
Rotor flux modulus (Wb)
Rotor speed (rd/s)
200
Backstepping
150 157.2
100
157 50 0
156.8 0.5 0
0.75
0.25
0.5
0.75
t (s) 1
Reference
1.001 1
0
0
0.25
Stator currents (A)
Stator currents (A)
−5 0.5
0.7
Isa 0
0.25
0.5
Isb
Isc
0.75
t (s) 1
0 0.5
−20 −30
Isa 0
0.25
0.5
Load
0
20
4
Torque (N.m)
Estimated stator currents (A)
25
0.7
0
−20 0.25
0.7
0
5
0.5
0
t (s) 1
Isb 0.75
Isc t (s) 1
(d) ISM
−5
20
0.75
−5
20
(c) Backstepping
40
0.75
0.5
5
40
0
0 −20 −30
0,5
(b)
5
20
Backstepping
1.002 0.5
(a)
40
ISM
1
0.5
0.75
6
(e)
0.5
Backstepping
0,6
0 −5
t (s) 1
0 −3
ISM
0
0.25
0.5
0.75
t (s) 1
(f)
1.5 0 0.3
Fc (A)
4
Fb (A)
Fa (A)
Fig. 2. Performance of the controlled induction machine when an ITSC fault in 12.5% of phase 𝑎 winding appears at 𝑡 = 0.5𝑠: (a) observed and reference amplitude of the rotor flux, (b) real and reference speed, (c) measured stator currents for the ISM control, (d) measured stator currents for the backstepping control ,(e) estimated stator currents, (f) electromagnetic torque.
4
1.5
1.5 0.5
0.75
t (s)
0 0.3
1
4
0,5
0.75
0 0.3
t (s) 1
0.5
0.75
t (s)
1
0,5
0.75
t (s)
1
4
1.5 0 0.3
Fc (A)
Fb (A)
Fa (A)
(a) Backstepping Control 4
1.5 0,5
0.75
t (s)
1
0 0.3
4
1.5 0,5
0.75
t (s)
1
0 0.3
(b) ISM Control Fig. 3. Evaluation of faulty IM operation with (a) ISM structure, (b) backstepping structure when an ITSC fault in 12.5% of phase 𝑎 winding appears at 𝑡 = 0.5𝑠.
[7] A. Guezmil, et al., High order sliding mode observer for inter-turn shortcircuit fault detection in induction machine., Systems, Signals & Devices (SSD), 2015 12th International Multi-Conference on. IEEE, 2015. [8] S. Cheng, P. Zhang and T. G. Habetler, An impedance identification approach to sensitive detection and location of stator turn-to-turn faults in a closed-loop multiple-motor drive., IEEE Transactions on Industrial Electronics, vol. 58, no. 5, pp. 1545-1554, 2011. [9] S.MA, Cruz and AJ.M, Cardoso, Diagnosis of stator inter-turn short circuits in DTC induction motor drives. IEEE Transactions on Industry Applications, vol. 40, no. 5, pp. 1349-1360, 2004. [10] M. Wolkiewicz, et al. Online stator interturn short circuits monitoring in the DFOC induction-motor drive. IEEE Transactions on Industrial Electronics, vol. 63, no. 4, pp. 2517-2528, 2016.
[11] A. Guezmil, et al. An Enhanced High Order Sliding Mode based Method for Detecting Inter-Turn Short-Circuit Fault in Induction Machine with Decoupled Current Control., Applications of Sliding Mode Control. Springer Singapore, pp. 299-329, 2017. [12] A. Guezmil, et al. Backstepping control for induction machine with high order sliding mode observer and unknown inputs observer: A comparative study., Electrical Sciences and Technologies in Maghreb (CISTEM), 2014 International Conference on. IEEE, 2014. [13] A. Guezmil, et al. High order sliding mode and an unknown input observers: Comparison with integral sliding mode control for induction machine drive., Modelling, Identification and Control (ICMIC), 2015 7th International Conference on. IEEE, 2015. [14] R. Krishnan, Electric motor drives: modeling, analysis, and control, Prentice Hall, 2001. [15] A. Bennassar, et al. Sensorless Backstepping Control Using an Adaptive Luenberger Observer with Three Levels NPC Inverter, International Journal of Electrical, Robotics, Electronics and Communications Engineering, vol. 7, No. 8, pp. 612-618, 2013. [16] A. Sabanovic, and B. I, Dmitrij, Application of sliding modes to induction motor control., IEEE Transactions on Industry Applications, vol. 1, pp. 41-49, 1981. [17] V. M, Panchade, RH. Chile and BM. Patre, A survey on sliding mode control strategies for induction motors, Annual Reviews in Control, vol. 37, no. 2, pp. 289-307, 2013. [18] M. Triki, et al, Unknown inputs observers for a class of nonlinear systems., International Journal of Science and Techniques of Automatic control and computer engineering, vol. 4, no. 1, pp. 1218-1229, 2010. [19] H. Chehimi, S. Hadj Sad and F. Msahli, A predictive controller for induction motors using an unknown input observer for online rotor resistance estimation and an update technique for load torque., Journal of Control Theory and Applications, vol. 11, no. 3, pp. 367-375, 2013.
